[PAS] Probability Abstracts 112
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Wed Nov 4 02:43:17 CST 2009
Probability Abstracts 112
This document contains abstracts 9029-9332
from Sep-1-2009 to October-31-2009.
They have been mailed on Nov 4th, 2009.
9029. Multivariate Log-Concave Distributions as a Nearly Parametric
Model
Author(s): Dominic Schuhmacher and Andre Huesler and Lutz Duembgen
Abstract: 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
9030. SDEs driven by a time-changed L\'evy process and their
associated time-fractional order pseudo-differential equations
Author(s): Marjorie G. Hahn and Kei Kobayashi and Sabir Umarov
Abstract: 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
9031. Brownian and fractional Brownian stochastic currents via
Malliavin calculus
Author(s): Franco Flandoli and Ciprian Tudor (CES and SAMOS)
Abstract: 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
9032. A min-type stochastic fixed-point equation related to the
smoothing transformation
Author(s): Gerold Alsmeyer and Matthias Meiners
Abstract: 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
9033. Interlacement percolation on transient weighted graphs
Author(s): Augusto Teixeira
Abstract: 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
9034. A functional combinatorial central limit theorem
Author(s): A. D. Barbour and Svante Janson
Abstract: 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
9035. Stability Properties of Linear File-Sharing Networks
Author(s): L. Leskel\"a (MSA) and Philippe Robert (INRIA) and Florian
Simatos
Abstract: 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
9036. Semimartingale decomposition of convex functions of continuous
semimartingales by Brownian perturbation
Author(s): Nastasiya F Grinberg
Abstract: 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
9037. Majority dynamics on trees and the dynamic cavity method
Author(s): Yashodhan Kanoria and Andrea Montanari
Abstract: 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
9038. A strong log-concavity property for measures on Boolean algebras
Author(s): Jeff Kahn and Michael Neiman
Abstract: 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
9039. A Cut-off Phenomenon in Location Based Random Access Games with
Imperfect Information
Author(s): Hazer Inaltekin and Mung Chiang and H. Vincent Poor
Abstract: 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
9040. Self-Intersections of Random Geodesics on Negatively Curved
Surfaces
Author(s): Steven P. Lalley
Abstract: 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
9041. Reducing the Ising model to matchings
Author(s): Mark Huber (Claremont McKenna College) and Jenny Law (Duke
University)
Abstract: 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
9042. Zeros of a two-parameter random walk
Author(s): Davar Khoshnevisan and Pal Revesz
Abstract: 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
9043. Branching Random Walks in Space-Time Random Environment:
Survival Probability, Global and Local Growth Rates
Author(s): Francis Comets and Nobuo Yoshida
Abstract: 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
9044. Uniform estimates for metastable transition times in a coupled
bistable system
Author(s): Florent Barret (CMAP) and Anton Bovier (IAM) and Sylvie M
\'el\'eard (CMAP)
Abstract: 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
9045. Upper large deviations for maximal flows through a tilted cylinder
Author(s): Marie Theret
Abstract: 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
9046. Central Limit Theorems for Multicolor Urns with Dominated Colors
Author(s): 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)
Abstract: 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
9047. D\'eviations mod\'er\'ees de la distance chimique
Author(s): Olivier Garet (IECN) and R\'egine Marchand (IECN)
Abstract: 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
9048. Moderate deviations for the chemical distance in Bernoulli
percolation
Author(s): Olivier Garet (IECN) and R\'egine Marchand (IECN)
Abstract: 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
9049. On the preservation of Gibbsianness under symbol amalgamation
Author(s): Jean-Rene Chazottes and Edgardo Ugalde
Abstract: 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
9050. Poincar\'e inequality and exponential integrability of hitting
times for linear diffusions
Author(s): D. Loukianova and O. Loukianov and Sh. Song
Abstract: 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
9051. Boundary Harnack Inequality for alpha-harmonic functions on the
Sierpi\'nski triangle
Author(s): Kamil Kaleta and Mateusz Kwa\'snicki
Abstract: 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
9052. Duality and Intertwining for discrete Markov kernels: a relation
and examples
Author(s): Thierry Huillet (LPTM) and Servet Martinez
Abstract: 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
9053. Diffusion approximation for the components in critical
inhomogeneous random graphs of rank 1
Author(s): Tatyana S. Turova
Abstract: 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
9054. Differentiability of quadratic BSDE generated by continuous
martingales and hedging in incomplete markets
Author(s): Peter Imkeller and Anthony Reveillac and Anja Richter
Abstract: 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
9055. On the orthogonal component of BSDEs in a Markovian setting
Author(s): Anthony R\'eveillac
Abstract: 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
9056. A constructive approach to the Monge-Kantorovich problem for
chains of infinite order
Author(s): Antonio Galves and Nancy L. Garcia and Clementine Prieur
Abstract: 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
9057. Hsu-Robbins and Spitzer's theorems for the variations of
fractional Brownian motion
Author(s): Ciprian Tudor (CES and Samos)
Abstract: 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
9058. Convergence to L\'evy stable processes under strong mixing
conditions
Author(s): Marta Tyran-Kaminska
Abstract: 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
9059. An application to credit risk of a hybrid Monte Carlo-Optimal
quantization method
Author(s): Giorgia Callegaro and Abass Sagna (PMA)
Abstract: 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
9060. Perimeter and Area of the Convex Hull of N Planar Brownian Motions
Author(s): Julien Randon-Furling and Satya N. Majumdar and Alain Comtet
Abstract: 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 = \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
9061. Distributed Random Access Algorithm: Scheduling and Congesion
Control
Author(s): Libin Jiang and Devavrat Shah and Jinwoo Shin and Jean
Walrand
Abstract: 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
9062. Dynkin's isomorphism theorem and the stochastic heat equation
Author(s): Nathalie Eisenbaum and Mohammud Foondun and Davar
Khoshnevisan
Abstract: 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
9063. On the discretization of backward doubly stochastic differential
equations
Author(s): Omar Aboura (CES and Samos)
Abstract: 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
9064. Existence and uniqueness of solutions for Fokker-Planck
equations on Hilbert spaces
Author(s): Vladimir Bogachev and Giuseppe Da Prato and Michael Roeckner
Abstract: 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
9065. Limit distributions for large P\'olya urns
Author(s): Brigitte Chauvin and Nicolas Pouyanne and R\'eda Sahnoun
Abstract: 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
http://arxiv.org/abs/0907.1477
9066. Inhomogeneity and universality: off-critical behavior of
interfaces
Author(s): Pierre Nolin
Abstract: 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
9067. Les Probabilit\'es D\'efaillance comme Indicateurs de
Performance des Barri\`eres Techniques de S\'ecurit\'e ? Approche
Analytique
Author(s): Florent Brissaud (INERIS and UTT) and Brice Lanternier
(INERIS)
Abstract: 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
9068. A Remark on Zeros of Brownian Motion
Author(s): Weber Michel
Abstract: 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
9069. Symmetrization of L\'evy processes and applications
Author(s): Rodrigo Banuelos and Pedro J. Mendez-Hernandez
Abstract: 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
9070. Are fractional Brownian motions predictable?
Author(s): Adam Jakubowski
Abstract: 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
9071. Random walks on discrete cylinders with large bases and random
interlacements
Author(s): David Windisch
Abstract: 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
9072. Fluctuations of the nodal length of random spherical harmonics
Author(s): Igor Wigman
Abstract: 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
9073. Some almost sure results for unbounded functions of intermittent
maps and their associated Markov chains
Author(s): Jerome Dedecker (LSTA) and Sebastien Gouezel (IRMAR) and
Florence Merlevede (LAMA)
Abstract: 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
9074. Almost sure invariance principle for dynamical systems by
spectral methods
Author(s): Sebastien Gouezel (IRMAR)
Abstract: 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
9075. Forest fires on $\Z_+$ with ignition only at 0
Author(s): Stanislav Volkov
Abstract: 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
9076. Queueing with neighbours
Author(s): Vadim Shcherbakov and Stanislav Volkov
Abstract: 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
9077. Estimates on the speedup and slowdown for a diffusion in a
drifted brownian potential
Author(s): Gabriel Faraud
Abstract: 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
9078. Hidden Markov processes in the context of symbolic dynamics
Author(s): Mike Boyle (University of Maryland) and Karl Petersen
(University of North Carolina)
Abstract: 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
9079. The triangle and the open triangle
Author(s): Gady Kozma
Abstract: We show that for percolation on any transitive graph, the
triangle condition implies the open triangle condition.
http://arxiv.org/abs/0907.1959
9080. Lp-solution of backward doubly stochastic differential equations
Author(s): Auguste Aman (LMAI)
Abstract: 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
9081. On the Optimal Amount of Experimentation in Sequential Decision
Problems
Author(s): Dinah Rosenberg and Eilon Solan and Nicolas Vieille
Abstract: 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
9082. New rates for exponential approximation and the theorems of R
\'enyi and Yaglom
Author(s): Erol Pek\"oz and Adrian R\"ollin
Abstract: 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
9083. L$^{p}$-solution of reflected generalized BSDEs with non-
Lipschitz coefficients
Author(s): Auguste Aman (LMAI)
Abstract: 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
9084. Numerical scheme for backward doubly stochastic differential
equations
Author(s): Auguste Aman (LMAI)
Abstract: 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
9085. Homeomorphism of solutions to backward doubly SDEs and
applications
Author(s): Auguste Aman (LMAI)
Abstract: 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
9086. Reflected generalized backward doubly SDEs driven by L\'evy
processes and Applications
Author(s): Auguste Aman (LMAI)
Abstract: 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
9087. Stochastic 2D hydrodynamical systems: Support theorem
Author(s): Igor Chueshov and Annie Millet (SAMOS and Ces and Pma)
Abstract: 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
9088. Perfect simulation for stochastic chains with unbounded variable
length memory
Author(s): Alexsandro Gallo
Abstract: 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
9089. On the Domination of Random Walk on a Discrete Cylinder by
Random Interlacements
Author(s): Alain-Sol Sznitman
Abstract: 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
9090. A Path Guessing Game with Wagering
Author(s): Marcus Pendergrass
Abstract: 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
9091. On the philosophy of Cram\'er-Rao-Bhattacharya Inequalities in
Quantum Statistics
Author(s): K. R. Parthasarathy
Abstract: 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
9092. Optimal investment on finite horizon with random discrete order
flow in illiquid markets
Author(s): Paul Gassiat (PMA) and Huyen Pham (PMA and CREST) and Mihai
Sirbu
Abstract: 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
9093. A Shape Theorem for Riemannian First-Passage Percolation
Author(s): Tom LaGatta and Jan Wehr
Abstract: 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
9094. Heavy tail phenomenon and convergence to stable laws iterated
Lipschitz maps
Author(s): Mariusz Mirek
Abstract: 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
9095. Uniform Modulus of Continuity of Random Fields
Author(s): Yimin Xiao
Abstract: 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
9096. Spectral Analysis of Multi-dimensional Self-similar Markov
Processes
Author(s): N. Modarresi and S. Rezakhah
Abstract: 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
9097. Heat Kernel Upper Bounds on Long Range Percolation Clusters
Author(s): Nicholas Crawford and Allan Sly
Abstract: 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
9098. A graph-based equilibrium problem for the limiting distribution
of non-intersecting Brownian motions at low temperature
Author(s): Steven Delvaux and Arno B.J. Kuijlaars
Abstract: 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
9099. 3-Connected Cores In Random Planar Graphs
Author(s): Nikolaos Fountoulakis and Konstantinos Panagiotou
Abstract: 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
9100. On divergence form SPDEs with growing coefficients in $W^{1}_
{2}$ spaces without weights
Author(s): N.V. Krylov
Abstract: 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
9101. On the structure of Gaussian random variables
Author(s): Ciprian Tudor (CES and Samos)
Abstract: 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
9102. Weak convergence for the stochastic heat equation driven by
Gaussian white noise
Author(s): Xavier Bardina and Maria Jolis and Lluis Quer-Sardanyons
Abstract: 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
9103. The diversity of a distributed genome in bacterial populations
Author(s): F. Baumdicker and W. R. Hess and P. Pfaffelhuber
Abstract: 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
9104. Extremal solutions for stochastic equations indexed by negative
integers and taking values in compact groups
Author(s): Takao Hirayama and Kouji Yano
Abstract: 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
9105. On a zero-one law for the norm process of transient random walk
Author(s): Ayako Matsumoto and Kouji Yano
Abstract: 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
9106. Local limit of packable graphs
Author(s): Itai Benjamini and Nicolas Curien
Abstract: 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
9107. A CLT for the third integrated moment of Brownian local time
increments
Author(s): Jay Rosen
Abstract: 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
9108. Stochastic Taylor expansions and heat kernel asymptotics
Author(s): Fabrice Baudoin
Abstract: 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
9109. Explicit solutions of G-heat equation with a class of initial
conditions by G-Brownian motion
Author(s): Mingshang Hu
Abstract: 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
9110. Generalized backward doubly stochastic differential equations
driven by L\'evy processes with non-Lipschitz coefficients
Author(s): Auguste Aman (LMAI) and Jean Marc Owo (LMAI)
Abstract: 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
9111. Sharpness of the percolation transition in the two-dimensional
contact process
Author(s): Jacob van den Berg
Abstract: 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
9112. Conditional limit theorems for ordered random walks
Author(s): D. Denisov and V. Wachtel
Abstract: 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
9113. On Sojourn Times in the Finite Capacity $M/M/1$ Queue with
Processor Sharing
Author(s): Qiang Zhen and Charles Knessl
Abstract: 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
9114. Correlation and Brascamp-Lieb inequalities for Markov semigroups
Author(s): F. Barthe and D. Cordero-Erausquin and M. Ledoux and B.
Maurey
Abstract: 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
9115. The geometry of Euclidean convolution inequalities and entropy
Author(s): Dario Cordero-Erausquin and Michel Ledoux
Abstract: 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
9116. Asymptotic Expansions for the Conditional Sojourn Time
Distribution in the $M/M/1$-PS Queue
Author(s): Qiang Zhen and Charles Knessl
Abstract: 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
9117. Weak approximation of fractional SDES: The Donsker setting
Author(s): Xavier Bardina and Samy Tindel and Carles Rovira
Abstract: 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
9118. Bootstrap percolation in high dimensions
Author(s): Jozsef Balogh and Bela Bollobas and Robert Morris
Abstract: 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
9119. The Maxwell-Boltzmann Distribution is not the Equilibrium on a
Hyperboloid
Author(s): S. G. Rajeev
Abstract: 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
9120. From a dichotomy for images to Haagerup's inequality
Author(s): Iosif Pinelis
Abstract: 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
9121. Large deviations for flows of interacting Brownian motions
Author(s): A.A.Dorogovtsev and O.V.Ostapenko
Abstract: 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
9122. Scaling limits of random planar maps with large faces
Author(s): Jean-Fran\c{c}ois Le Gall (LM-Orsay) and Gr\'egory Miermont
(DMA)
Abstract: 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
9123. q-Exchangeability via quasi-invariance
Author(s): Alexander Gnedin and Grigori Olshanski
Abstract: 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
9124. High level excursion set geometry for non-Gaussian infinitely
divisible random fields
Author(s): Robert J Adler and Gennady Samorodnitsky and Jonathan E
Taylor
Abstract: 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
9125. Disorder chaos and multiple valleys in spin glasses
Author(s): Sourav Chatterjee
Abstract: 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
9126. Spin Needlets Spectral Estimation
Author(s): Daryl Geller and Xiaohong Lan and Domenico Marinucci
Abstract: 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
9127. A bijection theorem for domino tiling with diagonal impurities
Author(s): Fumihiko Nakano and Taizo Sadahiro
Abstract: 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
9128. Optimal Execution Problem with Market Impact
Author(s): Takashi Kato
Abstract: 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
9129. De Finetti theorems for easy quantum groups
Author(s): Teodor Banica and Stephen Curran and Roland Speicher
Abstract: 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
9130. SRB Measures For Certain Markov Processes
Author(s): Wael Bahsoun and Pawel Gora
Abstract: 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
9131. Optimal Execution Problem with Market Impact
Author(s): Takashi Kato
Abstract: 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
9132. Fractional Normal Inverse Gaussian Process
Author(s): Arun Kumar and P. Vellaisamy
Abstract: 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
9133. Flow of diffeomorphisms for SDEs with unbounded H\"older
continuous drift
Author(s): F. Flandoli and M. Gubinelli and E. Priola
Abstract: 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
9134. Systems of one-dimensional random walks in a common random
environment
Author(s): Jonathon Peterson
Abstract: 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
9135. A Spectral Analysis of the Sequence of Firing Phases in
Stochastic Integrate-and-Fire Oscillators
Author(s): Peter Baxendale and John Mayberry
Abstract: 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
9136. Evolution in predator-prey systems
Author(s): Rick Durrett and John Mayberry
Abstract: 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
9137. High Moments of Large Wigner Random MAtrices and Asymptotic
Properties of the Spectral Norm
Author(s): O. Khorunzhiy
Abstract: 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
9138. On the One Dimensional Critical "Learning from Neighbours" Model
Author(s): Antar Bandyopadhyay and Rahul Roy and Anish Sarkar
Abstract: 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
9139. On Hele-Shaw problems arising as scaling limits
Author(s): Pavel Etingof
Abstract: 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
9140. Wright-Fisher Diffusion in One Dimension
Author(s): Charles L. Epstein and Rafe Mazzeo
Abstract: 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
9141. Hard Core entropy: lower bounds
Author(s): Kari Eloranta
Abstract: 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
9142. Binomial Approximations for Barrier Options of Israeli Style
Author(s): Yan Dolinsky and Yuri Kifer
Abstract: 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
9143. An Introduction to Stochastic PDEs
Author(s): Martin Hairer
Abstract: 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
9144. Localization for a Class of Linear Systems
Author(s): Yukio Nagahata and Nobuo Yoshida
Abstract: 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
9145. The rank of diluted random graphs
Author(s): Charles Bordenave and Marc Lelarge
Abstract: 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
9146. Hausdorff measure of arcs and Brownian motion on Brownian
spatial trees
Author(s): David A. Croydon
Abstract: 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
9147. Scaling limits for critical inhomogeneous random graphs with
finite third moments
Author(s): Shankar Bhamidi and Remco van der Hofstad and Johan van
Leeuwaarden
Abstract: 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
9148. Time-reversal and elliptic boundary value problems
Author(s): Zhen-Qing Chen and Tusheng Zhang
Abstract: 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
9149. Notes on Using Control Variates for Estimation with Reversible
MCMC Samplers
Author(s): Ioannis Kontoyiannis and Petros Dellaportas
Abstract: 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
9150. The scaling window for a random graph with a given degree sequence
Author(s): Hamed Hatami and Michael Molloy
Abstract: 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
9151. Dense packing on uniform lattices
Author(s): Kari Eloranta
Abstract: 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
9152. On the Distribution of a Second Class Particle in the Asymmetric
Simple Exclusion Process
Author(s): Craig A. Tracy and Harold Widom
Abstract: 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
9153. Stein's Method of Exchangeable Pairs with Application to the
Curie-Weiss Model
Author(s): Sourav Chatterjee and Qi-Man Shao
Abstract: 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
9154. Large Deviation in Harnack type Dirichlet spaces
Author(s): Ann-Kathrin Jarecki
Abstract: 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
9155. Upper Bound for Large Deviations of Reversible Diffusion Processes
Author(s): Ann-Kathrin Jarecki
Abstract: 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
9156. Bounding relative entropy by the relative entropy of local
specifications in product spaces
Author(s): Katalin Marton
Abstract: 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
9157. On Markov chains induced by partitioned transition probability
matrices
Author(s): Thomas Kaijser
Abstract: 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
9158. Return probabilities of random walks among polynomial lower tail
random conductances
Author(s): Omar Boukhadra
Abstract: 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
9159. Recurrence and transience of branching random walks are
dynamically stable
Author(s): Sebastian M\"uller
Abstract: 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
9160. The t-Martin boundary of reflected random walks on a half-space
Author(s): Irina Ignatiouk-Robert
Abstract: 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
9161. Invariant random fields in vector bundles and application to
cosmology
Author(s): Anatoliy Malyarenko
Abstract: 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
9162. Disjoint Hamilton cycles in the random geometric graph
Author(s): Xavier P\'erez-Gim\'enez and Nicholas C. Wormald
Abstract: 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
9163. Limit theorems for vertex-reinforced jump processes on regular
trees
Author(s): Andrea Collevecchio
Abstract: 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
9164. Stochastic Flows of SDEs with Irregular Drifts and Stochastic
Transport Equations
Author(s): Xicheng Zhang
Abstract: 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
9165. The Monotone Cumulants
Author(s): Takahiro Hasebe and Hayato Saigo
Abstract: 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
9166. Invariant Measures and Decay of Correlations of a Class of
Ergodic Probabilistic Cellular Automata
Author(s): Cristian Coletti (CMCC) and Pierre Tisseur (CMCC)
Abstract: 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
9167. Bayesian estimate of the zero-density frequency of a Cs fountain
Author(s): D Calonico and F Levi and L Lorini and G Mana
Abstract: 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
9168. Dirichlet polynomials: some old and recent results, and their
interplay in number theory
Author(s): Michel Weber
Abstract: 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
9169. An Analogue of the L\'Evy-Cram\'Er Theorem for Multi-Dimensional
Rayleigh Distributions
Author(s): Thu Nguyen
Abstract: 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
9170. The weak coupling limit of disordered copolymer models
Author(s): Francesco Caravenna and Giambattista Giacomin
Abstract: 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
9171. Law of large numbers for the maximal flow through tilted
cylinders in two-dimensional first passage percolation
Author(s): Rapha\"el Rossignol and Marie Th\'eret
Abstract: 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
9172. Standard deviation of the longest common subsequence
Author(s): J\"uri Lember and Heinrich Matzinger
Abstract: 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
9173. Brunet-Derrida particle systems, free boundary problems and
Wiener-Hopf equations
Author(s): Rick Durrett and Daniel Remenik
Abstract: 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
9174. On ASEP with Step Bernoulli Initial Condition
Author(s): Craig A. Tracy and Harold Widom
Abstract: 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
9175. On infinitely cohomologous to zero observables
Author(s): Amanda de Lima and Daniel Smania
Abstract: 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
9176. A Discussion on Mean Excess Plots
Author(s): Souvik Ghosh and Sidney I Resnick
Abstract: 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
9177. A historical law of large numbers for the Marcus Lushnikov process
Author(s): St\'ephanie Jacquot
Abstract: 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
9178. A metric analysis of critical Hamilton--Jacobi equations in the
stationary ergodic setting
Author(s): Andrea Davini and Antonio Siconolfi
Abstract: 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
9179. Weak KAM Theory topics in the stationary ergodic setting
Author(s): Andrea Davini and Antonio Siconolfi
Abstract: 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
9180. Profiles of permutations
Author(s): Michael Lugo
Abstract: 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
9181. Self-interacting diffusions IV: Rate of convergence
Author(s): Michel Benaim (UNINE) and Olivier Raimond (MODAL'X)
Abstract: 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
9182. Upper large deviations for the maximal flow through a domain of $
\mathbb{R}^d$ in first passage percolation
Author(s): Rapha\"el Cerf and Marie Th\'eret
Abstract: 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
9183. Lower large deviations for the maximal flow through a domain of $
\mathbb{R}^d$ in first passage percolation
Author(s): Rapha\"el Cerf and Marie Th\'eret
Abstract: 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
9184. Law of large numbers for the maximal flow through a domain of $
\mathbb{R}^d$ in first passage percolation
Author(s): Rapha\"el Cerf and Marie Th\'eret
Abstract: 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
9185. Monotonicity properties of the asymptotic relative efficiency
between common correlation statistics in the bivariate normal model
Author(s): Raymond Molzon and Iosif Pinelis
Abstract: 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
9186. Conditionally monotone independence
Author(s): Takahiro Hasebe
Abstract: 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
9187. Loss of memory of random functions of Markov chains and Lyapunov
exponents
Author(s): Pierre Collet and Florencia Leonardi
Abstract: 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
9188. Scaling limits of anisotropic Hastings-Levitov clusters
Author(s): Fredrik Johansson and Alan Sola and Amanda Turner
Abstract: 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
9189. A stochastic min-driven coalescence process and its
hydrodynamical limit
Author(s): Anne-Laure Basdevant (IMT) and Philippe Laurencot (IMT) and
James R. Norris (DPMMS), Clement Rau (IMT)
Abstract: 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
9190. Sampling Conditioned Hypoelliptic Diffusions
Author(s): Martin Hairer and Andrew M. Stuart and Jochen Voss
Abstract: 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
9191. On the Speed of Spread for Fractional Reaction-Diffusion Equations
Author(s): Hans Engler
Abstract: 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
9192. A strong pair correlation bound implies the CLT for Sinai
Billiards
Author(s): Mikko Stenlund
Abstract: 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
9193. Approximating Eigenvectors by Subsampling
Author(s): Noureddine El Karoui and Alexandre d'Aspremont
Abstract: 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
9194. On the Role of Sparsity in Compressed Sensing and Random Matrix
Theory
Author(s): Roman Vershynin
Abstract: 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
9195. Layering and wetting transitions for an SOS interface
Author(s): Kenneth S. Alexander and Fran\c{c}ois Dunlop and Salvador
Miracle-Sol\'e
Abstract: 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
9196. Universal Gaussian fluctuations of non-Hermitian matrix ensembles
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X)
Abstract: 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
9197. Optimal Transport and Tessellation
Author(s): Martin Huesmann
Abstract: 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
9198. The Statistical Mechanics of Stretched Polymers
Author(s): Dmitry Ioffe and Yvan Velenik
Abstract: 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
9199. On linear evolution equations with cylindrical L\'evy noise
Author(s): Enrico Priola and Jerzy Zabczyk
Abstract: 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
9200. On the short time asymptotic of the stochastic Allen-Cahn equation
Author(s): Hendrik Weber
Abstract: 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
9201. Upper and Lower Bounds in Exponential Tauberian Theorems
Author(s): Jochen Voss
Abstract: 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
9202. Exact solution of a two-type branching process: Clone size
distribution in cell division kinetics
Author(s): Tibor Antal and P. L. Krapivsky
Abstract: 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
9203. Recurrence and ergodicity of random walks on linear groups and
on homogeneous spaces
Author(s): Y. Guivarc'h and C. R. E. Raja
Abstract: 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
9204. A general strong law of large numbers for additive arithmetic
functions
Author(s): Istvan Berkes and Michel Weber
Abstract: 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
9205. Optimal scalings for local Metropolis--Hastings chains on
nonproduct targets in high dimensions
Author(s): Alexandros Beskos and Gareth Roberts and Andrew Stuart
Abstract: 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
9206. Asymptotic optimality of isoperimetric constants with respect to
$L^{2}(\pi)$-spectral gaps
Author(s): Achim Wuebker
Abstract: 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
9207. $L^{2}$-spectral gaps, weak-reversible and very weak-reversible
Markov chains
Author(s): Achim Wuebker and Zakhar Kabluchko
Abstract: 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
9208. $L^{2}$-spectral gaps for time discrete reversible Markov chains
Author(s): Achim Wuebker
Abstract: 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
9209. Robust mean-variance hedging in the single period model
Author(s): R. Tevzadze and T. Uzunashvili
Abstract: 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
9210. Efficient importance sampling for binary contingency tables
Author(s): Jose H. Blanchet
Abstract: 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
9211. A probabilistic study of neural complexity
Author(s): Jerome Buzzi (LM-Orsay) and Lorenzo Zambotti (PMA)
Abstract: 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
9212. Selling a stock at the ultimate maximum
Author(s): Jacques du Toit and Goran Peskir
Abstract: 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
9213. An operator approach for Markov chain weak approximations with
an application to infinite activity L\'{e}vy driven SDEs
Author(s): Hideyuki Tanaka and Arturo Kohatsu-Higa
Abstract: 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
9214. Asymptotic normality of plug-in level set estimates
Author(s): David M. Mason and Wolfgang Polonik
Abstract: 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
9215. Gaussian perturbations of circle maps: A spectral approach
Author(s): John Mayberry
Abstract: 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
9216. A continuous analogue of the invariance principle and its almost
sure version
Author(s): E.E. Permyakova
Abstract: 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
9217. Functional limit theorems for Levy processes and their almost-
sure versions
Author(s): E.E. Permyakova
Abstract: 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
9218. Total progeny in killed branching random walk
Author(s): Louigi Addario-Berry and Nicolas Broutin
Abstract: 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
9219. Asymptotic Behavior of the Finite-Size Magnetization as a
Function of the Speed of Approach to Criticality
Author(s): Richard S. Ellis and Jonathan Machta and Peter Tak-Hun Otto
Abstract: 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
9220. On the uniqueness of classical solutions of Cauchy problems
Author(s): Erhan Bayraktar and Hao Xing
Abstract: 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
9221. Cram\'{e}r Type Moderate Deviation for the Maximum of the
Periodogram with Application to Simultaneous Tests in Gene Expression
Time Series
Author(s): Weidong Liu and Qi Man Shao
Abstract: 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
9222. Absorbing-State Phase Transition for Stochastic Sandpiles and
Activated Random Walks
Author(s): Leonardo T. Rolla and Vladas Sidoravicius
Abstract: 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
9223. A Ciesielski-Taylor type identity for positive self-similar
Markov processes
Author(s): A.E. Kyprianou and P. Patie
Abstract: 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
9224. A sharp analysis of the mixing time for random walk on rooted
trees
Author(s): Jason Fulman
Abstract: 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
9225. Sharp Heat Kernel Estimates for Relativistic Stable Processes in
Open Sets
Author(s): Zhen-Qing Chen and Panki Kim and Renming Song
Abstract: 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
9226. The two-type continuum Richardson model: Non-dependence of the
survival of both types on the initial configuration
Author(s): Sebastian Carstens and Thomas Richthammer
Abstract: 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
9227. Boundary Harnack principle for $\Delta + \Delta^{\alpha/2}$
Author(s): Zhen-Qing Chen and Panki Kim and Renming Song and Zoran
Vondra\v{c}ek
Abstract: 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
9228. Conformal loop ensembles and the stress-energy tensor. II.
Construction of the stress-energy tensor
Author(s): Benjamin Doyon
Abstract: 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
9229. The uniqueness of symmetrizing measure and linear diffusions
Author(s): Xing Fang and Jiangang Ying and Minzhi Zhao
Abstract: 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
9230. Perfect simulation of Vervaat perpetuities
Author(s): James Allen Fill and Mark Huber
Abstract: 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
9231. Static large deviations of boundary driven exclusion processes
Author(s): Jonathan Farfan
Abstract: 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
9232. Lack of strong completeness for stochastic flows
Author(s): Xue-Mei Li and Michael Scheutzow
Abstract: 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
9233. Stein's method for dependent random variables occurring in
Statistical Mechanics
Author(s): Peter Eichelsbacher and Matthias L\"owe
Abstract: 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
9234. Replica Symmetry and Combinatorial Optimization
Author(s): Johan W\"astlund
Abstract: 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
9235. High order discretization schemes for stochastic volatility models
Author(s): Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
Abstract: 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
9236. Filtering equations for partially observable diffusion processes
with Lipschitz continuous coefficients
Author(s): N.V. Krylov
Abstract: 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
9237. A Characterization Theorem for the Distribution of a Continuous
Local Martingale and Related Limit Theorems
Author(s): Andriy Yurachkivsky
Abstract: 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
9238. An application of disc packing to statistical mechanics
Author(s): Matthew Kahle
Abstract: 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
9239. A comprehensive connection between the basic results and
properties derived from two kinds of topologies of a random locally
convex module
Author(s): Tiexin Guo
Abstract: 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
9240. Static large deviations of boundary driven exclusion processes
Author(s): Jonathan Farfan
Abstract: 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
9241. Lack of strong completeness for stochastic flows
Author(s): Xue-Mei Li and Michael Scheutzow
Abstract: 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
9242. Stein's method for dependent random variables occurring in
Statistical Mechanics
Author(s): Peter Eichelsbacher and Matthias L\"owe
Abstract: 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
9243. Replica Symmetry and Combinatorial Optimization
Author(s): Johan W\"astlund
Abstract: 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
9244. High order discretization schemes for stochastic volatility models
Author(s): Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
Abstract: 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
9245. Filtering equations for partially observable diffusion processes
with Lipschitz continuous coefficients
Author(s): N.V. Krylov
Abstract: 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
9246. A Characterization Theorem for the Distribution of a Continuous
Local Martingale and Related Limit Theorems
Author(s): Andriy Yurachkivsky
Abstract: 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
9247. An application of disc packing to statistical mechanics
Author(s): Matthew Kahle
Abstract: 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
9248. A comprehensive connection between the basic results and
properties derived from two kinds of topologies of a random locally
convex module
Author(s): Tiexin Guo
Abstract: 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
9249. Random matrices: Universality of local eigenvalue statistics up
to the edge
Author(s): Terence Tao and Van Vu
Abstract: 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
9250. Optimal co-adapted coupling for a random walk on the hyper-
complete-grap
Author(s): Stephen B. Connor
Abstract: 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
9251. Reconstruction on Trees: Exponential Moment Bounds for Linear
Estimators
Author(s): Yuval Peres and Sebastien Roch
Abstract: 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
9252. Sequence-Length Requirement of Distance-Based Phylogeny
Reconstruction: Breaking the Polynomial Barrier
Author(s): Sebastien Roch
Abstract: 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
9253. Sharp approximation for density dependent Markov chains
Author(s): Kamil Szczegot
Abstract: 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
9254. A Sharp Estimate for Divisors of Bernoulli Sums
Author(s): Michel Weber
Abstract: 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
9255. Simple Error Scattering Model for improved Information
Reconciliation
Author(s): Stefan Rass
Abstract: 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
9256. Probabilistic model associated with the pressureless gas dynamics
Author(s): Sergio Albeverio and Anastasia Korshunova and Olga Rozanova
Abstract: 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
9257. The Mahonian probability distribution on words is asymptotically
normal
Author(s): E. Rodney Canfield and Svante Janson and and Doron Zeilberger
Abstract: 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
9258. Random matrices: Universality of local eigenvalue statistics up
to the edge
Author(s): Terence Tao and Van Vu
Abstract: 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
9259. Optimal co-adapted coupling for a random walk on the hyper-
complete-grap
Author(s): Stephen B. Connor
Abstract: 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
9260. Reconstruction on Trees: Exponential Moment Bounds for Linear
Estimators
Author(s): Yuval Peres and Sebastien Roch
Abstract: 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
9261. Sequence-Length Requirement of Distance-Based Phylogeny
Reconstruction: Breaking the Polynomial Barrier
Author(s): Sebastien Roch
Abstract: 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
9262. Sharp approximation for density dependent Markov chains
Author(s): Kamil Szczegot
Abstract: 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
9263. A Sharp Estimate for Divisors of Bernoulli Sums
Author(s): Michel Weber
Abstract: 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
9264. Simple Error Scattering Model for improved Information
Reconciliation
Author(s): Stefan Rass
Abstract: 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
9265. Probabilistic model associated with the pressureless gas dynamics
Author(s): Sergio Albeverio and Anastasia Korshunova and Olga Rozanova
Abstract: 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
9266. The Mahonian probability distribution on words is asymptotically
normal
Author(s): E. Rodney Canfield and Svante Janson and and Doron Zeilberger
Abstract: 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
9267. High order discretization schemes for stochastic volatility models
Author(s): Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
Abstract: 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
9268. High order discretization schemes for stochastic volatility models
Author(s): Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
Abstract: 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
9269. On the Copula for multivariate Extreme Value distributions
Author(s): Glauco Valle and Marco Aurelio Sanfins
Abstract: 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
9270. Simulation reductions for the Ising model
Author(s): Mark L. Huber
Abstract: 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
9271. Connectivity Bounds for the Vacant Set of Random Interlacements
Author(s): Vladas Sidoravicius and Alain-Sol Sznitman
Abstract: 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
9272. Random permutations with cycle weights
Author(s): Volker Betz and Daniel Ueltschi and Yvan Velenik
Abstract: 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
9273. The tree length of an evolving coalescent
Author(s): Peter Pfaffelhuber and Anton Wakolbinger and Heinz Weisshaupt
Abstract: 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
9274. Stochastic integral representation of the $L^{2}$ modulus of
Brownian local time and a central limit theorem
Author(s): Yaozhong Hu and David Nualart
Abstract: 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
9275. Environmental Noise Variability in Population Dynamics Matrix
Models
Author(s): Michel De Lara (CERMICS)
Abstract: 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
9276. A Backward Particle Interpretation of Feynman-Kac Formulae
Author(s): Pierre Del Moral and Arnaud Doucet and Sumeetpal S. Singh
Abstract: 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
9277. Threshold graph limits and random threshold graphs
Author(s): Persi Diaconis and Susan Holmes and Svante Janson
Abstract: 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
9278. Phase Transition for the Mixing Time of the Glauber Dynamics for
Coloring Regular Trees
Author(s): Prasad Tetali and Juan C. Vera and Eric Vigoda and Linji Yang
Abstract: 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
9279. Stochastic Partial Differential Equations with Unbounded and
Degenerate Coefficients
Author(s): Xicheng Zhang
Abstract: 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
9280. Probabilistic representation for solutions of an irregular
porous media type equation: the degenerate case
Author(s): Viorel Barbu and Michael Roeckner (SFB 701) and Francesco
Russo (LAGA)
Abstract: 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
9281. Sharp interface limit for invariant measures of a stochastic
Allen-Cahn equation
Author(s): Hendrik Weber
Abstract: 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
9282. Hydrodynamic limit of move-to-front rules and search cost
probabilities
Author(s): Kumiko Hattori and Tetsuya Hattori
Abstract: 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
9283. Stochastic Evolutions of Point Processes
Author(s): Philippe Robert
Abstract: 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
9284. Reflected Brownian motion in Weyl chambers
Author(s): Nizar Demni
Abstract: 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
9285. A zero-one law for linear transformations of Levy noise
Author(s): Steven N. Evans
Abstract: 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
9286. Finite-time blowup and existence of global positive solutions of
a semi-linear SPDE
Author(s): Marco Dozzi (IECN) and Jos\'e Alfredo Lopez
Abstract: 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
9287. Limit theorems for random processes with random time substitution
Author(s): Permyakova Elena
Abstract: 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
9288. Poisson Splitting by Factors
Author(s): Alexander E. Holroyd and Russell Lyons and and Terry Soo
Abstract: 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
9289. A rule of thumb for riffle shuffling
Author(s): Sami Assaf and Persi Diaconis and K. Soundararajan
Abstract: 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
9290. Optimal transportation and monotonic quantities on evolving
manifolds
Author(s): Hong Huang
Abstract: 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
9291. Rank-based attachment leads to power law graphs
Author(s): Jeannette Janssen and Pawel Pralat
Abstract: 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
9292. Notes on Feige's gumball machines problem
Author(s): John H. Elton
Abstract: 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
9293. Limit theorems for projections of random walk on a hypersphere
Author(s): Max Skipper
Abstract: 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
9294. Can an infinite product of nonnegative matrices be expressed in
terms of infinite products of stochastic ones?
Author(s): Alain Thomas (LATP)
Abstract: 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
9295. Critical random graphs: limiting constructions and
distributional properties
Author(s): L. Addario-Berry and N. Broutin and C. Goldschmidt
Abstract: 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
9296. Harnack Inequalities and Applications for Multivalued Stochastic
Evolution Equations
Author(s): Shun-Xiang Ouyang
Abstract: 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
9297. Applications of Weak Convergence for Hedging of American and
Game Options
Author(s): Yan Dolinsky
Abstract: 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
9298. On the minimal penalty for Markov order estimation
Author(s): Ramon van Handel
Abstract: 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
9299. Zero-one laws for connectivity in random key graphs
Author(s): Osman Yagan and Armand M. Makowski
Abstract: 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
9300. Randomized Scheduling Algorithm for Queueing Networks
Author(s): Devavrat Shah and Jinwoo Shin
Abstract: 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
9301. Asymptotic regimes for the partition into colonies of a
branching process with emigration
Author(s): Jean Bertoin (PMA and Dma)
Abstract: 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
9302. On the absolute continuity of multidimensional Ornstein-
Uhlenbeck processes
Author(s): Thomas Simon (LPP)
Abstract: 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
9303. Extremal Subgraphs of Random Graphs: an Extended Version
Author(s): Graham Brightwell and Konstantinos Panagiotou and Angelika
Steger
Abstract: 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
9304. Mixing time of near-critical random graphs
Author(s): Jian Ding and Eyal Lubetzky and Yuval Peres
Abstract: 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
9305. Utility Optimization in Congested Queueing Networks
Author(s): Neil Stuart Walton
Abstract: 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
9306. Distributed Averaging via Lifted Markov Chains
Author(s): Kyomin Jung and Devavrat Shah and Jinwoo Shin
Abstract: 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
9307. Hydrodynamic limit of the exclusion process in inhomogeneous media
Author(s): Milton Jara
Abstract: 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
9308. Contact process in a wedge
Author(s): J. Theodore Cox and Nevena Maric and Rinaldo B. Schinazi
Abstract: 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
9309. Kolmogorov equation associated to the stochastic reflection
problem on a smooth convex set of a Hilbert space
Author(s): Viorel Barbu and Giuseppe Da Prato and Luciano Tubaro
Abstract: 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
9310. The survival of large dimensional threshold contact processes
Author(s): Thomas Mountford and Roberto H. Schonmann
Abstract: 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
9311. On the extendibility of partially and Markov exchangeable binary
sequences
Author(s): Davide Di Cecco
Abstract: 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
9312. Asymptotic properties of the columns in the products of
nonnegative matrices
Author(s): \'Eric Olivier (LATP) and Alain Thomas (LATP)
Abstract: 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
9313. On the inverse first-passage-time problem for a Wiener process
Author(s): Cristina Zucca and Laura Sacerdote
Abstract: 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
9314. Extremal shot noises, heavy tails and max-stable random fields
Author(s): Cl\'ement Dombry (LMA)
Abstract: 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
9315. Stochastic completeness and volume growth
Author(s): Christian Baer and G. Pacelli Bessa
Abstract: 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
9316. Matrix factorization identity for almost semi-continuous
processes on a Markov chain
Author(s): D.V. Gusak and E.V. Karnaukh
Abstract: 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
9317. Limit laws of transient excited random walks on integers
Author(s): Elena Kosygina and Thomas Mountford
Abstract: 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
9318. Poisson Dirichlet$(\alpha,\theta)$-Bridge Equations and
Coagulation-Fragmentation Duality
Author(s): Lancelot F. James
Abstract: 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
9319. Convergence of Numerical Time-Averaging and Stationary Measures
via Poisson Equations
Author(s): Jonathan C. Mattingly and Andrew M. Stuart and M.V. Tretyakov
Abstract: 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
9320. Time averages, recurrence and transience in the stochastic
replicator dynamics
Author(s): Josef Hofbauer and Lorens A. Imhof
Abstract: 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
9321. Bubbles, convexity and the Black--Scholes equation
Author(s): Erik Ekstr\"{o}m and Johan Tysk
Abstract: 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
9322. On convergence to stationarity of fractional Brownian storage
Author(s): Michel Mandjes and Ilkka Norros and Peter Glynn
Abstract: 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
9323. Random recurrence equations and ruin in a Markov-dependent
stochastic economic environment
Author(s): Jeffrey F. Collamore
Abstract: 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
9324. Non-Markov property of certain eigenvalue processes analogous to
Dyson's model
Author(s): Ryoki Fukushima and Atsushi Tanida and Kouji Yano
Abstract: 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
9325. Optimal reinsurance/investment problems for general insurance
models
Author(s): Yuping Liu and Jin Ma
Abstract: 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
9326. Recursive estimation of time-average variance constants
Author(s): Wei Biao Wu
Abstract: 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
9327. Asymptotic behavior of unstable INAR(p) processes
Author(s): Matyas Barczy and Marton Ispany and Gyula Pap
Abstract: 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
9328. Analysis of a Stochastic Predator-Prey Model with Applications
to Intrahost HIV Genetic Diversity
Author(s): Sivan Leviyang
Abstract: 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
9329. Stability of a spatial polling system with greedy myopic service
Author(s): Lasse Leskel\"a and Falk Unger
Abstract: 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
9330. Strict positivity of the density for non-linear spatially
homogeneous SPDEs
Author(s): Eulalia Nualart
Abstract: 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
9331. On the spectral dimension of causal triangulations
Author(s): Bergfinnur Durhuus and Thordur Jonsson and John F. Wheater
Abstract: 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
9332. Stochastic Cahn-Hilliard equation with double singular
nonlinearities and two reflections
Author(s): Arnaud Debussche (IRMAR) and Ludovic Gouden\`ege (IRMAR)
Abstract: 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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