[PAS] Probability Abstracts 111
Probability Abstract Service
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Tue Sep 1 02:20:45 CDT 2009
Probability Abstracts 111
This document contains abstracts 8725-9028
from July-1-2009 to August-31-2009.
They have been mailed on Sep 1st, 2009.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_111.shtml
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8725. MULTIVARIATE LOG-CONCAVE DISTRIBUTIONS AS A NEARLY PARAMETRIC
MODEL
Dominic Schuhmacher and Andre Huesler and Lutz Duembgen
In this paper we show that the family P_d of probability distributions
on R^d
with log-concave densities satisfies a strong continuity condition. In
particular, it turns out that weak convergence within this family
entails (i)
convergence in total variation distance, (ii) convergence of arbitrary
moments,
and (iii) pointwise convergence of Laplace transforms. Hence the
nonparametric
model P_d has similar properties as parametric models such as, for
instance,
the family of all d-variate Gaussian distributions.
http://arxiv.org/abs/0907.0250
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8726. SDES DRIVEN BY A TIME-CHANGED L\'EVY PROCESS AND THEIR
ASSOCIATED TIME-FRACTIONAL ORDER PSEUDO-DIFFERENTIAL EQUATIONS
Marjorie G. Hahn and Kei Kobayashi and Sabir Umarov
It is known that if a stochastic process is a solution to a classical
Ito
stochastic differential equation (SDE), then its transition
probabilities
satisfy in the weak sense the associated Cauchy problem for the forward
Kolmogorov equation. The forward Kolmogorov equation is a parabolic
partial
differential equation with coefficients determined by the
corresponding SDE.
Stochastic processes which are scaling limits of continuous time
random walks
have been connected with time-fractional differential equations.
However, the
class of SDEs that is associated with time-fractional Kolmogorov type
equations
is unknown. The present paper shows that in the cases of either time-
fractional
order or more general time-distributed order differential equations, the
associated class of SDEs can be described within the framework of SDEs
driven
by semimartingales. These semimartingales are time-changed Levy
processes where
the independent time-change is given respectively by the inverse of a
stable
subordinator or the inverse of a mixture of independent stable
subordinators.
http://arxiv.org/abs/0907.0253
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8727. BROWNIAN AND FRACTIONAL BROWNIAN STOCHASTIC CURRENTS VIA
MALLIAVIN CALCULUS
Franco Flandoli and Ciprian Tudor (CES and SAMOS)
By using Malliavin calculus and multiple Wiener-It\^o integrals, we
study the
existence and the regularity of stochastic currents defined as Skorohod
(divergence) integrals with respect to the Brownian motion and to the
fractional Brownian motion. We consider also the multidimensional
multiparameter case and we compare the regularity of the current as a
distribution in negative Sobolev spaces with its regularity in
Watanabe space.
http://arxiv.org/abs/0907.0292
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8728. A MIN-TYPE STOCHASTIC FIXED-POINT EQUATION RELATED TO THE
SMOOTHING TRANSFORMATION
Gerold Alsmeyer and Matthias Meiners
This paper is devoted to the study of the stochastic fixed-point
equation X
\stackrel{d}{=} \inf_{i \geq 1: T_i > 0} X_i/T_i and the connection
with its
additive counterpart $X \stackrel{d}{=} \sum_{i\ge 1}T_{i}X_{i}$
associated
with the smoothing transformation. Here $\stackrel{d}{=}$ means
equality in
distribution, $T := (T_i)_{i \geq 1}$ is a given sequence of
nonnegative random
variables and $X, X_1, ...$ is a sequence of nonnegative i.i.d. random
variables independent of $T$. We draw attention to the question of the
existence of nontrivial solutions and, in particular, of special
solutions
named $\alpha$-regular solutions $(\alpha>0)$. We give a complete
answer to the
question of when $\alpha$-regular solutions exist and prove that they
are
always mixtures of Weibull distributions or certain periodic variants.
We also
give a complete characterization of all fixed points of this kind. A
disintegration method which leads to the study of certain multiplicative
martingales and a pathwise renewal equation after a suitable transform
are the
key tools for our analysis. Finally, we provide corresponding results
for the
fixed points of the related additive equation mentioned above. To some
extent,
these results have been obtained earlier by Iksanov.
http://arxiv.org/abs/0907.0300
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8729. INTERLACEMENT PERCOLATION ON TRANSIENT WEIGHTED GRAPHS
Augusto Teixeira
In this article, we first extend the construction of random
interlacements,
introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general
setting
of transient weighted graphs. We prove the Harris-FKG inequality for
this model
and analyze some of its properties on specific classes of graphs. For
the case
of non-amenable graphs, we prove that the critical value u_* for the
percolation of the vacant set is finite. We also prove that, once G
satisfies
the isoperimetric inequality IS_6 (see (1.5)), u_* is positive for the
product
GxZ (where we endow Z with unit weights). When the graph under
consideration is
a tree, we are able to characterize the vacant cluster containing some
fixed
point in terms of a Bernoulli independent percolation process. For the
specific
case of regular trees, we obtain an explicit formula for the critical
value
u_*.
http://arxiv.org/abs/0907.0316
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8730. A FUNCTIONAL COMBINATORIAL CENTRAL LIMIT THEOREM
A. D. Barbour and Svante Janson
The paper establishes a functional version of the Hoeffding
combinatorial
central limit theorem. First, a pre-limiting Gaussian process
approximation is
defined, and is shown to be at a distance of the order of the
Lyapounov ratio
from the original random process. Distance is measured by comparison of
expectations of smooth functionals of the processes, and the argument
is by way
of Stein's method. The pre-limiting process is then shown, under weak
conditions, to converge to a Gaussian limit process. The theorem is
used to
describe the shape of random permutation tableaux.
http://arxiv.org/abs/0907.0347
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8731. STABILITY PROPERTIES OF LINEAR FILE-SHARING NETWORKS
L. Leskel\"a (MSA) and Philippe Robert (INRIA) and Florian Simatos
File-sharing networks are distributed systems used to disseminate
files among
a subset of the nodes of the Internet. A file is split into several
pieces
called chunks, the general simple principle is that once a node of the
system
has retrieved a chunk, it may become a server for this chunk. A
stochastic
model is considered for arrival times and durations of time to
download chunks.
One investigates the maximal arrival rate that such a network can
accommodate,
i.e., the conditions under which the Markov process describing this
network is
ergodic. Technical estimates related to the survival of interacting
branching
processes are key ingredients to establish the stability of these
systems.
Several cases are considered: networks with one and two chunks where a
complete
classification is obtained and several cases of a network with $n$
chunks.
http://arxiv.org/abs/0907.0375
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8732. SEMIMARTINGALE DECOMPOSITION OF CONVEX FUNCTIONS OF CONTINUOUS
SEMIMARTINGALES BY BROWNIAN PERTURBATION
Nastasiya F Grinberg
In this note we prove that the local martingale part of a convex
function f
of a d-dimensional semimartingale X=M+A can be wrtitten in terms of an
Ito
stochastic intergral of H(x), some measurable choice of subgradient of
fat x,
against M, the martingale part of X. This result was first proved by
Bouleau in
[2]. Here we present a new treatment of the problem.
http://arxiv.org/abs/0907.0382
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8733. MAJORITY DYNAMICS ON TREES AND THE DYNAMIC CAVITY METHOD
Yashodhan Kanoria and Andrea Montanari
An elector sits on each vertex of an infinite tree of degree $k$, and
has to
decide between two alternatives. At each time step, each elector
switches to
the opinion of the majority of her neighbors. We analyze this majority
process
when opinions are initialized to independent and identically
distributed random
variables.
In particular, we bound the threshold value of the initial bias
such that the
process converges to consensus. In order to prove an upper bound, we
characterize the process of a single node in the large $k$-limit. This
approach
is inspired by the theory of mean field spin-glass and can potentially
be
generalized to a wider class of models. We also derive a lower bound
that is
non-trivial for small, odd values of $k$.
http://arxiv.org/abs/0907.0449
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8734. A STRONG LOG-CONCAVITY PROPERTY FOR MEASURES ON BOOLEAN ALGEBRAS
Jeff Kahn and Michael Neiman
We introduce the antipodal pairs property for probability measures on
finite
Boolean algebras and prove that conditional versions imply strong
forms of
log-concavity. We give several applications of this fact, including
improvements of some results of Wagner; a new proof of a theorem of
Liggett
stating that ultra-log-concavity of sequences is preserved by
convolutions; and
some progress on a well-known log-concavity conjecture of J. Mason.
http://arxiv.org/abs/0907.0243
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8735. A CUT-OFF PHENOMENON IN LOCATION BASED RANDOM ACCESS GAMES WITH
IMPERFECT INFORMATION
Hazer Inaltekin and Mung Chiang and H. Vincent Poor
This paper analyzes the behavior of selfish transmitters under imperfect
location information. The scenario considered is that of a wireless
network
consisting of selfish nodes that are randomly distributed over the
network
domain according to a known probability distribution, and that are
interested
in communicating with a common sink node using common radio resources.
In this
scenario, the wireless nodes do not know the exact locations of their
competitors but rather have belief distributions about these locations.
Firstly, properties of the packet success probability curve as a
function of
the node-sink separation are obtained for such networks. Secondly, a
monotonicity property for the best-response strategies of selfish
nodes is
identified. That is, for any given strategies of competitors of a
node, there
exists a critical node-sink separation for this node such that its
best-response is to transmit when its distance to the sink node is
smaller than
this critical threshold, and to back off otherwise. Finally, necessary
and
sufficient conditions for a given strategy profile to be a Nash
equilibrium are
provided.
http://arxiv.org/abs/0907.0255
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8736. SELF-INTERSECTIONS OF RANDOM GEODESICS ON NEGATIVELY CURVED
SURFACES
Steven P. Lalley
We study the fluctuations of self-intersection counts of random geodesic
segments of length $t$ on a compact, negatively curved surface in the
limit of
large $t$. If the initial direction vector of the geodesic is chosen
according
to the \emph{Liouville measure}, then it is not difficult to show that
the
number $N (t)$ of self-intersections by time $t$ grows like $\kappa
t^{2}$,
where $\kappa =\kappa_{M}$ is a positive constant depending on the
surface $M$.
We show that (for a smooth modification of $N (t)$) the fluctuations
are of
size $t$, and the limit distribution is a weak limit of Gaussian
quadratic
forms. We also show that the fluctuations of \emph{localized} self-
intersection
counts (that is, only self-intersections in a fixed subset of $M$ are
counted)
are typically of size $t^{3/2}$, and the limit distribution is Gaussian.
http://arxiv.org/abs/0907.0259
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8737. REDUCING THE ISING MODEL TO MATCHINGS
Mark Huber (Claremont McKenna College) and Jenny Law (Duke University)
Canonical paths is one of the most powerful tools available to show
that a
Markov chain is rapidly mixing, thereby enabling approximate sampling
from
complex high dimensional distributions. Two success stories for the
canonical
paths method are chains for drawing matchings in a graph, and a chain
for a
version of the Ising model called the subgraphs world. In this paper,
it is
shown that a subgraphs world draw can be obtained by taking a draw from
matchings on a graph that is linear in the size of the original graph.
This
provides a partial answer to why canonical paths works so well for both
problems, as well as providing a new source of algorithms for the
Ising model.
For instance, this new reduction immediately yields a fully polynomial
time
approximation scheme for the Ising model on a bounded degree graph
when the
magnitization is bounded away from 0.
http://arxiv.org/abs/0907.0477
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8738. ZEROS OF A TWO-PARAMETER RANDOM WALK
Davar Khoshnevisan and Pal Revesz
We prove that the number gamma(N) of the zeros of a two-parameter simple
random walk in its first N-by-N time steps is almost surely equal to N
to the
power 1+o(1) as N goes to infinity. This is in contrast with our
earlier joint
effort with Z. Shi [4]; that work shows that the number of zero
crossings in
the first N-by-N time steps is N to the power (3/2)+o(1) as N goes to
infinity.
We prove also that the number of zeros on the diagonal in the first N
time
steps is (c+o(1)) log N as N goes to infinity, where c is 2\pi.
http://arxiv.org/abs/0907.0487
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8739. BRANCHING RANDOM WALKS IN SPACE-TIME RANDOM ENVIRONMENT:
SURVIVAL PROBABILITY, GLOBAL AND LOCAL GROWTH RATES
Francis Comets and Nobuo Yoshida
We study the survival probability and the growth rate for branching
random
walks in random environment (BRWRE). The particles perform simple
symmetric
random walks on the $d$-dimensional integer lattice, while at each
time unit,
they split into independent copies according to time-space i.i.d.
offspring
distributions. The BRWRE is naturally associated with the directed
polymers in
random environment (DPRE), for which the quantity called the free
energy is
well studied. We discuss the survival probability (both global and
local) for
BRWRE and give a criterion for its positivity in terms of the free
energy of
the associated DPRE. We also show that the global growth rate for the
number of
particles in BRWRE is given by the free energy of the associated DPRE,
though
the local growth rateis given by the directional free energy.
http://arxiv.org/abs/0907.0509
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8740. UNIFORM ESTIMATES FOR METASTABLE TRANSITION TIMES IN A COUPLED
BISTABLE SYSTEM
Florent Barret (CMAP) and Anton Bovier (IAM) and Sylvie M\'el\'eard
(CMAP)
We consider a coupled bistable N-particle system driven by a Brownian
noise,
with a strong coupling corresponding to the synchronised regime. Our
aim is to
obtain sharp estimates on the metastable transition times betwen the
two stable
states, both for fixed N and in the limit when N tends to infinity.
These
estimates would be the main step for a rigorous understanding of the
metastable
behavior of infinite dimensional systems, as the stochastically
perturbed
Ginzburg-Landau equation. The quantities of interest are objects of
potential
theory, as capacities and equilibrium measure. We prove estimates with
error
bounds that are uniform in the dimension of the system.
http://arxiv.org/abs/0907.0537
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8741. UPPER LARGE DEVIATIONS FOR MAXIMAL FLOWS THROUGH A TILTED CYLINDER
Marie Theret
We consider the standard first passage percolation model in $\ZZ^d$ for
$d\geq 2$ and we study the maximal flow from the upper half part to
the lower
half part (respectively from the top to the bottom) of a cylinder
whose basis
is a hyperrectangle of sidelength proportional to $n$ and whose height
is
$h(n)$ for a certain height function $h$. We denote this maximal flow by
$\tau_n$ (respectively $\phi_n$). We emphasize the fact that the
cylinder may
be tilted. We look at the probability that these flows, rescaled by
the surface
of the basis of the cylinder, are greater than $\nu(\vec{v})+\eps$ for
some
positive $\eps$, where $\nu(\vec{v})$ is the almost sure limit of the
rescaled
variable $\tau_n$ when $n$ goes to infinity. On one hand, we prove
that the
speed of decay of this probability in the case of the variable $\tau_n
$ depends
on the tail of the distribution of the capacities of the edges: it can
decays
exponentially fast with $n^{d-1}$, or with $n^{d-1} \min(n,h(n))$, or
at an
intermediate regime. On the other hand, we prove that this probability
in the
case of the variable $\phi_n$ decays exponentially fast with the
volume of the
cylinder as soon as the law of the capacity of the edges admits one
exponential
moment; the importance of this result is however limited by the fact
that
$\nu(\vec{v})$ is not in general the almost sure limit of the rescaled
maximal
flow $\phi_n$, but it is the case at least when the height $h(n)$ of the
cylinder is negligible compared to $n$.
http://arxiv.org/abs/0907.0614
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8742. CENTRAL LIMIT THEOREMS FOR MULTICOLOR URNS WITH DOMINATED COLORS
Patrizia Berti (Dip. di Matematica and Univ. Modena and Italy) and
Irene Crimaldi (Dip. Matematica, Univ. Bologna, Italy), Luca Pratelli
(Accademia
Navale, Livorno, Italy), Pietro Rigo (Dip. Economia politica e Metodi
quantitativi, Univ. Pavia, Italy)
An urn contains balls of d colors. At each time, a ball is drawn and
then
replaced together with a random number of balls of the same color.
Assuming
that some colors are dominated by others, we prove central limit
theorems. Some
statistical applications are discussed.
http://arxiv.org/abs/0907.0676
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8743. D\'EVIATIONS MOD\'ER\'EES DE LA DISTANCE CHIMIQUE
Olivier Garet (IECN) and R\'egine Marchand (IECN)
In this paper, we establish moderate deviations for the chemical
distance in
Bernoulli percolation. The chemical distance between two points is the
length
of the shortest open path between these two points. Thus, we study the
size of
random fluctuations around the mean value, and also the asymptotic
behavior of
this mean value. The estimates we obtain improve our knowledge of the
convergence to the asymptotic shape. Our proofs rely on concentration
inequalities proved by Boucheron, Lugosi and Massart, and also on the
approximation theory of subadditive functions initiated by Alexander.
http://arxiv.org/abs/0907.0697
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8744. MODERATE DEVIATIONS FOR THE CHEMICAL DISTANCE IN BERNOULLI
PERCOLATION
Olivier Garet (IECN) and R\'egine Marchand (IECN)
In this paper, we establish moderate deviations for the chemical
distance in
Bernoulli percolation. The chemical distance between two points is the
length
of the shortest open path between these two points. Thus, we study the
size of
random fluctuations around the mean value, and also the asymptotic
behavior of
this mean value. The estimates we obtain improve our knowledge of the
convergence to the asymptotic shape. Our proofs rely on concentration
inequalities proved by Boucheron, Lugosi and Massart, and also on the
approximation theory of subadditive functions initiated by Alexander.
http://arxiv.org/abs/0907.0698
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8745. ON THE PRESERVATION OF GIBBSIANNESS UNDER SYMBOL AMALGAMATION
Jean-Rene Chazottes and Edgardo Ugalde
Starting from the full-shift on a finite alphabet $A$, suppose we
confound
some symbols of $A$. This gives a new full shift on a new alphabet $B
$. The
amalgamation map, call it $\pi$, defines a `factor map', that is, a
continuous
transformation between $(A^\nn,T_A)$ and $(B^\nn,T_B)$ with the
property that
$\pi\circ T_A=T_B\circ \pi$, where $T_A$, resp. $T_B$, is the shift
map on
$A^\nn$, resp. $B^\nn$.
Given a regular function $\psi:A^\nn\to\rr$ (a `potential'), there
is a
unique Gibbs measure $\mu_\psi$. In this article, we prove that, for a
large
class of potentials, the pushforward measure $\mu_\psi\circ\pi^{-1}$
is still
Gibbsian for a potential $\phi:B^\nn\to\rr$ having a `bit less'
regularity than
$\psi$. In the special case where $\psi$ is a `2-symbol' potential,
the Gibbs
measure $\mu_\psi$ is none other than a Markov measure and the
amalgamation
$\pi$ defines a hidden Markov chain. In that special case, our theorem
can be
recast by saying that a hidden Markov chain is a Gibbs measure (for a H
\"older
potential).
http://arxiv.org/abs/0907.0528
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8746. POINCAR\'E INEQUALITY AND EXPONENTIAL INTEGRABILITY OF HITTING
TIMES FOR LINEAR DIFFUSIONS
D. Loukianova and O. Loukianov and Sh. Song
Let $X$ be a regular linear continuous positively recurrent Markov
process
with state space $\R$, scale function $S$ and speed measure $m$. For $a
\in \R$
denote B^+_a&=\sup_{x\geq a} \m(]x,+\infty[)(S(x)-S(a)) B^-_a&=\sup_{x
\leq a}
\m(]-\infty;x[)(S(a)-S(x)) We study some characteristic relations
between
$B^+_a$, $B^-_a$, the exponential moments of the hitting times $T_a$
of $X$,
the Hardy and Poincar\'e inequalities for the Dirichlet form
associated with
$X$. As a corollary, we establish the equivalence between the
existence of
exponential moments of the hitting times and the spectral gap of the
generator
of $X$.
http://arxiv.org/abs/0907.0762
---------------------------------------------------------------
8747. BOUNDARY HARNACK INEQUALITY FOR ALPHA-HARMONIC FUNCTIONS ON THE
SIERPI\'NSKI TRIANGLE
Kamil Kaleta and Mateusz Kwa\'snicki
We prove an uniform boundary Harnack inequality for nonnegative
functions
harmonic with respect to $\alpha$-stable process on the Sierpi{\'n}ski
triangle, where $\alpha \in (0, 1)$. Our result requires no regularity
assumptions on the domain of harmonicity.
http://arxiv.org/abs/0907.0793
---------------------------------------------------------------
8748. DUALITY AND INTERTWINING FOR DISCRETE MARKOV KERNELS: A RELATION
AND EXAMPLES
Thierry Huillet (LPTM) and Servet Martinez
We work out some relations between duality and intertwining in the
context of
discrete Markov chains, fixing up the background of previous relations
first
established for birth and death chains and their Siegmund duals. In
view of the
results, the monotone properties resulting from the Siegmund dual of
birth and
death chains are revisited in some detail, with emphasis on the non
neutral
Moran model. We also introduce an ultrametric type dual extending the
Siegmund
kernel. Finally we discuss the sharp dual, following closely the
Diaconis-Fill
study.
http://arxiv.org/abs/0907.0840
---------------------------------------------------------------
8749. DIFFUSION APPROXIMATION FOR THE COMPONENTS IN CRITICAL
INHOMOGENEOUS RANDOM GRAPHS OF RANK 1
Tatyana S. Turova
Consider the random graph on $n$ vertices $1, ..., n$. Each vertex $i$
is
assigned a type $X_i$ with $X_1, ..., X_n$ being independent identically
distributed as a nonnegative discrete random variable $X$. We assume
that ${\bf
E} X^3<\infty$. Given types of all vertices, an edge exists between
vertices
$i$ and $j$ independent of anything else and with probability $\min \{1,
\frac{X_iX_j}{n}(1+\frac{a}{n^{1/3}}) \}$. We study the critical
phase, which
is known to take place when ${\bf E} X^2=1$. We prove that normalized by
$n^{-2/3}$ the asymptotic joint distributions of component sizes of
the graph
equals the joint distribution of the excursions of a reflecting
Brownian motion
$B^a(s)$ with diffusion coefficient $\sqrt{{\bf E}X{\bf E}X^3}$ and
drift
$a-\frac{1}{({\bf E}X)^2}s$. This shows that finiteness of ${\bf
E}X^3$ is the
necessary condition for the diffusion limit. In particular, we
conclude that
the size of the largest connected component is of order $n^{2/3}$.
http://arxiv.org/abs/0907.0897
---------------------------------------------------------------
8750. DIFFERENTIABILITY OF QUADRATIC BSDE GENERATED BY CONTINUOUS
MARTINGALES AND HEDGING IN INCOMPLETE MARKETS
Peter Imkeller and Anthony Reveillac and Anja Richter
In this paper we consider a class of BSDE with drivers of quadratic
growth,
on a stochastic basis generated by continuous local martingales. We
first
derive the Markov property of a forward-backward system (FBSDE) if the
generating martingale is a strong Markov process. Then we establish the
differentiability of a FBSDE with respect to the initial value of its
forward
component. This enables us to obtain the main result of this article
which from
the perspective of a utility optimization interpretation of the
underlying
control problem on a financial market takes the following form. The
control
process of the BSDE steers the system into a random liability
depending on a
market external uncertainty and this way describes the optimal
derivative hedge
of the liability by investment in a capital market the dynamics of
which is
described by the forward component. This delta hedge is described in a
key
formula in terms of a derivative functional of the solution process
and the
correlation structure of the internal uncertainty captured by the
forward
process and the external uncertainty responsible for the market
incompleteness.
The formula largely extends the scope of validity of the results
obtained by
several authors in the Brownian setting, designed to give a genuinely
stochastic representation of the optimal delta hedge in the context of
cross
hedging insurance derivatives generalizing the derivative hedge in the
Black-Scholes model. Of course, Malliavin's calculus needed in the
Brownian
setting is not available in the general local martingale framework. We
replace
it by new tools based on stochastic calculus techniques.
http://arxiv.org/abs/0907.0941
---------------------------------------------------------------
8751. ON THE ORTHOGONAL COMPONENT OF BSDES IN A MARKOVIAN SETTING
Anthony R\'eveillac
In this Note we consider a quadratic backward stochastic differential
equation (BSDE) driven by a continuous martingale $M$ and whose
generator is a
deterministic function. We prove (in Theorem \ref{theorem:main}) that
if $M$ is
a strong homogeneous Markov process and if the BSDE has the form
\eqref{BSDE}
then the unique solution $(Y,Z,N)$ of the BSDE is reduced to $(Y,Z)$,
\textit{i.e.} the orthogonal martingale $N$ is equal to zero showing
that in a
Markovian setting the "usual" solution $(Y,Z)$ has not to be completed
by a
strongly orthogonal even if $M$ does not enjoy the martingale
representation
property.
http://arxiv.org/abs/0907.1071
---------------------------------------------------------------
8752. A CONSTRUCTIVE APPROACH TO THE MONGE-KANTOROVICH PROBLEM FOR
CHAINS OF INFINITE ORDER
Antonio Galves and Nancy L. Garcia and Clementine Prieur
We propose a constructive approach to solve the Monge-Kantorovich
problem for
chains of infinite order on a finite alphabet with an additive cost
function.
From this constructive description of the Kantorovich coupling we
obtain, for
any $\epsilon > 0$, a perfect simulation algorithm for sampling from an
$\epsilon$-approximating coupling which assigns to the cost function an
expectation which is $\epsilon$-close to the minimum cost. Our
approach is
based on a regenerative scheme which enable us to construct the
Kantorovich
coupling as a mixture of product measures.
http://arxiv.org/abs/0907.1113
---------------------------------------------------------------
8753. HSU-ROBBINS AND SPITZER'S THEOREMS FOR THE VARIATIONS OF
FRACTIONAL BROWNIAN MOTION
Ciprian Tudor (CES and Samos)
Using recent results on the behavior of multiple Wiener-It\^o
integrals based
on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for
sequences of
correlated random variables related to the increments of the fractional
Brownian motion.
http://arxiv.org/abs/0907.1116
---------------------------------------------------------------
8754. CONVERGENCE TO L\'EVY STABLE PROCESSES UNDER STRONG MIXING
CONDITIONS
Marta Tyran-Kaminska
For a strictly stationary sequence of random vectors in $\mathbb{R}^d$
we
study convergence of partial sums processes to L\'evy stable process
in the
Skorohod space with $J_1$-topology. We identify necessary and sufficient
conditions for such convergence and provide sufficient conditions when
the
stationary sequence is strongly mixing.
http://arxiv.org/abs/0907.1185
---------------------------------------------------------------
8755. AN APPLICATION TO CREDIT RISK OF A HYBRID MONTE CARLO-OPTIMAL
QUANTIZATION METHOD
Giorgia Callegaro and Abass Sagna (PMA)
In this paper we use a hybrid Monte Carlo-Optimal quantization method to
approximate the conditional survival probabilities of a firm, given a
structural model for its credit defaul, under partial information. We
consider
the case when the firm's value is a non-observable stochastic process $
(V_t)_{t
\geq 0}$ and inverstors in the market have access to a process $
(S_t)_{t \geq
0}$, whose value at each time t is related to $(V_s, s \leq t)$. We are
interested in the computation of the conditional survival
probabilities of the
firm given the "investor information". As a application, we analyse
the shape
of the credit spread curve for zero coupon bonds in two examples.
http://arxiv.org/abs/0907.0645
---------------------------------------------------------------
8756. PERIMETER AND AREA OF THE CONVEX HULL OF N PLANAR BROWNIAN MOTIONS
Julien Randon-Furling and Satya N. Majumdar and Alain Comtet
We compute exactly the mean perimeter and area of the convex hull of N
independent planar Brownian paths each of duration T, both for open
and closed
paths. We show that the mean perimeter < L_N > = \alpha_N, \sqrt{T}
and the
mean area <A_N> = \beta_N T for all T. The prefactors \alpha_N and
\beta_N,
computed exactly for all N, increase very slowly (logarithmically) with
increasing N. This slow growth is a consequence of extreme value
statistics and
has interesting implication in ecological context in estimating the
home range
of a herd of animals with population size N.
http://arxiv.org/abs/0907.0921
---------------------------------------------------------------
8757. DISTRIBUTED RANDOM ACCESS ALGORITHM: SCHEDULING AND CONGESION
CONTROL
Libin Jiang and Devavrat Shah and Jinwoo Shin and Jean Walrand
This paper provides proofs of the rate stability, Harris recurrence, and
epsilon-optimality of CSMA algorithms where the backoff parameter of
each node
is based on its backlog. These algorithms require only local
information and
are easy to implement.
The setup is a network of wireless nodes with a fixed conflict
graph that
identifies pairs of nodes whose simultaneous transmissions conflict.
The paper
studies two algorithms. The first algorithm schedules transmissions to
keep up
with given arrival rates of packets. The second algorithm controls the
arrivals
in addition to the scheduling and attempts to maximize the sum of the
utilities
of the flows of packets at the different nodes. For the first
algorithm, the
paper proves rate stability for strictly feasible arrival rates and
also Harris
recurrence of the queues. For the second algorithm, the paper proves the
epsilon-optimality. Both algorithms operate with strictly local
information in
the case of decreasing step sizes, and operate with the additional
information
of the number of nodes in the network in the case of constant step size.
http://arxiv.org/abs/0907.1266
---------------------------------------------------------------
8758. DYNKIN'S ISOMORPHISM THEOREM AND THE STOCHASTIC HEAT EQUATION
Nathalie Eisenbaum and Mohammud Foondun and Davar Khoshnevisan
Consider the stochastic heat equation $\partial_t u = \sL u + \dot{W}
$, where
$\sL$ is the generator of a [Borel right] Markov process in duality.
We show
that the solution is locally mutually absolutely continuous with
respect to a
smooth perturbation of the Gaussian process that is associated, via
Dynkin's
isomorphism theorem, to the local times of the replica-symmetric
process that
corresponds to $\sL$.In the case that $\sL$ is the generator of a L\'evy
process on $\R^d$, our result gives a probabilistic explanation of the
recent
findings of Foondun et al.
http://arxiv.org/abs/0907.1316
---------------------------------------------------------------
8759. ON THE DISCRETIZATION OF BACKWARD DOUBLY STOCHASTIC
DIFFERENTIAL EQUATIONS
Omar Aboura (CES and Samos)
In this paper, we are dealing with the approximation of the process
(Y,Z)
solution to the backward doubly stochastic differential equation with
the
forward process X . After proving the L2-regularity of Z, we use the
Euler
scheme to discretize X and the Zhang approach in order to give a
discretization
scheme of the process (Y,Z).
http://arxiv.org/abs/0907.1406
---------------------------------------------------------------
8760. EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR FOKKER-PLANCK
EQUATIONS ON HILBERT SPACES
Vladimir Bogachev and Giuseppe Da Prato and Michael Roeckner
We consider a stochastic differential equation in a Hilbert space with
time-dependent coefficients for which no general existence and
uniqueness
results are known. We prove, under suitable assumptions, existence and
uniqueness of a measure valued solution, for the corresponding Fokker--
Planck
equation. In particular, we verify the Chapman--Kolmogorov equations
and get an
evolution system of transition probabilities for the stochastic dynamics
informally given by the stochastic differential equation.
http://arxiv.org/abs/0907.1431
---------------------------------------------------------------
8761. LIMIT DISTRIBUTIONS FOR LARGE P\'OLYA URNS
Brigitte Chauvin and Nicolas Pouyanne and R\'eda Sahnoun
We consider a two colors P\'olya urn with balance $S$. Assume it is a
\emph{large} urn \emph{i.e.} the second eigenvalue $m$ of the
replacement
matrix satisfies $1/2<m/S\leq 1$. After $n$ drawings, the composition
vector
has asymptotically a first deterministic term of order $n$ and a
second random
term of order $n^{m/S}$. The object of interest is the limit
distribution of
this random term.
The method consists in embedding the discrete time urn in
continuous time,
getting a two type branching process. The dislocation equations
associated with
this process lead to a system of two differential equations satisfied
by the
Fourier transforms of the limit distributions. The resolution is
carried out
and it turns out that the Fourier transforms are explicitely related
to Abelian
integrals on the Fermat curve of degree $m$.
http://arxiv.org/abs/0907.1477
---------------------------------------------------------------
8762. INHOMOGENEITY AND UNIVERSALITY: OFF-CRITICAL BEHAVIOR OF
INTERFACES
Pierre Nolin
We further study the interfaces arising in a situation of
inhomogeneity. More
precisely, we identify a characteristic length for the gradient
percolation
model, that enables us to tighten previous estimates established for
it. This
allows to construct non-trivial scaling limits: the limiting objects
share some
properties with critical percolation interfaces, but locally, they
rather
behave like off-critical percolation interfaces.
http://arxiv.org/abs/0907.1495
---------------------------------------------------------------
8763. LES PROBABILIT\'ES D\'EFAILLANCE COMME INDICATEURS DE
PERFORMANCE DES BARRI\`ERES TECHNIQUES DE S\'ECURIT\'E ? APPROCHE
ANALYTIQUE
Florent Brissaud (INERIS and UTT) and Brice Lanternier (INERIS)
French environmental laws require industrialists to include probability
criteria in risk assessments, especially to define confidence levels
for risk
management measures. This paper presents the failure probabilities as
efficient
indicators for technical safety barrier performances. Generic formulas
are
proposed to evaluate these probabilities, including failure rate,
barrier
architecture, full and partial proof tests. In many cases, these
results can be
directly used to assess safety barrier confidence levels.
http://arxiv.org/abs/0907.1516
---------------------------------------------------------------
8764. A REMARK ON ZEROS OF BROWNIAN MOTION
Weber Michel
Let $ \{W(t), t\ge 0\}$ be a standard Brownian motion. If $I$ is a
bounded
interval on which $W $ has no zero, an almost sure lower bound to
$\inf\{|W(t)|, t\in I\}$ can be provided, when $I$ is taken from a given
countable family of intervals covering the positive half-line.
http://arxiv.org/abs/0907.1572
---------------------------------------------------------------
8765. SYMMETRIZATION OF L\'EVY PROCESSES AND APPLICATIONS
Rodrigo Banuelos and Pedro J. Mendez-Hernandez
It is shown that many of the classical generalized isoperimetric
inequalities
for the Laplacian when viewed in terms of Brownian motion extend to a
wide
class of Levy processes. The results are derived from the multiple
integral
inequalities of Brascamp, Lieb and Luttinger but the probabilistic
structure of
the processes plays a crucial role in the proofs.
http://arxiv.org/abs/0907.1598
---------------------------------------------------------------
8766. ARE FRACTIONAL BROWNIAN MOTIONS PREDICTABLE?
Adam Jakubowski
We provide a device, called the local predictor, which extends the
idea of
the predictable compensator. It is shown that a fBm with the Hurst index
greater than 1/2 coincides with its local predictor while fBm with the
Hurst
index smaller than 1/2 does not admit any local predictor. The local
predictor
of a martingale (in particular: Brownian motion) trivially exists and
equals 0.
http://arxiv.org/abs/0907.1618
---------------------------------------------------------------
8767. RANDOM WALKS ON DISCRETE CYLINDERS WITH LARGE BASES AND RANDOM
INTERLACEMENTS
David Windisch
Following the recent work of Sznitman (arXiv:0805.4516), we
investigate the
microscopic picture induced by a random walk trajectory on a cylinder
of the
form G_N x Z, where G_N is a large finite connected weighted graph,
and relate
it to the model of random interlacements on infinite transient
weighted graphs.
Under suitable assumptions, the set of points not visited by the
random walk
until a time of order |G_N|^2 in a neighborhood of a point with Z-
component of
order |G_N| converges in distribution to the law of the vacant set of
a random
interlacement on a certain limit model describing the structure of the
graph in
the neighborhood of the point. The level of the random interlacement
depends on
the local time of a Brownian motion. The result also describes the limit
behavior of the joint distribution of the local pictures in the
neighborhood of
several distant points with possibly different limit models. As
examples of
G_N, we treat the d-dimensional box of side length N, the Sierpinski
graph of
depth N and the d-ary tree of depth N, where d >= 2.
http://arxiv.org/abs/0907.1627
---------------------------------------------------------------
8768. FLUCTUATIONS OF THE NODAL LENGTH OF RANDOM SPHERICAL HARMONICS
Igor Wigman
Using the multiplicities of the Laplace eigenspace on the sphere (the
space
of spherical harmonics) we endow the space with Gaussian probability
measure.
This induces a notion of random Gaussian spherical harmonics of degree
$n$
having Laplace eigenvalue $E=n(n+1)$. We study the length distribution
of the
nodal lines of random spherical harmonics.
It is known that the expected length is of order $n$. It is natural
to
conjecture that the variance should be of order $n$, due to the natural
scaling. Our principal result is that, due to an unexpected
cancelation, the
variance of the nodal length of random spherical harmonics is of order
$\log{n}$. This behaviour is consistent to the one predicted by Berry
for nodal
lines on chaotic billiards (Random Wave Model). In addition we find
that a
similar result is applicable for "generic" linear statistics of the
nodal
lines.
http://arxiv.org/abs/0907.1648
---------------------------------------------------------------
8769. SOME ALMOST SURE RESULTS FOR UNBOUNDED FUNCTIONS OF INTERMITTENT
MAPS AND THEIR ASSOCIATED MARKOV CHAINS
Jerome Dedecker (LSTA) and Sebastien Gouezel (IRMAR) and Florence
Merlevede (LAMA)
We consider a large class of piecewise expanding maps T of [0,1] with a
neutral fixed point, and their associated Markov chain Y_i whose
transition
kernel is the Perron-Frobenius operator of T with respect to the
absolutely
continuous invariant probability measure. We give a large class of
unbounded
functions f for which the partial sums of f\circ T^i satisfy both a
central
limit theorem and a bounded law of the iterated logarithm. For the
same class,
we prove that the partial sums of f(Y_i) satisfy a strong invariance
principle.
When the class is larger, so that the partial sums of f\circ T^i may
belong to
the domain of normal attraction of a stable law of index p\in (1, 2),
we show
that the almost sure rates of convergence in the strong law of large
numbers
are the same as in the corresponding i.i.d. case.
http://arxiv.org/abs/0907.1403
---------------------------------------------------------------
8770. ALMOST SURE INVARIANCE PRINCIPLE FOR DYNAMICAL SYSTEMS BY
SPECTRAL METHODS
Sebastien Gouezel (IRMAR)
We prove the almost sure invariance principle for stationary R^d--valued
processes (with dimension-independent very precise error terms),
solely under a
strong assumption on the characteristic functions of these processes.
This
assumption is easy to check for large classes of dynamical systems or
Markov
chains, using strong or weak spectral perturbation arguments.
http://arxiv.org/abs/0907.1404
---------------------------------------------------------------
8771. FOREST FIRES ON $\Z_+$ WITH IGNITION ONLY AT 0
Stanislav Volkov
We consider a version of the forest fire model on graph $G$, where each
vertex of a graph becomes occupied with rate one. A fixed vertex $v_0$
is hit
by lightning with the same rate, and when this occurs, the whole
cluster of
occupied vertices containing $v_0$ is burnt out. We show that when
$G=Z_{+}$,
the times between consecutive burnouts at vertex $n$, divided by $\log
n$,
converge weakly as $n\to\infty$ to a random variable which
distribution is
$1-\rho(x)$ where $\rho(x)$ is the Dickman function.
We also show that on transitive graphs with a non-trivial site
percolation
threshold and one infinite cluster at most, the distributions of the
time till
the first burnout of {\it any} vertex have exponential tails.
Finally, we give an elementary proof of an interesting limit:
$\lim_{n\to\infty} \sum_{k=1}^n {n \choose k} (-1)^k \log k -\log\log
n=\gamma$.
http://arxiv.org/abs/0907.1821
---------------------------------------------------------------
8772. QUEUEING WITH NEIGHBOURS
Vadim Shcherbakov and Stanislav Volkov
In this paper we study asymptotic behaviour of a growth process
generated by
a semi-deterministic variant of cooperative sequential adsorption
model (CSA).
This model can also be viewed as a particular queueing system with local
interactions. We show that quite limited randomness of the model still
generates a rich collection of possible limiting behaviours.
http://arxiv.org/abs/0907.1826
---------------------------------------------------------------
8773. ESTIMATES ON THE SPEEDUP AND SLOWDOWN FOR A DIFFUSION IN A
DRIFTED BROWNIAN POTENTIAL
Gabriel Faraud
We study a model of diffusion in a brownian potential. This model was
firstly
introduced by T. Brox (1986) as a continuous time analogue of random
walk in
random environment. We estimate the deviations of this process above
or under
its typical behavior. Our results rely on different tools such as a
representation introduced by Y. Hu, Z. Shi and M. Yor, Kotani's lemma,
introduced at first by K. Kawazu and H. Tanaka (1997), and a
decomposition of
hitting times developed in a recent article by A. Fribergh, N. Gantert
and S.
Popov (2008). Our results are in agreement with their results in the
discrete
case.
http://arxiv.org/abs/0907.1864
---------------------------------------------------------------
8774. HIDDEN MARKOV PROCESSES IN THE CONTEXT OF SYMBOLIC DYNAMICS
Mike Boyle (University of Maryland) and Karl Petersen (University of
North Carolina)
In an effort to aid communication among different fields and perhaps
facilitate progress on problems common to all of them, this article
discusses
hidden Markov processes from several viewpoints, especially that of
symbolic
dynamics, where they are known as sofic measures, or continuous shift-
commuting
images of Markov measures. It provides background, describes known
tools and
methods, surveys some of the literature, and proposes several open
problems.
http://arxiv.org/abs/0907.1858
---------------------------------------------------------------
8775. THE TRIANGLE AND THE OPEN TRIANGLE
Gady Kozma
We show that for percolation on any transitive graph, the triangle
condition
implies the open triangle condition.
http://arxiv.org/abs/0907.1959
---------------------------------------------------------------
8776. LP-SOLUTION OF BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS
Auguste Aman (LMAI)
In this paper, our goal is solving backward doubly stochastic
differential
equation (BDSDE for short) under weak assumptions on the data. The
first part
of the paper is devoted to the development of some new technical
aspects of
stochastic calculus related to BDSDEs. Then we derive a priori
estimates and
prove existence and uniqueness of solutions, extending the results of
Pardoux
and Peng \cite{PP1} to the case where the solution is taked in $L^{p},
p>1$ and
the monotonicity conditions are satisfied. This study is limited to
deterministic terminal time.
http://arxiv.org/abs/0907.1983
---------------------------------------------------------------
8777. ON THE OPTIMAL AMOUNT OF EXPERIMENTATION IN SEQUENTIAL DECISION
PROBLEMS
Dinah Rosenberg and Eilon Solan and Nicolas Vieille
We provide a tight bound on the amount of experimentation under the
optimal
strategy in sequential decision problems. We show the applicability of
the
result by providing a bound on the cut-off in a one-arm bandit problem.
http://arxiv.org/abs/0907.2002
---------------------------------------------------------------
8778. NEW RATES FOR EXPONENTIAL APPROXIMATION AND THE THEOREMS OF R
\'ENYI AND YAGLOM
Erol Pek\"oz and Adrian R\"ollin
We introduce two abstract theorems that reduce a variety of complex
exponential distributional approximation problems to the construction of
couplings. These are applied to obtain rates of convergence with
respect to the
Wasserstein and Kolmogorov metrics for the theorem of R\'enyi on
random sums
and generalizations of it, hitting times for Markov chains, and to
obtain a new
rate for the classical theorem of Yaglom on the exponential asymptotic
behavior
of a critical Galton-Watson process conditioned on non-extinction. The
primary
tools are an adaptation of Stein's method, Stein couplings, as well as
the
equilibrium distributional transformation from renewal theory.
http://arxiv.org/abs/0907.2009
---------------------------------------------------------------
8779. L$^{P}$-SOLUTION OF REFLECTED GENERALIZED BSDES WITH NON-
LIPSCHITZ COEFFICIENTS
Auguste Aman (LMAI)
In this paper, we continue in solving reflected generalized backward
stochastic differential equations (RGBSDE for short) and fixed
terminal time
with use some new technical aspects of the stochastic calculus related
to the
reflected generalized BSDE. Here, existence and uniqueness of solution
is
proved under a non-Lipschitz condition on the coefficients.
http://arxiv.org/abs/0907.2032
---------------------------------------------------------------
8780. NUMERICAL SCHEME FOR BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL
EQUATIONS
Auguste Aman (LMAI)
We study a discrete-time approximation for solutions of systems of
decoupled
forward-backward doubly stochastic differential equations (FBDSDEs).
Assuming
that the coefficients are Lipschitz-continuous, we prove the
convergence of the
scheme when the step of time discretization, $|\pi|$ goes to zero. The
rate of
convergence is exactly equal to $|\pi|^{1/2}$. The proof is based on a
generalization of a remarkable result on the $^{2}$-regularity of the
solution
of the backward equation derived by J. Zhang
http://arxiv.org/abs/0907.2035
---------------------------------------------------------------
8781. HOMEOMORPHISM OF SOLUTIONS TO BACKWARD DOUBLY SDES AND
APPLICATIONS
Auguste Aman (LMAI)
In this paper we study the homeomorphic properties of the solutions to
one
dimensional backward doubly stochastic differential equations under
suitable
assumptions, where the terminal values depend on a real parameter.
Then, we
apply them to the solutions for a class of second order quasilinear
parabolic
stochastic partial differential equations.
http://arxiv.org/abs/0907.2036
---------------------------------------------------------------
8782. REFLECTED GENERALIZED BACKWARD DOUBLY SDES DRIVEN BY L\'EVY
PROCESSES AND APPLICATIONS
Auguste Aman (LMAI)
In this paper, a class of reflected generalized backward doubly
stochastic
differential equations (reflected GBDSDEs in short) driven by Teugels
martingales associated with L\'{e}vy process and the integral with
respect to
an adapted continuous increasing process is investigated. We obtain the
existence and uniqueness of solutions to these equations. A
probabilistic
interpretation for solutions to a class of reflected stochastic partial
differential integral equations (PDIEs in short) with a nonlinear
Neumann
boundary condition is given.
http://arxiv.org/abs/0907.2037
---------------------------------------------------------------
8783. STOCHASTIC 2D HYDRODYNAMICAL SYSTEMS: SUPPORT THEOREM
Igor Chueshov and Annie Millet (SAMOS and Ces and Pma)
We deal with a class of abstract nonlinear stochastic models with
multiplicative noise, which covers many 2D hydrodynamical models
including the
2D Navier-Stokes equation, 2D MHD models and 2D magnetic B\'enard
problems as
well as some shell models of turbulence. Our main result describes the
support
of the distribution of solutions. Both inclusions are proved by means
of a
general result of convergence in probability for non linear stochastic
PDEs
driven by a Hilbert-valued Brownian motion and some adapted finite
dimensional
approximation of this process.
http://arxiv.org/abs/0907.2100
---------------------------------------------------------------
8784. PERFECT SIMULATION FOR STOCHASTIC CHAINS WITH UNBOUNDED VARIABLE
LENGTH MEMORY
Alexsandro Gallo
We present a new perfect simulation algorithm for stationary chains
(indexed
by $\mathbb{Z}$) having unbounded variable length memory. This is the
class of
infinite memory chains for which the family of transition
probabilities is
represented through the form of a \emph{probabilistic context tree}. Our
condition is expressed in terms of the structure of the context tree. In
particular, we do not assume the continuity of the family of transition
probabilities. We give an explicit construction of the chain using a
sequence
of i.i.d. random variables uniformly distributed in $[0,1[$.
http://arxiv.org/abs/0907.2150
---------------------------------------------------------------
8785. ON THE DOMINATION OF RANDOM WALK ON A DISCRETE CYLINDER BY
RANDOM INTERLACEMENTS
Alain-Sol Sznitman
We consider simple random walk on a discrete cylinder with base a large
d-dimensional torus of side-length N, when d is two or more. We
develop a
stochastic domination control on the local picture left by the random
walk in
boxes of side-length almost of order N, at certain random times
comparable to
the square of the number of sites in the base. We show a domination
control in
terms of the trace left in similar boxes by random interlacements in the
infinite (d+1)-dimensional cubic lattice at a suitably adjusted level.
As an
application we derive a lower bound on the disconnection time of the
discrete
cylinder, which as a by-product shows the tightness of the laws of the
ratio of
the square of the number of sites in the base to the disconnection
time. This
fact had previously only been established when d is at least 17, in
arXiv:
math/0701414.
http://arxiv.org/abs/0907.2184
---------------------------------------------------------------
8786. A PATH GUESSING GAME WITH WAGERING
Marcus Pendergrass
We consider a two-player game in which the first player (the Guesser)
tries
to guess, edge-by-edge, the path that second player (the Chooser)
takes through
a directed graph. At each step, the Guesser makes a wager as to the
correctness
of her guess, and receives a payoff proportional to her wager if she is
correct. We derive optimal strategies for both players for various
classes of
graphs, and describe the Markov-chain dynamics of the game under
optimal play.
These results are applied to the infinite-duration Lying Oracle Game,
in which
the Guesser must use information provided by an unreliable Oracle to
predict
the outcome of a coin toss.
http://arxiv.org/abs/0907.2196
---------------------------------------------------------------
8787. ON THE PHILOSOPHY OF CRAM\'ER-RAO-BHATTACHARYA INEQUALITIES IN
QUANTUM STATISTICS
K. R. Parthasarathy
To any parametric family of states of a finite level quantum system we
associate a space of Fisher maps and introduce the natural notions of
Cram\'er-Rao-Bhattacharya tensor and Fisher information form. This
leads us to
an abstract Cram\'er-Rao-Bhattacharya lower bound for the covariance
matrix of
any finite number of unbiased estimators of parameteric functions. A
number of
illustrative examples is included. Modulo technical assumptions of
various
kinds our methods can be applied to infinite level quantum systems as
well as
parametric families of classical probability distributions on Borel
spaces.
http://arxiv.org/abs/0907.2210
---------------------------------------------------------------
8788. OPTIMAL INVESTMENT ON FINITE HORIZON WITH RANDOM DISCRETE ORDER
FLOW IN ILLIQUID MARKETS
Paul Gassiat (PMA) and Huyen Pham (PMA and CREST) and Mihai Sirbu
We study the problem of optimal portfolio selection in an illiquid
market
with discrete order flow. In this market, bids and offers are not
available at
any time but trading occurs more frequently near a terminal horizon. The
investor can observe and trade the risky asset only at exogenous
random times
corresponding to the order flow given by an inhomogenous Poisson
process. By
using a direct dynamic programming approach, we first derive and solve
the
fixed point dynamic programming equation satisfied by the value
function, and
then perform a verification argument which provides the existence and
characterization of optimal trading strategies. We prove the
convergence of the
optimal performance, when the deterministic intensity of the order flow
approaches infinity at any time, to the optimal expected utility for an
investor trading continuously in a perfectly liquid market model with
no-short
sale constraints.
http://arxiv.org/abs/0907.2203
---------------------------------------------------------------
8789. A SHAPE THEOREM FOR RIEMANNIAN FIRST-PASSAGE PERCOLATION
Tom LaGatta and Jan Wehr
Riemannian first-passage percolation (FPP) is a continuum analogue of
standard FPP on the lattice, where the discrete passage times of
standard FPP
are replaced by a random Riemannian metric. We prove a shape theorem
for this
model--that balls in this metric grow linearly in time--and from this
conclude
that the metric is complete.
http://arxiv.org/abs/0907.2228
---------------------------------------------------------------
8790. HEAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS ITERATED
LIPSCHITZ MAPS
Mariusz Mirek
We consider the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$
defined by
the stochastic recursion $X_{n}^{x}=\p_{\theta_{n}}(X_{n-1}^{x})$,
starting at
$x\in\R^d$, where $\theta_{1}, \theta_{2}, ...$ are i.i.d. random
variables
taking their values in a matric space $(\Theta, d)$ and
$\p_{\theta_{n}}:\R^d\mapsto\R^d$ are Lipschitz maps. Assume that the
Markov
chain has a unique stationary measure $\nu$. Under appropriate
assumptions on
$\p_{\theta_n}$ we will show that the measure $\nu$ has a heavy tail
with the
exponent $\alpha>0$ i.e. $\nu(\{x\in\R^d: |x|>t\})\asymp t^{-\alpha}$.
Using
this result we show that properly normalized Birkhoff sums $S_n^x=
\sum_{k=0}^n
X_k^x$, converge in law to an $\alpha$--stable laws for $\alpha\in(0,
2]$.
http://arxiv.org/abs/0907.2261
---------------------------------------------------------------
8791. UNIFORM MODULUS OF CONTINUITY OF RANDOM FIELDS
Yimin Xiao
A sufficient condition for the uniform modulus of continuity of a random
field $X = \{X(t), t \in \R^N\}$ is provided. The result is applicable
to
random fields with heavy-tailed distribution such as stable random
fields.
http://arxiv.org/abs/0907.2291
---------------------------------------------------------------
8792. SPECTRAL ANALYSIS OF MULTI-DIMENSIONAL SELF-SIMILAR MARKOV
PROCESSES
N. Modarresi and S. Rezakhah
In this paper we consider a wide sense discrete scale invariant
process with
scale $l>1$. We consider to have $T$ samples at each scale, and choose
$\alpha$
by the equality $l=\alpha^T$. Our special scheme of sampling is to
choose our
samples at discrete points $\alpha^k, k\in W$. So we provide a
discrete time
wide sense scale invariant(DT-SI) process. We find the spectral
representation
of the covariance function of such DT-SI process. By providing
harmonic like
representation of multi-dimensional self-similar processes, spectral
density
function of them are presented. We also consider a discrete time scale
invariance Markov(DT-SIM) process with the above scheme of sampling at
points
$\alpha^k, k\in {\bf W}$ and show that the spectral density matrix of
DT-SIM
process and its associated $T$-dimensional self-similar Markov process
is fully
specified by $\{R_{j}^H(1),R_{j}^H(0),j=0, 1, ..., T-1\}$ where
$R_{j}^H(\tau)=\mathrm{Cov}\big(X(\alpha^{j+\tau}),X(\alpha^j)\big)$
http://arxiv.org/abs/0907.2295
---------------------------------------------------------------
8793. HEAT KERNEL UPPER BOUNDS ON LONG RANGE PERCOLATION CLUSTERS
Nicholas Crawford and Allan Sly
In this paper, we derive upper bounds for the heat kernel of the simple
random walk on the infinite cluster of a supercritical long range
percolation
process. For any $d \geq 1$ and for any exponent $s \in (d, (d+2)
\wedge 2d)$
giving the rate of decay of the percolation process, we show that the
return
probability decays like $t^{-\ffrac{d}{s-d}}$ up to logarithmic
corrections,
where $t$ denotes the time the walk is run. Moreover, our methods also
yield
generalized bounds on the spectral gap of the dynamics and on the
diameter of
the largest component in a box.
Besides its intrinsic interest, the main result is needed for a
companion
paper studying the scaling limit of simple random walk on the infinite
cluster.
http://arxiv.org/abs/0907.2434
---------------------------------------------------------------
8794. A GRAPH-BASED EQUILIBRIUM PROBLEM FOR THE LIMITING DISTRIBUTION
OF NON-INTERSECTING BROWNIAN MOTIONS AT LOW TEMPERATURE
Steven Delvaux and Arno B.J. Kuijlaars
We consider n non-intersecting Brownian motion paths with p prescribed
starting positions at time t=0 and q prescribed ending positions at
time t=1.
The positions of the paths at any intermediate time are a
determinantal point
process, which in the case p=1 is equivalent to the eigenvalue
distribution of
a random matrix from the Gaussian unitary ensemble with external
source. For
general p and q, we show that if a temperature parameter is
sufficiently small,
then the distribution of the Brownian paths is characterized in the
large n
limit by a vector equilibrium problem with an interaction matrix that
is based
on a bipartite planar graph. Our proof is based on a steepest descent
analysis
of an associated (p+q) by (p+q) matrix valued Riemann-Hilbert problem
whose
solution is built out of multiple orthogonal polynomials. A new
feature of the
steepest descent analysis is a systematic opening of a large number of
global
lenses.
http://arxiv.org/abs/0907.2310
---------------------------------------------------------------
8795. 3-CONNECTED CORES IN RANDOM PLANAR GRAPHS
Nikolaos Fountoulakis and Konstantinos Panagiotou
The study of the structural properties of large random planar graphs has
become in recent years a field of intense research in computer science
and
discrete mathematics. Nowadays, a random planar graph is an important
and
challenging model for evaluating methods that are developed to study
properties
of random graphs from classes with structural side constraints. In
this paper
we focus on the structure of random biconnected planar graphs
regarding the
sizes of their 3-connected building blocks, which we call cores. In
fact, we
prove a general theorem regarding random biconnected graphs. If B_n is
a graph
drawn uniformly at random from a class B of labeled biconnected
graphs, then we
show that with probability 1-o(1) B_n belongs to exactly one of the
following
categories:
(i) Either there is a unique giant core in B_n, that is, there is a
0 < c < 1
such that the largest core contains ~ cn vertices, and every other core
contains at most n^a vertices, where 0 < a < 1; (ii) or all cores of B_n
contain O(log n) vertices.
Moreover, we find the critical condition that determines the
category to
which B_n belongs, and also provide sharp concentration results for
the counts
of cores of all sizes between 1 and n. As a corollary, we obtain that
a random
biconnected planar graph belongs to category (i), where in particular
c =
0.765... and a = 2/3.
http://arxiv.org/abs/0907.2326
---------------------------------------------------------------
8796. ON DIVERGENCE FORM SPDES WITH GROWING COEFFICIENTS IN
$W^{1}_{2}$ SPACES WITHOUT WEIGHTS
N.V. Krylov
We consider divergence form uniformly parabolic SPDEs with bounded and
measurable leading coefficients and possibly growing lower-order
coefficients
in the deterministic part of the equations. We look for solutions
which are
summable to the second power with respect to the usual Lebesgue
measure along
with their first derivatives with respect to the spatial variable.
http://arxiv.org/abs/0907.2467
---------------------------------------------------------------
8797. ON THE STRUCTURE OF GAUSSIAN RANDOM VARIABLES
Ciprian Tudor (CES and Samos)
We study when a given Gaussian random variable on a given probability
space
$(\Omega, {\cal{F}}, P) $ is equal almost surely to $\beta_{1}$ where $
\beta $
is a Brownian motion defined on the same (or possibly extended)
probability
space. As a consequences of this result, we prove that the
distribution of a
random variable (satisfying in addition a certain property) in a
finite sum of
Wiener chaoses cannot be normal. This result also allows to understand
better
some characterization of the Gaussian variables obtained via Malliavin
calculus.
http://arxiv.org/abs/0907.2501
---------------------------------------------------------------
8798. WEAK CONVERGENCE FOR THE STOCHASTIC HEAT EQUATION DRIVEN BY
GAUSSIAN WHITE NOISE
Xavier Bardina and Maria Jolis and Lluis Quer-Sardanyons
In this paper, we consider a quasi-linear stochastic heat equation on
$[0,1]$, with Dirichlet boundary conditions and controlled by the
space-time
white noise. We formally replace the random perturbation by a family
of noisy
inputs depending on a parameter $n\in \mathbb{N}$ such that
approximate the
white noise in some sense. Then, we provide sufficient conditions
ensuring that
the real-valued {\it mild} solution of the SPDE perturbed by this
family of
noises converges in law, in the space $\mathcal{C}([0,T]\times [0,1])$
of
continuous functions, to the solution of the white noise driven SPDE.
Making
use of a suitable continuous functional of the stochastic convolution
term, we
show that it suffices to tackle the linear problem. For this, we prove
that the
corresponding family of laws is tight and we identify the limit law by
showing
the convergence of the finite dimensional distributions. We have also
considered two particular families of noises to that our result
applies. The
first one involves a Poisson process in the plane and has been
motivated by a
one-dimensional result of Stroock, which states that the family of
processes $n
\int_0^t (-1)^{N(n^2 s)} ds$, where $N$ is a standard Poisson process,
converges in law to a Brownian motion. The second one is constructed
in terms
of the kernels associated to the extension of Donsker's theorem to the
plane.
http://arxiv.org/abs/0907.2508
---------------------------------------------------------------
8799. THE DIVERSITY OF A DISTRIBUTED GENOME IN BACTERIAL POPULATIONS
F. Baumdicker and W. R. Hess and P. Pfaffelhuber
The distributed genome hypothesis states that the set of genes in a
population of bacteria is distributed over all individuals that belong
to the
specific taxon. It implies that certain genes can be gained and lost
from
generation to generation. We use the random genealogy given by a Kingman
coalescent in order to superimpose events of gene gain and loss along
ancestral
lines. Gene gains occur at constant rate along ancestral lines. We
assume that
gained genes have never been present in the population before. Gene
losses
occur at a rate proportional to the number of genes present along the
ancestral
line. In this "infinitely many genes model" we derive moments for
several
statistics within a sample: the average number of genes per
individual, the
average number of genes differing between individuals, the number of
incongruent pairs of genes, the total number of different genes in the
sample
and the gene frequency spectrum. We demonstrate that the model gives a
reasonable fit with gene frequency data from marine cyanobacteria.
http://arxiv.org/abs/0907.2572
---------------------------------------------------------------
8800. EXTREMAL SOLUTIONS FOR STOCHASTIC EQUATIONS INDEXED BY NEGATIVE
INTEGERS AND TAKING VALUES IN COMPACT GROUPS
Takao Hirayama and Kouji Yano
Stochastic equations indexed by negative integers and taking values in
compact groups are studied. Extremal solutions of the equations are
characterized in terms of infinite products of independent random
variables.
This result is applied to characterize several properties of the set
of all
solutions in terms of the law of the driving noise.
http://arxiv.org/abs/0907.2587
---------------------------------------------------------------
8801. ON A ZERO-ONE LAW FOR THE NORM PROCESS OF TRANSIENT RANDOM WALK
Ayako Matsumoto and Kouji Yano
A zero-one law of Engelbert--Schmidt type is proven for the norm
process of a
transient random walk. An invariance principle for random walk local
times and
a limit version of Jeulin's lemma play key roles.
http://arxiv.org/abs/0907.2588
---------------------------------------------------------------
8802. LOCAL LIMIT OF PACKABLE GRAPHS
Itai Benjamini and Nicolas Curien
We adapt some of the planar results into higher dimensions. In
particular, it
is shown that every unbiased local limit of graphs sphere packed in
R^d is
d-parabolic (under some additional boundedness assumptions). We then
extend
parts of the circle packing theory into higher dimensions and derive few
geometric corollaries. E.g. every infinite graph ``well'' packed in
R^d has
either strictly positive isoperimetric (Cheeger) constant or admits
arbitrarily
large finite sets W with boundary size which satisfies |\partial W| <
|W|^{(d-1)/d + o(1)}, were "well" is a local bounded geometry
assumption. Some
open problems and conjectures are gathered at the end.
http://arxiv.org/abs/0907.2609
---------------------------------------------------------------
8803. A CLT FOR THE THIRD INTEGRATED MOMENT OF BROWNIAN LOCAL TIME
INCREMENTS
Jay Rosen
Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local
time of
Brownian motion. Our main result is to show that for each fixed $t$ $$
{\int
(L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t - L^x_t)L^x_t dx\over h^2}
\stackrel{\mathcal{L}}{\Longrightarrow}\sqrt{192}(\int
(L^x_t)^3dx)^{1/2}\eta$$
as $h\to 0$, where $\eta$ is a normal random variable with mean zero and
variance one that is independent of $L^{x}_{t}$. This generalizes our
previous
result for the second moment. We also explain why our approach will
not work
for higher moments
http://arxiv.org/abs/0907.2693
---------------------------------------------------------------
8804. STOCHASTIC TAYLOR EXPANSIONS AND HEAT KERNEL ASYMPTOTICS
Fabrice Baudoin
These notes focus on the applications of the stochastic Taylor
expansion of
solutions of stochastic differential equations to the study of heat
kernels in
small times. As an illustration of these methods we provide a new heat
kernel
proof of the Chern-Gauss-Bonnet theorem.
http://arxiv.org/abs/0907.2711
---------------------------------------------------------------
8805. EXPLICIT SOLUTIONS OF G-HEAT EQUATION WITH A CLASS OF INITIAL
CONDITIONS BY G-BROWNIAN MOTION
Mingshang Hu
We obtain the viscosity solution of G-heat equation with the initial
condition $\phi(x)=x^{n}$ for each integer $n\geq1$ using the method of
G-Brownian motion.
http://arxiv.org/abs/0907.2748
---------------------------------------------------------------
8806. GENERALIZED BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS
DRIVEN BY L\'EVY PROCESSES WITH NON-LIPSCHITZ COEFFICIENTS
Auguste Aman (LMAI) and Jean Marc Owo (LMAI)
We prove an existence and uniqueness result for generalized backward
doubly
stochastic differential equations driven by L\'evy processes with non-
Lipschitz
assumptions.
http://arxiv.org/abs/0907.2785
---------------------------------------------------------------
8807. SHARPNESS OF THE PERCOLATION TRANSITION IN THE TWO-DIMENSIONAL
CONTACT PROCESS
Jacob van den Berg
For ordinary (independent) percolation on a large class of lattices it
is
well-known that below the critical percolation parameter the cluster
size
distribution has exponential decay, and that power-law behaviour of this
distribution can only occur at the critical value. This behaviour is
often
called `sharpness of the percolation transition'.
For theoretical reasons as well as motivated by applied research,
there is an
increasing interest in percolation models with (weak) dependencies. For
instance, biologists and agricultural researchers have used (stationary
distributions of) certain two-dimensional contact-like processes to
model
vegetation patterns in an arid landscape. In that context, occupied
clusters
are interpreted as patches of vegetation. For some of these models it
has been
reported in the literature that computer simulations indicate power-law
behaviour in some interval of positive length of a model parameter.
This would
mean that in these models the percolation transition is not sharp.
This motivated us to investigate similar questions for the ordinary
('basic')
two-dimensional contact process with parameter the infection rate. We
show,
using techniques from papers on Voronoi and Johnson-Mehl tessellations
by
Bollob\'as and Riordan, that for the upper invariant measure of the
contact
process the percolation transition is sharp.
http://arxiv.org/abs/0907.2843
---------------------------------------------------------------
8808. CONDITIONAL LIMIT THEOREMS FOR ORDERED RANDOM WALKS
D. Denisov and V. Wachtel
In a recent paper of Eichelsbacher and Koenig (2008) the model of
ordered
random walks has been considered. There it has been shown that, under
certain
moment conditions, one can construct a k-dimensional random walk
conditioned to
stay in a strict order at all times. Moreover, they have shown that the
rescaled random walk converges to the Dyson Brownian motion. In the
present
paper we find the optimal moment assumptions for the construction of the
conditional random walk and generalise the limit theorem for this
conditional
process.
http://arxiv.org/abs/0907.2854
---------------------------------------------------------------
8809. ON SOJOURN TIMES IN THE FINITE CAPACITY $M/M/1$ QUEUE WITH
PROCESSOR SHARING
Qiang Zhen and Charles Knessl
We consider a processor shared $M/M/1$ queue that can accommodate at
most a
finite number $K$ of customers. We give an exact expression for the
sojourn
time distribution in the finite capacity model, in terms of a Laplace
transform. We then give the tail behavior, for the limit $K\to\infty$,
by
locating the dominant singularity of the Laplace transform.
http://arxiv.org/abs/0907.2908
---------------------------------------------------------------
8810. CORRELATION AND BRASCAMP-LIEB INEQUALITIES FOR MARKOV SEMIGROUPS
F. Barthe and D. Cordero-Erausquin and M. Ledoux and B. Maurey
This paper builds upon several recent works, where semigroup proofs of
Brascamp-Lieb inequalities are provided in various settings (Euclidean
space,
spheres and symmetric groups). Our aim is twofold. Firstly, we provide a
general, unifying, framework based on Markov generators, in order to
cover a
variety of examples of interest going beyond previous investigations.
Secondly,
we put forward the combinatorial reasons for which unexpected
exponents occur
in these inequalities.
http://arxiv.org/abs/0907.2858
---------------------------------------------------------------
8811. THE GEOMETRY OF EUCLIDEAN CONVOLUTION INEQUALITIES AND ENTROPY
Dario Cordero-Erausquin and Michel Ledoux
The goal of this note is to show that some convolution type
inequalities from
Harmonic Analysis and Information Theory, such as Young's convolution
inequality (with sharp constant), Nelson's hypercontractivity of the
Hermite
semi-group or Shannon's inequality, can be reduced to a simple
geometric study
of frames of $\R^2$. We shall derive directly entropic inequalities,
which were
recently proved to be dual to the Brascamp-Lieb convolution type
inequalities.
http://arxiv.org/abs/0907.2861
---------------------------------------------------------------
8812. ASYMPTOTIC EXPANSIONS FOR THE CONDITIONAL SOJOURN TIME
DISTRIBUTION IN THE $M/M/1$-PS QUEUE
Qiang Zhen and Charles Knessl
We consider the $M/M/1$ queue with processor sharing. We study the
conditional sojourn time distribution, conditioned on the customer's
service
requirement, in various asymptotic limits. These include large time
and/or
large service request, and heavy traffic, where the arrival rate is only
slightly less than the service rate. The asymptotic formulas relate
to, and
extend, some results of Morrison \cite{MO} and Flatto \cite{FL}.
http://arxiv.org/abs/0907.2910
---------------------------------------------------------------
8813. WEAK APPROXIMATION OF FRACTIONAL SDES: THE DONSKER SETTING
Xavier Bardina and Samy Tindel and Carles Rovira
In this note, we take up the study of weak convergence for stochastic
differential equations driven by a (Liouville) fractional Brownian
motion $B$
with Hurst parameter $H\in(1/3,1/2)$. In the current paper, we
approximate the
$d$-dimensional fBm by the convolution of a rescaled random walk with
Liouville's kernel. We then show that the corresponding differential
equation
converges in law to a fractional SDE driven by $B$.
http://arxiv.org/abs/0907.3030
---------------------------------------------------------------
8814. BOOTSTRAP PERCOLATION IN HIGH DIMENSIONS
Jozsef Balogh and Bela Bollobas and Robert Morris
In r-neighbour bootstrap percolation on a graph G, a set of initially
infected vertices A \subset V(G) is chosen independently at random, with
density p, and new vertices are subsequently infected if they have at
least r
infected neighbours. The set A is said to percolate if eventually all
vertices
are infected. Our aim is to understand this process on the grid,
[n]^d, for
arbitrary functions n = n(t), d = d(t) and r = r(t), as t -> infinity.
The main
question is to determine the critical probability p_c([n]^d,r) at which
percolation becomes likely, and to give bounds on the size of the
critical
window. In this paper we study this problem when r = 2, for all
functions n and
d satisfying d \gg log n.
The bootstrap process has been extensively studied on [n]^d when d
is a fixed
constant and 2 \le r \le d, and in these cases p_c([n]^d,r) has
recently been
determined up to a factor of 1 + o(1) as n -> infinity. At the other
end of the
scale, Balogh and Bollobas determined p_c([2]^d,2) up to a constant
factor, and
Balogh, Bollobas and Morris determined p_c([n]^d,d) asymptotically if
d > (log
log n)^{2+\eps}, and gave much sharper bounds for the hypercube.
Here we prove the following result: let x be the smallest positive
root of
the equation \sum_{k=0}^\infty (-1)^k x^k / (2^{k^2-k} k!) = 0, so x
\approx
1.166. Then
(16x/d^2 + (log d)/d^{5/2) 2^{-2\sqrt{d}} < p_c([2]^d,2) < (16x/d^2
+ 5(log
d)^2/d^{5/2}) 2^{-2\sqrt{d}} if d is sufficiently large, and moreover we
determine a sharp threshold for the critical probability p_c([n]^d,2)
for every
function n = n(d) with d \gg log n.
http://arxiv.org/abs/0907.3097
---------------------------------------------------------------
8815. THE MAXWELL-BOLTZMANN DISTRIBUTION IS NOT THE EQUILIBRIUM ON A
HYPERBOLOID
S. G. Rajeev
We give a geometric formulation of the Fokker-Planck-Kramer equations
for a
particle moving on a Lie algebra under the influence of a dissipative
and a
random force. Special cases of interest are fluid mechanics, the
Stochastic
Loewner Equation and the rigid body. We find that the Boltzmann
distribution,
although a static solution, is not normalizable when the algebra is not
unimodular. This is because the invariant measure of integration in
momentum
space is not the standard one. We solve the special case of the upper
half-plane (hyperboloid) explicitly: there is another equilibrium
solution to
the Fokker-Planck equation, which is integrable. It breaks rotation
invariance;
moreover, the most likely value for velocity is not zero.
http://arxiv.org/abs/0907.2401
---------------------------------------------------------------
8816. FROM A DICHOTOMY FOR IMAGES TO HAAGERUP'S INEQUALITY
Iosif Pinelis
Let X be a compact topological space, and let D be a subset of X. Let
Y be a
Hausdorff topological space. Let f be a continuous map of the closure
of D to Y
such that f(D) is open. Let E be any connected subset of the
complement (to Y)
of the boundary of D. Then f(D) either contains E or is contained in the
complement of E.
Applications of this dichotomy principle are given, in particular for
holomorphic maps, including maximum and minimum modulus principles, an
inverse
boundary correspondence, and a proof of Haagerup's inequality for the
absolute
power moments of linear combinations of independent Rademacher random
variables.
http://arxiv.org/abs/0907.2960
---------------------------------------------------------------
8817. LARGE DEVIATIONS FOR FLOWS OF INTERACTING BROWNIAN MOTIONS
A.A.Dorogovtsev and O.V.Ostapenko
We establish the large deviation principle (LDP) for stochastic flows of
interacting Brownian motions. In particular, we consider smoothly
correlated
flows, coalescing flows and Brownian motion stopped at a hitting moment.
http://arxiv.org/abs/0907.3207
---------------------------------------------------------------
8818. SCALING LIMITS OF RANDOM PLANAR MAPS WITH LARGE FACES
Jean-Fran\c{c}ois Le Gall (LM-Orsay) and Gr\'egory Miermont (DMA)
We discuss asymptotics for large random planar maps under the
assumption that
the distribution of the degree of a typical face is in the domain of
attraction
of a stable distribution with index $\alpha\in(1,2)$. When the number
$n$ of
vertices of the map tends to infinity, the asymptotic behavior of
distances
from a distinguished vertex is described by a random process called the
continuous distance process, which can be constructed from a centered
stable
process with no negative jumps and index $\alpha$. In particular, the
profile
of distances in the map, rescaled by the factor $n^{?1/2\alpha}$,
converges to
a random measure defined in terms of the distance process. With the same
rescaling of distances, the vertex set viewed as a metric space
converges in
distribution as $n\to\infty$, at least along suitable subsequences,
towards a
limiting random compact metric space whose Hausdorff dimension is
equal to
$2\alpha$.
http://arxiv.org/abs/0907.3262
---------------------------------------------------------------
8819. Q-EXCHANGEABILITY VIA QUASI-INVARIANCE
Alexander Gnedin and Grigori Olshanski
For positive q, the q-exchangeability is introduced as quasi-
invariance under
permutations, with a special cocycle. This allows us to extend the q-
analogue
of de Finetti's theorem for binary sequences (arXiv:0905.0367) to the
general
real-valued sequences. In contrast to the classical case with q=1, the
order on
the reals plays for the q-analogues a significant role. An explicit
construction of ergodic q-exchangeable measures involves a random
shuffling of
the set N={1,2,..} by iteration of the geometric choice. For q
distinct from 1,
the shuffling yields a probability measure Q that is supported by the
group of
bijections of N, and has the property of quasi-invariance under both
left and
right multiplications by finite permutations. We establish connections
of the
q-exchangeability to certain transient Markov chains on the q-Pascal
pyramids
and to invariant random flags over the Galois fields.
http://arxiv.org/abs/0907.3275
---------------------------------------------------------------
8820. HIGH LEVEL EXCURSION SET GEOMETRY FOR NON-GAUSSIAN INFINITELY
DIVISIBLE RANDOM FIELDS
Robert J Adler and Gennady Samorodnitsky and Jonathan E Taylor
We consider smooth, infinitely divisible random fields $X(t), t\in M)$,
$M\subset \real^d$, with regularly varying L\'evy measure, and are
interested
in the geometric characteristics of the excursion sets
\begin{eqnarray*} A_u =
\{t\in M: X(t) >u\} \end{eqnarray*} over high levels $u$.
For a large class of such random fields we compute the $u\to\infty$
asymptotic joint distribution of the numbers of critical points, of
various
types, of $X$ in $A_u$, conditional on $A_u$ being non-empty. This
allows us,
for example, to obtain the asymptotic conditional distribution of the
Euler
characteristic of the excursion set.
In a significant departure from the Gaussian situation, the high
level
excursion sets for these random fields can have quite a complicated
geometry.
Whereas in the Gaussian case non-empty excursion sets are, with high
probability, roughly ellipsoidal, in the more general infinitely
divisible
setting almost any shape is possible.
http://arxiv.org/abs/0907.3359
---------------------------------------------------------------
8821. DISORDER CHAOS AND MULTIPLE VALLEYS IN SPIN GLASSES
Sourav Chatterjee
We prove that the Sherrington-Kirkpatrick model of spin glasses is
chaotic
under small perturbations of the couplings at any temperature in the
absence of
an external field. The result is proved for two kinds of
perturbations: (a)
distorting the couplings via Ornstein-Uhlenbeck flows, and (b)
replacing a
small fraction of the couplings by independent copies. We further
prove that
the S-K model exhibits multiple valleys in its energy landscape, i.e.
there are
many states with near-minimal energy that are mutually nearly
orthogonal. We
show that the variance of the free energy of the S-K model is
unusually small
at any temperature. (By `unusually small' we mean that it is much
smaller than
the number of sites; in other words, it beats the classical Gaussian
concentration inequality, a phenomenon that we call
`superconcentration'.) We
prove that the bond overlap in the Edwards-Anderson model of spin
glasses is
not chaotic under perturbations of the couplings, even large
perturbations.
Lastly, we obtain sharp lower bounds on the variance of the free
energy in the
E-A model on any bounded degree graph, generalizing a result of Wehr and
Aizenman and establishing the absence of superconcentration in this
class of
models. Our techniques apply for the p-spin models and the Random
Field Ising
Model as well, although we do not work out the details in these cases.
http://arxiv.org/abs/0907.3381
---------------------------------------------------------------
8822. SPIN NEEDLETS SPECTRAL ESTIMATION
Daryl Geller and Xiaohong Lan and Domenico Marinucci
We consider the statistical analysis of random sections of a spin fibre
bundle over the sphere. These may be thought of as random fields that
at each
point p in $S^2$ take as a value a curve (e.g. an ellipse) living in the
tangent plane at that point $T_{p}S^2$, rather than a number as in
ordinary
situations. The analysis of such fields is strongly motivated by
applications,
for instance polarization experiments in Cosmology. To investigate
such fields,
spin needlets were recently introduced by Geller and Marinucci (2008)
and
Geller et al. (2008). We consider the use of spin needlets for spin
angular
power spectrum estimation, in the presence of noise and missing
observations,
and we provide Central Limit Theorem results, in the high frequency
sense; we
discuss also tests for bias and asymmetries with an asymptotic
justification.
http://arxiv.org/abs/0907.3369
---------------------------------------------------------------
8823. A BIJECTION THEOREM FOR DOMINO TILING WITH DIAGONAL IMPURITIES
Fumihiko Nakano and Taizo Sadahiro
We consider the dimer problem on a non-bipartite graph $G$, where
there are
two types of dimers one of which we regard impurities. Results of
simulations
using Markov chain seem to indicate that impurities are tend to
distribute on
the boundary, which we set as a conjecture. We first show that there
is a
bijection between the set of dimer coverings on $G$ and the set of
spanning
forests on two graphs which are made from $G$, with configuration of
impurities
satisfying a pairing condition. This bijection can be regarded as a
extension
of the Temperley bijection. We consider local move consisting of two
operations, and by using the bijection mentioned above, we prove local
move
connectedness. We further obtained some bound of the number of dimer
coverings
and the probability finding an impurity at given edge, by extending the
argument in our previous result.
http://arxiv.org/abs/0907.3252
---------------------------------------------------------------
8824. OPTIMAL EXECUTION PROBLEM WITH MARKET IMPACT
Takashi Kato
We study the optimal execution problem in the market model in
consideration
of market impact. First we study the discrete-time model and describe
the value
function with respect to the trader's optimization problem. Then, by
shortening
the intervals of execution times, we derive the value function of the
continuous-time model and study some properties of them (continuity,
semi-group
property and the characterization as the viscosity solution of HJB.)
We show
that the properties of the continuous-time value function vary by the
strength
of market impact. Moreover we introduce some examples of this model,
which tell
us that the forms of the optimal execution strategies entirely change
according
to the amount of the security holding.
http://arxiv.org/abs/0907.3282
---------------------------------------------------------------
8825. DE FINETTI THEOREMS FOR EASY QUANTUM GROUPS
Teodor Banica and Stephen Curran and Roland Speicher
We study sequences of noncommutative random variables which are
invariant
under ``quantum transformations'' coming from an orthogonal quantum
group
satisfying the ``easiness'' condition axiomatized in our previous
paper. For 10
easy quantum groups, we obtain de Finetti type theorems characterizing
the
joint distribution of any infinite, quantum invariant sequence. In
particular,
we give a new and unified proof of the classical results of de Finetti
and
Freedman for the easy groups S_n, O_n, which is based on the
combinatorial
theory of cumulants. We also recover the free de Finetti theorem of K
\"ostler
and Speicher, and the characterization of operator-valued free
semicircular
families due to Curran. We consider also finite sequences, and prove an
approximation result in the spirit of Diaconis and Freedman.
http://arxiv.org/abs/0907.3314
---------------------------------------------------------------
8826. SRB MEASURES FOR CERTAIN MARKOV PROCESSES
Wael Bahsoun and Pawel Gora
We study Markov processes generated by iterated function systems
(IFS). The
constituent maps of the IFS are monotonic transformations of the
interval with
common fixed points at 0 and 1. We first obtain an upper bound on the
number of
SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then theorems are given
to
analyze properties of the ergodic invariant measures $\delta_0$ and $
\delta_1$.
In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$
to be SRB
measures are given. We apply our results to asset market games.
http://arxiv.org/abs/0907.3372
---------------------------------------------------------------
8827. OPTIMAL EXECUTION PROBLEM WITH MARKET IMPACT
Takashi Kato
We study the optimal execution problem in the market model in
consideration
of market impact. First we study the discrete-time model and describe
the value
function with respect to the trader's optimization problem. Then, by
shortening
the intervals of execution times, we derive the value function of the
continuous-time model and study some properties of them (continuity,
semi-group
property and the characterization as the viscosity solution of HJB.)
We show
that the properties of the continuous-time value function vary by the
strength
of market impact. Moreover we introduce some examples of this model,
which tell
us that the forms of the optimal execution strategies entirely change
according
to the amount of the security holding.
http://arxiv.org/abs/0907.3282
---------------------------------------------------------------
8828. FRACTIONAL NORMAL INVERSE GAUSSIAN PROCESS
Arun Kumar and P. Vellaisamy
Normal inverse Gaussian (NIG) process was introduced by Barndorff-
Nielsen
(1997) by subordinating Brownian motion with drift to an inverse
Gaussian
process. Increments of NIG process are independent and stationary. In
this
paper, we introduce dependence between the increments of NIG process, by
subordinating fractional Brownian motion to an inverse Gaussian
process and
call it fractional normal inverse Gaussian (FNIG) process. The basic
properties
of this process are discussed. Its marginal distributions are scale
mixtures of
normal laws, infinitely divisible for the Hurst parameter 1/2<=H< 1
and are
heavy tailed. First order increments of the process are stationary and
possess
long-range dependence (LRD) property. It is shown that they have
persistence of
signs LRD property also. A generalization of the FNIG process called n-
FNIG
process is also discussed which allows Hurst parameter H in the
interval (n-1,
n). Possible applications to mathematical finance and hydraulics are
also
pointed out
http://arxiv.org/abs/0907.3637
---------------------------------------------------------------
8829. FLOW OF DIFFEOMORPHISMS FOR SDES WITH UNBOUNDED H\"OLDER
CONTINUOUS DRIFT
F. Flandoli and M. Gubinelli and E. Priola
We consider a SDE with a smooth multiplicative non-degenerate noise
and a
possibly unbounded Holder continuous drift term. We prove existence of
a global
flow of diffeomorphisms by means of a special transformation of the
drift of
Ito-Tanaka type. The proof requires non-standard elliptic estimates in
Holder
spaces. As an application of the stochastic flow, we obtain a
Bismut-Elworthy-Li type formula for the first derivatives of the
associated
diffusion semigroup.
http://arxiv.org/abs/0907.3668
---------------------------------------------------------------
8830. SYSTEMS OF ONE-DIMENSIONAL RANDOM WALKS IN A COMMON RANDOM
ENVIRONMENT
Jonathon Peterson
We consider a system of independent one-dimensional random walks in a
common
random environment under the condition that the random walks are
transient with
positive speed $v_P$. We give upper bounds on the quenched probability
that at
least one of the random walks started in the interval $[An, Bn]$ has
traveled a
distance of less than $(v_P - \epsilon)n$. This leads to both a
uniform law of
large numbers and a hydrodynamic limit. We also identify a family of
distributions on the configuration of particles (parameterized by
particle
density) which are stationary under the (quenched) dynamics of the
random walks
and show that these are the limiting distributions for the system when
started
from a certain natural collection of distributions.
http://arxiv.org/abs/0907.3680
---------------------------------------------------------------
8831. A SPECTRAL ANALYSIS OF THE SEQUENCE OF FIRING PHASES IN
STOCHASTIC INTEGRATE-AND-FIRE OSCILLATORS
Peter Baxendale and John Mayberry
Integrate and fire oscillators are widely used to model the generation
of
action potentials in neurons. In this paper, we discuss small noise
asymptotic
results for a class of stochastic integrate and fire oscillators
(SIFs) in
which the buildup of membrane potential in the neuron is governed by a
Gaussian
diffusion process. To analyze this model, we study the asymptotic
behavior of
the spectrum of the firing phase transition operator. We begin by
proving
strong versions of a law of large numbers and central limit theorem
for the
first passage-time of the underlying diffusion process across a
general time
dependent boundary. Using these results, we obtain asymptotic
approximations of
the transition operator's eigenvalues. We also discuss connections
between our
results and earlier numerical investigations of SIFs.
http://arxiv.org/abs/0907.3700
---------------------------------------------------------------
8832. EVOLUTION IN PREDATOR-PREY SYSTEMS
Rick Durrett and John Mayberry
We study the adaptive dynamics of predator prey systems modeled by a
dynamical system in which the characteristics are allowed to evolve by
small
mutations. When only the prey are allowed to evolve, and the size of the
mutational change tends to 0, the system does not exhibit long term prey
coexistence and the parameters of the resident prey type converges to
the
solution of an ODE. When only the predators are allowed to evolve,
coexistence
of predators occurs. In this case, depending on the parameters being
varied we
see (i) the number of coexisting predators remains tight and the
differences of
the parameters from a reference species converge in distribution to a
limit, or
(ii) the number of coexisting predators tends to infinity, and we
conjecture
that the differences converge to a deterministic limit.
http://arxiv.org/abs/0907.3702
---------------------------------------------------------------
8833. HIGH MOMENTS OF LARGE WIGNER RANDOM MATRICES AND ASYMPTOTIC
PROPERTIES OF THE SPECTRAL NORM
O. Khorunzhiy
We further modify the method proposed by Ya. Sinai and A. Soshnikov and
developed by A. Ruzmaikina to study the high moments of large Wigner
random
matrices. Our result concern the asymptotic estimates of the high
moments of
n-dimensional real symmetric random matrices whose elements have
symmetric
distribution such that the 12+delta-th moment exists.
http://arxiv.org/abs/0907.3743
---------------------------------------------------------------
8834. ON THE ONE DIMENSIONAL CRITICAL "LEARNING FROM NEIGHBOURS" MODEL
Antar Bandyopadhyay and Rahul Roy and Anish Sarkar
We consider a model of a discrete time "interacting particle system"
on the
integer line where infinitely many changes are allowed at each
instance of
time. We describe the model using chameleons of two different colours,
{\it
viz}., red ($R$) and blue ($B$). At each instance of time each chameleon
performs an independent but identical coin toss experiment with
probability
$\alpha$ to decide whether to change its colour or not. If the coin
lands head
then the creature retains its colour (this is to be interpreted as a
"success"), otherwise it observes the colours and coin tosses of its two
nearest neighbours and changes its colour only if, among its neighbors
and
including itself, the proportion of successes of the other colour is
larger
than the proportion of successes of its own colour. This produces a
Markov
chain with infinite state space ${R, B}^{\Zbold}$. This model was
first studied
by Chatterjee and Xu (2004) where different colours had different
success
probabilities. In this work we consider the "critical" case where the
success
probability, $\alpha$, is the same irrespective of the colour of the
chameleon.
We show that starting from any initial translation invariant
distribution of
colours the Markov chain converges to a limit of a single colour,
i.e., even at
the critical case there is no "coexistence" of the two colours at the
limit.
Moreover we show that starting with an i.i.d. colour distribution the
limiting
distribution gives some advantage to the "underdog".
http://arxiv.org/abs/0907.3828
---------------------------------------------------------------
8835. ON HELE-SHAW PROBLEMS ARISING AS SCALING LIMITS
Pavel Etingof
We discuss conjectural scaling limits of discrete 2-dimensional
aggregation
models conditioned on a semi-axis considered by Levine and Peres in
arXiv:0712.3378. These are certain problems about Hele-Show flows. We
study
moment properties of their solutions, and solve some of them using
conformal
mappings. In particular, we predict the exact formula for the
computer-generated shape on the left side of Fig. 4 in arXiv:0712.3378.
http://arxiv.org/abs/0907.3856
---------------------------------------------------------------
8836. WRIGHT-FISHER DIFFUSION IN ONE DIMENSION
Charles L. Epstein and Rafe Mazzeo
We analyze the diffusion processes associated to equations of Wright-
Fisher
type in one spatial dimension. These are defined by a degenerate
second order
operator on the interval [0, 1], where the coefficient of the second
order term
vanishes simply at the endpoints, and the first order term is an
inward-pointing vector field. We consider various aspects of this
problem,
motivated by applications in population genetics, including a sharp
regularity
theory for the zero flux boundary conditions, as well as a derivation
of the
precise asymptotics for solutions of this equation, both as t goes to
0 and
infinity, and as x goes to 0, 1.
http://arxiv.org/abs/0907.3881
---------------------------------------------------------------
8837. HARD CORE ENTROPY: LOWER BOUNDS
Kari Eloranta
We establish lower bounds for the entropy of the Hard Core Model on a
few 2d
lattices $\scriptstyle {\rm {\bf L}}.$ In this model the allowed
configurations
inside $\scriptstyle \{0,1\}^{{\rm {\bf L}}}$ are the one's in which the
nearest neighbor $\scriptstyle 1$'s are forbidden. Our method which is
based on
a sequential fill-in scheme is unbiassed and thereby yields in principle
arbitrarily good estimates for the topological entropy. The procedure
also
gives some detailed information on the support of the measure of maximal
entropy.
http://arxiv.org/abs/0907.4035
---------------------------------------------------------------
8838. BINOMIAL APPROXIMATIONS FOR BARRIER OPTIONS OF ISRAELI STYLE
Yan Dolinsky and Yuri Kifer
We show that prices and shortfall risks of game (Israeli) barrier
options in
a sequence of binomial approximations of the Black--Scholes (BS) market
converge to the corresponding quantities for similar game barrier
options in
the BS market with path dependent payoffs and the speed of convergence
is
estimated, as well. The results are new also for usual American style
options
and they are interesting from the computational point of view, as
well, since
in binomial markets these quantities can be obtained via dynamical
programming
algorithms. The paper continues the study of [11]and [7] but requires
substantial additional arguments in view of pecularities of barrier
options
which, in particular, destroy the regularity of payoffs needed in the
above
papers.
http://arxiv.org/abs/0907.4136
---------------------------------------------------------------
8839. AN INTRODUCTION TO STOCHASTIC PDES
Martin Hairer
These notes are based on a series of lectures given first at the
University
of Warwick in spring 2008 and then at the Courant Institute in spring
2009. It
is an attempt to give a reasonably self-contained presentation of the
basic
theory of stochastic partial differential equations, taking for
granted basic
measure theory, functional analysis and probability theory, but
nothing else.
The approach taken in these notes is to focus on semilinear parabolic
problems driven by additive noise. These can be treated as stochastic
evolution
equations in some infinite-dimensional Banach or Hilbert space that
usually
have nice regularising properties and they already form a very rich
class of
problems with many interesting properties. Furthermore, this class of
problems
has the advantage of allowing to completely pass under silence many
subtle
problems arising from stochastic integration in infinite-dimensional
spaces.
http://arxiv.org/abs/0907.4178
---------------------------------------------------------------
8840. LOCALIZATION FOR A CLASS OF LINEAR SYSTEMS
Yukio Nagahata and Nobuo Yoshida
We consider a class of continuous-time stochastic growth models on
$d$-dimensional lattice with non-negative real numbers as possible
values per
site. The class contains examples such as binary contact path process
and
potlatch process. We show the equivalence between the slow population
growth
and localization property that the time integral of the replica overlap
diverges. We also prove, under reasonable assumptions, a localization
property
in a stronger form that the spatial distribution of the population
does not
decay uniformly in space.
http://arxiv.org/abs/0907.4200
---------------------------------------------------------------
8841. THE RANK OF DILUTED RANDOM GRAPHS
Charles Bordenave and Marc Lelarge
We investigate the rank of the adjacency matrix of large diluted random
graphs: for a sequence of graphs converging locally to a tree, we give
new
formulas for the asymptotic of the multiplicity of the eigenvalue 0. In
particular, the result depends only on the limiting tree structure,
showing
that the normalized rank is 'continuous at infinity'. Our work also
gives a new
formula for the mass at zero of the spectral measure of a Galton-
Watson tree.
Our techniques of proofs borrow ideas from analysis of algorithms,
random
matrix theory, statistical physics and analysis of Schrodinger
operators on
trees.
http://arxiv.org/abs/0907.4244
---------------------------------------------------------------
8842. HAUSDORFF MEASURE OF ARCS AND BROWNIAN MOTION ON BROWNIAN
SPATIAL TREES
David A. Croydon
A Brownian spatial tree is defined to be a pair $(\mathcal{T},\phi)$,
where
$\mathcal{T}$ is the rooted real tree naturally associated with a
Brownian
excursion and $\phi$ is a random continuous function from $\mathcal{T}
$ into
$\mathbb{R}^d$ such that, conditional on $\mathcal{T}$, $\phi$ maps
each arc of
$\mathcal{T}$ to the image of a Brownian motion path in $\mathbb{R}^d$
run for
a time equal to the arc length. It is shown that, in high dimensions,
the
Hausdorff measure of arcs can be used to define an intrinsic metric
$d_{\mathcal{S}}$ on the set $\mathcal{S}:=\phi(\mathcal{T})$.
Applications of
this result include the recovery of the spatial tree $(\mathcal{T},
\phi)$ from
the set $\mathcal{S}$ alone, which implies in turn that a Dawson--
Watanabe
super-process can be recovered from its range. Furthermore,
$d_{\mathcal{S}}$
can be used to construct a Brownian motion on $\mathcal{S}$, which is
proved to
be the scaling limit of simple random walks on related discrete
structures. In
particular, a limiting result for the simple random walk on the
branching
random walk is obtained.
http://arxiv.org/abs/0907.4260
---------------------------------------------------------------
8843. SCALING LIMITS FOR CRITICAL INHOMOGENEOUS RANDOM GRAPHS WITH
FINITE THIRD MOMENTS
Shankar Bhamidi and Remco van der Hofstad and Johan van Leeuwaarden
We identify the scaling limits for the sizes of the largest components
at
criticality for inhomogeneous random graphs when the degree exponent $
\tau$
satisfies $\tau>4$. We see that the sizes of the (rescaled) components
converge
to the excursion lengths of an inhomogeneous Brownian motion,
extending results
of \cite{Aldo97}. We rely heavily on martingale convergence
techniques, and
concentration properties of (super)martingales. This paper is part of a
programme to study the critical behavior in inhomogeneous random
graphs of
so-called rank-1 initiated in \cite{Hofs09a}.
http://arxiv.org/abs/0907.4279
---------------------------------------------------------------
8844. TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS
Zhen-Qing Chen and Tusheng Zhang
In this paper, we prove that there exists a unique, bounded continuous
weak
solution to the Dirichlet boundary value problem for a general class of
second-order elliptic operators with singular coefficients, which does
not
necessarily have the maximum principle. Our method is probabilistic.
The time
reversal of symmetric Markov processes and the theory of Dirichlet
forms play a
crucial role in our approach.
http://arxiv.org/abs/0907.4301
---------------------------------------------------------------
8845. NOTES ON USING CONTROL VARIATES FOR ESTIMATION WITH REVERSIBLE
MCMC SAMPLERS
Ioannis Kontoyiannis and Petros Dellaportas
A general methodology is presented for the construction and effective
use of
control variates for reversible MCMC samplers. The values of the
coefficients
of the optimal linear combination of the control variates are
computed, and
adaptive, consistent MCMC estimators are derived for these optimal
coefficients. All methodological and asymptotic arguments are rigorously
justified. Numerous MCMC simulation examples from Bayesian inference
applications demonstrate that the resulting variance reduction can be
quite
dramatic.
http://arxiv.org/abs/0907.4160
---------------------------------------------------------------
8846. THE SCALING WINDOW FOR A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE
Hamed Hatami and Michael Molloy
We consider a random graph on a given degree sequence ${\cal D}$,
satisfying
certain conditions. We focus on two parameters $Q=Q({\cal D}),
R=R({\cal D})$.
Molloy and Reed proved that Q=0 is the threshold for the random graph
to have a
giant component. We prove that if $|Q|=O(n^{-1/3} R^{2/3})$ then, with
high
probability, the size of the largest component of the random graph
will be of
order $\Theta(n^{2/3}R^{-1/3})$. If $|Q|$ is asymptotically larger than
$n^{-1/3}R^{2/3}$ then the size of the largest component is
asymptotically
smaller or larger than $n^{2/3}R^{-1/3}$. Thus, we establish that the
scaling
window is $|Q|=O(n^{-1/3} R^{2/3})$.
http://arxiv.org/abs/0907.4211
---------------------------------------------------------------
8847. DENSE PACKING ON UNIFORM LATTICES
Kari Eloranta
We study the Hard Core Model on the graphs ${\rm {\bf \scriptstyle G}}$
obtained from Archimedean tilings i.e. configurations in $\scriptstyle
\{0,1\}^{{\rm {\bf G}}}$ with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the
geometry of
the densest packings in a uniform discrete set-up. We establish
density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal
configurations. Its
rule involves a parameter which can be naturally characterized as
packing
pressure. It can have a critical value but from packing point of view
just as
interesting are the noncritical cases. These phenomena are related to
the
exponential size of the set of densest packings and more specifically
whether
these packings are maximally symmetric, simple laminated or
essentially random
packings.
http://arxiv.org/abs/0907.4247
---------------------------------------------------------------
8848. ON THE DISTRIBUTION OF A SECOND CLASS PARTICLE IN THE ASYMMETRIC
SIMPLE EXCLUSION PROCESS
Craig A. Tracy and Harold Widom
We give an exact expression for the distribution of the position X(t)
of a
single second class particle in the asymmetric simple exclusion
process (ASEP)
where initially the second class particle is located at the origin and
the
first class particles occupy the sites {1,2,...}.
http://arxiv.org/abs/0907.4395
---------------------------------------------------------------
8849. STEIN'S METHOD OF EXCHANGEABLE PAIRS WITH APPLICATION TO THE
CURIE-WEISS MODEL
Sourav Chatterjee and Qi-Man Shao
Let $(W, W')$ be an exchangeable pair. Assume that $$E(W-W' | W) =
g(W)+r(W),$$ where $g(W)$ is a dominated term and $r(W)$ is
negligible. Let
$G(t) = \int_0^t g(s) ds$ and define $p(t) = c_1 e^{-c_0 G(t)}$, where
$c_0$ is
a properly chosen constant and $c_1 = 1/\int_{-\infty}^\infty p(t) dt
$. Let $Y$
be a random variable with the probability density function $p$. It is
proved
that $W$ converges to $Y$ in distribution when the conditional second
moment of
$(W-W')$ given $W$ satisfies a law of large numbers. A Berry-Esseen
type bound
is also given. We use this technique to obtain a Berry-Esseen error
bound of
order $1/\sqrt{n}$ in the non-central limit theorem for the
magnetization in
the Curie-Weiss ferromagnet at the critical temperature.
http://arxiv.org/abs/0907.4450
---------------------------------------------------------------
8850. LARGE DEVIATION IN HARNACK TYPE DIRICHLET SPACES
Ann-Kathrin Jarecki
In the framework of Harnack type Dirichlet forms, we prove a large
deviation
principle for the asymptotics of reversible Markov processes with rate
function
given by the energy of the paths.
http://arxiv.org/abs/0907.4479
---------------------------------------------------------------
8851. UPPER BOUND FOR LARGE DEVIATIONS OF REVERSIBLE DIFFUSION PROCESSES
Ann-Kathrin Jarecki
For a Markov process associated with a diffusion type Dirichlet form
an upper
bound is shown for the law of the finite dimensional distributions of
the
process. Under some more assumptions on the underlaying space this is
also
shown for the law of the Markov process itself. In the last section we
want to
give an application to the Wasserstein diffusion.
http://arxiv.org/abs/0907.4483
---------------------------------------------------------------
8852. BOUNDING RELATIVE ENTROPY BY THE RELATIVE ENTROPY OF LOCAL
SPECIFICATIONS IN PRODUCT SPACES
Katalin Marton
For a class of density functions $q^n(x^n)$ on $\Bbb R^n$ we prove an
inequality between relative entropy and the sum of average conditional
relative
entropies of the following form: For any density function $p^n(x^n)$
on $\Bbb
R^n$, $D(p^n||q^n)\leq Const. \sum_{i=1}^n \Bbb E D(p_i(\cdot|Y_1,...,
Y_{i-1},Y_{i+1},..., Y_n) || Q_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},...,
Y_n)),$
where $p_i(\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot|
x_1,...,
x_{i-1},x_{i+1},..., x_n)$ denote the local specifications for $p^n$
resp.
$q^n$, i.e., the conditional density functions of the $i$'th
coordinate, given
the other coordinates. The constant depends on the properties of the
local
specifications of $q^n$. The above inequality implies a logarithmic
Sobolev
inequality for $q^n$. We get an explicit lower bound for the logarithmic
Sobolev constant of $q^n$ under the assumptions that: (i) the local
specifications of $q^n$ satisfy logarithmic Sobolev inequalities with
constants
$\rho_i$, and (ii) they also satisfy some condition expressing that
the mixed
partial derivatives of the Hamiltonian of $q^n$ are not too large
relative to
the logarithmic Sobolev constants $\rho_i$. Condition (ii) may be
weaker than
that used in Otto and Reznikoff's recent paper on the estimation of
logarithmic
Sobolev constants of spin systems.
http://arxiv.org/abs/0907.4491
---------------------------------------------------------------
8853. ON MARKOV CHAINS INDUCED BY PARTITIONED TRANSITION PROBABILITY
MATRICES
Thomas Kaijser
Let S be a denumerable state space and let P be a transition probability
matrix on S. If a denumerable set M of nonnegative matrices is such
that the
sum of the matrices is equal to P, then we call M a partition of P.
Let K
denote the set of probability vectors on S. To every partition M of P
we can
associate a transition probability function on K defined in such a way
that if
p in K and m in M are such that ||pm|| > 0, then, with probability ||
pm|| the
vector p is transferred to the vector pm/||pm||. Here ||.|| denotes the
l_1-norm. In this paper we investigate convergence in distribution for
Markov
chains generated by transition probability functions induced by
partitions of
transition probability matrices. An important application of the
convergence
results obtained is to filtering processes of partially observed
Markov chains.
http://arxiv.org/abs/0907.4502
---------------------------------------------------------------
8854. RETURN PROBABILITIES OF RANDOM WALKS AMONG POLYNOMIAL LOWER TAIL
RANDOM CONDUCTANCES
Omar Boukhadra
We study models of continuous-time, symmetric, $\Z^{d}$-valued random
walks
in random environments, driven by a field of i.i.d. random nearest-
neighbor
conductances $\omega_{xy}\in[0,1]$ with a power law with an exponent $
\gamma$
near 0. We are interested in estimating the quenched decay of the return
probability $P_\omega^{t}(0,0)$, as $t$ tends to $+\infty$. We show
that for
$\gamma> \frac{d}{2}$, the standard bound turns out to be of the correct
logarithmic order. As an expected concequence, the same result holds
for the
discrete-time case.
http://arxiv.org/abs/0907.4525
---------------------------------------------------------------
8855. RECURRENCE AND TRANSIENCE OF BRANCHING RANDOM WALKS ARE
DYNAMICALLY STABLE
Sebastian M\"uller
Consider a sequence of i.i.d. random variables $X_n$ where each random
variable is refreshed independently according to a Poisson clock. At
any fixed
time $t$ the law of the sequence is the same as for the sequence at
time 0 but
at random times almost sure properties of the sequence may be
violated. If
there are such \emph{exceptional times} we say that the property is
\emph{dynamically sensitive}, otherwise we call it \emph{dynamically
stable}.
In this note we consider branching random walks on Cayley graphs and
prove that
recurrence and transience are dynamically stable. Our proof combines
techniques
from the theory of branching random walks with those of dynamical
percolation.
http://arxiv.org/abs/0907.4557
---------------------------------------------------------------
8856. THE T-MARTIN BOUNDARY OF REFLECTED RANDOM WALKS ON A HALF-SPACE
Irina Ignatiouk-Robert
The t-Martin boundary of a random walk on a half-space with reflected
boundary conditions is identified. It is shown in particular that the
t-Martin
boundary of such a random walk is not stable in the following sense :
for
different values of t, the t-Martin compactifications are not
homeomorphic to
each other.
http://arxiv.org/abs/0907.4592
---------------------------------------------------------------
8857. INVARIANT RANDOM FIELDS IN VECTOR BUNDLES AND APPLICATION TO
COSMOLOGY
Anatoliy Malyarenko
We develop the theory of invariant random fields in vector bundles. The
spectral decomposition of an invariant random field in a homogeneous
vector
bundle generated by an induced representation of a compact connected
Lie group
$G$ is obtained. We discuss an application to the theory of cosmic
microwave
background, where $G=SO(3)$. A theorem about equivalence of two
different
groups of assumptions in cosmological theories is proved.
http://arxiv.org/abs/0907.4620
---------------------------------------------------------------
8858. DISJOINT HAMILTON CYCLES IN THE RANDOM GEOMETRIC GRAPH
Xavier P\'erez-Gim\'enez and Nicholas C. Wormald
We prove a conjecture of Penrose about the standard random geometric
graph
process, in which n vertices are placed at random on the unit square
and edges
are sequentially added in increasing order of lengths taken in the l_p
norm. We
show that the first edge that makes the random geometric graph
Hamiltonian is
a.a.s. exactly the same one that gives 2-connectivity. We also extend
this
result to arbitrary connectivity, by proving that the first edge in
the process
that creates a k-connected graph coincides a.a.s. with the first edge
that
causes the graph to contain k/2 pairwise edge-disjoint Hamilton cycles
(for
even k), or (k-1)/2 Hamilton cycles plus one perfect matching, all of
them
pairwise edge-disjoint (for odd k).
http://arxiv.org/abs/0907.4459
---------------------------------------------------------------
8859. LIMIT THEOREMS FOR VERTEX-REINFORCED JUMP PROCESSES ON REGULAR
TREES
Andrea Collevecchio
Consider a vertex-reinforced jump process defined on a regular tree,
where
each vertex has exactly $b$ children, with $b \ge 3$. We prove the
strong law
of large numbers and the central limit theorem for the distance of the
process
from the root. Notice that it is still unknown if vertex-reinforced jump
process is transient on the binary tree.
http://arxiv.org/abs/0907.4854
---------------------------------------------------------------
8860. STOCHASTIC FLOWS OF SDES WITH IRREGULAR DRIFTS AND STOCHASTIC
TRANSPORT EQUATIONS
Xicheng Zhang
In this article we study (possibly degenerate) stochastic differential
equations (SDE) with irregular (or discontiuous) drifts, and prove
that under
certain conditions on the coefficients, there exists a unique almost
everywhere
stochastic invertible flow associated with the SDE in the sense of
Lebesgue
measure. In the case of constant diffusions and BV drifts, we obtain
such a
result by studying the related stochastic transport equation. In the
case of
non-constant diffusions and Sobolev drifts, we use a direct method. In
particular, we extend the recent results on ODEs with non-smooth
vector fields
to SDEs.
http://arxiv.org/abs/0907.4866
---------------------------------------------------------------
8861. THE MONOTONE CUMULANTS
Takahiro Hasebe and Hayato Saigo
In the present paper we define the notion of generalized cumulants which
gives a universal framework for commutative, free, Boolean, and
especially,
monotone probability theories. The uniqueness of generalized cumulants
holds
for each independence, and hence, generalized cumulants are equal to
the usual
cumulants in commutative, free and Boolean cases. The way we define
(generalized) cumulants is so elementary that we need neither partition
lattices nor generating functions. This new approach open the way to
introduce
monotone cumulants and we obtain quite simple proof of central limit
theorem
and Poisson's law of small numbers in monotone probability theory.
http://arxiv.org/abs/0907.4896
---------------------------------------------------------------
8862. INVARIANT MEASURES AND DECAY OF CORRELATIONS OF A CLASS OF
ERGODIC PROBABILISTIC CELLULAR AUTOMATA
Cristian Coletti (CMCC) and Pierre Tisseur (CMCC)
Using an extended version of the duality concept between two stochastic
processes, we give new ergodicity conditions for two states
probabilistic
cellular automata (PCA) of any dimensions and any radius. Under these
assumptions, in the one dimensional case, we study some properties of
the
unique invariant measure and show that it is shift mixing. Also, the
decay of
correlation is studied in detail. In this sense, the extended concept of
duality gives exponential decay of correlation. When the extended
concept of
duality can not be applied we are able to get, once again, exponential
decay of
correlation using well known results from the theory of branching
processes.
http://arxiv.org/abs/0907.4841
---------------------------------------------------------------
8863. BAYESIAN ESTIMATE OF THE ZERO-DENSITY FREQUENCY OF A CS FOUNTAIN
D Calonico and F Levi and L Lorini and G Mana
Caesium fountain frequency-standards realize the second in the
International
System of Units with a relative uncertainty approaching 10^-16. Among
the main
contributions to the accuracy budget, cold collisions play an
important role
because of the atomic density shift of the reference atomic
transition. This
paper describes an application of the Bayesian analysis of the clock
frequency
to estimate the density shift and describes how the Bayes theorem
allows the a
priori knowledge of the sign of the collisional coefficient to be
rigourously
embedded into the analysis. As an application, data from the INRIM
caesium
fountain are used and the Bayesian and orthodox analyses are compared.
The
Bayes theorem allows the orthodox uncertainty to be reduced by 28% and
demonstrates to be an important tool in primary frequency-metrology.
http://arxiv.org/abs/0907.4849
---------------------------------------------------------------
8864. DIRICHLET POLYNOMIALS: SOME OLD AND RECENT RESULTS, AND THEIR
INTERPLAY IN NUMBER THEORY
Michel Weber
In the first part of the paper, we present and discuss the interplay of
Dirichlet polynomials in some classical problems of number theory,
notably the
Lindel\"of Hypothesis. We review some typical properties of their
means and
continue with some investigations concerning their supremum
properties. Their
random counterpart is next considered in the second part of the paper.
An
analysis of their supremum properties, which is entirely based on
methods of
stochastic processes, is presented. Some complementary results and
related
questions are included in the last section of the paper.
http://arxiv.org/abs/0907.4931
---------------------------------------------------------------
8865. AN ANALOGUE OF THE L\'EVY-CRAM\'ER THEOREM FOR MULTI-
DIMENSIONAL RAYLEIGH DISTRIBUTIONS
Thu Nguyen
In the present paper we prove that every k-dimensional Cartesian
product of
Kingman convolutions can be embedded into a k-dimensional symmetric
convolution
(k=1, 2, ...) and obtain an analogue of the Cram\'er-L\'evy theorem for
multi-dimensional Rayleigh distributions. A new and more general class
of
multi-dimensional Rayleygh distributions and associated higher
dimensional
Bessel processes are introduced and studied. This class of processes
inherits
the well-known characteristics of Brownian motions: They are independent
stationary "increments" processes with continuous sample paths.
http://arxiv.org/abs/0907.5035
---------------------------------------------------------------
8866. THE WEAK COUPLING LIMIT OF DISORDERED COPOLYMER MODELS
Francesco Caravenna and Giambattista Giacomin
A copolymer is a chain of repetitive units (monomers) that are almost
identical, but they differ in their degree of affinity for certain
solvents.
This difference leads to striking phenomena when the polymer
fluctuates in a
non-homogeneous medium, for example made up by two solvents separated
by an
interface. One may observe, for instance, the localization of the
polymer at
the interface between the two solvents. A discrete model of such
system, based
on the simple symmetric random walk on Z, has been investigated in
[Bolthausen
and den Hollander, Ann. Probab. 25 (1997), 1334-1366], notably in the
weak
polymer-solvent coupling limit, where the convergence of the discrete
model
toward a continuum model, based on Brownian motion, has been
established. This
result is remarkable because it strongly suggests a universal feature of
copolymer models. In this work we prove that this is indeed the case.
More
precisely, we determine the weak coupling limit for a general class of
discrete
copolymer models, obtaining as limits a one-parameter (\alpha \in
(0,1)) family
of continuum models, based on \alpha-stable regenerative sets.
http://arxiv.org/abs/0907.5076
---------------------------------------------------------------
8867. LAW OF LARGE NUMBERS FOR THE MAXIMAL FLOW THROUGH TILTED
CYLINDERS IN TWO-DIMENSIONAL FIRST PASSAGE PERCOLATION
Rapha\"el Rossignol and Marie Th\'eret
Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random
capacities.
We prove a law of large numbers for the maximal flow crossing a
rectangle in
$\mathbb{R}^2$ when the side lengths of the rectangle go to infinity.
The value
of the limit depends on the asymptotic behaviour of the ratio of the
height of
the cylinder over the length of its basis. This law of large numbers
extends
the law of large numbers obtained by Grimmett and Kesten (1984) for
rectangles
of particular orientation.
http://arxiv.org/abs/0907.5112
---------------------------------------------------------------
8868. STANDARD DEVIATION OF THE LONGEST COMMON SUBSEQUENCE
J\"uri Lember and Heinrich Matzinger
Let $L_n$ be the length of the longest common subsequence of two
independent
i.i.d. sequences of Bernoulli variables of length $n$. We prove that
the order
of the standard deviation of $L_n$ is $\sqrt{n}$, provided the
parameter of the
Bernoulli variables is small enough. This validates Waterman's
conjecture in
this situation [Philos. Trans. R. Soc. Lond. Ser. B 344 (1994)
383--390]. The
order conjectured by Chvatal and Sankoff [J. Appl. Probab. 12 (1975)
306--315],
however, is different.
http://arxiv.org/abs/0907.5137
---------------------------------------------------------------
8869. BRUNET-DERRIDA PARTICLE SYSTEMS, FREE BOUNDARY PROBLEMS AND
WIENER-HOPF EQUATIONS
Rick Durrett and Daniel Remenik
We consider a branching-selection system in $\rr$ with $N$ particles
which
give birth independently at rate 1 and where after each birth the
leftmost
particle is erased, keeping the number of particles constant. We show
that, as
$N\to\infty$, the empirical measure process associated to the system
converges
in distribution to a deterministic measure-valued process whose
densities solve
a free boundary integro-differential equation. We also show that this
equation
has a unique traveling wave solution traveling at speed $c$ or no such
solution
depending on whether $c>a$ or $c\leq a$, where $a$ is the asymptotic
speed of
the branching random walk obtained by ignoring the removal of the
leftmost
particles in our process. The traveling wave solutions correspond to
solutions
of Wiener-Hopf equations.
http://arxiv.org/abs/0907.5180
---------------------------------------------------------------
8870. ON ASEP WITH STEP BERNOULLI INITIAL CONDITION
Craig A. Tracy and Harold Widom
This paper extends results of earlier work on ASEP to the case of step
Bernoulli initial condition. The main results are a representation in
terms of
a Fredholm determinant for the probability distribution of a fixed
particle,
and asymptotic results which in particular establish KPZ universality
for this
probability in one regime. (And, as a corollary, for the current
fluctuations.)
http://arxiv.org/abs/0907.5192
---------------------------------------------------------------
8871. ON INFINITELY COHOMOLOGOUS TO ZERO OBSERVABLES
Amanda de Lima and Daniel Smania
We show that for a large class of piecewise expanding maps T, the
bounded
p-variation observables u_0 that admits an infinite sequence of bounded
p-variation observables u_i satisfying u_i(x)= u_{i+1}(Tx) -u_{i+1}(x)
are
constant. The method of the proof consists in to find a suitable
Hilbert basis
for L^2(hm), where hm is the unique absolutely continuous invariant
probability
of T. In terms of this basis, the action of the Perron-Frobenious and
the
Koopan operator on L^2(hm) can be easily understood. This result
generalizes
earlier results by Bamon, Kiwi, Rivera-Letelier and Urzua in the case
T(x)= n x
mod 1, n in N-{0,1} and Lipchitizian observables u_0.
http://arxiv.org/abs/0907.5013
---------------------------------------------------------------
8872. A DISCUSSION ON MEAN EXCESS PLOTS
Souvik Ghosh and Sidney I Resnick
A widely used tool in the study of risk, insurance and extreme values
is the
mean excess plot. One use is for validating a Generalized Pareto model
for the
excess distribution. This paper investigates some theoretical and
practical
aspects of the use of the mean excess plot.
http://arxiv.org/abs/0907.5236
---------------------------------------------------------------
8873. A HISTORICAL LAW OF LARGE NUMBERS FOR THE MARCUS LUSHNIKOV PROCESS
St\'ephanie Jacquot
The Marcus-Lushnikov process is a finite stochastic particle system,
in which
each particle is entirely characterized by its mass. Each pair of
particles
with masses $x$ and $y$ merges into a single particle at a given rate
$K(x,y)$.
Under certain assumptions, this process converges to the solution to
Smoluchowski equation, as the number of particles increases to
infinity. The
Marcus-Lushnikov process gives at each time the distribution of masses
of the
particles present in the system, but does not retain the history of
formation
of the particles. In this paper, we set up a historical analogue of the
Marcus-Lushnikov process (built according the rules of construction of
the
usual Markov-Lushnikov process) each time giving what we call the
historical
tree of a particle. The historical tree of a particle present in the
Marcus-Lushnikov process at a given time $t$ encodes information about
the
times and masses of the coagulation events that have formed that
particle. We
prove a law of large numbers for the empirical distribution of such
historical
trees. The limit is a natural measure on trees which is constructed
from a
solution to Smoluchowski coagulation equation.
http://arxiv.org/abs/0907.5305
---------------------------------------------------------------
8874. A METRIC ANALYSIS OF CRITICAL HAMILTON--JACOBI EQUATIONS IN THE
STATIONARY ERGODIC SETTING
Andrea Davini and Antonio Siconolfi
We adapt the metric approach to the study of stationary ergodic
Hamilton-Jacobi equations, for which a notion of admissible random
(sub)solution is defined. For any level of the Hamiltonian greater
than or
equal to a distinguished critical value, we define an intrinsic random
semidistance and prove that an asymptotic norm does exist. Taking as
source
region a suitable class of closed random sets, we show that the Lax
formula
provides admissible subsolutions. This enables us to relate the
degeneracies of
the critical stable norm to the existence/nonexistence of exact or
approximate
critical admissible solutions.
http://arxiv.org/abs/0907.5332
---------------------------------------------------------------
8875. WEAK KAM THEORY TOPICS IN THE STATIONARY ERGODIC SETTING
Andrea Davini and Antonio Siconolfi
We perform a qualitative analysis of the critical equation associated
with a
stationary ergodic Hamiltonian through a stochastic version of the
metric
method, where the notion of closed random stationary set, issued from
stochastic geometry, plays a major role. Our purpose is to give an
appropriate
notion of random Aubry set, to single out characterizing conditions
for the
existence of exact or approximate correctors, and write down
representation
formulae for them. For the last task, we make use of a Lax--type
formula,
adapted to the stochastic environment. This material can be regarded
as a first
step of a long--term project to develop a random analog of Weak KAM
Theory,
generalizing what done in the periodic case or, more generally, when the
underlying space is a compact manifold.
http://arxiv.org/abs/0907.5334
---------------------------------------------------------------
8876. PROFILES OF PERMUTATIONS
Michael Lugo
This paper develops an analogy between the cycle structure of, on the
one
hand, random permutations with cycle lengths restricted to lie in an
infinite
set $S$ with asymptotic density $\sigma$ and, on the other hand,
permutations
selected according to the Ewens distribution with parameter $\sigma$. In
particular we show that the asymptotic expected number of cycles of
random
permutations of $[n]$ with all cycles even, with all cycles odd, and
chosen
from the Ewens distribution with parameter 1/2 are all ${1 \over 2}
\log n +
O(1)$, and the variance is of the same order. Furthermore, we show
that in
permutations of $[n]$ chosen from the Ewens distribution with parameter
$\sigma$, the probability of a random element being in a cycle longer
than
$\gamma n$ approaches $(1-\gamma)^\sigma$ for large $n$. The same
limit law
holds for permutations with cycles carrying multiplicative weights
with average
$\sigma$. We draw parallels between the Ewens distribution and the
asymptotic-density case and explain why these parallels should exist
using
permutations drawn from weighted Boltzmann distributions.
http://arxiv.org/abs/0907.5351
---------------------------------------------------------------
8877. SELF-INTERACTING DIFFUSIONS IV: RATE OF CONVERGENCE
Michel Benaim (UNINE) and Olivier Raimond (MODAL'X)
Self-interacting diffusions are processes living on a compact Riemannian
manifold defined by a stochastic differential equation with a drift term
depending on the past empirical measure of the process. The
asymptotics of this
measure is governed by a deterministic dynamical system and under
certain
conditions it converges almost surely towards a deterministic measure
(see
Bena\"im, Ledoux, Raimond (2002) and Bena\"im, Raimond (2005)). We are
interested here in the rate of this convergence. A central limit
theorem is
proved. In particular, this shows that greater is the interaction
repelling
faster is the convergence.
http://arxiv.org/abs/0907.5468
---------------------------------------------------------------
8878. UPPER LARGE DEVIATIONS FOR THE MAXIMAL FLOW THROUGH A DOMAIN OF
$\MATHBB{R}^D$ IN FIRST PASSAGE PERCOLATION
Rapha\"el Cerf and Marie Th\'eret
We consider the standard first passage percolation model in the rescaled
graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary
$\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $\Gamma^2$ be two
disjoint open
subsets of $\Gamma$, representing the parts of $\Gamma$ through which
some
water can enter and escape from $\Omega$. We investigate the asymptotic
behaviour of the flow $\phi_n$ through a discrete version $\Omega_n$ of
$\Omega$ between the corresponding discrete sets $\Gamma^1_n$ and $
\Gamma^2_n$.
We prove that under some conditions on the regularity of the domain
and on the
law of the capacity of the edges, the upper large deviations of $\phi_n/
n^{d-1}$ above a certain constant are of volume order.
http://arxiv.org/abs/0907.5499
---------------------------------------------------------------
8879. LOWER LARGE DEVIATIONS FOR THE MAXIMAL FLOW THROUGH A DOMAIN OF
$\MATHBB{R}^D$ IN FIRST PASSAGE PERCOLATION
Rapha\"el Cerf and Marie Th\'eret
We consider the standard first passage percolation model in the rescaled
graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary
$\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $\Gamma^2$ be two
disjoint open
subsets of $\Gamma$, representing the parts of $\Gamma$ through which
some
water can enter and escape from $\Omega$. We investigate the asymptotic
behaviour of the flow $\phi_n$ through a discrete version $\Omega_n$ of
$\Omega$ between the corresponding discrete sets $\Gamma^1_n$ and $
\Gamma^2_n$.
We prove that under some conditions on the regularity of the domain
and on the
law of the capacity of the edges, the lower large deviations of $\phi_n/
n^{d-1}$ below a certain constant are of surface order.
http://arxiv.org/abs/0907.5501
---------------------------------------------------------------
8880. LAW OF LARGE NUMBERS FOR THE MAXIMAL FLOW THROUGH A DOMAIN OF $
\MATHBB{R}^D$ IN FIRST PASSAGE PERCOLATION
Rapha\"el Cerf and Marie Th\'eret
We consider the standard first passage percolation model in the rescaled
graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary
$\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $\Gamma^2$ be two
disjoint open
subsets of $\Gamma$, representing the parts of $\Gamma$ through which
some
water can enter and escape from $\Omega$. We investigate the asymptotic
behaviour of the flow $\phi_n$ through a discrete version $\Omega_n$ of
$\Omega$ between the corresponding discrete sets $\Gamma^1_n$ and $
\Gamma^2_n$.
We prove that under some conditions on the regularity of the domain
and on the
law of the capacity of the edges, $\phi_n$ converges almost surely
towards a
constant $\phi_{\Omega}$, which is the solution of a continuous non-
random
min-cut problem. Moreover, we give a necessary and sufficient
condition on the
law of the capacity of the edges to ensure that $\phi_{\Omega} >0$.
http://arxiv.org/abs/0907.5504
---------------------------------------------------------------
8881. MONOTONICITY PROPERTIES OF THE ASYMPTOTIC RELATIVE EFFICIENCY
BETWEEN COMMON CORRELATION STATISTICS IN THE BIVARIATE NORMAL MODEL
Raymond Molzon and Iosif Pinelis
Pearson's is the most common correlation statistic, used mainly in
parametric
settings. Most common among nonparametric correlation statistics are
Spearman's
and Kendall's. We show that for bivariate normal i.i.d. samples the
pairwise
asymptotic relative efficiency between these three statistics depends
monotonically on the population correlation coefficient. This
monotonicity is a
corollary to a stronger result. The proofs rely on the use of
l'Hospital-type
rules for monotonicity patterns.
http://arxiv.org/abs/0907.5448
---------------------------------------------------------------
8882. CONDITIONALLY MONOTONE INDEPENDENCE
Takahiro Hasebe
We define the notion of conditionally monotone product as a part of
conditionally free product, which naturally includes monotone and
Boolean
products. Then we define conditionally monotone cumulants which are
useful to
calculate the limit distributions in central limit theorem and
Poisson's law of
small numbers. Moreover, we introduce deformed convolutions arising
from the
conditionally monotone convolution of probability measures and compute
the
limit distributions. In order to understand the validity of cumulants,
we
discuss what are cumulants of a given convolution product in general.
http://arxiv.org/abs/0907.5473
---------------------------------------------------------------
8883. LOSS OF MEMORY OF RANDOM FUNCTIONS OF MARKOV CHAINS AND
LYAPUNOV EXPONENTS
Pierre Collet and Florencia Leonardi
In this paper we prove that the asymptotic rate of exponential loss of
memory
of a random function of a Markov chain $(Z_{t})_{t\in\Z}$ is bounded
above by
the difference of the first two Lyapunov exponents of a certain
product of
matrices. We also show that this bound is in fact realized, namely for
almost
all realization of the process $(Z_{t})_{t\in\Z}$, we can find symbols
where
the asymptotic exponential rate of loss of memory attains the
difference of the
first two Lyapunov exponents. This shows that the process has infinite
memory
and leads to a lower bound on the asymptotic exponential loss of
memory which
is saturated (and equal to the upper bound for an adequate choice of the
symbols) on a set of full measure.
http://arxiv.org/abs/0908.0077
---------------------------------------------------------------
8884. SCALING LIMITS OF ANISOTROPIC HASTINGS-LEVITOV CLUSTERS
Fredrik Johansson and Alan Sola and Amanda Turner
We consider a variation of the standard Hastings-Levitov model HL(0), in
which growth is anisotropic. Two natural scaling limits are
established and we
give precise descriptions of the effects of the anisotropy. We show
that the
limit shapes can be realised as Loewner hulls and that the evolution of
harmonic measure on the cluster boundary can be described by the
solution to a
deterministic ordinary differential equation related to the Loewner
equation.
We also characterise the stochastic fluctuations around the
deterministic limit
flow.
http://arxiv.org/abs/0908.0086
---------------------------------------------------------------
8885. A STOCHASTIC MIN-DRIVEN COALESCENCE PROCESS AND ITS
HYDRODYNAMICAL LIMIT
Anne-Laure Basdevant (IMT) and Philippe Laurencot (IMT) and James R.
Norris (DPMMS), Clement Rau (IMT)
A stochastic system of particles is considered in which the sizes of the
particles increase by successive binary mergers with the constraint
that each
coagulation event involves a particle with minimal size. Convergence
of a
suitably renormalised version of this process to a deterministic
hydrodynamical
limit is shown and the time evolution of the minimal size is studied
for both
deterministic and stochastic models.
http://arxiv.org/abs/0908.0129
---------------------------------------------------------------
8886. SAMPLING CONDITIONED HYPOELLIPTIC DIFFUSIONS
Martin Hairer and Andrew M. Stuart and Jochen Voss
A series of recent articles introduced a method to construct stochastic
partial differential equations (SPDEs) which are invariant with
respect to the
distribution of a given conditioned diffusion. These works are
restricted to
the case of elliptic diffusions where the drift has a gradient
structure, and
the resulting SPDE is of second order parabolic type.
The present article extends this methodology to allow the
construction of
SPDEs which are invariant with respect to the distribution of a class of
hypoelliptic diffusion processes, subject to a bridge conditioning.
This allows
the treatment of more realistic physical models, for example one can
use the
resulting SPDE to study transitions between meta-stable states in
mechanical
systems with friction and noise. In this situation the restriction of
the drift
being a gradient can also be lifted.
http://arxiv.org/abs/0908.0162
---------------------------------------------------------------
8887. ON THE SPEED OF SPREAD FOR FRACTIONAL REACTION-DIFFUSION EQUATIONS
Hans Engler
The fractional reaction diffusion equation u_t + Au = g(u) is discussed,
where A is a fractional differential operator on the real line with
order
\alpha between 0 and 2, the C^1 function g vanishes at 0 and 1, and
either g is
non-negative on (0,1) or g < 0 near 0. In the case of non-negative g,
it is
shown that solutions with initial support on the positive half axis
spread into
the left half axis with unbounded speed if g satisfies some weak growth
condition near 0 in the case \alpha > 1, or if g is merely positive on a
sufficiently large interval near 1 in the case \alpha < 1. On the
other hand,
it shown that solutions spread with finite speed if g'(0) < 0. The
proofs use
comparison arguments and a new family of traveling wave solutions for
this
class of problems.
http://arxiv.org/abs/0908.0024
---------------------------------------------------------------
8888. A STRONG PAIR CORRELATION BOUND IMPLIES THE CLT FOR SINAI
BILLIARDS
Mikko Stenlund
For Dynamical Systems, a strong bound on multiple correlations implies
the
Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a
bound is
derived for dynamically Holder continuous observables of dispersing
Billiards.
Here we weaken the regularity assumption and subsequently show that
the bound
on multiple correlations follows directly from the bound on pair
correlations.
Thus, a strong bound on pair correlations alone implies the CLT, for a
wider
class of observables. The result is extended to Anosov diffeomorphisms
in any
dimension.
http://arxiv.org/abs/0908.0027
---------------------------------------------------------------
8889. APPROXIMATING EIGENVECTORS BY SUBSAMPLING
Noureddine El Karoui and Alexandre d'Aspremont
We show that averaging eigenvectors of randomly sampled submatrices
efficiently approximates the true eigenvectors of the original matrix
under
certain conditions on the incoherence of the spectral decomposition.
This
incoherence assumption is typically milder than those made in matrix
completion
and allows eigenvectors to be sparse. We discuss applications to
spectral
methods in dimensionality reduction and information retrieval.
http://arxiv.org/abs/0908.0137
---------------------------------------------------------------
8890. ON THE ROLE OF SPARSITY IN COMPRESSED SENSING AND RANDOM MATRIX
THEORY
Roman Vershynin
We discuss applications of some concepts of Compressed Sensing in the
recent
work on invertibility of random matrices due to Rudelson and the
author. We
sketch an argument leading to the optimal bound N^{-1/2} on the median
of the
smallest singular value of an N by N matrix with random independent
entries. We
highlight the parts of the argument where sparsity ideas played a key
role.
http://arxiv.org/abs/0908.0257
---------------------------------------------------------------
8891. LAYERING AND WETTING TRANSITIONS FOR AN SOS INTERFACE
Kenneth S. Alexander and Fran\c{c}ois Dunlop and Salvador Miracle-Sol
\'e
We study the solid-on-solid interface model above a horizontal wall in
three
dimensional space, with an attractive interaction when the interface
is in
contact with the wall, at low temperatures. There is no bulk external
field.
The system presents a sequence of layering transitions, whose levels
increase
with the temperature, before reaching the wetting transition.
http://arxiv.org/abs/0908.0321
---------------------------------------------------------------
8892. UNIVERSAL GAUSSIAN FLUCTUATIONS OF NON-HERMITIAN MATRIX ENSEMBLES
Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X)
We prove multi-dimensional central limit theorems for the spectral
moments
(of arbitrary degrees) associated with random matrices with real-
valued i.i.d.
entries, satisfying some appropriate moment conditions. Our techniques
rely on
a universality principle for the Gaussian Wiener chaos, recently
proved by the
authors together with Gesine Reinert, as well as on some combinatorial
estimates. Unlike other related results in the probabilistic
literature, we do
not require that the law of the entries has a density with respect to
the
Lebesgue measure. In particular, our results apply to the ensemble of
Bernoulli
random matrices.
http://arxiv.org/abs/0908.0391
---------------------------------------------------------------
8893. OPTIMAL TRANSPORT AND TESSELLATION
Martin Huesmann
Optimal transport from the volume measure to a convex combination of
Dirac
measures yields a tessellation of a Riemannian manifold into pieces of
arbitrary relative size. This tessellation is studied for the cost
functions
$c_p(z,y)=\frac{1}{p}d^p(z,y)$ and $1\leq p<\infty$. Geometric
descriptions of
the tessellations for all $p$ is obtained for compact subsets of the
Euclidean
space and the sphere. For $p=1$ this approach yields Laguerre
tessellations for
all compact Riemannian manifolds.
http://arxiv.org/abs/0908.0442
---------------------------------------------------------------
8894. THE STATISTICAL MECHANICS OF STRETCHED POLYMERS
Dmitry Ioffe and Yvan Velenik
We describe some recent results concerning the statistical properties
of a
self-interacting polymer stretched by an external force. We
concentrate mainly
on the cases of purely attractive or purely repulsive self-
interactions, but
our results are stable under suitable small perturbations of these
pure cases.
We provide in particular a precise description of the stretched phase
(local
limit theorems for the end-point and local observables, invariance
principle,
microscopic structure). Our results also characterize precisely the
(non-trivial, direction-dependent) critical force needed to trigger the
collapsed/stretched phase transition in the attractive case. We also
describe
some recent progress: first, the determination of the order of the phase
transition in the attractive case; second, a proof that a semi-
directed polymer
in quenched random environment is diffusive in dimensions 4 and higher
when the
temperature is high enough. In addition, we correct an incomplete
argument from
one of our earlier works.
http://arxiv.org/abs/0908.0452
---------------------------------------------------------------
8895. ON LINEAR EVOLUTION EQUATIONS WITH CYLINDRICAL L\'EVY NOISE
Enrico Priola and Jerzy Zabczyk
We study an infinite-dimensional
Ornstein-Uhlenbeck process $(X_t)$ in a given
Hilbert space $H$. This is driven by a cylindrical symmetric L\'evy
process
without a Gaussian component and taking values in a Hilbert space $U$
which
usually contains $H$. We give if and only if conditions under which
$X_t$ takes
values in $H$ for some $t>0$ or for all $t>0$. Moreover, we prove
irreducibility for $(X_t)$.
http://arxiv.org/abs/0908.0356
---------------------------------------------------------------
8896. ON THE SHORT TIME ASYMPTOTIC OF THE STOCHASTIC ALLEN-CAHN EQUATION
Hendrik Weber
A description of the short time behavior of solutions of the Allen-Cahn
equation with a smoothened additive noise is presented. The key result
is that
in the sharp interface limit solutions move according to motion by mean
curvature with an additional stochastic forcing. This extends a
similar result
of Funaki in spatial dimension $n=2$ to arbitrary dimensions.
http://arxiv.org/abs/0908.0580
---------------------------------------------------------------
8897. UPPER AND LOWER BOUNDS IN EXPONENTIAL TAUBERIAN THEOREMS
Jochen Voss
In this text we study, for positive random variables, the relation
between
the behaviour of the Laplace transform near infinity and the
distribution near
zero. A result of de Bruijn shows that $E(e^{-\lambda X}) \sim
e^{r\sqrt{\lambda}}$ for $\lambda\to\infty$ and $P(X\leq\eps) \sim
\e^{s/\eps}$
for $\eps\downarrow0$ are in some sense equivalent and gives a
relation between
the constants $r$ and $s$. We illustrate how this result can be used
to obtain
simple large deviation results. For use in more complex situations we
also give
a generalisation of de Bruijn's result to the case when the upper and
lower
limits are different from each other.
http://arxiv.org/abs/0908.0642
---------------------------------------------------------------
8898. EXACT SOLUTION OF A TWO-TYPE BRANCHING PROCESS: CLONE SIZE
DISTRIBUTION IN CELL DIVISION KINETICS
Tibor Antal and P. L. Krapivsky
We study a two-type branching process which provides excellent
description of
experimental data on cell dynamics in skin tissue (Clayton et al.,
2007). The
model involves only a single type of progenitor cell, and does not
require
support from a self-renewed population of stem cells. The progenitor
cells
divide and may differentiate into post-mitotic cells. We derive an exact
solution of this model in terms of generating functions for the total
number of
cells, and for the number of cells of different types. We also deduce
large
time asymptotic behaviors drawing on our exact results, and on an
independent
diffusion approximation.
http://arxiv.org/abs/0908.0484
---------------------------------------------------------------
8899. RECURRENCE AND ERGODICITY OF RANDOM WALKS ON LINEAR GROUPS AND
ON HOMOGENEOUS SPACES
Y. Guivarc'h and C. R. E. Raja
We discuss recurrence and ergodicity properties of random walks and
associated skew products for large classes of locally compact groups and
homogeneous spaces. In particular we show that a closed subgroup of a
product
of finitely many linear groups over local fields supports a recurrent
random
walk if and only if it has at most quadratic growth. We give also a
detailed
analysis of ergodicity properties for special classes of random walks on
homogeneous spaces. The structure of closed subgroups of linear groups
over
local fields and the properties of group actions with respect to
stationary
measures play an important role in the proofs.
http://arxiv.org/abs/0908.0637
---------------------------------------------------------------
8900. A GENERAL STRONG LAW OF LARGE NUMBERS FOR ADDITIVE ARITHMETIC
FUNCTIONS
Istvan Berkes and Michel Weber
Let $f(n)$ be a strongly additive complex valued arithmetic function.
Under
mild conditions on $f$, we prove the following weighted strong law of
large
numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d.
random
variables, then $$ \lim_{N\to \infty} {\sum_{n=1}^N f(n) X_n \over
\sum_{n=1}^N
f(n)} \buildrel{a.s.}\over{=} \E X . $$
http://arxiv.org/abs/0908.0680
---------------------------------------------------------------
8901. OPTIMAL SCALINGS FOR LOCAL METROPOLIS--HASTINGS CHAINS ON
NONPRODUCT TARGETS IN HIGH DIMENSIONS
Alexandros Beskos and Gareth Roberts and Andrew Stuart
We investigate local MCMC algorithms, namely the random-walk
Metropolis and
the Langevin algorithms, and identify the optimal choice of the local
step-size
as a function of the dimension $n$ of the state space, asymptotically as
$n\to\infty$. We consider target distributions defined as a change of
measure
from a product law. Such structures arise, for instance, in inverse
problems or
Bayesian contexts when a product prior is combined with the
likelihood. We
state analytical results on the asymptotic behavior of the algorithms
under
general conditions on the change of measure. Our theory is motivated by
applications on conditioned diffusion processes and inverse problems
related to
the 2D Navier--Stokes equation.
http://arxiv.org/abs/0908.0865
---------------------------------------------------------------
8902. ASYMPTOTIC OPTIMALITY OF ISOPERIMETRIC CONSTANTS WITH RESPECT
TO $L^{2}(\PI)$-SPECTRAL GAPS
Achim Wuebker
In this paper we investigate the existence of $L^{2}(\pi)$-spectral
gaps for
$\pi$-irreducible, positive recurrent Markov chains on general state
space. We
obtain necessary and sufficient conditions for the existence of
$L^{2}(\pi)$-spectral gaps in terms of a sequence of isoperimetric
constants
and establish their asymptotic behavior. It turns out that in some
cases the
spectral gap can be understood in terms of convergence of an induced
probability flow to the uniform flow. The obtained theorems can be
interpreted
as mixing results and yield sharp estimates for the spectral gap of
some Markov
chains.
http://arxiv.org/abs/0908.0867
---------------------------------------------------------------
8903. $L^{2}$-SPECTRAL GAPS, WEAK-REVERSIBLE AND VERY WEAK-REVERSIBLE
MARKOV CHAINS
Achim Wuebker and Zakhar Kabluchko
The theory of $L^2$-spectral gaps for reversible Markov chains has been
studied by many authors. In this paper we consider positive recurrent
general
state space Markov chains with stationary transition probabilities.
Replacing
the assumption of reversibility by a less strong one, we still obtain
a simple
necessary and sufficient condition for the spectral gap property of the
associated Markov operator in terms of isoperimetric constant.
Moreover, we
define a new sequence of isoperimetric constants which provides a
necessary and
sufficient condition for the existence of a spectral gap in a very
general
setting. Finally, these results are used to obtain simple sufficient
conditions
for the existence of a spectral gap in terms of the first and second
order
transition probabilities.
http://arxiv.org/abs/0908.0888
---------------------------------------------------------------
8904. $L^{2}$-SPECTRAL GAPS FOR TIME DISCRETE REVERSIBLE MARKOV CHAINS
Achim Wuebker
In this paper we study the spectral properties of Markov-operator on
$L^{2}$-spaces. Lawler and Sokal (Trans. Amer. Math. Soc., 1988, 309,
pp.
557-580) used isoperimetric constants for discrete and continuous time
Markov
chains to obtain a spectral gap at 1. For time discrete Markov chains
this does
not exclude periodic behavior. We define a new constant measuring the
distance
from periodicity and give necessary and sufficient conditions for the
existence
of a global spectral gap in terms of this constant.
http://arxiv.org/abs/0908.0897
---------------------------------------------------------------
8905. ROBUST MEAN-VARIANCE HEDGING IN THE SINGLE PERIOD MODEL
R. Tevzadze and T. Uzunashvili
We give an explicit solution of robust mean-variance hedging problem
in the
single period model for some type of contingent claims. The alternative
approach is also considered.
http://arxiv.org/abs/0908.0840
---------------------------------------------------------------
8906. EFFICIENT IMPORTANCE SAMPLING FOR BINARY CONTINGENCY TABLES
Jose H. Blanchet
Importance sampling has been reported to produce algorithms with
excellent
empirical performance in counting problems. However, the theoretical
support
for its efficiency in these applications has been very limited. In
this paper,
we propose a methodology that can be used to design efficient importance
sampling algorithms for counting and test their efficiency rigorously.
We apply
our techniques after transforming the problem into a rare-event
simulation
problem--thereby connecting complexity analysis of counting problems
with
efficiency in the context of rare-event simulation. As an illustration
of our
approach, we consider the problem of counting the number of binary
tables with
fixed column and row sums, $c_j$'s and $r_i$'s, respectively, and total
marginal sums $d=\sum_jc_j$. Assuming that $\max_jc_j=o(d^{1/2})$, $\sum
c_j^2=O(d)$ and the $r_j$'s are bounded, we show that a suitable
importance
sampling algorithm, proposed by Chen et al. [J. Amer. Statist. Assoc.
100
(2005) 109--120], requires $O(d^3\varepsilon^{-2}\delta^{-1})$
operations to
produce an estimate that has $\varepsilon$-relative error with
probability
$1-\delta$. In addition, if $\max_jc_j=o(d^{1/4-\delta_0})$ for some
$\delta_0>0$, the same coverage can be guaranteed with
$O(d^3\varepsilon^{-2}\log(\delta^{-1}))$ operations.
http://arxiv.org/abs/0908.0999
---------------------------------------------------------------
8907. A PROBABILISTIC STUDY OF NEURAL COMPLEXITY
Jerome Buzzi (LM-Orsay) and Lorenzo Zambotti (PMA)
G. Edelman, O. Sporns, and G. Tononi have introduced in theoretical
biology
the neural complexity of a family of random variables. They have
defined it as
a specific average of mutual information over subsystems. We show that
their
choice of weights satisfies two natural properties, namely
exchangeability and
additivity. This paper classifies all functionals satisfying these two
properties (which we call intricacies) in terms of probability laws on
the unit
interval and studies the growth rate of maximal intricacies when the
size of
the system goes to infinity. For systems of a fixed size, we show that
the
maximizers are non-unique and that the maximal value is not approached
by
exchangeable laws.
http://arxiv.org/abs/0908.1006
---------------------------------------------------------------
8908. SELLING A STOCK AT THE ULTIMATE MAXIMUM
Jacques du Toit and Goran Peskir
Assuming that the stock price $Z=(Z_t)_{0\leq t\leq T}$ follows a
geometric
Brownian motion with drift $\mu\in\mathbb{R}$ and volatility $
\sigma>0$, and
letting $M_t=\max_{0\leq s\leq t}Z_s$ for $t\in[0,T]$, we consider the
optimal
prediction problems \[V_1=\inf_{0\leq\tau\leq
T}\mathsf{E}\biggl(\frac{M_T}{Z_{\tau}}\biggr)\quadand\quad
V_2=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_T}\biggr),
\] where
the infimum and supremum are taken over all stopping times $\tau$ of $Z
$. We
show that the following strategy is optimal in the first problem: if $
\mu\leq0$
stop immediately; if $\mu\in (0,\sigma^2)$ stop as soon as $M_t/Z_t$
hits a
specified function of time; and if $\mu\geq\sigma^2$ wait until the
final time
$T$. By contrast we show that the following strategy is optimal in the
second
problem: if $\mu\leq\sigma^2/2$ stop immediately, and if $\mu>
\sigma^2/2$ wait
until the final time $T$. Both solutions support and reinforce the
widely held
financial view that ``one should sell bad stocks and keep good ones.''
The
method of proof makes use of parabolic free-boundary problems and local
time--space calculus techniques. The resulting inequalities are
unusual and
interesting in their own right as they involve the future and as such
have a
predictive element.
http://arxiv.org/abs/0908.1014
---------------------------------------------------------------
8909. AN OPERATOR APPROACH FOR MARKOV CHAIN WEAK APPROXIMATIONS WITH
AN APPLICATION TO INFINITE ACTIVITY L\'{E}VY DRIVEN SDES
Hideyuki Tanaka and Arturo Kohatsu-Higa
Weak approximations have been developed to calculate the expectation
value of
functionals of stochastic differential equations, and various numerical
discretization schemes (Euler, Milshtein) have been studied by many
authors. We
present a general framework based on semigroup expansions for the
construction
of higher-order discretization schemes and analyze its rate of
convergence. We
also apply it to approximate general L\'{e}vy driven stochastic
differential
equations.
http://arxiv.org/abs/0908.1021
---------------------------------------------------------------
8910. ASYMPTOTIC NORMALITY OF PLUG-IN LEVEL SET ESTIMATES
David M. Mason and Wolfgang Polonik
We establish the asymptotic normality of the $G$-measure of the
symmetric
difference between the level set and a plug-in-type estimator of it
formed by
replacing the density in the definition of the level set by a kernel
density
estimator. Our proof will highlight the efficacy of Poissonization
methods in
the treatment of large sample theory problems of this kind.
http://arxiv.org/abs/0908.1045
---------------------------------------------------------------
8911. GAUSSIAN PERTURBATIONS OF CIRCLE MAPS: A SPECTRAL APPROACH
John Mayberry
In this work, we examine spectral properties of Markov transition
operators
corresponding to Gaussian perturbations of discrete time dynamical
systems on
the circle. We develop a method for calculating asymptotic expressions
for
eigenvalues (in the zero noise limit) and show that changes to the
number or
period of stable orbits for the deterministic system correspond to
changes in
the number of limiting modulus 1 eigenvalues of the transition
operator for the
perturbed process. We call this phenomenon a $\lambda$-bifurcation.
Asymptotic
expressions for the corresponding eigenfunctions and eigenmeasures are
also
derived and are related to Hermite functions.
http://arxiv.org/abs/0908.1058
---------------------------------------------------------------
8912. A CONTINUOUS ANALOGUE OF THE INVARIANCE PRINCIPLE AND ITS ALMOST
SURE VERSION
E.E. Permyakova
We deal with random processes obtained from a homogeneous random
process with
independent increments by replacement of the time scale and by
multiplication
by a norming constant. We prove the convergence in distribution of these
processes to Wiener process in Skorokhod space endowed by the topology
of
uniform convergence. An integral type almost sure version of this
theorem is
obtained.
http://arxiv.org/abs/0908.1072
---------------------------------------------------------------
8913. FUNCTIONAL LIMIT THEOREMS FOR LEVY PROCESSES AND THEIR ALMOST-
SURE VERSIONS
E.E. Permyakova
In this paper we prove a criterion of convergence in distribution in
Skorokhod space. We apply this criterion to some special Levy
processes and
obtain almost-sure versions of limit theorems for these processes.
http://arxiv.org/abs/0908.1074
---------------------------------------------------------------
8914. TOTAL PROGENY IN KILLED BRANCHING RANDOM WALK
Louigi Addario-Berry and Nicolas Broutin
We consider a branching random walk for which the maximum position of a
particle in the n'th generation, M_n, has zero speed on the linear
scale: M_n/n
--> 0 as n --> infinity. We further remove (``kill'') any particle whose
displacement is negative, together with its entire descendence. The
size $Z$ of
the set of un-killed particles is almost surely finite. In this paper,
we
confirm a conjecture of Aldous that Exp[Z] < infinity while Exp[Z log
Z]=infinity. The proofs rely on precise large deviations estimates and
ballot
theorem-style results for the sample paths of random walks.
http://arxiv.org/abs/0908.1083
---------------------------------------------------------------
8915. ASYMPTOTIC BEHAVIOR OF THE FINITE-SIZE MAGNETIZATION AS A
FUNCTION OF THE SPEED OF APPROACH TO CRITICALITY
Richard S. Ellis and Jonathan Machta and Peter Tak-Hun Otto
The main focus of this paper is to determine whether the thermodynamic
magnetization is a physically relevant estimator of the finite-size
magnetization. This is done by comparing the asymptotic behaviors of
these two
quantities along parameter sequences converging to either a second-
order point
or the tricritical point in the mean-field Blume-Capel model. We show
that the
thermodynamic magnetization and the finite-size magnetization are
asymptotic
when the parameter alpha governing the speed at which the sequence
approaches
criticality is below a certain threshold alpha_0. However, when alpha
exceeds
alpha_0, the thermodynamic magnetization converges to 0 much faster
than the
finite-size magnetization. The asymptotic behavior of the finite-size
magnetization is proved via a moderate deviation principle when 0 <
alpha <
alpha_0 and via a weak-convergence limit when alpha > alpha_0. To the
best of
our knowledge, our results are the first rigorous confirmation of the
statistical mechanical theory of finite-size scaling for a mean-field
model.
http://arxiv.org/abs/0908.1103
---------------------------------------------------------------
8916. ON THE UNIQUENESS OF CLASSICAL SOLUTIONS OF CAUCHY PROBLEMS
Erhan Bayraktar and Hao Xing
Given that the terminal condition is of at most linear growth, it is
well
known that a Cauchy problem admits a unique classical solution when the
coefficient multiplying the second derivative (i.e., the volatility)
is also a
function of at most linear growth. In this note, we give a condition
on the
volatility that is necessary and sufficient for a Cauchy problem to
admit a
unique solution.
http://arxiv.org/abs/0908.1086
---------------------------------------------------------------
8917. CRAM\'{E}R TYPE MODERATE DEVIATION FOR THE MAXIMUM OF THE
PERIODOGRAM WITH APPLICATION TO SIMULTANEOUS TESTS IN GENE EXPRESSION
TIME SERIES
Weidong Liu and Qi Man Shao
In this paper, Cram\'{e}r type moderate deviations for the maximum of
the
periodogram and its studentized version are derived. The results are
then
applied to a simultaneous testing problem in gene expression time
series. It is
shown that the level of the simultaneous tests is accurate provided
that the
number of genes $G$ and the sample size $n$ satisfy $G=\exp(o(n^{1/3}))
$.
http://arxiv.org/abs/0908.1145
---------------------------------------------------------------
8918. ABSORBING-STATE PHASE TRANSITION FOR STOCHASTIC SANDPILES AND
ACTIVATED RANDOM WALKS
Leonardo T. Rolla and Vladas Sidoravicius
We study the long-time behavior of conservative interacting particle
systems
in $\mathbb Z$: The Activated Random Walk Model for reaction-diffusion
systems
and the Stochastic Sandpile. Our main result states that both systems
locally
fixate when the initial density of particles is small enough,
establishing the
existence of a non-trivial phase transition in the density parameter.
This fact
is predicted by theoretical physics arguments and supported by numerical
analysis.
http://arxiv.org/abs/0908.1152
---------------------------------------------------------------
8919. A CIESIELSKI-TAYLOR TYPE IDENTITY FOR POSITIVE SELF-SIMILAR
MARKOV PROCESSES
A.E. Kyprianou and P. Patie
The aim of this note is to give a straightforward proof of a general
version
of the Ciesielski-Taylor identity for positive self-similar Markov
processes of
the spectrally negative type which umbrellas all previously known
Ciesielski-Taylor identities within the latter class. The approach
makes use of
three fundamental features. Firstly a new transformation which maps a
subset of
the family of Laplace exponents of spectrally negative L\'evy
processes into
itself. Secondly some classical features of fluctuation theory for
spectrally
negative L\'evy processes as well as more recent fluctuation
identities for
positive self-similar Markov processes.
http://arxiv.org/abs/0908.1157
---------------------------------------------------------------
8920. A SHARP ANALYSIS OF THE MIXING TIME FOR RANDOM WALK ON ROOTED
TREES
Jason Fulman
We define an analog of Plancherel measure for the set of rooted
unlabeled
trees on n vertices, and a Markov chain which has this measure as its
stationary distribution. Using the combinatorics of commutation
relations, we
show that order n^2 steps are necessary and suffice for convergence to
the
stationary distribution.
http://arxiv.org/abs/0908.1141
---------------------------------------------------------------
8921. SHARP HEAT KERNEL ESTIMATES FOR RELATIVISTIC STABLE PROCESSES IN
OPEN SETS
Zhen-Qing Chen and Panki Kim and Renming Song
In this paper, we establish sharp two-sided estimates for the transition
densities of relativistic stable processes (or equivalently, for the
heat
kernels of the operators $m-(m^{2/\alpha}-\Delta)^{\alpha/2}$) in
$C^{1, 1}$
open sets. The estimates are uniform in $m\in (0, M]$ for each fixed
$M>0$.
Letting $m\downarrow 0$, the estimates given in this paper recover the
Dirichlet heat kernel estimates for $-(-\Delta)^{\alpha/2}$ in
$C^{1,1}$-open
sets obtained in \cite{CKS}. Sharp two-sided estimates are also
obtained for
Green functions of relativistic stable processes in half-space-like
$C^{1,1}$
open sets and bounded $C^{1,1}$ open sets.
http://arxiv.org/abs/0908.1509
---------------------------------------------------------------
8922. THE TWO-TYPE CONTINUUM RICHARDSON MODEL: NON-DEPENDENCE OF THE
SURVIVAL OF BOTH TYPES ON THE INITIAL CONFIGURATION
Sebastian Carstens and Thomas Richthammer
We consider the model of Deijfen et al. for the competing growth of two
infection types in R^d, based on the Richardson model on Z^d. Stochastic
ball-shaped infection outbursts transmit the infection type of the
center of
the ball to all points of the ball that are not yet infected. Relevant
parameters of the model are the initial infection configuration, the
(type-dependent) growth rates and the radius distribution of the
infection
outbursts. The main question is that of coexistence: For what values
of the
parameters is there a positive probability that both types grow
unboundedly? It
is known that for this question the initial configuration basically is
irrelevant, provided certain technical assumptions on the radius
distribution
are satisfied. Here we show how to get rid of these assumptions,
introducing a
slight generalization of the model, where immune regions and delayed
initial
infection configurations are allowed.
http://arxiv.org/abs/0908.1551
---------------------------------------------------------------
8923. BOUNDARY HARNACK PRINCIPLE FOR $\DELTA + \DELTA^{\ALPHA/2}$
Zhen-Qing Chen and Panki Kim and Renming Song and Zoran Vondra\v{c}ek
For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family of pseudo
differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$
on $\R^d$
that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/
2}$. In
this paper, we establish a uniform boundary Harnack principle (BHP) with
explicit boundary decay rate for nonnegative functions which are
harmonic with
respect to $\Delta +b \Delta^{\alpha/2}$ (or equivalently, the sum of a
Brownian motion and an independent symmetric $\alpha$-stable process
with
constant multiple $b^{1/\alpha}$) in $C^{1, 1}$ open sets. Here a
"uniform" BHP
means that the comparing constant in the BHP is independent of $b\in
[0, 1]$.
Along the way, a uniform Carleson type estimate is established for
nonnegative
functions which are harmonic with respect to $\Delta + b
\Delta^{\alpha/2}$ in
Lipschitz open sets. Our method employs a combination of probabilistic
and
analytic techniques.
http://arxiv.org/abs/0908.1559
---------------------------------------------------------------
8924. CONFORMAL LOOP ENSEMBLES AND THE STRESS-ENERGY TENSOR. II.
CONSTRUCTION OF THE STRESS-ENERGY TENSOR
Benjamin Doyon
This is the second part of a work aimed at constructing the stress-
energy
tensor of conformal field theory (CFT) as a local "object" in
conformal loop
ensembles (CLE). This work lies in the wider context of re-
constructing quantum
field theory from mathematically well-defined ensembles of random
objects. In
the present paper, based on results of the first part, we identify the
stress-energy tensor in the dilute regime of CLE. This is done by
deriving both
its conformal Ward identities for single insertion in CLE probability
functions, and its properties under conformal transformations
involving the
Schwarzian derivative. We also give the one-point function of the
stress-energy
tensor in terms of a notion of partition function, and we show that
this agrees
with standard CFT arguments. The construction is in the same spirit as
that
found in the context of SLE(8/3) by the author, Riva and Cardy (2006),
which
had to do with the case of zero central charge. The present construction
generalises this to all central charges between 0 and 1, including all
minimal
models. This generalisation is non-trivial: the application of these
ideas to
the CLE context requires the introduction of a renormalised
probability, and
the derivation of the transformation properties and of the one-point
function
do not have counterparts in the SLE context.
http://arxiv.org/abs/0908.1511
---------------------------------------------------------------
8925. THE UNIQUENESS OF SYMMETRIZING MEASURE AND LINEAR DIFFUSIONS
Xing Fang and Jiangang Ying and Minzhi Zhao
In this short article, we shall study one-dimensional local Dirichlet
spaces.
One result, which has its independent interest, is to prove that
irreducibility
implies the uniqueness of symmetrizing measure for right Markov
processes. The
other result is to give a representation for any 1-dim local,
irreducible and
regular Dirichlet space and a necessary and sufficient condition for a
Dirichlet space to be regular subspace of another Dirichlet space.
http://arxiv.org/abs/0908.1607
---------------------------------------------------------------
8926. PERFECT SIMULATION OF VERVAAT PERPETUITIES
James Allen Fill and Mark Huber
We use coupling into and from the past to sample perfectly in a simple
and
provably fast fashion from the Vervaat family of perpetuities. The
family
includes the Dickman distribution, which arises both in number theory
and in
the analysis of the Quickselect algorithm, which was the motivation
for our
work.
http://arxiv.org/abs/0908.1733
---------------------------------------------------------------
8927. STATIC LARGE DEVIATIONS OF BOUNDARY DRIVEN EXCLUSION PROCESSES
Jonathan Farfan
We prove that the stationary measure associated to a boundary driven
exclusion process in any dimension satisfies a large deviation
principle with
rate function given by the quasi potential of the Freidlin and
Wentzell theory.
http://arxiv.org/abs/0908.1798
---------------------------------------------------------------
8928. LACK OF STRONG COMPLETENESS FOR STOCHASTIC FLOWS
Xue-Mei Li and Michael Scheutzow
It is well-known that a stochastic differential equation (SDE) on a
Euclidean
space driven by a Brownian motion with Lipschitz coefficients
generates a
stochastic flow of homeomorphisms. When the coefficients are only
locally
Lipschitz, then a maximal continuous flow still exists but explosion
in finite
time may occur. If -- in addition -- the coefficients grow at most
linearly,
then this flow has the property that for each fixed initial condition
$x$, the
solution exists for all times almost surely. If the exceptional set of
measure
zero can be chosen independently $x$, then the maximal flow is called
{\em
strongly complete}. The question, whether an SDE with locally Lipschitz
continuous coefficients satisfying a linear growth condition is strongly
complete was open for many years. In this paper, we construct a 2-
dimensional
SDE with coefficients which are even bounded (and smooth) and which is
{\em
not} strongly complete thus answering the question in the negative.
http://arxiv.org/abs/0908.1839
---------------------------------------------------------------
8929. STEIN'S METHOD FOR DEPENDENT RANDOM VARIABLES OCCURRING IN
STATISTICAL MECHANICS
Peter Eichelsbacher and Matthias L\"owe
We obtain rates of convergence in limit theorems of partial sums $S_n$
for
certain sequences of dependent, identically distributed random
variables, which
arise naturally in statistical mechanics, in particular, in the
context of the
Curie-Weiss models. Under appropriate assumptions there exists a real
number
$\alpha$, a positive real number $\mu$, and a positive integer $k$
such that
$(S_n- n \alpha)/n^{1 - 1/2k}$ converges weakly to a random variable
with
density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. We develop
Stein's method
for exchangeable pairs for a rich class of distributional approximations
including the Gaussian distributions as well as the non-Gaussian limit
distributions with density proportional to $\exp(-\mu |x|^{2k} /(2k)!)
$. Our
results include the optimal Berry-Esseen rate in the Central Limit
Theorem for
the total magnetization in the classical Curie-Weiss model, for high
temperatures as well as at the critical temperature $\beta_c=1$, where
the
Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds
as the
temperature $1/ \beta_n$ converges to one and obtain a threshold for
the speed
of this convergence. Single spin distributions satisfying the
Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or
continuous Curie-Weiss models are considered.
http://arxiv.org/abs/0908.1909
---------------------------------------------------------------
8930. REPLICA SYMMETRY AND COMBINATORIAL OPTIMIZATION
Johan W\"astlund
We establish the soundness of the replica symmetric ansatz (introduced
by M.
Mezard and G. Parisi) for minimum matching and the traveling salesman
problem
in the pseudo-dimension d mean field model for d\geq 1. The case d=1
of minimum
matching corresponds to the pi^2/6 limit for the assignment problem
established
by D. Aldous in 2001, and the analogous limit for the d=1 case of TSP
was
recently established by the author with a different method. We
introduce a
game-theoretical framework by which we prove the correctness of the
replica-cavity prediction of the corresponding limits also for d>1.
http://arxiv.org/abs/0908.1920
---------------------------------------------------------------
8931. HIGH ORDER DISCRETIZATION SCHEMES FOR STOCHASTIC VOLATILITY MODELS
Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
In usual stochastic volatility models, the process driving the
volatility of
the asset price evolves according to an autonomous one-dimensional
stochastic
differential equation. We assume that the coefficients of this
equation are
smooth. Using It\^o's formula, we get rid, in the asset price
dynamics, of the
stochastic integral with respect to the Brownian motion driving this
SDE.
Taking advantage of this structure, we propose - a scheme, based on the
Milstein discretization of this SDE, with order one of weak trajectorial
convergence for the asset price, - a scheme, based on the Ninomiya-
Victoir
discretization of this SDE, with order two of weak convergence for the
asset
price. We also propose a specific scheme with improved convergence
properties
when the volatility of the asset price is driven by an Orstein-Uhlenbeck
process. We confirm the theoretical rates of convergence by numerical
experiments and show that our schemes are well adapted to the
multilevel Monte
Carlo method introduced by Giles [2008a,b].
http://arxiv.org/abs/0908.1926
---------------------------------------------------------------
8932. FILTERING EQUATIONS FOR PARTIALLY OBSERVABLE DIFFUSION PROCESSES
WITH LIPSCHITZ CONTINUOUS COEFFICIENTS
N.V. Krylov
We present several results on smoothness in $L_{p}$ sense of filtering
densities under the Lipschitz continuity assumption on the
coefficients of a
partially observable diffusion processes. We obtain them by rewriting in
divergence form filtering equation which are usually considered in
terms of
formally adjoint to operators in nondivergence form.
http://arxiv.org/abs/0908.1935
---------------------------------------------------------------
8933. A CHARACTERIZATION THEOREM FOR THE DISTRIBUTION OF A CONTINUOUS
LOCAL MARTINGALE AND RELATED LIMIT THEOREMS
Andriy Yurachkivsky
The main result of the article reads: the distribution of a continuous
starting from zero local martingale whose quadratic characteristic is
almost
surely absolutely continuous with respect to some non-random increasing
continuous function is determined by the distribution of the quadratic
characteristic. Functional limit theorem based on this
characterization are
proved.
http://arxiv.org/abs/0908.1939
---------------------------------------------------------------
8934. AN APPLICATION OF DISC PACKING TO STATISTICAL MECHANICS
Matthew Kahle
We construct stable configurations of n overlapping discs of radius r
in a
unit square, with r = O(1/n). By a result of Diaconis, Lebeau, and
Michel, this
result is best possible, up to a constant factor. A consequence is
that the
Metropolis algorithm, a well-studied Markov chain on the hardcore
model, is not
irreducible in this range of parameters.
http://arxiv.org/abs/0908.1830
---------------------------------------------------------------
8935. A COMPREHENSIVE CONNECTION BETWEEN THE BASIC RESULTS AND
PROPERTIES DERIVED FROM TWO KINDS OF TOPOLOGIES OF A RANDOM LOCALLY
CONVEX MODULE
Tiexin Guo
The purpose of this paper is to make a comprehensive connection
between the
basic results and properties derived from the two kinds of topologies
(namely
the $(\epsilon,\lambda)-$topology introduced by the author and locally
$L^{0}-$convex topology recently introduced by Filipovi$\acute{c}$ et.
al) of a
random locally convex module. First, we give an extremely simple proof
of the
known Hahn-Banach extension theorem of $L^{0}-$linear functions as
well as its
continuous variants. Then we give the essential relations between the
hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J. Funct.
Anal.256(2009)3996--4029] and a basic strict separation theorem in
[Guo et. al,
Nonlinear Anal. 71(2009)3794--3804], and in the process obtain a
useful and
surprising fact that a random locally convex module with the countable
concatenation property must have the same completeness under the two
topologies! Based on the relation between the two kinds of
completeness, we
further present the central part of this paper: we prove that most of
the
previously established deep results of random conjugate spaces of
random normed
modules under the $(\epsilon,\lambda)-$topology are still valid under
the
locally $L^{0}-$convex topology, which considerably enriches financial
applications of random normed modules.
http://arxiv.org/abs/0908.1843
---------------------------------------------------------------
8936. STATIC LARGE DEVIATIONS OF BOUNDARY DRIVEN EXCLUSION PROCESSES
Jonathan Farfan
We prove that the stationary measure associated to a boundary driven
exclusion process in any dimension satisfies a large deviation
principle with
rate function given by the quasi potential of the Freidlin and
Wentzell theory.
http://arxiv.org/abs/0908.1798
---------------------------------------------------------------
8937. LACK OF STRONG COMPLETENESS FOR STOCHASTIC FLOWS
Xue-Mei Li and Michael Scheutzow
It is well-known that a stochastic differential equation (SDE) on a
Euclidean
space driven by a Brownian motion with Lipschitz coefficients
generates a
stochastic flow of homeomorphisms. When the coefficients are only
locally
Lipschitz, then a maximal continuous flow still exists but explosion
in finite
time may occur. If -- in addition -- the coefficients grow at most
linearly,
then this flow has the property that for each fixed initial condition
$x$, the
solution exists for all times almost surely. If the exceptional set of
measure
zero can be chosen independently $x$, then the maximal flow is called
{\em
strongly complete}. The question, whether an SDE with locally Lipschitz
continuous coefficients satisfying a linear growth condition is strongly
complete was open for many years. In this paper, we construct a 2-
dimensional
SDE with coefficients which are even bounded (and smooth) and which is
{\em
not} strongly complete thus answering the question in the negative.
http://arxiv.org/abs/0908.1839
---------------------------------------------------------------
8938. STEIN'S METHOD FOR DEPENDENT RANDOM VARIABLES OCCURRING IN
STATISTICAL MECHANICS
Peter Eichelsbacher and Matthias L\"owe
We obtain rates of convergence in limit theorems of partial sums $S_n$
for
certain sequences of dependent, identically distributed random
variables, which
arise naturally in statistical mechanics, in particular, in the
context of the
Curie-Weiss models. Under appropriate assumptions there exists a real
number
$\alpha$, a positive real number $\mu$, and a positive integer $k$
such that
$(S_n- n \alpha)/n^{1 - 1/2k}$ converges weakly to a random variable
with
density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. We develop
Stein's method
for exchangeable pairs for a rich class of distributional approximations
including the Gaussian distributions as well as the non-Gaussian limit
distributions with density proportional to $\exp(-\mu |x|^{2k} /(2k)!)
$. Our
results include the optimal Berry-Esseen rate in the Central Limit
Theorem for
the total magnetization in the classical Curie-Weiss model, for high
temperatures as well as at the critical temperature $\beta_c=1$, where
the
Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds
as the
temperature $1/ \beta_n$ converges to one and obtain a threshold for
the speed
of this convergence. Single spin distributions satisfying the
Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or
continuous Curie-Weiss models are considered.
http://arxiv.org/abs/0908.1909
---------------------------------------------------------------
8939. REPLICA SYMMETRY AND COMBINATORIAL OPTIMIZATION
Johan W\"astlund
We establish the soundness of the replica symmetric ansatz (introduced
by M.
Mezard and G. Parisi) for minimum matching and the traveling salesman
problem
in the pseudo-dimension d mean field model for d\geq 1. The case d=1
of minimum
matching corresponds to the pi^2/6 limit for the assignment problem
established
by D. Aldous in 2001, and the analogous limit for the d=1 case of TSP
was
recently established by the author with a different method. We
introduce a
game-theoretical framework by which we prove the correctness of the
replica-cavity prediction of the corresponding limits also for d>1.
http://arxiv.org/abs/0908.1920
---------------------------------------------------------------
8940. HIGH ORDER DISCRETIZATION SCHEMES FOR STOCHASTIC VOLATILITY MODELS
Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
In usual stochastic volatility models, the process driving the
volatility of
the asset price evolves according to an autonomous one-dimensional
stochastic
differential equation. We assume that the coefficients of this
equation are
smooth. Using It\^o's formula, we get rid, in the asset price
dynamics, of the
stochastic integral with respect to the Brownian motion driving this
SDE.
Taking advantage of this structure, we propose - a scheme, based on the
Milstein discretization of this SDE, with order one of weak trajectorial
convergence for the asset price, - a scheme, based on the Ninomiya-
Victoir
discretization of this SDE, with order two of weak convergence for the
asset
price. We also propose a specific scheme with improved convergence
properties
when the volatility of the asset price is driven by an Orstein-Uhlenbeck
process. We confirm the theoretical rates of convergence by numerical
experiments and show that our schemes are well adapted to the
multilevel Monte
Carlo method introduced by Giles [2008a,b].
http://arxiv.org/abs/0908.1926
---------------------------------------------------------------
8941. FILTERING EQUATIONS FOR PARTIALLY OBSERVABLE DIFFUSION PROCESSES
WITH LIPSCHITZ CONTINUOUS COEFFICIENTS
N.V. Krylov
We present several results on smoothness in $L_{p}$ sense of filtering
densities under the Lipschitz continuity assumption on the
coefficients of a
partially observable diffusion processes. We obtain them by rewriting in
divergence form filtering equation which are usually considered in
terms of
formally adjoint to operators in nondivergence form.
http://arxiv.org/abs/0908.1935
---------------------------------------------------------------
8942. A CHARACTERIZATION THEOREM FOR THE DISTRIBUTION OF A CONTINUOUS
LOCAL MARTINGALE AND RELATED LIMIT THEOREMS
Andriy Yurachkivsky
The main result of the article reads: the distribution of a continuous
starting from zero local martingale whose quadratic characteristic is
almost
surely absolutely continuous with respect to some non-random increasing
continuous function is determined by the distribution of the quadratic
characteristic. Functional limit theorem based on this
characterization are
proved.
http://arxiv.org/abs/0908.1939
---------------------------------------------------------------
8943. AN APPLICATION OF DISC PACKING TO STATISTICAL MECHANICS
Matthew Kahle
We construct stable configurations of n overlapping discs of radius r
in a
unit square, with r = O(1/n). By a result of Diaconis, Lebeau, and
Michel, this
result is best possible, up to a constant factor. A consequence is
that the
Metropolis algorithm, a well-studied Markov chain on the hardcore
model, is not
irreducible in this range of parameters.
http://arxiv.org/abs/0908.1830
---------------------------------------------------------------
8944. A COMPREHENSIVE CONNECTION BETWEEN THE BASIC RESULTS AND
PROPERTIES DERIVED FROM TWO KINDS OF TOPOLOGIES OF A RANDOM LOCALLY
CONVEX MODULE
Tiexin Guo
The purpose of this paper is to make a comprehensive connection
between the
basic results and properties derived from the two kinds of topologies
(namely
the $(\epsilon,\lambda)-$topology introduced by the author and locally
$L^{0}-$convex topology recently introduced by Filipovi$\acute{c}$ et.
al) of a
random locally convex module. First, we give an extremely simple proof
of the
known Hahn-Banach extension theorem of $L^{0}-$linear functions as
well as its
continuous variants. Then we give the essential relations between the
hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J. Funct.
Anal.256(2009)3996--4029] and a basic strict separation theorem in
[Guo et. al,
Nonlinear Anal. 71(2009)3794--3804], and in the process obtain a
useful and
surprising fact that a random locally convex module with the countable
concatenation property must have the same completeness under the two
topologies! Based on the relation between the two kinds of
completeness, we
further present the central part of this paper: we prove that most of
the
previously established deep results of random conjugate spaces of
random normed
modules under the $(\epsilon,\lambda)-$topology are still valid under
the
locally $L^{0}-$convex topology, which considerably enriches financial
applications of random normed modules.
http://arxiv.org/abs/0908.1843
---------------------------------------------------------------
8945. RANDOM MATRICES: UNIVERSALITY OF LOCAL EIGENVALUE STATISTICS UP
TO THE EDGE
Terence Tao and Van Vu
This is a continuation of our earlier paper on the universality of the
eigenvalues of Wigner random matrices. The main new results of this
paper are
an extension of the results in that paper from the bulk of the
spectrum up to
the edge. In particular, we prove a variant of the universality
results of
Soshnikov for the largest eigenvalues, assuming moment conditions
rather than
symmetry conditions. The main new technical observation is that there
is a
significant bias in the Cauchy interlacing law near the edge of the
spectrum
which allows one to continue ensuring the delocalization of
eigenvectors.
http://arxiv.org/abs/0908.1982
---------------------------------------------------------------
8946. OPTIMAL CO-ADAPTED COUPLING FOR A RANDOM WALK ON THE HYPER-
COMPLETE-GRAP
Stephen B. Connor
Let $G_d$ be the complete graph with d vertices, and let X and Y be two
simple symmetric continuous-time random walks on the vertices of $G_d^n
$. When
d=2, X and Y are random walks on the hypercube, for which a
stochastically
fastest co-adapted coupling is described by Connor & Jacka (2008).
Here we
extend this result to random walks on $G_d^n$, once again producing a
stochastically optimal coupling: as d tends to infinity we show that
this
optimal co-adapted coupling tends to a maximal coupling.
http://arxiv.org/abs/0908.2038
---------------------------------------------------------------
8947. RECONSTRUCTION ON TREES: EXPONENTIAL MOMENT BOUNDS FOR LINEAR
ESTIMATORS
Yuval Peres and Sebastien Roch
Consider a Markov chain $(\xi_v)_{v \in V} \in [k]^V$ on the infinite
$b$-ary
tree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b
\geq 2$,
$k \geq 2$ and $[k] = \{1,...,k\}$. We denote by $L_n$ the level-$n$
vertices
of $T$. Assume $M$ has a real second-largest (in absolute value)
eigenvalue
$\lambda$ with corresponding real eigenvector $\nu \neq 0$. Letting $
\sigma_v =
\nu_{\xi_v}$, we consider the following root-state estimator, which was
introduced by Mossel and Peres (2003) in the context of the
``recontruction
problem'' on trees: \begin{equation*} S_n = (b\lambda)^{-n} \sum_{x\in
L_n}
\sigma_x. \end{equation*} As noted by Mossel and Peres, when $b
\lambda^2 > 1$
(the so-called Kesten-Stigum reconstruction phase) the quantity $S_n$
has
uniformly bounded variance. Here, we give bounds on the moment-
generating
functions of $S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results have
implications for the inference of evolutionary trees.
http://arxiv.org/abs/0908.2056
---------------------------------------------------------------
8948. SEQUENCE-LENGTH REQUIREMENT OF DISTANCE-BASED PHYLOGENY
RECONSTRUCTION: BREAKING THE POLYNOMIAL BARRIER
Sebastien Roch
We introduce a new distance-based phylogeny reconstruction technique
which
provably achieves, at sufficiently short branch lengths, a
polylogarithmic
sequence-length requirement -- improving significantly over previous
polynomial
bounds for distance-based methods. The technique is based on an
averaging
procedure that implicitly reconstructs ancestral sequences.
In the same token, we extend previous results on phase transitions in
phylogeny reconstruction to general time-reversible models. More
precisely, we
show that in the so-called Kesten-Stigum zone (roughly, a region of the
parameter space where ancestral sequences are well approximated by
``linear
combinations'' of the observed sequences) sequences of length $
\poly(\log n)$
suffice for reconstruction when branch lengths are discretized. Here $n
$ is the
number of extant species.
Our results challenge, to some extent, the conventional wisdom that
estimates
of evolutionary distances alone carry significantly less information
about
phylogenies than full sequence datasets.
http://arxiv.org/abs/0908.2061
---------------------------------------------------------------
8949. SHARP APPROXIMATION FOR DENSITY DEPENDENT MARKOV CHAINS
Kamil Szczegot
Consider a sequence (indexed by n) of Markov chains Z^n in R^d
characterized
by transition kernels that approximately (in n) depend only on the
rescaled
state n^{-1} Z^n. Subject to a smoothness condition, such a family can
be
closely coupled on short time intervals to a Brownian motion with
quadratic
drift. This construction is used to determine the first two terms in the
asymptotic (in n) expansion of the probability that the rescaled chain
exits a
convex polytope. The constant term and the first correction of size
n^{-1/6}
admit sharp characterization by solutions to associated differential
equations
and an absolute constant. The error is smaller than O(n^{-b}) for any
b < 1/4.
These results are directly applied to the analysis of randomized
algorithms
at phase transitions. In particular, the `peeling' algorithm in large
random
hypergraphs, or equivalently the iterative decoding scheme for low-
density
parity-check codes over the binary erasure channel is studied to
determine the
finite size scaling behavior for irregular hypergraph ensembles.
http://arxiv.org/abs/0908.2088
---------------------------------------------------------------
8950. A SHARP ESTIMATE FOR DIVISORS OF BERNOULLI SUMS
Michel Weber
Let $S_n=\e_1+...+\e_n$, where $ \e_i $ are i.i.d. Bernoulli r.v.'s. Let
$0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, $\bar r_d(n)= 2d
-r_d(n)$ and $\b(n,d)=\max ({1\over d}, {1\over \sqrt n})[e^{-
{r_d(n)^2/2 n}}
+e^{- {\bar r_d(n)^2/2 n}}]$. We show that $$\sup_{2\le d\le n}
\big|\P\big\{d|S_n\big\}- E(n,d) \big|= {\cal O}\big({\log^{5/2} n \over
n^{3/2}}\big), $$ where $E(n,d) $ verifies $c_1\b(n,d)\le E(n,d)\le
c_2\b(n,d)
$ and $c_1,c_2 $ are numerical constants.
http://arxiv.org/abs/0908.2047
---------------------------------------------------------------
8951. SIMPLE ERROR SCATTERING MODEL FOR IMPROVED INFORMATION
RECONCILIATION
Stefan Rass
Implementations of quantum key distribution as available nowadays
suffer from
inefficiencies due to post processing of the raw key that severely
cuts down
the final secure key rate. We present a simple model for the error
scattering
across the raw key and derive "closed form" expressions for the
probability of
a parity check failure, or experiencing more than some fixed number of
errors.
Our results can serve for improvement for key establishment, as
information
reconciliation via interactive error correction and privacy
amplification rests
on mostly unproven assumptions. We support those hypotheses on
statistical
grounds.
http://arxiv.org/abs/0908.2069
---------------------------------------------------------------
8952. PROBABILISTIC MODEL ASSOCIATED WITH THE PRESSURELESS GAS DYNAMICS
Sergio Albeverio and Anastasia Korshunova and Olga Rozanova
Using a method of stochastic perturbation of a Langevin system
associated
with the non-viscous Burgers equation we construct a solution to the
Riemann
problem for the pressureless gas dynamics describing sticky particles
dynamics.
As a bridging step we consider a medium consisting of noninteracting
particles.
We analyze the difference in the behavior of discontinuous solutions
for these
two models and the relations between them. In our framework in 1D case
we
obtain a unique entropy solution to the Riemann problem. Moreover, we
describe
how starting from smooth data a $\delta$ - singularity arises in one
component
of the solution.
http://arxiv.org/abs/0908.2084
---------------------------------------------------------------
8953. THE MAHONIAN PROBABILITY DISTRIBUTION ON WORDS IS ASYMPTOTICALLY
NORMAL
E. Rodney Canfield and Svante Janson and and Doron Zeilberger
The Mahonian statistic is the number of inversions in a permutation of a
multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting
function
for this statistic is the $q$ analog of the multinomial coefficient
$\binom{a_1+...+a_m}{a_1,... a_m}$, and the probability generating
function is
the normalization of the latter. We give two proofs that the
distribution is
asymptotically normal. The first is {\it computer-assisted}, based on
the
method of moments. The Maple package {\tt MahonianStat}, available
from the
webpage of this article, can be used by the reader to perform
experiments and
calculations. Our second proof uses characteristic functions. We then
take up
the study of a local limit theorem to accompany our central limit
theorem. Here
our result is less general, and we must be content with a conjecture
about
further work. Our local limit theorem permits us to conclude that the
coeffiecients of the $q$-multinomial are log-concave, provided one
stays near
the center (where the largest coefficients reside.)
http://arxiv.org/abs/0908.2089
---------------------------------------------------------------
8954. RANDOM MATRICES: UNIVERSALITY OF LOCAL EIGENVALUE STATISTICS UP
TO THE EDGE
Terence Tao and Van Vu
This is a continuation of our earlier paper on the universality of the
eigenvalues of Wigner random matrices. The main new results of this
paper are
an extension of the results in that paper from the bulk of the
spectrum up to
the edge. In particular, we prove a variant of the universality
results of
Soshnikov for the largest eigenvalues, assuming moment conditions
rather than
symmetry conditions. The main new technical observation is that there
is a
significant bias in the Cauchy interlacing law near the edge of the
spectrum
which allows one to continue ensuring the delocalization of
eigenvectors.
http://arxiv.org/abs/0908.1982
---------------------------------------------------------------
8955. OPTIMAL CO-ADAPTED COUPLING FOR A RANDOM WALK ON THE HYPER-
COMPLETE-GRAP
Stephen B. Connor
Let $G_d$ be the complete graph with d vertices, and let X and Y be two
simple symmetric continuous-time random walks on the vertices of $G_d^n
$. When
d=2, X and Y are random walks on the hypercube, for which a
stochastically
fastest co-adapted coupling is described by Connor & Jacka (2008).
Here we
extend this result to random walks on $G_d^n$, once again producing a
stochastically optimal coupling: as d tends to infinity we show that
this
optimal co-adapted coupling tends to a maximal coupling.
http://arxiv.org/abs/0908.2038
---------------------------------------------------------------
8956. RECONSTRUCTION ON TREES: EXPONENTIAL MOMENT BOUNDS FOR LINEAR
ESTIMATORS
Yuval Peres and Sebastien Roch
Consider a Markov chain $(\xi_v)_{v \in V} \in [k]^V$ on the infinite
$b$-ary
tree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b
\geq 2$,
$k \geq 2$ and $[k] = \{1,...,k\}$. We denote by $L_n$ the level-$n$
vertices
of $T$. Assume $M$ has a real second-largest (in absolute value)
eigenvalue
$\lambda$ with corresponding real eigenvector $\nu \neq 0$. Letting $
\sigma_v =
\nu_{\xi_v}$, we consider the following root-state estimator, which was
introduced by Mossel and Peres (2003) in the context of the
``recontruction
problem'' on trees: \begin{equation*} S_n = (b\lambda)^{-n} \sum_{x\in
L_n}
\sigma_x. \end{equation*} As noted by Mossel and Peres, when $b
\lambda^2 > 1$
(the so-called Kesten-Stigum reconstruction phase) the quantity $S_n$
has
uniformly bounded variance. Here, we give bounds on the moment-
generating
functions of $S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results have
implications for the inference of evolutionary trees.
http://arxiv.org/abs/0908.2056
---------------------------------------------------------------
8957. SEQUENCE-LENGTH REQUIREMENT OF DISTANCE-BASED PHYLOGENY
RECONSTRUCTION: BREAKING THE POLYNOMIAL BARRIER
Sebastien Roch
We introduce a new distance-based phylogeny reconstruction technique
which
provably achieves, at sufficiently short branch lengths, a
polylogarithmic
sequence-length requirement -- improving significantly over previous
polynomial
bounds for distance-based methods. The technique is based on an
averaging
procedure that implicitly reconstructs ancestral sequences.
In the same token, we extend previous results on phase transitions in
phylogeny reconstruction to general time-reversible models. More
precisely, we
show that in the so-called Kesten-Stigum zone (roughly, a region of the
parameter space where ancestral sequences are well approximated by
``linear
combinations'' of the observed sequences) sequences of length $
\poly(\log n)$
suffice for reconstruction when branch lengths are discretized. Here $n
$ is the
number of extant species.
Our results challenge, to some extent, the conventional wisdom that
estimates
of evolutionary distances alone carry significantly less information
about
phylogenies than full sequence datasets.
http://arxiv.org/abs/0908.2061
---------------------------------------------------------------
8958. SHARP APPROXIMATION FOR DENSITY DEPENDENT MARKOV CHAINS
Kamil Szczegot
Consider a sequence (indexed by n) of Markov chains Z^n in R^d
characterized
by transition kernels that approximately (in n) depend only on the
rescaled
state n^{-1} Z^n. Subject to a smoothness condition, such a family can
be
closely coupled on short time intervals to a Brownian motion with
quadratic
drift. This construction is used to determine the first two terms in the
asymptotic (in n) expansion of the probability that the rescaled chain
exits a
convex polytope. The constant term and the first correction of size
n^{-1/6}
admit sharp characterization by solutions to associated differential
equations
and an absolute constant. The error is smaller than O(n^{-b}) for any
b < 1/4.
These results are directly applied to the analysis of randomized
algorithms
at phase transitions. In particular, the `peeling' algorithm in large
random
hypergraphs, or equivalently the iterative decoding scheme for low-
density
parity-check codes over the binary erasure channel is studied to
determine the
finite size scaling behavior for irregular hypergraph ensembles.
http://arxiv.org/abs/0908.2088
---------------------------------------------------------------
8959. A SHARP ESTIMATE FOR DIVISORS OF BERNOULLI SUMS
Michel Weber
Let $S_n=\e_1+...+\e_n$, where $ \e_i $ are i.i.d. Bernoulli r.v.'s. Let
$0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, $\bar r_d(n)= 2d
-r_d(n)$ and $\b(n,d)=\max ({1\over d}, {1\over \sqrt n})[e^{-
{r_d(n)^2/2 n}}
+e^{- {\bar r_d(n)^2/2 n}}]$. We show that $$\sup_{2\le d\le n}
\big|\P\big\{d|S_n\big\}- E(n,d) \big|= {\cal O}\big({\log^{5/2} n \over
n^{3/2}}\big), $$ where $E(n,d) $ verifies $c_1\b(n,d)\le E(n,d)\le
c_2\b(n,d)
$ and $c_1,c_2 $ are numerical constants.
http://arxiv.org/abs/0908.2047
---------------------------------------------------------------
8960. SIMPLE ERROR SCATTERING MODEL FOR IMPROVED INFORMATION
RECONCILIATION
Stefan Rass
Implementations of quantum key distribution as available nowadays
suffer from
inefficiencies due to post processing of the raw key that severely
cuts down
the final secure key rate. We present a simple model for the error
scattering
across the raw key and derive "closed form" expressions for the
probability of
a parity check failure, or experiencing more than some fixed number of
errors.
Our results can serve for improvement for key establishment, as
information
reconciliation via interactive error correction and privacy
amplification rests
on mostly unproven assumptions. We support those hypotheses on
statistical
grounds.
http://arxiv.org/abs/0908.2069
---------------------------------------------------------------
8961. PROBABILISTIC MODEL ASSOCIATED WITH THE PRESSURELESS GAS DYNAMICS
Sergio Albeverio and Anastasia Korshunova and Olga Rozanova
Using a method of stochastic perturbation of a Langevin system
associated
with the non-viscous Burgers equation we construct a solution to the
Riemann
problem for the pressureless gas dynamics describing sticky particles
dynamics.
As a bridging step we consider a medium consisting of noninteracting
particles.
We analyze the difference in the behavior of discontinuous solutions
for these
two models and the relations between them. In our framework in 1D case
we
obtain a unique entropy solution to the Riemann problem. Moreover, we
describe
how starting from smooth data a $\delta$ - singularity arises in one
component
of the solution.
http://arxiv.org/abs/0908.2084
---------------------------------------------------------------
8962. THE MAHONIAN PROBABILITY DISTRIBUTION ON WORDS IS ASYMPTOTICALLY
NORMAL
E. Rodney Canfield and Svante Janson and and Doron Zeilberger
The Mahonian statistic is the number of inversions in a permutation of a
multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting
function
for this statistic is the $q$ analog of the multinomial coefficient
$\binom{a_1+...+a_m}{a_1,... a_m}$, and the probability generating
function is
the normalization of the latter. We give two proofs that the
distribution is
asymptotically normal. The first is {\it computer-assisted}, based on
the
method of moments. The Maple package {\tt MahonianStat}, available
from the
webpage of this article, can be used by the reader to perform
experiments and
calculations. Our second proof uses characteristic functions. We then
take up
the study of a local limit theorem to accompany our central limit
theorem. Here
our result is less general, and we must be content with a conjecture
about
further work. Our local limit theorem permits us to conclude that the
coeffiecients of the $q$-multinomial are log-concave, provided one
stays near
the center (where the largest coefficients reside.)
http://arxiv.org/abs/0908.2089
---------------------------------------------------------------
8963. HIGH ORDER DISCRETIZATION SCHEMES FOR STOCHASTIC VOLATILITY MODELS
Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
In usual stochastic volatility models, the process driving the
volatility of
the asset price evolves according to an autonomous one-dimensional
stochastic
differential equation. We assume that the coefficients of this
equation are
smooth. Using It\^o's formula, we get rid, in the asset price
dynamics, of the
stochastic integral with respect to the Brownian motion driving this
SDE.
Taking advantage of this structure, we propose - a scheme, based on the
Milstein discretization of this SDE, with order one of weak trajectorial
convergence for the asset price, - a scheme, based on the Ninomiya-
Victoir
discretization of this SDE, with order two of weak convergence for the
asset
price. We also propose a specific scheme with improved convergence
properties
when the volatility of the asset price is driven by an Orstein-Uhlenbeck
process. We confirm the theoretical rates of convergence by numerical
experiments and show that our schemes are well adapted to the
multilevel Monte
Carlo method introduced by Giles [2008a,b].
http://arxiv.org/abs/0908.1926
---------------------------------------------------------------
8964. HIGH ORDER DISCRETIZATION SCHEMES FOR STOCHASTIC VOLATILITY MODELS
Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
In usual stochastic volatility models, the process driving the
volatility of
the asset price evolves according to an autonomous one-dimensional
stochastic
differential equation. We assume that the coefficients of this
equation are
smooth. Using It\^o's formula, we get rid, in the asset price
dynamics, of the
stochastic integral with respect to the Brownian motion driving this
SDE.
Taking advantage of this structure, we propose - a scheme, based on the
Milstein discretization of this SDE, with order one of weak trajectorial
convergence for the asset price, - a scheme, based on the Ninomiya-
Victoir
discretization of this SDE, with order two of weak convergence for the
asset
price. We also propose a specific scheme with improved convergence
properties
when the volatility of the asset price is driven by an Orstein-Uhlenbeck
process. We confirm the theoretical rates of convergence by numerical
experiments and show that our schemes are well adapted to the
multilevel Monte
Carlo method introduced by Giles [2008a,b].
http://arxiv.org/abs/0908.1926
---------------------------------------------------------------
8965. ON THE COPULA FOR MULTIVARIATE EXTREME VALUE DISTRIBUTIONS
Glauco Valle and Marco Aurelio Sanfins
We show that all multivariate Extreme Value distributions, which are the
possible weak limits of the $K$ largest order statistics of iid
sequences, have
the same copula, the so called K-extremal copula. This copula is
described
through exact expressions for its density and distribution functions.
We also
study measures of dependence, we obtain a weak convergence result and we
propose a simulation algorithm for the K-extremal copula.
http://arxiv.org/abs/0908.2144
---------------------------------------------------------------
8966. SIMULATION REDUCTIONS FOR THE ISING MODEL
Mark L. Huber
Polynomial time reductions between problems have long been used to
delineate
problem classes. Simulation reductions also exist, where an oracle for
simulation from some probability distribution can be employed together
with an
oracle for Bernoulli draws in order to obtain a draw from a different
distribution. Here linear time simulation reductions are given for:
the Ising
spins world to the Ising subgraphs world and the Ising subgraphs world
to the
Ising spins world. This answers a long standing question of whether
such a
direct relationship between these two versions of the Ising model
existed.
Moreover, these reductions result in the first method for perfect
simulation
from the subgraphs world and a new Swendsen-Wang style Markov chain
for the
Ising model. The method used is to write the desired distribution with
set
parameters as a mixture of distributions where the parameters are at
their
extreme values.
http://arxiv.org/abs/0908.2151
---------------------------------------------------------------
8967. CONNECTIVITY BOUNDS FOR THE VACANT SET OF RANDOM INTERLACEMENTS
Vladas Sidoravicius and Alain-Sol Sznitman
The model of random interlacements on Z^d, d bigger or equal to 3, was
recently introduced in arXiv:0704.2560. A non-negative parameter u
parametrizes
the density of random interlacements on Z^d. In the present note we
investigate
the connectivity properties of the vacant set left by random
interlacements at
level u, in the non-percolative regime, where u is bigger than the
non-degenerate critical parameter for percolation of the vacant set, see
arXiv:0704.2560, arXiv:0808.3344. We prove a stretched exponential
decay of the
connectivity function for the vacant set at level u, when u is bigger
than an
other critical parameter. It is presently an open problem whether
these two
critical parameters actually coincide.
http://arxiv.org/abs/0908.2206
---------------------------------------------------------------
8968. RANDOM PERMUTATIONS WITH CYCLE WEIGHTS
Volker Betz and Daniel Ueltschi and Yvan Velenik
We study the distribution of cycle lengths in models of nonuniform
random
permutations with cycle weights. We identify several regimes.
Depending on the
weights, the length of typical cycles grows like the total number n of
elements, or a fraction of n, or a logarithmic power of n.
http://arxiv.org/abs/0908.2217
---------------------------------------------------------------
8969. THE TREE LENGTH OF AN EVOLVING COALESCENT
Peter Pfaffelhuber and Anton Wakolbinger and Heinz Weisshaupt
A well-established model for the genealogy of a large population in
equilibrium is Kingman's coalescent. For the population together with
its
genealogy evolving in time, this gives rise to a time-stationary tree-
valued
process. We study the sum of the branch lengths, briefly denoted as tree
length, and prove that the (suitably compensated) sequence of tree
length
processes converges, as the population size tends to infinity, to a
limit
process with cadlag paths, infinite infinitesimal variance, and a Gumbel
distribution as its equilibrium.
http://arxiv.org/abs/0908.2444
---------------------------------------------------------------
8970. STOCHASTIC INTEGRAL REPRESENTATION OF THE $L^{2}$ MODULUS OF
BROWNIAN LOCAL TIME AND A CENTRAL LIMIT THEOREM
Yaozhong Hu and David Nualart
The purpose of this note is to prove a central limit theorem for the
$L^2$-modulus of continuity of the Brownian local time obtained in
\cite{CLMR},
using techniques of stochastic analysis. The main ingredients of the
proof are
an asymptotic version of Knight's theorem and the Clark-Ocone formula
for the
$L^2$-modulus of the Brownian local time.
http://arxiv.org/abs/0908.2473
---------------------------------------------------------------
8971. ENVIRONMENTAL NOISE VARIABILITY IN POPULATION DYNAMICS MATRIX
MODELS
Michel De Lara (CERMICS)
The impact of environmental variability on population size growth rate
in
dynamic models is a recurrent issue in the theoretical ecology
literature. In
the scalar case, R. Lande pointed out that results are ambiguous
depending on
whether the noise is added at arithmetic or logarithmic scale, while
the matrix
case has been investigated by S. Tuljapurkar. Our contribution
consists first
in introducing another notion of variability than the widely used
variance or
coefficient of variation, namely the so-called convex orders. Second, in
population dynamics matrix models, we focus on how matrix components
depend
functionaly on uncertain environmental factors. In the log-convex
case, we show
that, in a sense, environmental variability increases both mean
population size
and mean log-population size and makes them more variable. Our main
result is
that specific analytical dependence coupled with appropriate notion of
variability lead to wide generic results, valid for all times and not
only
asymptotically, and requiring no assumptions of stationarity, of
normality, of
independency, etc. Though the approach is different, our conclusions are
consistent with previous results in the literature. However, they make
it clear
that the analytical dependence on environmental factors cannot be
overlooked
when trying to tackle the influence of variability.
http://arxiv.org/abs/0908.2499
---------------------------------------------------------------
8972. A BACKWARD PARTICLE INTERPRETATION OF FEYNMAN-KAC FORMULAE
Pierre Del Moral and Arnaud Doucet and Sumeetpal S. Singh
We design a particle interpretation of Feynman-Kac measures on path
spaces
based on a backward Markovian representation combined with a
traditional mean
field particle interpretation of the flow of their final time
marginals. In
contrast to traditional genealogical tree based models, these new
particle
algorithms can be used to compute normalized additive functionals "on-
the-fly"
as well as their limiting occupation measures with a given precision
degree
that does not depend on the final time horizon.
We provide uniform convergence results w.r.t. the time horizon
parameter as
well as functional central limit theorems and exponential concentration
estimates. We also illustrate these results in the context of
computational
physics and imaginary time Schroedinger type partial differential
equations,
with a special interest in the numerical approximation of the
invariant measure
associated to $h$-processes.
http://arxiv.org/abs/0908.2556
---------------------------------------------------------------
8973. THRESHOLD GRAPH LIMITS AND RANDOM THRESHOLD GRAPHS
Persi Diaconis and Susan Holmes and Svante Janson
We study the limit theory of large threshold graphs and apply this to a
variety of models for random threshold graphs. The results give a nice
set of
examples for the emerging theory of graph limits.
http://arxiv.org/abs/0908.2448
---------------------------------------------------------------
8974. PHASE TRANSITION FOR THE MIXING TIME OF THE GLAUBER DYNAMICS
FOR COLORING REGULAR TREES
Prasad Tetali and Juan C. Vera and Eric Vigoda and Linji Yang
We prove that the mixing time of the Glauber dynamics for random
$k$-colorings of the complete tree with branching factor $b$ undergoes
a phase
transition at $k=b(1+o_b(1))/\ln{b}$. Our main result shows nearly
sharp bounds
on the mixing time of the dynamics on the complete tree with $n$
vertices for
$k=Cb/\ln{b}$ colors with constant $C$. For $C\geq 1$ we prove the
mixing time
is $O(n^{1+o_b(1)}\ln^2{n})$. On the other side, for $C< 1$ the mixing
time
experiences a slowing down, in particular, we prove it is $O(n^{1/C +
o_b(1)}\ln^2{n})$ and $\Omega(n^{1/C-o_b(1)})$. The critical point C=1
is
interesting since it coincides (at least up to first order) to the so-
called
reconstruction threshold which was recently established by Sly. The
reconstruction threshold has been of considerable interest recently
since it
appears to have close connections to the efficiency of certain local
algorithms, and this work was inspired by our attempt to understand
these
connections in this particular setting.
http://arxiv.org/abs/0908.2665
---------------------------------------------------------------
8975. STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED AND
DEGENERATE COEFFICIENTS
Xicheng Zhang
In this article, using DiPerna-Lions theory \cite{Di-Li}, we investigate
linear second order stochastic partial differential equations with
unbounded
and degenerate non-smooth coefficients, and obtain several conditions
for
existence and uniqueness. Moreover, we also prove the $L^1$-
integrability and a
general maximal principle for generalized solutions of SPDEs. As
applications,
we study nonlinear filtering problem and also obtain the existence and
uniqueness of generalized solutions for a degenerate nonlinear SPDE.
http://arxiv.org/abs/0908.2695
---------------------------------------------------------------
8976. PROBABILISTIC REPRESENTATION FOR SOLUTIONS OF AN IRREGULAR
POROUS MEDIA TYPE EQUATION: THE DEGENERATE CASE
Viorel Barbu and Michael Roeckner (SFB 701) and Francesco Russo (LAGA)
We consider a possibly degenerate porous media type equation over all of
$\R^d$ with $d = 1$, with monotone discontinuous coefficients with
linear
growth and prove a probabilistic representation of its solution in
terms of an
associated microscopic diffusion. This equation is motivated by some
singular
behaviour arising in complex self-organized critical systems. The main
idea
consists in approximating the equation by equations with monotone
non-degenerate coefficients and deriving some new analytical
properties of the
solution.
http://arxiv.org/abs/0908.2701
---------------------------------------------------------------
8977. SHARP INTERFACE LIMIT FOR INVARIANT MEASURES OF A STOCHASTIC
ALLEN-CAHN EQUATION
Hendrik Weber
The invariant measure of a one-dimensional Allen-Cahn equation with an
additive space-time white noise is studied. This measure is absolutely
continuous with respect to a Brownian bridge with a density which can be
interpreted as a potential energy term. We consider the sharp
interface limit
in this setup. In the right scaling this corresponds to a Gibbs type
measure on
a growing interval with decreasing temperature. Our main result is
that in the
limit we still see exponential convergence towards a curve of
minimizers of the
energy if the interval does not grow too fast. In the original scaling
the
limit measure is concentrated on configurations with precisely one
jump. This
jump is distributed uniformly.
http://arxiv.org/abs/0908.2717
---------------------------------------------------------------
8978. HYDRODYNAMIC LIMIT OF MOVE-TO-FRONT RULES AND SEARCH COST
PROBABILITIES
Kumiko Hattori and Tetsuya Hattori
We study a hydrodynamic limit approach to move-to-front rules, namely, a
scaling limit as the number of items tends to infinity, of the joint
distribution of jump rate and position of items. As an application of
the limit
formula, we present asymptotic formulas on search cost probability
distributions, applicable for general jump rate distributions.
http://arxiv.org/abs/0908.3222
---------------------------------------------------------------
8979. STOCHASTIC EVOLUTIONS OF POINT PROCESSES
Philippe Robert
The asymptotic behavior of birth and death processes of particles in a
compact space is analyzed. Births: Particles are created at rate $
\lambda_+$
and their location is independent of the current configuration. Deaths
are due
to negative particles arriving at rate $\lambda_-$. The death of a
particle
occurs when a negative particle arrives in its neighborhood and kills
it.
Several killing schemes are considered. The arriving locations of
positive and
negative particles are assumed to have the same distribution. By using a
combination of monotonicity properties and invariance relations it is
shown
that the configurations of particles converge in distribution for
several
models. The problems of uniqueness of invariant measures and of the
existence
of accumulation points for the limiting configurations are also
investigated.
It is shown for several natural models that if $\lambda_+<\lambda_-$
then the
asymptotic configuration has a finite number of points with
probability 1.
Examples with $\lambda_+<\lambda_-$ and an infinite number of
particles in the
limit are also presented.
http://arxiv.org/abs/0908.3256
---------------------------------------------------------------
8980. REFLECTED BROWNIAN MOTION IN WEYL CHAMBERS
Nizar Demni
We supply two different descriptions of the pushing process driving the
reflected Brownian motion in Weyl chambers, when the latter domains are
simplexes. The first one shows that a simple root lies in one and only
one
orbit if and only if the pushing process in the direction of that
simple root
increases as the sum of all the Brownian local times in the directions
of the
orbit's positive elements. The last one shows that the pushing process
may be
written as the sum of an inward normal vector at the chamber's
boundary and an
inward normal vector at the origin, yielding a kind of a multivoque
stochastic
differential equation for the reflected process. We finally give a
particles
system interpretation of the reflected process and construct a
multidimensional
skew Brownian motion.
http://arxiv.org/abs/0908.3302
---------------------------------------------------------------
8981. A ZERO-ONE LAW FOR LINEAR TRANSFORMATIONS OF LEVY NOISE
Steven N. Evans
A L\'evy noise on $\mathbb{R}^d$ assigns a random real "mass" $\Pi(B)$
to
each Borel subset $B$ of $\mathbb{R}^d$ with finite Lebesgue measure.
The
distribution of $\Pi(B)$ only depends on the Lebesgue measure of $B$,
and if
$B_1, ..., B_n$ is a finite collection of pairwise disjoint sets, then
the
random variables $\Pi(B_1), ..., \Pi(B_n)$ are independent with $
\Pi(B_1 \cup
>... \cup B_n) = \Pi(B_1) + ... + \Pi(B_n)$ almost surely. In
particular, the
distribution of $\Pi \circ g$ is the same as that of $\Pi$ when $g$ is a
bijective transformation of $\mathbb{R}^d$ that preserves Lebesgue
measure. It
follows from the Hewitt--Savage zero--one law that any event which is
almost
surely invariant under the mappings $\Pi \mapsto \Pi \circ g$ for every
Lebesgue measure preserving bijection $g$ of $\mathbb{R}^d$ must have
probability 0 or 1. We investigate whether certain smaller groups of
Lebesgue
measure preserving bijections also possess this property. We show that
if $d
\ge 2$, the L\'evy noise is not purely deterministic, and the group
consists of
linear transformations and is closed, then the invariant events all have
probability 0 or 1 if and only if the group is not compact.
http://arxiv.org/abs/0908.3339
---------------------------------------------------------------
8982. FINITE-TIME BLOWUP AND EXISTENCE OF GLOBAL POSITIVE SOLUTIONS OF
A SEMI-LINEAR SPDE
Marco Dozzi (IECN) and Jos\'e Alfredo Lopez
We consider stochastic equations of the prototype $du(t,x) =(\Delta
u(t,x)+u(t,x)^{1+\beta})dt+\kappa u(t,x) dW_{t}$ on a smooth domain $D
\subset
\mathord{\rm I\mkern-3.6mu R\:}^d$, with Dirichlet boundary condition,
where
$\beta$, $\kappa$ are positive constants and $\{W_t $, $t\ge0\}$ is a
one-dimensional standard Wiener process. We estimate the probability
of finite
time blowup of positive solutions, as well as the probability of
existence of
non-trivial positive global solutions.
http://arxiv.org/abs/0908.3364
---------------------------------------------------------------
8983. LIMIT THEOREMS FOR RANDOM PROCESSES WITH RANDOM TIME SUBSTITUTION
Permyakova Elena
In this paper the sufficient conditions for convergence in Skorokhod
space
$D[0,1]$ of sequence of random processes with random time substitution
are
obtained.
http://arxiv.org/abs/0908.3395
---------------------------------------------------------------
8984. POISSON SPLITTING BY FACTORS
Alexander E. Holroyd and Russell Lyons and and Terry Soo
Given a homogeneous Poisson process on R^d with intensity L, we prove
that it
is possible to partition the points into two sets, as a deterministic
function
of the process, and in an isometry-equivariant way, so that each set
of points
forms a homogeneous Poisson process, with any given pair of
intensities summing
to L. In particular, this answers a question of Ball, who proved that
in d=1,
the Poisson points may be similarly partitioned (via a translation-
equivariant
function) so that one set forms a Poisson process of lower intensity,
and asked
whether the same was possible for all d. We do not know whether it is
possible
similarly to add points (again chosen as a deterministic function of a
Poisson
process) to obtain a Poisson process of higher intensity, but we prove
that
this is not possible under an additional finitariness condition.
http://arxiv.org/abs/0908.3409
---------------------------------------------------------------
8985. A RULE OF THUMB FOR RIFFLE SHUFFLING
Sami Assaf and Persi Diaconis and K. Soundararajan
We study how many riffle shuffles are required to mix n cards if only
certain
features of the deck are of interest, e.g. suits disregarded or only
the colors
of interest. For these features, the number of shuffles drops from 3/2
log_2(n)
to log_2(n). We derive closed formulae and an asymptotic `rule of thumb'
formula which is remarkably accurate.
http://arxiv.org/abs/0908.3462
---------------------------------------------------------------
8986. OPTIMAL TRANSPORTATION AND MONOTONIC QUANTITIES ON EVOLVING
MANIFOLDS
Hong Huang
In this note we adapt Topping's $\mathcal{L}$-optimal transportation
theory
for Ricci flow to a more general situation, i.e. to a closed manifold
$(M,g_{ij}(t)$ evolving by $\partial_tg_{ij}=-2S_{ij}$, where $S_{ij}$
is a
symmetric tensor field of (2,0)-type on $M$. We extend some of
Topping's and
Lott's recent results, generalize the monotonicity of List's (and
hence also of
Perelman's) $\mathcal{W}$-entropy, and recover the monotonicity of
M$\ddot{u}$ller's (and hence also of Perelman's) reduced volume.
http://arxiv.org/abs/0908.3293
---------------------------------------------------------------
8987. RANK-BASED ATTACHMENT LEADS TO POWER LAW GRAPHS
Jeannette Janssen and Pawel Pralat
We investigate the degree distribution resulting from graph generation
models
based on rank-based attachment. In rank-based attachment, all vertices
are
ranked according to a ranking scheme. The link probability of a given
vertex is
proportional to its rank raised to the power -a, for some a in (0,1).
Through a
rigorous analysis, we show that rank-based attachment models lead to
graphs
with a power law degree distribution with exponent 1+1/a whenever
vertices are
ranked according to their degree, their age, or a randomly chosen
fitness
value. We also investigate the case where the ranking is based on the
initial
rank of each vertex; the rank of existing vertices only changes to
accommodate
the new vertex. Here, we obtain a sharp threshold for power law
behaviour. Only
if initial ranks are biased towards lower ranks, or chosen uniformly
at random,
we obtain a power law degree distribution with exponent 1+1/a. This
indicates
that the power law degree distribution often observed in nature can be
explained by a rank-based attachment scheme, based on a ranking scheme
that can
be derived from a number of different factors; the exponent of the
power law
can be seen as a measure of the strength of the attachment.
http://arxiv.org/abs/0908.3436
---------------------------------------------------------------
8988. NOTES ON FEIGE'S GUMBALL MACHINES PROBLEM
John H. Elton
We give a detailed proof, in the identically distributed case, of a
conjecture of Feige about the maximum probability that the sum of n
independent
non-negative integer valued random variables, each of mean 1, exceeds
n. The
general case is reduced to two-point distributions.
http://arxiv.org/abs/0908.3528
---------------------------------------------------------------
8989. LIMIT THEOREMS FOR PROJECTIONS OF RANDOM WALK ON A HYPERSPHERE
Max Skipper
We show that almost any one-dimensional projection of a suitably scaled
random walk on a hypercube, inscribed in a hypersphere, converges
weakly to an
Ornstein-Uhlenbeck process as the dimension of the sphere tends to
infinity. We
also observe that the same result holds when the random walk is
replaced with
spherical Brownian motion. This latter result can be viewed as a
"functional"
generalisation of Poincar\'e's observation for projections of uniform
measure
on high dimensional spheres; the former result is an analogous
generalisation
of the Bernoulli-Laplace central limit theorem. Given the relation of
these two
classic results to the central limit theorem for convex bodies, the
modest
results provided here would appear to motivate a functional
generalisation.
http://arxiv.org/abs/0908.3536
---------------------------------------------------------------
8990. CAN AN INFINITE PRODUCT OF NONNEGATIVE MATRICES BE EXPRESSED IN
TERMS OF INFINITE PRODUCTS OF STOCHASTIC ONES?
Alain Thomas (LATP)
It is known that if the product $M_n... M_1$ converges to a nonnull
limit
when $n\to\infty$ and if the $M_n$ belong to a finite set of complex
matrices,
then the $M_n$ for $n\ge n_0$ have a common right eigenvector $V$ for
the
eigenvalue 1. In case the $M_n$ are nonnegative and $V$ is positive,
$\Delta^{-1}M_{n_0}... M_n\Delta$ is the product of the stochastic
matrices
$\Delta^{-1}M_n\Delta$, where the diagonal matrix $\Delta$ has on its
diagonal
the same entries as $V$. In the last section we examine what happen
when we
remove the hypothesis that $V$ is positive.
http://arxiv.org/abs/0908.3538
---------------------------------------------------------------
8991. CRITICAL RANDOM GRAPHS: LIMITING CONSTRUCTIONS AND
DISTRIBUTIONAL PROPERTIES
L. Addario-Berry and N. Broutin and C. Goldschmidt
We consider the Erdos--Renyi random graph G(n,p) inside the critical
window,
where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a
previous
paper (arXiv:0903.4730) that considering the connected components of
G(n,p) as
a sequence of metric spaces with the graph distance rescaled by
n^{-1/3} and
letting n go to infinity yields a non-trivial sequence of limit metric
spaces C
= (C_1, C_2, ...). These limit metric spaces can be constructed from
certain
random real trees with vertex-identifications. We give here equivalent
constructions using standard Brownian continuum random trees, their
recursive
construction from inhomogeneous Poisson point processes, and Polya's urn
scheme. We also characterize the distributions of the masses and
lengths in the
constituant parts of a limit component when it is decomposed according
to its
cycle structure.
http://arxiv.org/abs/0908.3629
---------------------------------------------------------------
8992. HARNACK INEQUALITIES AND APPLICATIONS FOR MULTIVALUED
STOCHASTIC EVOLUTION EQUATIONS
Shun-Xiang Ouyang
By the method of coupling and Girsanov transformation, Harnack
inequalities
[F.-Y. Wang, 1997] and strong Feller property are proved for the
transition
semigroup associated with the multivalued stochastic evolution
equation on a
Gelfand triple. The concentration property of the invariant measure
for the
semigroup is investigated. As applications of Harnack inequalities,
explicit
upper bounds of the $L^p$-norm of the density, contractivity,
compactness and
entropy-cost inequality for the semigroup are also presented.
http://arxiv.org/abs/0908.3630
---------------------------------------------------------------
8993. APPLICATIONS OF WEAK CONVERGENCE FOR HEDGING OF AMERICAN AND
GAME OPTIONS
Yan Dolinsky
This paper studies stability of Dynkin's games value under weak
convergence.
We use these results to approximate game options prices with path
dependent
payoffs in continuous time models by sequence of game options prices in
discrete time models which can be calculated by dynamical programming
algorithms. We also show that shortfall risks of American options in a
sequence
of multinomial approximations of the multidimensional BS market
converge to the
corresponding quantities for similar American options in the
multidimensional
BS market with path dependent payoffs. In comparison to previous
papers we work
under more general convergence of underlying processes, as well, as
weaker
condition on the payoffs.
http://arxiv.org/abs/0908.3661
---------------------------------------------------------------
8994. ON THE MINIMAL PENALTY FOR MARKOV ORDER ESTIMATION
Ramon van Handel
We show that large-scale typicality of Markov sample paths implies
that the
likelihood ratio statistic satisfies a law of iterated logarithm
uniformly to
the same scale. As a consequence, the penalized likelihood Markov order
estimator is strongly consistent for penalties growing as slowly as
log log n
when an upper bound is imposed on the order which may grow as rapidly
as log n.
Our method of proof, using techniques from empirical process theory,
does not
rely on the explicit expression for the maximum likelihood estimator
in the
Markov case and could therefore be applicable in other settings.
http://arxiv.org/abs/0908.3666
---------------------------------------------------------------
8995. ZERO-ONE LAWS FOR CONNECTIVITY IN RANDOM KEY GRAPHS
Osman Yagan and Armand M. Makowski
The random key graph is a random graph naturally associated with the
random
key predistribution scheme of Eschenauer and Gligor for wireless sensor
networks. For this class of random graphs we establish a new version
of a
conjectured zero-one law for graph connectivity as the number of nodes
becomes
unboundedly large. The results reported here complement and strengthen
recent
work on this conjecture by Blackburn and Gerke. In particular, the
results are
given under conditions which are more realistic for applications to
wireless
sensor networks.
http://arxiv.org/abs/0908.3644
---------------------------------------------------------------
8996. RANDOMIZED SCHEDULING ALGORITHM FOR QUEUEING NETWORKS
Devavrat Shah and Jinwoo Shin
There has recently been considerable interest in design of low-
complexity,
myopic, distributed and stable scheduling policies for constrained
queueing
network models that arise in the context of emerging communication
networks.
Here, we consider two representative models. One, a model for the
collection of
wireless nodes communicating through a shared medium, that represents
randomly
varying number of packets in the queues at the nodes of networks. Two, a
buffered circuit switched network model for an optical core of future
Internet,
to capture the randomness in calls or flows present in the network.
The maximum
weight scheduling policy proposed by Tassiulas and Ephremide in 1992
leads to a
myopic and stable policy for the packet-level wireless network model.
But
computationally it is very expensive (NP-hard) and centralized. It is
not
applicable to the buffered circuit switched network due to the
requirement of
non-premption of the calls in the service. As the main contribution of
this
paper, we present a stable scheduling algorithm for both of these
models. The
algorithm is myopic, distributed and performs few logical operations
at each
node per unit time.
http://arxiv.org/abs/0908.3670
---------------------------------------------------------------
8997. ASYMPTOTIC REGIMES FOR THE PARTITION INTO COLONIES OF A
BRANCHING PROCESS WITH EMIGRATION
Jean Bertoin (PMA and Dma)
We consider a spatial branching process with emigration in which
children
either remain at the same site as their parents or migrate to new
locations and
then found their own colonies. We are interested in asymptotics of the
partition of the total population into colonies for large populations
with rare
migrations. Under appropriate regimes, we establish weak convergence
of the
rescaled partition to some random measure that is constructed from the
restriction of a Poisson point measure to a certain random region, and
whose
cumulant solves a simple integral equation.
http://arxiv.org/abs/0908.3735
---------------------------------------------------------------
8998. ON THE ABSOLUTE CONTINUITY OF MULTIDIMENSIONAL ORNSTEIN-
UHLENBECK PROCESSES
Thomas Simon (LPP)
Let $X$ be a $n$-dimensional Ornstein-Uhlenbeck process, solution of the
S.D.E. $$\d X_t = AX_t \d t + \d B_t$$ where $A$ is a real $n\times n$
matrix
and $B$ a L\'evy process without Gaussian part. We show that when $A$ is
non-singular, the law of $X_1$ is absolutely continuous in $\r^n$ if
and only
if the jumping measure of $B$ fulfils a certain geometric condition with
respect to $A,$ which we call the exhaustion property. This optimal
criterion
is much weaker than for the background driving L\'evy process $B$,
which might
be very singular and sometimes even have a one-dimensional discrete
jumping
measure. It also solves a difficult problem for a certain class of
multivariate
Non-Gaussian infinitely divisible distributions.
http://arxiv.org/abs/0908.3736
---------------------------------------------------------------
8999. EXTREMAL SUBGRAPHS OF RANDOM GRAPHS: AN EXTENDED VERSION
Graham Brightwell and Konstantinos Panagiotou and Angelika Steger
We prove that there is a constant $c >0$, such that whenever $p \ge
n^{-c}$,
with probability tending to 1 when $n$ goes to infinity, every maximum
triangle-free subgraph of the random graph $G_{n,p}$ is bipartite.
This answers
a question of Babai, Simonovits and Spencer (Journal of Graph Theory,
1990).
The proof is based on a tool of independent interest: we show, for
instance,
that the maximum cut of almost all graphs with $M$ edges, where $M >> n
$, is
``nearly unique''. More precisely, given a maximum cut $C$ of $G_{n,M}
$, we can
obtain all maximum cuts by moving at most $O(\sqrt{n^3/M})$ vertices
between
the parts of $C$.
http://arxiv.org/abs/0908.3778
---------------------------------------------------------------
9000. MIXING TIME OF NEAR-CRITICAL RANDOM GRAPHS
Jian Ding and Eyal Lubetzky and Yuval Peres
Let $C_1$ be the largest component of the Erd\H{o}s-R\'enyi random graph
$G(n,p)$. The mixing time of random walk on $C_1$ in the strictly
supercritical
regime, $p=c/n$ with fixed $c>1$, was shown to have order $\log^2 n$ by
Fountoulakis and Reed, and independently by Benjamini, Kozma and
Wormald. In
the critical window, $p=(1+\epsilon)/n$ where $\lambda=\epsilon^3 n$ is
bounded, Nachmias and Peres proved that the mixing time on $C_1$ is of
order
$n$. However, it was unclear how to interpolate between these results,
and
estimate the mixing time as the giant component emerges from the
critical
window. Indeed, even the asymptotics of the diameter of $C_1$ in this
regime
were only recently obtained by Riordan and Wormald, as well as the
present
authors and Kim.
In this paper we show that for $p=(1+\epsilon)/n$ with $\lambda=
\epsilon^3
n\to\infty$ and $\lambda=o(n)$, the mixing time on $C_1$ is with high
probability of order $(n/\lambda)\log^2 \lambda$. In addition, we show
that
this is the order of the largest mixing time over all components, both
in the
slightly supercritical and in the slightly subcritical regime (i.e.,
$p=(1-\epsilon)/n$ with $\lambda$ as above).
http://arxiv.org/abs/0908.3870
---------------------------------------------------------------
9001. UTILITY OPTIMIZATION IN CONGESTED QUEUEING NETWORKS
Neil Stuart Walton
We consider a multi-class single server queueing network as a model of a
packet switching network. The rates packets are sent into this network
are
controlled by queues which act as congestion windows. By considering a
sequence
of such congestion windows we allow the network to become congested.
We show
the stationary throughput of routes on this sequence of networks
converges to
an allocation that maximizes aggregate utility subject to the network's
capacity constraints. To perform this analysis we require that our
utility
functions satisfy an exponential concavity condition. This family of
utilities
includes weighted $\alpha$-fair utilities for $\alpha >1$.
http://arxiv.org/abs/0908.3787
---------------------------------------------------------------
9002. DISTRIBUTED AVERAGING VIA LIFTED MARKOV CHAINS
Kyomin Jung and Devavrat Shah and Jinwoo Shin
Motivated by applications of distributed linear estimation, distributed
control and distributed optimization, we consider the question of
designing
linear iterative algorithms for computing the average of numbers in a
network.
Specifically, our interest is in designing such an algorithm with the
fastest
rate of convergence given the topological constraints of the network.
As the
main result of this paper, we design an algorithm with the fastest
possible
rate of convergence using a non-reversible Markov chain on the given
network
graph. We construct such a Markov chain by transforming the standard
Markov
chain, which is obtained using the Metropolis-Hastings method. We call
this
novel transformation pseudo-lifting. We apply our method to graphs with
geometry, or graphs with doubling dimension. Specifically, the
convergence time
of our algorithm (equivalently, the mixing time of our Markov chain) is
proportional to the diameter of the network graph and hence optimal.
As a
byproduct, our result provides the fastest mixing Markov chain given the
network topological constraints, and should naturally find their
applications
in the context of distributed optimization, estimation and control.
http://arxiv.org/abs/0908.4073
---------------------------------------------------------------
9003. HYDRODYNAMIC LIMIT OF THE EXCLUSION PROCESS IN INHOMOGENEOUS MEDIA
Milton Jara
We obtain the hydrodynamic limit of a simple exclusion process in an
inhomogeneous environment of divergence form. Our main assumption is a
suitable
version of Gamma-convergence for the environment. In this way we
obtain an
unified approach to recent works on the field.
http://arxiv.org/abs/0908.4120
---------------------------------------------------------------
9004. CONTACT PROCESS IN A WEDGE
J. Theodore Cox and Nevena Maric and Rinaldo B. Schinazi
We prove that the supercritical one-dimensional contact process
survives in
certain wedge-like space-time regions, and that when it survives it
couples
with the unrestricted contact process started from its upper invariant
measure.
As an application we show that a type of weak coexistence is possible
in the
nearest-neighbor ``grass-bushes-trees'' successional model introduced in
Durrett and Swindle (1991).
http://arxiv.org/abs/0908.4125
---------------------------------------------------------------
9005. KOLMOGOROV EQUATION ASSOCIATED TO THE STOCHASTIC REFLECTION
PROBLEM ON A SMOOTH CONVEX SET OF A HILBERT SPACE
Viorel Barbu and Giuseppe Da Prato and Luciano Tubaro
We consider the stochastic reflection problem associated with a self-
adjoint
operator $A$ and a cylindrical Wiener process on a convex set $K$ with
nonempty
interior and regular boundary $\Sigma$ in a Hilbert space $H$. We
prove the
existence and uniqueness of a smooth solution for the corresponding
elliptic
infinite-dimensional Kolmogorov equation with Neumann boundary
condition on
$\Sigma$.
http://arxiv.org/abs/0908.4139
---------------------------------------------------------------
9006. THE SURVIVAL OF LARGE DIMENSIONAL THRESHOLD CONTACT PROCESSES
Thomas Mountford and Roberto H. Schonmann
We study the threshold $\theta$ contact process on $\mathbb{Z}^d$ with
infection parameter $\lambda$. We show that the critical point
$\lambda_{\mathrm{c}}$, defined as the threshold for survival starting
from
every site occupied, vanishes as $d\to\infty$. This implies that the
threshold
$\theta$ voter model on $\mathbb{Z}^d$ has a nondegenerate extremal
invariant
measure, when $d$ is large.
http://arxiv.org/abs/0908.4146
---------------------------------------------------------------
9007. ON THE EXTENDIBILITY OF PARTIALLY AND MARKOV EXCHANGEABLE
BINARY SEQUENCES
Davide Di Cecco
In [Fortini et al., Stoch. Proc. Appl. 100 (2002), 147--165] it is
demonstrated that a recurrent Markov exchangeable process in the sense
of
Diaconis and Freedman is essentially a partially exchangeable process
in the
sense of de Finetti. In case of finite sequences there is not such an
equivalence. We analyze both finite partially exchangeable and finite
Markov
exchangeable binary sequences and formulate necessary and sufficient
conditions
for extendibility in both cases.
http://arxiv.org/abs/0908.4158
---------------------------------------------------------------
9008. ASYMPTOTIC PROPERTIES OF THE COLUMNS IN THE PRODUCTS OF
NONNEGATIVE MATRICES
\'Eric Olivier (LATP) and Alain Thomas (LATP)
We consider the sequence of column-vectors $R_n=A_1... A_nR$
associated to a
sequence $(A_n)$ of nonnegative $d\times d$ matrices and to a positive
$d$-dimensional column-vector $R$. The problem to know the necessary and
sufficient conditions -- on the sequence $(A_n)$ -- for
$\displaystyle{R_n\over\Vert R_n\Vert}$ to converge is yet not solved.
Nevertheless we prove this convergence in case the $A_n$ are -- in a
sense --
echeloned and fulfill certain boundness conditions. If the $A_n$ do
not fulfill
the conditions and even if they are sparse, it may exist a sequence of
integers
$(n_k)$ such that the matrices $A'_k=A_{n_k+1}... A_{n_{k+1}}$ do: we
see in
some other paper how to proceed in one example, and how to use the
obtained
result to study some continuous singular measure.
http://arxiv.org/abs/0908.4171
---------------------------------------------------------------
9009. ON THE INVERSE FIRST-PASSAGE-TIME PROBLEM FOR A WIENER PROCESS
Cristina Zucca and Laura Sacerdote
The inverse first-passage problem for a Wiener process $(W_t)_{t\ge0}$
seeks
to determine a function $b{}:{}\mathbb{R}_+\to\mathbb{R}$ such that
\[\tau=\inf\{t>0| W_t\ge b(t)\}\] has a given law. In this paper two
methods
for approximating the unknown function $b$ are presented. The errors
of the two
methods are studied. A set of examples illustrates the methods. Possible
applications are enlighted.
http://arxiv.org/abs/0908.4213
---------------------------------------------------------------
9010. EXTREMAL SHOT NOISES, HEAVY TAILS AND MAX-STABLE RANDOM FIELDS
Cl\'ement Dombry (LMA)
Extremal shot noises naturally appear in extreme value theory as a
model for
spatial extremes and serve as basic models for annual maxima of
rainfall or for
coverage field in telecommunication. In this work, we examine their
properties
such as boundedness, regularity, ergodicity ... Connexions with max-
stable
random fields are established: we prove a limit theorem when the
distribution
of the weights is heavy tailed and the intensity of points goes to
infinity. We
use a point process approach strongly connected to the Peak Over
Threshold
method used by hydrologists. Properties of the limit max-stable random
fields
are also investigated.
http://arxiv.org/abs/0908.4221
---------------------------------------------------------------
9011. STOCHASTIC COMPLETENESS AND VOLUME GROWTH
Christian Baer and G. Pacelli Bessa
It has been suggested in 1999 that a certain volume growth condition for
geodesically complete Riemannian manifolds might imply that the
manifold is
stochastically complete. This is motivated by a large class of
examples and by
a known analogous criterion for recurrence of Brownian motion. We show
that the
suggested implication is not true in general. We also give counter-
examples to
a converse implication.
http://arxiv.org/abs/0908.4222
---------------------------------------------------------------
9012. MATRIX FACTORIZATION IDENTITY FOR ALMOST SEMI-CONTINUOUS
PROCESSES ON A MARKOV CHAIN
D.V. Gusak and E.V. Karnaukh
In this article almost semi-continuous processes with stationary
independent
increments on a finite irreducible Markov chain are considered. For
these
processes the components of matrix factorization identity are concretely
defined. On the basis of this concrete definition the relations for the
distributions of extrema and distributions of their complements for
the almost
upper semi-continuous processes are established.
http://arxiv.org/abs/0908.4326
---------------------------------------------------------------
9013. LIMIT LAWS OF TRANSIENT EXCITED RANDOM WALKS ON INTEGERS
Elena Kosygina and Thomas Mountford
We consider excited random walks (ERWs) on integers with a bounded
number of
i.i.d. cookies per site without the non-negativity assumption on the
drifts
induced by the "cookies". E. Kosygina and M.P.W. Zerner have shown
that when
the total expected drift per site, delta, is larger than 1 then ERW is
transient to the right and, moreover, for delta>4 under the averaged
measure it
obeys the Central Limit Theorem. We show that when delta is in (2,4] the
limiting behavior of an appropriately centered and scaled excited
random walk
is described by a strictly stable law with parameter delta/2. Our
method also
extends the results obtained by A.-L. Basdevant and A. Singh for delta
in (1,2]
under the non-negativity assumption to the setting which allows both
positive
and negative cookies.
http://arxiv.org/abs/0908.4356
---------------------------------------------------------------
9014. POISSON DIRICHLET$(\ALPHA,\THETA)$-BRIDGE EQUATIONS AND
COAGULATION-FRAGMENTATION DUALITY
Lancelot F. James
This paper derives distributional properties of a class of exchangeable
bridges closely related to the Poisson-Dirichlet $(\alpha,\theta)$
family of
bridges. We then show that various stochastic equations derived for
these
bridges lead to constructions of a new large class of coagulation and
fragmentation operators that satisfy a duality property, and are
otherwise
easily manipulated. This class, builds on, and includes the duality
relations
developed in Pitman (1999), Bertoin and Goldschmidt (2004), and Dong,
Goldschmidt and Martin (2006),which we can treat in a unified way. Our
exposition also suggests an approach to obtain other dualities and
related
results.
http://arxiv.org/abs/0908.4436
---------------------------------------------------------------
9015. CONVERGENCE OF NUMERICAL TIME-AVERAGING AND STATIONARY MEASURES
VIA POISSON EQUATIONS
Jonathan C. Mattingly and Andrew M. Stuart and M.V. Tretyakov
Numerical approximation of the long time behavior of a stochastic
differential equation (SDE) is considered. Error estimates for time-
averaging
estimators are obtained and then used to show that the stationary
behavior of
the numerical method converges to that of the SDE. The error analysis
is based
on using an associated Poisson equation for the underlying SDE. The main
advantage of this approach is its simplicity and universality. It
works equally
well for a range of explicit and implicit schemes including those with
simple
simulation of random variables, and for general hypoelliptic SDEs. An
analogy
between this approach and Stein's method is indicated. Some practical
implications of the results are discussed.
http://arxiv.org/abs/0908.4450
---------------------------------------------------------------
9016. TIME AVERAGES, RECURRENCE AND TRANSIENCE IN THE STOCHASTIC
REPLICATOR DYNAMICS
Josef Hofbauer and Lorens A. Imhof
We investigate the long-run behavior of a stochastic replicator process,
which describes game dynamics for a symmetric two-player game under
aggregate
shocks. We establish an averaging principle that relates time averages
of the
process and Nash equilibria of a suitably modified game. Furthermore, a
sufficient condition for transience is given in terms of mixed
equilibria and
definiteness of the payoff matrix. We also present necessary and
sufficient
conditions for stochastic stability of pure equilibria.
http://arxiv.org/abs/0908.4467
---------------------------------------------------------------
9017. BUBBLES, CONVEXITY AND THE BLACK--SCHOLES EQUATION
Erik Ekstr\"{o}m and Johan Tysk
A bubble is characterized by the presence of an underlying asset whose
discounted price process is a strict local martingale under the pricing
measure. In such markets, many standard results from option pricing
theory do
not hold, and in this paper we address some of these issues. In
particular, we
derive existence and uniqueness results for the Black--Scholes
equation, and we
provide convexity theory for option pricing and derive related
ordering results
with respect to volatility. We show that American options are convexity
preserving, whereas European options preserve concavity for general
payoffs and
convexity only for bounded contracts.
http://arxiv.org/abs/0908.4468
---------------------------------------------------------------
9018. ON CONVERGENCE TO STATIONARITY OF FRACTIONAL BROWNIAN STORAGE
Michel Mandjes and Ilkka Norros and Peter Glynn
With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running maximum of a
fractional Brownian motion $A(\cdot)$ with negative drift, this paper
studies
the rate of convergence of $\mathbb {P}(M(t)>x)$ to $\mathbb{P}(M>x)$.
We
define two metrics that measure the distance between the (complementary)
distribution functions $\mathbb{P}(M(t)>\cdot)$ and $\mathbb{P}(M>
\cdot)$. Our
main result states that both metrics roughly decay as $\exp(-\vartheta
t^{2-2H})$, where $\vartheta$ is the decay rate corresponding to the
tail
distribution of the busy period in an fBm-driven queue, which was
computed
recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs
extensively rely on application of the well-known large deviations
theorem for
Gaussian processes. We also show that the identified relation between
the decay
of the convergence metrics and busy-period asymptotics holds in other
settings
as well, most notably when G\"artner--Ellis-type conditions are
fulfilled.
http://arxiv.org/abs/0908.4472
---------------------------------------------------------------
9019. RANDOM RECURRENCE EQUATIONS AND RUIN IN A MARKOV-DEPENDENT
STOCHASTIC ECONOMIC ENVIRONMENT
Jeffrey F. Collamore
We develop sharp large deviation asymptotics for the probability of
ruin in a
Markov-dependent stochastic economic environment and study the
extremes for
some related Markovian processes which arise in financial and insurance
mathematics, related to perpetuities and the $\operatorname {ARCH}(1)$
and
$\operatorname {GARCH}(1,1)$ time series models. Our results build
upon work of
Goldie [Ann. Appl. Probab. 1 (1991) 126--166], who has developed tail
asymptotics applicable for independent sequences of random variables
subject to
a random recurrence equation. In contrast, we adopt a general approach
based on
the theory of Harris recurrent Markov chains and the associated theory
of
nonnegative operators, and meanwhile develop certain recurrence
properties for
these operators under a nonstandard "G\"artner--Ellis" assumption on the
driving process.
http://arxiv.org/abs/0908.4479
---------------------------------------------------------------
9020. NON-MARKOV PROPERTY OF CERTAIN EIGENVALUE PROCESSES ANALOGOUS TO
DYSON'S MODEL
Ryoki Fukushima and Atsushi Tanida and Kouji Yano
It is proven that the eigenvalue process of Dyson's random matrix
process of
size two becomes non-Markov if the common coefficient $1/\sqrt{2}$ in
the
non-diagonal entries is replaced by a different positive number.
http://arxiv.org/abs/0908.4481
---------------------------------------------------------------
9021. OPTIMAL REINSURANCE/INVESTMENT PROBLEMS FOR GENERAL INSURANCE
MODELS
Yuping Liu and Jin Ma
In this paper the utility optimization problem for a general insurance
model
is studied. The reserve process of the insurance company is described
by a
stochastic differential equation driven by a Brownian motion and a
Poisson
random measure, representing the randomness from the financial market
and the
insurance claims, respectively. The random safety loading and stochastic
interest rates are allowed in the model so that the reserve process is
non-Markovian in general. The insurance company can manage the
reserves through
both portfolios of the investment and a reinsurance policy to optimize a
certain utility function, defined in a generic way. The main feature
of the
problem lies in the intrinsic constraint on the part of reinsurance
policy,
which is only proportional to the claim-size instead of the current
level of
reserve, and hence it is quite different from the optimal
investment/consumption problem with constraints in finance. Necessary
and
sufficient conditions for both well posedness and solvability will be
given by
modifying the ``duality method'' in finance and with the help of the
solvability of a special type of backward stochastic differential
equations.
http://arxiv.org/abs/0908.4538
---------------------------------------------------------------
9022. RECURSIVE ESTIMATION OF TIME-AVERAGE VARIANCE CONSTANTS
Wei Biao Wu
For statistical inference of means of stationary processes, one needs to
estimate their time-average variance constants (TAVC) or long-run
variances.
For a stationary process, its TAVC is the sum of all its covariances
and it is
a multiple of the spectral density at zero. The classical TAVC
estimate which
is based on batched means does not allow recursive updates and the
required
memory complexity is O(n). We propose a faster algorithm which
recursively
computes the TAVC, thus having memory complexity of order O(1) and the
computational complexity scales linearly in $n$. Under short-range
dependence
conditions, we establish moment and almost sure convergence of the
recursive
TAVC estimate. Convergence rates are also obtained.
http://arxiv.org/abs/0908.4540
---------------------------------------------------------------
9023. ASYMPTOTIC BEHAVIOR OF UNSTABLE INAR(P) PROCESSES
Matyas Barczy and Marton Ispany and Gyula Pap
In this paper the asymptotic behavior of an unstable integer-valued
autoregressive model of order p (INAR(p)) is described. Under a natural
assumption it is proved that the sequence of appropriately scaled
random step
functions formed from an unstable INAR(p) process converges weakly
towards a
squared Bessel process. We note that this limit behavior is quite
different
from that of familiar unstable autoregressive processes of order p.
http://arxiv.org/abs/0908.4560
---------------------------------------------------------------
9024. ANALYSIS OF A STOCHASTIC PREDATOR-PREY MODEL WITH APPLICATIONS
TO INTRAHOST HIV GENETIC DIVERSITY
Sivan Leviyang
During an infection, HIV experiences strong selection by immune system T
cells. Recent experimental work has shown that MHC escape mutations
form an
important pathway for HIV to avoid such selection. In this paper, we
study a
model of MHC escape mutation. The model is a predator-prey model with
two prey,
composed of two HIV variants, and one predator, the immune system CD8
cells. We
assume that one HIV variant is visible to CD8 cells and one is not.
The model
takes the form of a system of stochastic differential equations.
Motivated by
well-known results concerning the short life-cycle of HIV intrahost,
we assume
that HIV population dynamics occur on a faster time scale then CD8
population
dynamics. This separation of time scales allows us to analyze our
model using
an asymptotic approach.
Using this model we study the impact of an MHC escape mutation on the
population dynamics and genetic evolution of the intrahost HIV
population. From
the perspective of population dynamics, we show that the competition
between
the visible and invisible HIV variants can reach steady states in
which either
a single variant exists or in which coexistence occurs depending on the
parameter regime. We show that in some parameter regimes the end state
of the
system is stochastic. From a genetics perspective, we study the impact
of the
population dynamics on the lineages of HIV samples taken after an escape
mutation occurs. We show that the lineages go through severe
bottlenecks and
that the lineage distribution can be characterized by a Kingman
coalescent.
http://arxiv.org/abs/0908.4569
---------------------------------------------------------------
9025. STABILITY OF A SPATIAL POLLING SYSTEM WITH GREEDY MYOPIC SERVICE
Lasse Leskel\"a and Falk Unger
This paper studies a spatial queueing system on a circle, polled at
random
locations by a myopic server that can only observe customers in a
bounded
neighborhood. The server operates according to a greedy policy, always
serving
the nearest customer in its neighborhood, and leaving the system
unchanged at
polling instants where the neighborhood is empty. This system is
modeled as a
measure-valued random process, which is shown to be positive recurrent
under a
natural stability condition that does not depend on the server's scan
range.
When the interpolling times are light-tailed, the stable system is
shown to be
geometrically ergodic. We also briefly discuss how the stationary mean
number
of customers behaves in light and heavy traffic.
http://arxiv.org/abs/0908.4585
---------------------------------------------------------------
9026. STRICT POSITIVITY OF THE DENSITY FOR NON-LINEAR SPATIALLY
HOMOGENEOUS SPDES
Eulalia Nualart
In this paper, we consider a system of $k$ second order non-linear
stochastic
differential equations with spatial dimension $d \geq 1$, driven by a
$k$-dimensional Gaussian noise, which is white in time and with some
spatially
homogeneous covariance. We prove existence, smoothness, and strict
positivity
of the density of the law of the solution of this system of equations,
on the
set where the diffusion matrix is invertible, under sufficient
conditions on
the fundamental solution $\Gamma$ of the deterministic equation. For
this, we
apply techniques of Malliavin calculus. We apply this result to the
case of the
stochastic heat equation in any space dimension and the the stochastic
wave
equation in dimension $d\in \{1,2,3\}$, with a spatial covariance
given by a
Riesz kernel. We then study the strict positivity of the density for
the case
of a single equation ($k=1$), and apply it to the stochastic heat
equation in
any space dimension, and the stochastic wave equation in dimension $d\in
\{1,2,3\}$, with a general spatial covariance.
http://arxiv.org/abs/0908.4587
---------------------------------------------------------------
9027. ON THE SPECTRAL DIMENSION OF CAUSAL TRIANGULATIONS
Bergfinnur Durhuus and Thordur Jonsson and John F. Wheater
We introduce an ensemble of infinite causal triangulations, called the
uniform infinite causal triangulation, and show that it is equivalent
to an
ensemble of infinite trees, the uniform infinite planar tree. It is
proved that
in both cases the Hausdorff dimension almost surely equals 2. The
infinite
causal triangulations are shown to be almost surely recurrent or,
equivalently,
their spectral dimension is almost surely less than or equal to 2. We
also
establish that for certain reduced versions of the infinite causal
triangulations the spectral dimension equals 2 both for the ensemble
average
and almost surely. The triangulation ensemble we consider is
equivalent to the
causal dynamical triangulation model of two-dimensional quantum
gravity and
therefore our results apply to that model.
http://arxiv.org/abs/0908.3643
---------------------------------------------------------------
9028. STOCHASTIC CAHN-HILLIARD EQUATION WITH DOUBLE SINGULAR
NONLINEARITIES AND TWO REFLECTIONS
Arnaud Debussche (IRMAR) and Ludovic Gouden\`ege (IRMAR)
We consider a stochastic partial differential equation with two
logarithmic
nonlinearities, with two reflections at 1 and -1 and with a constraint
of
conservation of the space average. The equation, driven by the
derivative in
space of a space-time white noise, contains a bi-Laplacian in the
drift. The
lack of the maximum principle for the bi-Laplacian generates
difficulties for
the classical penalization method, which uses a crucial monotonicity
property.
Being inspired by the works of Debussche, Gouden\`ege and Zambotti, we
obtain
existence and uniqueness of solution for initial conditions in the
interval
$(-1,1)$. Finally, we prove that the unique invariant measure is
ergodic, and
we give a result of exponential mixing.
http://arxiv.org/abs/0908.4295
-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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