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Probability Abstracts 99
This document contains abstracts 5757-5996 from
July-1-2007 to August-31-2007.
They have been mailed on September 14th, 2007.
Author(s): Michael Aizenman and Francois Germinet and Abel Klein and Simone Warzel
Abstract: As was noted already by A. N. Kolmogorov, any random variable has
a Bernoulli
component. This observation provides a tool for the extension of
results which
are known for Bernoulli random variables to arbitrary distributions. Two
applications are provided here: i. an anti-concentration bound for a
class of
functions of independent random variables, where probabilistic bounds
are
extracted from combinatorial results, and ii. a proof, based on the
Bernoulli
case, of spectral localization for random Schroedinger operators with
arbitrary
probability distributions for the single site coupling constants. For
a general
random variable, the Bernoulli component may be defined so that its
conditional
variance is uniformly positive. The natural maximization problem is
an optimal
transport question which is also addressed here.
http://arxiv.org/abs/0707.0095
Author(s): Kurt Johansson
Abstract: We show that the transition probability of the Markoc chain
$(G(j,1),...,G(j,n))_{j\ge 1}$, where the $G(i,j)'s$ are certain
directed
last-passage times, is given by a determinant of a special form. An
analogous
formula has recently been obtained by Warren in a Brownian motion model.
Furthermore we demonstrate that this formula leads to the Meixner
ensemble when
we compute the distribution function for $G(m,n)$. We also obtain the
Fredholm
determinant representation of this distribution, where the kernel has
a double
contour integral representation.
http://arxiv.org/abs/0707.0098
Author(s): Thomas Cass and Peter Friz and Nicolas Victoir
Abstract: Malliavin Calculus is about Sobolev-type regularity of functionals
on Wiener
space, the main example being the Ito map obtained by solving stochastic
differential equations. Rough path analysis is about strong
regularity of
solution to (possibly stochastic) differential equations. We combine
arguments
of both theories and discuss existence of a density for solutions to
stochastic
differential equations driven by a general class of non-degenerate
Gaussian
processes, including processes with sample path regularity worse than
Brownian
motion.
http://arxiv.org/abs/0707.0154
Author(s): Ilya Molchanov
Abstract: It is known that each symmetric stable distribution in $R^d$ is
related to a
norm on $R^d$ that makes $R^d$ embeddable in $L_p([0,1])$. In case of a
multivariate Cauchy distribution the unit ball in this norm
corresponds is the
polar set to a convex set in $R^d$ called a zonoid. This work
exploits recent
advances in convex geometry in order to come up with new
probabilistic results
for multivariate stable distributions. In particular, it provides
expressions
for moments of the Euclidean norm of a stable vector, mixed moments
and various
integrals of the density function. It is shown how to use geometric
inequalities in order to bound important parameters of stable laws.
It is shown
that each symmetric stable laws appears as the limit for the sum of
sub-Gaussian laws and an estimate for the probability distance to a
sub-Gaussian law is given. Operations with convex sets induce the
well-known
and new operations with stable vectors. Furthermore, covariation,
regression
and orthogonality concepts for stable laws acquire geometric
interpretations. A
similar collection of results is presented for one-sided stable laws.
http://arxiv.org/abs/0707.0221
Author(s): Arnaud Gloter (LAMA) and Emmanuel Gobet (LJK)
Abstract: In this paper we prove the Local Asymptotic Mixed Normality (LAMN)
property
for the statistical model given by the observation of local means of a
diffusion process $X$. Our data are given by $ \int_0^1 X_{\frac{s+i}
{n}} \dd
\mu (s)$ for $i=0,...,n-1$ and the unknown parameter appears in the
diffusion
coefficient of the process $X$ only. Although the data are nor Markovian
neither Gaussian we can write down, with help of Malliavin calculus, an
explicit expression for the log-likelihood of the model, and then
study the
asymptotic expansion. We actually find that the asymptotic
information of this
model is the same one as for a usual discrete sampling of $X$.
http://arxiv.org/abs/0707.0257
Author(s): Pavel Chigansky
Abstract: The paper studies large sample asymptotic properties of the Maximum
Likelihood Estimator (MLE) for the parameter of a continuous time
Markov chain,
observed in white noise. Using the method of weak convergence of
likelihoods
due to I.Ibragimov and R.Khasminskii, consistency, asymptotic
normality and
convergence of moments are established for MLE under certain strong
ergodicity
conditions of the chain.
http://arxiv.org/abs/0707.0271
Author(s): Xinjia Chen and Guoxiang Gu and Kemin Zhou
Abstract: In this paper, we propose a statistical theory on measurement and
estimation
of Rayleigh fading channels in wireless communications and provide
complete
solutions to the fundamental problems: What is the optimum estimator
for the
statistical parameters associated with the Rayleigh fading channel,
and how
many measurements are sufficient to estimate these parameters with the
prescribed margin of error and confidence level? Our proposed
statistical
theory suggests that two testing signals of different strength be
used. The
maximum likelihood (ML) estimator is obtained for estimation of the
statistical
parameters of the Rayleigh fading channel that is both sufficient and
complete
statistic. Moreover, the ML estimator is the minimum variance (MV)
estimator
that in fact achieves the Cramer-Rao lower bound.
http://arxiv.org/abs/0707.0284
Author(s): Itaru Mitoma and Seiki Nishikawa
Abstract: In an abstract Wiener space setting, we constract a rigorous
mathematical
model of the one-loop approximation of the perturbative Chern-Simons
integral,
and derive its explicit asymptotic expansion for stochastic Wilson
lines.
http://arxiv.org/abs/0707.0047
Author(s): Erhan Bayraktar and Masahiko Egami
Abstract: We characterize the optimal switching problem as coupled optimal
stoping
problems. We then use the optimal stopping theory to provide a
solution. As
opposed to the methods using quasi-variational inequalities and
verification
theorem we directly work with the value function.
http://arxiv.org/abs/0707.0100
Author(s): Peter Friz and Nicolas Victoir
Abstract: We consider multi-dimensional Gaussian processes and give a new
condition on
the covariance, simple and sharp, for the existence of stochastic area
(s).
Gaussian rough paths are constructed with a variety of weak and strong
approximation results. Together with a new RKHS embedding, we obtain
a powerful
- yet conceptually simple - framework in which to analysize differential
equations driven by Gaussian signals in the rough paths sense.
http://arxiv.org/abs/0707.0313
Author(s): Piotr Milos
Abstract: Limit theorems are presented for the rescaled occupation time
fluctuation
process of a critical finite variance branching particle system in
$\mathbb{R}^{d}$ with symmetric $\alpha$-stable motion starting off
from either
a standard Poisson random field or the equilibrium distribution for
critical
$d=2\alpha$ and large $d>2\alpha$ dimensions. The limit processes are
generalised Wiener processes. The obtained convergence is in space-time,
finite-dimensional distributions sense. With the addtional assumption
on the
branching law we obtain functional convergence.
http://arxiv.org/abs/0707.0316
Author(s): Jos\'e Trashorras
Abstract: In this paper we prove a Large Deviation Principle for the
sequence of
symmetrised empirical measures $\frac{1}{n} \sum_{i=1}^{n}
\delta_{(X^n_i,X^n_{\sigma_n(i)})}$ where $\sigma_n$ is a random
permutation
and $((X_i^n)_{1 \leq i \leq n})_{n \geq 1}$ is a triangular array of
random
variables with suitable properties. As an application we show how
this result
allows to improve the Large Deviation Principles for symmetrised
initial-terminal conditions bridge processes recently established by
Adams,
Dorlas and K\"{o}nig.
http://arxiv.org/abs/0707.0344
Author(s): Nizar Demni (PMA)
Abstract: We begin with the study of some properties of the radial Dunkl
process
associated to a reduced root system $R$. It is shown that this
diffusion is the
unique strong solution for all $t \geq 0$ of a SDE with singular
drift. Then,
we study $T_0$, the first hitting time of the positive Weyl chamber :
we prove,
via stochastic calculus, a result already obtained by Chybiryakov on the
finiteness of $T_0$. The second and new part deals with the law of
$T_0$ for
which we compute the tail distribution, as well as some insight via
stochastic
calculus on how root systems are connected with eigenvalues of standard
matrix-valued processes. This gives rise to the so-called $\beta$-
processes.
The ultraspherical $\beta$-Jacobi case still involves a reduced root
system
while the general case is closely connected to a non reduced one.
This process
lives in a convex bounded domain known as principal Weyl alcove and
the strong
uniqueness result remains valid. The last part deals with the first
hitting
time of the alcove's boundary and the semi group density which
enables us to
answer some open questions.
http://arxiv.org/abs/0707.0367
Author(s): Mark Adler and Jonathan Delepine and Pierre van Moerbeke
Abstract: Consider n non-intersecting Brownian particles on the real line
(Dyson
Brownian motions), all starting from the origin at time t=0, and
evolving up to
time t=1. Assume that, among those particles, r are forced to reach a
given
final target a >0 (outliers), while the (n-r) remaining ones return
to the
position x=0. Letting n tend to infinity, view this cloud of
particles from the
edge (i.e., near the largest particle), with the space and time
rescaling given
by the edge statistics of GUE. Also let the target point a go to
infinity with
n at the rate a=rho\sqrt{n/2} for rho between 0 and 1. Then a phase
transition
takes place at rho=1.
Indeed, for rho<1, the limit cloud is described by the Airy
process, which in
effect is rho-independent and also independent of the number r of
outlying
particles; it is as if rho were =0. For rho=1, the process depends on
the
number r of outliers, and leads to a new process: an Airy process with r
outliers (in short: r-Airy process), which is a kind of interpolation
between
the Airy and Pearcey processes. The log of the probability that at
time tau
(the new rescaled time) the cloud does not exceed x is given by the
Fredholm
determinant of a new kernel (extending the Airy kernel) and it
satisfies a
non-linear PDE in x and tau, from which the asymptotic behavior of
the process
can be deduced for tau tending to -infinity (remote past). This
kernel is
closely related to one found by Baik, Ben Arous and Peche in the
context of
multivariate statistics.
http://arxiv.org/abs/0707.0442
Author(s): Laurent Decreusefond and Nicolas Savy
Abstract: By a method inspired of the Stein's method, we derive an upper-
bound of the
Rubinstein distance between two absolutely continuous probability
measures on
configurations space. As an application, we show that the best way to
approximate a Modulated Poisson Process (see below for the
definition) by a
Poisson process is to equate their intensity.
http://arxiv.org/abs/0707.0445
Author(s): Guillaume Aubrun (ICJ) and Ion Nechita (ICJ)
Abstract: We study how iterated convolutions of probability measures compare
under
stochastic domination. We give necessary and sufficient conditions
for the
existence of an integer $n$ such that $\mu^{*n}$ is stochastically
dominated by
$\nu^{*n}$ for two given probability measures $\mu$ and $\nu$. As a
consequence
we obtain a similar theorem on the majorization order for vectors in $
\R^d$. In
particular we prove results about catalysis in quantum information
theory.
http://arxiv.org/abs/0707.0211
Author(s): Peter Elbau
Abstract: We show that in the large matrix limit, the eigenvalues of the
normal matrix
model for matrices with spectrum inside a compact domain with a
special class
of potentials homogeneously fill the interior of a polynomial curve
uniquely
defined by the area of its interior domain and its exterior harmonic
moments
which are all given as parameters of the potential.
Then we consider the orthogonal polynomials corresponding to this
matrix
model and show that, under certain assumptions, the density of the
zeros of the
highest relevant orthogonal polynomial in the large matrix limit is
(up to some
constant factor) given by the discontinuity of the Schwarz function
of this
polynomial curve.
http://arxiv.org/abs/0707.0425
Author(s): Mireille Chaleyat-Maurel (MAP5 and PMA) and Valentine Genon- Catalot (MAP5)
Abstract: We consider a Wright-Fisher diffusion (x(t)) whose current state
cannot be
observed directly. Instead, at times t1 < t2 < . . ., the
observations y(ti)
are such that, given the process (x(t)), the random variables (y(ti))
are
independent and the conditional distribution of y(ti) only depends on
x(ti).
When this conditional distribution has a specific form, we prove that
the model
((x(ti), y(ti)), i 1) is a computable filter in the sense that all
distributions involved in filtering, prediction and smoothing are
exactly
computable. These distributions are expressed as finite mixtures of
parametric
distributions. Thus, the number of statistics to compute at each
iteration is
finite, but this number may vary along iterations.
http://arxiv.org/abs/0707.0537
Author(s): Ken-iti Sato
Abstract: Let $X^{(\mu)}(ds)$ be an $\mathbb{R}^d$-valued homogeneous
independently
scattered random measure over $\mathbb{R}$ having $\mu$ as the
distribution of
$X^{(\mu)}((t,t+1])$. Let $f(s)$ be a nonrandom measurable function
on an open
interval $(a,b)$ where $-\infty\leqslant a
http://arxiv.org/abs/0707.0538
Author(s): Giuseppina Guatteri and Federica Masiero
Abstract: We study quadratic optimal stochastic control problems with
control dependent
noise state equation perturbed by an affine term and with stochastic
coefficients. Both infinite horizon case and ergodic case are
treated. To this
purpose we introduce a Backward Stochastic Riccati Equation and a
dual backward
stochastic equation, both considered in the whole time line. Besides
some
stabilizability conditions we prove existence of a solution for the two
previous equations defined as limit of suitable finite horizon
approximating
problems. This allows to perform the synthesis of the optimal control.
http://arxiv.org/abs/0707.0606
Author(s): Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
Abstract: We prove strong theorems for the local time at infinity of a
nearest neighbor
transient random walk. First, laws of the iterated logarithm are
given for the
large values of the local time. Then we investigate the length of
intervals
over which the walk runs through (always from left to right) without
ever
returning.
http://arxiv.org/abs/0707.0734
Author(s): Amine Asselah
Abstract: We obtain a large deviations principle for the self-intersection
local times
for a simple random walk in dimension d>4. As an application, we obtain
moderate deviations for random walk in random sceneries in some
region of
parameters.
http://arxiv.org/abs/0707.0813
Author(s): Francis Comets and Serguei Popov and Marina Vachkovskaia
Abstract: We study the asymptotic properties of the number of open paths of
length $n$
in an oriented $\rho$-percolation model. We show that this number is
$e^{n\alpha(\rho)(1+o(1))}$ as $n \to \infty$. The exponent $\alpha$ is
deterministic, it can be expressed in terms of the free energy of a
polymer
model, and it can be explicitely computed in some range of the
parameters.
Moreover, in a restricted range of the parameters, we even show that
the number
of such paths is $n^{-1/2} W e^{n\alpha(\rho)}(1+o(1))$ for some
nondegenerate
random variable $W$. We build on connections with the model of directed
polymers in random environment, and we use techniques and results
developed in
this context.
http://arxiv.org/abs/0707.0818
Author(s): Xinjia Chen
Abstract: In this article, we derive a new generalization of Chebyshev
inequality for
random vectors. We demonstrate that the new generalization is much less
conservative than the classical generalization.
http://arxiv.org/abs/0707.0805
Author(s): Xinjia Chen and Kemin Zhou and Jorge L. Aravena
Abstract: In this paper, we derive an explicit formula for constructing the
confidence
interval of binomial parameter with guaranteed coverage probability. The
formula overcomes the limitation of normal approximation which is
asymptotic in
nature and thus inevitably introduce unknown errors in applications.
Moreover,
the formula is very tight in comparison with classic Clopper-
Pearson's approach
from the perspective of interval width. Based on the rigorous
formula, we also
obtain approximate formulas with excellent performance of coverage
probability.
http://arxiv.org/abs/0707.0837
Author(s): Jean-Luc Marichal
Abstract: We give the cumulative distribution functions, the expected
values, and the
moments of weighted lattice polynomials when regarded as real
functions of
independent random variables. Since weighted lattice polynomial
functions
include ordinary lattice polynomial functions and, particularly, order
statistics, our results encompass the corresponding formulas for these
particular functions. We also provide an application to the reliability
analysis of coherent systems.
http://arxiv.org/abs/0707.0953
Author(s): Svante Janson and Niclas Petersson
Abstract: In this paper we study the integral of the supremum process of
standard
Brownian motion. We present an explicit formula for the moments of
the integral
(or area) A(T), covered by the process in the time interval [0,T].
The Laplace
transform of A(T) follows as a consequence. The main proof involves a
double
Laplace transform of A(T) and is based on excursion theory and local
time for
Brownian motion.
http://arxiv.org/abs/0707.0989
Author(s): Svante Janson and Guy Louchard
Abstract: Several Brownian areas are considered in this paper: the Brownian
excursion
area, the Brownian bridge area, the Brownian motion area, the
Brownian meander
area, the Brownian double meander area, the positive part of Brownian
bridge
area, the positive part of Brownian motion area. We are interested in
the
asymptotics of the right tail of their density function. Inverting a
double
Laplace transform, we can derive, in a mechanical way, all terms of an
asymptotic expansion. We illustrate our technique with the
computation of the
first four terms. We also obtain asymptotics for the right tail of the
distribution function and for the moments. Our main tool is the two-
dimensional
saddle point method.
http://arxiv.org/abs/0707.0991
Author(s): Ton\'ci Antunovi\'c and Ivan Veseli\'c
Abstract: We study homogeneous, independent percolation on general quasi-
transitive
graphs. We prove that in the disorder regime where all clusters are
finite
almost surely, in fact the expectation of the cluster size is finite.
This
extends a well-known theorem by Menshikov and Aizenman & Barsky to all
quasi-transitive graphs. Moreover we deduce that in this disorder
regime the
cluster size distribution decays exponentially, extending a result of
Aizenman
& Newman. Our results apply to both edge and site percolation, as
well as long
range (edge) percolation. The proof is based on a modification of the
Aizenman
& Barsky method.
http://arxiv.org/abs/0707.1089
Author(s): Giovanni Peccati (LSTA)
Abstract: Fix an integer k, and let I(l), l=1,2,..., be a sequence of k-
dimensional
vectors of multiple Wiener-It\^o integrals with respect to a general
Gaussian
process. We establish necessary and sufficient conditions to have
that, as l
diverges, the law of I(l) is asymptotically close (for example, in
the sense of
Prokhorov's distance) to the law of a k-dimensional Gaussian vector
having the
same covariance matrix as I(l). The main feature of our results is
that they
require minimal assumptions (basically, boundedness of variances) on the
asymptotic behaviour of the variances and covariances of the elements
of I(l).
In particular, we will not assume that the covariance matrix of I(l) is
convergent. This generalizes the results proved in Nualart and
Peccati (2005),
Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As
shown in
Marinucci and Peccati (2007b), the criteria established in this paper
are
crucial in the study of the high-frequency behaviour of stationary
fields
defined on homogeneous spaces.
http://arxiv.org/abs/0707.1220
Author(s): Julien Guyon (CERMICS)
Abstract: Given a smooth R^d-valued diffusion, we study how fast the Euler
scheme with
time step 1/n converges in law. To be precise, we look for which
class of test
functions f the approximate expectation E[f(X^{n,x}_1)] converges
with speed
1/n to E[f(X^x_1)]. If X is uniformly elliptic, we show that this class
contains all tempered distributions, and all measurable functions with
exponential growth. We give applications to option pricing and
hedging, proving
numerical convergence rates for prices, deltas and gammas.
http://arxiv.org/abs/0707.1243
Author(s): Jean-Baptiste Gouere (MAPMO) and Regine Marchand (IECN)
Abstract: We study a random growth model on $\R^d$ introduced by Deijfen.
This is a
continuous first-passage percolation model. The growth occurs by
means of
spherical outbursts with random radii in the infected region. We aim
at finding
conditions on the distribution of the random radii to determine
whether the
growth of the process is linear or not. To do so, we compare this
model with a
continuous analogue of the greedy lattice paths model and transpose
results in
the lattice setting to the continuous setting.
http://arxiv.org/abs/0707.1395
Author(s): Fabrice Gamboa (IMT) and Thierry Klein (IMT) and Cl\'ementine Prieur (IMT)
Abstract: We prove large and moderate deviation principles for the
distribution of an
empirical mean conditioned by the value of the sum of discrete i.i.d.
random
variables. Some applications for combinatoric problems are discussed.
http://arxiv.org/abs/0707.1461
Author(s): Volker Betz and Olaf Wittich
Abstract: We consider Gibbs measures relative to Brownian motion of Feynman-
Kac type,
with single site potential V. We show that for a large class of V,
including
the Coulomb potential, there exist infinitely many infinite volume Gibbs
measures.
http://arxiv.org/abs/0707.1462
Author(s): O. Khorunzhiy
Abstract: Regarding the adjacency matrices of n-vertex graphs and related graph
Laplacian, we introduce two families of discrete matrix models
constructed both
with the help of the Erdos-Renyi ensemble of random graphs.
Corresponding
matrix sums represent the characteristic functions of the average
number of
walks and closed walks over the random graph. These sums can be
considered as
discrete analogs of the matrix integrals of random matrix theory.
We study the diagram structure of the cumulant expansions of
logarithms of
these matrix sums and analyze the limiting expressions in the cases
of constant
and vanishing edge probabilities as n tends to infinity.
http://arxiv.org/abs/0707.0997
Author(s): Rui Dong
Abstract: Kingman derived the Ewens sampling formula for random partitions
from the
genealogy model defined by a Poisson process of mutations along lines of
descent governed by a simple coalescent process. M\"ohle described the
recursion which determines the generalization of the Ewens sampling
formula
when the lines of descent are governed by a coalescent with multiple
collisions. In a recent work by Dong, Gnedin and Pitman, authors
exploit an
analogy with the theory of regenerative composition and partition
structures,
and provide various characterizations of the associated exchangeable
random
partitions. This paper gives parallel results for the further
generalized model
with lines of descent following a coalescent with simultaneous multiple
collisions.
http://arxiv.org/abs/0707.1606
Author(s): Jean Bertoin (PMA and Dma)
Abstract: In the classical occupancy scheme, one considers a fixed discrete
probability
measure ${\bf p}=(p_i: {i\in{\cal I}})$ and throws balls
independently at
random in boxes labeled by ${\cal I}$, such that p_i is the
probability that a
given ball falls into the box i. In this work, we are interested in
asymptotic
regimes of this scheme in the situation induced by a refining
sequence $({\bf
p}(k) : k\in\N)$ of random probability measures which arise from some
multiplicative cascade. Our motivation comes from the study of the
asymptotic
behavior of certain fragmentation chains
http://arxiv.org/abs/0707.1640
Author(s): Yusuke Ide and Norio Konno and and Naoki Masuda
Abstract: The threshold network model is a type of finite random graphs. In
this paper,
we introduce a generalized threshold network model. A pair of
vertices with
random weights is connected by an edge when real-valued functions of
the pair
of weights belong to given Borel sets. We extend several known limit
theorems
for the number of prescribed subgraphs to show that the strong law of
large
numbers can be uniform convergence. We also prove two limit theorems
for the
local and global clustering coefficients.
http://arxiv.org/abs/0707.1744
Author(s): Mikhail Menshikov and Dimitri Petritis and Stanislav Volkov
Abstract: In this paper we study a regular rooted coloured tree with random
labels
assigned to its edges, where the distribution of the label assigned
to an edge
depends on the colours of its endpoints. We obtain some new results
relevant to
this model and also show how our model generalizes many other
probabilistic
models, including random walk in random environment on trees, recursive
distributional equations, and multi-type branching random walk on $
\mathbb{R}$.
http://arxiv.org/abs/0707.1746
Author(s): Abass Sagna (PMA)
Abstract: Let $ r, s>0 $. For a given probability measure $P$ on $\mathbb{R}
^d$, let
$(\alpha_n)_{n \geq 1}$ be a sequence of (asymptotically) $L^r(P)$-
optimal
quantizers. For all $\mu \in \mathbb{R}^d $ and for every $\theta >0
$, one
defines the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ by : $
\forall n \geq
1, \alpha_n^{\theta, \mu} = \mu + \theta(\alpha_n - \mu) = \{\mu +
\theta(a-
\mu), a \in \alpha_n \} $. In this paper, we are interested in the
asymptotics
of the $L^s$-quantization error induced by the sequence $(\alpha_n^
{\theta,
\mu})_{n \geq 1}$. We show that for a wide family of distributions, the
sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ is $L^s$-rate-optimal.
For the
Gaussian and the exponential distributions, one shows how to choose the
parameter $\theta$ such that $(\alpha_n^{\theta, \mu})_{n \geq 1}$
satisfies
the empirical measure theorem.
http://arxiv.org/abs/0707.1808
Author(s): Alex Gamburd and Shlomo Hoory and Mehrdad Shahshahani and Aner Shalev, Balint
Virag
Abstract: We prove that random d-regular Cayley graphs of the symmetric group
asymptotically almost surely have girth at least (log_{d-1}|G|)^{1/2}/
2 and
that random d-regular Cayley graphs of simple algebraic groups over F_q
asymptotically almost surely have girth at least log_{d-1}|G|/dim(G).
For the
symmetric p-groups the girth is between log log |G| and (log|G|)
^alpha with
alpha<1. Several conjectures and open questions are presented.
http://arxiv.org/abs/0707.1833
Author(s): A. B. Dieker and J. Warren
Abstract: We find the transition kernels for four Markovian interacting
particle
systems on the line, by proving that each of these kernels is
intertwined with
a Karlin-McGregor type kernel. The resulting kernels all inherit the
determinantal structure from the Karlin-McGregor formula, and have a
similar
form to Schutz's kernel for the totally asymmetric simple exclusion
process.
http://arxiv.org/abs/0707.1843
Author(s): Ming Yang
Abstract: We prove a theorem on additive Levy processes and give applications
http://arxiv.org/abs/0707.1845
Author(s): Ming Yang
Abstract: We prove a new theorem on additive Levy processes and show that
this theorem
implies several proved theorems and a hard conjectured theorem.
http://arxiv.org/abs/0707.1847
Author(s): Ming Yang
Abstract: We compute the Hausdorff dimension of the zero set of an additive
Levy
process.
http://arxiv.org/abs/0707.1849
Author(s): Svante Janson and Malwina Luczak
Abstract: We study the largest component of a random (multi)graph on n
vertices with a
given degree sequence. We let n tend to infinity. Then, under some
regularity
conditions on the degree sequences, we give conditions on the
asymptotic shape
of the degree sequence that imply that with high probability all the
components
are small, and other conditions that imply that with high probability
there is
a giant component and the sizes of its vertex and edge sets satisfy a
law of
large numbers; under suitable assumptions these are the only two
possibilities.
In particular, we recover the results by Molloy and Reed on the size
of the
largest component in a random graph with a given degree sequence.
We further obtain a new sharp result for the giant component just
above the
threshold, generalizing the case of G(n,p) with np=1+omega(n)n^
{-1/3}, where
omega(n) tends to infinity arbitrarily slowly.
Our method is based on the properties of empirical distributions of
independent random variables, and leads to simple proofs.
http://arxiv.org/abs/0707.1786
Author(s): Guang-hui Cai and Hang Wu
Abstract: Based on a law of the iterated logarithm for independent random
variables
sequences, an iterated logarithm theorem for NA sequences with non-
identical
distributions is obtained. The proof is based on a Kolmogrov-type
exponential
inequality.
http://arxiv.org/abs/0707.1968
Author(s): Yan Doumerc and John Moriarty
Abstract: We give the distribution of the first exit time of Brownian motion
from the
alcove of an affine Weyl group, in terms of the distributions of
first exit
times from simpler domains such as orthants. Applications are
explicitly given
in the different type cases. The results extend to any process for
which the
reflection arguments are valid. We also give the real eigenfunctions
of the
Laplacian for alcoves with Dirichlet and Neumann boundary conditions.
http://arxiv.org/abs/0707.2009
Author(s): Rick Durrett and Deena Schmidt and and Jason Schweinsberg
Abstract: We consider the population genetics problem: How long does it take
before
some member of the population has m specified mutations? The case m=2 is
relevant to onset of cancer due to the inactivation of both copies of
a tumor
suppressor gene. Models for larger m are needed for colon cancer and
other
diseases where a sequence of mutations leads to cells with
uncontrolled growth.
http://arxiv.org/abs/0707.2057
Author(s): Jacob van den Berg (CWI and VUA)
Abstract: One of the most well-known classical results for site percolation
on the
square lattice is the equation p_c + p_c^* = 1. In words, this
equation means
that for all values different from p_c of the parameter p the
following holds:
Either a.s. there is an infinite open cluster or a.s. there is an
infinite
closed `star' cluster. This result is closely related to the percolation
transition being sharp: Below p_c the size of the open cluster of a
given
vertex is not only (a.s.) finite, but has a distrubtion with an
exponential
tail. The analog of this result has been proved by Higuchi in 1993 for
two-dimensional Ising percolation, with fixed inverse temparature
beta
http://arxiv.org/abs/0707.2077
Author(s): V P Belavkin and L Gregory
Abstract: We study homogeneous quantum L\'{e}vy processes and fields with
independent
additive increments over a noncommutative *-monoid. These are
described by
infinitely divisible generating state functionals, invariant with
respect to an
endomorphic injective action of a symmetry semigroup. A strongly
covariant GNS
representation for the conditionally positive logarithmic functionals
of these
states is constructed in the complex Minkowski space in terms of
canonical
quadruples and isometric representations on the underlying pre-
Hilbert field
space. This is of much use in constructing quantum stochastic
representations
of homogeneous quantum L\'{e}vy fields on It\^{o} monoids, which is a
natural
algebraic way of defining dimension free, covariant quantum stochastic
integration over a space-time indexing set.
http://arxiv.org/abs/0707.2142
Author(s): Remi Leandre
Abstract: We translate in semigroup theory Bismut's way of the Malliavin
calculus.
http://arxiv.org/abs/0707.2143
Author(s): Un Cig Ji
Abstract: In this paper a quantum stochastic integral representation theorem is
obtained for unbounded regular martingales with respect to
multidimensional
quantum noise. This simultaneously extends results of Parthasarathy
and Sinha
to unbounded martingales and those of the author to multidimensions.
http://arxiv.org/abs/0707.2144
Author(s): Gerard Ben Arous and Alice Guionnet
Abstract: Let $X_N$ be an $N\ts N$ random symmetric matrix with independent
equidistributed entries. If the law $P$ of the entries has a finite
second
moment, it was shown by Wigner \cite{wigner} that the empirical
distribution of
the eigenvalues of $X_N$, once renormalized by $\sqrt{N}$, converges
almost
surely and in expectation to the so-called semicircular distribution
as $N$
goes to infinity. In this paper we study the same question when $P$
is in the
domain of attraction of an $\alpha$-stable law. We prove that if we
renormalize
the eigenvalues by a constant $a_N$ of order $N^{\frac{1}{\alpha}}$, the
corresponding spectral distribution converges in expectation towards
a law
$\mu_\alpha$ which only depends on $\alpha$. We characterize $\mu_
\alpha$ and
study some of its properties; it is a heavy-tailed probability
measure which is
absolutely continuous with respect to Lebesgue measure except
possibly on a
compact set of capacity zero.
http://arxiv.org/abs/0707.2159
Author(s): Matyas Barczy and Gyula Pap
Abstract: On the torus group, on the group of p-adic integers and on the p-adic
solenoid we give a construction of an arbitrary weakly infinitely
divisible
probability measure using real random variables. As a special case of
our
results, we have a new construction of the Haar measure on the p-adic
solenoid.
http://arxiv.org/abs/0707.2186
Author(s): Xing M. Wang
Abstract: In this paper, we continue to explore the consistence and
usability of
Probability Bracket Notation (PBN) proposed in our previous articles.
After a
brief review of PBN with dimensional analysis, we investigate
probability
spaces in terms of PBN by introducing probability spaces associated
with random
variables (R.V) or associated with stochastic processes (S.P). Next,
we express
several important properties of conditional expectation (CE) and some
their
proofs in PBN. Then, we introduce martingales based on sequence of
R.V or based
on filtration in PBN. In the process, we see PBN can be used to
investigate
some probability problems, which otherwise might need explicit usage
of Measure
theory. Whenever applicable, we use dimensional analysis to validate our
formulas and use graphs for visualization of concepts in PBN. We hope
this
study shows that PBN, stimulated by and adapted from Dirac notation
in Quantum
Mechanics (QM), may have the potential to be a useful tool in
probability
modeling, at least for those who are already familiar with Dirac
notation in
QM.
http://arxiv.org/abs/0707.2236
Author(s): Katrin Hofmann-Credner and Michael Stolz
Abstract: It is a classical result of Wigner that for an hermitian matrix with
independent entries on and above the diagonal, the mean empirical
eigenvalue
distribution converges weakly to the semicircle law as matrix size
tends to
infinity. In this paper, we prove analogs of Wigner's theorem for random
matrices taken from all infinitesimal versions of classical symmetric
spaces.
This is a class of models which contains those studied by Wigner and
Dyson,
along with seven others arising in condensed matter physics. Like
Wigner's, our
results are universal in that they only depend on certain assumptions
about the
moments of the matrix entries, but not on the specifics of their
distributions.
What is more, we allow for a certain amount of dependence among the
matrix
entries, in the spirit of a recent generalization of Wigner's
theorem, due to
Schenker and Schulz-Baldes. As a byproduct, we obtain a universality
result for
sample covariance matrices with dependent entries.
http://arxiv.org/abs/0707.2333
Author(s): Julius Borcea and Petter Br\"and\'en and Thomas M. Liggett
Abstract: We introduce the class of {\em strongly Rayleigh} probability
measures by
means of geometric properties of their generating polynomials that
amount to
the stability of the latter. This class contains e.g. product
measures, uniform
random spanning tree measures, and large classes of determinantal
probability
measures and distributions for symmetric exclusion processes. We show
that
strongly Rayleigh measures enjoy all virtues of negative dependence
and we also
prove a series of conjectures due to Liggett, Pemantle, and Wagner,
respectively. Moreover, we extend Lyons' recent results on determinantal
probability measures and we construct counterexamples to several
conjectures of
Pemantle and Wagner on negative dependence and ultra log-concave rank
sequences.
http://arxiv.org/abs/0707.2340
Author(s): Martin Hairer and Grigorios Pavliotis
Abstract: The long-time/large-scale, small-friction asymptotic for the one
dimensional
Langevin equation with a periodic potential is studied in this paper.
It is
shown that the Freidlin-Wentzell and central limit theorem
(homogenization)
limits commute. We prove that, in the combined small friction,
long-time/large-scale limit the particle position converges weakly to a
Brownian motion with a singular diffusion coefficient which we compute
explicitly. We show that the same result is valid for a whole one
parameter
family of space/time rescalings. The proofs of our main results are
based on
some novel estimates on the resolvent of a hypoelliptic operator.
http://arxiv.org/abs/0707.2352
Author(s): Xinjia Chen
Abstract: It is a common contention that it is an ``impossible mission'' to
exactly
determine the minimum sample size for the estimation of a binomial
parameter
with prescribed margin of error and confidence level. In this paper, we
investigate such a very old but also extremely important problem and
demonstrate that the difficulty for obtaining the exact solution is not
insurmountable. Unlike the classical approximate sample size method
based on
the central limit theorem, we develop a new approach for computing
the minimum
sample size that does not require any approximation. Moreover, our
approach
overcomes the conservatism of existing rigorous sample size methods
derived
from Bernoulli's theorem or Chernoff bounds.
Our computational machinery consists of two essential ingredients.
First, we
prove that the minimum of coverage probability with respect to a
binomial
parameter bounded in an interval is attained at a discrete set of
finite many
values of the binomial parameter. This allows for reducing infinite many
evaluations of coverage probability to finite many evaluations.
Second, a
recursive bounding technique is developed to further improve the
efficiency of
computation.
http://arxiv.org/abs/0707.2113
Author(s): Xinjia Chen
Abstract: In this paper, we develop an exact method for the determination of
the
minimum sample size for the estimation of the proportion of a finite
population
with prescribed margin of error and confidence level. By
characterizing the
behavior of the coverage probability with respect to the proportion,
we show
that the computational complexity can be significantly reduced and
bounded
regardless population size.
http://arxiv.org/abs/0707.2115
Author(s): Xinjia Chen
Abstract: In this paper, we develop an approach for the exact determination
of the
minimum sample size for the estimation of a Poisson parameter with
prescribed
margin of error and confidence level. The exact computation is made
possible by
reducing infinite many evaluations of coverage probability to finite
many
evaluations. Such reduction is based on our discovery that the
minimum of
coverage probability with respect to a Poisson parameter bounded in
an interval
is attained at a discrete set of finite many values.
http://arxiv.org/abs/0707.2116
Author(s): Benny Sudakov and Jan Vondrak
Abstract: In this paper we show how to find nearly optimal embeddings of
large trees in
several natural classes of graphs. The size of the tree T can be as
large as a
constant fraction of the size of the graph G, and the maximum degree
of T can
be close to the minimum degree of G. For example, we prove that any
graph of
minimum degree d without 4-cycles contains every tree of size
\epsilon d^2 and
maximum degree at most (1-2\epsilon)d - 2. As there exist d-regular
graphs
without 4-cycles of size O(d^2), this result is optimal up to
constant factors.
We prove similar nearly tight results for graphs of given girth,
graphs with no
complete bipartite subgraph K_{s,t}, random and certain pseudorandom
graphs.
These results are obtained using a simple and very natural randomized
embedding
algorithm, which can be viewed as a "self-avoiding tree-indexed
random walk".
http://arxiv.org/abs/0707.2079
Author(s): Rainer Buckdahn and Marc Quincampoix and Catherine Rainer and Josef Teichmann
Abstract: We provide a short and elementary proof for the recently proved
result by G.
da Prato and H. Frankowska that a closed set is stochastically
invariant if and
only if it is deterministically invariant.
http://arxiv.org/abs/0707.2353
Author(s): Litan Yan and Yu Sun and Yunsheng Lu
Abstract: In this paper, we introduce the linear fractional self-attracting
diffusion
driven by a fractional Brownian motion with Hurst index 1/2
http://arxiv.org/abs/0707.2627
Author(s): D. Marquez-Carreras and C. Rovira
Abstract: We prove a functional law of iterated logarithm for the following
kind of
anticipating stochastic differential equations
$$\xi^u_t=X_0^u+\frac{1}{\sqrt{\log\log u}}\sum_{j=1}^k \int_0^{t}
A_j^u(\xi^u_s)\circ dW_{s}^j+ \int_0^{t} A_0^u(\xi^u_s)ds,$$ where
$u>e$,
$W=\{(W_t^1,...,W_t^k), 0\le t\le 1\}$ is a standard $k$-dimensional
Wiener
process, $A_0^u,A_1^u,..., A_k^u:\mathbb{R}^d\longrightarrow \mathbb
{R}^d$ are
functions of class $\mathcal{C}^2$ with bounded partial derivatives
up to order
2, $X_0^u$ is a random vector not necessarily adapted and the first
integral is
a generalized Stratonovich integral .
http://arxiv.org/abs/0707.2650
Author(s): Boualem Djehiche and Said Hamadene and Alexandre Popier
Abstract: We consider the problem of optimal multiple switching in finite
horizon, when
the state of the system, including the switching costs, is a general
adapted
stochastic process. The problem is formulated as an extended impulse
control
problem and completely solved using probabilistic tools such as the
Snell
envelop of processes and reflected backward stochastic differential
equations.
Finally, when the state of the system is a Markov diffusion process,
we show
that the vector of value functions of the optimal problem is a viscosity
solution to a system of variational inequalities with inter-connected
obstacles.
http://arxiv.org/abs/0707.2663
Author(s): Benjamin Jourdain (CERMICS) and Sylvie M\'el\'eard (CMAP) and Wojbor Woyczynski
Abstract: In this paper we study general nonlinear stochastic differential
equations,
where the usual Brownian motion is replaced by a L\'evy process. We also
suppose that the coefficient multiplying the increments of this
process is
merely Lipschitz continuous and not necessarily linear in the time-
marginals of
the solution as is the case in the classical McKean-Vlasov model. We
first
study existence, uniqueness and particle approximations for these
stochastic
differential equations. When the driving process is a pure jump L
\'evy process
with a smooth but unbounded L\'evy measure, we develop a stochastic
calculus of
variations to prove that the time-marginals of the solutions are
absolutely
continuous with respect to the Lebesgue measure. In the case of a
symmetric
stable driving process, we deduce the existence of a function
solution to a
nonlinear integro-differential equation involving the fractional
Laplacian.
http://arxiv.org/abs/0707.2723
Author(s): Hiroki Sumi and Mariusz Urbanski
Abstract: We consider the dynamics of skew product maps associated with
finitely
generated semigroups of rational maps on the Riemann sphere. We show
that under
some conditions on the dynamics and the potential function \psi,
there exists a
unique equilibrium state for \psi and a unique $\exp(\P(\psi)-\psi)$-
conformal
measure, where P(\psi) denotes the topological pressure of \psi.
http://arxiv.org/abs/0707.2444
Author(s): Hiroki Sumi and Mariusz Urbanski
Abstract: We consider the dynamics of expanding semigroups generated by
finitely many
rational maps on the Riemann sphere. We show that for an analytic
family of
such semigroups, the Bowen parameter function is real-analytic and
plurisubharmonic. Combining this with a result obtained by the first
author, we
show that if for each semigroup of such an analytic family of expanding
semigroups satisfies the open set condition, then the function of the
Hausdorff
dimension of the Julia set is real-analytic and plurisubharmonic.
Moreover, we
provide an extensive collection of classes of examples of analytic
families of
semigroups satisfying all the above conditions and we analyze in
detail the
corresponding Bowen's parameters and Hausdorff dimension function.
http://arxiv.org/abs/0707.2447
Author(s): Pierre Collet and Antonio Galves and Florencia G. Leonardi
Abstract: We consider binary infinite order stochastic chains perturbed by a
random
noise. This means that at each time step, the value assumed by the
chain can be
randomly and independently flipped with a small fixed probability. We
show that
the transition probabilities of the perturbed chain are uniformly
close to the
corresponding transition probabilities of the original chain. As a
consequence,
in the case of stochastic chains with unbounded but otherwise finite
variable
length memory, we show that it is possible to recover the context
tree of the
original chain, using a suitable version of the algorithm Context,
provided
that the noise is small enough.
http://arxiv.org/abs/0707.2796
Author(s): Nathael Gozlan (LAMA)
Abstract: In this paper, we consider Poincar\'e inequalities for non
euclidean metrics
on $\mathbb{R}^d$. These inequalities enable us to derive precise
dimension
free concentration inequalities for product measures. This technique is
appropriate for a large scope of concentration rate: between
exponential and
gaussian and beyond. We give different equivalent functional forms of
these
Poincar\'e type inequalities in terms of transportation-cost
inequalities and
infimum convolution inequalities. Workable sufficient conditions are
given and
a comparison is made with generalized Beckner-Latala-Oleszkiewicz
inequalities.
http://arxiv.org/abs/0707.2834
Author(s): Asaf Nachmias and Yuval Peres
Abstract: We describe the component sizes in critical independent p-bond
percolation on
a random d-regular graph on n vertices, where d is fixed and n grows.
We prove
mean-field behavior around the critical probability p_c=1/(d-1).
In particular, we show that there is a scaling window of width n^
{-1/3}
around p_c in which the sizes of the largest components are roughly n^
{2/3} and
we describe their limiting joint distribution. We also show that for the
subcritical regime, i.e. p = (1-eps(n))p_c where eps(n)=o(1) but \eps
(n)n^{1/3}
tends to infinity, the sizes of the largest components are
concentrated around
an explicit function of n and eps(n) which is of order o(n^{2/3}). In
the
supercritical regime, i.e. p = (1+\eps(n))p_c where eps(n)=o(1) but
eps(n)n^{1/3} tends to infinity, the size of the largest component is
concentrated around the value (2d/(d-2))\eps(n)n and a duality
principle holds:
other component sizes are distributed as in the subcritical regime.
http://arxiv.org/abs/0707.2839
Author(s): David Coupier
Abstract: A $d$-dimensional ferromagnetic Ising model on a lattice torus is
considered.
As the size $n$ of the lattice tends to infinity, the magnetic field
$a=a(n)$
and the pair potential $b=b(n)$ depend on $n$. Precise bounds for the
probability for local configurations to occur in a large ball are
given. Under
some conditions bearing on potentials $a(n)$ and $b(n)$, the distance
between
copies of different local configurations is estimated according to their
weights. Finally, a sufficient condition ensuring that a given local
configuration occurs everywhere in the lattice is suggested.
http://arxiv.org/abs/0707.2889
Author(s): Sebastien Chambeu and Aline Kurtzmann
Abstract: This paper is concerned with some self-interacting diffusions $
(X_t,t\geq 0)$
living on $\mathbb{R}^d$. These diffusions are solutions to stochastic
differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t - g(t)\nabla V
(X_t -
\bar{\mu}_t) \mathrm{d}t,$$ where $\bar{\mu}_t$ is the mean of the
empirical
measure of the process $X$, $V$ is an asymptotically strictly convex
potential
and $g$ is a given function. We study the ergodic behavior of $X$ and
prove
that it is strongly related to $g$. Actually, we will show that $X$ and
$\bar{\mu}_t$ have the same asymptotic behavior and we will give
necessary and
sufficient conditions (on $g$ and $V$) for the almost sure
convergence of $X$.
http://arxiv.org/abs/0707.2908
Author(s): Sebastien Chambeu and Aline Kurtzmann
Abstract: The present paper is concerned with some self-interacting diffusions
$(X_t,t\geq 0)$ living on $\mathbb{R}^d$. These diffusions are
solutions to
stochastic differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t - g
(t)\nabla
V(X_t - \bar{\mu}_t) \mathrm{d}t$$ where $\bar{\mu}_t$ is the
empirical mean of
the process $X$, $V$ is an asymptotically strictly convex potential
and $g$ is
a given function. The authors have still studied the ergodic behavior
of $X$
and proved that it is strongly related to $g$. We go further and give
necessary
and sufficient conditions (for small $g$'s) in order that $X$
converges in
probability to $X_\infty$ (which is related to the global minima of $V
$).
http://arxiv.org/abs/0707.2910
Author(s): Xinjia Chen
Abstract: By our recently developed techniques, we have shown that the
minimum coverage
probability of an open binomial confidence interval with respect to the
corresponding binomial parameter is achieved at a discrete set of
finite many
values. Moreover, we have obtained similar results for the case of
Poisson
confidence interval and the case of confidence interval for the
proportion of
finite population.
http://arxiv.org/abs/0707.2814
Author(s): Xing M. Wang
Abstract: Dirac notation has been widely used for vectors in Hilbert spaces
of Quantum
Theories. It now has also been introduced to Information Retrieval.
In this
paper, we propose a new set of symbols, the Probability Bracket
Notation (PBN),
for representation of probability theories. The new are defined
similarly (but
not identically) as their counterparts in Dirac notation, which we
refer as
Vector Bracket Notation (VBN). By using PBN to represent fundamental
definitions and theorems for discrete and continuous random
variables, we show
that PBN could play a similar role in probability sample space as Dirac
notation in Hilbert space. We also find that there is a close
relation between
our probability state kets and probability vectors in Markov chains.
In the
end, we apply PBN to some important stochastic processes, and present
the time
evolution differential equations (TEDE) of time-continuous Markov
chains in
both Heisenberg and Schrodinger pictures. We summarize the
similarities and
differences between PBN and VBN in the two tables of Appendix A.
http://arxiv.org/abs/cs/0702021
Author(s): Xing M. Wang
Abstract: In this article, we continue to explore Probability Bracket
Notation (PBN),
proposed in our previous article. Using both Dirac vector bracket
notation
(VBN) and PBN, we define induced Hilbert space and induced sample
space, and
propose that there exists an equivalence relation between a Hilbert
space and a
probability sample space constructed from the same base observable
(s). Then we
investigate Markov transition matrices and their eigenvectors to make
diffusion
maps with two examples: a simple graph theory example, to serve as a
prototype
of bidirectional transition operator; a famous text document example
in IR
literature, to serve as a tutorial of diffusion map in text document
space. We
notice that, in both examples, the sample space of the Markov chain
and the
Hilbert space spanned by the eigenvectors of the transition matrix
are not
equivalent. At the end, we apply our PBN and equivalence proposal to
Thermophysics by associating phase space with Hilbert space or Fock
space of
many-particle systems.
http://arxiv.org/abs/cs/0702121
Author(s): Alexei Borodin and Eugene Kanzieper
Abstract: Two alternative, fairly compact proofs are presented of the Pfaffian
integration theorem that surfaced in the recent studies of spectral
properties
of Ginibre's Orthogonal Ensemble. The first proof is based on a
concept of the
Fredholm Pfaffian; the second proof is purely linear-algebraic.
http://arxiv.org/abs/0707.2784
Author(s): Omer Angel and Yuval Peres and David B. Wilson
Abstract: The ``overlapping-cycles shuffle'' mixes a deck of n cards by
moving either
the nth card or the (n-k)th card to the top of the deck, with
probability half
each. We determine the spectral gap for the location of a single
card, which,
as a function of k and n, has surprising behavior. For example,
suppose k is
the closest integer to alpha n for a fixed real alpha in (0,1). Then for
rational alpha the spectral gap is Theta(n^{-2}), while for poorly
approximable
irrational numbers alpha, such as the reciprocal of the golden ratio,
the
spectral gap is Theta(n^{-3/2}).
http://arxiv.org/abs/0707.2994
Author(s): S. Gerhold and R. Warnung
Abstract: We derive recursions for the probability distribution of random
sums by
computer algebra. Unlike the well-known Panjer-type recursions, they
are of
finite order and thus allow for computation in linear time. This
efficiency is
bought by the assumption that the probability generating function of
the claim
size be algebraic. The probability generating function of the claim
number is
supposed to be from the rather general class of D-finite functions.
http://arxiv.org/abs/0707.3028
Author(s): Frank Aurzada and Steffen Dereich
Abstract: We investigate the high resolution coding problem for general real-
valued
L\'evy processes under L^p[0,1]-norm distortion. Tight asymptotic
formulas are
found under mild regularity assumptions.
http://arxiv.org/abs/0707.3040
Author(s): S. V. Lototsky
Abstract: A change of variables is introduced to reduce certain nonlinear
stochastic
evolution equations with multiplicative noise to the corresponding
deterministic equation. The result is then used to investigate a
stochastic
porous medium equation.
http://arxiv.org/abs/0707.3155
Author(s): L. V. Bogachev
Abstract: Random walks provide a simple conventional model to describe various
transport processes, for example propagation of heat or diffusion of
matter
through a medium. However, in many practical cases the medium is highly
irregular due to defects, impurities, fluctuations etc., and it is
natural to
model this as random environment. In the random walks context, such
models are
referred to as Random Walks in Random Environments (RWRE). This is a
relatively
new chapter in applied probability and physics of disordered systems,
initiated
in the 1970s. Early interest was motivated by some problems in biology,
crystallography and metal physics, but later applications have spread
through
numerous areas. After 30 years of extensive work, RWRE remain a very
active
area of research, which has already led to many surprising
discoveries. The
goal of this article is to give a brief introduction to the beautiful
area of
RWRE. The principal model to be discussed is a random walk with
nearest-neighbor jumps in independent identically distributed
(i.i.d.) random
environment in one dimension, although we shall also comment on some
extensions
and generalizations. The focus is on rigorous results; however,
heuristics is
used freely to motivate the ideas and explain the approaches and
proofs. In a
few cases, sketches of the proofs have been included, which should
help the
reader to appreciate the flavor of results and methods.
http://arxiv.org/abs/0707.3160
Author(s): Tom Alberts (New York University) and Michael J. Kozdron (University of
Regina)
Abstract: We derive an estimate for the diameter of a chordal SLE path in
the upper
half plane H between two real boundary points 0 and x>0. In
particular, we
prove that if 0 < kappa < 8 and gamma:[0,1] to closure(H) is a
chordal SLE in H
from 0 to x, then P(gamma[0,1] cap C_R neq emptyset) asymp R^(1-4a)
where
a=2/kappa and C_R denotes the circle of radius Rx centred at 0 in the
upper
half plane. As an application of our result, we derive an estimate
that two
nearby points, one on the boundary and one in the interior, are
swallowed
together by a chordal SLE path, 4 < kappa <8.
http://arxiv.org/abs/0707.3163
Author(s): Ashkan Nikeghbali and Marc Yor
Abstract: We give a probabilistic interpretation for the Barnes G-function
which
appears in random matrix theory and in analytic number theory in the
important
moments conjecture due to Keating-Snaith for the Riemann zeta
function, via the
analogy with the characteristic polynomial of random unitary
matrices. We show
that the Mellin transform of the characteristic polynomial of random
unitary
matrices and the Barnes G-function are intimately related with
products and
sums of gamma, beta and log-gamma variables. In particular, we show
that the
law of the modulus of the characteristic polynomial of random unitary
matrices
can be expressed with the help of products of gamma or beta
variables, and that
the reciprocal of the Barnes G-function has a L\'{e}vy-Khintchin type
representation. These results lead us to introduce the so called
generalized
gamma convolution variables.
http://arxiv.org/abs/0707.3187
Author(s): Jan Palczewski and Lukasz Stettner
Abstract: This paper studies a portfolio optimization problem in a discrete-
time
Markovian model of a financial market, in which asset price dynamics
depend on
an external process of economic factors. There are transaction costs
with a
structure that covers, in particular, the case of fixed plus
proportional
costs. We prove that there exists a self-financing trading strategy
maximizing
the average growth rate of the portfolio wealth. We show that this
strategy has
a Markovian form. Our result is obtained by large deviations
estimates on
empirical measures of the price process and by a generalization of the
vanishing discount method to discontinuous transition operators.
http://arxiv.org/abs/0707.3198
Author(s): Elchanan Mossel and Allan Sly
Abstract: Gibbs sampling also known as Glauber dynamics is a popular
technique for
sampling high dimensional distributions defined on graphs. Of special
interest
is the behavior of Gibbs sampling on the Erd\H{o}s-R\'enyi random graph
G(n,d/n). While the average degree in G(n,d/n) is d(1-o(1)), it
contains many
nodes of degree of order $\log n / \log \log n$.
The existence of nodes of almost logarithmic degrees implies that
for many
natural distributions defined on G(n,p) such as uniform coloring or
the Ising
model, the mixing time of Gibbs sampling is at least $n^{1 + \Omega
(1 / \log
\log n)}$. High degree nodes pose a technical challenge in proving
polynomial
time mixing of the dynamics for many models including coloring.
In this work consider sampling q-colorings and show that for every
$d <
\infty$ there exists $q(d) < \infty$ such that for all $q \geq q(d)$
the mixing
time of Gibbs sampling on G(n,d/n) is polynomial in $n$ with high
probability.
Our results are the first polynomial time mixing results proven for the
coloring model on G(n,d/n) for d > 1 where the number of colors does
not depend
on n. They extend to much more general families of graphs which are
sparse in
some average sense and to much more general interactions. The results
also
generalize to the hard-core model at low fugacity and to general
models of soft
constraints at high temperatures.
http://arxiv.org/abs/0707.3241
Author(s): G. Giacomin (1) and F. L. Toninelli (2) ((1) Universite' de Paris 7, (2)
Laboratoire de Physique, ENS Lyon and CNRS)
Abstract: Recent results have lead to substantial progress in understanding
the role of
disorder in the (de)localization transition of polymer pinning
models. Notably,
there is an understanding of the crucial issue of disorder relevance and
irrelevance that, albeit still partial, is now rigorous. In this work we
exploit interpolation and replica coupling methods to get sharper
results on
the irrelevant disorder regime of pinning models. In particular, we
compute in
this regime the first order term in the expansion of the free energy
close to
criticality, which coincides with the first order of the formal
expansion
obtained by field theory methods. We also show that the quenched and the
quenched averaged correlation length exponents coincide, while in
general they
are expected to be different. Interpolation and replica coupling
methods in
this class of models naturally lead to studying the behavior of the
intersection of certain renewal sequences and one of the main tools
in this
work is precisely renewal theory and the study of these intersection
renewals.
http://arxiv.org/abs/0707.3340
Author(s): Annalisa Cerquetti
Abstract: We show that a Gibbs characterization of normalized generalized Gamma
processes, recently obtained in Lijoi, Pr\"unster and Walker (2007), can
alternatively be derived by exploiting a characterization of
exponentially
tilted Poisson-Kingman models stated in Pitman (2003). We also provide a
completion of this result investigating the existence of normalized
random
measures inducing exchangeable Gibbs partitions of type $\alpha \in (-
\infty,
0]$.
http://arxiv.org/abs/0707.3408
Author(s): Eben Kenah and Marc Lipsitch and James M. Robins
Abstract: The serial interval may be defined as the time between the onset
of symptoms
in an infectious person and the onset of symptoms in a person he or she
infects. Several methods of analyzing epidemic data, such as
estimates of
reproductive numbers, are based on a probability distribution for the
serial
interval. In this paper, we specify a general SIR epidemic model and
prove that
the mean serial interval must contract when susceptible persons are
at risk of
multiple infectious contacts. In an epidemic, the mean serial interval
contracts as the prevalence of infection increases. We illustrate two
mechanisms through which serial interval contraction can occur: In
global
competition among infectious contacts, risk of multiple infectious
contacts
results from a high global prevalence of infection. In local
competition among
infectious contacts, clustering of contacts places susceptible
persons at risk
of multiple infectious contacts even when the global prevalence of
infection is
low. We illustrate these patterns with simulations. We also find that
the
minimum mean serial interval in a compartmental SIR model becomes
arbitrarily
small with sufficiently high R_{0}. We conclude that the serial interval
distribution is not a stable characteristic of an infectious disease.
http://arxiv.org/abs/0706.2024
Author(s): Luigi Manca
Abstract: Given a real and separable Hilbert space H we consider the measure-
valued
equation \begin{equation*} \int_H\phi(x)\mu_t(dx)- \int_H\phi(x)\mu(dx)=
\int_0^t(\int_HK_0\phi(x)\mu_s(dx))ds, \end{equation*} where K_0
is the
Kolmogorov differential operator \[
K_0\phi(x)=\frac12\textrm{Trace}\big[BB^*D^2\phi(x)\big]+< x,A^*D
\phi(x)>+<
D\phi(x),F(x)>, \] $x\in H$, $\phi:H\to \Rset$ is a suitable smooth
function,
$A:D(A)\subset H\to H $ is linear, $F:H\to H$ is a globally Lipschitz
function
and $B:H\to H$ is linear and continuous. In order prove existence and
uniqueness of a solution for the above equation, we show that $K_0$
is a core,
in a suitable way, of the infinitesimal generator associated to the
solution of
a certain stochastic differential equation in H.
We also extend the above results to a reaction-diffusion operator
with
polinomial nonlinearities.
http://arxiv.org/abs/0707.3233
Author(s): Jiun-Chau Wang
Abstract: We study the distributional behavior for products, and for sums of
boolean
independent random variables in an infinitesimal triangular array. We
show that
the limit laws of boolean convolutions are determined by the limit
laws of free
convolutions, and vice versa. We further use these results to show
several
connections between the limiting distributional behavior of classical
convolutions and that of boolean convolutions. The proof of our
results is
based on the analytical apparatus developed for free convolutions.
http://arxiv.org/abs/0707.3401
Author(s): Peter Hegarty and Steven J. Miller
Abstract: We investigate the relationship between the sizes of the sum and
difference
sets attached to a subset of {0,1,...,N}, chosen randomly according to a
binomial model with parameter p(N), with N^{-1} = o(p(N)). We show
that the
random subset is almost surely difference dominated, as $N \to \infty
$, for any
choice of p(N) tending to zero, confirming a conjecture of Martin and
O'Bryant.
We exhibit a threshold phenomenon regarding the ratio of the size
of the
difference- to the sumset. If p(N) = o(N^{-1/2}) then almost all sums
and
differences in the random subset are almost surely distinct, and the
difference
set is almost surely about twice as large as the sumset. If N^{-1/2}
= o(p(N))
then both the sum and difference sets almost surely have size $(2N+1) -
(p(N)^{-2})$, and so the ratio in question is almost surely very
close to one.
If $p(N) = c \cdot N^{-1/2}$ then as c increases from zero to
infinity (i.e.:
as the threshold is crossed), the same ratio almost surely decreases
continuously from two to one according to an explicitly given
function of c.
We extend our results to the comparison of the generalized
difference sets
attached to an arbitrary pair of binary linear forms. For certain
pairs of
forms we show that there is a sharp threshold such that one form
almost surely
dominates below the threshold, and the other almost surely above it.
The heart of our approach involves proving strong concentration of
the sizes
of the sum and difference sets about their mean values.
http://arxiv.org/abs/0707.3417
Author(s): Ivan Nourdin (PMA) and David Nualart
Abstract: By means of Malliavin calculus, we prove the convergence in law
for certain
weighted quadratic variations of a fractional Brownian motion B with
Hurst
index H between 1/4 and 1/2.
http://arxiv.org/abs/0707.3448
Author(s): Nicolas Champagnat (INRIA Sophia Antipolis / INRIA Lorraine / IECN),
Sylvie Roelly
Abstract: A multitype Dawson-Watanabe process is conditioned, in subcritical
and
critical cases, on non-extinction in the remote future. On every
finite time
interval, its distribution is absolutely continuous with respect to
the law of
the unconditioned process. A martingale problem characterization is
also given.
The explicit form of the Laplace functional of the conditioned
process is used
to obtain several results on the long time behaviour of the mass of the
conditioned and unconditioned processes. The general case is
considered first,
where the mutation matrix which modelizes the interaction between the
types, is
irreducible. Several two-type models with decomposable mutation
matrices are
also analysed.
http://arxiv.org/abs/0707.3504
Author(s): Laurent Decreusefond (LTCI) and Eduardo Ferraz (LTCI) and Philippe Martins
(LTCI)
Abstract: For OFDMA systems, we find a rough but easily computed upper bound
for the
probability of loosing communications by insufficient number of sub-
channels on
downlink. We consider as random the positions of receiving users in
the system
as well as the number of sub-channels dedicated to each one. We use
recent
results of the theory of point processes which reduce our
calculations to the
first and second moments of the total required number of sub-carriers.
http://arxiv.org/abs/0707.3509
Author(s): F. Bassetti and M. Cosentino Lagomarsino and S. Mandr\'a
Abstract: We introduce and study a class of exchangeable random graph
ensembles. They
can be used as statistical null models for empirical networks, and as
a tool
for theoretical investigations. We provide general theorems that
carachterize
the degree distribution of the ensemble graphs, together with some
features
that are important for applications, such as subgraph distributions
and kernel
of the adjacency matrix. These results are used to compare to other
models of
simple and complex networks. A particular case of directed networks with
power-law out--degree is studied in more detail, as an example of the
flexibility of the model in applications.
http://arxiv.org/abs/0707.3545
Author(s): Akhtam Dzhalilov and Isabelle Liousse and Dieter Mayer
Abstract: Let $T_{f}$ be a circle homeomorphism with two break points $a_
{b},c_{b}$ and
irrational rotation number $\varrho_{f}$. Suppose that the derivative
$Df$ of
its lift $f$ is absolutely continuous on every connected interval of
the set
$S^{1}\backslash\{a_{b},c_{b}\}$, that $DlogDf \in L^{1}$ and the
product of
the jump ratios of $ Df $ at the break points is nontrivial, i.e.
$\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\frac{Df_{-}(c_{b})}{Df_{+}(c_
{b})}\neq1$.
We prove that the unique $T_{f}$- invariant probability measure $\mu_
{f}$ is
then singular with respect to Lebesgue measure $l$ on $S^{1}$.
http://arxiv.org/abs/0707.3528
Author(s): Joanna Jaroszewska and Michal Rams
Abstract: We consider measures which are invariant under a measurable
iterated function
system with positive, place-dependent probabilities in a separable
metric
space. We provide an upper bound of the Hausdorff dimension of such a
measure
if it is ergodic. We also prove that it is ergodic iff the related
skew product
is.
http://arxiv.org/abs/0707.3532
Author(s): Ingemar Kaj and Lasse Leskel\"a and Ilkka Norros and Volker Schmidt
Abstract: This paper studies the limits of a spatial random field generated by
uniformly scattered random sets, as the density $\lambda$ of the sets
grows to
infinity and the mean volume $\rho$ of the sets tends to zero.
Assuming that
the volume distribution has a regularly varying tail with infinite
variance, we
show that the centered and renormalized random field can have three
different
limits, depending on the relative speed at which $\lambda$ and $\rho$
are
scaled. If $\lambda$ grows much faster than $\rho$ shrinks, the limit is
Gaussian with long-range dependence, while in the opposite case, the
limit is
independently scattered with infinite second moments. In a special
intermediate
scaling regime, there exists a nontrivial limiting random field that
is not
stable.
http://arxiv.org/abs/0707.3729
Author(s): Jean-Fran\c{c}ois Renaud and Bruno R\'emillard
Abstract: In this paper, we construct a Malliavin derivative for functionals of
square-integrable L\'evy processes and derive a Clark-Ocone formula. The
Malliavin derivative is defined via chaos expansions involving
stochastic
integrals with respect to Brownian motion and Poisson random measure.
As an
illustration, we compute the explicit martingale representation for
the maximum
of a L\'evy process.
http://arxiv.org/abs/0707.3734
Author(s): Emmanuel Roy
Abstract: We show that a stationary IDp process (i.e., an infinitely divisible
stationary process without Gaussian part) can be written as the
independent sum
of four stationary IDp processes, each of them belonging to a
different class
characterized by its L\'{e}vy measure. The ergodic properties of each
class
are, respectively, nonergodicity, weak mixing, mixing of all order and
Bernoullicity. To obtain these results, we use the representation of
an IDp
process as an integral with respect to a Poisson measure, which, more
generally, has led us to study basic ergodic properties of these
objects.
http://arxiv.org/abs/0707.3746
Author(s): Bernard Bercu (IMB) and Abderrahmen Touati (IMB)
Abstract: We propose several exponential inequalities for self-normalized
martingales
similar to those established by De la Pena. The keystone is the
introduction of
a new notion of random variable heavy on left or right. Applications
associated
with linear regressions, autoregressive and branching processes are also
provided.
http://arxiv.org/abs/0707.3715
Author(s): Steven Lalley and Xinghua Zheng
Abstract: We consider a critical nearest neighbor branching random walk on the
$d-$dimensional integer lattice. Denote by $V_m$ the maximal number of
particles at a single site at time $m$, and by $G_{m}$ the event that
the
branching random walk survives to generation $m$. We show that if the
offspring
distribution has finite $n$-th moment, then in dimensions $d\geq 3$,
conditional on $G_{m}$, $V_m=O_p(m^{\frac{1}{n}})$; and if the offspring
distribution has exponentially decaying tail, then, conditional on $G_
{m}$, (a)
$V_m=O_p(\log m)$ in dimensions $d\geq 3$, and (b) $V_m=O_p((\log m)
^2)$ in
dimension $d=2$. On the other hand, we show that if the offspring
distribution
is non-degenerate then $P(V_m\geq \delta \log m | G_{m})\to 1$ for
some $\delta
> 0$. Therefore, in dimensions $d\geq 3$, if the offspring
> distribution has
exponentially decaying tail then conditional on $G_{m}$, the
distribution of
${V_m}/{\log m}$ must converge to a nontrivial limit as $m \to \infty$.
Furthermore, we show that, conditional on $G_{m}$, in dimensions $d
\geq 3$, the
number of multiplicity-$j$ sites, $j\geq 1$, and the number of
occupied sites,
normalized by $m$, converge jointly to multiples of an exponential
random
variable; in dimension $d=2$, however, the number of particles on a
`typical'
site is $O_p(\log m)$, and the number of occupied sites is $O_p(m/
\log m).$
http://arxiv.org/abs/0707.3829
Author(s): Michel Benaim and Itai Benjamini and Raphael Rossignol
Abstract: Take a big graph and make a random electrical network of it by
assigning
independent resistances on its edges. Now, ask for the behaviour of the
effective resistance between two vertices (two ``poles'') far apart.
We assume
in general that resistances are bounded away from 0 and infinity. In
this
paper, we study three cases of effective resistance in such random
electrical
networks: from one side to another in a box of $Z^d$, between two
points in
$Z^2$, and between two points on a cylinder graph $GxZ$. For all
these cases,
we obtain the right order of the fluctuations when the poles move
apart from
each other, and give corresponding subgaussian concentration
inequalities. For
the cylinder graphs, we prove two additional results: a central limit
theorem
and a result of uniform stability with respect to noise.
http://arxiv.org/abs/0707.3837
Author(s): Anna Amirdjanova and Matthew Linn
Abstract: The problem of nonlinear filtering of a random field observed in
the presence
of a noise, modeled by a persistent fractional Brownian sheet of
Hurst index
$(H_1,H_2)$ with $0.5
http://arxiv.org/abs/0707.3856
Author(s): Jean Brossard and Christophe Leuridan
Abstract: Nous \'{e}tudions les cha\^{{\i}}nes de Markov $(X_n)_{n\in\mathbf
{Z}}$
gouvern\'{e}es par une relation de r\'{e}currence de la forme
$X_{n+1}=f(X_n,V_{n+1})$, o\`{u} $(V_n)_{n\in\mathbf{Z}}$ est une
suite de
variables al\'{e}atoires ind\'{e}pendantes et de m\^{e}me loi telle
pour tout
$n\in \mathbf{Z}$, $V_{n+1}$ est ind\'{e}pendante de la suite
$((X_k,V_k))_{k\le n}$. L'objet de l'article est de donner une condition
n\'{e}cessaire et suffisante pour que les innovations $(V_n)_{n\in
\mathbf{Z}}$
d\'{e}terminent compl\`{e}tement la suite $(X_n)_{n\in \mathbf{Z}}$
et de
d\'{e}crire l'information manquante dans le cas contraire.
http://arxiv.org/abs/0707.3860
Author(s): F. Nazarov and M. Sodin and A. Volberg
Abstract: By random complex zeroes we mean the zero set of a random entire
function
whose Taylor coefficients are independent complex-valued Gaussian
variables,
and the variance of the k-th coefficient is 1/k!. This zero set is
distribution
invariant with respect to isometries of the complex plane. We study
large
fluctuations of random complex zeroes and show that they obey the
asymptotic
law that was discovered some time ago by Jancovici, Lebowitz and
Magnificat for
charge fluctuations of a Coulomb system of particles.
http://arxiv.org/abs/0707.3863
Author(s): A. Deniz Sezer
Abstract: We develop the mathematics of a filtration shrinkage model that
has recently
been considered in the credit risk modeling literature. Given a finite
collection of points $x_1<...
http://arxiv.org/abs/0707.3866
Author(s): Ming Yang
Abstract: Let $X_t$ be any additive process in $\mathbb{R}^d.$ There are
finite indices
$\delta_i, \beta_i, i=1,2$ and a function $u$, all of which are
defined in
terms of the characteristics of $X_t$, such that
\liminf_{t\to0}u(t)^{-1/\eta}X_t^*= \cases{0, \quad if $\eta>
\delta_1$,
\cr\infty, \quad if $\eta<\delta_2$,}
\limsup_{t\to0}u(t)^{-1/\eta}X_t^*= \cases{0, \quad if $\eta>
\beta_2$,
\cr\infty, \quad if $\eta<\beta_1$,}\qquad {a.s.},
where $X_t^*=\sup_{0\le s\le t}|X_s|.$ When $X_t$ is a L\'{e}vy
process with
$X_0=0$, $\delta_1=\delta_2$, $\beta_1=\beta_2$ and $u(t)=t.$ This is
a special
case obtained by Pruitt. When $X_t$ is not a L\'{e}vy process, its
characteristics are complicated functions of $t$. However, there are
interesting conditions under which $u$ becomes sharp to achieve
$\delta_1=\delta_2$, $\beta_1=\beta_2.$
http://arxiv.org/abs/0707.3886
Author(s): Itai Benjamini and Ariel Yadin and Ofer Zeitouni
Abstract: Let U(N) denote the maximal length of arithmetic progressions in a
random
uniform subset of {0,1}^N. By an application of the Chen-Stein
method, we show
that U(N)- 2 log(N)/log(2) converges in law to an extreme type
(asymmetric)
distribution. The same result holds for the maximal length W(N) of
arithmetic
progressions (mod N). When considered in the natural way on a common
probability space, we observe that U(N)/log(N) converges almost
surely to
2/log(2), while W(N)/log(N) does not converge almost surely (and in
particular,
limsup W(N)/log(N) is at least 3/log(2)).
http://arxiv.org/abs/0707.3888
Author(s): Mathew D. Penrose and Andrew R. Wade
Abstract: Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the
sum is
over points $x$ of a Poisson point process of intensity $\lambda$ on
a bounded
region in $d$-space, and $\xi_x$ is a functional determined by the
Poisson
points near to $x$, i.e. satisfying an exponential stabilization
condition,
along with a moments condition (examples include statistics for
proximity
graphs, germ-grain models and random sequential deposition models). A
known
general result says the $\mu_\lambda$-measures (suitably scaled and
centred) of
disjoint sets in $R^d$ are asymptotically independent normals as $
\lambda \to
\infty$; here we give an $O(\lambda^{-1/(2d + \epsilon)})$ bound on
the rate of
convergence. We illustrate our result with an explicit multivariate
central
limit theorem for the nearest-neighbour graph on Poisson points on a
finite
collection of disjoint intervals.
http://arxiv.org/abs/0707.3898
Author(s): Michael Marcus and Jay Rosen
Abstract: Let G be a mean zero Gaussian processes with stationary increments
and set
\si ^2(|x-y|)= E(G(x)-G(y))^2. Let f be a function with Ef^{2}(\eta)<
\ff, where
\eta=N(0,1). When \si^2 is convex and regularly varying at zero and
\lim_{h\to
0} \si(h)/h=\ff \quad {but} \quad ({d\over ds^2}\si^2(s))^{j_0} \mbox{is
locally integrable} for some integer j_0\ge 1, and satisfies some
additional
regularity conditions, then \int_a^bf(\frac{G(x+h)-G(x)}{\si (h)}) dx =
\sum_{j=0}^{j_0} (h/\si(h))^{j} {E(H_{j}(\eta) f(\eta))\over\sqrt {j!}}
:(G')^{j}:(I_{[a,b]}) +o({h\over\si (h)})^{j_0}\nn in L^2.
Here H_j is the j-th Hermite polynomial in the Hermite polynomial
expansion
of f. Also :(G')^{j}:(I_{[a,b]}) is a j-th order Wick power Gaussian
chaos
constructed from the Gaussian field G'(g)=\int g(x) dG(x) with
covariance
E(G'(g)G'(\wt g)) = \int \int \rho (x-y)g(x)\wt g(y) dx dy where
\rho(s)={1/2}{d^{2}\over ds^2}\si^2(s). Moreover, under the same
conditions
\lim_{h\downarrow0}\int_a^b :(\frac{G(x+h)-G(x)}{h})^{j_0}: dx =
:(G')^{j_0}:(I_{[a,b]}) \qquad {a.s.}
http://arxiv.org/abs/0707.3928
Author(s): Bruno Cernuschi-Frias (IRISA)
Abstract: New theoretical results are presented here on the recently
introduced model
called mixed states MRF. Such models were introduced in the context
of image
motion analysis and are useful to represent information which can
take both
discrete values accounting for symbolic states, and real values
corresponding
to continuous measurements. In particular, results are given when the
global
energy for the Gibbs formulation expressing the mixed states model,
can be
decomposed into one term accounting for the discrete part of the
model, and a
second term related to the continuous part. This decomposition
theorem permits
to define conditional mixed states models in a very simple way.
http://arxiv.org/abs/0707.3986
Author(s): Bojan Basrak and Johan Segers
Abstract: A multivariate, stationary time series is said to be jointly
regularly
varying if all its finite-dimensional distributions are multivariate
regularly
varying. This property is shown to be equivalent to weak convergence
of the
conditional distribution of the rescaled series given that, at a
fixed time
instant, its distance to the origin exceeds a threshold tending to
infinity.
The limit object, called the tail process, admits a decomposition in
independent radial and angular components. Under an appropriate mixing
condition, this tail process allows for a concise and explicit
description of
the limit of a sequence of point processes recording both the times
and the
positions of the time series when it is far away from the origin. The
theory is
applied to multivariate moving averages of finite order with random
coefficient
matrices.
http://arxiv.org/abs/0707.3989
Author(s): Nasir Ganikhodjaev and Fatimah Abdul Razak
Abstract: In this paper, correlation inequalities which have been considered
on Ising
model are extended to q-Potts model. It is considered on generalized
Potts
model with interaction of any number of spins. We replace the set of
spin
values $F=\{1,2,..., q\}$ by the centered set $F=\{-(q-1)/2,-(q-3)/2,...
,(q-3)/2,(q-1)/2\}$. Let $N$ be the subset of one-dimensional lattice
with $n$
vertices, $\g=(\s_1,\s_2,...,\s_n):N \to F^c$ be a configuration where
${(\s_i)}_\g$ is the number which appears as the ith spin (component)
in $\g$
and $\s_i$ be a random variable whose value at $\g$ is ${(\s_i)}_\g$.
Define
$\s^R=\prod_{i \in R}\s_i$ for any list $R$ where any $i \in R$
implies that $i
\in N$. We first prove that $<\s^R > \ge 0$ then we prove that for
any two
lists $R$ and $S$, we have $<\s^R \s^S >- < \s^R > < \s^S > \ge 0$.
http://arxiv.org/abs/0707.3848
Author(s): James Allen Fill
Abstract: A well-known theorem usually attributed to Keilson states that,
for an
irreducible continuous-time birth-and-death chain on the nonnegative
integers
and any d, the passage time from state 0 to state d is distributed as
a sum of
d independent exponential random variables. Until now, no
probabilistic proof
of the theorem has been known. In this paper we use the theory of strong
stationary duality to give a stochastic proof of a similar result for
discrete-time birth-and-death chains and geometric random variables,
and the
continuous-time result (which can also be given a direct stochastic
proof) then
follows immediately. In both cases we link the parameters of the
distributions
to eigenvalue information about the chain.
Intimately related to the passage-time theorem is a theorem of
Fill that any
fastest strong stationary time T for an ergodic birth-and-death chain
on {0,
> ..., d} in continuous time with generator G, started in state 0, is
distributed as a sum of d independent exponential random variables
whose rate
parameters are the nonzero eigenvalues of the negative of G. Our
approach
yields the first (sample-path) construction of such a T for which
individual
such exponentials summing to T can be explicitly identified.
http://arxiv.org/abs/0707.4042
Author(s): George P. Yanev and M. Ahsanullah and and M.I. Beg
Abstract: Bairamov et al. (Aust N Z J Stat 47:543-547, 2005) characterize the
exponential distribution in terms of the regression of a function of
a record
value with its adjacent record values as covariates. We extend these
results to
the case of non-adjacent covariates. We also consider a more general
setting
involving monotone transformations. As special cases, we present
characterizations involving weighted arithmetic, geometric, and
harmonic means.
http://arxiv.org/abs/0707.4121
Author(s): Alexei Borodin (1) and Patrik L. Ferrari (2) ((1) Caltech and (2) WIAS
Berlin)
Abstract: We consider a new interacting particle system on the one-
dimensional lattice
that interpolates between TASEP and Toom's model: A particle cannot
jump to the
right if the neighboring site is occupied, and when jumping to the
left it
simply pushes all the neighbors that block its way.
We prove that for flat and step initial conditions, the large time
fluctuations of the height function of the associated growth model
along any
space-like path are described by the Airy_1 and Airy_2 processes.
This includes
fluctuations of the height profile for a fixed time and fluctuations
of a
tagged particle's trajectory as special cases.
http://arxiv.org/abs/0707.2813
Author(s): Pablo A. Ferrari and James B. Martin
Abstract: In the Hammersley-Aldous-Diaconis process infinitely many
particles sit in R
and at most one particle is allowed at each position. A particle at x
$ whose
nearest neighbor to the right is at y, jumps at rate y-x to a position
uniformly distributed in the interval (x,y). The basic coupling between
trajectories with different initial configuration induces a process with
different classes of particles. We show that the invariant measures
for the
two-class process can be obtained as follows. First, a stationary M/M/
1 queue
is constructed as a function of two homogeneous Poisson processes,
the arrivals
with rate \lambda and the (attempted) services with rate \rho>
\lambda. Then put
the first class particles at the instants of departures (effective
services)
and second class particles at the instants of unused services. The
procedure is
generalized for the n-class case by using n-1 queues in tandem with n-1
priority-types of customers. A multi-line process is introduced; it
consists of
a coupling (different from Liggett's basic coupling), having as
invariant
measure the product of Poisson processes. The definition of the multi-
line
process involves the dual points of the space-time Poisson process
used in the
graphical construction of the system. The coupled process is a
transformation
of the multi-line process and its invariant measure the transformation
described above of the product measure.
http://arxiv.org/abs/0707.4202
Author(s): Marco Fuhrman (Dipartimento Di Matematica) and Ying Hu (IRMAR) and Gianmario
Tessitore (Dipartimento Di Matematica E Applicazioni)
Abstract: In this paper we introduce a new kind of Backward Stochastic
Differential
Equations, called ergodic BSDEs, which arise naturally in the study
of optimal
ergodic control. We study the existence, uniqueness and regularity of
solution
to ergodic BSDEs. Then we apply these results to the optimal ergodic
control of
a Banach valued stochastic state equation. We also establish the link
between
the ergodic BSDEs and the associated Hamilton-Jacobi-Bellman equation.
Applications are given to ergodic control of stochastic partial
differential
equations.
http://arxiv.org/abs/0707.4214
Author(s): Marton Balazs and Timo Seppalainen
Abstract: Take a random variable X with some finite exponential moments.
Define an
exponentially weighted expectation by E^t(f) = E(e^{tX}f)/E(e^{tX}) for
admissible values of the parameter t. Denote the weighted expectation
of X
itself by r(t) = E^t(X), with inverse function t(r). We prove that
for a convex
function f the expectation E^{t(r)}(f) is a convex function of the
parameter r.
Along the way we develop correlation inequalities for convex functions.
Motivation for this result comes from equilibrium investigations of some
stochastic interacting systems with stationary product distributions. In
particular, convexity of the hydrodynamic flux function follows in
some cases.
http://arxiv.org/abs/0707.4273
Author(s): R. Ba\~nuelos and T. Kulczycki and B. Siudeja
Abstract: We prove that the diagonal of the transition probabilities for the
d-dimensional Bessel processes on (0, 1], reflected at 1, which we
denote by
$p_R^N(t, r,r)$, is an increasing function of r for d>2 and that this
is false
for d=2.
http://arxiv.org/abs/0707.4299
Author(s): Xia Chen
Abstract: We study the upper tail behaviors of the local times of the
additive L\'{e}vy
processes and additive random walks. The limit forms we establish are
the
moderate deviations and the laws of the iterated logarithm for the
L_2-norms of
the local times and for the local times at a fixed site.
http://arxiv.org/abs/0707.4355
Author(s): Toshiro Watanabe
Abstract: A necessary and sufficient condition for the almost sure existence
of an
absolutely continuous (with respect to the branching measure) exact
Hausdorff
measure on the boundary of a Galton--Watson tree is obtained. In the
case where
the absolutely continuous exact Hausdorff measure does not exist
almost surely,
a criterion which classifies gauge functions $\phi$ according to whether
$\phi$-Hausdorff measure of the boundary minus a certain exceptional
set is
zero or infinity is given. Important examples are discussed in four
additional
theorems. In particular, Hawkes's conjecture in 1981 is solved.
Problems of
determining the exact local dimension of the branching measure at a
typical
point of the boundary are also solved.
http://arxiv.org/abs/0707.4358
Author(s): A. Popier
Abstract: In this paper we are concerned with one-dimensional backward
stochastic
differential equations (BSDE in short) of the following type: \[Y_t=\xi
-\int_{t\wedge \tau}^{\tau}Y_r|Y_r|^q dr-\int_{t\wedge \tau}^{\tau}Z_r
dB_r,\qquad t\geq 0,\] where $\tau$ is a stopping time, $q$ is a
positive
constant and $\xi$ is a $\mathcal{F}_{\tau}$-measurable random
variable such
that $\mathbf{P}(\xi =+\infty)>0$. We study the link between these
BSDE and the
Dirichlet problem on a domain $D\subset \mathbb{R}^d$ and with boundary
condition $g$, with $g=+\infty$ on a set of positive Lebesgue
measure. We also
extend our results for more general BSDE.
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