|
|
|
|
 |
|
|
|
|
|
|
|
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
http://arxiv.org/abs/0904.2689
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Author(s): Yoav Freund
Abstract: We present a method for hedging in continuous time.
http://arxiv.org/abs/0904.3356
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
|
|
|
|
|
|
