|
|
|
|
 |
|
|
|
|
|
|
|
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
|
|
|
|
|
|
