[PAS] Probability Abstracts 109

[Probability Abstract Service]

Probability Abstracts 109
This document contains abstracts 8213-8462
from Mar-1-2009 to April-30-2009.
They have been mailed on May 11, 2009.


This letter can be also found on line at
http://pas.imstat.org/Letters/letter_109.shtml


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8213. Martin boundary of a killed random walk on a quadrant
Author(s): Irina Ignatiouk-Robert and Christophe Loree

Abstract: A complete representation of the Martin boundary of killed  
random walks on the quadrant NxN is obtained. It is proved that the  
corresponding full Martin compactification of the quadrant NxN is  
homeomorphic to the closure of the set {w =z/(1+|z|): z in NxN}$ in  
R2. The method is based on a ratio limit theorem for local processes  
and large deviation techniques.

http://arxiv.org/abs/0903.0070


8214. Poisson asymptotics for random projections of points on a high- 
dimensional sphere
Author(s): Itai Benjamini and Oded Schramm and and Sasha Sodin

Abstract: Project a collection of points on the high-dimensional  
sphere onto a random direction. If most of the points are sufficiently  
far from one another in an appropriate sense, the projection is  
locally close in distribution to the Poisson point process.

http://arxiv.org/abs/0903.0107


8215. Large dimensional random k circulants
Author(s): Arup Bose and Joydip Mitra and Arnab Sen

Abstract: Circulant matrices with general shift by k places have been  
studied in the literature and formula for their eigenvalues is known.  
We first reestablish this formula and some further properties of these  
eigenvalues in a manner suitable for our use. We then consider random  
k=k(n) circulants A_{k,n} with $n \to \infty$ and whose input sequence  
{a_i} is independent with mean zero and variance one and $\sup_n  
n^{-1}\sum_{i=1}^n E|a_i|^{2+\delta}< \infty$ for some $\delta > 0$.  
Under suitable restrictions on {k(n)},we show that the limiting  
spectral distribution (LSD) of the empirical distribution of suitably  
scaled eigenvalues exists and identify the limits. As examples, (i) if  
k^g = -1+ s n where $g \ge 1 $ fixed and $s=o(n^{1/3})$, then the LSD  
is $U_1(\prod_{i=1}^g E_i)^{1/2g}$ where E_i are i.i.d. Exp(1) and U_1  
is uniformly distributed over the (2g)th roots of unity, independent  
of the {E_i}, and (ii) if k^g = 1+ sn where $g \ge 2$ is fixed and  
$s=o(n^{\frac{g+1}{g-1}})$ or $s=o(n)$ according as $g \ge 2$ is odd  
or even, then the LSD is $U_2(\prod_{i=1}^g E_i)^{1/2g}$ where {E_i}  
are i.i.d. Exp(1) and U_2 is uniformly distributed over the unit  
circle, independent of the {E_i}. We then consider the limit  
distribution of the spectral norm of A_{k,n}. We show that when  
$n=k^2+1\to \infty$, the spectral norm, with appropriate scaling and  
centering, which we provide explicitly, converges to the Gumbel  
distribution.

http://arxiv.org/abs/0903.0128


8216. Conditioning of quadratic harnesses
Author(s): W. Bryc and J. Wesolowski

Abstract: We describe quadratic harnesses that arise through the  
double sided conditioning of an already known quadratic harness and we  
characterize quadratic harnesses that arise by this construction from  
bridges of Levy processes. We also analyze a construction that  
produces quadratic harnesses by "gluing together" two conditionally- 
independent quadratic harnesses and we show that the only q-Meixner  
processes that can be used in this construction are pairs of Poisson  
processes or pairs of negative binomial processes. Our main tool is a  
deterministic time and space transformation of quadratic harnesses.

http://arxiv.org/abs/0903.0150


8217. Reaching the best possible rate of convergence to equilibrium  
for solutions of Kac's equation via central limit theorem
Author(s): Emanuele Dolera and Ester Gabetta and Eugenio Regazzini

Abstract: Let $f(\cdot,t)$ be the probability density function which  
represents the solution of Kac's equation at time $t$, with initial  
data $f_0$, and let $g_{\sigma}$ be the Gaussian density with zero  
mean and variance $\sigma^2$, $\sigma^2$ being the value of the second  
moment of $f_0$. This is the first study which proves that the total  
variation distance between $f(\cdot,t)$ and $g_{\sigma}$ goes to zero,  
as $t\to +\infty$, with an exponential rate equal to -1/4. In the  
present paper, this fact is proved on the sole assumption that $f_0$  
has finite fourth moment and its Fourier transform $\varphi_0$  
satisfies $|\varphi_0(\xi)|=o(|\xi|^{-p})$ as $|\xi|\to+\infty$, for  
some $p>0$. These hypotheses are definitely weaker than those  
considered so far in the state-of-the-art literature, which in any  
case, obtains less precise rates.

http://arxiv.org/abs/0903.0255


8218. Fluid limits for networks with bandwidth sharing and general  
document size distributions
Author(s): H. Christian Gromoll and Ruth J. Williams

Abstract: We consider a stochastic model of Internet congestion  
control, introduced by Massouli\'{e} and Roberts [Telecommunication  
Systems 15 (2000) 185--201], that represents the randomly varying  
number of flows in a network where bandwidth is shared among document  
transfers. In contrast to an earlier work by Kelly and Williams [Ann.  
Appl. Probab. 14 (2004) 1055--1083], the present paper allows  
interarrival times and document sizes to be generally distributed,  
rather than exponentially distributed. Furthermore, we allow a fairly  
general class of bandwidth sharing policies that includes the weighted  
$\alpha$-fair policies of Mo and Walrand [IEEE/ACM Transactions on  
Networking 8 (2000) 556--567], as well as certain other utility based  
scheduling policies. To describe the evolution of the system, measure  
valued processes are used to keep track of the residual document sizes  
of all flows through the network. We propose a fluid model (or formal  
functional law of large numbers approximation) associated with the  
stochastic flow level model. Under mild conditions, we show that the  
appropriately rescaled measure valued processes corresponding to a  
sequence of such models (with fixed network structure) are tight, and  
that any weak limit point of the sequence is almost surely a fluid  
model solution. For the special case of weighted $\alpha$-fair  
policies, we also characterize the invariant states of the fluid model.

http://arxiv.org/abs/0903.0291


8219. Modified discrete random walk with absorption
Author(s): Theo van Uem

Abstract: We obtain expected number of arrivals, probability of  
arrival, absorption probabilities and expected time before absorption  
for a modified discrete random walk on the (sub)set of integers. In a  
[pqrs] random walk the particle can move one step forward or backward,  
stay for a moment in the same state or it can be absorbed immediately  
in the current state. M[pqrs] is a modified version, where  
probabilities on both sides of a multiple function barrier M are of  
different [pqrs] type.

http://arxiv.org/abs/0903.0364


8220. The Generalized Road Coloring Problem and periodic digraphs
Author(s): Greg Budzban and Philip Feinsilver

Abstract: A proof of the Generalized Road Coloring Problem,  
independent of the recent work by Beal and Perrin, is presented, using  
both semigroup methods and Trakhtman's algorithm. Algebraic properties  
of periodic, strongly connected digraphs are studied in the semigroup  
context. An algebraic condition which characterizes periodic, strongly  
connected digraphs is determined in the context of periodic Markov  
chains.

http://arxiv.org/abs/0903.0192


8221. On the equality of the quenched and averaged large deviation  
rate functions for high-dimensional ballistic random walk in a random  
environment
Author(s): Atilla Yilmaz

Abstract: We consider large deviations for nearest-neighbor random  
walk in a uniformly elliptic i.i.d. environment. It is easy to see  
that the quenched and averaged rate functions are not identically  
equal. When the dimension is at least four and Sznitman's transience  
condition (T) is satisfied, we prove that these rate functions are  
finite and equal on a closed set whose interior contains every nonzero  
velocity at which the rate functions vanish.

http://arxiv.org/abs/0903.0410


8222. Motion in a Random Force Field
Author(s): Dmitry Dolgopyat and Leonid Koralov

Abstract: We consider the motion of a particle in a random isotropic  
force field. Assuming that the force field arises from a Poisson field  
in $\mathbb{R}^d$, $d \geq 4$, and the initial velocity of the  
particle is sufficiently large, we describe the asymptotic behavior of  
the particle.

http://arxiv.org/abs/0903.0425


8223. Nonlinear Stochastic Perturbations of Dynamical Systems and  
Quasi-linear Parabolic PDE's with a Small Parameter
Author(s): M. Freidlin and L. Koralov

Abstract: In this paper we describe the asymptotic behavior, in the  
exponential time scale, of solutions to quasi-linear parabolic  
equations with a small parameter at the second order term and the long  
time behavior of corresponding diffusion processes. In particular, we  
discuss the exit problem and metastability for the processes  
corresponding to quasi-linear initial-boundary value problems.

http://arxiv.org/abs/0903.0428


8224. Metastability for Non-Linear Random Perturbations of Dynamical  
Systems
Author(s): M. Freidlin and L. Koralov

Abstract: In this paper we describe the long time behavior of  
solutions to quasi-linear parabolic equations with a small parameter  
at the second order term and the long time behavior of corresponding  
diffusion processes.

http://arxiv.org/abs/0903.0430


8225. Random Perturbations of 2-dimensional Hamiltonian Flows
Author(s): L. Koralov

Abstract: We consider the motion of a particle in a periodic two  
dimensional flow perturbed by small (molecular) diffusion. The flow is  
generated by a divergence free zero mean vector field. The long time  
behavior corresponds to the behavior of the homogenized process - that  
is diffusion process with the constant diffusion matrix (effective  
diffusivity). We obtain the asymptotics of the effective diffusivity  
when the molecular diffusion tends to zero.

http://arxiv.org/abs/0903.0436


8226. Coupled paraxial wave equations in random media in the white- 
noise regime
Author(s): Josselin Garnier and Knut S{\o}lna

Abstract: In this paper the reflection and transmission of waves by a  
three-dimensional random medium are studied in a white-noise and  
paraxial regime. The limit system derives from the acoustic wave  
equations and is described by a coupled system of random Schr 
\"{o}dinger equations driven by a Brownian field whose covariance is  
determined by the two-point statistics of the fluctuations of the  
random medium. For the reflected and transmitted fields the associated  
Wigner distributions and the autocorrelation functions are determined  
by a closed system of transport equations. The Wigner distribution is  
then used to describe the enhanced backscattering phenomenon for the  
reflected field.

http://arxiv.org/abs/0903.0449


8227. Adaptive independent Metropolis--Hastings
Author(s): Lars Holden and Ragnar Hauge and Marit Holden

Abstract: We propose an adaptive independent Metropolis--Hastings  
algorithm with the ability to learn from all previous proposals in the  
chain except the current location. It is an extension of the  
independent Metropolis--Hastings algorithm. Convergence is proved  
provided a strong Doeblin condition is satisfied, which essentially  
requires that all the proposal functions have uniformly heavier tails  
than the stationary distribution. The proof also holds if proposals  
depending on the current state are used intermittently, provided the  
information from these iterations is not used for adaption. The  
algorithm gives samples from the exact distribution within a finite  
number of iterations with probability arbitrarily close to 1. The  
algorithm is particularly useful when a large number of samples from  
the same distribution is necessary, like in Bayesian estimation, and  
in CPU intensive applications like, for example, in inverse problems  
and optimization.

http://arxiv.org/abs/0903.0483


8228. Macroscopic stability for nonfinite range kernels
Author(s): Tom S. Mountford (EPFL) and K. Ravishankar (SUNY) and Ellen  
Saada (LMRS)

Abstract: We extend the strong macroscopic stability introduced in  
Bramson & Mountford (2002) for one-dimensional asymmetric exclusion  
processes with finite range to a large class of one-dimensional  
conservative attractive models (including misanthrope process) for  
which we relax the requirement of finite range kernels. A key  
motivation is extension of constructive hydrodynamics result of  
Bahadoran et al. (2002, 2006, 2008) to nonfinite range kernels.

http://arxiv.org/abs/0903.0498


8229. Crested products of Markov chains
Author(s): Daniele D'Angeli and Alfredo Donno

Abstract: In this work we define two kinds of crested product for  
reversible Markov chains, which naturally appear as a generalization  
of the case of crossed and nested product, as in association schemes  
theory, even if we do a construction that seems to be more general and  
simple. Although the crossed and nested product are inspired by the  
study of Gelfand pairs associated with the direct and the wreath  
product of two groups, the crested products are a more general  
construction, independent from the Gelfand pairs theory, for which a  
complete spectral theory is developed. Moreover, the $k$-step  
transition probability is given. It is remarkable that these Markov  
chains describe some classical models (Ehrenfest diffusion model,  
Bernoulli--Laplace diffusion model, exclusion model) and give some  
generalization of them. As a particular case of nested product, one  
gets the classical Insect Markov chain on the ultrametric space.  
Finally, in the context of the second crested product, we present a  
generalization of this Markov chain to the case of many insects and  
give the corresponding spectral decomposition.

http://arxiv.org/abs/0903.0513


8230. ROC and the bounds on tail probabilities via theorems of Dubins  
and F. Riesz
Author(s): Eric Clarkson and J. L. Denny and Larry Shepp

Abstract: For independent $X$ and $Y$ in the inequality $P(X\leq Y+\mu) 
$, we give sharp lower bounds for unimodal distributions having finite  
variance, and sharp upper bounds assuming symmetric densities bounded  
by a finite constant. The lower bounds depend on a result of Dubins  
about extreme points and the upper bounds depend on a symmetric  
rearrangement theorem of F. Riesz. The inequality was motivated by  
medical imaging: find bounds on the area under the Receiver Operating  
Characteristic curve (ROC).

http://arxiv.org/abs/0903.0518


8231. Random matrices: The distribution of the smallest singular values
Author(s): Terence Tao and Van Vu

Abstract: Let $\a$ be a real-valued random variable of mean zero and  
variance 1. Let $M_n(\a)$ denote the $n \times n$ random matrix whose  
entries are iid copies of $\a$ and $\sigma_n(M_n(\a))$ denote the  
least singular value of $M_n(\a)$. ($\sigma_n(M_n(\a))^2$ is also  
usually interpreted as the least eigenvalue of the Wishart matrix $M_n  
M_n^{\ast}$.) We show that (under a finite moment assumption) the  
probability distribution $n \sigma_n(M_n(\a))^2$ is {\it universal} in  
the sense that it does not depend on the distribution of $\a$. In  
particular, it converges to the same limiting distribution as in the  
special case when $a$ is real gaussian. (The limiting distribution was  
computed explicitly in this case by Edelman.) We also proved a similar  
result for complex-valued random variables of mean zero, with real and  
imaginary parts having variance 1/2 and covariance zero. Similar  
results are also obtained for the joint distribution of the bottom $k$  
singular values of $M_n(\a)$ for any fixed $k$ (or even for $k$  
growing as a small power of $n$) and for rectangular matrices. Our  
approach is motivated by the general idea of ``property testing'' from  
combinatorics and theoretical computer science. This seems to be a new  
approach in the study of spectra of random matrices and combines tools  
from various areas of mathematics.

http://arxiv.org/abs/0903.0614


8232. Coupling, Attractiveness and Hydrodynamics for Conservative  
Particle Systems
Author(s): Thierry Gobron (LPTM) and Ellen Saada (LMRS)

Abstract: Attractiveness is a fundamental tool to study interacting  
particle systems and the basic coupling construction is a usual route  
to prove this property, as for instance in simple exclusion. The  
derived Markovian coupled process $(\xi_t,\zeta_t)_{t\geq 0}$  
satisfies: (A) if $\xi_0\leq\zeta_0$ (coordinate-wise), then for all $t 
\geq 0$, $\xi_t\leq\zeta_t$ a.s. In this paper, we consider  
generalized misanthrope models which are conservative particle systems  
on $\Z^d$ such that, in each transition, $k$ particles may jump from a  
site $x$ to another site $y$, with $k\geq 1$. These models include  
simple exclusion for which $k=1$, but, beyond that value, the basic  
coupling construction is not possible and a more refined one is  
required. We give necessary and sufficient conditions on the rates to  
insure attractiveness; we construct a Markovian coupled process which  
both satisfies (A) and makes discrepancies between its two marginals  
non-increasing. We determine the extremal invariant and translation  
invariant probability measures under general irreducibility  
conditions. We apply our results to examples including a two-species  
asymmetric exclusion process with charge conservation (for which $k\le  
2$) which arises from a Solid-on-Solid interface dynamics, and a stick  
process (for which $k$ is unbounded) in correspondence with a  
generalized discrete Hammersley-Aldous-Diaconis model. We derive the  
hydrodynamic limit of these two one-dimensional models.

http://arxiv.org/abs/0903.0316


8233. The Existence of Pair Potential Corresponding to Specified  
Density and Pair Correlation
Author(s): L. Koralov

Abstract: Given a potential of pair interaction and a value of  
activity, one can consider the Gibbs distribution in a finite domain $ 
\Lambda \subset \mathbb{Z}^d$. It is well known that for small values  
of activity there exist the infinite volume ($\Lambda \to \mathbb{Z}^d 
$) limiting Gibbs distribution and the infinite volume correlation  
functions. In this paper we consider the converse problem - we show  
that given $\rho_1$ and $\rho_2(x)$, where $\rho_1$ is a constant and $ 
\rho_2(x)$ is a function on $\mathbb{Z}^d$, which are sufficiently  
small, there exist a pair potential and a value of activity, for which  
$\rho_1$ is the density and $\rho_2(x)$ is the pair correlation  
function.

http://arxiv.org/abs/0903.0432


8234. An Inverse Problem for Gibbs Fields with Hard Core Potential
Author(s): L. Koralov

Abstract: It is well known that for a regular stable potential of pair  
interaction and a small value of activity one can define the  
corresponding Gibbs field (a measure on the space of configurations of  
points in $\mathbb{R}^d$). In this paper we consider a converse  
problem. Namely, we show that for a sufficiently small constant $ 
\overline{\rho}_1$ and a sufficiently small function $ 
\overline{\rho}_2(x)$, $x \in \mathbb{R}^d$, that is equal to zero in  
a neighborhood of the origin, there exist a hard core pair potential,  
and a value of activity, such that $\overline{\rho}_1$ is the density  
and $\overline{\rho}_2$ is the pair correlation function of the  
corresponding Gibbs field.

http://arxiv.org/abs/0903.0433


8235. Some Diffusion Processes Associated With Two Parameter Poisson- 
Dirichlet Distribution and Dirichlet Process
Author(s): Shui Feng and Wei Sun

Abstract: The two parameter Poisson-Dirichlet distribution $PD(\alpha, 
\theta)$ is the distribution of an infinite dimensional random  
discrete probability. It is a generalization of Kingman's Poisson- 
Dirichlet distribution. The two parameter Dirichlet process $ 
\Pi_{\alpha,\theta,\nu_0}$ is the law of a pure atomic random measure  
with masses following the two parameter Poisson-Dirichlet  
distribution. In this article we focus on the construction and the  
properties of the infinite dimensional symmetric diffusion processes  
with respective symmetric measures $PD(\alpha,\theta)$ and $ 
\Pi_{\alpha,\theta,\nu_0}$. The methods used come from the theory of  
Dirichlet forms.

http://arxiv.org/abs/0903.0623


8236. Products of random matrices: Dimension and growth in norm
Author(s): Vladislav Kargin

Abstract: Suppose that X_1, X_2, ... are independent, identically- 
distributed, rotationally invariant N-by-N matrices. Let P_n be the  
product X_n...X_1. It is known that log|P_n|/n converges to a non- 
random limit. We prove that under certain additional assumptions on  
matrices X_i the speed of convergence to this limit does not decrease  
when the size of matrices, N, grows.

http://arxiv.org/abs/0903.0632


8237. Loss networks
Author(s): Stan Zachary and Ilze Ziedins

Abstract: We review the theory of loss networks, including recent  
results on their dynamical behaviour. We give also some new results.

http://arxiv.org/abs/0903.0640


8238. SPDEs in divergence form with VMO coefficients and filtering  
theory of partially observable diffusion processes with Lipschitz  
coefficients
Author(s): N.V. Krylov

Abstract: We present several results on the 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/0903.0877


8239. Optimal investment with counterparty risk: a default-density  
modeling approach
Author(s): Ying Jiao (PMA) and Huyen Pham (PMA)

Abstract: We consider a financial market with a stock exposed to a  
counterparty risk inducing a drop in the price, and which can still be  
traded after this default time. We use a default-density modeling  
approach, and address in this incomplete market context the expected  
utility maximization from terminal wealth. We show how this problem  
can be suitably decomposed in two optimization problems in complete  
market framework: an after-default utility maximization and a global  
before-default optimization problem involving the former one. These  
two optimization problems are solved explicitly, respectively by  
duality and dynamic programming approaches, and provide a fine  
understanding of the optimal strategy. We give some numerical results  
illustrating the impact of counterparty risk and the loss given  
default on optimal trading strategies, in particular with respect to  
the Merton portfolio selection problem.

http://arxiv.org/abs/0903.0909


8240. Zero bias transformation and asymptotic expansions
Author(s): Ying Jiao (PMA)

Abstract: We apply the zero bias transformation to deduce a recursive  
asymptotic expansion formula for expectation of functions of sum of  
independent random variables in terms of normal expectations and we  
discuss the remainder term estimations.

http://arxiv.org/abs/0903.0910


8241. Convergence, Strong Law of Large Numbers, and Measurement Theory  
in the Language of Fuzzy Variables
Author(s): Adam Bzowski and Michal K. Urbanski

Abstract: In the paper we define the convergence of compact fuzzy sets  
as a convergence of alpha-cuts in the topology of compact subsets of a  
metric space. Furthermore we define typical convergences of fuzzy  
variables and show relations with convergence of their fuzzy  
distributions. In this context we prove a general formulation of the  
Strong Law of Large Numbers for fuzzy sets and fuzzy variables with  
Archimedean t-norms. Next we dispute a structure of fuzzy logics and  
postulate a new definition of necessity measures. Finally, we prove  
fuzzy version of the Glivenko-Cantelli theorem and use it for a  
construction of a complete fuzzy measure theory.

http://arxiv.org/abs/0903.0959


8242. Transformations des lois multivari\'ees avec queues r\'eguli\`eres
Author(s): Youri Davydov and Shuyan Liu

Abstract: Let $X$ be a random vector in $\rd$ with a regularly varying  
tail. We consider two transformations $\|X\|f(\frac{X}{\|X\|})$, $f:  
\sd\to\sd$, and $Xf(\frac{X}{\|X\|})$, $f: \sd\to \mathbb{R}_+$. Some  
sufficient conditions for preserving the property of regularity of the  
tail for this kind of transformations are given.

http://arxiv.org/abs/0903.1005


8243. Strong Convergence on Weakly Logarithmic Combinatorial Assemblies
Author(s): Eugenijus Manstavi\v{c}ius

Abstract: We deal with the random combinatorial structures called  
assemblies. By weakening the logarithmic condition which assures  
regularity of the number of components of a given order, we extend the  
notion of logarithmic assemblies. Using the author's analytic  
approach, we generalize the so-called Fundamental Lemma giving  
independent process approximation in the total variation distance of  
the component structure of an assembly. To evaluate the influence of  
strongly dependent large components, we obtain estimates of the  
appropriate conditional probabilities by unconditioned ones. These  
estimates are applied to examine additive functions defined on such a  
class of structures. Some analogs of Major's and Feller's theorems  
which concern almost sure behavior of sums of independent random  
variables are proved.

http://arxiv.org/abs/0903.1051


8244. A functional approach for random walks in random sceneries
Author(s): Cl\'ement Dombry (LMA) and Nadine Guillotin-Plantard (UCB  
and ICJ)

Abstract: A functional approach for the study of the random walks in  
random sceneries (RWRS) is proposed. Under fairly general assumptions  
on the random walk and on the random scenery, functional limit  
theorems are proved. The method allows to study separately the  
convergence of the walk and of the scenery: on the one hand, a general  
criterion for the convergence of the local time of the walk is  
provided, on the other hand, the convergence of the random measures  
associated with the scenery is studied. This functional approach is  
robust enough to recover many of the known results on RWRS as well as  
new ones, including the case of many walkers evolving in the same  
scenery.

http://arxiv.org/abs/0903.1071


8245. On the Traces of symmetric stable processes on Lipschitz domains
Author(s): Rodrigo Banuelos and Tadeusz Kulczycki and Bartlomiej Siudeja

Abstract: It is shown that the second term in the asymptotic expansion  
as $t\to 0$ of the trace of the semigroup of symmetric stable  
processes (fractional powers of the Laplacian) of order $\alpha$, for  
any $0<\alpha<2$, in Lipschitz domains is given by the surface area of  
the boundary of the domain. This brings the asymptotics for the trace  
of stable processes in domains of Euclidean space on par with those of  
Brownian motion (the Laplacian), as far as boundary smoothness is  
concerned.

http://arxiv.org/abs/0903.1198


8246. Power law Polya's urn and fractional Brownian motion
Author(s): Alan Hammond and Scott Sheffield

Abstract: We introduce a natural family of random walks on the set of  
integers that scale to fractional Brownian motion. The increments X_n  
have the property that given {X_k: k < n}, the conditional law of X_n  
is that of X_{n-k_n}, where k_n is sampled independently from a fixed  
law \mu on the positive integers. When \mu has a roughly power law  
decay (precisely, when it lies in the domain of attraction of an  
\alpha stable subordinator, for 0 < \alpha < 1/2) the walk scales to  
fractional Brownian motion with Hurst parameter \alpha + 1/2. The  
walks are easy to simulate and their increments satisfy an FKG  
inequality. In a sense we describe, they are the natural "fractional"  
analogs of simple random walk on Z.

http://arxiv.org/abs/0903.1284


8247. Stochastic ordering of classical discrete distributions
Author(s): Achim Klenke and Lutz Mattner

Abstract: For several pairs $(P,Q)$ of classical distributions on $ 
\N_0$, we show that their stochastic ordering $P\leq_{st} Q$ can be  
characterized by their extreme tail ordering equivalent to $ P(\{k_ 
\ast \})/Q(\{k_\ast\}) \le 1 \le \lim_{k\to k^\ast} P(\{k\})/Q(\{k\}) 
$, with $k_\ast$ and $k^\ast$ denoting the minimum and the supremum of  
the support of $P+Q$, and with the limit to be read as $P(\{k^\ast\})/ 
Q(\{k^\ast\})$ for $k^\ast$ finite. This includes in particular all  
pairs where $P$ and $Q$ are both binomial ($b_{n_1,p_1} \leq_{st}  
b_{n_2,p_2}$ if and only if $n_1\le n_2$ and $(1-p_1)^{n_1}\ge(1- 
p_2)^{n_2}$, or $p_1=0$), both negative binomial ($b^- 
_{r_1,p_1}\leq_{st} b^-_{r_2,p_2}$ if and only if $p_1\geq p_2$ and  
$p_1^{r_1}\geq p_2^{r_2}$), or both hypergeometric with the same  
sample size parameter. The binomial case is contained in a known  
result about Bernoulli convolutions, the other two cases appear to be  
new. The emphasis of this paper is on providing a variety of different  
methods of proofs: (i) half monotone likelihood ratios, (ii) explicit  
coupling, (iii) Markov chain comparison, (iv) analytic calculation,  
and (v) comparison of Levy measures. We give four proofs in the  
binomial case (methods (i)-(iv)) and three in the negative binomial  
case (methods (i), (iv) and (v)). The statement for hypergeometric  
distributions is proved via method (i).

http://arxiv.org/abs/0903.1361


8248. Positive definite functions and multidimensional versions of  
random variables
Author(s): Alexander Koldobsky

Abstract: We say that a random vector $X=(X_1,...,X_n)$ in $R^n$ is an  
$n$-dimensional version of a random variable $Y$ if for any $a\in R^n$  
the random variables $\sum a_iX_i$ and $\gamma(a) Y$ are identically  
distributed, where $\gamma:R^n\to [0,\infty)$ is called the standard  
of $X.$ An old problem is to characterize those functions $\gamma$  
that can appear as the standard of an $n$-dimensional version. In this  
paper, we prove the conjecture of Lisitsky that every standard must be  
the norm of a space that embeds in $L_0.$ This result is almost  
optimal, as the norm of any finite dimensional subspace of $L_p$ with  
$p\in (0,2]$ is the standard of an $n$-dimensional version ($p$-stable  
random vector) by the classical result of P.L\`evy. An equivalent  
formulation is that if a function of the form $f(\|\cdot\|_K)$ is  
positive definite on $R^n,$ where $K$ is an origin symmetric star body  
in $R^n$ and $f:R\to R$ is an even continuous function, then either  
the space $(R^n,\|\cdot\|_K)$ embeds in $L_0$ or $f$ is a constant  
function. Combined with known facts about embedding in $L_0,$ this  
result leads to several generalizations of the solution of  
Schoenberg's problem on positive definite functions.

http://arxiv.org/abs/0903.1433


8249. Smoothness of scale functions for spectrally negative Levy  
processes
Author(s): Terence Chan and Andreas Kyprianou and Mladen Savov

Abstract: Scale functions play a central role in the fluctuation  
theory of spectrally negative L\'evy processes and often appear in the  
context of martingale relations. These relations are often complicated  
to establish requiring excursion theory in favour of It\^o calculus.  
The reason for the latter is that standard It\^o calculus is only  
applicable to functions with a sufficient degree of smoothness and  
knowledge of the precise degree of smoothness of scale functions is  
seemingly incomplete. The aim of this article is to offer new results  
concerning properties of scale functions in relation to the smoothness  
of the underlying L\'evy measure. We place particular emphasis on  
spectrally negative L\'evy processes with a Gaussian component and  
processes of bounded variation. An additional motivation is the very  
intimate relation of scale functions to renewal functions of  
subordinators. The results obtained for scale functions have direct  
implications offering new results concerning the smoothness of such  
renewal functions for which there seems to be very little existing  
literature on this topic.

http://arxiv.org/abs/0903.1467


8250. Sharp thresholds for the random-cluster and Ising models
Author(s): Benjamin Graham and Geoffrey Grimmett

Abstract: A sharp-threshold theorem is proved for box-crossing  
probabilities on the square lattice. The models in question are the  
random-cluster model near the self-dual point $\psd(q)=\sqrt q/(1+ 
\sqrt q)$, the Ising model with external field, and the coloured  
random-cluster model. The principal technique is an extension of the  
influence theorem for monotonic probability measures applied to  
increasing events with no assumption of symmetry.

http://arxiv.org/abs/0903.1501


8251. Discrete approximation of stable white noise - Application to  
spatial linear filtering
Author(s): Cl\'ement Dombry (LMA)

Abstract: Motivated by the simulation of stable random fields, we  
consider the issue of discrete approximations of independently  
scattered stable noise. Two approaches are proposed: grid  
approximations available when the underlying space is $\bbR^d$ and  
shot noise approximations available on more general spaces. Limit  
theorems stating the convergence of discrete random noises to stable  
white noise are proved. These results are then applied to study moving  
average spatial random fields with heavy-tailed innovations and  
related limit theorems. A second application deals with discrete  
approximation for Brownian L\'evy motion on the sphere or on the  
euclidean space.

http://arxiv.org/abs/0903.1552


8252. Deducing the Density Hales-Jewett Theorem from an infinitary  
removal lemma
Author(s): Tim Austin (UCLA)

Abstract: We offer a new proof of Furstenberg and Katznelson's density  
version of the Hales-Jewett Theorem: For any \delta > 0 there is some  
N_0 \geq 1 such that whenever A \subseteq [k]^N with N \geq N_0 and |A| 
\geq \delta k^N, A contains a combinatorial line: that is, for some I  
\subseteq [N] nonempty and w_0 \in [k]^{[N]\setminus I} we have A  
\supseteq \{w: w|_{[N]\setminus I} = w_0, w|_I = \rm{const.}\}.  
Following Furstenberg and Katznelson, we first show that this result  
is equivalent to a `multiple recurrence' assertion for a class of  
probability measures enjoying a certain kind of stationarity. However,  
we then give a quite different proof of this latter assertion through  
a reduction to an infinitary removal lemma in the spirit of recent  
work of Tao (and also its recent re-interpretation by the author to  
give a proof of the multidimensional Szemeredi Theorem), and resting  
crucially on an observation that arose during ongoing work by a  
collaborative team of authors to give a purely finitary proof of the  
above theorem.

http://arxiv.org/abs/0903.1633


8253. The Central Limit Theorem for uniformly strong mixing measures
Author(s): Nicolai T A Haydn

Abstract: The theorem of Shannon-McMillan-Breiman states that for  
every generating partition on an ergodic system, the exponential decay  
rate of the measure of cylinder sets equals the metric entropy almost  
everywhere (provided the entropy is finite). In this paper we prove  
that the measure of cylinder sets are lognormally distributed for  
strongly mixing systems and infinite partitions and show that the rate  
of convergence is polynomial provided the fourth moment of the  
information function is finite. Also, unlike previous results by  
Ibragimov and others which only apply to finite partitions, here we do  
not require any regularity of the conditional entropy function. We  
also obtain the law of the iterated logarithm and the weak invariance  
principle for the information function.

http://arxiv.org/abs/0903.1325


8254. A Lower Bound on Arbitrary $f$--Divergences in Terms of the  
Total Variation
Author(s): Jochen Br\"ocker

Abstract: An important tool to quantify the likeness of two  
probability measures are f-divergences, which have seen widespread  
application in statistics and information theory. An example is the  
total variation, which plays an exceptional role among the f- 
divergences. It is shown that every f-divergence is bounded from below  
by a monotonous function of the total variation. Under appropriate  
regularity conditions, this function is shown to be monotonous.  
Remark: The proof of the main proposition is relatively easy, whence  
it is highly likely that the result is known. The author would be very  
grateful for any information regarding references or related work.

http://arxiv.org/abs/0903.1765


8255. Definition of evidence fusion rules on the basis of Referee  
Functions
Author(s): Frederic Dambreville (DGA/Cta/DT/Gip)

Abstract: This chapter defines a new concept and framework for  
constructing fusion rules for evidences. This framework is based on a  
referee function, which does a decisional arbitrament conditionally to  
basic decisions provided by the several sources of information. A  
simple sampling method is derived from this framework. The purpose of  
this sampling approach is to avoid the combinatorics which are  
inherent to the definition of fusion rules of evidences. This  
definition of the fusion rule by the means of a sampling process makes  
possible the construction of several rules on the basis of an  
algorithmic implementation of the referee function, instead of a  
mathematical formulation. Incidentally, it is a versatile and  
intuitive way for defining rules. The framework is implemented for  
various well known evidence rules. On the basis of this framework, new  
rules for combining evidences are proposed, which takes into account a  
consensual evaluation of the sources of information.

http://arxiv.org/abs/0903.1451


8256. Laws of Large Numbers for the Occupation Time of an Age- 
Dependent Critical Binary Branching System
Author(s): Jos\'e Alfredo L\'opez-Mimbela and Antonio Murillo Salas

Abstract: The occupation time of an age-dependent branching particle  
system in $\Rd$ is considered, where the initial population is a  
Poisson random field and the particles are subject to symmetric $\alpha 
$-stable migration, critical binary branching and random lifetimes.  
Two regimes of lifetime distributions are considered: lifetimes with  
finite mean and lifetimes belonging to the normal domain of attraction  
of a $\gamma$-stable law, $\gamma\in(0,1)$. It is shown that in  
dimensions $d>\alpha\gamma$ for the heavy-tailed lifetimes case, and  
$d>\alpha$ for finite mean lifetimes, the occupation time proccess  
satisfies a strong law of large numbers.

http://arxiv.org/abs/0903.1871


8257. Invariance principles for linear processes. Application to  
isotonic regression
Author(s): J. Dedecker and F. Merlev\`ede and M. Peligrad

Abstract: In this paper we prove maximal inequalities and study the  
functional central limit theorem for the partial sums of linear  
processes generated by dependent innovations. Due to the general  
weights these processes can exhibit long range dependence and the  
limiting distribution is a fractional Brownian motion. The proofs are  
based on new approximations by a linear process with martingale  
difference innovations. The results are then applied to study an  
estimator of the isotonic regression when the error process is a  
(possibly long range dependent) time series.

http://arxiv.org/abs/0903.1951


8258. $\kappa$-exponential models from the geometrical viewpoint
Author(s): Giovanni Pistone

Abstract: We discuss the use of Kaniadakis' $\kappa$-exponential in  
the construction of a statistical manifold modelled on Lebesgue spaces  
of real random variables. Some algebraic features of the deformed  
exponential models are considered. A chart is defined for each  
strictly positive densities; every other strictly positive density in  
a suitable neighborhood of the reference probability is represented by  
the centered $\Kln$ likelihood

http://arxiv.org/abs/0903.2012


8259. Numerical method for optimal stopping of piecewise deterministic  
Markov processes
Author(s): B. de Saporta and F. Dufour and K. Gonzalez

Abstract: We propose a numerical method to approximate the value  
function for the optimal stopping problem of a piecewise deterministic  
Markov process (PDMP). Our approach is based on quantization of the  
post jump location -- inter-arrival time Markov chain naturally  
embedded in the PDMP, and path-adapted time discretization grids. It  
allows us to derive bounds for the convergence rate of the algorithm  
and to provide a computable epsilon-optimal stopping time. The paper  
is illustrated by a numerical example.

http://arxiv.org/abs/0903.2114


8260. Heat kernel of fractional Laplacian in cones
Author(s): Krzysztof Bogdan and Tomasz Grzywny

Abstract: We give sharp estimates for the transition density of the  
isotropic stable L\'evy process killed when leaving a right circular  
cone.

http://arxiv.org/abs/0903.2269


8261. Quantitative estimates of the convergence of the empirical  
covariance matrix in Log-concave Ensembles
Author(s): Rados{\l}aw Adamczak and Alexander E. Litvak and Alain  
Pajor and Nicole Tomczak-Jaegermann

Abstract: Let $K$ be an isotropic convex body in $\R^n$. Given $ 
\eps>0$, how many independent points $X_i$ uniformly distributed on $K 
$ are needed for the empirical covariance matrix to approximate the  
identity up to $\eps$ with overwhelming probability? Our paper answers  
this question posed by Kannan, Lovasz and Simonovits. More precisely,  
let $X\in\R^n$ be a centered random vector with a log-concave  
distribution and with the identity as covariance matrix. An example of  
such a vector $X$ is a random point in an isotropic convex body. We  
show that for any $\eps>0$, there exists $C(\eps)>0$, such that if $N 
\sim C(\eps) n$ and $(X_i)_{i\le N}$ are i.i.d. copies of $X$, then $  
\Big\|\frac{1}{N}\sum_{i=1}^N X_i\otimes X_i - \Id\Big\| \le \epsilon,  
$ with probability larger than $1-\exp(-c\sqrt n)$.

http://arxiv.org/abs/0903.2323


8262. Large deviations for singular and degenerate diffusion models in  
adaptive evolution
Author(s): Nicolas Champagnat (INRIA Sophia Antipolis / INRIA  
Lorraine / IECN)

Abstract: In the course of Darwinian evolution of a population,  
punctualism is an important phenomenon whereby long periods of genetic  
stasis alternate with short periods of rapid evolutionary change. This  
paper provides a mathematical interpretation of punctualism as a  
sequence of change of basin of attraction for a diffusion model of the  
theory of adaptive dynamics. Such results rely on large deviation  
estimates for the diffusion process. The main difficulty lies in the  
fact that this diffusion process has degenerate and non-Lipschitz  
diffusion part at isolated points of the space and non-continuous  
drift part at the same points. Nevertheless, we are able to prove  
strong existence and the strong Markov property for these diffusions,  
and to give conditions under which pathwise uniqueness holds. Next, we  
prove a large deviation principle involving a rate function which has  
not the standard form of diffusions with small noise, due to the  
specific singularities of the model. Finally, this result is used to  
obtain asymptotic estimates for the time needed to exit an attracting  
domain, and to identify the points where this exit is more likely to  
occur.

http://arxiv.org/abs/0903.2345


8263. A Mean Field Approach for Optimization in Particles Systems and  
Applications
Author(s): Nicolas Gast (INRIA Rh\^one-Alpes / LIG laboratoire  
d'Informatique de Grenoble), Bruno Gaujal (INRIA Rh\^one-Alpes / LIG  
laboratoire d'Informatique de Grenoble)

Abstract: This paper investigates the limit behavior of Markov  
Decision Processes (MDPs) made of independent particles evolving in a  
common environment, when the number of particles goes to infinity. In  
the finite horizon case or with a discounted cost and an infinite  
horizon, we show that when the number of particles becomes large, the  
optimal cost of the system converges almost surely to the optimal cost  
of a discrete deterministic system (the "optimal mean field").  
Convergence also holds for optimal policies. We further provide  
insights on the speed of convergence by proving several central limits  
theorems for the cost and the state of the Markov decision process  
with explicit formulas for the variance of the limit Gaussian laws.  
Then, our framework is applied to a brokering problem in grid  
computing. The optimal policy for the limit deterministic system is  
computed explicitly. Several simulations with growing numbers of  
processors are reported. They compare the performance of the optimal  
policy of the limit system used in the finite case with classical  
policies (such as Join the Shortest Queue) by measuring its asymptotic  
gain as well as the threshold above which it starts outperforming  
classical policies.

http://arxiv.org/abs/0903.2352


8264. Random Marked Sets
Author(s): Felix Ballani and Zakhar Kabluchko and Martin Schlather

Abstract: We introduce a new class of stochastic processes which are  
defined on a random set in R^d. These processes can be seen as a link  
between random fields and marked point processes. Unlike for random  
fields, the mark covariance function need in general not be positive  
definite. This implies that in many situations the use of simple  
geostatistical methods appears to be questionable. Surprisingly, for a  
special class of processes based on Gaussian random fields, we do have  
positive definiteness for the corresponding mark covariance function  
and mark correlation function.

http://arxiv.org/abs/0903.2388


8265. Polynomial bounds in the Ergodic Theorem for positive recurrent  
one-dimensional diffusions and integrability of hitting times
Author(s): Dasha Loukianova and Oleg Loukianov and Eva Loecherbach

Abstract: Let $X$ be a one dimensional positive recurrent diffusion  
with invariant measure $\mu.$ We say that the degree of recurrence of  
$X$ is polynomial of order $p\geq 1$, if for all $x,a$ we have $ 
\E_xT_a^p<\infty$ and $\E_xT_a^{p+1}=\infty$, where $T_a$ is the  
hitting time of $a.$ We give sufficient conditions on the coefficients  
of $X$ in order to have a degree of recurrence at least equal to $p$.  
For such a diffusion, we derive non asymptotic deviation bounds $$ 
\P_{\nu} (|\frac1t\int_0^tf(X_s)ds-\mu(f)|\geq\ge)\leq K(p)\frac1{t^{p/ 
2}}\frac 1{\ge^p}A(f)^p$$ where $\nu$ is an initial distribution, $f$  
bounded or bounded and compactly supported and $A(f)=\|f\|_{\infty}$  
when $f$ is bounded and $A(f)=\mu(|f|)$ when $f$ is bounded and  
compactly supported. We also give a polynomial control of $\E_xT_a^p$  
from above and below.

http://arxiv.org/abs/0903.2405


8266. Moderate deviations for centered additive functionals of  
recurrent Harris processes having general state space
Author(s): Dasha Loukianova and Eva Loecherbach

Abstract: Let $X$ be a Harris recurrent strong Markov process with  
general Polish state space $E,$ having invariant measure $\mu .$ In  
this paper we derive non asymptotic deviation bounds for $$P_{x} (| 
\int_0^tf(X_s)ds|\geq t^{\frac12 + \eta} \ge)$$ in the positive  
recurrent case, for nice functions $f$ with $\mu (f) =0 .$ We  
generalize these bounds to the fully null-recurrent case where we  
obtain an exponential rate of convergence which is expressed in terms  
of the deterministic equivalent of the process. The main ingredient of  
the proof is Nummelin splitting in continuous time which allows to  
introduce regeneration times for the process.

http://arxiv.org/abs/0903.2408


8267. Outliers in INAR(1) models
Author(s): Matyas Barczy and Marton Ispany and Gyula Pap and Manuel  
Scotto and Maria Eduarda Silva

Abstract: In this paper the integer-valued autoregressive model of  
order one, contaminated with additive or innovational outliers is  
studied in some detail, parameter estimation is also addressed. In  
particular, the asymptotic behavior of conditional least squares (CLS)  
estimators is analyzed. We suppose that the time points of the  
outliers are known, but their sizes are unknown. It is proved that the  
CLS estimators of the offspring and innovation means are strongly  
consistent, but the CLS estimators of the sizes of the outliers are  
not strongly consistent; nevertheless, they converge to a random limit  
with probability 1. This random limit depends on the values of the  
process at the outliers' time points and on the values at the  
preceding time points and in case of additive outliers also on the  
values at the following time points. We also prove that the joint CLS  
estimator of the offspring and innovation means is asymptotically  
normal. Conditionally on the above described values of the process,  
the joint CLS estimator of the sizes of the outliers is also  
asymptotically normal.

http://arxiv.org/abs/0903.2421


8268. Non uniqueness of stationary measures for self-stabilizing  
processes
Author(s): Samuel Herrmann Julian Tugaut

Abstract: We investigate the existence of invariant measures for self- 
stabilizing diffusions. These stochastic processes represent roughly  
the behavior of some Brownian particle moving in a double-well  
landscape and attracted by its own law. This specific self-interaction  
leads to nonlinear stochastic differential equations and permits to  
point out singular phenomenons like non uniqueness of associated  
stationary measures. The existence of several invariant measures is  
essentially based on the non convex environment and requires  
generalized Laplace's method approximations.

http://arxiv.org/abs/0903.2460


8269. On the usefulness of persistent excitation in ARX adaptive  
tracking
Author(s): Bernard Bercu and Victor Vazquez

Abstract: The usefulness of persistent excitation is well-known in the  
control community. Thanks to a persistently excited adaptive tracking  
control, we show that it is possible to avoid the strong  
controllability assumption recently proposed in the multidimensional  
ARX framework. We establish the almost sure convergence for both least  
squares and weighted least squares estimators of the unknown  
parameters. A central limit theorem and a law of iterated logarithm  
are also provided. All this asymptotical analysis is related to the  
Schur complement of a suitable limiting matrix.

http://arxiv.org/abs/0903.2572


8270. A Quantitative Arrow Theorem
Author(s): Elchanan Mossel

Abstract: Arrow's Impossibility Theorem states that any constitution  
which satisfies Independence of Irrelevant Alternatives (IIA) and  
Unanimity and is not a Dictator has to be non-transitive. In this  
paper we study quantitative versions of Arrow theorem. Consider $n$  
voters who vote independently at random, each following the uniform  
distribution over the 6 rankings of 3 alternatives. Arrow's theorem  
implies that any constitution which satisfies IIA and Unanimity and is  
not a dictator has a probability of at least $6^{-n}$ for a non- 
transitive outcome. When $n$ is large, $6^{-n}$ is a very small  
probability, and the question arises if for large number of voters it  
is possible to avoid paradoxes with probability close to 1. Here we  
give a negative answer to this question by proving that for every $ 
\eps > 0$, there exists a $\delta = \delta(\eps) > 0$, which depends  
on $\eps$ only, such that for all $n$, and all constitutions on 3  
alternatives, if the constitution satisfies: The IIA condition. For  
every pair of alternatives $a,b$, the probability that the  
constitution ranks $a$ above $b$ is at least $\eps$. For every voter $i 
$, the probability that the social choice function agrees with a  
dictatorship on $i$ at most $1-\eps$. Then the probability of a non- 
transitive outcome is at least $\delta$.

http://arxiv.org/abs/0903.2574


8271. A Polynomial Number of Random Points does not Determine the  
Volume of a Convex Body
Author(s): Ronen Eldan

Abstract: We show that there is no algorithm which, provided a  
polynomial number of random points uniformly distributed over a convex  
body in R^n, can approximate the volume of the body up to a constant  
factor with high probability.

http://arxiv.org/abs/0903.2634


8272. Free point processes and free extreme values
Author(s): G. Ben Arous and V. Kargin

Abstract: We continue here the study of free extreme values begun in  
Ben Arous and Voiculescu (2006). We study the convergence of the free  
point processes associated with free extreme values to a free Poisson  
random measure (Voiculescu (1998), Barndorff-Nielsen and Thorbjornsen  
(2005)). We relate this convergence to the free extremal laws  
introduced in Ben Arous and Voiculescu (2006) and give the limit laws  
for free order statistics.

http://arxiv.org/abs/0903.2672


8273. Random Walks on Dicyclic Group
Author(s): Songzi Du

Abstract: This paper works out the rate of convergence of two  
"natural" random walks on the dicyclic group.

http://arxiv.org/abs/0903.2692


8274. The local time of a random walk on growing hypercubes
Author(s): Pierre Andreoletti (MAPMO)

Abstract: We study a random walk in a random environment (RWRE) on $ 
\Z^d$, $1 \leq d < +\infty$. The main assumptions are that  
conditionned on the environment the random walk is reversible.  
Moreover we construct our environment in such a way that the walk  
can't be trapped on a single point like in some particular RWRE but in  
some specific d-1 surfaces. These surfaces are basic surfaces with  
deterministic geometry. We prove that the local time in the  
neighborhood of these surfaces is driven by a function of the (random)  
reversible measure. As an application we get the limit law of the  
local time as a process on these surfaces.

http://arxiv.org/abs/0903.2696


8275. An explicit rough path construction for continuous paths with  
arbitrary H\"older exponent
Author(s): J. Unterberger

Abstract: We construct in this article an explicit geometric rough  
path over arbitrary $d$-dimensional paths with finite $1/\alpha$- 
variation for any $\alpha\in(0,1)$. The method is a rather  
straightforward extension of that used in a previous article  
\cite{Unt09} for multi-dimensional fractional Brownian motion. It may  
be coined as 'Fourier normal ordering' since it consists in a  
regularization obtained after permuting the order of integration in  
iterated integrals so that innermost integrals have highest Fourier  
frequencies. In doing so, there appear non-trivial tree combinatorics,  
which are best understood by using the Hopf algebra structure of  
decorated rooted trees. The algorithm of regularization follows very  
closely the BPHZ algorithm for the renormalization of Feynmann  
diagrams in quantum field theory. The new feature here (compared to  
\cite{Unt09}) is the use of Besov norms to prove H\"older continuity.

http://arxiv.org/abs/0903.2716


8276. Stationary systems of Gaussian processes
Author(s): Zakhar Kabluchko

Abstract: We describe all countable particle systems on $\mathbb R$  
which have the following three properties: independence, Gaussianity,  
and stationarity. More precisely, we consider particles on the real  
line starting at the points of a Poisson point process with intensity  
measure $m$ and moving independently of each other according to the  
law of some Gaussian process $\xi$. We describe all pairs $(m,\xi)$  
generating a stationary particle system, obtaining three families of  
examples. One of these families appeared in connection with extremes  
of independent Gaussian processes in [Z. Kabluchko, M. Schlather, L.  
de Haan, Stationary max-stable fields associated to negative definite  
functions, Ann. Probab. (2009), in press].

http://arxiv.org/abs/0903.2738


8277. Sharp thresholds for constraint satisfaction problems and  
homomorphisms
Author(s): Hamed Hatami and Michael Molloy

Abstract: We determine under which conditions certain natural models  
of random constraint satisfaction problems have sharp thresholds of  
satisfiability. These models include graph and hypergraph  
homomorphism, the $(d,k,t)$-model, and binary constraint satisfaction  
problems with domain size three.

http://arxiv.org/abs/0903.2579


8278. Exact Thresholds for Ising-Gibbs Samplers on General Graphs
Author(s): Elchanan Mossel and Allan Sly

Abstract: We establish tight results for rapid mixing of Gibbs  
Samplers for the Ferromagnetic Ising model on general graphs. We show  
that if $(d-1) \tanh \beta < 1$, then there exists a constant $C$ such  
that the discrete time mixing time of Gibbs Samplers for the  
Ferromagnetic Ising model on {\em any} graph of $n$ vertices and  
maximal degree $d$, where all interactions are bounded by $\beta$, and  
arbitrary external fields is bounded by $C n \log n$. We further show  
the when $d \tanh \beta < 1$, with high probability over the Erd\H{o}s- 
R\'enyi random graph on $n$ vertices with average degree $d$, it holds  
that the mixing time of Gibbs Samplers is $n^{1+\Theta(\frac{1}{\log  
\log n})}$. Both result are tight as it is known that the mixing time  
for random regular and Erd\H{o}s-R\'enyi random graphs is, with high  
probability, exponential in $n$ when if $(d-1) \tanh \beta > 1$ and $d  
\tanh \beta > 1$ respectively.

http://arxiv.org/abs/0903.2906


8279. A symmetric entropy bound on the non-reconstruction regime of  
Markov chains on Galton-Watson trees
Author(s): M. Formentin and C. Kuelske

Abstract: We give a criterion of the form Q(d)c(M)<1 for the non- 
reconstructability of tree-indexed q-state Markov chains obtained by  
broadcasting a signal from the root with a given transition matrix M.  
Here c(M) is an explicit constant defined in terms of a (q-1)- 
dimensional variational problem over symmetric entropies, and Q(d) is  
the expected number of offspring on the Galton-Watson tree. This  
result is equivalent to proving the extremality of the free boundary  
condition-Gibbs measure within the corresponding Gibbs-simplex. Our  
theorem holds for possibly non-reversible M and its proof is based on  
a general 'Magic Recursion Formula' for expectations of a symmetrized  
relative entropy function, which invites their use as a Lyapunov  
function. In the case of the Potts model, the present theorem  
reproduces earlier results of the authors, with a simplified proof. In  
the case of the Ising model (where the method produces the correct  
reconstruction threshold) the argument becomes similar to the approach  
of Pemantle and Peres.

http://arxiv.org/abs/0903.2962


8280. Knights, spies, games and ballot sequences
Author(s): Mark Wildon

Abstract: This paper presents a solution to the Knights and Spies  
Problem: In a room there are n people, each labelled with a unique  
number between 1 and n. A person may either be a knight or a spy.  
Knights always tell the truth, while spies may either lie or tell the  
truth, as they see fit. Each person in the room knows the identity of  
everyone else. Apart from this, all that is known is that strictly  
more knights than spies are present. Asking only questions of the  
form: `Person i, what is the identity of person j?', what is the least  
number of questions that will guarantee to find the true identities of  
all n people? The analysis of a related two-player game is critical to  
the proof. Some probabilistic aspects are also explored. The paper  
ends by presenting three open questions concerned with generalisations  
of the problem.

http://arxiv.org/abs/0903.2869


8281. Metastability in the generalized Hopfield model with finitely  
many patterns
Author(s): Mykhaylo Shkolnikov

Abstract: This paper continues the study of metastable behaviour in  
disordered mean field models initiated in [2], [3]. We consider the  
generalized Hopfield model with finitely many independent patterns $ 
\xi_1,...,\xi_P$ where the patterns have i.i.d. components and the  
components of patterns $\xi_1,...\xi_p$ have absolutely continuous  
distributions on $[-1,1]$ whereas the components of patterns $\xi_{p 
+1},...,\xi_P$ have discrete distributions on $[-1,1]$ with no atom at  
0. We show that metastable behaviour occurs if there is at least one  
pattern of each type and $2p+7

http://arxiv.org/abs/0903.3050


8282. On normal approximations to $U$-statistics
Author(s): V. Bentkus and B.-Y. Jing and W. Zhou

Abstract: Let ${X_1,...,X_n}$ be i.i.d. random observations. Let $ 
{\Sta =\Lr+\T}$ be a $U$-statistic of order $k \ge 2$, where $\Lr$ is  
a linear statistic having asymptotic normal distribution, and $\T$ is  
a stochastically smaller statistic. We show that the rate of  
convergence to normality for $\Sta$ can be simply expressed as the  
rate of convergence to normality for the linear part $\Lr$ plus a  
correction term, $(\var \T) \ln^2 (\var \T)$, under the condition ${\E  
\T^2 < \infty}$. An optimal bound without this $\log$ factor is  
obtained under a lower moment assumption ${\E |\T |^\alpha < \infty}$  
for ${\alpha<2}$. Some other related results are also obtained in the  
paper. Our results extend, refine and yield a number of related known  
results in the literature.

http://arxiv.org/abs/0903.3081


8283. Amenability of horocyclic products of percolation trees
Author(s): Florian Sobieczky

Abstract: For horocyclic products of percolation subtrees of regular  
trees, we show almost sure amenability. Under a symmetry condition  
concerning the growth of the two percolation trees, we show the  
existence of an increasing Foelner sequence (which we call strong  
amenability).

http://arxiv.org/abs/0903.3140


8284. Note on the Heat-Kernel Decay for Random Walk among Random  
Conductances with Heavy Tail
Author(s): Omar Boukhadra

Abstract: We study models of discrete-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  
polynomial tail near 0 with exponent $\gamma>0$. We study the decay of  
the $2n$-step return probability $P_\omega^{2n}(0,0)$. For all $d 
\geq4$, we prove that the decay of $P^{2n}_\omega(0,0)$ is as close as  
we want to the standard decay $n^{-d/2}$ for large values of the  
parameter $\gamma$.

http://arxiv.org/abs/0903.3157


8285. Entropy of Random Walk Range
Author(s): Itai Benjamini and Gady Kozma and Ariel Yadin and Amir  
Yehudayoff

Abstract: We study the entropy of the set traced by an $n$-step random  
walk on $\Z^d$. We show that for $d \geq 3$, the entropy is of order $n 
$. For $d = 2$, the entropy is of order $n/\log^2 n$. These values are  
essentially governed by the size of the boundary of the trace.

http://arxiv.org/abs/0903.3179


8286. Time Allocation of a Set of Radars in a Multitarget Environment
Author(s): Emmanuel Duflos (INRIA Futurs) and Marie De Vilmorin  
(LGI2A) and Philippe Vanheeghe (INRIA Futurs)

Abstract: The question tackled here is the time allocation of radars  
in a multitarget environment. At a given time radars can only observe  
a limited part of the space; it is therefore necessary to move their  
axis with respect to time, in order to be able to explore the overall  
space facing them. Such sensors are used to detect, to locate and to  
identify targets which are in their surrounding aerial space. In this  
paper we focus on the detection schema when several targets need to be  
detected by a set of delocalized radars. This work is based on the  
modelling of the radar detection performances in terms of probability  
of detection and on the optimization of a criterion based on detection  
probabilities. This optimization leads to the derivation of allocation  
strategies and is made for several contexts and several hypotheses  
about the targets locations.

http://arxiv.org/abs/0903.3100


8287. Continuity of large closed queueing networks with bottlenecks
Author(s): Vyacheslav M. Abramov

Abstract: This paper studies a closed queueing network containing a  
hub (a state dependent queueing system with service depending on the  
number of units residing here) and $k$ satellite stations, which are  
$GI/M/1$ queueing systems. The number of units in the system, $N$, is  
assumed to be a large number. After service completion in the hub, a  
unit visit the satellite station $j$ with probability $p_j$, and after  
the service completion returns to the hub. The parameters of service  
times in the satellite stations and in the hub are proportional to $ 
\frac{1}{N}$. One of the satellite stations is assumed to be a  
bottleneck station, while others are non-bottleneck. The paper  
establishes the continuity of the queue-length processes in non- 
bottleneck satellite stations of the network when the service times in  
the hub are close in certain sense (exactly defined in the paper) to  
the exponential distribution.

http://arxiv.org/abs/0903.3259


8288. Well-posedness and ergodicity for stochastic reaction-diffusion  
equations with multiplicative Poisson noise
Author(s): Carlo Marinelli and Michael R\"ockner

Abstract: We establish well-posedness in the mild sense for a class of  
stochastic semilinear evolution equations with a polynomially growing  
quasi-monotone nonlinearity and multiplicative Poisson noise. We also  
study existence and uniqueness of invariant measures for the  
associated semigroup in the Markovian case. A key role is played by a  
new maximal inequality for stochastic convolutions in $L_p$ spaces.

http://arxiv.org/abs/0903.3299


8289. New Maximally Stable Gaussian Partitions with Discrete  
Applications
Author(s): Marcus Isaksson and Elchanan Mossel

Abstract: Gaussian noise stability results have recently played an  
important role in proving fundamental results in hardness of  
approximation in computer science and in the study of voting schemes  
in social choice. We propose two Gaussian noise stability conjectures  
and derive consequences of the conjectures in hardness of  
approximation and social choice. Both conjectures generalize  
isoperimetric results by Borell on the heat kernel. One of the  
conjectures may be also be viewed as a generalization of the "Double  
Bubble" theorem. The applications of the conjectures include an  
optimality result for majority in the context of Condorcet voting and  
a proof that the Frieze-Jerrum SDP for MAX-q-CUT achieves the optimal  
approximation factor assuming the Unique Games Conjecture. We finally  
derive a short proof of the first conjecture based on the extended  
Riesz inequality.

http://arxiv.org/abs/0903.3362


8290. Constrained Backward SDEs with Jumps: Application to Optimal  
Switching
Author(s): Romuald Elie (CREST and Ceremade) and Idris Kharroubi  
(CREST and Pma)

Abstract: In this paper, we introduce a new class of BSDE generalizing  
and offering a unifying framework to represent the constrained ones  
presented in [16] or [12] as well as the oblique reflected ones  
studied by [11] and [9]. Via a penalization procedure, we provide an  
existence and uniqueness result for this new class of so-called  
constrained BSDEs with jumps. Remarkably, these BSDEs appear to be  
very convenient to represent the solution to eventually non-Markovian  
switching problems. As a by-product, we enlarge the class of obliquely  
reflected BSDE's, allowing to represent switching problems with  
controlled underlined diffusion.

http://arxiv.org/abs/0903.3372


8291. Off-Critical SLE(2) and SLE(4): a Field Theory Approach
Author(s): Michel Bauer and Denis Bernard and Luigi Cantini

Abstract: Using their relationship with the free boson and the free  
symplectic fermion, we study the off-critical perturbation of SLE(4)  
and SLE(2) obtained by adding a mass term to the action. We compute  
the off-critical statistics of the source in the Loewner equation  
describing the two dimensional interfaces. In these two cases we show  
that ratios of massive by massless partition functions, expressible as  
ratios of regularised determinants of massive and massless Laplacians,  
are (local) martingales for the massless interfaces. The off-critical  
drifts in the stochastic source of the Loewner equation are  
proportional to the logarithmic derivative of these ratios. We also  
show that massive correlation functions are (local) martingales for  
the massive interfaces. In the case of massive SLE(4), we use this  
property to prove a factorisation of the free boson measure.

http://arxiv.org/abs/0903.1023


8292. Exactly Solvable Birth and Death Processes
Author(s): Ryu Sasaki

Abstract: Many examples of exactly solvable birth and death processes,  
a typical stationary Markov chain, are presented together with the  
explicit expressions of the transition probabilities. They are derived  
by similarity transforming exactly solvable `matrix' quantum  
mechanics, which is recently proposed by Odake and the author. The ($q 
$-)Askey-scheme of hypergeometric orthogonal polynomials of a discrete  
variable and their dual polynomials play a central role. The most  
generic solvable birth/death rates are rational functions of $q^x$ ($x 
$ being the population) corresponding to the $q$-Racah polynomial.

http://arxiv.org/abs/0903.3097


8293. The heat semigroup and Brownian motion on strip complexes
Author(s): Alexander Bendikov and Laurent Saloff-Coste and Maura  
Salvatori and and Wolfgang Woess

Abstract: We introduce the notion of strip complex. A strip complex is  
a special type of complex obtained by gluing "strips" along their  
natural boundaries according to a given graph structure. The most  
familiar example is the one dimensional complex classically associated  
with a graph, in which case the strips are simply copies of the unit  
interval (our setup actually allows for variable edge length). A  
leading key example is treebolic space, a geometric object studied in  
a number of recent articles, which arises as a horocyclic product of a  
metric tree with the hyperbolic plane. In this case, the graph is a  
regular tree, the strips are the closed unit interval times the real  
line, and each strip is equipped with the hyperbolic geometry of a  
specific strip in upper half plane. We consider natural families of  
Dirichlet forms on a general strip complex and show that the  
associated heat kernels and harmonic functions have very strong  
smoothness properties. We study questions such as essential  
selfadjointness of the underlying differential operator acting on a  
suitable space of smooth functions satisfying a Kirchoff type  
condition at points where the strip complex bifurcates. Compatibility  
with projections that arise from proper group actions is also  
considered.

http://arxiv.org/abs/0903.3518


8294. Spectrum of large random reversible Markov chains - heavy-tailed  
weights on the complete graph
Author(s): Charles Bordenave (IMT) and Pietro Caputo and Djalil Chafai  
(IMT and UPTE)

Abstract: We consider the random reversible Markov kernel K on the  
complete graph with n vertices obtained by putting i.i.d. positive  
weights of law L on the n(n+1)/2 edges of the graph and normalizing  
each weight by the corresponding row sum. We have already shown in a  
previous work that if L has finite second moment then, as n goes to  
infinity, the limiting spectral distribution of n^{1/2} K is Wigner's  
semi-circle law. In the present work, we consider the case where L  
belongs to the domain of attraction of a stable law of index a. When  
1< a <2, we show that for a suitable regularly varying sequence k_n of  
index 1 - 1/a, the limiting spectral distribution of k_n K coincides  
with the one of the random symmetric matrix of the un-normalized  
weights (i.i.d. entries). In contrast, when 0< a <1, we show that the  
empirical spectral distribution of K converges, without any rescaling,  
to a non-trivial law supported on [-1,1], whose moments are the return  
probabilities of the random walk on a suitable Poisson weighted  
infinite tree of Aldous. The limiting operator is naturally linked  
with the Poisson-Dirichlet distribution PD(a,0). The "critical" cases  
a=1 and a=2 are not solved here.

http://arxiv.org/abs/0903.3528


8295. Spectra of large random trees
Author(s): Shankar Bhamidi and Steven N. Evans and Arnab Sen

Abstract: We analyze the eigenvalues of the adjacency matrices of a  
wide variety of random trees. Using general, broadly applicable  
arguments based on the interlacing inequalities for the eigenvalues of  
a principal submatrix of a Hermitian matrix and a suitable notion of  
local weak convergence for an ensemble of random trees, we show that  
the empirical spectral distributions for each of a number of random  
tree models converge to a deterministic (model dependent) limit as the  
number of vertices goes to infinity. We conclude for ensembles such as  
the linear preferential attachment models, random recursive trees, and  
the uniform random trees that the limiting spectral distribution has a  
set of atoms that is dense in the real line. We obtain precise  
asymptotics on the mass assigned to zero by the empirical spectral  
measures via the connection with the cardinality of a maximal  
matching. Moreover, we show that the total weight of a weighted  
matching is asymptotically equivalent to a constant multiple of the  
number of vertices when the edge weights are independent, identically  
distributed, non-negative random variables with finite expected value.  
We greatly extend a celebrated result obtained by Schwenk for the  
uniform random trees by showing that, under mild conditions, with  
probability converging to one, the spectrum of a realization is shared  
by at least one other tree. For the the linear preferential attachment  
model with parameter $a > -1$, we show that the suitably rescaled $k$  
largest eigenvalues converge jointly.

http://arxiv.org/abs/0903.3589


8296. On the Structure and Representations of Max--Stable Processes
Author(s): Yizao Wang and Stilian A. Stoev

Abstract: We develop classification results for max--stable processes,  
based on their spectral representations. The structure of max--linear  
isometries and minimal spectral representations play important roles.  
We propose a general classification strategy for measurable max-- 
stable processes based on the notion of co--spectral functions. In  
particular, we discuss the spectrally continuous--discrete, the  
conservative--dissipative, and positive--null decompositions. For  
stationary max--stable processes, the latter two decompositions arise  
from connections to non--singular flows and are closely related to the  
classification of stationary sum--stable processes. The interplay  
between the introduced decompositions of max--stable processes is  
further explored. As an example, the Brown--Resnick stationary  
processes, driven by fractional Brownian motions, are shown to be  
dissipative. A result on general Gaussian processes with stationary  
increments and continuous paths is obtained.

http://arxiv.org/abs/0903.3594


8297. Adversarial Smoothed Analysis
Author(s): Felipe Cucker and Raphael Hauser and Martin Lotz

Abstract: The purpose of this note is to extend the results on uniform  
smoothed analysis of condition numbers from \cite{BuCuLo:07} to the  
case where the perturbation follows a radially symmetric probability  
distribution. In particular, we will show that the bounds derived in  
\cite{BuCuLo:07} still hold in the case of distributions whose density  
has a singularity at the center of the perturbation, which we call  
{\em adversarial}.

http://arxiv.org/abs/0903.3499


8298. Some annealed bounds for renewal pinning polymer models with  
weakly dependent disorder
Author(s): Julien Poisat (ICJ)

Abstract: The aim of this paper is to provide some estimates on the  
critical curve of a renewal pinning polymer model in the general case  
of ergodic disorder. More precisely, annealed bounds are given when  
the disorder sequence is no longer i.i.d but has still some nice  
mixing properties.

http://arxiv.org/abs/0903.3704


8299. Invariance principles for local times at the supremum of random  
walks and L\'evy processes
Author(s): Lo\"ic Chaumont (LAREMA) and Ron Arthur Doney

Abstract: We prove that when a sequence of L\'evy processes $X^{(n)}$  
or a normed sequence of random walks $S^{(n)}$ converges a.s. on the  
Skorokhod space toward a L\'evy process $X$, the sequence $L^{(n)}$ of  
local times at the supremum of $X^{(n)}$ converges uniformly on  
compact sets in probability toward the local time at the supremum of $X 
$. A consequence of this result is that the sequence of  
(quadrivariate) ladder processes (both ascending and descending)  
converges jointly in law towards the ladder processes of $X$. As an  
application, we show that in general, the sequence $S^{(n)}$  
conditioned to stay positive converges weakly, jointly with its local  
time at the future minimum, towards the corresponding functional for  
the limiting process $X$. From this we deduce an invariance principle  
for the meander which extends known results for the case of attraction  
to a stable law.

http://arxiv.org/abs/0903.3705


8300. Num\'eraire-invariant preferences in financial modeling
Author(s): Constantinos Kardaras

Abstract: We provide an axiomatic foundation for the representation of  
numeraire-invariant preferences of agents acting in a financial  
market. In a static environment, the simple axioms turn out to be  
equivalent to the following choice rule: the agent prefers one outcome  
over another if and only if the expected (under the agent's subjective  
probability) relative rate of return of the latter outcome with  
respect to the former is nonpositive. With the addition of a  
transitivity requirement, this last preference relation is extended to  
expected logarithmic utility maximization. We also discuss the  
previous in a dynamic environment, where consumption streams are the  
objects of choice. There, a novel result concerning a canonical  
representation of optional measures with unit mass enables one to  
explicitly solve the investment-consumption problem by completely  
separating the two aspects of investment and consumption. Finally, we  
give an application to the problem of optimal numeraire investment  
with a random-time horizon.

http://arxiv.org/abs/0903.3736


8301. Two-parameter stochastic calculus and Malliavin's integration-by- 
parts formula on Wiener space
Author(s): J. R. Norris

Abstract: The integration-by-parts formula discovered by Malliavin for  
the Ito map on Wiener space is proved using the two-parameter  
stochastic calculus. It is also shown that the solution of a one- 
parameter stochastic differential equation driven by a two-parameter  
semimartingale is itself a two-parameter semimartingale.

http://arxiv.org/abs/0903.3855


8302. Entropy, Invertibility and Variational Calculus of the Adapted  
Shifts on Wiener space
Author(s): Ali S\"uleyman \"Ust\"unel

Abstract: In this work we study the necessary and sufficient  
conditions for a positive random variable whose expectation under the  
Wiener measure is one, to be represented as the Radon-Nikodym  
derivative of the image of the Wiener measure under an adapted  
perturbation of identity with the help of the associated innovation  
process. We prove that the innovation conjecture holds if and only if  
the original process is almost surely invertible. We also give  
variational characterizations of the invertibility of the  
perturbations of identity and the representability of a positive  
random variable whose total mass is equal to unity. We prove in  
particular that an adapted perturbation of identity $U=I_W+u$  
satisfying the Girsanov theorem, is invertible if and only if the  
kinetic energy of $u$ is equal to the entropy of the measure induced  
with the action of $U$ on the Wiener measure $\mu$, in other words $U$  
is invertible iff $$ \half \int_W|u|_H^2d\mu=\int_W \frac{dU\mu}{d\mu} 
\log\frac{dU\mu}{d\mu}d\mu >. $$ otherwise the l.h.s. is always  
strictly greater than the r.h.s. The relations with the Monge- 
Kantorovitch measure transportation are also studied. An application  
of these results to a variational problem related to large deviations  
is also given.

http://arxiv.org/abs/0903.3891


8303. Exit time for anchored expansion
Author(s): T. Delmotte and C. Rau

Abstract: Let $(X_n)_{n\geq 0}$ be a reversible random walk on a graph  
$G$ satisfying an anchored isoperimetric inequality. We give upper  
bounds for exit time (and occupation time in transient case) by X of  
any set which contains the root. As an application, we consider random  
environments of $\Z^d$.

http://arxiv.org/abs/0903.3892


8304. Exponential rate of L_p-convergence of intrinsic martingales in  
supercritical branching random walks
Author(s): Gerold Alsmeyer and Alex Iksanov and Sergej Polotsky and  
Uwe Roesler

Abstract: Let $W_n, n\in\mn_{0}$ be an intrinsic martingale with  
almost sure limit $W$ in a supercritical branching random walk. We  
provide criteria for the $L_p$-convergence of the series $\sum_{n\ge  
0} e^{an}(W-W_n)$ for $p>1$ and $a>0$. The result may be viewed as a  
statement about the exponential rate of convergence of $\me |W-W_n|^p$  
to zero.

http://arxiv.org/abs/0903.3935


8305. Fixed point theorems on partial randomness
Author(s): Kohtaro Tadaki

Abstract: In our former work [K. Tadaki, Local Proceedings of CiE  
2008, pp.425-434, 2008], we developed a statistical mechanical  
interpretation of algorithmic information theory by introducing the  
notion of thermodynamic quantities at temperature T, such as free  
energy F(T), energy E(T), and statistical mechanical entropy S(T),  
into the theory. These quantities are real functions of real argument  
T>0. We then discovered that, in the interpretation, the temperature T  
equals to the partial randomness of the values of all these  
thermodynamic quantities, where the notion of partial randomness is a  
stronger representation of the compression rate by program-size  
complexity. Furthermore, we showed that this situation holds for the  
temperature itself as a thermodynamic quantity. Namely, the  
computability of the value of partition function Z(T) gives a  
sufficient condition for T in (0,1) to be a fixed point on partial  
randomness. In this paper, we show that the computability of each of  
all the thermodynamic quantities above gives the sufficient condition  
also. Moreover, we show that the computability of F(T) gives  
completely different fixed points from the computability of Z(T).

http://arxiv.org/abs/0903.3433


8306. Statistical RIP and Semi-Circle Distribution of Incoherent  
Dictionaries
Author(s): Shamgar Gurevich (Berkeley) and Ronny Hadani (Chicago)

Abstract: In this paper we formulate and prove a statistical version  
of the Candes-Tao restricted isometry property (SRIP for short) which  
holds in general for any incoherent dictionary which is a disjoint  
union of orthonormal bases. In addition, we prove that, under  
appropriate normalization, the eigenvalues of the associated Gram  
matrix fluctuate around 1 according to the Wigner semicircle  
distribution. The result is then applied to various dictionaries that  
arise naturally in the setting of finite harmonic analysis, giving, in  
particular, a better understanding on a remark of Applebaum-Howard- 
Searle-Calderbank concerning RIP for the Heisenberg dictionary of  
chirp like functions.

http://arxiv.org/abs/0903.3627


8307. Mass Transportation Proofs of Free Functional Inequalities, and  
Free Poincare Inequalities
Author(s): Michel Ledoux and Ionel Popescu

Abstract: This work is devoted to direct mass transportation proofs of  
families of functional inequalities in the context of one-dimensional  
free probability, avoiding random matrix approximation. The  
inequalities include the free form of the transportation, Log-Sobolev,  
HWI interpolation and Brunn-Minkowski inequalities for strictly convex  
potentials. Sharp constants and some extended versions are put  
forward. The paper also addresses two versions of free Poincar\'e  
inequalities and their interpretation in terms of spectral properties  
of Jacobi operators. The last part establishes the corresponding  
inequalities for measures on $\R_{+}$ with the reference example of  
the Marcenko-Pastur distribution.

http://arxiv.org/abs/0903.3761


8308. Dislocation measure of the fragmentation of a general L\'evy tree
Author(s): Guillaume Voisin (MAPMO)

Abstract: Given a general critical or sub-critical branching mechanism  
and its associated L\'evy continuum random tree, we consider a pruning  
procedure on this tree using a Poisson snake. It defines a  
fragmentation process on the tree. We compute the family of  
dislocation measures associated with this fragmentation. This work  
generalizes the work made for a Brownian tree [Abraham, Serlet] and  
for a tree without Brownian part [Abraham, Delmas].

http://arxiv.org/abs/0903.4024


8309. On the Stability and Ergodicity of an Adaptive Scaling  
Metropolis Algorithm
Author(s): Matti Vihola

Abstract: This paper considers the stability and ergodicity of an  
adaptive random walk Metropolis algorithm. The algorithm adjusts the  
scale of the symmetric proposal distribution continuously, based on  
the observed acceptance probability. A strong law of large numbers is  
shown to hold for functionals bounded on compact sets and growing at  
most exponentially as $\|x\|\to\infty$, assuming that the target  
density is smooth enough and has either compact support or super- 
exponentially decaying tails.

http://arxiv.org/abs/0903.4061


8310. Asymptotic exponential bounds for MLE deviation under minimal  
conditions via classical and generic chaining methods
Author(s): E. Ostrovsky and E. Rogover

Abstract: In this paper non-asymptotic exact exponential estimates are  
derived (under minimal conditions) for the tail of deviation of the  
MLE distribution in the so-called natural terms: natural function,  
natural distance, metric entropy, Banach spaces of random variables,  
contrast function, majorizing measures or, equally, generic chaining.

http://arxiv.org/abs/0903.4062


8311. Random graph asymptotics on high-dimensional tori. II. Volume,  
diameter and mixing time
Author(s): Markus Heydenreich and Remco van der Hofstad

Abstract: For critical (bond-) percolation on general high-dimensional  
torus, this paper answers the following questions: What is the  
diameter of the largest cluster? What is the mixing time of simple  
random walk on the largest cluster? The answer is the same as for  
critical Erdos-Renyi random graphs, and extends an earlier result by  
Nachmias and Peres (2008). We further improve our bound on the size of  
the largest cluster in Heydenreich and van der Hofstad (2007), and  
extend the results on the largest clusters in Borgs, Chayes, van der  
Hofstad, Slade and Spencer (2005a,b) to any finite number of the  
largest clusters. Finally, we show that any weak limit of the largest  
connected component is non-degenerate, which can be viewed as a  
significant sign of critical behavior. This result further justifies  
that the critical value defined in Borgs et al. is appropriate in our  
rather general setting of random subgraphs of high-dimensional tori.

http://arxiv.org/abs/0903.4279


8312. A note on the distribution of the maximum of a set of Poisson  
random variables
Author(s): K. M. Briggs and L. Song and T. Prellberg

Abstract: Given a set of independent Poisson random variables with  
common mean, we study the distribution of their maximum and obtain an  
accurate asymptotic formula to locate the most probable value of the  
maximum. We verify our analytic results with very precise numerical  
computations.

http://arxiv.org/abs/0903.4373


8313. Ballisticity conditions for random walk in random environment
Author(s): Alexander Drewitz and Alejandro F. Ram\'irez

Abstract: Consider a random walk in a uniformly elliptic i.i.d. random  
environment in dimensions $d\ge 2$. In 2002, Sznitman introduced for  
each $\gamma\in (0,1)$ the ballisticity conditions $(T)_\gamma$ and $ 
(T'),$ the latter being defined as the fulfilment of $(T)_\gamma$ for  
all $\gamma\in (0,1).$ He proved that $(T')$ implies ballisticity and  
that for each $\gamma\in (0.5,1),$ $(T)_\gamma$ is equivalent to $(T') 
$. It is conjectured that this equivalence holds for all $\gamma\in  
(0,1).$ Here we prove that for $\gamma\in (\gamma_d,1),$ where $ 
\gamma_d$ is a dimension dependent constant taking values in the  
interval $(0.366,0.388),$ $(T)_\gamma$ is equivalent to $(T').$ This  
is achieved by a detour along the effective criterion, the fulfilment  
of which we establish by a combination of techniques developed by  
Sznitman giving a control on the occurrence of atypical quenched exit  
distributions through boxes.

http://arxiv.org/abs/0903.4465


8314. Outlets of 2D invasion percolation and multiple-armed incipient  
infinite clusters
Author(s): Michael Damron and Artem Sapozhnikov

Abstract: We study invasion percolation in two dimensions, focusing on  
properties of the outlets of the invasion and their relation to  
critical percolation and to incipient infinite clusters (IICs). First  
we compute the exact decay rate of the distribution of both the weight  
of the kth outlet and the volume of the kth pond. Next we prove bounds  
for all moments of the distribution of the number of outlets in an  
annulus. This result leads to almost sure bounds for the number of  
outlets in a box B(2^n) and for the decay rate of the weight of the  
kth outlet to p_c. We then prove existence of multiple-armed IIC  
measures for any number of arms and for any color sequence. We use  
these measures to study the invaded region near outlets and near edges  
in the invasion backbone far from the origin.

http://arxiv.org/abs/0903.4496


8315. Existence, uniqueness and convergence of a particle  
approximation for the Adaptive Biasing Force process
Author(s): Benjamin Jourdain (CERMICS) and Tony Leli\`evre (CERMICS)  
and Rapha\"el Roux (CERMICS)

Abstract: We prove existence and uniqueness for some non linear  
stochastic differential equation used in molecular dynamics, whose non  
linearity comes from a conditional expectation term. We also introduce  
an interacting particle system in order to approximate this  
conditional expectation, providing a discretization scheme for this  
equation.

http://arxiv.org/abs/0903.4518


8316. Phase Transitions in Gravitational Allocation
Author(s): Sourav Chatterjee and Ron Peled and Yuval Peres and Dan Romik

Abstract: Given a Poisson point process of unit masses (``stars'') in  
dimension d>=3, Newtonian gravity partitions space into domains of  
attraction (cells) of equal volume. In earlier work, we showed the  
diameters of these cells have exponential tails. Here we analyze the  
quantitative geometry of the cells and show that their large  
deviations occur at the stretched-exponential scale. More precisely,  
the probability that mass exp(-R^gamma) in a cell travels distance R  
decays like exp(-R^f_d(gamma)) where we identify the functions f_d  
exactly. These functions are piecewise smooth and the discontinuities  
of f_d' represent phase transitions. In dimension d=3, the large  
deviation is due to a ``distant attracting galaxy'' but a phase  
transition occurs when f_3(gamma)=1 (at that point, the fluctuations  
due to individual stars dominate). When d>=5, the large deviation is  
due to a thin tube (a ``wormhole'') along which the star density  
increases monotonically, until the point f_d(gamma)=1 (where again  
fluctuations due to individual stars dominate). In dimension 4 we find  
a double phase transition, where the transition between low- 
dimensional behavior (attracting galaxy) and high-dimensional behavior  
(wormhole) occurs at gamma=4/3. As consequences, we determine the tail  
behavior of the distance from a star to a uniform point in its cell,  
and prove a sharp lower bound for the tail probability of the cell's  
diameter, matching our earlier upper bound.

http://arxiv.org/abs/0903.4647


8317. On the Goodness-of-Fit Testing for Ergodic Diffusion Processes
Author(s): Yury A. Kutoyants

Abstract: We consider the goodness of fit testing problem for ergodic  
diffusion processes. The basic hypothesis is supposed to be simple.  
The diffusion coefficient is known and the alternatives are described  
by the different trend coefficients. We study the asymptotic  
distribution of the Cramer-von Mises type tests based on the empirical  
distribution function and local time estimator of the invariant  
density. At particularly, we propose a transformation which makes  
these tests asymptotically distribution free. We discuss the  
modifications of this test in the case of composite basic hypothesis.

http://arxiv.org/abs/0903.4550


8318. Goodness-of-Fit Tests for Perturbed Dynamical Systems
Author(s): Yury A. Kutoyants

Abstract: We consider the goodness of fit testing problem for  
stochastic differential equation with small diffiusion coefficient.  
The basic hypothesis is always simple and it is described by the known  
trend coefficient. We propose several tests of the type of Cramer-von  
Mises, Kolmogorov-Smirnov and Chi-Square. The power functions of these  
tests we study for a special classes of close alternatives. We discuss  
the construction of the goodness of fit test based on the local time  
and the possibility of the construction of asymptotically distribution  
free tests in the case of composite basic hypothesis.

http://arxiv.org/abs/0903.4612


8319. Correlations, Scale Invariance, and the Riemann Hypothesis
Author(s): B. Holdom

Abstract: Negative correlations in the distribution of prime numbers  
are found to display a scale invariance. There are similarities and  
differences when compared to the scale invariant correlations of  
fractional Brownian motion. We conjecture that a violation of the  
Riemann hypothesis is equivalent to a breakdown of the scale invariance.

http://arxiv.org/abs/0903.2592


8320. Simple Universal Bounds for Chebyshev-Type Quadratures
Author(s): Ron Peled

Abstract: A Chebyshev-type quadrature for a probability measure sigma  
is a distribution which is uniform on n points and has the same first  
k moments as sigma. We give bounds for the smallest possible n  
required to achieve a certain degree k. In contrast to previous  
results of this type, our bounds use only simple properties of sigma  
and are thus applicable in wide generality. In particular, it is shown  
that whenever sigma has bounded density on a finite interval, n may  
increase at most exponentially with k. Examples are given illustrating  
the tightness of our bounds, and applications are given to special  
local constructions on the sphere and cylinder and to an apparently  
new result on Gaussian quadrature. We also introduce the concept of  
random Chebyshev-type quadratures, the case in which nodes are chosen  
by independent random samples from sigma. The concept is discussed and  
some preliminary results are proven. These results were recently  
applied to understand how well can a Poisson process approximate  
certain continuous distributions. We conclude with a list of open  
questions.

http://arxiv.org/abs/0903.4625


8321. On the moments of the meeting time of independent random walks  
in random environment
Author(s): Christophe Gallesco

Abstract: We consider, in the continuous time version, $\gamma$  
independent random walks on $\mathbb{Z_+}$ in random environment in  
the Sinai's regime. Let $T_\gam$ be the first meeting time of one pair  
of the $\gamma$ random walks starting at different positions. We first  
show that the tail of the quenched distribution of $T_\gamma$, after a  
suitable rescaling, converges in probability, to some functional of  
the Brownian motion. Then we compute the law of this functional.  
Eventually, we obtain results about the moments of this meeting time.  
Being $\Eo$ the quenched expectation, we show that, for almost all  
environments $\omega$, $\Eo[T_\gamma^{c}]$ is finite for $c< 
\gamma(\gamma-1)/2$ and infinite for $c>\gamma(\gamma-1)/2$.

http://arxiv.org/abs/0903.4697


8322. The continuum limit of critical random graphs
Author(s): Louigi Addario-Berry and Nicolas Broutin and Christina  
Goldschmidt

Abstract: We consider the Erdos--Renyi random graph G(n,p) inside the  
critical window, that is when p=1/n+\lambda n^{-4/3}, for some fixed  
\lambda in R. Then, as a metric space with the graph distance rescaled  
by n^{1/3}, the sequence of connected components G(n,p) converges  
towards a sequence of continuous compact metric spaces. The result  
relies on a bijection between graphs and certain marked random walks,  
and the theory of continuum random trees. Our result gives access to  
the answers to a great many questions about distances in critical  
random graphs. In particular, we deduce that the diameter of G(n,p)  
rescaled by n^{1/3} converges in distribution to an absolutely  
continuous random variable with finite mean.

http://arxiv.org/abs/0903.4730


8323. Central limit theorems for eigenvalues of deformations of Wigner  
matrices
Author(s): Mireille Capitaine and Catherine Donati-Martin (PMA) and  
Delphine F\'eral (IMB)

Abstract: In this paper, we explain the dependance of the fluctuations  
of the largest eigenvalues of a Deformed Wigner model with respect to  
the eigenvectors of the perturbation matrix. We exhibit quite general  
situations that will give rise to universality or non universality of  
the fluctuations.

http://arxiv.org/abs/0903.4740


8324. Three problems for the clairvoyant demon
Author(s): Geoffrey Grimmett

Abstract: A number of tricky problems in probability are discussed,  
having in common one or more infinite sequences of coin tosses, and a  
representation as a problem in dependent percolation. Three of these  
problems are of `Winkler' type, that is, they ask about what can be  
achieved by a clairvoyant demon.

http://arxiv.org/abs/0903.4749


8325. The Arcsine law as the limit of the internal DLA cluster  
generated by Sinai's walk
Author(s): N. Enriquez and C. Lucas and F. Simenhaus

Abstract: We identify the limit of the internal DLA cluster generated  
by Sinai's walk as the law of a functional of a Brownian motion which  
turns out to be a new interpretation of the Arcsine law.

http://arxiv.org/abs/0903.4831


8326. Recovering a time-homogeneous stock price process from perpetual  
option prices
Author(s): Erik Ekstrom and David Hobson

Abstract: It is well-known how to determine the price of perpetual  
American options if the underlying stock price is a time-homogeneous  
diffusion. In the present paper we consider the inverse problem, i.e.  
given prices of perpetual American options for different strikes we  
show how to construct a time-homogeneous model for the stock price  
which reproduces the given option prices.

http://arxiv.org/abs/0903.4833


8327. Lyapunov exponents of Green's functions for random potentials  
tending to zero
Author(s): Elena Kosygina and Thomas S. Mountford and Martin P. W.  
Zerner

Abstract: We consider quenched and annealed Lyapunov exponents for the  
Green's function of $-\Delta+\gamma V$, where the potentials $V(x), x 
\in\Z^d$, are i.i.d. nonnegative random variables and $\gamma>0$ is a  
scalar. We present a probabilistic proof that both Lyapunov exponents  
scale like $c\sqrt{\gamma}$ as $\gamma$ tends to 0. Here the constant  
$c$ is the same for the quenched as for the annealed exponent and is  
computed explicitly. This improves results obtained previously by Wei- 
Min Wang. We also consider other ways to send the potential to zero  
than multiplying it by a small number.

http://arxiv.org/abs/0903.4928


8328. Hydrodynamic limit of gradient exclusion processes with  
conductances on $\bb Z^d$
Author(s): Fabio J. Valentim

Abstract: Fix a smooth function $\Phi : [l,r] \to \bb R$, defined on  
some interval $[l,r]$ of $\bb R$, such that $0

http://arxiv.org/abs/0903.4993


8329. A Probabilistic Characterization of Random Proximity Catch  
Digraphs and the Associated Tools
Author(s): Elvan Ceyhan

Abstract: Proximity catch digraphs (PCDs) are based on proximity maps  
which yield proximity regions and are special types of proximity  
graphs. PCDs are based on the relative allocation of points from two  
or more classes in a region of interest and have applications in  
various fields. In this article, we provide auxiliary tools for and  
various characterizations of PCDs based on their probabilistic  
behavior. We consider the cases in which the vertices of the PCDs come  
from uniform and non-uniform distributions in the region of interest.  
We also provide some of the newly defined proximity maps as  
illustrative examples.

http://arxiv.org/abs/0903.5005


8330. Convergence to equilibrium of biased plane partitions
Author(s): Pietro Caputo and Fabio Martinelli and Fabio Lucio Toninelli

Abstract: We study a single-flip dynamics for the monotone surface in  
(2+1) dimensions obtained from a boxed plane partition. The surface is  
analyzed as a system of non-intersecting simple paths. When the flips  
have a non-zero bias we prove that there is a positive spectral gap  
uniformly in the boundary conditions and in the size of the system.  
Under the same assumptions, for a system of size $M$, the mixing time  
is shown to be of order $M$ up to logarithmic corrections.

http://arxiv.org/abs/0903.5079


8331. Regularity Properties for a System of Interacting Bessel Processes
Author(s): Sebastian Andres and Max-K. von Renesse

Abstract: We study the regularity of a diffusion on a simplex with  
singular drift and reflecting boundary condition which describes a  
finite system of particles on an interval with Coulomb interaction and  
reflection between nearest neighbors. As our main result we establish  
the Feller property for the process in both cases of repulsion and  
attraction. In particular the system can be started from any initial  
state, including multiple point configurations. Moreover we show that  
the process is a Euclidean semi-martingale if and only if the  
interaction is repulsive. Hence, contrary to classical results about  
reflecting Brownian motion in smooth domains, in the attractive regime  
a construction via a system of Skorokhod SDEs is impossible. Finally,  
we establish exponential heat kernel gradient estimates in the  
repulsive regime. The main proof for the attractive case is based on  
potential theory in Sobolev spaceswith Muckenhoupt weights.

http://arxiv.org/abs/0903.5085


8332. First passage percolation on random graphs with finite mean  
degrees
Author(s): S. Bhamidi and R. van der Hofstad and G. Hooghiemstra

Abstract: We study first passage percolation on the configuration  
model. Assuming that each edge has an independent exponentially  
distributed edge weight, we derive explicit distributional asymptotics  
for the minimum weight between two randomly chosen connected vertices  
in the network, as well as for the number of edges on the least weight  
path, the so-called hopcount. We analyze the configuration model with  
degree power-law exponent \tau >2, in which the degrees are assumed to  
be i.i.d. with a tail distribution which is either of power-law form  
with exponent \tau-1>1, or has even thinner tails (\tau=\infty). In  
this model, the degrees have a finite first moment, while the variance  
is finite for \tau>3, but infinite for \tau\in (2,3). We prove a  
central limit theorem for the hopcount, with asymptotically equal  
means and variances equal to \alpha\log{n}, where \alpha\in (0,1) for  
\tau\in (2,3), while \alpha>1 for \tau>3. Here n denotes the size of  
the graph. For \tau\in (2,3), it is known that the graph distance  
between two randomly chosen connected vertices is proportional to \log 
\log{n} (van der Hofstad, Hooghiemtra and Znamenski (2007), i.e.,  
distances are ultra small. Thus, the addition of edge weights causes a  
marked change in the geometry of the network. We further study the  
weight of the least weight path, and prove convergence in distribution  
of an appropriately centered version.

http://arxiv.org/abs/0903.5136


8333. Approximation of Stable-dominated Semigroups
Author(s): Pawe/l Sztonyk

Abstract: We consider Feller semigroups of operators determinated by  
systems of jumps dominated by the rotation invariant stable L\'evy  
measure. Using an approximation schema we prove the existence and  
obtain estimates of corresponding heat kernels.

http://arxiv.org/abs/0903.5294


8334. Asymptotics of The Hole Probability for Zeros of Random Entire  
Functions
Author(s): Alon Nishry

Abstract: We study the hole probability of Gaussian random entire  
functions. More specifically, we work with the flat model (the zero  
set of this function has a distribution which is invariant with  
respect to the plane isometries). A hole is the event where the  
function has no zeros in a disc of radius r. We show that the  
logarithm of the probability of the hole event decays asymptotically  
like -3/4 * e^2 * r^4 + o(r^4). We also study the behavior of the hole  
probability with other types of random coefficients.

http://arxiv.org/abs/0903.4970


8335. Maximum entropy Gaussian approximation for the number of integer  
points and volumes of polytopes
Author(s): Alexander Barvinok and John Hartigan

Abstract: We describe a maximum entropy approach for computing volumes  
and counting integer points in polyhedra. To estimate the number of  
points from a particular set X from R^n in a polyhedron P in R^n we  
construct a probability distribution on the set X by solving a certain  
entropy maximization problem such that a) the probability mass  
function is constant on the intersection of P and X and b) the  
expectation of the distribution lies in P. This allows us to apply  
Central Limit Theorem type arguments to deduce computationally  
efficient approximations for the number of integer points, volumes,  
and the number of 0-1 vectors in the polytope in a number of cases.  
Examples include polytopes of doubly stochastic matrices and  
polystochastic tensors, polytopes defined by totally unimodular  
matrices of constraints, and polytopes associated to some covering  
problems.

http://arxiv.org/abs/0903.5223


8336. Unspecified distribution in single disorder problem
Author(s): Wojciech Sarnowski and Krzysztof Szajowski

Abstract: We register a stochastic sequence affected by one disorder.  
Monitoring of the sequence is made in the circumstances when not full  
information about distributions before and after the change is  
available. The initial problem of disorder detection is transformed to  
optimal stopping of observed sequence. Formula for optimal decision  
functions is derived.

http://arxiv.org/abs/0903.5341


8337. Exact Non-Parametric Bayesian Inference on Infinite Trees
Author(s): Marcus Hutter

Abstract: Given i.i.d. data from an unknown distribution, we consider  
the problem of predicting future items. An adaptive way to estimate  
the probability density is to recursively subdivide the domain to an  
appropriate data-dependent granularity. A Bayesian would assign a data- 
independent prior probability to "subdivide", which leads to a prior  
over infinite(ly many) trees. We derive an exact, fast, and simple  
inference algorithm for such a prior, for the data evidence, the  
predictive distribution, the effective model dimension, moments, and  
other quantities. We prove asymptotic convergence and consistency  
results, and illustrate the behavior of our model on some prototypical  
functions.

http://arxiv.org/abs/0903.5342


8338. Analytic and asymptotic properties of multivariate generalized  
Linnik's probability densities
Author(s): S.C. Lim and L.P. Teo

Abstract: This paper studies the properties of the probability density  
function $p_{\alpha,\nu, n}(\mathbf{x})$ of the $n$-variate  
generalized Linnik distribution whose characteristic function $ 
\varphi_{\alpha,\nu,n}(\boldsymbol{t})$ is given by \varphi_{\alpha, 
\nu,n}(\boldsymbol{t})=\frac{1} {(1+\Vert\boldsymbol{t} 
\Vert^{\alpha})^{\nu}}, \alpha\in (0,2], \nu>0, \boldsymbol{t}\in  
\mathbb{R}^n, where $\Vert\boldsymbol{t}\Vert$ is the Euclidean norm  
of $\boldsymbol{t}\in\mathbb{R}^n$. Integral representations of  
$p_{\alpha,\nu, n}(\mathbf{x})$ are obtained and used to derive the  
asymptotic expansions of $p_{\alpha,\nu, n}(\mathbf{x})$ when $\Vert 
\mathbf{x}\Vert\to 0$ and $\Vert\mathbf{x}\Vert\to \infty$  
respectively. It is shown that under certain conditions which are  
arithmetic in nature, $p_{\alpha,\nu, n}(\mathbf{x})$ can be  
represented in terms of entire functions.

http://arxiv.org/abs/0903.5344


8339. Dutch Books and Combinatorial Games
Author(s): Peter Harremoes

Abstract: The theory of combinatorial game (like board games) and the  
theory of social games (where one looks for Nash equilibria) are  
normally considered as two separate theories. Here we shall see what  
comes out of combining the ideas. The central idea is Conway's  
observation that real numbers can be interpreted as special types of  
combinatorial games. Therefore the payoff function of a social game is  
a combinatorial game. Probability theory should be considered as a  
safety net that prevents inconsistent decisions via the Dutch Book  
Argument. This result can be extended to situations where the payoff  
function is a more general game than a real number. The main  
difference between number valued payoff and game valued payoff is that  
a probability distribution that gives non-negative mean payoff does  
not ensure that the game will be lost due to the existence of  
infinitisimal games. Also the Ramsay/de Finetti theorem on exchangable  
sequences is discussed.

http://arxiv.org/abs/0903.5429


8340. Random walks in $(\mathbb{Z}_+)^2$ with non-zero drift absorbed  
at the axes
Author(s): Irina Kurkova and Kilian Raschel

Abstract: Spatially homogeneous random walks in $(\mathbb{Z}_{+})^{2}$  
with non-zero jump probabilities at distance at most 1, with non-zero  
drift in the interior of the quadrant and absorbed when reaching the  
axes are studied. Absorption probabilities generating functions are  
obtained and the asymptotic of absorption probabilities along the axes  
is made explicit. The asymptotic of the Green functions is computed  
along all different infinite paths of states, in particular along  
those approaching the axes.

http://arxiv.org/abs/0903.5486


8341. Convergence of delay differential equations driven by fractional  
Brownian motion
Author(s): Marco Ferrante Carles Rovira

Abstract: In this note we prove an existence and uniqueness result of  
solution for stochastic differential delay equations with hereditary  
drift driven by a fractional Brownian motion with Hurst parameter $H >  
1/2$. Then, we show that, when the delay goes to zero, the solutions  
to these equations converge, almost surely and in $L^p$, to the  
solution for the equation without delay. The stochastic integral with  
respect to the fractional Brownian motion is a pathwise Riemann- 
Stieltjes integral.

http://arxiv.org/abs/0903.5498


8342. Hydrostatics and dynamical large deviations of boundary driven  
gradient symmetric exclusion
Author(s): Jonathan Farfan and Claudio Landim and Mustapha Mourragui

Abstract: We prove hydrostatics of boundary driven gradient exclusion  
processes, Fick's law and we present a simple proof of the dynamical  
large deviations principle which holds in any dimension

http://arxiv.org/abs/0903.5526


8343. Exact Tail Asymptotics of Dirichlet Distributions
Author(s): Enkelejd Hashorva

Abstract: Let X be a generalised symmetrised Dirichlet random vector  
in R^k, and let t_n be thresholds such that P{X> t_n} tends to 0 as n  
goes infinity. In this paper we derive an exact asymptotic expansion  
of P{X> t_n} assuming that the associated random radius of X has  
distribution function in the Gumbel max-domain of attraction

http://arxiv.org/abs/0904.0144


8344. Noise Correlation Bounds for Uniform Low Degree Functions
Author(s): Per Austrin and Elchanan Mossel

Abstract: We study correlation bounds under pairwise independent  
distributions for functions with no large Fourier coefficients.  
Functions in which all Fourier coefficients are bounded by $\delta$  
are called $\delta$-{\em uniform}. The search for such bounds is  
motivated by their potential applicability to hardness of  
approximation, derandomization, and additive combinatorics. In our  
main result we show that $\E[f_1(X_1^1,...,X_1^n) ...  
f_k(X_k^1,...,X_k^n)]$ is close to 0 under the following assumptions:  
1. The vectors $\{(X_1^j,...,X_k^j) : 1 \leq j \leq n\}$ are i.i.d,  
and for each $j$ the vector $(X_1^j,...,X_k^j)$ has a pairwise  
independent distribution. 2. The functions $f_i$ are uniform. 3. The  
functions $f_i$ are of low degree. We compare our result with recent  
results by the second author for low influence functions and to recent  
results in additive combinatorics using the Gowers norm. Our proofs  
extend some techniques from the theory of hypercontractivity to a  
multilinear setup.

http://arxiv.org/abs/0904.0157


8345. Pointwise ergodic theorems with rate and application to limit  
theorems for stationary processes
Author(s): Christophe Cuny

Abstract: We obtain pointwise ergodic theorems with rate under  
conditions expressed in terms of the convergence of series involving $ 
\|\sum_{k=1} ^nf\circ \theta^k\|_2$, improving previous results. Then,  
using known results on martingale approximation, we obtain some LIL  
for stationary ergodic processes and quenched central limit theorems  
for functional of Markov chains. The proofs are based on the use of  
the spectral theorem and, on a recent work of Zhao-Woodroofe extending  
a method of Derriennic-Lin.

http://arxiv.org/abs/0904.0185


8346. A new model for evolution in a spatial continuum
Author(s): N.H. Barton and A.M. Etheridge and A. Veber

Abstract: We introduce a new model for populations evolving in a  
spatial continuum. This model can be thought of as a spatial version  
of the Lambda-Fleming-Viot process. It explicitly incorporates both  
small scale reproduction events and large scale extinction- 
recolonisation events. The lineages ancestral to a sample from a  
population evolving according to this model can be described in terms  
of a spatial version of the Lambda-coalescent. Using a technique of  
Evans(1997), we prove existence and uniqueness in law for the model.  
We then investigate the asymptotic behaviour of the genealogy of a  
finite number of individuals sampled uniformly at random (or more  
generally `far enough apart') from a two-dimensional torus of side L  
as L tends to infinity. Under appropriate conditions (and on a  
suitable timescale), we can obtain as limiting genealogical processes  
a Kingman coalescent, a more general Lambda-coalescent or a system of  
coalescing Brownian motions (with a non-local coalescence mechanism).

http://arxiv.org/abs/0904.0210


8347. Percolation and Connectivity in AB Random Geometric Graphs
Author(s): Srikanth K. Iyer (INRIA Rocquencourt) and D. Yogeshwaran  
(INRIA Rocquencourt)

Abstract: We study a generalization to the continuum of the $AB$  
percolation model on discrete lattices. Let $\Pl,\Pm$ be independent  
Poisson point processes in $\mR^d$, $d \geq 2,$ of intensities $ 
\lambda, \mu$ respectively. The $AB$ random geometric graph $G(\lam,  
\mu, r)$ is a graph whose vertex set is $\Pl$ with edges between any  
two points $X_i, X_j \in \Pl$ provided there exists a $Y \in \Pm$ such  
that $|X_k - Y| \leq r$, $k=i, j$. We investigate percolation and  
connectivity in $AB$ random geometric graphs.

http://arxiv.org/abs/0904.0223


8348. Self-similarity and random walks
Author(s): Vadim A. Kaimanovich

Abstract: This is an introductory level survey of some topics from a  
new branch of fractal analysis -- the theory of self-similar groups.  
We discuss recent works on random walks on self-similar groups and  
their applications to the problem of amenability for these groups.

http://arxiv.org/abs/0904.0047


8349. Step Size in Stein's Method of Exchangeable Pairs
Author(s): Nathan Ross

Abstract: Stein's method of exchangeable pairs is examined through  
five examples in relation to Poisson and normal distribution  
approximation. In particular, in the case where the exchangeable pair  
is constructed from a reversible Markov chain, we analyze how  
modifying the step size of the chain in a natural way affects the  
error term in the approximation acquired through Stein's method. It  
has been noted for the normal approximation that smaller step sizes  
may yield better bounds, and we obtain the first rigorous results that  
verify this intuition. For the examples associated to the normal  
distribution, the bound on the error is expressed in terms of the  
spectrum of the underlying chain, a characteristic of the chain  
related to convergence rates. The Poisson approximation using  
exchangeable pairs is less studied than the normal, but in the  
examples presented here the same principles hold.

http://arxiv.org/abs/0904.0284


8350. Backward stochastic dynamics on a filtered probability space
Author(s): G. Liang and T. Lyons and Z. Qian (Mathematical Institute  
and University of Oxford) (Oxford-Man Institute, University of Oxford)

Abstract: We consider the following backward stochastic dynamics based  
on a general filtered probability space (\Omega, F, {F_t}_{t\geq  
0},P): dY_t=-f_0(t,Y_t,L(M)_t)dt-\sum_{i=1}^{N}f_i(t,Y_t)dB_t^i+dM_t,  
Y_T=\xi \in F_T where B is an N-dimensional Brownian motion as given,  
and M, a correction term, is a square-integrable martingale to be  
determined. Under adapteness constraints on Y, we prove that the  
equation admits a solution pair (Y,M) which is unique in the sense of  
strict solutions to be introduced in the main text. The martingale  
representation is not required, and in order to prove the existence  
and uniqueness, we establish the existence and uniqueness of a  
functional differential equation, in a form V=\mathbb{L}(V), where  
\mathbb{L} is a non-linear functional. Finally we indicate a  
connection between the backward stochastic equations discussed here  
and a class of non-linear PDE, namely semi-linear parabolic PDE with  
non-local integral term.

http://arxiv.org/abs/0904.0377


8351. Hamilton cycles in 3-out
Author(s): Tom Bohman and Alan Frieze

Abstract: Let G_{\rm 3-out} denote the random graph on vertex set [n]  
in which each vertex chooses 3 neighbors uniformly at random. Note  
that G_{\rm 3-out} has minimum degree 3 and average degree 6. We prove  
that the probability that G_{\rm 3-out} is Hamiltonian goes to 1 as n  
tends to infinity.

http://arxiv.org/abs/0904.0431


8352. Ces\'aro summation for random fields
Author(s): Allan Gut (Uppsala University) and Ulrich Stadtmueller (Ulm  
University)

Abstract: Various methods of summation for divergent series of real  
numbers have been generalized to analogous results for sums of iid  
random variables. The natural extension of results corresponding to Ces 
\`aro summation amounts to proving almost sure convergence of the Ces 
\`aro means. In the present paper we extend such results as well as  
weak laws and results on complete convergence to random fields, more  
specifically to random variables indexed by $\mathbb{Z}_+^2$, the  
positive two-dimensional integer lattice points.

http://arxiv.org/abs/0904.0538


8353. A Large Deviation Principle for Martingales over Brownian  
Filtration
Author(s): Z. Qian and C. Xu (Mathematical Institute and University of  
Oxford)

Abstract: In this article we establish a large deviation principle for  
the family {\nu_{\epsilon}:\epsilon \in (0,1)} of distributions of the  
scaled stochastic processes {P_{-\log\sqrt{\epsilon}}Z_t}_{t\leq 1},  
where (Z_t)_{t\in \lbrack 0,1]} is a square-integrable martingale over  
Brownian filtration and (P_t)_{t\geq 0} is the Ornstein-Uhlenbeck  
semigroup. The rate function is identified as well in terms of the  
Wiener-It\^{o} chaos decomposition of the terminal value Z_{1}. The  
result is established by developing a continuity theorem for large  
deviations, together with two essential tools, the hypercontractivity  
of the Ornstein-Uhlenbeck semigroup and Lyons' continuity theorem for  
solutions of Stratonovich type stochastic differential equations.

http://arxiv.org/abs/0904.0547


8354. A new approach to LIBOR modeling
Author(s): Martin Keller-Ressel and Antonis Papapantoleon and Josef  
Teichmann

Abstract: We provide a general and flexible approach to LIBOR modeling  
based on the class of affine factor processes. Our approach respects  
the basic economic requirement that LIBOR rates are non-negative, and  
the basic requirement from mathematical finance that LIBOR rates are  
analytically tractable martingales with respect to their own forward  
measure. Additionally, and most importantly, our approach also leads  
to analytically tractable expressions of multi-LIBOR payoffs. This  
approach unifies therefore the advantages of well-known forward price  
models with those of classical LIBOR rate models. Several examples are  
added and prototypical volatility smiles are shown. We believe that  
the CIR-process based LIBOR model might be of particular interest for  
applications, since closed form valuation formulas for caps and  
swaptions are derived.

http://arxiv.org/abs/0904.0555


8355. Limit conditional distributions for bivariate vectors with polar  
representation
Author(s): Anne-Laure Foug\`eres (ICJ) and Philippe Soulier (MODAL'X)

Abstract: We investigate conditions for the existence of the limiting  
conditional distribution of a bivariate random vector when one  
component becomes large. We revisit the existing literature on the  
topic, and present some new sufficient conditions. We concentrate on  
the case where the conditioning variable belongs to the maximum domain  
of attraction of the Gumbel law, and we study geometric conditions on  
the joint distribution of the vector. We show that these conditions  
are of a local nature and imply asymptotic independence when both  
variables belong to the domain of attraction of an extreme value  
distribution. The new model we introduce can also be useful for  
simulations.

http://arxiv.org/abs/0904.0580


8356. A limit theorem for trees of alleles in branching processes with  
rare neutral mutations
Author(s): Jean Bertoin (DMA and Pma)

Abstract: We are interested in the genealogical structure of alleles  
for a Bienaym\'e-Galton-Watson branching process with neutral  
mutations (infinite alleles model), in the situation where the initial  
population is large and the mutation rate small. We shall establish  
that for an appropriate regime, the process of the sizes of the  
allelic sub-families converges in distribution to a certain continuous  
state branching process (i.e. a Jirina process) in discrete time. It 
\^o's excursion theory and the L\'eevy-It\^o decomposition of  
subordinators provide fundamental insights for the results.

http://arxiv.org/abs/0904.0581


8357. Prime chains and Pratt trees
Author(s): Kevin Ford and Sergei V. Konyagin and Florian Luca

Abstract: We study the distribution of prime chains, which are  
sequences p_1,...,p_k of primes for which p_{j+1}\equiv 1\pmod{p_j}  
for each j. We first give conditional upper bounds on the length of  
Cunningham chains, chains with p_{j+1}=2p_j+1 for each j. We give  
estimates for the number of chains with p_k\le x (k variable), and the  
number of chains with p_1=p and p_k \le px. The majority of the paper  
concerns the distribution of H(p), the length of the longest chain  
with p_k=p, which is also the height of the Pratt tree for p. We show  
H(p)\ge c\log\log p and H(p)\le (\log p)^{1-c'} for almost all p, with  
c,c' explicit positive constants. We can take, for any \epsilon>0, c=e- 
\epsilon assuming the Elliott-Halberstam conjecture. A stochastic  
model of the Pratt tree, based on a branching random walk, is  
introduced and analyzed. The model suggests that for most p, H(p)  
stays very close to e \log\log p.

http://arxiv.org/abs/0904.0473


8358. Analysis of the market weights under the Volatility-Stabilized  
Market models
Author(s): Soumik Pal

Abstract: We derive the joint density of market weights, at fixed  
times and suitable stopping times, of the Volatility-stabilized market  
models introduced by Fernholz & Karatzas in 2005. The key argument  
involves computing the exit density of a collection of independent  
Bessel-square processes of different dimensions from the unit simplex  
in n-dimension. As a side result, we furnish a novel proof of the  
transition density function of the multi-allele Wright-Fisher model  
which was originally derived by Griffiths by orthogonal series  
expansion.

http://arxiv.org/abs/0904.0656


8359. Optimal Multi-Modes Switching Problem in Infinite Horizon
Author(s): Brahim El Asri

Abstract: This paper studies the problem of the deterministic version  
of the Verification Theorem for the optimal m-states switching in  
infinite horizon under Markovian framework with arbitrary switching  
cost functions. The problem is formulated as an extended impulse  
control problem and solved by means of probabilistic tools such as the  
Snell envelop of processes and reflected backward stochastic  
differential equations. A viscosity solutions approach is employed to  
carry out a finne analysis on the associated system of m variational  
inequalities with inter-connected obstacles. We show that the vector  
of value functions of the optimal problem is the unique viscosity  
solution to the system. This problem is in relation with the valuation  
of firms in a financial market.

http://arxiv.org/abs/0904.0707


8360. On the reversal of radial SLE, I: Commutation Relations in Annuli
Author(s): Dapeng Zhan

Abstract: We aim at finding the reversal of radial SLE and proving the  
reversibility of whole-plane SLE. For this purpose, we define annulus  
SLE$(\kappa,\Lambda)$ processes in doubly connected domains with one  
marked boundary point. We derive some partial differential equation  
for $\Lambda$, which is sufficient for the annulus SLE$(\kappa,\Lambda) 
$ process to satisfy commutation relation. If $\Lambda$ satisfies this  
PDE, then using a coupling technique, we are able to construct a  
global commutation coupling of two annulus SLE$(\kappa,\Lambda)$  
processes. If more conditions are satisfied, the coupling exists in  
the degenerate case, which becomes a coupling of two whole-plane SLE$_ 
\kappa$ processes. The reversibility of whole-plane SLE$_\kappa$  
follows from this coupling together with the assumption that such  
annulus SLE$(\kappa,\Lambda)$ trace ends at the marked point. We then  
conclude that the limit of such annulus SLE$(\kappa,\Lambda)$ trace is  
the reversal of radial SLE$_\kappa$ trace. In the end, we derive some  
particular solutions to the PDE for $\Lambda$.

http://arxiv.org/abs/0904.0808


8361. A Central Limit Theorem and its Applications to Multicolor  
Randomly Reinforced Urns
Author(s): Patrizia Berti and Irene Crimaldi and Luca Pratelli and  
Pietro Rigo

Abstract: We give a central limit theorem, which has applications to  
Bayesian statistics and urn problems. The latter are investigated, by  
paying special attention to multicolor randomly reinforced generalized  
Polya urns.

http://arxiv.org/abs/0904.0932


8362. Regularity of Intersection Local Times of Fractional Brownian  
Motions
Author(s): Dongsheng Wu (University of Alabama in Huntsville) and  
Yimin Xiao (Michigan State University)

Abstract: Let $B^{\alpha_i}$ be an $(N_i,d)$-fractional Brownian  
motion with Hurst index ${\alpha_i}$ ($i=1,2$), and let $B^{\alpha_1}$  
and $B^{\alpha_2}$ be independent. We prove that, if $\frac{N_1} 
{\alpha_1}+\frac{N_2}{\alpha_2}>d$, then the intersection local times  
of $B^{\alpha_1}$ and $B^{\alpha_2}$ exist, and have a continuous  
version. We also establish H\"{o}lder conditions for the intersection  
local times and determine the Hausdorff and packing dimensions of the  
sets of intersection times and intersection points. One of the main  
motivations of this paper is from the results of Nualart and Ortiz- 
Latorre ({\it J. Theor. Probab.} {\bf 20} (2007)), where the existence  
of the intersection local times of two independent $(1,d)$-fractional  
Brownian motions with the same Hurst index was studied by using a  
different method. Our results show that anisotropy brings subtle  
differences into the analytic properties of the intersection local  
times as well as rich geometric structures into the sets of  
intersection times and intersection points.

http://arxiv.org/abs/0904.0949


8363. Exact Asymptotics of Bivariate Scale Mixture Distributions
Author(s): Enkelejd Hashorva

Abstract: Let (RU_1, R U_2) be a given bivariate scale mixture random  
vector, with R>0 being independent of the bivariate random vector  
(U_1,U_2). In this paper we derive exact asymptotic expansions of the  
tail probability P{RU_1> x, RU_2> ax}, a \in (0,1] as x tends infintiy  
assuming that R has distribution function in the Gumbel max-domain of  
attraction and (U_1,U_2) has a specific tail behaviour around some  
absorbing point. As a special case of our results we retrieve the  
exact asymptotic behaviour of bivariate polar random vectors. We apply  
our results to investigate the asymptotic independence and the  
asymptotic behaviour of conditional excess for bivariate scale mixture  
distributions.

http://arxiv.org/abs/0904.0966


8364. On the stability of call/put option's prices in incomplete  
models under statistical estimations
Author(s): L. Vostrikova

Abstract: In exponential semi-martingale setting for risky asset we  
estimate the difference of prices of options when initial physical  
measure $P$ and corresponding martingale measure $Q$ change to $ 
\tilde{P}$ and $\tilde{Q}$ respectively. Then, we estimate $L_1$- 
distance of option's prices for corresponding parametric models with  
known and estimated parameters. The results are applied to exponential  
Levy models with special choice of martingale measure as Esscher  
measure, minimal entropy measure and $f^q$-minimal martingale measure.  
We illustrate our results by considering GMY and CGMY models.

http://arxiv.org/abs/0904.0984


8365. Breaking through the Thresholds: an Analysis for Iterative  
Reweighted $\ell_1$ Minimization via the Grassmann Angle Framework
Author(s): Weiyu Xu and M. Amin Khajehnejad and Salman Avestimehr and  
Babak Hassibi

Abstract: It is now well understood that $\ell_1$ minimization  
algorithm is able to recover sparse signals from incomplete  
measurements [2], [1], [3] and sharp recoverable sparsity thresholds  
have also been obtained for the $\ell_1$ minimization algorithm.  
However, even though iterative reweighted $\ell_1$ minimization  
algorithms or related algorithms have been empirically observed to  
boost the recoverable sparsity thresholds for certain types of  
signals, no rigorous theoretical results have been established to  
prove this fact. In this paper, we try to provide a theoretical  
foundation for analyzing the iterative reweighted $\ell_1$ algorithms.  
In particular, we show that for a nontrivial class of signals, the  
iterative reweighted $\ell_1$ minimization can indeed deliver  
recoverable sparsity thresholds larger than that given in [1], [3].  
Our results are based on a high-dimensional geometrical analysis  
(Grassmann angle analysis) of the null-space characterization for $ 
\ell_1$ minimization and weighted $\ell_1$ minimization algorithms.

http://arxiv.org/abs/0904.0994


8366. Thin Partitions: Isoperimetric Inequalities and Sampling  
Algorithms for some Nonconvex Families
Author(s): Karthekeyan Chandrasekaran and Daniel Dadush and Santosh  
Vempala

Abstract: Star-shaped bodies are an important nonconvex generalization  
of convex bodies (e.g., linear programming with violations). Here we  
present an efficient algorithm for sampling a given star-shaped body.  
The complexity of the algorithm grows polynomially in the dimension  
and inverse polynomially in the fraction of the volume taken up by the  
kernel of the star-shaped body. The analysis is based on a new  
isoperimetric inequality. Our main technical contribution is a tool  
for proving such inequalities when the domain is not convex. As a  
consequence, we obtain a polynomial algorithm for computing the volume  
of such a set as well. In contrast, linear optimization over star- 
shaped sets is NP-hard.

http://arxiv.org/abs/0904.0583


8367. Comportement asymptotique des polyn\^omes orthogonaux associ\'es  
\`a un poids ayant un z\'ero d'ordre fractionnaire sur le cercle.  
Applications aux valeurs propres d'une classe de matrices al\'eatoires  
unitaires
Author(s): Philippe Rambour (LM-Orsay) and Abdellatif Seghier (LM-Orsay)

Abstract: Asymptotic behavior of orthogonal polynomials on the circle,  
with respect to a weight having a fractional zero on the torus.  
Applications to the eigenvalues of certain unitary random matrices.  
This paper is devoted to the orthogonal polynomial on the circle, with  
respect to a weight of type $ f=(1-\cos \theta )^\alpha c$ where $c$  
is a sufficiently smooth function and $\alpha \in ]-{1/2}, {1/2}[$. We  
obtain an asymptotic expansion of the coefficients of this polynomial  
and of $\Phi^{(p)}_{N}(1)$ for all integer $p$. These results allow us  
to obtain an asymptotic expansion of the associated Christofel-Darboux  
kernel, and to compute the distribution of the eigenvalues of a family  
of random unitary matrices. The proof of the resuts related with the  
orthogonal polynomials are essentialy based on the inversion of  
Toeplitz matice associated to the symbol $f$.

http://arxiv.org/abs/0904.0777


8368. Strong law of large numbers on graphs and groups with  
applications -- I
Author(s): Natalia Mosina and Alexander Ushakov

Abstract: We introduce the notion of the mean-set (expectation) of a  
graph-(group-)valued random element $\xi$ and prove a generalization  
of the strong law of large numbers on graphs and groups. Furthermore,  
we prove an analogue of the classical Chebyshev's inequality for $\xi 
$. We show that our generalized law of large numbers, as a new  
theoretical tool, provides a framework for practical applications;  
namely, it has implications for cryptanalysis of group-based  
authentication protocols. In addition, we prove several results about  
configurations of mean-sets in graphs and their applications. In  
particular, we discuss computational problems and methods of computing  
of mean-sets in practice and propose an algorithm for such computation.

http://arxiv.org/abs/0904.1005


8369. Invariance Principles for Homogeneous Sums: Universality of  
Gaussian Wiener Chaos
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X) and  
Gesine Reinert

Abstract: We compute explicit bounds in the normal and chi-square  
approximations of multilinear homogenous sums (of arbitrary order) of  
general centered independent random variables with unit variance. Our  
techniques combine an invariance principle by Mossel, O'Donnell and  
Oleszkiewicz with a refinement of some recent results by Nourdin and  
Peccati, about the approximation of laws of random variables belonging  
to a fixed (Gaussian) Wiener chaos. In particular, we show that  
chaotic random variables enjoy the following form of  
\textsl{universality}: (a) the normal and chi-square approximations of  
any homogenous sum can be completely characterized and assessed by  
first switching to its Wiener chaos counterpart, and (b) the simple  
upper bounds and convergence criteria available on the Wiener chaos  
extend almost verbatim to the class of homogeneous sums. These results  
partially rely on the notion of "low influences" for functions defined  
on product spaces, and provide a generalization of central and non- 
central limit theorems proved by Nourdin, Nualart and Peccati. They  
also imply a further drastic simplification of the method of moments  
and cumulants -- as applied to the proof of probabilistic limit  
theorems -- and yield substantial generalizations, new proofs and new  
insights into some classic findings by de Jong and Rotar'. Our tools  
involve the use of Malliavin calculus, and of both the Stein's method  
and the Lindeberg invariance principle for probabilistic approximations.

http://arxiv.org/abs/0904.1153


8370. Martingales and Rates of Presence in Homogeneous Fragmentations
Author(s): Nathalie Krell (MAP5) and Alain Rouault (LM-Versailles)

Abstract: In mass-conservative homogeneous fragmentations, sizes of  
the fragments decrease at {\bf asymptotic} exponential rates. Like in  
branching processes, two situations occur: either the number of such  
fragments is exponentially growing - the rate is effective -, or the  
probability of presence of such fragments is exponentially decreasing.  
In a recent paper, N. Krell considers fragments whose sizes decrease  
at {\bf exact} exponential rates. In this new setting, she  
characterizes the effective rates and studies Hausdorff dimension. The  
present paper carries out a detailed analysis of this model and focus  
on presence probabilities, using the spine method and a suitable  
martingale. For the sake of completeness, we compare our results with  
results and methods of the classical model.

http://arxiv.org/abs/0904.1167


8371. Space-time duality for fractional diffusion
Author(s): Boris Baeumer and Mark M. Meerschaert and Erkan Nane

Abstract: Zolotarev proved a duality result that relates stable  
densities with different indices. In this paper, we show how Zolotarev  
duality leads to some interesting results on fractional diffusion.  
Fractional diffusion equations employ fractional derivatives in place  
of the usual integer order derivatives. They govern scaling limits of  
random walk models, with power law jumps leading to fractional  
derivatives in space, and power law waiting times between the jumps  
leading to fractional derivatives in time. The limit process is a  
stable L\'evy motion that models the jumps, subordinated to an inverse  
stable process that models the waiting times. Using duality, we relate  
the density of a spectrally negative stable process with index $1< 
\alpha<2$ to the density of the hitting time of a stable subordinator  
with index $1/\alpha$, and thereby unify some recent results in the  
literature. These results also provide a concrete interpretation of  
Zolotarev duality in terms of the fractional diffusion model.

http://arxiv.org/abs/0904.1176


8372. Optimal Holder exponent for the SLE path
Author(s): Fredrik Johansson and Gregory F. Lawler

Abstract: We prove an upper bound on the optimal H\"older exponent for  
the chordal SLE path parameterized by capacity and thereby establish  
the optimal exponent as conjectured by J. Lind. We also give a new  
proof of the lower bound. Our proofs are based on the sharp estimates  
of moments of the derivative of the inverse map. In particular, we  
improve an estimate of the second author.

http://arxiv.org/abs/0904.1180


8373. Curvature, concentration, and error estimates for Markov chain  
Monte Carlo
Author(s): Ald\'eric Joulin and Yann Ollivier

Abstract: Under a "positive curvature" assumption expressing a kind of  
metric ergodicity, we provide explicit non-asymptotic estimates for  
the rate of convergence of empirical means of Markov chains, together  
with a Gaussian or exponential control on the deviations of empirical  
means.

http://arxiv.org/abs/0904.1312


8374. Limit theorems for nonlinear functionals of Volterra processes  
via white noise analysis
Author(s): S\'ebastien Darses and Ivan Nourdin (PMA) and David Nualart

Abstract: By means of white noise analysis, we prove some limit  
theorems for nonlinear functionals of a given Volterra process. In  
particular, our results apply to fractional Brownian motion (fBm), and  
should be compared with the classical convergence results of the  
eighties by Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu,  
as well as the recent advances concerning the construction of a L\'evy  
area for fBm by Coutin, Qian and Unterberger

http://arxiv.org/abs/0904.1401


8375. The Engel algorithm for absorbing Markov chains
Author(s): J. Laurie Snell

Abstract: In this module, suitable for use in an introductory  
probability course, we present Engel's chip-moving algorithm for  
finding the basic descriptive quantities for an absorbing Markov  
chain, and prove that it works. The tricky part of the proof involves  
showing that the initial distribution of chips recurs. At the time of  
writing (circa 1979) no published proof of this was available, though  
Engel had stated that such a proof had been found by L. Scheller.

http://arxiv.org/abs/0904.1413


8376. Asymptotic Normality of Statistics on Permutation Tableaux
Author(s): Pawel Hitczenko and Svante Janson

Abstract: In this paper we use a probabilistic approach to derive the  
expressions for the characteristic functions of basic statistics  
defined on permutation tableaux. Since our expressions are exact, we  
can identify the distributions of basic statistics (like the number of  
unrestricted rows, the number of rows, and the number of 1s in the  
first row) exactly. In all three cases the distributions are known to  
be asymptotically normal after a suitable normalization. We also  
establish the asymptotic normality of the number of superfluous 1s.  
The latter result relies on a bijection between permutation tableaux  
and permutations and on a rather general sufficient condition for the  
central limit theorem for the sums of random variables in terms of  
dependency graph of the summands.

http://arxiv.org/abs/0904.1222


8377. Risk-averse asymptotics for reservation prices
Author(s): Laurence Carassus (PMA) and Miklos Rasonyi (MTA-SZTAKI)

Abstract: An investor's risk aversion is assumed to tend to infinity.  
In a fairly general setting, we present conditions ensuring that the  
respective utility indifference prices of a given contingent claim  
converge to its super replication price.

http://arxiv.org/abs/0904.1480


8378. Interacting Poisson processes and applications to neuronal  
modeling
Author(s): Stefano Cardanobile and Stefan Rotter

Abstract: A family of interacting Poisson processes is introduced.  
Events from a process are assumed to act multiplicatively on the rate  
of the processes to which they are connected. The family can be seen  
as a multivariate Cox process with both excitatory and inhibitory  
connections. The expected intensities of the process are approximated  
by a differential system of first-order and the stability of the  
solutions of this equation is studied. We discuss the applications in  
the neuroscience and the relations to the generalised linear model  
used for the analysis of spike trains.

http://arxiv.org/abs/0904.1505


8379. Kingman's coalescent and Brownian motion
Author(s): J. Berestycki and N. Berestycki

Abstract: We describe a simple construction of Kingman's coalescent in  
terms of a Brownian excursion. This construction is closely related  
to, and sheds some new light on, earlier work by Aldous and Warren.  
Our approach also yields some new results: for instance, we obtain the  
full multifractal spectrum of Kingman's coalescent. This complements  
earlier work on Beta-coalescents by the authors and Schweinsberg.  
Surprisingly, the thick part of the spectrum is not obtained by taking  
the limit as $\alpha \to 2$ in the result for Beta-coalescents  
mentioned above. Other analogies and differences between the case of  
Beta-coalescents and Kingman's coalescent are discussed.

http://arxiv.org/abs/0904.1526


8380. On the Numerical Evaluation of Distributions in Random Matrix  
Theory: A Review with an Invitation to Experimental Mathematics
Author(s): Folkmar Bornemann

Abstract: In this paper we review and compare the numerical evaluation  
of those probability distributions in random matrix theory that are  
analytically represented in terms of Painleve transcendents or  
Fredholm determinants. Concrete examples for the Gaussian and Laguerre  
(Wishart) beta-ensembles and their various scaling limits are  
discussed. We argue that the numerical approximation of Fredholm  
determinants is the conceptually more simple and efficient of the two  
approaches, easily generalized to the computation of joint  
probabilities and correlations. Having the means for extensive  
numerical explorations at hand, we discovered new and surprising  
determinantal formulae for the k-th largest level in the edge scaling  
limit of the Gaussian Orthogonal and Symplectic Ensembles; formulae  
that in turn led to improved numerical evaluations. The paper comes  
with a toolbox of Matlab functions that facilitates further  
mathematical experiments by the reader.

http://arxiv.org/abs/0904.1581


8381. A statistical mechanical interpretation of algorithmic  
information theory
Author(s): Kohtaro Tadaki

Abstract: We develop a statistical mechanical interpretation of  
algorithmic information theory by introducing the notion of  
thermodynamic quantities, such as free energy, energy, statistical  
mechanical entropy, and specific heat, into algorithmic information  
theory. We investigate the properties of these quantities by means of  
program-size complexity from the point of view of algorithmic  
randomness. It is then discovered that, in the interpretation, the  
temperature plays a role as the compression rate of the values of all  
these thermodynamic quantities, which include the temperature itself.  
Reflecting this self-referential nature of the compression rate of the  
temperature, we obtain fixed point theorems on compression rate.

http://arxiv.org/abs/0801.4194


8382. A statistical mechanical interpretation of algorithmic  
information theory III: Composite systems and fixed points
Author(s): Kohtaro Tadaki

Abstract: The statistical mechanical interpretation of algorithmic  
information theory (AIT, for short) was introduced and developed by  
our former works [K. Tadaki, Local Proceedings of CiE 2008, pp. 
425-434, 2008] and [K. Tadaki, Proceedings of LFCS'09, Springer's  
LNCS, vol.5407, pp.422-440, 2009], where we introduced the notion of  
thermodynamic quantities, such as partition function Z(T), free energy  
F(T), energy E(T), and statistical mechanical entropy S(T), into AIT.  
We then discovered that, in the interpretation, the temperature T  
equals to the partial randomness of the values of all these  
thermodynamic quantities, where the notion of partial randomness is a  
stronger representation of the compression rate by means of program- 
size complexity. Furthermore, we showed that this situation holds for  
the temperature itself as a thermodynamic quantity, namely, for each  
of all the thermodynamic quantities above, the computability of its  
value at temperature T gives a sufficient condition for T in (0,1) to  
be a fixed point on partial randomness. In this paper, we develop the  
statistical mechanical interpretation of AIT further and pursue its  
formal correspondence to normal statistical mechanics. The  
thermodynamic quantities in AIT are defined based on the halting set  
of an optimal computer, which is a universal decoding algorithm used  
to define the notion of program-size complexity. We show that there  
are infinitely many optimal computers which give completely different  
sufficient conditions in each of the thermodynamic quantities in AIT.  
We do this by introducing the notion of composition of computers to  
AIT, which corresponds to the notion of composition of systems in  
normal statistical mechanics.

http://arxiv.org/abs/0904.0973


8383. Spatial and Temporal Correlation of the Interference in ALOHA Ad  
Hoc Networks
Author(s): Radha Krishna Ganti and Martin Haenggi

Abstract: Interference is a main limiting factor of the performance of  
a wireless ad hoc network. The temporal and the spatial correlation of  
the interference makes the outages correlated temporally (important  
for retransmissions) and spatially correlated (important for routing).  
In this letter we quantify the temporal and spatial correlation of the  
interference in a wireless ad hoc network whose nodes are distributed  
as a Poisson point process on the plane when ALOHA is used as the  
multiple-access scheme.

http://arxiv.org/abs/0904.1444


8384. Average and deviation for slow-fast stochastic partial  
differential equations
Author(s): W.Wang and A.J. Roberts

Abstract: Averaging is an important method to extract effective  
macroscopic dynamics from complex systems with slow modes and fast  
modes. This article derives an averaged equation for a class of  
stochastic partial differential equations without any Lipschitz  
assumption on the slow modes. The rate of convergence in probability  
is obtained as a byproduct. Importantly, the deviation between the  
original equation and the averaged equation is also studied. A  
martingale approach proves that the deviation is described by a  
Gaussian process. This gives an approximation to errors of $\mathcal{O} 
(\e)$ instead of $\mathcal{O}(\sqrt{\e})$ attained in previous  
averaging.

http://arxiv.org/abs/0904.1462


8385. First hitting time law for some jump-diffusion processes :  
existence of a density
Author(s): Laure Coutin (MAP5) and Diana Dorobantu (SAF - EA2429)

Abstract: Let (Xt, t >= 0) be a diffusion process with jumps, sum of a  
Brownian motion with drift and a compound Poisson process. We consider  
T_x the first hitting time of a fixed level x > 0 by (Xt, t >= 0). We  
prove that the law of T_x has a density (defective when E(X1) < 0)  
with respect to the Lebesgue measure.

http://arxiv.org/abs/0904.1669


8386. Central Limit Theorems for the Brownian motion on large unitary  
groups
Author(s): Florent Benaych-Georges (CMAP and PMA)

Abstract: In this paper, we are concerned with the large N limit of  
linear combinations of entries of Brownian motions on the group of N  
by N unitary matrices. We prove that the process of such a linear  
combination converges to a Gaussian one. Various scales of time are  
concerned, giving rise to various limit processes, in relation to the  
geometric construction of the unitary Brownian motion. As an  
application, we recover certain results about linear combinations of  
the entries of Haar distributed random unitary matrices.

http://arxiv.org/abs/0904.1681


8387. The threshold function for vanishing of the top homology group  
of random $d$-complexes
Author(s): Dmitry N. Kozlov

Abstract: For positive integers $n$ and $d$, and the probability  
function $0\leq p(n)\leq 1$, we let $Y_{n,p,d}$ denote the probability  
space of all at most $d$-dimensional simplicial complexes on $n$  
vertices, which contain the full $(d-1)$-dimensional skeleton, and  
whose $d$-simplices appear with probability $p(n)$. In this paper we  
determine the threshold function for vanishing of the top homology  
group in $Y_{n,p,d}$, for all $d\geq 1$.

http://arxiv.org/abs/0904.1652


8388. Market viability via absence of arbitrages of the first kind
Author(s): Constantinos Kardaras

Abstract: The absence of arbitrages of the first kind, a weakening of  
the "No Free Lunch with Vanishing Risk" condition, is analyzed in a  
general semimartingale financial market model. In the spirit of the  
Fundamental Theorem of Asset Pricing (FTAP), it is shown that there is  
absence of arbitrages of the first kind in the market if and only if  
an equivalent local martingale deflator (ELMD) exists. An ELMD is a  
strictly positive process that, when deflated by it, discounted  
nonnegative wealth processes become local martingales. In terms of  
measures, absence of arbitrages of the first kind is shown to be  
equivalent to the existence of a finitely additive probability, weakly  
equivalent to the original and locally countably additive, under which  
the discounted asset-price process is a "local martingale". Finally,  
the aforementioned results are used to obtain an independent proof of  
the FTAP.

http://arxiv.org/abs/0904.1798


8389. Hitting half-spaces by Bessel-Brownian diffusions
Author(s): T. Byczkowski and J. Malecki and M. Ryznar

Abstract: The purpose of the paper is to find explicit formulas  
describing the joint distributions of the first hitting time and place  
for half-spaces of codimension one for a diffusion in $\R^{n+1}$,  
composed of one-dimensional Bessel process and independent n- 
dimensional Brownian motion. The most important argument is carried  
out for the two-dimensional situation. We show that this amounts to  
computation of distributions of various integral functionals with  
respect to a two-dimensional process with independent Bessel  
components. As a result, we provide a formula for the Poisson kernel  
of a half-space or of a strip for the operator $(I-\Delta)^{\alpha/ 
2}$, $0<\alpha<2$. In the case of a half-space, this result was  
recently found, by different methods, in [6]. As an application of our  
method we also compute various formulas for first hitting places for  
the isotropic stable L\'evy process.

http://arxiv.org/abs/0904.1803


8390. Random Walks on Strict Partitions
Author(s): Leonid Petrov

Abstract: We consider a certain sequence of random walks. The state  
space of the n-th random walk is the set of all strict partitions of n  
(that is, partitions without equal parts). We prove that, as n goes to  
infinity, these random walks converge to a continuous-time Markov  
process. The state space of this process is the infinite-dimensional  
simplex consisting of all nonincreasing infinite sequences of  
nonnegative numbers with sum less than or equal to one. The main  
result about the limit process is the expression of its the pre- 
generator as a formal second order differential operator in a  
polynomial algebra. Of separate interest is the generalization of  
Kerov interlacing coordinates to the case of shifted Young diagrams.

http://arxiv.org/abs/0904.1823


8391. Coupled perfect simulation of infinite range Gibbs measures and  
their finite range approximations
Author(s): Antonio Galves and Eva Loecherbach and Enza Orlandi

Abstract: Consider a Gibbs measure with a pairwise infinite range  
potential and its finite range approximation obtained by truncating  
the pairwise interaction at a certain range. If we make a local  
inspection of a perfect sampling of the finite range approximation,  
how often does it coincide with a sample from the original infinite  
range measure? We address this question by introducing a new coupled  
perfect simulation algorithm for these measures.

http://arxiv.org/abs/0904.1845


8392. On the distribution of the integral of the exponential Brownian  
motion
Author(s): Leonid Tolmatz

Abstract: The density distribution function of the integral of the  
exponential Brownian motion is determined explicitly in the form of a  
rapidly convergent series.

http://arxiv.org/abs/0904.1870


8393. Uniform bounds for norms of sums of independent random functions
Author(s): A. Goldenshluger and O.Lepski

Abstract: In this paper we study a collection of random processes $\ 
{\psi_w, w\in \cW\}$ determined by a sequence of independent random  
elements and parameterized by a set of weight functions $w\in \cW$. We  
develop uniform concentration--type inequalities for a norm $\|\psi_w\| 
$, i.e., we present an explicit upper bound $U_\psi(w)$ on $\|\psi_w\| 
$ and study behavior of \[ \sup_{w\in \cW} \{\|\psi_w\|-U_\psi(w)\}.  
\] Several probability and moment inequalities for this random  
variable are derived and used in order to get some asymptotic results.  
We also consider applications of obtained bounds to many important  
problems arising in modern nonparametric statistics including  
bandwidth selection in multivariate density and regression estimation.

http://arxiv.org/abs/0904.1950


8394. A multiple stochastic integral criterion for almost sure limit  
theorems
Author(s): Bernard Bercu and Ivan Nourdin and Murad S. Taqqu

Abstract: In this paper, we study almost sure central limit theorems  
for multiple stochastic integrals and provide a criterion based on the  
kernel of these multiple integrals. We apply our result to normalized  
partial sums of Hermite polynomials of increments of fractional  
Brownian motion. We obtain almost sure central limit theorems for  
these normalized sums when they converge in law to a normal  
distribution.

http://arxiv.org/abs/0904.2094


8395. Fonctions de Mittag-Leffler et processus de L\'evy stables sans  
saut n\'egatif
Author(s): Thomas Simon

Abstract: It is noticed that a certain transform of the Mittag-Leffler  
function Ea is completely monotone for a in [1,2]. Using the explicit  
expressions of its Bernstein density, an identity in law between  
suprema of completely asymmetric Levy a-stable processes. In the  
spectrally positive case, we retrieve the exact expression of a  
unilateral small deviation constant which had been previously obtained  
by a different method by Bernyk, Dalang and Peskir.

http://arxiv.org/abs/0904.2191


8396. Asymptotics of a Brownian ratchet for Protein Translocation
Author(s): Andrej Depperschmidt and Peter Pfaffelhuber

Abstract: Protein translocation in cells has been modelled by  
\emph{Brownian ratchets}. In such models, the protein diffuses through  
the nanopore by thermal fluctuations. On one side of the pore  
ratcheting molecules bind to the protein and hinder it to diffuse out  
of the pore. We study a simple Brownian ratchet by means of a  
reflected Brownian motion $(X_t)_{t\geq 0}$ with a changing reflection  
point $(R_t)_{t\geq 0}$. The rate of change of $R_t$ is $\gamma(X_t- 
R_t)$ and is distributed uniformly on $[R_t;X_t]$. We show that the  
asymptotic speed of the ratchet scales with $\gamma^{1/3}$ and the  
asymptotic variance is independent of $\gamma$.

http://arxiv.org/abs/0904.2276


8397. Correction to: Branching-coalescing particle systems
Author(s): Siva R. Athreya and Jan M. Swart

Abstract: In the article titled "Branching-Coalescing Particle  
Systems" published in Probability Theory and Related Fields 131(3),  
pages 376-414, (2005), Theorem 7 as stated there is incorrect. Indeed,  
we show by counterexample that the equality that we claimed there to  
hold for all time, in general holds only for almost every time with  
respect to Lebesgue measure. We prove a weaker version of the theorem  
that is still sufficient for our applications in the mentioned paper.

http://arxiv.org/abs/0904.2288


8398. Supremum of Random Dirichlet Polynomials with Sub-multiplicative  
Coefficients
Author(s): Michel Weber

Abstract: We study the supremum of random Dirichlet polynomials  
$D_N(t)=\sum_{n=1}^N\varepsilon_n d(n) n^{- s}$, where $(\varepsilon_n) 
$ is a sequence of independent Rademacher random variables, and $ d $  
is a sub-multiplicative function. The approach is gaussian and  
entirely based on comparison properties of Gaussian processes, with no  
use of the metric entropy method.

http://arxiv.org/abs/0904.2316


8399. Tridiagonal realization of the anti-symmetric Gaussian $\beta$- 
ensemble
Author(s): Ioana Dumitriu and Peter J. Forrester

Abstract: The Householder reduction of a member of the anti-symmetric  
Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix  
with all independent elements. The random variables permit the  
introduction of a positive parameter $\beta$, and the eigenvalue  
probability density function of the corresponding random matrices can  
be computed explicitly, as can the distribution of $\{q_i\}$, the  
first components of the eigenvectors. Three proofs are given. One  
involves an inductive construction based on bordering of a family of  
random matrices which are shown to have the same distributions as the  
anti-symmetric tridiagonal matrices. This proof uses the Dixon- 
Anderson integral from Selberg integral theory. A second proof  
involves the explicit computation of the Jacobian for the change of  
variables between real anti-symmetric tridiagonal matrices, its  
eigenvalues and $\{q_i\}$. The third proof, which is restricted to $n$  
even, maps matrices from the anti-symmetric Gaussian $\beta$-ensemble  
to those realizing particular examples of the Laguerre $\beta$- 
ensemble. In addition to these proofs, we note some simple properties  
of the shooting eigenvector and associated Pr\"ufer phases of the  
random matrices.

http://arxiv.org/abs/0904.2216


8400. Taylor expansions of solutions of stochastic partial  
differential equations
Author(s): Arnulf Jentzen

Abstract: The solutions of parabolic and hyperbolic stochastic partial  
differential equations (SPDEs) driven by an infinite dimensional  
Brownian motion, which is a martingale, are in general not semi- 
martingales any more and therefore do not satisfy an It\^o formula  
like the solutions of finite dimensional stochastic differential  
equations (SODEs). In particular, it is not possible to derive  
stochastic Taylor expansions as for the solutions of SODEs using an  
iterated application of the It\^o formula. However, in this article we  
introduce Taylor expansions of solutions of SPDEs via an alternative  
approach, which avoids the need of an It\^o formula. The main idea  
behind these Taylor expansions is to use first classical Taylor  
expansions for the nonlinear coefficients of the SPDE and then to  
insert recursively the mild presentation of the solution of the SPDE.  
The iteration of this idea allows us to derive stochastic Taylor  
expansions of arbitrarily high order. Combinatorial concepts of trees  
and woods provide a compact formulation of the Taylor expansions.

http://arxiv.org/abs/0904.2232


8401. Random walk versus random line
Author(s): Joel De Coninck and Francois Dunlop and Thierry Huillet

Abstract: We consider random walks X_n in Z+, obeying a detailed  
balance condition, with a weak drift towards the origin when X_n tends  
to infinity. We reconsider the equivalence in law between a random  
walk bridge and a 1+1 dimensional Solid-On-Solid bridge with a  
corresponding Hamiltonian. Phase diagrams are discussed in terms of  
recurrence versus wetting. A drift -delta/X_n of the random walk  
yields a Solid-On-Solid potential with an attractive well at the  
origin and a repulsive tail delta(delta+2)/(8X_n^2) at infinity,  
showing complete wetting for delta<=1 and critical partial wetting for  
delta>1.

http://arxiv.org/abs/0904.2440


8402. Statistical analysis of single-server loss queueing systems
Author(s): Vyacheslav M. Abramov

Abstract: In this article statistical bounds for certain output  
characteristics of the $M/GI/1/n$ and $GI/M/1/n$ loss queueing systems  
are derived on the basis of large samples of an input characteristic  
of these systems.

http://arxiv.org/abs/0904.2426


8403. Joint Range of R\'enyi Entropies
Author(s): Peter Harremo\"es

Abstract: The exact range of the joined values of several R\'{e}nyi  
entropies is determined. The method is based on topology with special  
emphasis on the orientation of the objects studied. Like in the case  
when only two orders of R\'{e}nyi entropies are studied one can  
parametrize upper and lower bounds but an explicit formula for a tight  
upper or lower bound cannot be given.

http://arxiv.org/abs/0904.2477


8404. A Class of degenerate Stochastic differential equations with non- 
Lipschitz coefficients
Author(s): K. Suresh Kumar

Abstract: We obtain sufficient condition for SDEs to evolve in the  
positive orthant. We use comparison theorem arguments to achieve this.  
As a result we prove the existence of a unique strong solution for a  
class of multidimensional degenerate SDEs with non-Lipschitz diffusion  
coefficients.

http://arxiv.org/abs/0904.2629


8405. Boundary crossing identities for diffusions having the time  
inversion property
Author(s): Larbi Alili and Pierre Patie

Abstract: We review and study a one-parameter family of functional  
transformations, denoted by $(S^{(\beta)})_{\beta\in \R}$, which, in  
the case $\beta<0$, provides a path realization of bridges associated  
to the family of diffusion processes enjoying the time inversion  
property. This family includes the Brownian motion, Bessel processes  
with a positive dimension and their conservative $h$-transforms. By  
means of these transformations, we derive an explicit and simple  
expression which relates the law of the boundary crossing times for  
these diffusions over a given function $f$ to those over the image of  
$f$ by the mapping $S^{(\beta)}$, for some fixed $\beta\in \mathbb{R} 
$. We give some new examples of boundary crossing problems for the  
Brownian motion and the family of Bessel processes. We also provide,  
in the Brownian case, an interpretation of the results obtained by the  
standard method of images and establish connections between the exact  
asymptotics for large time of the densities corresponding to various  
curves of each family.

http://arxiv.org/abs/0904.2680


8406. On the It\^o-Wentzell formula for distribution-valued processes  
and related topics
Author(s): N.V. Krylov

Abstract: We prove the It\^o-Wentzell formula for processes with  
values in the space of generalized functions by using the stochastic  
Fubini theorem and the It\^o-Wentzell formula for real-valued  
processes, appropriate versions of which are also proved.

http://arxiv.org/abs/0904.2752


8407. Horizontal diffusion in $C^1$ path space
Author(s): Marc Arnaudon (LMA) and Abdoulaye Kol\'eh\`e Coulibaly- 
Pasquier (LMA) and Anton Thalmaier

Abstract: We define horizontal diffusion in $C^1$ path space over a  
Riemannian manifold and prove its existence. If the metric on the  
manifold is developing under the forward Ricci flow, horizontal  
diffusion along Brownian motion turns out to be length preserving. As  
application, we prove contraction properties in the Monge-Kantorovich  
minimization problem for probability measures evolving along the heat  
flow. For constant rank diffusions, differentiating a family of  
coupled diffusions gives a derivative process with a covariant  
derivative of finite variation. This construction provides an  
alternative method to filtering out redundant noise.

http://arxiv.org/abs/0904.2762


8408. Random surface growth with a wall and Plancherel measures for  
O(infinity)
Author(s): Alexei Borodin and Jeffrey Kuan

Abstract: We consider a Markov evolution of lozenge tilings of a  
quarter-plane and study its asymptotics at large times. One of the  
boundary rays serves as a reflecting wall. We observe frozen and  
liquid regions, prove convergence of the local correlations to  
translation-invariant Gibbs measures in the liquid region, and obtain  
new discrete Jacobi and symmetric Pearcey determinantal point  
processes near the wall. The model can be viewed as the one-parameter  
family of Plancherel measures for the infinite-dimensional orthogonal  
group, and we use this interpretation to derive the determinantal  
formula for the correlation functions at any finite time moment.

http://arxiv.org/abs/0904.2607


8409. Asymptotic properties of resolvents of large dilute Wigner  
matrices
Author(s): S. Ayadi and O. Khorunzhiy

Abstract: We study the spectral properties of the dilute Wigner random  
real symmetric n-dimensional matrices H such that the entries H(i,j)  
take zero value with probability 1-p/n. We prove that under rather  
general conditions on the probability distribution of H(i,j) the  
semicircle law is valid for the dilute Wigner ensemble in the limit of  
infinite n and p. In the second part of the paper we study the leading  
term of the correlation function of the resolvent G(z) of H with large  
enough Im z in the limit of infinite n and p such that 3/5 log n < log  
n. We show that this leading term, when considered in the local  
spectral scale, converges to the same limit as that of the resolvent  
correlation function of the Wigner ensemble of random matrices. This  
shows that the moderate dilution of the Wigner ensemble does not alter  
its universality class.

http://arxiv.org/abs/0904.2689


8410. Symmetric Jump Processes and their Heat Kernel Estimates
Author(s): Zhen-Qing Chen

Abstract: We survey the recent development of the DeGiorgi-Nash-Moser- 
Aronson type theory for a class of symmetric jump processes(or  
equivalently, a class of symmetric integro-differential operators). We  
focus on the sharp two-sided estimates for the transition density  
functions (or heat kernels) of the processes, a priori Holder estimate  
and parabolic Harnack inequalities for their parabolic functions. In  
contrast to the second order elliptic differential operator case, the  
methods to establish these properties for symmetric integro- 
differential operators are mainly probabilistic.

http://arxiv.org/abs/0904.2796


8411. The Optimal Filtering of Markov Jump Processes in Additive White  
Noise
Author(s): M. Zakai

Abstract: This note is based on Wonham \cite{Wonham}. The differences  
between this note and [Wonham] are discussed in Section VIII.

http://arxiv.org/abs/0904.2888


8412. Viability of infinite-asset financial models where constrained  
agents with limited information act
Author(s): Constantinos Kardaras

Abstract: A study of the boundedness in probability of the set of  
possible wealth outcomes of an economic agent facing constraints, and  
with limited access to information, is undertaken. The wealth-process  
set is abstractly structured with reasonable economic properties,  
instead of the usual practice of taking it to consist of stochastic  
integrals against a semimartingale integrator. We obtain the  
equivalence of (a) the boundedness in probability of wealth outcomes  
with (b) the existence of at least one deflator that make the deflated  
wealth processes have a generalized supermartingale property.  
Specializing in the case of full information, we obtain as a  
consequence that in a viable market all wealth processes have versions  
that are semimartingales.

http://arxiv.org/abs/0904.2913


8413. Limit Distributions for Random Hankel, Toeplitz Matrices and  
Independent Products
Author(s): Dang-Zheng Liu and Zheng-Dong Wang

Abstract: For random selfadjoint (real symmetric, complex Hermitian,  
or quaternion self-dual) Toeplitz matrices and real symmetric Hankel  
matrices, the existence of universal limit distributions for  
eigenvalues and products of several independent matrices is proved.  
The joint moments are the integral sums related to certain pair  
partitions. Our method can apply to random Hankel and Toeplitz band  
matrices, and the similar results are given. In particular, when the  
band width grows slowly as the dimension $N\ra \iy$, the exact limit  
distribution functions are given (N(0,1) for Toeplitz band matrices)  
and some asymptotic commutativity is observed.

http://arxiv.org/abs/0904.2958


8414. Law of the exponential functional of one-sided L\'evy processes  
and Asian options
Author(s): Pierre Patie

Abstract: The purpose of this note is to describe, in terms of a power  
series, the distribution function of the exponential functional, taken  
at some independent exponential time, of a spectrally negative L\'evy  
process \xi with unbounded variation. We also derive a Geman-Yor type  
formula for Asian options prices in a financial market driven by e^\xi.

http://arxiv.org/abs/0904.3000


8415. Quasi-stationary distributions and Fleming-Viot processes for  
finite state Markov processes
Author(s): Amine Asselah and Pablo A. Ferrari and Pablo Groisman

Abstract: Consider a continuous time Markov chain with rates $Q$ in  
the state space $\Lambda\cup\{0\}$ with 0 as an absorbing state. In  
the associated Fleming-Viot process $N$ particles evolve independently  
in $\Lambda$ with rates $Q$ until one of them attempts to jump to the  
absorbing state 0. At this moment the particle comes back to $\Lambda$  
instantaneously, by jumping to one of the positions of the other  
particles, chosen uniformly at random. When $\Lambda$ is finite, we  
show that the empirical distribution of the particles at a fixed time  
converges as $N\to\infty$ to the distribution of a single particle at  
the same time conditioned on non absorption. Furthermore, the  
empirical profile of the unique invariant measure for the Fleming-Viot  
process with $N$ particles converges as $N\to\infty$ to the unique  
quasi-stationary distribution of the one-particle motion. A key  
element of the approach is to show that the two-particle correlations  
is of order $1/N$.

http://arxiv.org/abs/0904.3039


8416. Hoeffding spaces and Specht modules
Author(s): Giovanni Peccati (LSTA and MODAL'X) and Jean-Renaud Pycke  
(DP)

Abstract: It is proved that each Hoeffding space associated with a  
random permutation (or, equivalently, with extractions without  
replacement from a finite population) carries an irreducible  
representation of the symmetric group, equivalent to a two-block  
Specht module.

http://arxiv.org/abs/0904.3086


8417. L1-Penalized Quantile Regression in High-Dimensional Sparse Models
Author(s): Alexandre Belloni and Victor Chernozhukov

Abstract: We consider median regression and, more generally, quantile  
regression in high-dimensional sparse models. In these models the  
overall number of regressors $p$ is very large, possibly larger than  
the sample size $n$, but only $s$ of these regressors have non-zero  
impact on the conditional quantile of the response variable, where $s$  
grows slower than $n$. Since in this case the ordinary quantile  
regression is not consistent, we consider quantile regression  
penalized by the $\ell_1$-norm of coefficients ($\ell_1$-QR). First,  
we show that $\ell_1$-QR is consistent at the rate $\sqrt{s/n}  
\sqrt{\log p}$, which is close to the oracle rate $\sqrt{s/n}$,  
achievable when the minimal true model is known. The overall number of  
regressors $p$ affects the rate only through the $\log p$ factor, thus  
allowing nearly exponential growth in the number of zero-impact  
regressors. The rate result holds under relatively weak conditions,  
requiring that $s/n$ converges to zero at a super-logarithmic speed  
and that regularization parameter satisfies certain theoretical  
constraints. Second, we propose a pivotal, data-driven choice of the  
regularization parameter and show that it satisfies these theoretical  
constraints. Third, we show that $\ell_1$-QR correctly selects the  
true minimal model as a valid submodel, when the non-zero coefficients  
of the true model are well separated from zero. We also show that the  
number of non-zero coefficients in $\ell_1$-QR is of same stochastic  
order as $s$, the number of non-zero coefficients in the minimal true  
model. Fourth, we analyze the rate of convergence of a two-step  
estimator that applies ordinary quantile regression to the selected  
model. Fifth, we evaluate the performance of $\ell_1$-QR in a Monte- 
Carlo experiment, and illustrate its use on an international economic  
growth application.

http://arxiv.org/abs/0904.2931


8418. Asymptotic Properties of Random Matrices of Long-Range  
Percolation Model
Author(s): Slim Ayadi

Abstract: We study the spectral properties of matrices of long-range  
percolation model. These are N\times N random real symmetric matrices  
H=\{H(i,j)\}_{i,j} whose elements are independent random variables  
taking zero value with probability 1-\psi((i-j)/b), b\in  
\mathbb{R}^{+}, where $\psi$ is an even positive function with \psi(t) 
\le{1} and vanishing at infinity. We study the resolvent G(z)=(H- 
z)^{-1}, Imz\neq{0} in the limit N,b\to\infty, b=O(N^{\alpha}), 1/3< 
\alpha<1 and obtain the explicit expression T(z_{1},z_{2}) for the  
leading term of the correlation function of the normalized trace of  
resolvent g_{N,b}(z)=N^{-1}Tr G(z). We show that in the scaling limit  
of local correlations, this term leads to the expression  
(Nb)^{-1}T(\lambda+r_{1}/N+i0,\lambda+r_{2}/N-i0)= b^{-1}\sqrt{N}| 
r_{1}-r_{2}|^{-3/2}(1+o(1)) found earlier by other authors for band  
random matrix ensembles. This shows that the ratio $b^{2}/N$ is the  
correct scale for the eigenvalue density correlation function and that  
the ensemble we study and that of band random matrices belong to the  
same class of spectral universality.

http://arxiv.org/abs/0904.2837


8419. Gibbs random fields with unbounded spins on unbounded degree  
graphs
Author(s): Yuri Kondratiev and Yuri Kozitsky and Tanja Pasurek

Abstract: Gibbs random fields corresponding to systems of real-valued  
spins (e.g. systems of interacting anharmonic oscillators) indexed by  
the vertices of unbounded degree graphs with a certain summability  
property are constructed. It is proven that the set of tempered Gibbs  
random fields is non-void and weakly compact, and that they obey  
uniform exponential integrability estimates. In the second part of the  
paper, a class of graphs is described in which the mentioned  
summability is obtained as a consequence of a property, by virtue of  
which vertices of large degree are located at large distances from  
each other. The latter is a stronger version of a metric property,  
introduced in [Bassalygo, L. A. and Dobrushin, R. L. (1986).  
\textrm{Uniqueness of a Gibbs field with a random potential--an  
elementary approach.}\textit{Theory Probab. Appl.} {\bf 31} 572--589].

http://arxiv.org/abs/0904.3207


8420. Weak Solutions of stochastic recursions: an explicit construction
Author(s): Pascal Moyal

Abstract: We propose an explicit construction of the solution of a  
stationary stochastic recursion of the form $X\circ\theta=\phi(X)$ on  
a semi-ordered Polish space, when the monotonicity of $\phi$ is not  
assumed. This solution exists on an enriched probability space (it is  
said \emph{weak}), provided the recursion is lattice-valued, and  
dominated by a proper monotonic stochastic recursion.

http://arxiv.org/abs/0904.3240


8421. Computations of Greeks in stochastic volatility models via the  
Malliavin calculus
Author(s): Youssef El-Khatib

Abstract: We compute Greeks for stochastic volatility models driven by  
Brownian informations. We use the Malliavin method introduced for  
deterministic volatility models.

http://arxiv.org/abs/0904.3247


8422. On continuity properties for option prices in exponential L\'evy  
models
Author(s): S. Cawston and L. Vostrikova

Abstract: For a converging sequence of exponential L\'evy models, we  
give conditions under which the associated sequence of option prices  
converges. We also study the behaviour of the prices when no such  
convergence holds. We then consider two special cases, first when the  
martingale measure is chosen by minimisation of entropy and then when  
it minimises Hellinger integrals.

http://arxiv.org/abs/0904.3274


8423. Large Deviation Principle for Semilinear Stochastic Evolution  
Equations with Monotone Nonlinearity and Multiplicative Noise
Author(s): Hassan Dadashi-Arani and Bijan Z. Zangeneh

Abstract: Using a recently developed method, weak convergence method,  
in dealing with the large deviation principle, we demonstrate the  
large deviation principle property for mild solutions of stochastic  
evolution equations with monotone nonlinearity and multiplicative  
noise. An It^o-type inequality is a main tool in the proofs. We also  
give two examples to illustrate the applications of the theorems.

http://arxiv.org/abs/0904.3305


8424. Posterior Inference in Curved Exponential Families under  
Increasing Dimensions
Author(s): Alexandre Belloni and Victor Chernozhukov

Abstract: The goal of this work is to study the large sample  
properties of the posterior-based inference in the curved exponential  
family under increasing dimension. The curved structure arises from  
the imposition of various restrictions, such as moment restrictions,  
on the model, and plays a fundamental role in various branches of data  
analysis. We establish conditions under which the posterior  
distribution is approximately normal, which in turn implies various  
good properties of estimation and inference procedures based on the  
posterior. In the process we revisit and improve upon previous results  
for the exponential family under increasing dimension by making use of  
concentration of measure. We also discuss a variety of applications  
including the multinomial model with moment restrictions, seemingly  
unrelated regression equations, and single structural equation models.  
In our analysis, both the parameter dimension and the number of  
moments are increasing with the sample size.

http://arxiv.org/abs/0904.3132


8425. Quasi-stationary distributions for structured birth and death  
processes with mutations
Author(s): Pierre Collet (CPHT) and Servet Martinez and Sylvie M\'el 
\'eard (CMAP) and Jaime San Martin

Abstract: We study the probabilistic evolution of a birth and death  
continuous time measure-valued process with mutations and ecological  
interactions. The individuals are characterized by (phenotypic) traits  
that take values in a compact metric space. Each individual can die or  
generate a new individual. The birth and death rates may depend on the  
environment through the action of the whole population. The offspring  
can have the same trait or can mutate to a randomly distributed trait.  
We assume that the population will be extinct almost surely. Our goal  
is the study, in this infinite dimensional framework, of quasi- 
stationary distributions when the process is conditioned on non- 
extinction. We firstly show in this general setting, the existence of  
quasi-stationary distributions. This result is based on an abstract  
theorem proving the existence of finite eigenmeasures for some  
positive operators. We then consider a population with constant birth  
and death rates per individual and prove that there exists a unique  
quasi-stationary distribution with maximal exponential decay rate. The  
proof of uniqueness is based on an absolute continuity property with  
respect to a reference measure.

http://arxiv.org/abs/0904.3468


8426. An excursion approach to maxima of the Brownian Bridge
Author(s): Mihael Perman (Institute for Mathematics and Physics and  
Mechanics and Ljubljana, Slovenia) Jon A. Wellner (University of  
Washington, Seattle)

Abstract: Functionals of Brownian bridge arise as limiting  
distributions in nonparametric statistics. In this paper we will give  
a derivation of distributions of extrema of the Brownian bridge based  
on excursion theory for Brownian motion. Only the Poisson character of  
the excursion process will be used. Particular cases of calculations  
include the distributions of the Kolmogorov-Smirnov statistic, the  
Kuiper statistic, and the ratio of the maximum positive ordinate to  
the minumum negative ordinate.

http://arxiv.org/abs/0904.3473


8427. Regularity of harmonic functions for a class of singular stable- 
like processes
Author(s): Richard F. Bass and Zhen-Qing Chen

Abstract: We consider the system of stochastic differential equations  
dX_t=A(X_{t-}) dZ_t, where Z_t^1, ..., Z^d_t are independent one- 
dimensional symmetric stable processes of order \alpha, and the matrix- 
valued function A is bounded, continuous and everywhere non- 
degenerate. We show that bounded harmonic functions associated with X  
are Holder continuous, but a Harnack inequality need not hold. The  
Levy measure associated with the vector-valued process Z is highly  
singular.

http://arxiv.org/abs/0904.3518


8428. A method for Hedging in continuous time
Author(s): Yoav Freund

Abstract: We present a method for hedging in continuous time.

http://arxiv.org/abs/0904.3356


8429. Application of the lent particle method to Poisson driven SDE's
Author(s): Nicolas Bouleau (CERMICS) and Laurent Denis (DP)

Abstract: We apply the Dirichlet forms version of Malliavin calculus  
to stochastic differential equations with jumps. As in the continuous  
case this weakens signi?cantly the assumptions on the coefficients of  
the SDE. In spite of the use of the Dirichlet forms theory, this  
approach brings also an important simpli?cation which was not  
available nor visible previously : an explicit formula giving the carr 
\'e du champ matrix, i.e. the Malliavin matrix. Following this formula  
a new procedure appears, called the lent particle method which  
shortens the computations both theoretically and in concrete examples.

http://arxiv.org/abs/0904.3613


8430. A spatially explicit Markovian individual-based model for  
terrestrial plant dynamics
Author(s): Fabien Campillo and Marc Joannides

Abstract: An individual-based model (IBM) of a spatiotemporal  
terrestrial ecological population is proposed. This model is spatially  
explicit and features the position of each individual together with  
another characteristic, such as the size of the individual, which  
evolves according to a given stochastic model. The population is  
locally regulated through an explicit competition kernel. The IBM is  
represented as a measure-valued branching/diffusing stochastic  
process. The approach allows (i) to describe the associated Monte  
Carlo simulation and (ii) to analyze the limit process under large  
initial population size asymptotic. The limit macroscopic model is a  
deterministic integro-differential equation.

http://arxiv.org/abs/0904.3632


8431. On adding a list of numbers (and other one-dependent  
determinantal processes)
Author(s): Alexei Borodin and Persi Diaconis and and Jason Fulman

Abstract: Adding a column of numbers produces "carries" along the way.  
We show that random digits produce a pattern of carries with a neat  
probabilistic description: the carries form a one-dependent  
determinantal point process. This makes it easy to answer natural  
questions: How many carries are typical? Where are they located? We  
show that many further examples, from combinatorics, algebra and group  
theory, have essentially the same neat formulae, and that any one- 
dependent point process on the integers is determinantal. The examples  
give a gentle introduction to the emerging fields of one-dependent and  
determinantal point processes.

http://arxiv.org/abs/0904.3740


8432. Levy solutions of a randomly forced Burgers equation
Author(s): Marie-Line Chabanol and Jean Duchon

Abstract: We consider the one dimensional Burgers equation forced by a  
brownian in space and white noise in time process $\partial_t u + u  
\partial_x u = f(x,t)$, with $2E(f(x,t)f(y,s)) = (|x|+|y|-|x-y|) 
\delta(t-s)$ and we show that there are Levy processes solutions, for  
which we give the evolution equation of the characteristic exponent.  
In particular we give the explicit solution in the case $u_0(x)=0$.

http://arxiv.org/abs/0904.3397


8433. New critical exponents for percolation and the random-cluster  
model
Author(s): Youjin Deng and Wei Zhang and Timothy M. Garoni and Alan D.  
Sokal and Andrea Sportiello

Abstract: We introduce several infinite families of new critical  
exponents for the random-cluster model, and give heuristic scaling  
arguments determining all but one of these exponents as a function of  
q in the two-dimensional case. We then give Monte Carlo simulations  
confirming these predictions. For the shortest-path fractal dimension  
we give the conjectured exact formula d_min = (g+2)(g+18)/(32g) where  
g is the Coulomb-gas coupling. Finally, we apply these exponents to  
provide a radically improved implementation of the Sweeny Monte Carlo  
algorithm.

http://arxiv.org/abs/0904.3448


8434. Remarks on Pickands theorem
Author(s): Zbigniew Michna

Abstract: In this article we present Pickands theorem and his double  
sum method. We follow Piterbarg's proof of this theorem. Since his  
proof relies on general lemmas we present a complete proof of Pickands  
theorem using Borell inequality and Slepian lemma. The original  
Pickands proof is rather complicated and is mixed with upcrossing  
probabilities for stationary Gaussian processes. We give a lower bound  
for Pickands constant.

http://arxiv.org/abs/0904.3832


8435. Matrix measures, random moments and Gaussian ensembles
Author(s): Jan Nagel and Holger Dette

Abstract: We consider the moment space $\mathcal{M}_n$ corresponding  
to $p \times p$ real or complex matrix measures defined on the  
interval $[0,1]$. The asymptotic properties of the first $k$  
components of a uniformly distributed vector $(S_{1,n}, ...,  
S_{n,n})^* \sim \mathcal{U} (\mathcal{M}_n)$ are studied if $n \to  
\infty$. In particular, it is shown that an appropriately centered and  
standardized version of the vector $(S_{1,n}, ..., S_{k,n})^*$  
converges weakly to a vector of $k$ independent $p \times p$ Gaussian  
ensembles. For the proof of our results we use some new relations  
between ordinary moments and canonical moments of matrix measures  
which are of own interest. In particular, it is shown that the first $k 
$ canonical moments corresponding to the uniform distribution on the  
real or complex moment space $\mathcal{M}_n$ are independent  
multivariate Beta distributed random variables and that each of these  
random variables converge in distribution (if the parameters converge  
to infinity) to the Gaussian orthogonal ensemble or to the Gaussian  
unitary ensemble, respectively.

http://arxiv.org/abs/0904.3847


8436. The Gapeev-K\"uhn stochastic game driven by a spectrally  
positive L\'evy process
Author(s): E.J. Baurdoux and A.E. Kyprianou and J.C. Pardo

Abstract: In Gapeev and K\"uhn (2005), the stochastic game  
corresponding to perpetual convertible bonds was considered when  
driven by a Brownian motion and a compound Poisson process with  
exponential jumps. We consider the same stochastic game but driven by  
a spectrally positive L\'evy process. We establish a complete solution  
to the game indicating four principle parameter regimes as well as  
characterizing the occurence of continuous and smooth fit. In Gapeev  
and K\"uhn (2005), the method of proof was mainly based on solving a  
free boundary value problem. In this paper, we instead use fluctuation  
theory and an auxiliary optimal stopping problem to find a solution to  
the game.

http://arxiv.org/abs/0904.3871


8437. On Marginal Markov Processes of Quantum Quadratic Stochastic  
Processes
Author(s): Farrukh Mukhamdov

Abstract: In the paper it is defined two marginal Markov processes on  
von Neumann algebras $\cm$ and $\cm\o\cm$, respectively, corresponding  
to given quantum quadratic stochastic process (q.q.s.p.). It is proved  
that such marginal processes uniquely determines the q.q.s.p.  
Moreover, certain ergodic relations between them are established as  
well.

http://arxiv.org/abs/0904.3790


8438. Metastable behavior for bootstrap percolation on regular trees
Author(s): Marek Biskup and Roberto H. Schonmann

Abstract: We examine bootstrap percolation on a regular (b+1)-ary tree  
with initial law given by Bernoulli(p). The sites are updated  
according to the usual rule: a vacant site becomes occupied if it has  
at least theta occupied neighbors, occupied sites remain occupied  
forever. It is known that, when b>theta>1, the limiting density q=q(p)  
of occupied sites exhibits a jump at some p_t=p_t(b,theta) in (0,1)  
from q_t:=q(p_t)<1 to q(p)=1 when p>p_t. We investigate the metastable  
behavior associated with this transition. Explicitly, we pick p=p_t+h  
with h>0 and show that, as h decreases to 0, the system lingers around  
the "critical" state for time order h^{-1/2} and then passes to fully  
occupied state in time O(1). The law of the entire configuration  
observed when the occupation density is q in (q_t,1) converges, as h  
tends to 0, to a well-defined measure.

http://arxiv.org/abs/0904.3965


8439. Some asymptotic properties of the spectrum of the Jacobi ensemble
Author(s): Holger Dette and Jan Nagel

Abstract: For the random eigenvalues with density corresponding to the  
Jacobi ensemble $$c \cdot \prod_{i < j} | \lambda_i - \lambda_j |^ 
\beta \prod^n_{i=1} (2 - \lambda_i)^a (2 + \lambda_i)^b I_{(-2,2)}  
(\lambda_i) $$ $(a, b > -1, \beta > 0) $ a strong uniform  
approximation by the roots of the Jacobi polynomials is derived if the  
parameters $a, b,$ $\beta$ depend on $n$ and $n \to \infty$. Roughly  
speaking, the eigenvalues can be uniformly approximated by roots of  
Jacobi polynomials with parameters $((2a+2)/\beta -1, (2b+2)/ 
\beta-1)$, where the error is of order $\{\log n/(a+b) \}^{1/4}$.  
These results are used to investigate the asymptotic properties of the  
corresponding spectral distribution if $n \to \infty$ and the  
parameters $a, b$ and $\beta$ vary with $n$. We also discuss further  
applications in the context of multivariate random $F$-matrices.

http://arxiv.org/abs/0904.4091


8440. Zero bias transformation and asymptotic expansions II : the  
Poisson case
Author(s): Ying Jiao (PMA)

Abstract: We apply a discrete version of the methodology in  
\cite{gauss} to obtain a recursive asymptotic expansion for $\esp[h(W)] 
$ in terms of Poisson expectations, where $W$ is a sum of independent  
integer-valued random variables and $h$ is a polynomially growing  
function. We also discuss the remainder estimations.

http://arxiv.org/abs/0904.4115


8441. Limiting Distributions for Sums of Independent Random Products
Author(s): Zakhar Kabluchko

Abstract: Let $\{X_{i,j}:(i,j)\in\mathbb N^2\}$ be a two-dimensional  
array of independent copies of a random variable $X$, and let $\{N_n 
\}_{n\in\mathbb N}$ be a sequence of natural numbers such that $\lim_{n 
\to\infty}e^{-cn}N_n=1$ for some $c>0$. Our main object of interest is  
the sum of independent random products $$Z_n=\sum_{i=1}^{N_n}  
\prod_{j=1}^{n}e^{X_{i,j}}.$$ It is shown that the limiting properties  
of $Z_n$, as $n\to\infty$, undergo phase transitions at two critical  
points $c=c_1$ and $c=c_2$. Namely, if $c>c_2$, then $Z_n$ satisfies  
the central limit theorem with the usual normalization, whereas for  
$cc_1$. If the random variable $X$ is Gaussian, we recover the results  
of Bovier, Kurkova, and L\"owe [Fluctuations of the free energy in the  
REM and the $p$-spin SK models. Ann. Probab. 30(2002), 605-651].

http://arxiv.org/abs/0904.4127


8442. A continuum-tree-valued Markov process
Author(s): Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS)

Abstract: We present a construction of a L\'evy continuum random tree  
(CRT) associated with a super-critical continuous state branching  
process using the so-called exploration process and a Girsanov's  
theorem. We also extend the pruning procedure to this super-critical  
case. Let $\psi$ be a critical branching mechanism. We set $\psi_ 
\theta(\cdot)=\psi(\cdot+\theta)-\psi(\theta)$. Let $\Theta=(\theta_ 
\infty,+\infty)$ or $\Theta=[\theta_\infty,+\infty)$ be the set of  
values of $\theta$ for which $\psi_\theta$ is a branching mechanism.  
The pruning procedure allows to construct a decreasing L\'evy-CRT- 
valued Markov process $(\ct_\theta,\theta\in\Theta)$, such that $ 
\mathcal{T}_\theta$ has branching mechanism $\psi_\theta$. It is sub- 
critical if $\theta>0$ and super-critical if $\theta<0$. We then  
consider the explosion time $A$ of the CRT: the smaller (negative)  
time $\theta$ for which $\mathcal{T}_\theta$ has finite mass. We  
describe the law of $A$ as well as the distribution of the CRT just  
after this explosion time. The CRT just after explosion can be seen as  
a CRT conditioned not to be extinct which is pruned with an  
independent intensity related to $A$. We also study the evolution of  
the CRT-valued process after the explosion time. This extends results  
from Aldous and Pitman on Galton-Watson trees. For the particular case  
of the quadratic branching mechanism, we show that after explosion the  
total mass of the CRT behaves like the inverse of a stable  
subordinator with index 1/2. This result is related to the size of the  
tagged fragment for the fragmentation of Aldous' CRT.

http://arxiv.org/abs/0904.4175


8443. A uniqueness theorem for the martingale problem describing a  
diffusion in media with membranes
Author(s): Olga V. Aryasova and Mykola I. Portenko

Abstract: We formulate a martingale problem that describes a diffusion  
process in a multidimensional Euclidean space with a membrane located  
on a given smooth surface and having the properties of skewing and  
delaying. The theorem on the existence of no more than one solution to  
the problem is proved.

http://arxiv.org/abs/0904.4223


8444. Numerical Computation of First-Passage Times of Increasing Levy  
Processes
Author(s): Mark S. Veillette; Murad S. Taqqu

Abstract: Let $\{D(s), s \geq 0\}$ be a non-decreasing L\'evy process.  
The first-hitting time process $\{E(t) t \geq 0\}$ (which is sometimes  
referred to as an inverse subordinator) defined by $E(t) = \inf \{s:  
D(s) > t \}$ is a process which has arisen in many applications. Of  
particular interest is the mean first-hitting time $U(t)=\mathbb{E}E(t) 
$. This function characterizes all finite-dimensional distributions of  
the process $E$. The function $U$ can be calculated by inverting the  
Laplace transform of the function $\widetilde{U}(\lambda) = (\lambda  
\phi(\lambda))^{-1}$, where $\phi$ is the L\'evy exponent of the  
subordinator $D$. In this paper, we give two methods for computing  
numerically the inverse of this Laplace transform. The first is based  
on the Bromwich integral and the second is based on the Post-Widder  
inversion formula. The software written to support this work is  
available from the authors and we illustrate its use at the end of the  
paper.

http://arxiv.org/abs/0904.4232


8445. Exact maximum likelihood estimators for drift fractional  
Brownian motions
Author(s): Hu Yaozhong and Xiao Weilin and Zhang Weiguo

Abstract: This paper deals with the problems of consistence and strong  
consistence of the maximum likelihood estimators of the mean and  
variance of the drift fractional Brownian motions observed at discrete  
time instants. A central limit theorem for these estimators is also  
obtained by using the Malliavin calculus.

http://arxiv.org/abs/0904.4186


8446. On Limit theorems in $JW$- algebras
Author(s): Abdusalom Karimov and Farrukh Mukhamedov

Abstract: In the present paper, we study bundle convergence in $JW$-  
algebra and prove some ergodic theorems with respect to such  
convergence. Moreover, conditional expectations of $JW$-algebras are  
considered. Using such expectations, the convergence of  
supermartingales in $JW$- algebras is established.

http://arxiv.org/abs/0904.4070


8447. Large deviations of empirical zero point measures on Riemann  
surfaces,
Author(s): O. Zeitouni and S. Zelditch

Abstract: We prove an LDP for the empirical measure of complex zeros  
of a Gaussian random complex polynomial of degree N of one variable as  
N tends to infinity. The Gaussian measure is induced by an inner  
product defined by a smooth weight (Hermitian metric) $h$ and a  
Bernstein-Markov measure $\nu$. The speed is N^2 and the the unique  
minimizer of the rate function $I$ is the weighted equilibrium measure  
$\nu_{h, K}$ with respect to $h$ on the support $K$ of $\nu$.

http://arxiv.org/abs/0904.4271


8448. Continuous-time trading and the emergence of probability
Author(s): Vladimir Vovk

Abstract: This paper establishes a non-stochastic analogue of the  
celebrated result by Dubins and Schwarz about reduction of continuous  
martingales to Brownian motion via time change. We consider an  
idealized financial security with continuous price process, without  
making any stochastic assumptions. It is shown that almost all sample  
paths of the price process possess quadratic variation, where "almost  
all" is understood in the following game-theoretic sense: there exists  
a trading strategy that earns infinite capital without risking more  
than one monetary unit if the process of quadratic variation does not  
exist. Replacing time by the quadratic variation process, we show that  
the price process becomes Brownian motion. This is essentially the  
same conclusion as in the Dubins-Schwarz result, except that the  
probabilities (constituting the Wiener measure) emerge instead of  
being postulated. We also give an elegant statement, inspired by Peter  
McCullagh's unpublished work, of this result in terms of game- 
theoretic probability.

http://arxiv.org/abs/0904.4364


8449. Intrinsic ultracontractivity for Schrodinger operators based on  
fractional Laplacians
Author(s): Kamil Kaleta and Tadeusz Kulczycki

Abstract: We study the Feynman-Kac semigroup generated by the  
Schr{\"o}dinger operator based on the fractional Laplacian $-(- 
\Delta)^{\alpha/2} - q$ in $\Rd$, for $q \ge 0$, $\alpha \in (0,2)$.  
We obtain sharp estimates of the first eigenfunction $\phi_1$ of the  
Schr{\"o}dinger operator and conditions equivalent to intrinsic  
ultracontractivity of the Feynman-Kac semigroup. For potentials $q$  
such that $\lim_{|x| \to \infty} q(x) = \infty$ and comparable on unit  
balls we obtain that $\phi_1(x)$ is comparable to $(|x| + 1)^{-d -  
\alpha} (q(x) + 1)^{-1}$ and intrinsic ultracontractivity holds iff $ 
\lim_{|x| \to \infty} q(x)/\log|x| = \infty$. Proofs are based on  
uniform estimates of $q$-harmonic functions.

http://arxiv.org/abs/0904.4386


8450. Adaptive sampling for linear state estimation
Author(s): Maben Rabi and George V. Moustakides and John S. Baras

Abstract: State estimation under sampling rate constraints is  
important for Networked control. To obtain the lowest possible  
estimator distortion under such constraints, the samples must be  
chosen adaptively based on the trajectory of the signal being sampled,  
rather than deterministically. We treat the case of perfect  
observations at the sensor in which it measures a diffusion state  
process perfectly. The sensor has to choose causally, exactly N  
sampling times when it transmits samples to a supervisor which  
receives the samples without delay or distortion. Based on the causal  
sequence of samples it receives, the supervisor maintains a continuous  
MMSE estimate. In this paper we provide the optimal adaptive sampling  
rules to be used by the sensor that minimize the aggregate, finite- 
horizon, mean-square error distortion for scalar linear estimation. We  
also characterize the performance of the suboptimal class of Delta  
sampling schemes which uses fixed thresholds as sampling envelopes.  
The results of these calculations are surprising. Delta sampling  
performs worse than even the periodic sampling scheme, except possibly  
when the sample budget is quite small.

http://arxiv.org/abs/0904.4358


8451. Rotor Walks and Markov Chains
Author(s): Alexander E. Holroyd and James Propp

Abstract: The rotor walk is a derandomized version of the random walk  
on a graph. On successive visits to any given vertex, the walker is  
routed to each of the neighboring vertices in some fixed cyclic order,  
rather than to a random sequence of neighbors. The concept generalizes  
naturally to Markov chains on a countable state space. Subject to  
general conditions, we prove that many natural quantities associated  
with the rotor walk (including normalized hitting frequencies, hitting  
times and occupation frequencies) concentrate around their expected  
values for the random walk. Furthermore, the concentration is stronger  
than that associated with repeated runs of the random walk, with  
discrepancy at most C/n after n runs (for an explicit constant C),  
rather than constant/sqrt n.

http://arxiv.org/abs/0904.4507


8452. On the Representation Theorem of G-Expectations and Paths of G-- 
Brownian Motion
Author(s): Mingshang Hu and Shige Peng

Abstract: We give a very simple and elementary proof of the existence  
of a weakly compact family of probability measures $\{P_{\theta}: 
\theta \in \Theta \}$ to represent an important sublinear expectation-- 
G-expectation $\mathbb{E}[\cdot]$. We also give a concrete  
approximation of a bounded continuous function $X(\omega)$ by an  
increasing sequence of cylinder functions $L_{ip}(\Omega)$ in order to  
prove that $C_{b}(\Omega)$ belongs to the $\mathbb{E}[|\cdot|]$- 
completion of the $L_{ip}(\Omega)$.

http://arxiv.org/abs/0904.4519


8453. Metric properties of discrete time exclusion type processes in  
continuum
Author(s): Michael Blank

Abstract: A new class of exclusion type processes acting in continuum  
with synchronous updating is introduced and studied. Ergodic averages  
of particle velocities are obtained and their connections to other  
statistical quantities, in particular to the particle density (the so  
called Fundamental Diagram) is analyzed rigorously. The main technical  
tool is a "dynamical" coupling applied in a nonstandard fashion: we do  
not prove the existence of the successful coupling (which even might  
not hold) but instead use its presence/absence as an important  
diagnostic tool. Despite that this approach cannot be applied to  
lattice systems directly, it allows to obtain new results for the  
lattice systems embedding them to the systems in continuum.  
Applications to the traffic flows modelling are discussed as well.

http://arxiv.org/abs/0904.4585


8454. On random topological Markov chains with big images and preimages
Author(s): Manuel Stadlbauer

Abstract: We introduce a relative notion of the 'big images and  
preimages'-property for random topological Markov chains. This then  
implies that a relative version of the Ruelle-Perron-Frobenius theorem  
holds with respect to summable and locally Hoelder continuous  
potentials.

http://arxiv.org/abs/0904.4630


8455. VRRW on complete-like graphs: almost sure behavior
Author(s): Vlada Limic and Stanislav Volkov

Abstract: By a theorem of Volkov (2001) we know that on most graphs,  
with positive probability, the linearly vertex-reinforced random walk  
(VRRW) stays within a finite "trapping" subgraph at all large times.  
The question of whether this tail behavior occurs with probability one  
is open in general. R. Pemantle (1988) in his thesis proved, via a  
dynamical system approach, that for a VRRW on any complete graph the  
asymptotic frequency of visits is uniform over vertices. These  
techniques do not easily extend even to the setting of complete-like  
graphs, that is, complete graphs ornamented with finitely many leaves  
at each vertex. In this work we combine martingale and large deviation  
techniques to prove that almost surely the VRRW on any such graph  
spends positive (and equal) proportions of time on each of its non- 
leaf vertices. This behavior was previously shown to occur only up to  
event of positive probability, cf. Volkov (2001). We believe that our  
approach can be used as a building block in studying related questions  
on more general graphs. The same set of techniques is used to obtain  
explicit bounds on the speed of convergence of the empirical  
occupation measure.

http://arxiv.org/abs/0904.4722


8456. Restricted isometry property of matrices with independent  
columns and neighborly polytopes by random sampling
Author(s): Rados{\l}aw Adamczak and Alexander E. Litvak and Alain  
Pajor and Nicole Tomczak-Jaegermann

Abstract: This paper considers compressed sensing matrices and  
neighborliness of a centrally symmetric convex polytope generated by  
vectors $\pm X_1,...,\pm X_N\in\R^n$, ($N\ge n$). We introduce a class  
of random sampling matrices and show that they satisfy a restricted  
isometry property (RIP) with overwhelming probability. In particular,  
we prove that matrices with i.i.d. centered and variance 1 entries  
that satisfy uniformly a sub-exponential tail inequality possess this  
property RIP with overwhelming probability. We show that such  
"sensing" matrices are valid for the exact reconstruction process of $m 
$-sparse vectors via $\ell_1$ minimization with $m\le Cn/\log^2 (cN/n) 
$. The class of sampling matrices we study includes the case of  
matrices with columns that are independent isotropic vectors with log- 
concave densities. We deduce that if $K\subset \R^n$ is a convex body  
and $X_1,..., X_N\in K$ are i.i.d. random vectors uniformly  
distributed on $K$, then, with overwhelming probability, the symmetric  
convex hull of these points is an $m$-centrally-neighborly polytope  
with $m\sim n/\log^2 (cN/n)$.

http://arxiv.org/abs/0904.4723


8457. Current fluctuations of a system of one-dimensional random walks  
in random environment
Author(s): Jonathon Peterson and Timo Seppalainen

Abstract: We study the current of particles that move independently in  
a common static random environment on the one-dimensional integer  
lattice. A two-level fluctuation picture appears. On the central limit  
scale the quenched mean of the current process converges to a Brownian  
motion. On a smaller scale the current process centered at its  
quenched mean converges to a mixture of Gaussian process. These  
Gaussian processes are similar to those arising from classical random  
walks, but the environment makes itself felt through an additional  
Brownian random shift in the spatial argument of the limiting current  
process.

http://arxiv.org/abs/0904.4768


8458. Right Inverses of Levy processes
Author(s): R. Doney and M. Savov

Abstract: We call a right continuous increasing process K(x) a partial  
right inverse (PRI) of a given Levy process X if X(K{x))=x at least  
for all x in some random interval [0,c) of of positive length. In this  
paper we give a necessary and sufficient condition for the existence  
of a PRI in terms of the Levy triplet.

http://arxiv.org/abs/0904.4871


8459. Remarks on the fractional Brownian motion
Author(s): Denis Feyel and Arnaud De La Pradelle (Institut math jussieu)

Abstract: We study the fBm by use of convolution of the standard white  
noise with a certain distribution. This brings some simplifications  
and new results.

http://arxiv.org/abs/0904.4923


8460. A Supplement to the Paper Poisson Approximation in a Poisson  
Limit Theorem Inspired by Coupon Collecting
Author(s): Anna P\'osfai

Abstract: In this note we give a proof for the result stated as  
Theorem 4 in Poisson Approximation in a Poisson Limit Theorem Inspired  
by Coupon Collecting.

http://arxiv.org/abs/0904.4924


8461. Maximizing the probability of attaining a target prior to  
extinction
Author(s): Debasish Chatterjee and Eugenio Cinquemani and John Lygeros

Abstract: We present a dynamic programming-based solution to the  
problem of maximizing the probability of attaining a target set before  
hitting a cemetery set for a discrete-time Markov control process.  
Under mild hypotheses we establish that there exists a deterministic  
stationary policy that achieves the maximum value of this probability.  
We demonstrate how the maximization of this probability can be  
computed through the maximization of an expected total reward until  
the first hitting time to either the target or the cemetery set.  
Martingale characterizations of thrifty, equalizing, and optimal  
policies in the context of our problem are also established.

http://arxiv.org/abs/0904.4143


8462. Two speed TASEP
Author(s): Alexei Borodin (1) and Patrik L. Ferrari (2) and Tomohiro  
Sasamoto (3) ((1) Caltech, (2) Bonn University, (3) Chiba University  
and TU Munich)

Abstract: We consider the TASEP on Z with two blocks of particles  
having different jump rates. We study the large time behavior of  
particles' positions. It depends both on the jump rates and the region  
we focus on, and we determine the complete process diagram. In  
particular, we discover a new transition process in the region where  
the influence of the random and deterministic parts of the initial  
condition interact. Slow particles may create a shock, where the  
particle density is discontinuous and the distribution of a particle's  
position is asymptotically singular. We determine the diffusion  
coefficient of the shock without using second class particles. We also  
analyze the case where particles are effectively blocked by a wall  
moving with speed equal to their intrinsic jump rate.

http://arxiv.org/abs/0904.4655




-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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