[PAS] Probability Abstracts 110
Probability Abstract Service
pas at lists.imstat.org
Mon Jul 27 12:03:08 CDT 2009
Probability Abstracts 110
This document contains abstracts 8463-8724
from May-1-2009 to June-30-2009.
They have been mailed on July 27, 2009.
Apologizes for this delay. Next PAS letter will
be distributed regularly on early september.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_110.shtml
---------------------------------------------------------------
8463. POLYGONAL WEB REPRESENTATION FOR HIGHER ORDER CORRELATION
FUNCTIONS OF CONSISTENT POLYGONAL MARKOV FIELDS IN THE PLANE
Tomasz Schreiber
We consider polygonal Markov fields originally introduced by Arak and
Surgailis (1982,1989).
Our attention is focused on fields with nodes of order two, which
can be
regarded as continuum ensembles of non-intersecting contours in the
plane,
sharing a number of salient features with the two-dimensional Ising
model. The
purpose of this paper is to establish an explicit stochastic
representation for
the higher-order correlation functions of polygonal Markov fields in
their
consistency regime. The representation is given in terms of the so-
called crop
functionals (defined by a Moebius-type formula) of polygonal webs
which arise
in a graphical construction dual to that giving rise to polygonal
fields. The
proof of our representation formula goes by constructing a martingale
interpolation between the correlation functions of polygonal fields
and crop
functionals of polygonal webs.
http://arxiv.org/abs/0905.0208
---------------------------------------------------------------
8464. LEVY'S ZERO-ONE LAW IN GAME-THEORETIC PROBABILITY
Glenn Shafer and Vladimir Vovk and and Akimichi Takemura
We prove a game-theoretic version of Levy's zero-one law, and deduce
several
corollaries from it, including Kolmogorov's zero-one law, the
ergodicity of
Bernoulli shifts, and a zero-one law for dependent trials. Our
secondary goal
is to explore the basic definitions of game-theoretic probability
theory, with
Levy's zero-one law serving a useful role.
http://arxiv.org/abs/0905.0254
---------------------------------------------------------------
8465. A Q-ANALOGUE OF DE FINETTI'S THEOREM
Alexander Gnedin and Grigori Olshanski
A q-analogue of de Finetti's theorem is obtained in terms of a boundary
problem for the q-Pascal graph. For q a power of prime this leads to a
characterisation of random spaces over the Galois field F_q that are
invariant
under the natural action of the infinite group of invertible matrices
with
coefficients from F_q.
http://arxiv.org/abs/0905.0367
---------------------------------------------------------------
8466. SUSCEPTIBILITY IN INHOMOGENEOUS RANDOM GRAPHS
Svante Janson and Oliver Riordan
We study the susceptibility, i.e., the mean size of the component
containing
a random vertex, in a general model of inhomogeneous random graphs.
This is one
of the fundamental quantities associated to (percolation) phase
transitions; in
practice one of its main uses is that it often gives a way of
determining the
critical point by solving certain linear equations. Here we relate the
susceptibility of suitable random graphs to a quantity associated to the
corresponding branching process, and study both quantities in various
natural
examples.
http://arxiv.org/abs/0905.0437
---------------------------------------------------------------
8467. ON ERGODIC TWO-ARMED BANDITS
Pierre Tarr\`es and Pierre Vandekerkhove
A device has two arms with unknown deterministic payoffs, and the aim
is to
asymptotically identify the best one without spending too much time on
the
other. The Narendra algorithm offers a stochastic procedure to this
end. We
show under weak ergodic assumptions on these deterministic payoffs
that the
procedure eventually chooses the best arm (i.e. with greatest Cesaro
limit)
with probability one, for appropriate step sequences of the algorithm.
In the
case of i.i.d. payoffs, this implies a "quenched" version of the
"annealed"
result of Lamberton, Pages and Tarres in 2004 by the law of iterated
logarithm,
thus generalizing it.
More precisely, if $(\eta_{l,i})_{i\in\N}\in\{0,1\}^\N$, $l\in\{A,B
\}$, are
the deterministic reward sequences we would get if we played at time $i
$, we
obtain infallibility with the same assumption on nonincreasing step
sequences
on the payoffs as in the result mentioned above, replacing the i.i.d.
assumption by the hypothesis that the empirical averages
$\sum_{i=1}^n\eta_{A,i}/n$ and$\sum_{i=1}^n\eta_{B,i}/n$ converge, as
$n$ tends
to infinity, respectively to $\theta_A$ and $\theta_B$, with rate at
least
$1/(\log n)^{1+\e}$, for some $\e>0$.
http://arxiv.org/abs/0905.0463
---------------------------------------------------------------
8468. NON-GLOBALLY LIPSCHITZ COUNTEREXAMPLES FOR THE STOCHASTIC EULER
SCHEME
Martin Hutzenthaler and Arnulf Jentzen
The stochastic Euler scheme is known to converge to the exact solution
of a
stochastic differential equation with globally Lipschitz coefficients
and even
with coefficients which grow at most linearly. For super-linearly
growing
coefficients convergence in the strong and numerically weak sense
remained an
open question. In this article we prove for many stochastic differential
equations with super-linearly growing coefficients that Euler's
approximation
does not converge neither in the strong sense nor in the numerically
weak sense
to the exact solution. Even worse, the difference of the exact
solution and of
the numerical approximation diverges to infinity in the strong sense
and in the
numerically weak sense.
http://arxiv.org/abs/0905.0273
---------------------------------------------------------------
8469. MOD-POISSON CONVERGENCE IN PROBABILITY AND NUMBER THEORY
E. Kowalski and A. Nikeghbali
Building on earlier work introducing the notion of "mod-Gaussian"
convergence
of sequences of random variables, which arises naturally in Random
Matrix
Theory and number theory, we discuss the analogue notion of "mod-
Poisson"
convergence. We show in particular how it occurs naturally in analytic
number
theory in the classical Erd\H{o}s-K\'ac Theorem. In fact, this case
reveals
deep connections and analogies with conjectures concerning the
distribution of
L-functions on the critical line, which belong to the mod-Gaussian
framework,
and with analogues over finite fields, where it can be seen as a
zero-dimensional version of the Katz-Sarnak philosophy in the large
conductor
limit.
http://arxiv.org/abs/0905.0318
---------------------------------------------------------------
8470. DUALITY IN INHOMOGENEOUS RANDOM GRAPHS, AND THE CUT METRIC
Svante Janson and Oliver Riordan
The classical random graph model $G(n,\lambda/n)$ satisfies a `duality
principle', in that removing the giant component from a supercritical
instance
of the model leaves (essentially) a subcritical instance. Such
principles have
been proved for various models; they are useful since it is often much
easier
to study the subcritical model than to directly study small components
in the
supercritical model. Here we prove a duality principle of this type
for a very
general class of random graphs with independence between the edges,
defined by
convergence of the matrices of edge probabilities in the cut metric.
http://arxiv.org/abs/0905.0434
---------------------------------------------------------------
8471. WHAT HAPPENS AFTER A DEFAULT: THE CONDITIONAL DENSITY APPROACH
Nicole El Karoui (PMA and CMAP) and Monique Jeanblanc (DP) and Ying
Jiao (PMA)
We present a general model for default time, making precise the role
of the
intensity process, and showing that this process allows for a
knowledge of the
conditional distribution of the default only "before the default".
This lack of
information is crucial while working in a multi-default setting. In a
single
default case, the knowledge of the intensity process does not allow to
compute
the price of defaultable claims, except in the case where immersion
property is
satisfied. We propose in this paper the density approach for default
time. The
density process will give a full characterization of the links between
the
default time and the reference filtration, in particular "after the
default
time". We also investigate the description of martingales in the full
filtration in terms of martingales in the reference filtration, and
the impact
of Girsanov transformation on the density and intensity processes, and
also on
the immersion property.
http://arxiv.org/abs/0905.0559
---------------------------------------------------------------
8472. LARGE CLIQUES IN A POWER-LAW RANDOM GRAPH
Svante Janson and Tomasz {\L}uczak and Ilkka Norros
We study the size of the largest clique $\omega(G(n,\alpha))$ in a
random
graph $G(n,\alpha)$ on $n$ vertices which has power-law degree
distribution
with exponent $\alpha$. We show that for `flat' degree sequences with
$\alpha>2$ whp the largest clique in $G(n,\alpha)$ is of a constant
size, while
for the heavy tail distribution, when $0<\alpha<2$, $\omega(G(n,
\alpha))$ grows
as a power of $n$. Moreover, we show that a natural simple algorithm
whp finds
in $G(n,\alpha)$ a large clique of size $(1+o(1))\omega(G(n,\alpha))$ in
polynomial time.
http://arxiv.org/abs/0905.0561
---------------------------------------------------------------
8473. COUNTING NONDECREASING INTEGER SEQUENCES THAT LIE BELOW A BARRIER
Robin Pemantle and Herbert S. Wilf
Given a barrier $0 \leq b_0 \leq b_1 \leq ...$, let $f(n)$ be the
number of
nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq ... \leq a_n
$ for
which $a_j \leq b_j$ for all $0 \leq j \leq n$. Known formul\ae for
$f(n)$
include an $n \times n$ determinant whose entries are binomial
coefficients
(Kreweras, 1965) and, in the special case of $b_j = rj+s$, a short
explicit
formula (Proctor, 1988, p.320). A relatively easy bivariate recursion,
decomposing all sequences according to $n$ and $a_n$, leads to a
bivariate
generating function, then a univariate generating function, then a
linear
recursion for $\{f(n) \}$. Moreover, the coefficients of the bivariate
generating function have a probabilistic interpretation, leading to an
analytic
inequality which is an identity for certain values of its argument.
http://arxiv.org/abs/0905.0609
---------------------------------------------------------------
8474. RIGOROUS DERIVATION OF THE LANDAU EQUATION IN THE WEAK COUPLING
LIMIT
Kay Kirkpatrick
We examine a family of microscopic models of plasmas, with a parameter
$\alpha$ comparing the typical distance between collisions to the
strength of
the grazing collisions. These microscopic models converge in
distribution, in
the weak coupling limit, to a velocity diffusion described by the
linear Landau
equation (also known as the Fokker-Planck equation). The present work
extends
and unifies previous results that handled the extremes of the parameter
$\alpha$, for the whole range (0, 1/2], by showing that clusters of
overlapping
obstacles are negligible in the limit. Additionally, we study the
diffusion
coefficient of the Landau equation and show it to be independent of the
parameter.
http://arxiv.org/abs/0905.0649
---------------------------------------------------------------
8475. Q-DISTRIBUTIONS ON BOXED PLANE PARTITIONS
Alexei Borodin and Vadim Gorin and Eric M. Rains
We introduce elliptic weights of boxed plane partitions and prove that
they
give rise to a generalization of MacMahon's product formula for the
number of
plane partitions in a box. We then focus on the most general positive
degenerations of these weights that are related to orthogonal
polynomials; they
form three two-dimensional families. For distributions from these
families we
prove two types of results.
First, we construct explicit Markov chains that preserve these
distributions.
In particular, this leads to a relatively simple exact sampling
algorithm.
Second, we consider a limit when all dimensions of the box grow and
plane
partitions become large, and prove that the local correlations
converge to
those of ergodic translation invariant Gibbs measures. For fixed
proportions of
the box, the slopes of the limiting Gibbs measures (that can also be
viewed as
slopes of tangent planes to the hypothetical limit shape) are encoded
by a
single quadratic polynomial.
http://arxiv.org/abs/0905.0679
---------------------------------------------------------------
8476. MOMENT ESTIMATES FOR SOLUTIONS OF LINEAR STOCHASTIC
DIFFERENTIAL EQUATIONS DRIVEN BY ANALYTIC FRACTIONAL BROWNIAN MOTION
J\'er\'emie Unterberger (IECN)
As a general rule, differential equations driven by a multi-dimensional
irregular path $\Gamma$ are solved by constructing a rough path over $
\Gamma$.
The domain of definition ? and also estimates ? of the solutions
depend on
upper bounds for the rough path; these general, deterministic
estimates are too
crude to apply e.g. to the solutions of stochastic differential
equations with
linear coefficients driven by a Gaussian process with H\"older
regularity
$\alpha < 1/2$. We prove here (by showing convergence of Chen's
series) that
linear stochastic differential equations driven by analytic fractional
Brownian
motion [7, 8] with arbitrary Hurst index $\alpha \in (0, 1)$ may be
solved on
the closed upper halfplane, and that the solutions have finite variance.
http://arxiv.org/abs/0905.0782
---------------------------------------------------------------
8477. PATH REGULARITY AND EXPLICIT CONVERGENCE RATE FOR BSDE WITH
TRUNCATED QUADRATIC GROWTH
Peter Imkeller and Goncalo dos Reis
We consider backward stochastic differential equations with drivers of
quadratic growth (qgBSDE). We prove several statements concerning path
regularity and stochastic smoothness of the solution processes of the
qgBSDE,
in particular we prove an extension of Zhang's path regularity theorem
to the
quadratic growth setting. We give explicit convergence rates for the
difference
between the solution of a qgBSDE and its truncation, filling an
important gap
in numerics for qgBSDE. We give an alternative proof of second order
Malliavin
differentiability for BSDE with drivers that are Lipschitz continuous
(and
differentiable), and then derive the same result for qgBSDE.
http://arxiv.org/abs/0905.0788
---------------------------------------------------------------
8478. RANDOM WALK ON THE INTEGERS WITH EQUIDISTANT MULTIPLE FUNCTION
BARRIERS
Theo van Uem
We obtain expected number of arrivals, probability of arrival,
absorption
probabilities and expected time before absorption for a discrete
random walk on
the integers with an infinite set of equidistant multiple function
barriers
http://arxiv.org/abs/0905.0823
---------------------------------------------------------------
8479. STABILITY OF A GROWTH PROCESS GENERATED BY MONOMER FILLING WITH
NEAREST-NEIGHBOR COOPERATIVE EFFECTS
Vadim Shcherbakov and Stanislav Volkov
In this paper we study stability of a growth process generated by a
cooperative sequential adsorption model (CSA) on the lattice. The
lattice CSA
can be regarded as a variant of Polya urn scheme with interaction and
the
growth process is formed by the numbers of adsorbed (allocated)
particles at
lattice sites, called heights. In our paper stability of the growth
process,
loosely speaking, means that its components grow at approximately the
same
rate. To assess stability quantitatively we study a stochastic process
formed
by differences of heights.
http://arxiv.org/abs/0905.0835
---------------------------------------------------------------
8480. UPPER TAILS FOR COUNTING OBJECTS IN RANDOMLY INDUCED
SUBHYPERGRAPHS AND ROOTED RANDOM GRAPHS
Svante Janson and Andrzej Rucinski
General upper tail estimates are given for counting edges in a random
induced
subhypergraph of a fixed hypergraph H, with an easy proof by
estimating the
moments. As an application we consider the numbers of arithmetic
progressions
and Schur triples in random subsets of integers. In the second part of
the
paper we return to the subgraph counts in random graphs and provide
upper tail
estimates in the rooted case.
http://arxiv.org/abs/0905.0972
---------------------------------------------------------------
8481. ON A STOCHASTIC WAVE EQUATION DRIVEN BY A NON-GAUSSIAN LEVY
PROCESS
Lijun Bo (XIDIAN) and Kehua Shi (NANKAI) and Yongjin Wang (NANKAI)
This paper investigates a damped stochastic wave equation driven by a
non-Gaussian Levy noise. The weak solution is proved to exist and be
unique.
Moreover we show the existence of a unique invariant measure
associated with
the transition semigroup under mild conditions.
http://arxiv.org/abs/0905.0992
---------------------------------------------------------------
8482. INTERMITTENCY AND AGING FOR THE SYMBIOTIC BRANCHING MODEL
Frank Aurzada and Leif D\"oring
For the symbiotic branching model introduced by Etheridge/Fleischmann
(2004),
it is shown that aging and intermittency exhibit different behaviour for
negative, zero, and positive correlations. Our approach also provides an
alternative, elementary proof and refinements of classical results
concerning
second moments of the parabolic Anderson model with Brownian
potential. Some
refinements to more general (also infinite range) kernels of recent
aging
results of Dembo/Deuschel (2007) for interacting diffusions are given.
http://arxiv.org/abs/0905.1003
---------------------------------------------------------------
8483. STRONG MIXING PROPERTY FOR STIT TESSELLATION
Rapha\"el Lachi\`eze-Rey
The so-called STIT tessellations form the class of homogeneous
(spatially
stationary) tessellations of $\mathbb{R}^d$ which are stable under the
nesting/iteration operation. In this paper, we establish the strong
mixing
property for these tessellations and give the optimal form of the rate
of decay
for the quantity $|\mathbb{P}({A}\cap Y=\emptyset,T_h B \cap
Y=\emptyset)-\mathbb{P}({A}\cap Y=\emptyset)\mathbb{P}({B}\cap Y=
\emptyset)|$
when $A$ and $B$ are two compact sets, $h$ a vector of $\mathbb{R}^d$,
$T_{h}$
the corresponding translation operator and $Y$ a STIT Tessellation.
http://arxiv.org/abs/0905.1145
---------------------------------------------------------------
8484. $L^P$ BOUNDS FOR A COMBINATORIAL CENTRAL LIMIT THEOREM WITH
INVOLUTIONS
Subhankar Ghosh
Let $E=((e_{ij}))_{n\times n}$ be a fixed array of real numbers such
that
$e_{ij}=e_{ji}, e_{ii}=0$ for $1\le i,j \le n$. Let the symmetric
group be
denoted by $S_n$ and the collection of involutions with no fixed
points by
$\Pi_n$, that is, $\Pi_n=\{\pi\in S_n: \pi^2=id, \pi(i)\neq i \forall i
\}$. For
$\pi$ uniformly chosen from $\Pi_n$, let $Y_E=\sum_{i=1}^n e_{i\pi(i)}
$ and
$W=(Y_E-\mu_E)/\sigma_E$ where $\mu_E=E(Y_E)$ and $\sigma_E^2={Var}
(Y_E)$.
Denoting by $F_W$ and $\Phi$ the distribution functions of $W$ and a
$\mathcal{N}(0,1)$ variate respectively, we bound $||F_W-\Phi||_p$ for
$ 1\le
p\le \infty$ using Stein's method and zero bias transformations. The
resulting
bound obtained is the product of a third moment type quantity
multiplied by an
explicit constant, and in particular for $p=\infty$ is of the same
form as the
one obtained by Bolthausen for Hoeffdings combinatorial central limit
theorem
when $\pi$ is chosen uniformly from $S_n$. The approximation developed
here for
involutions has applications in testing whether there is a significant
degree
of similarity in certain matched pairs experiments.
http://arxiv.org/abs/0905.1150
---------------------------------------------------------------
8485. TRANSITION PHENOMENA FOR LADDER EPOCHS OF RANDOM WALKS WITH
SMALL NEGATIVE DRIFT
Vitali Wachtel
For a family of random walks $\{S^{(a)}\}$ satisfying
$\mathbf{E}S_1^{(a)}=-a<0$ we consider ladder epochs $\tau^{(a)}=\min\
{k\geq1:
S_k^{(a)}<0\}$. We study the asymptotic, as $a\to0$, behaviour of
$\mathbf{P}(\tau^{(a)}>n)$ in the case when $n=n(a)\to\infty$. As a
consequence
we obtain also the growth rates of the moments of $\tau^{(a)}$.
http://arxiv.org/abs/0905.1186
---------------------------------------------------------------
8486. ERRATUM: PERCOLATION ON RANDOM JOHNSON-MEHL TESSELLATIONS AND
RELATED MODELS
Bela Bollobas and Oliver Riordan
We correct a simple error in Percolation on random Johnson-Mehl
tessellations
and related models, Probability Theory and Related Fields 140 (2008),
417-468.
(See also arXiv:math/0610716)
http://arxiv.org/abs/0905.1275
---------------------------------------------------------------
8487. CENTRAL LIMIT THEOREMS FOR GROMOV HYPERBOLIC GROUPS
Michael Bjorklund
In this paper we study asymptotic properties of symmetric and non-
degenerate
random walks on transient hyperbolic groups. We prove a central limit
theorem
and a law of iterated logarithm for the drift of a random walk,
extending
previous results by S. Sawyer and T. Steger and F. Ledrappier for
certain CAT
minus one groups. The proofs use a result by A. Ancona on the
identification of
the Martin boundary of a hyperbolic group with its Gromov boundary. We
also
give a new interpretation, in terms of Hilbert metrics, of the Green
metric,
first introduced by S. Brofferio and S. Blachere.
http://arxiv.org/abs/0905.1297
---------------------------------------------------------------
8488. DIAMOND AGGREGATION
Wouter Kager and Lionel Levine
Internal diffusion-limited aggregation is a growth model based on
random walk
in Z^d. We study how the shape of the aggregate depends on the law of
the
underlying walk, focusing on a family of walks in Z^2 for which the
limiting
shape is a diamond. Certain of these walks -- those with a directional
bias
toward the origin -- have at most logarithmic fluctuations around the
limiting
shape. This contrasts with the simple random walk, where the limiting
shape is
a disk and the best known bound on the fluctuations, due to Lawler, is
a power
law. Our walks enjoy a uniform layering property which simplifies many
of the
proofs.
http://arxiv.org/abs/0905.1361
---------------------------------------------------------------
8489. TERM STRUCTURE MODELS DRIVEN BY WIENER PROCESS AND POISSON
MEASURES: EXISTENCE AND POSITIVITY
Damir Filipovic and Stefan Tappe and Josef Teichmann
In the spirit of Bj\"ork-DiMasi-Kabanov-Runggaldier, we investigate term
structure models driven by Wiener process and Poisson measures with
forward
curve dependent volatilities. This includes a full existence and
uniqueness
proof for the corresponding Heath--Jarrow--Morton type term structure
equation.
Furthermore, we characterize positivity preserving models by means of
the
characteristic coefficients, which was open for jump-diffusions.
Additionally
we treat existence, uniqueness and positivity of the Brody-Hughston
equation of
interest rate theory with jumps, an equation which we believe to be
very useful
for applications. A key role in our investigation is played by the
method of
the moving frame, which allows to transform the Heath--Jarrow--Morton--
Musiela
equation to a time-dependent SDE.
http://arxiv.org/abs/0905.1413
---------------------------------------------------------------
8490. ESTIMATION OF THE DRIFT OF FRACTIONAL BROWNIAN MOTION
Es-Sebaiy Khalifa (SAMOS) and Idir Ouassou and Youssef Ouknine
We consider the problem of efficient estimation for the drift of
fractional
Brownian motion $B^H:=(B^H_t)_{t\in[0,T]}$ with hurst parameter $H$
less than
1/2. We also construct superefficient James-Stein type estimators which
dominate, under the usual quadratic risk, the natural maximum likelihood
estimator.
http://arxiv.org/abs/0905.1419
---------------------------------------------------------------
8491. ON THE DOVBYSH-SUDAKOV REPRESENTATION RESULT
Dmitry Panchenko
We present a detailed proof of the Dovbysh-Sudakov representation for
symmetric positive definite weakly exchangeable infinite random
arrays, called
Gram-de Finetti matrices, which is based on the representation result
of Aldous
and Hoover for arbitrary (not necessarily positive definite) symmetric
weakly
exchangeable arrays.
http://arxiv.org/abs/0905.1524
---------------------------------------------------------------
8492. TOTAL VARIATION MIXING TIME OF KAC'S RANDOM WALK
Yunjiang Jiang
We show that the classical Kac's random walk on $S^{n-1}$ starting
from the
point mass at $e_1$ mixes in $\mathcal{O}(n^5 \log n)$ steps in total
variation
distance. This improves a previous bound by Diaconis and Saloff-Coste of
$\mathcal{O}(n^{2n})$.
http://arxiv.org/abs/0905.1539
---------------------------------------------------------------
8493. ON SMALL BALLS PROBLEM FOR STABLE MEASURES IN A HILBERT SPACE
Vygantas Paulauskas
In the paper the old results on probabilities of small balls for stable
measures in a Hilbert space, obtained in 1977 and remaining
unpublished, are
presented. Apart of historical value these results are interesting
even now,
since they are comparable with recently obtained ones.
http://arxiv.org/abs/0905.1658
---------------------------------------------------------------
8494. LOCALLY MOST POWERFUL SEQUENTIAL TESTS OF A SIMPLE HYPOTHESIS
VS ONE-SIDED ALTERNATIVES
Andrey Novikov and Petr Novikov
Let $X_1,X_2,...$ be a discrete-time stochastic process with a
distribution
$P_\theta$, $\theta\in\Theta$, where $\Theta$ is an open subset of the
real
line. We consider the problem of testing a simple hypothesis $H_0:$
$\theta=\theta_0$ versus a composite alternative $H_1:$ $\theta>
\theta_0$,
where $\theta_0\in\Theta$ is some fixed point. The main goal of this
article is
to characterize the structure of locally most powerful sequential
tests in this
problem.
For any sequential test $(\psi,\phi)$ with a (randomized) stopping
rule
$\psi$ and a (randomized) decision rule $\phi$ let $\alpha(\psi,\phi)$
be the
type I error probability, $\dot \beta_0(\psi,\phi)$ the derivative, at
$\theta=\theta_0$, of the power function, and $\mathscr N(\psi)$ an
average
sample number of the test $(\psi,\phi)$. Then we are concerned with
the problem
of maximizing $\dot \beta_0(\psi,\phi)$ in the class of all sequential
tests
such that $$ \alpha(\psi,\phi)\leq \alpha\quad{and}\quad \mathscr
N(\psi)\leq
\mathscr N, $$ where $\alpha\in[0,1]$ and $\mathscr N\geq 1$ are some
restrictions. It is supposed that $\mathscr N(\psi)$ is calculated
under some
fixed (not necessarily coinciding with one of $P_\theta$) distribution
of the
process $X_1,X_2...$.
The structure of optimal sequential tests is characterized.
http://arxiv.org/abs/0905.1437
---------------------------------------------------------------
8495. INFORMATION RANKING AND POWER LAWS ON TREES
Predrag R. Jelenkovic and Mariana Olvera-Cravioto
We study the situations when the solution to a weighted stochastic
recursion
has a power law tail. To this end, we develop two complementary
approaches, the
first one extends Goldie's (1991) implicit renewal theorem to cover
recursions
on trees; and the second one is based on a direct sample path large
deviations
analysis of weighted recursive random sums. We believe that these
methods may
be of independent interest in the analysis of more general weighted
branching
processes as well as in the analysis of algorithms.
http://arxiv.org/abs/0905.1738
---------------------------------------------------------------
8496. LARGE DEVIATION PRINCIPLE AND INVISCID SHELL MODELS
Hakima Bessaih and Annie Millet (CES and SAMOS and PMA)
A LDP is proved for the inviscid shell model of turbulence. As the
viscosity
coefficient converges to 0 and the noise intensity is multiplied by
the square
root of the viscosity, we prove that some shell models of turbulence
with a
multiplicative stochastic perturbation driven by a H-valued Brownian
motion
satisfy a LDP in C([0,T],V) for the topology of uniform convergence on
[0,T],
but where V is endowed with a topology weaker than the natural one.
The initial
condition has to belong to V and the proof is based on the weak
convergence of
a family of stochastic control equations. The rate function is
described in
terms of the solution to the inviscid equation.
http://arxiv.org/abs/0905.1854
---------------------------------------------------------------
8497. STOCHASTIC APPROXIMATIONS OF SET-VALUED DYNAMICAL SYSTEMS:
CONVERGENCE WITH POSITIVE PROBABILITY TO AN ATTRACTOR
Mathieu Faure (UNINE) and Roth Gregory (UNINE)
A succesful method to describe the asymptotic behavior of a discrete
time
stochastic process governed by some recursive formula is to relate it
to the
limit sets of a well chosen mean differential equation. Under an
attainability
condition, convergence to a given attractor of the flow induced by this
dynamical system was proved to occur with positive probability (Bena
\"im, 1999)
for a class of Robbins Monro algorithms. Bena\"im et al. (2005)
generalised
this approach for stochastic approximation algorithms whose average
behavior is
related to a differential inclusion instead. We pursue the analogy by
extending
to this setting the result of convergence with positive probability to
an
attractor.
http://arxiv.org/abs/0905.1858
---------------------------------------------------------------
8498. A PROBABILISTIC NUMERICAL METHOD FOR FULLY NONLINEAR PARABOLIC
PDES
Arash Fahim and Nizar Touzi and Xavier Warin
We consider the probabilistic numerical scheme for fully nonlinear PDEs
suggested in [10], and show that it can be introduced naturally as a
combination of Monte Carlo and finite differences scheme without
appealing to
the theory of backward stochastic differential equations. Our first
main result
provides the convergence of the discrete-time approximation and
derives a bound
on the discretization error in terms of the time step. An explicit
implementable scheme requires to approximate the conditional expectation
operators involved in the discretization. This induces a further Monte
Carlo
error. Our second main result is to prove the convergence of the latter
approximation scheme, and to derive an upper bound on the
approximation error.
Numerical experiments are performed for the approximation of the
solution of
the mean curvature flow equation in dimensions two and three, and for
two and
five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman
equations
arising in the theory of portfolio optimization in financial
mathematics.
http://arxiv.org/abs/0905.1863
---------------------------------------------------------------
8499. CONCENTRATION OF RANDOM DETERMINANTS AND PERMANENT ESTIMATORS
Kevin P. Costello and Van Vu
We show that the absolute value of the determinant of a matrix with
random
independent (but not necessarily iid) entries is strongly concentrated
around
its mean. As an application, we show that the Godsil-Gutman and Barvinok
estimators for the permanent of a strictly positive matrix give sub-
exponential
approximation ratios with high probability.
http://arxiv.org/abs/0905.1909
---------------------------------------------------------------
8500. COERCIVE INEQUALITIES ON METRIC MEASURE SPACES
W. Hebisch and B. Zegarlinski
We study coercive inequalities on finite dimensional metric spaces with
probability measures which do not have volume doubling property. This
class of
inequalities includes Poincar\'e and Log-Sobolev inequality. Our main
result is
proof of Log-Sobolev inequality on Heisenberg group equipped with
either heat
kernel measure or "gaussian" density build from optimal control
distance. As
intermediate results we prove so called U-bounds.
http://arxiv.org/abs/0905.1713
---------------------------------------------------------------
8501. ON SCHROEDINGER'S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES, AND
PASSAGE TIME PROBLEMS
Gerardo Hernandez-del-Valle
We obtain explicit solutions for the density $\varphi_T$ of the first-
time
$T$ that a one-dimensional Brownian process $B$ reaches the twice,
continuously
differentiable moving boundary $f$ and such that $f''(t)\geq 0$ for
all $t\in
\mathbb{R}^+$. We do so by finding the expected value of some
functionals of a
3-dimensional Bessel bridge $\tilde{X}$ and exploiting its
relationship with
first-passage time problems as pointed out by Kardaras (2007). It
turns out
that this problem is related to Schr\"odinger's equation with time-
dependent
linear potential, see Feng (2001).
http://arxiv.org/abs/0905.1971
---------------------------------------------------------------
8502. ON THE FIRST PASSAGE TIME DENSITY OF A CONTINUOUS MARTINGALE
OVER A MOVING BOUNDARY
Gerardo Hernandez-del-Valle
In this paper we derive the density $\varphi$ of the first time $T$
that a
continuous martingale $M$ with non-random quadratic variation
$<M>_\cdot:=\int_0^\cdot h^2(u)du$ hits a moving boundary $f$ which is
twice
continuously differentiable, and $f'/h\in\mathbb{C}^2[0,\infty)$.
Thus, this work is an extension to case in which $M$ is in fact a
one-dimensional standard Brownian motion $B$, as studied in Hernandez-
del-Valle
(2007).
http://arxiv.org/abs/0905.1975
---------------------------------------------------------------
8503. NON-EXTINCTION OF A FLEMING-VIOT PARTICLE MODEL
Mariusz Bieniek and Krzysztof Burdzy and Sam Finch
We consider a branching particle model in which particles move inside a
Euclidean domain according to the following rules. The particles move as
independent Brownian motions until one of them hits the boundary. This
particle
is killed but another randomly chosen particle branches into two
particles, to
keep the population size constant. We prove that the particle
population does
not approach the boundary simultaneously in a finite time in some
Lipschitz
domains. This is used to prove a limit theorem for the empirical
distribution
of the particle family.
http://arxiv.org/abs/0905.1999
---------------------------------------------------------------
8504. $L_P$-THEORY FOR THE STOCHASTIC HEAT EQUATION WITH INFINITE-
DIMENSIONAL FRACTIONAL NOISE
Raluca Balan
In this article, we consider the stochastic heat equation $du=(\Delta
u+f(t,x))dt+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in
[0,T]$,
with random coefficients $f$ and $g^k$, driven by a sequence $
(\beta^k)_k$ of
i.i.d. fractional Brownian motions of index $H>1/2$. Using the Malliavin
calculus techniques and a $p$-th moment maximal inequality for the
infinite sum
of Skorohod integrals with respect to $(\beta^k)_k$, we prove that the
equation
has a unique solution (in a Banach space of summability exponent $p
\geq 2$),
and this solution is H\"older continuous in both time and space.
http://arxiv.org/abs/0905.2150
---------------------------------------------------------------
8505. MARKOVIAN BRIDGES: WEAK CONTINUITY AND PATHWISE CONSTRUCTIONS
Lo\"ic Chaumont and Ger\'onimo Uribe Bravo
A Markovian bridge is a probability measure taken from a
disintegration of
the law of an initial part of the path of a Markov process given its
terminal
value. As such, Markovian bridges admit a natural parameterization in
terms of
the state space of the process. In the context of Feller processes with
continuous transition densities, we construct by weak convergence
considerations the only versions of Markovian bridges which are weakly
continuous with respect to their parameter. We use this weakly
continuous
construction to provide an extension of the strong Markov property in
which the
flow of time is reversed. In the context of self-similar Feller
process, the
last result is shown to be useful in the construction of Markovian
bridges out
of the trajectories of the original process.
http://arxiv.org/abs/0905.2155
---------------------------------------------------------------
8506. NON-MARKOVIAN LIMITS OF ADDITIVE FUNCTIONALS OF MARKOV PROCESSES
Milton Jara and Tomasz Komorowski
In this paper we consider an additive functional of an observable $V(x)
$ of a
Markov jump process. We assume that the law of the expected jump time
$t(x)$
under the invariant probability measure $\pi$ of the skeleton chain
belongs to
the domain of attraction of a subordinator. Then, the scaled limit of
the
functional is a Mittag-Leffler proces, provided that $\Psi(x):=V(x)t(x)
$ is
square integrable w.r.t. $\pi$. When the law of $\Psi(x)$ belongs to a
domain
of attraction of a stable law the resulting process can be described
by a
composition of a stable process and the inverse of a subordinator and
these
processes are not necessarily independent. On the other hand when the
singularities of $\Psi(x)$ and $t(x)$ do not overlap with large
probability the
law of the resulting process has some scaling invariance property. We
provide
an application of the results to a process that arises in quantum
transport
theory.
http://arxiv.org/abs/0905.2163
---------------------------------------------------------------
8507. AMENABILITY OF LINEAR-ACTIVITY AUTOMATON GROUPS
Gideon Amir and Omer Angel and Balint Virag
We prove that every linear-activity automaton group is amenable.
The proof is based on showing that a sufficiently symmetric random
walk on a
specially constructed degree 1 automaton group -- the mother group --
has
asymptotic entropy 0.
Our result answers an open question by Nekrashevich in the Kourovka
notebook,
and gives a partial answer to a question of Sidki.
http://arxiv.org/abs/0905.2007
---------------------------------------------------------------
8508. A GLOBAL VIEW OF BROWNIAN PENALISATIONS
Joseph Najnudel and Bernard Roynette and Marc Yor
In this monograph, we construct and study a sigma-finite measure on
continuous functions from R_+ to R, strongly related to many probability
measures obtained by penalisation of Brownian motion, i.e. as limits of
probabilities which are absolutely continuous with respect to Wiener
measure.
This remarkable sigma-finite measure can be generalized in three other
cases:
one can start from a two-dimensional Brownian motion, from a recurrent
diffusion with values in R_+, and from a discrete, recurrent Markov
chain.
http://arxiv.org/abs/0905.2220
---------------------------------------------------------------
8509. THE MEAN PERIMETER OF SOME RANDOM PLANE CONVEX SETS GENERATED BY
A BROWNIAN MOTION
Philippe Biane G\'erard Letac
If $C_1$ is the convex hull of the curve of the standard Brownian
motion in
the complex plane watched from 0 to 1, we consider the convex hulls of
$C_1$
and several rotations of it and we compute the mean of the length of
their
perimeter by elementary calculations.
http://arxiv.org/abs/0905.2256
---------------------------------------------------------------
8510. A SCALING ANALYSIS OF A CAT AND MOUSE MARKOV CHAIN
Nelly Litvak and Philippe Robert (INRIA)
Motivated by an original on-line page-ranking algorithm, starting from
an
arbitrary Markov chain (C_n) on a discrete state space S, a Markov chain
(C_n,M_n) on the product space S^2, the cat and mouse Markov chain, is
constructed. The first coordinate of this Markov chain behaves like the
original Markov chain and the second component changes only when both
coordinates are equal. The asymptotic properties of this Markov chain
are
investigated. A representation of its invariant measure is in particular
obtained. When the state space is infinite it is shown that this
Markov chain
is in fact null recurrent if the initial Markov chain (C_n) is positive
recurrent and reversible. In this context, the scaling properties of the
location of the second component, the mouse, are investigated in various
situations: simple random walks in Z and Z^2, reflected simple random
walk in N
and also in a continuous time setting. For several of these processes,
a time
scaling with rapid growth gives an interesting asymptotic behavior
related to
limit results for occupation times and rare events of Markov processes.
http://arxiv.org/abs/0905.2259
---------------------------------------------------------------
8511. PRODUCT FORMULA FOR JACOBI POLYNOMIALS, SPHERICAL HARMONICS AND
GENERALIZED BESSEL FUNCTION OF DIHEDRAL TYPE
Nizar Demni
We work out the expression of the generalized Bessel function of type
B in
the two-rank case. This is done using Dijskma and Koornwinder's
product formula
for Jacobi polynomials and the obtained expression is given by multiple
integrals involving only a normalized modified Bessel function and two
symmetric Beta distributions. We think of that expression as the major
step
toward the explicit expression of the Dunkl's intertwining V operator
reflections-invariant functions. Finally, we give in the same setting an
explicit formula for the action of V on a product of a power of the
norm and a
spherical harmonic. The obtained formula extends to all dihedral
systems and it
improves the one derived by Y.Xu.
http://arxiv.org/abs/0905.2265
---------------------------------------------------------------
8512. PASSAGE TIME FROM FOUR TO TWO BLOCKS IN THE VOTER MODEL
Kilian Raschel
We consider a voter model in which there are two candidates and
initially, in
the population $\mathbb{Z}$, four connected blocks of same opinions.
We assume
that a citizen changes his mind at a rate proportional to the number
of its
neighbors that disagree with him, and we study the passage from four
to two
connected blocks of same opinions. More precisely we make explicit the
generating function of the probabilities to go from four to two blocks
in time
$k$ and we find the asymptotic of these probabilities when $k$ goes to
infinity.
http://arxiv.org/abs/0905.2310
---------------------------------------------------------------
8513. A STOCHASTIC MODEL FOR PHYLOGENETIC TREES
T. M. Liggett and R. B. Schinazi
We propose the following simple stochastic model for phylogenetic
trees. New
types are born and die according to a birth and death chain. At each
birth we
associate a fitness to the new type sampled from a fixed distribution.
At each
death the type with the smallest fitness is killed. We show that if
the birth
(i.e. mutation) rate is subcritical we get a phylogenetic tree
consistent with
an influenza tree (few types at any given time and one dominating type
lasting
a long time). When the birth rate is supercritical we get a
phylogenetic tree
consistent with an HIV tree (many types at any given time, none
lasting very
long).
http://arxiv.org/abs/0905.2349
---------------------------------------------------------------
8514. EXTENDING THE SUPPORT THEOREM TO INFINITE DIMENSIONS
Jeremy J. Becnel
The Radon transform is one of the most useful and applicable tools in
functional analysis. First constructed by John Radon in 1917 it has
now been
adapted to several settings. One of the principle theorems involving
the Radon
transform is the Support Theorem. In this paper, we discuss how the
Radon
transform can be constructed in the white noise setting. We also
develop a
Support Theorem in this setting.
http://arxiv.org/abs/0905.2372
---------------------------------------------------------------
8515. GLOBAL EXISTENCE FOR ROUGH DIFFERENTIAL EQUATIONS UNDER LINEAR
GROWTH CONDITIONS
Massimiliano Gubinelli (CEREMADE) and Antoine Lejay (IECN and INRIA
Sophia Antipolis / INRIA Lorraine / IECN)
We prove existence of global solutions for differential equations
driven by a
geometric rough path under the condition that the vector fields have
linear
growth. We show by an explicit counter-example that the linear growth
condition
is not sufficient if the driving rough path is not geometric. This
settle a
long-standing open question in the theory of rough paths. So in the
geometric
setting we recover the usual sufficient condition for differential
equation.
The proof rely on a simple mapping of the differential equation from the
Euclidean space to a manifold to obtain a rough differential equation
with
bounded coefficients.
http://arxiv.org/abs/0905.2399
---------------------------------------------------------------
8516. TAIL ASYMPTOTICS FOR EXPONENTIAL FUNCTIONALS OF LEVY PROCESSES:
THE CONVOLUTION EQUIVALENT CASE
V'\ictor Rivero
We determine the rate of decrease of the right tail distribution of the
exponential functional of a Levy process with a convolution equivalent
Levy
measure. Our main result establishes that it decreases as the right
tail of the
image under the exponential function of the Levy measure of the
underlying Levy
process. The method of proof relies on fluctuation theory of Levy
processes and
an explicit path-wise representation of the exponential functional as
the
exponential functional of a bivariate subordinator. Our techniques
allow us to
establish rather general estimates of the measure of the excursions
out from
zero for the underlying Levy process reflected in its past infimum,
whose area
under the exponential of the excursion path exceed a given value.
http://arxiv.org/abs/0905.2401
---------------------------------------------------------------
8517. THE DISTRIBUTION ROUTE FROM ANCESTORS TO DESCENDANTS
Baruch Fischer and Moshe Zakai
We study the distribution of descendants of a known personality, or of
anybody else, as it propagates along generations from father or mother
through
any of their children. We ask for the ratio of the descendants to the
total
population and construct a model for the route of Distribution from
Ancestors
to Descendants (DAD). The population ratio $r_n$ is found to be given
by the
recursive equation $ r_{n+1} \approx (2-r_n) r_n ,$ that provides the
transition from the $n-$th to the $(n+1)$th generation. $ r_0 =1/N_0$
and $N_0$
is the total relevant population at the first generation. The number of
generations it takes to make half the population descendants is $\log
N_0/\log
2$ and additional $\sim 4$ generations make everyone a descendent
(=the full
descendant spreading time). These results are independent of the
population
growth factor even if it changes along generations. As a running
example we
consider the offspring of King David. Assuming a population between
$N_0 =
10^6$ and $5 \cdot 10^6$ of Israelites at King David's time ($\sim
1000$ BC),
it took 24 to 26 generations (about 600-650 years, when taking 25
years for a
generation) to make every Israelite a King David descendent. The
conclusion is
that practically every Israelite living today (and in fact already at
350-400
BC), and probably also many others beyond them, are descendants of
King David.
We note that this work doesn't deal with any genetical aspect. We also
didn't
take into account here any geo-social-demographic factor.
Nevertheless, along
tens of generations, about 120 from King David's time till today, the
DAD route
is likely to govern the distribution in communities that are not very
isolated.
http://arxiv.org/abs/0904.4792
---------------------------------------------------------------
8518. STATIONARY MAP COLORING
Omer Angel and Itai Benjamini and Ori Gurel-Gurevich and Tom
Meyerovitch and Ron Peled
We consider a planar Poisson process and its associated Voronoi map.
We show
that there is a proper coloring with 6 colors of the map which is a
deterministic isometry-equivariant function of the Poisson process. As
part of
the proof we show that the 6-core of the corresponding Delaunay
triangulation
is empty.
Generalizations, extensions and some open questions are discussed.
http://arxiv.org/abs/0905.2563
---------------------------------------------------------------
8519. RANDOM QUANTUM CHANNELS I: GRAPHICAL CALCULUS AND THE BELL
STATE PHENOMENON
Beno\^it Collins (ICJ) and Ion Nechita (ICJ)
This paper is the first of a series where we study quantum channels
from the
random matrix point of view. We develop a graphical tool that allows
us to
compute the expected moments of the output of a random quantum
channel. As an
application, we study variations of random matrix models introduced by
Hayden
\cite{hayden}, and show that their eigenvalues converge almost surely.
In
particular we obtain for some models sharp improvements on the value
of the
largest eigenvalue, and this is shown in a further work to have new
applications to minimal output entropy inequalities.
http://arxiv.org/abs/0905.2313
---------------------------------------------------------------
8520. QUANTUM STOCHASTIC CONVOLUTION COCYCLES III
J. Martin Lindsay and Adam G. Skalski
The theory of quantum Levy processes on a compact quantum group, and
more
generally quantum stochastic convolution cocycles on a C*-bialgebra, is
extended to locally compact quantum groups and multiplier C*-
bialgebras. Strict
extension results obtained by Kustermans, and automatic strictness
properties
developed here, are exploited to obtain existence and uniqueness for
coalgebraic quantum stochastic differential equations in this setting.
Working
in the universal enveloping von Neumann bialgebra, the stochastic
generators of
Markov-regular, completely positive, respectively *-homomorphic, quantum
stochastic convolution cocycles are characterised. Every Markov-
regular quantum
Levy process on a multiplier C*-bialgebra is shown to be equivalent to
one
governed by a quantum stochastic differential equation, and the
generating
functionals of norm-continuous convolution semigroups on a multiplier
C*-bialgebra are characterised. Applying a recent result of Belton's,
we give a
thorough treatment of the approximation of quantum stochastic
convolution
cocycles by quantum random walks.
http://arxiv.org/abs/0905.2410
---------------------------------------------------------------
8521. HEAT KERNEL ESTIMATES FOR THE FRACTIONAL LAPLACIAN
Krzysztof Bogdan and Tomasz Grzywny and Michal Ryznar
We give sharp estimates for the heat kernel of the fractional
Laplacian with Dirichlet exterior condition for a general class of
domains
including Lipschitz domains. The estimates are sharp and explicit for
smooth
domains.
http://arxiv.org/abs/0905.2626
---------------------------------------------------------------
8522. A NOTE ON A FENYMAN-KAC-TYPE FORMULA
Raluca Balan
In this article, we establish a probabilistic representation for the
second-order moment of the solution of stochastic heat equation in
$[0,1]
\times \bR^d$, with multiplicative noise, which is fractional in time
and
colored in space. This representation is similar to the one given in
Dalang,
Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a
Gaussian noise,
which is white in time. Unlike the formula of Dalang, Mueller and
Tribe (2008),
which is based on the usual Poisson process, our representation is
based on the
planar Poisson process, due to the fractional component of the noise.
http://arxiv.org/abs/0905.2698
---------------------------------------------------------------
8523. A STRONG LAW OF LARGE NUMBERS FOR MARTINGALE ARRAYS
Yves F. Atchade
We prove a martingale triangular array generalization of the
Chow-Birnbaum-Marshall's inequality. The result is used to derive a
strong law
of large numbers for martingale triangular arrays whose rows are
asymptotically
stable in a certain sense. To illustrate, we derive a simple proof,
based on
martingale arguments, of the consistency of kernel regression with
dependent
data. Another application can be found in \cite{atchadeetfort08} where
the new
inequality is used to prove a strong law of large numbers for adaptive
Markov
Chain Monte Carlo methods.
http://arxiv.org/abs/0905.2761
---------------------------------------------------------------
8524. STATIONARY STOCHASTIC VISCOSITY SOLUTIONS OF SPDES
Qi Zhang
In this paper we aim to obtain the stationary stochastic viscosity
solutions
of a parabolic type SPDEs through the infinite horizon backward doubly
stochastic differential equations (BDSDEs). For this, we study the
existence,
uniqueness and regularity of solutions of infinite horizon BDSDEs as
well as
the "perfection procedure" applied to the solutions of BDSDEs to
derive the
"perfect" stationary stochastic viscosity solutions of SPDEs.
http://arxiv.org/abs/0905.2806
---------------------------------------------------------------
8525. CYLINDRICAL LEVY PROCESSES IN BANACH SPACES
David Applebaum and Markus Riedle
Cylindrical probability measures are finitely additive measures on
Banach
spaces that have sigma-additive projections to Euclidean spaces of all
dimensions. They are naturally associated to notions of weak
(cylindrical)
random variable and hence weak (cylindrical) stochastic processes. In
this
paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito
decompositions and an associated Levy-Khintchine formula. If the
process is
weakly square integrable, its covariance operator can be used to
construct a
reproducing kernel Hilbert space in which the process has a
decomposition as an
infinite series built from a sequence of uncorrelated bona fide one-
dimensional
Levy processes. This series is used to define cylindrical stochastic
integrals
from which cylindrical Ornstein-Uhlenbeck processes may be constructed
as
unique solutions of the associated Cauchy problem. We demonstrate that
such
processes are cylindrical Markov processes and study their (cylindrical)
invariant measures.
http://arxiv.org/abs/0905.2858
---------------------------------------------------------------
8526. A NOTE ON CORRELATIONS IN RANDOMLY ORIENTED GRAPHS
Svante Linusson
Two models are compared on a given graph G. On the one hand regular edge
percolation with probability 1/2 and on the other hand orienting G by
giving
each edge a random direction. Two lemmas are presented proving that
for a given
vertex v the probability distribution for the open cluster C_v in edge
percolation is equal to the distribution for the directed out-cluster
$\hpil{C}_v$ and then a similar statement for the joint distribution of
disjoint clusters around two different vertices.
One application of the two lemmas is then to prove correlation
inequalities
of the existence of directed paths. It is proven that for vertices
a,b,s in G,
the events {s\to a} and {s\to b} are positively correlated. This is
proven to
be true also if we first condition on that there does not exist a path
from s
to t for any vertex t\neq s. With this conditioning it is also true
that {s\to
b} and {a\to t} are negatively correlated.
A concept of increasing events in random orientations is defined
and a
general inequality corresponding to Harris inequality is given.
The lemmas and applications are true also for another model of
randomly
directed graphs.
http://arxiv.org/abs/0905.2881
---------------------------------------------------------------
8527. SPECTRAL ANALYSIS OF 1D NEAREST-NEIGHBOR RANDOM WALKS WITH
APPLICATIONS TO SUBDIFFUSIVE RANDOM TRAP AND BARRIER MODELS
A. Faggionato
Given a family $X^{(n)}(t)$ of continuous--time nearest--neighbor random
walks on the one dimensional lattice $\bbZ$, parameterized by $n \in
\bbN_+$,
we show that the spectral analysis of the Markov generator of $X^{(n)}
$ with
Dirichlet conditions outside $(0,n)$ reduces to the analysis of the
eigenvalues
and eigenfunctions of a suitable generalized second order differential
operator
$-D_{m_n} D_x$ with Dirichlet conditions outside $(0,1)$. If in
addition the
measures $dm_n$ weakly converge to some measure $dm$, similarly to
Krein's
correspondence we prove a limit theorem of the eigenvalues and
eigenfunctions
of $-D_{m_n}D_x$ to the corresponding spectral quantities of $-D_mD_x$.
Applying the above result together with the Dirichlet--Neumann
bracketing, we
investigate the limiting behavior of the small eigenvalues of
subdiffusive
random trap and barrier models and establish lower and upper bounds
for the
asymptotic annealed eigenvalue counting functions.
http://arxiv.org/abs/0905.2900
---------------------------------------------------------------
8528. EXPONENTIAL DEFICIENCY OF CONVOLUTIONS OF DENSITIES
Iosif Pinelis
If a probability density p(\x) (\x\in\R^k) is bounded and R(t) := \int
\exp(t\ell(\x)) \d\x < \infty for some linear functional \ell and all
t\in(0,1), then, for each t\in(0,1) and all large enough n, the n-fold
convolution of the t-tilted density p_t(\x) := \exp(t\ell(\x)) p(\x)/
R(t) is
bounded. This is a corollary of a general, "non-i.i.d." result, which
is also
shown to enjoy a certain optimality property. Such results are useful
for
saddle-point approximations.
http://arxiv.org/abs/0905.2944
---------------------------------------------------------------
8529. USING THE SCHRAMM-LOEWNER EVOLUTION TO EXPLAIN CERTAIN NON-
LOCAL OBSERVABLES IN THE 2D CRITICAL ISING MODEL
Michael J. Kozdron (University of Regina)
We present a mathematical proof of theoretical predictions made by
Arguin and
Saint-Aubin, as well as by Bauer, Bernard, and Kytola, about certain
non-local
observables for the two-dimensional Ising model at criticality by
combining
Smirnov's recent proof of the fact that the scaling limit of critical
Ising
interfaces can be described by chordal SLE(3) with Kozdron and Lawler's
configurational measure on mutually avoiding chordal SLE paths. As an
extension
of this result, we also compute the probability that an SLE(k) path, k
in
(0,4], and a Brownian motion excursion do not intersect.
http://arxiv.org/abs/0905.2430
---------------------------------------------------------------
8530. CURRENT FLUCTUATIONS FOR TASEP: A PROOF OF THE PR\"{A}HOFER-
SPOHN CONJECTURE
G. Ben Arous and I. Corwin
We consider the family of two-sided initial conditions for TASEP
which, as
the left and right densities (\rho_-,\rho_+) are varied, give rise to
shock
waves and rarefaction fans -- the two phenomena which are typical to
TASEP. We
provide a proof of Conjecture 7.1 of Pr\"{a}hofer and Spohn which
characterizes
the order of and scaling functions for the fluctuations of the height
function
of two-sided TASEP in terms of the two densities \rho_-,\rho_+ and the
speed y
around which the height is observed. In proving this theorem for TASEP
we also
prove a fluctuation theorem for a class of corner growth processes with
external sources, or equivalently for the last passage time in a
directed last
passage percolation model with two-sided boundary conditions: \rho_- and
1-\rho_+. We provide a complete characterization of the order of and the
scaling functions for the fluctuations of this model's last passage
time L(N,M)
as a function of three parameters: the two boundary/source rates
\rho_- and
1-\rho_+, and the scaling ratio gamma^2=M/N. The proof of this theorem
draws on
the results of P.L. Ferrari and Spohn and extensively on the work of
Baik, Ben
Arous and P\'{e}ch\'{e} on finite rank perturbations of Wishart
ensembles in
random matrix theory.
http://arxiv.org/abs/0905.2993
---------------------------------------------------------------
8531. A POINCAR\'E INEQUALITY ON LOOP SPACES
Xin Chen and Xue-Mei Li and Bo Wu
We investigate properties of measures in infinite dimensional spaces
in terms
of Poincar\'e inequalities.
A Poincar\'e inequality states that the $L^2$ variance of an
admissible
function is controlled by the homogeneous $H^1$ norm. In the case of
Loop
spaces, it was observed by L. Gross that the homogeneous $H^1$ norm
alone may
not control the $L^2$ norm and a potential term involving the end
value of the
Brownian bridge is introduced. Aida, on the other hand, introduced a
weight on
the Dirichlet form. We show that Aida's modified Logarithmic Sobolev
inequality
implies weak Logarithmic Sobolev Inequalities and weak Poincar\'e
inequalities
with precise estimates on the order of convergence. The order of
convergence in
the weak Sobolev inequalities are related to weak $L^1$ estimates on
the weight
function. This and a relation between Logarithmic Sobolev inequalities
and weak
Poincar\'e inequalities lead to a Poincar\'e inequality on the loop
space over
certain manifolds.
http://arxiv.org/abs/0905.3007
---------------------------------------------------------------
8532. ON THE MIXING TIME OF THE 2D STOCHASTIC ISING MODEL WITH "PLUS"
BOUNDARY CONDITIONS AT LOW TEMPERATURE
F. Martinelli (Matematica and Roma 3) and F. Toninelli (CNRS and ENS
Lyon)
We consider the Glauber dynamics for the 2D Ising model in a box of
side L,
at inverse temperature $\beta$ and random boundary conditions $\tau$
whose
distribution P either stochastically dominates the extremal plus phase
(hence
the quotation marks in the title) or it is stochastically dominated by
the
extremal minus phase. A particular case is when P is concentrated on the
homogeneous configuration identically equal to + (equal to -). For $
\beta$
large enough we show that for any $\epsilon$ there exists $c=c(\beta,
\epsilon)$
such that the corresponding mixing time $T_{mix}$ satisfies
$\lim_{L\to\infty}P(T_{mix}> \exp({cL^\epsilon})) =0$. In the non-
random case
$\tau\equiv +$ (or $\tau\equiv -$), this implies that $T_{mix}<
\exp({cL^\epsilon})$. The same bound holds when the boundary
conditions are all
+ on three sides and all - on the remaining one. The result, although
still
very far from the expected Lifshitz behaviour $T_{mix}=O(L^2)$,
considerably
improves upon the previous known estimates of the form $T_{mix}\le
\exp({c
L^{1/2 + \epsilon}})$. The techniques are based on induction over length
scales, combined with a judicious use of the so-called "censoring
inequality"
of Y. Peres and P. Winkler, which in a sense allows us to guide the
dynamics to
its equilibrium measure.
http://arxiv.org/abs/0905.3040
---------------------------------------------------------------
8533. CENTRAL LIMIT THEOREM FOR THE HEAT KERNEL MEASURE ON THE UNITARY
GROUP
Thierry L\'evy (DMA) and Myl\`ene Ma\"ida (LM-Orsay)
We prove that for a finite collection of real-valued functions
$f_{1},...,f_{n}$ on the group of complex numbers of modulus 1 which are
derivable with Lipschitz continuous derivative, the distribution of $
(\tr
f_{1},...,\tr f_{n})$ under the properly scaled heat kernel measure at
a given
time on the unitary group $\U(N)$ has Gaussian fluctuations as $N$
tends to
infinity, with a covariance for which we give a formula and which is
of order
$N^{-1}$. In the limit where the time tends to infinity, we prove that
this
covariance converges to that obtained by P. Diaconis and S. Evans in a
previous
work on uniformly distributed unitary matrices. Finally, we discuss some
combinatorial aspects of our results.
http://arxiv.org/abs/0905.3282
---------------------------------------------------------------
8534. A FUNCTIONAL EQUATION WHOSE UNKNOWN IS P([0,1]) VALUED
Giacomo Aletti and Caterina May and Piercesare Secchi
We study a functional equation whose unknown maps a euclidean space
into the
space of probability distributions on [0,1]. We prove existence and
uniqueness
of its solution under suitable regularity and boundary conditions and we
characterize solutions that are diffuse on [0,1]. A canonical solution
is
obtained by means of a Randomly Reinforced Urn with different
reinforcement
distributions having equal means.
http://arxiv.org/abs/0905.3310
---------------------------------------------------------------
8535. PATH REGULARITY OF GAUSSIAN PROCESSES VIA SMALL DEVIATIONS
Frank Aurzada
We study the a.s. sample path regularity of Gaussian processes. To
this end
we relate the path regularity directly to the theory of small
deviations. In
particular, we show that if the process is $n$-times differentiable
then the
exponential rate of decay of its small deviations is at most
$\varepsilon^{-1/n}$. We also show a similar result if $n$ is not an
integer.
http://arxiv.org/abs/0905.3358
---------------------------------------------------------------
8536. CUMULANTS AS ITERATED INTEGRALS
Franz Lehner
A formula expressing cumulants in terms of iterated integrals of the
distribution function is derived. It generalizes results of Jones and
Balakrishnan who computed expressions for cumulants up to order 4.
http://arxiv.org/abs/0905.3375
---------------------------------------------------------------
8537. GLOBALLY OPTIMAL PARAMETER ESTIMATES FOR NON-LINEAR DIFFUSIONS
A. Mijatovi\'c and P. Schneider
This paper studies an approximation method for the log likelihood
function of
a non-linear diffusion process using the bridge of the diffusion. The
main
result (Theorem 1) shows that this approximation converges uniformly
to the
unknown likelihood function and can therefore be used efficiently with
any
algorithm for sampling from the law of the bridge. We also introduce an
expected maximum likelihood (EML) algorithm for inferring the
parameters of
discretely observed diffusion processes. The approach is applicable to a
subclass of non-linear SDEs with constant volatility and drift that is
linear
in the model parameters. In this setting globally optimal parameters are
obtained in a single step by solving a square linear system whose
dimension
equals the number of parameters in the model. Simulation studies to
test the
EML algorithm show that it performs well when compared with algorithms
based on
the exact maximum likelihood as well as closed-form likelihood
expansions.
http://arxiv.org/abs/0905.3321
---------------------------------------------------------------
8538. CONVERGENCE OF THE STRUCTURE FUNCTION OF A MULTIFRACTAL RANDOM
WALK IN A MIXED ASYMPTOTIC SETTING
Laurent Duvernet
Some asymptotic properties of a Brownian motion in multifractal time,
also
called multifractal random walk, are established. We show the almost
sure and
$L^1$ convergence of its structure function. This is an issue directly
connected to the scale invariance and multifractal property of the
sample
paths. We place ourselves in a mixed asymptotic setting where both the
observation length and the sampling frequency may go together to
infinity at
different rates. The results we obtain are similar to the ones that
were given
by Ossiander and Waymire and Bacry \emph{et al.} in the simpler
framework of
Mandelbrot cascades.
http://arxiv.org/abs/0905.3405
---------------------------------------------------------------
8539. CUMULANTS OF A CONVOLUTION AND APPLICATIONS TO MONOTONE
PROBABILITY THEORY
Takahiro Hasebe
In non-commutative probability theory, cumulants and their generating
function are defined once a notion of independence is given. In this
paper, we
give a definition of cumulants of a (non-commutative) convolution and
prove the
uniqueness of cumulants. Then we define cumulants of monotone
convolution and
prove limit theorems as applications.
http://arxiv.org/abs/0905.3446
---------------------------------------------------------------
8540. A PHASE TRANSITION FOR THE HEIGHTS OF A FRAGMENTATION TREE
Adrien Joseph (PMA)
We provide information about the asymptotic regimes for a homogeneous
fragmentation of a finite set. We establish a phase transition for the
asymptotic behaviours of the shattering times, defined as the first
instants
when all the blocks of the partition process have cardinality less
than a fixed
integer. Our results may be applied to the study of certain random
split trees.
http://arxiv.org/abs/0905.3545
---------------------------------------------------------------
8541. A STOCHASTIC OPTIMAL CONTROL PROBLEM FOR THE HEAT EQUATION ON
THE HALFLINE WITH DIRICHLET BOUNDARY-NOISE AND BOUNDARY-CONTROL
Federica Masiero
We consider a controlled state equation of parabolic type on the
halfline
$(0,+\infty)$ with boundary conditions of Dirichlet type in which the
unknown
is equal to the sum of the control and of a white noise in time. We
study
finite horizon and infinite horizon optimal control problem related by
menas of
backward stochastic differential equations.
http://arxiv.org/abs/0905.3628
---------------------------------------------------------------
8542. APPROXIMATION OF QUASI-STATIONARY DISTRIBUTIONS FOR 1-
DIMENSIONAL KILLED DIFFUSIONS WITH UNBOUNDED DRIFTS
Denis Villemonais (CMAP)
The long time behavior of an absorbed Markov process is well described
by the
limiting distribution of the process conditioned to not be killed when
it is
observed. Our aim is to give an approximation's method of this limit,
when the
process is a 1-dimensional It\^o diffusion whose drift is allowed to
explode at
the boundary. In a first step, we show how to restrict the study to
the case of
a diffusion with values in a bounded interval and whose drift is
bounded. In a
second step, we show an approximation method of the limiting conditional
distribution of such diffusions, based on a Fleming-Viot type
interacting
particle system. We end the paper with two numerical applications : to
the
logistic Feller diffusion and to the Wright-Fisher diffusion with
values in
$]0,1[$ conditioned to be killed at 0.
http://arxiv.org/abs/0905.3636
---------------------------------------------------------------
8543. ON A PROCESSOR SHARING QUEUE THAT MODELS BALKING
Qiang Zhen and Johan S. H. van Leeuwaarden and Charles Knessl
We consider the processor sharing $M/M/1$-PS queue which also models
balking.
A customer that arrives and sees $n$ others in the system "balks" (i.e.,
decides not to enter) with probability $1-b_n$. If $b_n$ is inversely
proportional to $n+1$, we obtain explicit expressions for a tagged
customer's
sojourn time distribution. We consider both the conditional
distribution,
conditioned on the number of other customers present when the tagged
customer
arrives, as well as the unconditional distribution. We then evaluate the
results in various asymptotic limits. These include large time (tail
behavior)
and/or large $n$, lightly loaded systems where the arrival rate $
\lambda\to 0$,
and heavily loaded systems where $\lambda\to\infty$. We find that the
asymptotic structure for the problem with balking is much different
from the
standard $M/M/1$-PS queue. We also discuss a perturbation method for
deriving
the asymptotics, which should apply to more general balking functions.
http://arxiv.org/abs/0905.3700
---------------------------------------------------------------
8544. ON THE MARTINGALE PROPERTY OF CERTAIN LOCAL MARTINGALES:
CRITERIA AND APPLICATIONS
Aleksandar Mijatovic and Mikhail Urusov
The stochastic exponential $Z_t=\exp[M_t-M_0-(1/2)< M,M>_t]$ of a
continuous
local martingale $M$ is itself a continuous local martingale. We give a
necessary and sufficient condition for the process $Z$ to be a true
martingale
in the case where $M_t=\int_0^t b(Y_u) dW_u$ and $Y$ is a one-
dimensional
diffusion driven by a Brownian motion $W$. Furthermore, we provide a
necessary
and sufficient condition for $Z$ to be a uniformly integrable
martingale in the
same setting. These conditions are deterministic and expressed only in
terms of
the function $b$ and the drift and diffusion coefficients of $Y$. We
also
classify, via deterministic necessary and sufficient conditions, when
the
process $Z$ is a.s. strictly positive, when its limit $Z_\infty$ is a.s.
strictly positive, and when $Z_\infty$ is a.s. zero. This allows us to
obtain a
deterministic necessary and sufficient condition in the one-
dimensional setting
for a discounted stock price to be a true martingale under the risk-
neutral
measure, and for it to be a uniformly integrable martingale. These
results
enable us to ascertain the existence of financial bubbles in diffusion-
based
models. Finally, we obtain a deterministic characterisation of the
\emph{no
free lunch with vanishing risk}, the \emph{no generalised arbitrage},
and the
\emph{no relative arbitrage} conditions in the one-dimensional setting
and
examine how these notions of no-arbitrage relate to each other.
http://arxiv.org/abs/0905.3701
---------------------------------------------------------------
8545. OPTIMAL STOPPING FOR NON-LINEAR EXPECTATIONS
Erhan Bayraktar and Song Yao
We develop a theory for solving continuous time optimal stopping
problems for
non-linear expectations. Our motivation is to consider problems in
which the
stopper uses risk measures to evaluate future rewards.
http://arxiv.org/abs/0905.3601
---------------------------------------------------------------
8546. A NOTE ON FURSTENBERG'S FILTERING PROBLEM
Rodolphe Garbit (LMJL)
This short note gives a positive answer to an elementary question in
probability theory that arose in Furstenberg's famous article
"Disjointness in
Ergodic Theory". As a consequence, Furstenberg's filtering theorem holds
without any integrability assumption.
http://arxiv.org/abs/0905.3879
---------------------------------------------------------------
8547. REVERSED DIRICHLET ENVIRONMENT AND DIRECTIONAL TRANSIENCE OF
RANDOM WALKS IN DIRICHLET RANDOM ENVIRONMENT
Christophe Sabot (ICJ) and Laurent Tournier (ICJ)
We consider random walks in a random environment that is given by i.i.d.
Dirichlet distributions at each vertex of Z^d or, equivalently,
oriented edge
reinforced random walks on Z^d. The parameters of the distribution are a
2d-uplet of positive real numbers indexed by the unit vectors of Z^d.
We prove
that, as soon as these weights are nonsymmetric, the random walk in
this random
environment is transient in a direction with positive probability. In
dimension
2, this result can be strenghened to an almost sure directional
transience
thanks to the 0-1 law from [ZM01]. Our proof relies on the property of
stability of Dirichlet environment by time reversal proved in [Sa09].
In a
first part of this paper, we also give a probabilistic proof of this
property
as an alternative to the change of variable computation used in that
article.
http://arxiv.org/abs/0905.3917
---------------------------------------------------------------
8548. ON QUADRATIC G-EVALUATIONS/EXPECTATIONS AND RELATED ANALYSIS
Jin Ma and Song Yao
In this paper we extend the notion of g-evaluation, in particular
g-expectation, to the case where the generator g is allowed to have a
quadratic
growth. We show that some important properties of the g-expectations,
including
a representation theorem between the generator and the corresponding
g-expectation, and consequently the reverse comparison theorem of
quadratic
BSDEs as well as the Jensen inequality, remain true in the quadratic
case. Our
main results also include a Doob-Meyer type decomposition, the optional
sampling theorem, and the up-crossing inequality. The results of this
paper are
important in the further development of the general quadratic nonlinear
expectations.
http://arxiv.org/abs/0905.3941
---------------------------------------------------------------
8549. DISCRETE TIME SCALE INVARIANT MARKOV PROCESSES
N. Modarresi and S. Rezakhah
In this paper we consider a discrete scale invariant Markov process with
scale $l$ which by a scheme of sampling at discrete points we provide
discrete
time scale invariant Markov(DT-SIM) process. We also define quasi
Lamperti
transformation as a basic tool in relation with such sampling. We
study the
properties of a DT-SIM process and find the covariance function of it
which is
specified by the values of $\{R_{j}^H(1),R_{j}^H(0),j\in {\bf Z^+},
0\leq j\leq
{T-1}\}$, where $R_j^H(k)$ is the covariance function $j$th and $(j+k)
$th
observations of DT-SIM and $T$ is the number of observations in each
scale. We
also define T-dimensional self-similar Markov process corresponding to
DT-SIM
process and characterize its covariance matrix.
http://arxiv.org/abs/0905.3959
---------------------------------------------------------------
8550. TAGGED PARTICLE PROCESSES AND THEIR NON-EXPLOSION CRITERIA
Hirofumi Osada
We give a derivation of tagged particle processes from unlabeled
interacting
Brownian motions. We give a criteria of the non-explosion property of
tagged
particle processes. We prove the quasi-regularity of Dirichlet forms
describing
the environment seen from the tagged particle, which were used in
previous
papers to prove the invariance principle of tagged particles of
interacting
Brownian motions.
http://arxiv.org/abs/0905.3973
---------------------------------------------------------------
8551. MAXIMUM OF DYSON BROWNIAN MOTION AND NON-COLLIDING SYSTEMS WITH
A BOUNDARY
Alexei Borodin and Patrik L. Ferrari and Michael Praehofer and
Tomohiro Sasamoto, Jon Warren
We prove an equality-in-law relating the maximum of GUE Dyson's Brownian
motion and the non-colliding systems with a wall. This generalizes the
well
known relation between the maximum of a Brownian motion and a reflected
Brownian motion.
http://arxiv.org/abs/0905.3989
---------------------------------------------------------------
8552. LIMITS OF RANDOMLY GROWN GRAPH SEQUENCES
C. Borgs and J. Chayes and L. Lov\'asz and V.T. S\'os and K.
Vesztergombi
Motivated in part by various sequences of graphs growing under random
rules
(like internet models), convergent sequences of dense graphs and their
limits
were introduced by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi and
by
Lov\'asz and Szegedy. In this paper we use this framework to study one
of the
motivating class of examples, namely randomly growing graphs. We prove
the
(almost sure) convergence of several such randomly growing graph
sequences, and
determine their limit. The analysis is not always straightforward: in
some
cases the cut distance from a limit object can be directly estimated,
in other
case densities of subgraphs can be shown to converge.
http://arxiv.org/abs/0905.3806
---------------------------------------------------------------
8553. ERGODIC PROPERTIES OF MAX-INFINITELY DIVISIBLE PROCESSES
Zakhar Kabluchko and Martin Schlather
We prove that a stationary max--infinitely divisible process is mixing
(ergodic) iff its dependence function converges to 0 (is Cesaro
summable to 0).
These criteria are applied to some classes of max--infinitely divisible
processes.
http://arxiv.org/abs/0905.4196
---------------------------------------------------------------
8554. ON CHARACTERIZATIONS BASED ON REGRESSION OF LINEAR COMBINATIONS
OF RECORDS
George P. Yanev and M. Ahsanullah
We characterize the exponential distribution in terms of the
regression of a
record value with non-adjacent records as covariates. We also study
characterizations based on the regression of linear combinations of
records.
http://arxiv.org/abs/0905.4230
---------------------------------------------------------------
8555. ON ASYMPTOTIC EXPANSION IN THE RANDOM ALLOCATION OF PARTICLES BY
SETS
Saidbek S.Mirakhmedov and Sherzod M.Mirakhmedov
We consider a scheme of equiprobable allocation of particles into
cells by
sets. The Edgeworth type asymptotic expansion in the local central limit
theorem for a number of empty cells left after allocation of all sets of
particles is derived.
http://arxiv.org/abs/0905.4247
---------------------------------------------------------------
8556. FIXED TRACE $\BETA$-HERMITE ENSEMBLES: ASYMPTOTIC EIGENVALUE
DENSITY AND THE EDGE OF THE DENSITY
Da-Sheng Zhou and Dang-Zheng Liu and Tao Qian
In the present paper, fixed trace $\beta$-Hermite ensembles
generalizing the
fixed trace Gaussian Hermite ensemble are considered. For all $\beta$,
we prove
the Wigner semicircle law for these ensembles by using two different
methods:
one is the moment equivalence method with the help of the matrix model
for
general $\beta$, the other is to use asymptotic analysis tools. At the
edge of
the density, we prove that the edge scaling limit for $\beta$-HE
implies the
same limit for fixed trace $\beta$-Hermite ensembles. Consequently,
explicit
limit can be given for fixed trace GOE, GUE and GSE. Furthermore, for
even
$\beta$, analogous to $\beta$-Hermite ensembles, a multiple integral
of the
Konstevich type can be obtained.
http://arxiv.org/abs/0905.4255
---------------------------------------------------------------
8557. BOUNDARY BEHAVIOUR OF HARMONIC FUNCTIONS ON HYPERBOLIC GROUPS
Camille Petit (IF)
We consider random walks with finite support on non-elementary Gromov
hyperbolic groups. For a given harmonic function on such a group, we
prove that
asymptotic properties of non-tangential boundedness and non-tangential
convergence are almost everywhere equivalent. The proof is inspired
from works
of F. Mouton in the cases of Riemannian manifolds of pinched negative
curvature
and infinite trees. It involves geometric and probabilitistic methods.
http://arxiv.org/abs/0905.4118
---------------------------------------------------------------
8558. ON THE INFLUENCES OF VARIABLES ON BOOLEAN FUNCTIONS IN PRODUCT
SPACES
Nathan Keller
In this paper we consider the influences of variables on Boolean
functions in
general product spaces. Unlike the case of functions on the discrete
cube where
there is a clear definition of influence, in the general case at least
three
definitions were presented in different papers. We propose a family of
definitions for the influence, that contains all the known
definitions, as well
as other natural definitions, as special cases. We prove a
generalization of
the BKKKL theorem, which is tight in terms of the definition of
influence used
in the assertion, and use it to generalize several known results on
influences
in general product spaces.
http://arxiv.org/abs/0905.4216
---------------------------------------------------------------
8559. LDP APPLICATION FOR BILLINGSLEY'S EXAMPLE
R. Liptser
We consider a classical model discussed in Theorem 16.4 (Billingsley
\cite{Bil}) concerning to an empirical distribution function $$
F_n(t)=\frac{1}{n}\sum_{k=1}^nI_{\{\xi_k\le t\}}, $$ of $(\xi_k)_{i\ge
1}$ -
i.i.d. sequence of random variables, supported on the interval
$[0,1]$, with
$F(t)=\mathsf{P}(\xi_1\le t)$ the continuous distribution function. We
give a
proof of Kolmogorov's exponential estimate
\mathsf{P}\Big(\sup_{t\in[0,1]}|F_n(t)-F(t)|\ge \varepsilon\Big) \le
2\exp\Big\{-n[\varepsilon/8\log(1+\varepsilon^2/32)-\varepsilon/8
+(4/\varepsilon)\log(1+\varepsilon^2/32\Big)]\} with the help of which
jointly
with the large deviations technique, we establish a logarithmic
asymptotic: for
any $T\in[F^{-1}({1/2}),1)$ and any $\alpha\in\big(0,{1/2}\big)$:
\lim_{n\to\infty}\frac{1}{n^{1-2\alpha}}
\log\mathsf{P}\bigg(\sup_{t\in[0,T]}n^\alpha\Big|F_n(t)-F(t)\Big|\ge
\varepsilon\bigg)=-2\varepsilon^2.
http://arxiv.org/abs/0905.4334
---------------------------------------------------------------
8560. A DECOMPOSITION AND WEAK APPROXIMATION OF THE SUB-FRACTIONAL
BROWNIAN MOTION
Xavier Bardina and David Bascompte
We present a decomposition of the sub-fractional Brownian motion into
the sum
of a fractional Brownian motion plus a stochastic process with
absolutely
continuous trajectories. The first application we show of this
decomposition is
the relation between the spaces of integrable functions with respect
each one
of these three processes. A general result of weak convergence to
integrals of
$L^2(\mathbb R^{+})$ functions with respect to standard Brownian
motion is
proved, and this result permits us to obtain approximations in law of
the
fractional Brownian motion and the sub-fractional Brownian motion with
parameter $H\in(0,1)$.
http://arxiv.org/abs/0905.4360
---------------------------------------------------------------
8561. FROM THE LIFSHITZ TAIL TO THE QUENCHED SURVIVAL ASYMPTOTICS IN
THE TRAPPING PROBLEM
Ryoki Fukushima
The survival problem for a diffusing particle moving among random
traps is
considered. We introduce a simple argument to derive the quenched
asymptotics
of the survival probability from the Lifshitz tail effect for the
associated
operator. In particular, the upper bound is proved in fairly general
settings
and is shown to be sharp in the case of the Brownian motion in the
Poissonian
obstacles. As an application, we derive the quenched asymptotics for the
Brownian motion in traps distributed according to a random
perturbation of the
lattice.
http://arxiv.org/abs/0905.4436
---------------------------------------------------------------
8562. EXTREME VALUE THEORY, POISSON-DIRICHLET DISTRIBUTIONS AND FPP ON
RANDOM NETWORKS
Shankar Bhamidi and Remco van der Hofstad and Gerard Hooghiemstra
We study first passage percolation on the configuration model (CM)
having
power-law degrees with exponent $\tau\in [1,2)$. To this end we equip
the edges
with exponential weights. We derive the distributional limit of the
minimal
weight of a path between typical vertices in the network and the
number of
edges on the minimal weight path, which can be computed in terms of the
Poisson-Dirichlet distribution. We explicitly describe these limits
via the
construction of an infinite limiting object describing the FPP problem
in the
densely connected core of the network. We consider two separate cases,
namely,
the {\it original CM}, in which each edge, regardless of its
multiplicity,
receives an independent exponential weight, as well as the {\it erased
CM}, for
which there is an independent exponential weight between any pair of
direct
neighbors. While the results are qualitatively similar, surprisingly the
limiting random variables are quite different.
Our results imply that the flow carrying properties of the network
are
markedly different from either the mean-field setting or the locally
tree-like
setting, which occurs as $\tau>2$, and for which the hopcount between
typical
vertices scales as $\log{n}$. In our setting the hopcount is tight and
has an
explicit limiting distribution, showing that one can transfer
information
remarkably quickly between different vertices in the network. This
efficiency
has a down side in that such networks are remarkably fragile to directed
attacks. These results continue a general program by the authors to
obtain a
complete picture of how random disorder changes the inherent geometry of
various random network models.
http://arxiv.org/abs/0905.4438
---------------------------------------------------------------
8563. DEDUCING VERTEX WEIGHTS FROM EMPIRICAL OCCUPATION TIMES
Joshua N. Cooper
We consider the following problem arising from the study of human
problem
solving: Let $G$ be a vertex-weighted graph with marked "in" and "out"
vertices. Suppose a random walker begins at the in-vertex, steps to
neighbors
of vertices with probability proportional to their weights, and stops
upon
reaching the out-vertex. Could one deduce the weights from the paths
that many
such walkers take? We analyze an iterative numerical solution to this
reconstruction problem, in particular, given the empirical mean
occupation
times of the walkers. In the process, a result concerning the
differentiation
of a matrix pseudoinverse is given, which may be of independent
interest. We
then consider the existence of a choice of weights for the given
occupation
times, formulating a natural conjecture to the effect that -- barring
obvious
obstructions -- a solution always exists. It is shown that the
conjecture holds
for a class of graphs that includes all trees and complete graphs.
Several open
problems are discussed.
http://arxiv.org/abs/0905.4391
---------------------------------------------------------------
8564. L^P ESTIMATES FOR FEYNMAN-KAC PROPAGATORS WITH TIME-DEPENDENT
REFERENCE MEASURES
Andreas Eberle and Carlo Marinelli
We introduce a class of time-inhomogeneous transition operators of
Feynman-Kac type that can be considered as a generalization of
symmetric Markov
semigroups to the case of a time-dependent reference measure. Applying
weighted
Poincar\'e and logarithmic Sobolev inequalities, we derive L^p-L^p and
L^p-L^q
estimates for the transition operators. Since the operators are not
Markovian,
the estimates depend crucially on the value of p. Our studies are
motivated by
applications to sequential Markov Chain Monte Carlo methods.
http://arxiv.org/abs/0905.4411
---------------------------------------------------------------
8565. QUANTUM PROBABILITY, RENORMALIZATION AND INFINITE-DIMENSIONAL *-
LIE ALGEBRAS
Luigi Accardi and Andreas Boukas
The present paper reviews some intriguing connections which link
together a
new renormalization technique, the theory of *-representations of
infinite
dimensional *-Lie algebras, quantum probability, white noise and
stochastic
calculus and the theory of classical and quantum infinitely divisible
processes.
http://arxiv.org/abs/0905.4491
---------------------------------------------------------------
8566. OMNIBUS SEQUENCES, COUPON COLLECTION, AND MISSING WORD COUNTS
Sunil Abraham and Greg Brockman and Stephanie Sapp and Anant P.
Godbole
An {\it Omnibus Sequence} of length $n$ is one that has each possible
"message" of length $k$ embedded in it as a subsequence. We study
various
properties of Omnibus Sequences in this paper, making connections,
whenever
possible, to the classical coupon collector problem.
http://arxiv.org/abs/0905.4517
---------------------------------------------------------------
8567. PERCOLATION OF WORDS ON $\Z^D$ WITH LONG RANGE CONNECTIONS
Bernardo N. B. de Lima and Remy Sanchis and Roger W. C. Silva
Consider an independent site percolation model on $\Z^d$, with
parameter $p
\in (0,1)$, where all long range connections in the axes directions are
allowed. In this work we show that given any parameter $p$, there
exists and
integer $K(p)$ such that all binary sequences (words) $\xi \in
\{0,1\}^{\N}$
can be seen simultaneously, almost surely, even if all connections
whose length
is bigger than $K(p)$ are suppressed. We also show some results
concerning the
question how $K(p)$ should scale with $p$ when $p$ goes to zero. Related
results are also obtained for the question of whether or not almost
all words
are seen.
http://arxiv.org/abs/0905.4615
---------------------------------------------------------------
8568. HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS
Jozsef Balogh and Bela Bollobas and Mark Walters
We prove that, in the Gilbert model for a random geometric graph, almost
every graph becomes Hamiltonian exactly when it first becomes 2-
connected. This
proves a conjecture of Penrose.
We also show that in the $k$-nearest neighbour model, there is a
constant
$\kappa$ such that almost every $\kappa$-connected graph has a
Hamilton cycle.
http://arxiv.org/abs/0905.4650
---------------------------------------------------------------
8569. BULK UNIVERSALITY FOR WIGNER MATRICES
Laszlo Erdos and Jose A. Ramirez and Benjamin Schlein and Horng-
Tzer Yau
We consider $N\times N$ Hermitian Wigner random matrices $H$ where the
probability density for each matrix element is given by the density $
\nu(x)=
e^{- U(x)}$. We prove that the eigenvalue statistics in the bulk is
given by
Dyson sine kernel provided that $U \in C^6(\RR)$ with at most
polynomially
growing derivatives and $\nu(x) \le C e^{- C |x|}$ for $x$ large. The
proof is
based upon an approximate time reversal of the Dyson Brownian motion
combined
with the convergence of the eigenvalue density to the Wigner
semicircle law on
short scales.
http://arxiv.org/abs/0905.4176
---------------------------------------------------------------
8570. CHARACTERIZING PREDICTABLE CLASSES OF PROCESSES
Daniil Ryabko (INRIA Futurs and LIFL and INRIA Lille - Nord Europe)
The problem is sequence prediction in the following setting. A sequence
$x_1,...,x_n,...$ of discrete-valued observations is generated
according to
some unknown probabilistic law (measure) $\mu$. After observing each
outcome,
it is required to give the conditional probabilities of the next
observation.
The measure $\mu$ belongs to an arbitrary class $\C$ of stochastic
processes.
We are interested in predictors $\rho$ whose conditional probabilities
converge
to the "true" $\mu$-conditional probabilities if any $\mu\in\C$ is
chosen to
generate the data. We show that if such a predictor exists, then a
predictor
can also be obtained as a convex combination of a countably many
elements of
$\C$. In other words, it can be obtained as a Bayesian predictor whose
prior is
concentrated on a countable set. This result is established for two very
different measures of performance of prediction, one of which is very
strong,
namely, total variation, and the other is very weak, namely,
prediction in
expected average Kullback-Leibler divergence.
http://arxiv.org/abs/0905.4341
---------------------------------------------------------------
8571. RIFFLE SHUFFLES OF A DECK WITH REPEATED CARDS
Sami Assaf and Persi Diaconis and K. Soundararajan
We study the Gilbert-Shannon-Reeds model for riffle shuffles and ask
'How many times must a deck of cards be shuffled for the deck to be
in close
to random order?'. In 1992, Bayer and Diaconis gave a solution which
gives
exact and asymptotic results for all decks of practical interest, e.g.
a deck
of 52 cards. But what if one only cares about the colors of the cards or
disregards the suits focusing solely on the ranks? More generally, how
does the
rate of convergence of a Markov chain change if we are interested in
only
certain features? Our exploration of this problem takes us through
random walks
on groups and their cosets, discovering along the way exact formulas
leading to
interesting combinatorics, an 'amazing matrix', and new analytic
methods which
produce a completely general asymptotic solution that is remarkable
accurate.
http://arxiv.org/abs/0905.4698
---------------------------------------------------------------
8572. ASYMPTOTICS OF THE VISIBILITY FUNCTION IN THE BOOLEAN MODEL
Pierre Calka (MAP5) and Julien Michel (UMPA-ENSL) and Sylvain Porret-
Blanc (UMPA-ENSL)
The aim of this paper is to give a precise estimate on the tail
probability
of the visibility function in a germ-grain model: this function is
defined as
the length of the longest ray starting at the origin that does not
intersect an
obstacle in a Boolean model. We proceed in two or more dimensions using
coverage techniques. Moreover, convergence results involving a type I
extreme
value distribution are shown in the two particular cases of small
obstacles or
a large obstacle-free region.
http://arxiv.org/abs/0905.4874
---------------------------------------------------------------
8573. A CRITERION FOR HYPOTHESIS TESTING FOR STATIONARY PROCESSES
Daniil Ryabko (INRIA Futurs and LIFL and INRIA Lille - Nord Europe)
Given a discrete-valued sample X_1... X_n we wish to test whether it was
generated by a process belonging to a family H_0, or it was generated
by a
process outside H_0. All process distributions are assumed stationary
ergodic,
and no further probabilistic or parametric assumptions are made. We
require the
Type I error of the test to be uniformly bounded, while the
probability of Type
II error has to tend to zero as the sample size increases. For this
notion of
consistency we provide necessary and sufficient conditions on the
family H_0
for the existence of a consistent test. This criterion is illustrated
with
applications to testing for a membership to parametric families,
generalizing
some existing results.
http://arxiv.org/abs/0905.4937
---------------------------------------------------------------
8574. LONG AND SHORT PATHS IN UNIFORM RANDOM RECURSIVE DAGS
Luc Devroye and Svante Janson
In a uniform random recursive k-dag, there is a root, 0, and each node
in
turn, from 1 to n, chooses k uniform random parents from among the
nodes of
smaller index. If S_n is the shortest path distance from node n to the
root,
then we determine the constant \sigma such that S_n/log(n) tends to
\sigma in
probability as n tends to infinity. We also show that max_{1 \le i \le
n}
S_i/log(n) tends to \sigma in probability.
http://arxiv.org/abs/0906.0152
---------------------------------------------------------------
8575. COMPUTATIONAL METHODS FOR STOCHASTIC RELATIONS AND MARKOVIAN
COUPLINGS
Lasse Leskel\"a (Helsinki University of Technology)
Order-preserving couplings are elegant tools for obtaining robust
estimates
of the time-dependent and stationary distributions of Markov processes
that are
too complex to be analyzed exactly. The starting point of this paper
is to
study stochastic relations, which may be viewed as natural
generalizations of
stochastic orders. This generalization is motivated by the observation
that for
the stochastic ordering of two Markov processes, it suffices that the
generators of the processes preserve some, not necessarily reflexive or
transitive, subrelation of the order relation. The main contributions
of the
paper are an algorithmic characterization of stochastic relations
between
finite spaces, and a truncation approach for comparing infinite-state
Markov
processes. The methods are illustrated with applications to loss
networks and
parallel queues.
http://arxiv.org/abs/0906.0153
---------------------------------------------------------------
8576. QUASI-MARTINGALES WITH A LINEARLY ORDERED INDEX SET
Gianluca Cassese
We prove a version of Rao decomposition for quasi-martingales indexed
by a
linearly ordered set.
http://arxiv.org/abs/0906.0183
---------------------------------------------------------------
8577. A COUNTER-INTUITIVE CORRELATION IN A RANDOM TOURNAMENT
Sven Erick Alm and Svante Linusson
Consider a randomly oriented graph $G=(V,E)$ and let $a$, $s$ and $b$ be
three distinct vertices in $V$. We study the correlation between the
events
$\{a\to s\}$ and $\{s\to b\}$. We show that, when $G$ is the complete
graph
$K_n$, the correlation is negative for $n=3$, zero for $n=4$, and that,
counter-intuitively, it is positive for $n\ge 5$. We also show that the
correlation is always negative when $G$ is a cycle, $C_n$, and
negative or zero
when $G$ is a tree (or a forest).
http://arxiv.org/abs/0906.0240
---------------------------------------------------------------
8578. APPROXIMATING A DIFFUSION BY A HIDDEN MARKOV MODEL
Ioannis Kontoyiannis and Sean P. Meyn
For a wide class of continuous-time Markov processes, including all
irreducible hypoelliptic diffusions evolving on an open, connected
subset of
$\RL^d$, the following are shown to be equivalent: (i) The process
satisfies (a
slightly weaker version of) the classical Donsker-Varadhan conditions;
(ii) The
transition semigroup of the process can be approximated by a finite-
state
hidden Markov model, in a strong sense in terms of an associated
operator norm;
(iii) The resolvent kernel of the process is `$v$-separable', that is,
it can
be approximated arbitrarily well in operator norm by finite-rank
kernels. Under
any (hence all) of the above conditions, the Markov process is shown
to have a
purely discrete spectrum on a naturally associated weighted $L_\infty$
space.
http://arxiv.org/abs/0906.0259
---------------------------------------------------------------
8579. BRANCHING BROWNIAN MOTION: ALMOST SURE GROWTH ALONG SCALED PATHS
Simon Harris and Matthew Roberts
We give a proof of a result on the growth of the number of particles
along
chosen paths in a branching Brownian motion. The work follows the
approach of
classical large deviations results, in which paths in $C[0,1]$ are
rescaled
onto $C[0,T]$ for large $T$. The methods used are probabilistic and take
advantage of modern spine techniques.
http://arxiv.org/abs/0906.0291
---------------------------------------------------------------
8580. BERRY-ESSEEN BOUNDS FOR GENERAL NONLINEAR STATISTICS, WITH
APPLICATIONS TO PEARSON'S AND NON-CENTRAL STUDENT'S AND HOTELLING'S
Iosif Pinelis and Raymond Molzon
Recently Chen and Shao developed a Stein-type method to obtain bounds
on the
closeness of the distribution of a general nonlinear statistic to that
of a
linear approximation. We generalize these results so as to allow one
to use
lesser moment restrictions when applied to nonlinear statistics
expressed as
smooth enough functions of sums of independent random vectors. Our main
innovation in the method is the use of a Cramer-type of tilt
transform. Other
techniques used to obtain improvements include exponential and
Rosenthal-type
inequalities for sums of random vectors established by Pinelis and
Sakhanenko.
As applications, Berry-Esseen type bounds are obtained for concrete
nonlinear
statistics such as the Pearson correlation coefficient and the non-
central
Student and Hotelling statistics.
http://arxiv.org/abs/0906.0177
---------------------------------------------------------------
8581. HAMILTONICITY OF THE RANDOM GEOMETRIC GRAPH
Michael Krivelevich and Tobias Muller
Let $X_1,X_2,...$ be independent, uniformly random points from
$[0,1]^2$. For
$r\geq 0$ the {\em random geometric graph} $G(n,r)$ has vertex set
$V_n :=
\{X_1,...,X_n\}$ and an edge $X_iX_j \in E_n$ iff. $\norm{X_i-X_j}
\leq r$. The
"hitting radius" $\rho_n(\Pcal)$ of an increasing graph property $\Pcal
$ is the
least $r$ such that $G(n,r)$ satisfies $\Pcal$, i.e. $\rho_n(\Pcal) :=
\inf\{r
\geq 0 : G(n,r) \text{satisfies} \Pcal \}$. Here we prove that $
{\mathcal P}[
\rho_n(\text{min. degree}\geq 2) = \rho_n(\text{Hamiltonian}) ] \to 1$
as $n
\to \infty$. This answers an open question of Penrose in the
affirmative and
provides an analogue for the random geometric graph of a celebrated
result of
Ajtai, Koml{\'o}s and Szemer{\'e}di on the Erd\H{o}s-R\'enyi random
graph. The
proof generalises to uniform random points on the $d$-dimensional
hypercube
with $\norm{.}$ any $l_p$-norm.
http://arxiv.org/abs/0906.0071
---------------------------------------------------------------
8582. INTRINSIC VOLUMES OF INSCRIBED RANDOM POLYTOPES IN SMOOTH CONVEX
BODIES
Imre B\'ar\'any and Ferenc Fodor and Viktor V\'igh
Let $K$ be a $d$ dimensional convex body with a twice continuously
differentiable boundary and everywhere positive Gauss-Kronecker
curvature.
Denote by $K_n$ the convex hull of $n$ points chosen randomly and
independently
from $K$ according to the uniform distribution. Matching lower and
upper bounds
are obtained for the orders of magnitude of the variances of the $s$-th
intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$.
Furthermore,
strong laws of large numbers are proved for the intrinsic volumes of
$K_n$. The
essential tools are the Economic Cap Covering Theorem of B\'ar\'any
and Larman,
and the Efron-Stein jackknife inequality.
http://arxiv.org/abs/0906.0309
---------------------------------------------------------------
8583. LARGE DEVIATIONS OF U-EMPIRICAL KOLMOGOROV-SMIRNOV TESTS, AND
THEIR EFFICIENCY
Yakov Nikitin
Non-degenerate U-empirical Kolmogorov-Smirnov tests are studied and
their
large deviation asymptotics under the null-hypothesis is described.
Several
examples of such statistics used for testing goodness-of-fit and
symmetry are
considered. It is shown how to calculate their local Bahadur efficiency.
http://arxiv.org/abs/0906.0428
---------------------------------------------------------------
8584. RANDOM MATRICES: UNIVERSALITY OF LOCAL EIGENVALUE STATISTICS
Terence Tao and Van Vu
In this paper, we consider the universality of the local eigenvalue
statistics of random matrices. Our main result shows that these
statistics are
determined by the first four moments of the distribution of the
entries. As a
consequence, we derive the universality of eigenvalue gap distribution
and
$k$-point correlation and many other statistics (under some mild
assumptions)
for both Wigner Hermitian matrices and Wigner real symmetric matrices.
http://arxiv.org/abs/0906.0510
---------------------------------------------------------------
8585. RATE OF CONVERGENCE OF STOCHASTIC PROCESSES WITH VALUES IN $
\MATHBB{R}$-TREES AND HADAMARD MANIFOLDS
Kei Funano
Under K.-T. Sturm's formulation, we obtain a Gaussian upper bound for
tail
probability of mean value of independent, identically distributed random
variables with values in $\mathbb{R}$-trees and Hadamard manifolds.
http://arxiv.org/abs/0906.0649
---------------------------------------------------------------
8586. CORRELATIONS FOR PATHS IN RANDOM ORIENTATIONS OF G(N,P)
Sven Erick Alm and Svante Linusson
We study the random graph G(n,p) with a random orientation. For three
fixed
vertices s,a,b in G(n,p) we study the correlation of the events {a\to
s} and
{s\to b}. We prove that for a fixed p<1/2 the correlation is negative
for large
enough n and for p>1/2 the correlation is positive for large enough n.
We
present exact recursions to compute P(a\to s) and P(a\to s, s\to b). We
conjecture that for a fixed n>26 the correlation changes sign three
times for
three critical values of p.
http://arxiv.org/abs/0906.0720
---------------------------------------------------------------
8587. ON THE UNIQUENESS OF SOLUTIONS TO QUADRATIC BSDES WITH CONVEX
GENERATORS AND UNBOUNDED TERMINAL CONDITIONS
Freddy Delbaen (Department of Mathematics) and Ying Hu (IRMAR) and
Adrien Richou (IRMAR)
In a previous work, P. Briand and Y. Hu proved the uniqueness among the
solutions which admit every exponential moments. In this paper, we
prove that
uniqueness holds among solutions which admit some given exponential
moments.
These exponential moments are natural as they are given by the existence
theorem. Thanks to this uniqueness result we can strengthen the
nonlinear
Feynman-Kac formula proved by P. Briand and Y. Hu.
http://arxiv.org/abs/0906.0752
---------------------------------------------------------------
8588. EXPONENTIAL AND GAUSSIAN CONCENTRATION OF 1-LIPSCHITZ MAPS
Kei Funano
In this paper, we prove an exponential and Ganssian concentration
inequality
for 1-Lipschitz maps from mm-spaces to Hadamard manifolds. In
particular, we
give a complete answer to a question by M. Gromov.
http://arxiv.org/abs/0906.0648
---------------------------------------------------------------
8589. THINNING, ENTROPY AND THE LAW OF THIN NUMBERS
Peter Harremoes and Oliver Johnson and Ioannis Kontoyiannis
Renyi's "thinning" operation on a discrete random variable is a natural
discrete analog of the scaling operation for continuous random
variables. The
properties of thinning are investigated in an information-theoretic
context,
especially in connection with information-theoretic inequalities
related to
Poisson approximation results. The classical Binomial-to-Poisson
convergence
(sometimes referred to as the "law of small numbers" is seen to be a
special
case of a thinning limit theorem for convolutions of discrete
distributions. A
rate of convergence is provided for this limit, and nonasymptotic
bounds are
also established. This development parallels, in part, the development
of
Gaussian inequalities leading to the information-theoretic version of
the
central limit theorem. In particular, a "thinning Markov chain" is
introduced,
and it is shown to play a role analogous to that of the Ornstein-
Uhlenbeck
process in connection to the entropy power inequality.
http://arxiv.org/abs/0906.0690
---------------------------------------------------------------
8590. A MARKOVIAN SLOT MACHINE AND PARRONDO'S PARADOX
S. N. Ethier and Jiyeon Lee
The antique Mills Futurity slot machine has two unusual features.
First, if a
player loses 10 times in a row, the 10 lost coins are returned.
Second, the
payout distribution varies from coup to coup in a manner that is
nonrandom and
periodic with period 10. It follows that the machine is driven by a
100-state
irreducible period-10 Markov chain. Here we evaluate the stationary
distribution of the Markov chain, and this leads to a strong law of
large
numbers and a central limit theorem for the sequence of payouts.
Following a
suggestion of Pyke (2003), we address the question of whether there
exists a
two-armed version of this "one-armed bandit" that obeys Parrondo's
paradox.
More precisely, is there such a machine with the property that the
casino can
honestly advertise that both arms are fair, yet when players alternate
arms in
certain random or nonrandom ways, the casino makes money in the long
run? The
answer is a qualified yes. Although this "history-dependent" game is
conceptually simpler than the original such games of Parrondo, Harmer,
and
Abbott (2000), it is nearly as complicated analytically, and open
problems
remain.
http://arxiv.org/abs/0906.0792
---------------------------------------------------------------
8591. BANDIT PROBLEMS WITH LEVY PAYOFF PROCESSES
Asaf Cohen and Eilon Solan
We study two-armed Levy bandits in continuous-time, which have one
safe arm
that yields a constant payoff s, and one risky arm that can be either
of type
High or Low; both types yield stochastic payoffs generated by a Levy
process.
The expectation of the Levy process when the arm is High is greater
than s, and
lower than s if the arm is Low.
The decision maker (DM) has to choose, at any given time t, the
fraction of
resource to be allocated to each arm over the time interval [t,t+dt).
We show
that under proper conditions on the Levy processes, there is a unique
optimal
strategy, which is a cut-off strategy, and we provide an explicit
formula for
the cut-off and the optimal payoff, as a function of the data of the
problem.
We also examine the case where the DM has incorrect prior over the
type of the
risky arm, and we calculate the expected payoff gained by a DM who
plays the
optimal strategy that corresponds to the incorrect prior.
In addition, we study two applications of the results: (a) we show
how to
price information in two-armed Levy bandit problem, and (b) we
investigate who
fares better in two-armed bandit problems: an optimist who assigns to
High a
probability higher than the true probability, or a pessimist who
assigns to
High a probability lower than the true probability.
http://arxiv.org/abs/0906.0835
---------------------------------------------------------------
8592. FUNCTIONAL INTEGRAL REPRESENTATIONS FOR SELF-AVOIDING WALK
David C. Brydges and John Z. Imbrie and Gordon Slade
We give a survey and unified treatment of functional integral
representations
for both simple random walk and some self-avoiding walk models,
including
models with strict self-avoidance, with weak self-avoidance, and a
model of
walks and loops. Our representation for the strictly self-avoiding
walk is new.
The representations have recently been used as the point of departure
for
rigorous renormalization group analyses of self-avoiding walk models in
dimension 4. For the models without loops, the integral
representations involve
fermions, and we also provide an introduction to fermionic integrals.
The
fermionic integrals are in terms of anti-commuting Grassmann
variables, which
can be conveniently interpreted as differential forms.
http://arxiv.org/abs/0906.0922
---------------------------------------------------------------
8593. DIRECTED POLYMERS ON HIERARCHICAL LATTICES WITH SITE DISORDER
Hubert Lacoin (PMA) and Gregorio Moreno Flores (PMA)
We study a polymer model on hierarchical lattices very close to the one
introduced and studied in \cite{DGr, CD}. For this model, we prove the
existence of free energy and derive the necessary and sufficient
condition for
which very strong disorder holds for all $\gb$, and give some accurate
results
on the behavior of the free energy at high-temperature. We obtain
these results
by using a combination of fractional moment method and change of
measure over
the environment to obtain an upper bound, and second moment method to
get a
lower bound. We also get lower bounds on the fluctuation exponent of $
\log
Z_n$, and study the infinite polymer measure in the weak disorder phase.
http://arxiv.org/abs/0906.0992
---------------------------------------------------------------
8594. APPLICATIONS OF STEIN'S METHOD FOR CONCENTRATION INEQUALITIES
Sourav Chatterjee and Partha S. Dey
Stein's method for concentration inequalities was introduced to prove
concentration of measure in problems involving complex dependencies
such as
random permutations and Gibbs measures. In this paper, we provide some
extensions of the theory and three applications: (1) We obtain a
concentration
inequality for the magnetization in the Curie-Weiss model at critical
temperature (where it obeys a non-standard normalization and super-
Gaussian
concentration). (2) We derive exact large deviation asymptotics for
the number
of triangles in the Erdos-Renyi random graph G(n,p) when p \ge 0.31.
Similar
results are derived also for general subgraph counts. (3) We obtain some
interesting concentration inequalities for the Ising model on lattices
that
hold at all temperatures.
http://arxiv.org/abs/0906.1034
---------------------------------------------------------------
8595. ON ASYMPTOTIC EFFICIENCY OF MULTIVARIATE VERSION OF SPEARMAN'S RHO
Alexander Nazarov and Natalia Stepanova
A multivariate version of Spearman's rho for testing independence is
considered. Its asymptotic efficiency is calculated under a general
distribution model specified by the dependence function. The efficiency
comparison study that involves other multivariate Spearman-type test
statistics
is made. Conditions for Pitman optimality of the test are established.
Examples
that illustrate the quality of the multivariate Spearman's test are
included.
http://arxiv.org/abs/0906.1059
---------------------------------------------------------------
8596. A SYMBOLIC COMPUTATIONAL APPROACH TO A PROBLEM INVOLVING
MULTIVARIATE POISSON DISTRIBUTIONS
Eduardo Sontag and Doron Zeilberger
Multivariate Poisson random variables subject to linear integer
constraints
arise in several application areas, such as queuing and biomolecular
networks.
This note shows how to compute conditional statistics in this context,
by
employing WF Theory and associated algorithms. A symbolic computation
package
has been developed and is made freely available. A discussion of
motivating
biomolecular problems is also provided.
http://arxiv.org/abs/0906.1141
---------------------------------------------------------------
8597. CONFORMAL INVARIANCE AND UNIVERSAL CRITICAL EXPONENTS IN THE
TWO-DIMENSIONAL PERCOLATION MODEL
Yu Zhang
For most two-dimensional critical percolation models, we show the
existence
of a scaling limit for the crossing probabilities in an isosceles right
triangle. Furthermore, by justifying the lattice, the scaling limit is a
conformal invariance satisfying Cardy's formula in Carleson's form.
Together
with the standard results of the $SLE_6$ process, we show that most
critical
exponents exist in the sense of universality predicted by physics on
most
two-dimensional lattices.
http://arxiv.org/abs/0906.1203
---------------------------------------------------------------
8598. A NOTE ON WIENER-HOPF FACTORIZATION FOR MARKOV ADDITIVE PROCESSES
Przemyslaw Klusik and Zbigniew Palmowski
We prove the Wiener-Hopf factorization for Markov Additive processes. We
derive also Spitzer-Rogozin theorem for this class of processes which
serves
for obtaining Kendall's formula and Fristedt representation of the
cumulant
matrix of the ladder epoch process. Finally, we also obtain the so-
called
ballot theorem.
http://arxiv.org/abs/0906.1223
---------------------------------------------------------------
8599. GLOBAL HEAT KERNEL ESTIMATES FOR FRACTIONAL LAPLACIANS IN
UNBOUNDED OPEN SETS
Zhen-Qing Chen and Joshua Tokle
In this paper, we derive global sharp heat kernel estimates for
symmetric
alpha-stable processes (or equivalently, for the fractional Laplacian
with zero
exterior condition) in two classes of unbounded C^{1,1} open sets in
R^d:
half-space-like open sets and exterior open sets. These open sets can be
disconnected. We focus in particular on explicit estimates for
p_D(t,x,y) for
all t>0 and x, y\in D. Our approach is based on the idea that for x
and y in
$D$ far from the boundary and t sufficiently large, we can compare
p_D(t,x,y)
to the heat kernel in a well understood open set: either a half-space
or R^d;
while for the general case we can reduce them to the above case by
pushing $x$
and $y$ inside away from the boundary. As a consequence, sharp Green
functions
estimates are obtained for the Dirichlet fractional Laplacian in these
two
types of open sets. Global sharp heat kernel estimates and Green
function
estimates are also obtained for censored stable processes (or
equivalently, for
regional fractional Laplacian) in exterior open sets.
http://arxiv.org/abs/0906.1234
---------------------------------------------------------------
8600. A RECURSIVE APPROACH FOR ALDOUS' SPECTRAL GAP CONJECTURE
Pietro Caputo and Thomas M. Liggett and Thomas Richthammer
Aldous' spectral gap conjecture asserts that on any graph the random
walk
process and the random transposition (or interchange) process have the
same
spectral gap. We describe a recursive strategy which could potentially
settle
the conjecture for arbitrary weighted graphs. The approach is a natural
extension of the recursive method already used to prove the validity
of the
conjecture on trees. The novelty is an idea based on electric network
reduction
which reduces the proof of the conjecture to the proof of an explicit
inequality for a random transposition operator involving both positive
and
negative rates. At present we are able to check the latter inequality
only in
some cases. This already allows us to prove the conjecture for the
class of
weighted graphs that can be reduced to a single edge by composing one-
vertex
network reductions where at each step at most 3 edges are removed. In
particular, this includes all weighted trees, cycles and many more
graphs with
small degree.
http://arxiv.org/abs/0906.1238
---------------------------------------------------------------
8601. COMPUTING EXPECTATIONS WITH CONTINUOUS P-BOXES: UNIVARIATE CASE
L. Utkin and S. Destercke
Given an imprecise probabilistic model over a continuous space,
computing
lower/upper expectations is often computationally hard to achieve,
even in
simple cases. Because expectations are essential in decision making
and risk
analysis, tractable methods to compute them are crucial in many
applications
involving imprecise probabilistic models. We concentrate on p-boxes (a
simple
and popular model), and on the computation of lower expectations of
non-monotone functions. This paper is devoted to the univariate case,
that is
where only one variable has uncertainty. We propose and compare two
approaches
: the first using general linear programming, and the second using the
fact
that p-boxes are special cases of random sets. We underline the
complementarity
of both approaches, as well as the differences.
http://arxiv.org/abs/0906.1260
---------------------------------------------------------------
8602. A L\'EVY AREA BY FOURIER NORMAL ORDERING FOR MULTIDIMENSIONAL
FRACTIONAL BROWNIAN MOTION WITH SMALL HURST INDEX
Jeremie Unterberger (IECN)
The main tool for stochastic calculus with respect to a multidimensional
process $B$ with small H\"older regularity index is rough path theory.
Once $B$
has been lifted to a rough path, a stochastic calculus -- as well as
solutions
to stochastic differential equations driven by $B$ -- follow by standard
arguments. Although such a lift has been proved to exist by abstract
arguments
\cite{LyoVic07}, a first general, explicit construction has been
proposed in
\cite{Unt09,Unt09bis} under the name of Fourier normal ordering. The
purpose of
this short note is to convey the main ideas of the Fourier normal
ordering
method in the particular case of the iterated integrals of lowest
order of
fractional Brownian motion with arbitrary Hurst index.
http://arxiv.org/abs/0906.1416
---------------------------------------------------------------
8603. TREND TO EQUILIBRIUM AND PARTICLE APPROXIMATION FOR A WEAKLY
SELFCONSISTENT VLASOV-FOKKER-PLANCK EQUATION
Francois Bolley (CEREMADE) and Arnaud Guillin and Florent Malrieu
(IRMAR)
We consider a Vlasov-Fokker-Planck equation governing the evolution of
the
density of interacting and diffusive matter in the space of positions
and
velocities. We use a probabilistic interpretation to obtain
convergence towards
equilibrium in Wasserstein distance with an explicit exponential rate.
We also
prove a propagation of chaos property for an associated particle
system, and
give rates on the approximation of the solution by the particle system.
Finally, a transportation inequality for the distribution of the
particle
system leads to quantitative deviation bounds on the approximation of
the
equilibrium solution of the equation by an empirical mean of the
particles at
given time.
http://arxiv.org/abs/0906.1417
---------------------------------------------------------------
8604. A WORLD RECORD AT AN ATLANTIC CITY CASINO AND THE DISTRIBUTION
OF THE LENGTH OF THE CRAPSHOOTER'S HAND
S. N. Ethier and Fred M. Hoppe
It was widely reported in the media that, on 23 May 2009, at the Borgata
Hotel Casino & Spa in Atlantic City, Patricia DeMauro, playing craps
for only
the second time, rolled the dice for four hours and 18 minutes, finally
sevening out at the 154th roll, a world record. Initial estimates of the
probability of this event were erroneous, but consensus was reached
within
days: one chance in 5.6 billion. More generally, what is P(L \ge n),
where the
random variable L denotes the length of the crapshooter's hand (154 in
Ms.
DeMauro's case) and n is a positive integer? It is well known that these
probabilities can be derived recursively or by Markov chain methods.
Our aim
here is to give an explicit closed-form expression for them, showing
that the
distribution of L is a linear combination (not a convex combination)
of four
geometric distributions.
http://arxiv.org/abs/0906.1545
---------------------------------------------------------------
8605. A $Q$-ANALOGUE OF THE FKG INEQUALITY AND SOME APPLICATIONS
Anders Bj\"orner
Let $L$ be a finite distributive lattice and $\mu : L \to {\mathbb
R}^{+}$ a
log-supermodular function. For functions $k: L \to {\mathbb R}^{+}$ let
$$E_{\mu} (k; q) \defeq \sum_{x\in L} k(x) \mu (x) q^{{\mathrm rank}
(x)} \in
{\mathbb R}^{+}[q].$$ We prove for any pair $g,h: L\to {\mathbb R}^{+}
$ of
monotonely increasing functions, that $$E_{\mu} (g; q)\cdot E_{\mu}
(h; q) \ll
E_{\mu} (1; q)\cdot E_{\mu} (gh; q), $$ where ``$ \ll $'' denotes
coefficientwise inequality of real polynomials. The FKG inequality of
Fortuin,
Kasteleyn and Ginibre (1971) is the real number inequality obtained by
specializing to $q=1$.
The polynomial FKG inequality has applications to $f$-vectors of
joins and
intersections of simplicial complexes, to Betti numbers of
intersections of
certain Schubert varieties, and to the following kind of correlation
inequality
for power series weighted by Young tableaux. Let $Y$ be the set of all
integer
partitions. Given functions $k, \mu: Y \rarr \R^+$, define the formal
power
series $$F_{\mu}(k ; z) \defeq \sum_{\la\in Y} k(\la) \mu(\la) f_{\la}
\frac{z^{|\la|}}{|\la| !} \in \R^+ [[z]],$$ where $f_{\la}$ is the
number of
standard Young tableaux of shape $\la$. Assume that $\mu: Y\rarr \R^+$
is
log-supermodular, and that $g, h: Y \rarr \R^+$ are monotonely
increasing with
respect to containment order of partition shapes. Then $$F_{\mu}(g;z)
\cdot
F_{\mu}(h;z) \ll F_{\mu}(1;z) \cdot F_{\mu}(gh;z). $$
http://arxiv.org/abs/0906.1389
---------------------------------------------------------------
8606. DYNAMIC RISK DIVERSIFICATION AND INSURANCE PREMIUM PRINCIPLES
Kei Fukuda and Akihiko Inoue and Yumiharu Nakano
We present an approach to the dynamic valuation of exposure risks in the
multi-period setting, which incorporates a dynamic and multiple
diversification
of risks in Pareto optimal sense. This approach extends classical
indifference
premium principles and can be applied for the valuation of insurance
risks. In
particular, our method produces explicit computation formulas for the
dynamic
version of the exponential premium principles. Moreover, we show limit
theorems
asserting that the risk loading for our valuation decreases to zero
when the
number of divisions of a risk goes to infinity.
http://arxiv.org/abs/0906.1632
---------------------------------------------------------------
8607. WEIGHTED POINCAR\'{E}-TYPE INEQUALITIES FOR CAUCHY AND OTHER
CONVEX MEASURES
Sergey G. Bobkov and Michel Ledoux
Brascamp--Lieb-type, weighted Poincar\'{e}-type and related analytic
inequalities are studied for multidimensional Cauchy distributions and
more
general $\kappa$-concave probability measures (in the hierarchy of
convex
measures). In analogy with the limiting (infinite-dimensional log-
concave)
Gaussian model, the weighted inequalities fully describe the measure
concentration and large deviation properties of this family of measures.
Cheeger-type isoperimetric inequalities are investigated similarly,
giving rise
to a common weight in the class of concave probability measures under
consideration.
http://arxiv.org/abs/0906.1651
---------------------------------------------------------------
8608. INFINITE PATHS IN RANDOM SHIFT GRAPHS
Matteo Novaga and Pietro Majer
We determine the probability tresholds for the existence of infinite
paths in
random shift graphs.
http://arxiv.org/abs/0906.1689
---------------------------------------------------------------
8609. ON SIMULTANEOUS HITTING OF MEMBRANES BY TWO SKEW BROWNIAN MOTIONS
Olga Aryasova and Andrey Pilipenko
We consider two depending Wiener processes which have membranes at
zero with
different permeability coefficients. Starting from different points, the
processes almost surely do not meet at any fixed point except that where
membranes are situated. The necessary and sufficient conditions for
the meeting
of the processes are found. It is shown that the probability of
meeting is
equal to zero or one.
http://arxiv.org/abs/0906.1695
---------------------------------------------------------------
8610. AN EXTENSION OF THE YAMADA-WATANABE CONDITION FOR PATHWISE
UNIQUENESS TO STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS
Reinhard Hoepfner
We extend the Yamada-Watanabe condition for pathwise uniqueness to
stochastic
differential equations with jumps, in the special case where small
jumps are
summable.
http://arxiv.org/abs/0906.1699
---------------------------------------------------------------
8611. INTERLACINGS FOR RANDOM WALKS ON WEIGHTED GRAPHS AND THE
INTERCHANGE PROCESS
A. B. Dieker
We study Aldous' conjecture that the spectral gap of the interchange
process
on a weighted undirected graph equals the spectral gap of the random
walk on
this graph. We present a conjecture in the form of an inequality, and
prove
that this inequality implies Aldous' conjecture by combining an
interlacing
result for Laplacians of random walks on weighted graphs with
representation
theory. We prove the conjectured inequality for several important
instances. As
an application of the developed theory, we prove Aldous' conjecture
for a large
class of weighted graphs, which includes all wheel graphs, all graphs
with four
vertices, certain nonplanar graphs, certain graphs with several
weighted cycles
of arbitrary length, as well as all trees.
http://arxiv.org/abs/0906.1716
---------------------------------------------------------------
8612. CLARK--OCONE FORMULA AND VARIATIONAL REPRESENTATION FOR POISSON
FUNCTIONALS
Xicheng Zhang
In this paper we first prove a Clark--Ocone formula for any bounded
measurable functional on Poisson space. Then using this formula, under
some
conditions on the intensity measure of Poisson random measure, we
prove a
variational representation formula for the Laplace transform of
bounded Poisson
functionals, which has been conjectured by Dupuis and Ellis [A Weak
Convergence
Approach to the Theory of Large Deviations (1997) Wiley], p. 122.
http://arxiv.org/abs/0906.1721
---------------------------------------------------------------
8613. ON SOME UNIVERSAL SIGMA FINITE MEASURES AND SOME EXTENSIONS OF
DOOB'S OPTIONAL STOPPING THEOREM
Joseph Najnudel and Ashkan Nikeghbali
In this paper, we associate, to any submartingale of class $(\Sigma)$,
defined on a filtered probability space $(\Omega, \mathcal{F},
\mathbb{P},
(\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical
conditions, a
$\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such
that for
all $t \geq 0$, and for all events $\Lambda_t
\in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] =
\mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is
the last
hitting time of zero of the process $X$. This measure $\mathcal{Q}$
has already
been defined in several particular cases, some of them are involved in
the
study of Brownian penalisation, and others are related with problems in
mathematical finance. More precisely, the existence of $\mathcal{Q}$
in the
general case solves a problem stated by D. Madan, B. Roynette and M.
Yor, in a
paper studying the link between Black-Scholes formula and last passage
times of
certain submartingales. Moreover, the equality defining $\mathcal{Q}$
remains
true if one replaces the fixed time $t$ by any bounded stopping time.
This
generalization can be viewed as an extension of Doob's optional stopping
theorem.
http://arxiv.org/abs/0906.1782
---------------------------------------------------------------
8614. UNIQUE DECOMPOSITIONS, FACES, AND AUTOMORPHISMS OF SEPARABLE
STATES
Erik Alfsen and Fred Shultz
We show that the set of separable states of length at most max(m,n) on
B(C^m
otimes C^n) admits an open dense set of states with unique
decomposition as a
convex combination of pure product states, and we describe all
possible convex
decompositions for a larger set of separable states. In both cases we
describe
the associated faces of the space of separable states, which in the
first case
are simplexes, and in the second case are direct convex sums of faces
that are
isomorphic to state spaces of full matrix algebras. As an application
of these
results, we characterize all affine automorphisms of the convex set of
separable states, and all automorphisms of the state space of B(C^m
otimes C^n)
that preserve entanglement and separability.
http://arxiv.org/abs/0906.1761
---------------------------------------------------------------
8615. A MODEL FOR SEXED COAGULATION
Raoul Normand
We consider in this work a model for aggregation, where the coalescing
particles initially have a certain number of potential links (called
arms)
which are used to perform coagulations. This model is sexed, is the
sense that
there are male and female arms: two particles may coagulate only if
one has an
available male arm, and the other has an available female arm. After a
coagulation, the used arms are no longer available. We are interested
in the
concentrations of the different types of particles, which are governed
by a
modification of Smoluchowski's coagulation equation -- that is, an
infinite
system of nonlinear differential equations. Using generating functions
and
solving a nonlinear PDE, we show that, up to some critical time, there
is a
unique solution to this equation. The Lagrange Inversion Formula
allows in some
cases to obtain explicit solutions, and to relate our model to two
recent
models for limited aggregation. We also show that, whenever the
critical time
is infinite, the concentrations converge to a state where all arms have
disappeared, and the distribution of the masses is related to the law
of the
size of some two-type Galton-Watson tree. Finally, we consider a
microscopic
model for coagulation: we construct a sequence of Marcus-Luschnikov
processes,
and show that it converges, before the critical time, to the solution
of our
modified Smoluchowski's equation.
http://arxiv.org/abs/0906.1773
---------------------------------------------------------------
8616. A TIME INHOMOGENEOUS COX-INGERSOLL-ROSS DIFFUSION WITH JUMPS
Reinhard Hoepfner
We consider a time inhomogeneous Cox-Ingersoll-Ross diffusion with
positive
jumps. We exploit a branching property to prove existence of a unique
strong
solution under a restrictive condition on the jump measure. We give
Laplace
transforms for the transition probabilities, with an interpretation in
terms of
limits of mixtures over Gamma laws.
http://arxiv.org/abs/0906.1856
---------------------------------------------------------------
8617. RANDOM QUANTUM CHANNELS II: ENTANGLEMENT OF RANDOM SUBSPACES,
RENYI ENTROPY ESTIMATES AND ADDITIVITY PROBLEMS
Beno\^it Collins (ICJ) and Ion Nechita (ICJ)
In this paper we obtain new bounds for the minimum output entropies of
random
quantum channels. These bounds rely on random matrix techniques
arising from
free probability theory. We then revisit the counterexamples developed
by
Hayden and Winter to get violations of the additivity equalities for
minimum
output R\'enyi entropies. We show that random channels obtained by
randomly
coupling the input to a qubit violate the additivity of the $p$-R\'enyi
entropy. For some sequences of random quantum channels, we compute
almost
surely the limit of their Schatten $S_1 \to S_p$ norms.
http://arxiv.org/abs/0906.1877
---------------------------------------------------------------
8618. RIESZ EXPONENTIAL FAMILIES ON HOMOGENEOUS CONES
Imen Boutouria and Abdelhamid Hassairi
In this paper, we introduce, for a multiplier $\chi$, a notion of
generalized
power function $x\mapsto \Delta_{\chi}(x),$ defined on the homogeneous
cone
${\mathcal{P}}$ of a Vinberg algebra ${\mathcal{A}}$. We then extend to
${\mathcal{A}}$ the famous Gindikin result, that is we determine the
set of
multipliers $\chi$ such that the map $\theta \mapsto \Delta_{\chi}
(\theta
^{-1})$, defined on ${\mathcal{P}}^{\ast}$, is the Laplace transform
of a
positive measure $R_{\chi}$. We also determine the set of $\chi $ such
that
$R_{\chi}$ generates an exponential family, and we calculate the
variance
function of this family
http://arxiv.org/abs/0906.1892
---------------------------------------------------------------
8619. THE REAL ZEROS OF A RANDOM POLYNOMIAL WITH DEPENDENT COEFFICIENTS
Jeffrey Matayoshi
Mark Kac gave one of the first results analyzing random polynomial
zeros. He
considered the case of independent standard normal coefficients and
was able to
show that the expected number of real zeros for a degree n polynomial
is on the
order of (2/pi)log(n), as n goes to infinity. Several years later,
Sambandham
considered two cases with some dependence assumed among the
coefficients. The
first case looked at coefficients with an exponentially decaying
covariance
function, while the second assumed a constant covariance. He showed
that the
expectation of the number of real zeros for an exponentially decaying
covariance matches the independent case, while having a constant
covariance
reduces the expected number of zeros in half. In this paper we will
apply
techniques similar to Sambandham's and extend his results to a wider
class of
covariance functions. Under certain restrictions on the spectral
density, we
will show that the order of the expected number of real zeros remains
the same
as in the independent case.
http://arxiv.org/abs/0906.1996
---------------------------------------------------------------
8620. BESSEL PROCESS AND CONFORMAL QUANTUM MECHANICS
M. A. Rajabpour
Different aspects of the connection between the Bessel process and the
conformal quantum mechanics (CQM) are discussed. The meaning of the
possible
generalizations of both models is investigated with respect to the
other model,
including self adjoint extension of the CQM. Some other
generalizations such as
the Bessel process in the wide sense and radial Ornstein- Uhlenbeck
process are
discussed with respect to the underlying conformal group structure.
http://arxiv.org/abs/0906.1728
---------------------------------------------------------------
8621. ANATOMY OF A YOUNG GIANT COMPONENT IN THE RANDOM GRAPH
Jian Ding and Jeong Han Kim and Eyal Lubetzky and Yuval Peres
We provide a complete description of the giant component of the
Erd\H{o}s-R\'enyi random graph $G(n,p)$ as soon as it emerges from the
scaling
window, i.e., for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to \infty$
and
$\epsilon=o(1)$.
Our description is particularly simple for $\epsilon = o(n^{-1/4})
$, where
the giant component $C_1$ is contiguous with the following model
(i.e., every
graph property that holds with high probability for this model also
holds
w.h.p. for $C_1$). Let $Z$ be normal with mean $\frac23 \epsilon^3 n$
and
variance $\epsilon^3 n$, and let $K$ be a random 3-regular graph on
$2\lfloor
Z\rfloor$ vertices. Replace each edge of $K$ by a path, where the path
lengths
are i.i.d. geometric with mean $1/\epsilon$. Finally, attach an
independent
Poisson($1-\epsilon$)-Galton-Watson tree to each vertex.
A similar picture is obtained for larger $\epsilon=o(1)$, in which
case the
random 3-regular graph is replaced by a random graph with $N_k$
vertices of
degree $k$ for $k\geq 3$, where $N_k$ has mean and variance of order
$\epsilon^k n$.
This description enables us to determine fundamental
characteristics of the
supercritical random graph. Namely, we can infer the asymptotics of the
diameter of the giant component for any rate of decay of $\epsilon$,
as well as
the mixing time of the random walk on $C_1$.
http://arxiv.org/abs/0906.1839
---------------------------------------------------------------
8622. DIAMETERS IN SUPERCRITICAL RANDOM GRAPHS VIA FIRST PASSAGE
PERCOLATION
Jian Ding and Jeong Han Kim and Eyal Lubetzky and Yuval Peres
We study the diameter of $C_1$, the largest component of the
Erd\H{o}s-R\'enyi random graph $\cG(n,p)$ emerging from the critical
window,
i.e., for $p = \frac{1+\epsilon}n$ where $\epsilon^3 n \to \infty$ and
$\epsilon=o(1)$. This parameter was extensively studied for fixed $
\epsilon >
0$, yet results for $\epsilon=o(1)$ outside the critical window were
only
obtained very recently: Riordan and Wormald gave precise estimates on
the
diameter, however these do not cover the entire supercritical regime
(namely,
when $\epsilon^3 n\to\infty$ arbitrarily slowly); {\L}uczak and
Seierstad
estimated its order throughout this regime, yet their upper and lower
bounds
differ by a factor of $\frac{1000}7$.
We show that for any $\epsilon=o(1)$ with $\epsilon^3n\to\infty$, the
diameter of $C_1$ is with high probability asymptotic to $D
(\epsilon,n)=(3/\epsilon)\log(\epsilon^3 n)$. We also prove that, in
this
regime, the diameter of the 2-core of $C_1$ is w.h.p. asymptotic to $
\frac23
D(\epsilon,n)$, and the maximal distance in it between any pair of
kernel
vertices is w.h.p. asymptotic to $\frac{5}9D(\epsilon,n)$.
The proofs rely on a recent structure result for the supercritical
giant
component, which reduces the problem of estimating distances between its
vertices to the study of passage times in first-passage percolation.
http://arxiv.org/abs/0906.1840
---------------------------------------------------------------
8623. VECTOR MEASURES OF BOUNDED GAMMA-VARIATION AND STOCHASTIC
INTEGRALS
Jan van Neerven and Lutz Weis
We introduce the class of vector measures of bounded $\gamma$-
variation and
study its relationship with vector-valued stochastic integrals with
respect to
Brownian motions.
http://arxiv.org/abs/0906.1883
---------------------------------------------------------------
8624. THE SUBELLIPTIC HEAT KERNEL ON SL(2,R): AN INTEGRAL
REPRESENTATION AND SOME FUNCTIONAL INEQUALITIES
Michel Bonnefont
In this paper, we study a subelliptic heat kernel on the Lie group
SL(2,R).
The subelliptic structure on SL(2,R) comes from the fibration $SO(2) -
> SL(2,R)
-> H^2$. First, we derive an integral representation for this heat
kernel. This
expression allows us to obtain some asymptotics in small time of this
heat
kernel and gives a way to compute the subriemannian distance. Then, we
establish some gradient estimates and some functional inequalities
like a
Li-Yau type estimate and a reverse Poincar\'e inequality.
http://arxiv.org/abs/0906.1977
---------------------------------------------------------------
8625. INVARIANT TRANSPORTS OF STATIONARY RANDOM MEASURES AND MASS-
STATIONARITY
G\"unter Last and Hermann Thorisson
We introduce and study invariant (weighted) transport-kernels balancing
stationary random measures on a locally compact Abelian group. The
first main
result is an associated fundamental invariance property of Palm
measures,
derived from a generalization of Neveu's exchange formula. The second
main
result is a simple sufficient and necessary criterion for the
existence of
balancing invariant transport-kernels. We then introduce (in a
nonstationary
setting) the concept of mass-stationarity with respect to a random
measure,
formalizing the intuitive idea that the origin is a typical location
in the
mass. The third main result of the paper is that a measure is a Palm
measure if
and only if it is mass-stationary.
http://arxiv.org/abs/0906.2062
---------------------------------------------------------------
8626. RATIONAL BEHAVIOUR IN THE PRESENCE OF STOCHASTIC PERTURBATIONS
Panayotis Mertikopoulos and Aris L. Moustakas
We study repeated games where players employ an exponential learning
scheme
in order to adapt to an ever-changing environment. If the game's
payoffs are
subject to random perturbations, this scheme leads to a new stochastic
version
of the replicator dynamics that is quite different from the "aggregate
shocks"
approach of evolutionary game theory. Irrespective of the perturbations'
magnitude, we find that strategies which are dominated (even
iteratively)
eventually become extinct and that the game's strict Nash equilibria are
stochastically asymptotically stable. We complement our analysis by
illustrating these results in the case of congestion games.
http://arxiv.org/abs/0906.2094
---------------------------------------------------------------
8627. DE FINETTI'S DIVIDEND PROBLEM AND IMPULSE CONTROL FOR A TWO-
DIMENSIONAL INSURANCE RISK PROCESS
Irmina Czarna and Zbigniew Palmowski
Consider two insurance companies (or two branches of the same company)
that
have the same claims and they divide premia in some specified
proportions. We
model the occurrence of claims according to a Poisson process. The
ruin is
achieved if the corresponding two-dimensional risk process first leave
the
positive quadrant. We consider different kinds of linear barriers. We
will
consider two scenarios of controlled process. In first one when two-
dimensional
risk process hits the barrier the minimal amount of dividends is payed
out to
keep the risk process within the region bounded by the barrier. In the
second
scenario whenever process hits horizontal line, the risk process is
reduced by
paying dividend to some fixed point in the positive quadrant and waits
there
for the first claim to arrive. In both models we calculate discounted
cumulative dividend payments until the ruin time.
http://arxiv.org/abs/0906.2100
---------------------------------------------------------------
8628. VARIATIONAL CHARACTERISATION OF GIBBS MEASURES WITH DELAUNAY
TRIANGLE INTERACTION
David Dereudre and Hans-Otto Georgii
This paper deals with stationary Gibbsian point processes on the plane
with
an interaction that depends on the tiles of the Delaunay triangulation
of
points via a bounded triangle potential. It is shown that the class of
these
Gibbs processes includes all minimisers of the associated free energy
density
and is therefore nonempty. Conversely, each such Gibbs process
minimises the
free energy density, provided the potential satisfies a weak long-range
assumption.
http://arxiv.org/abs/0906.2153
---------------------------------------------------------------
8629. INFINITESIMAL NON-CROSSING CUMULANTS AND FREE PROBABILITY OF
TYPE B
Maxime Fevrier and Alexandru Nica
Free probabilistic considerations of type B first appeared in a paper by
Biane, Goodman and Nica in 2003. Recently, connections between type B
and
infinitesimal free probability were put into evidence by Belinschi and
Shlyakhtenko (arXiv:0903.2721). The interplay between "type B" and
"infinitesimal" is also the object of the present paper. We study
infinitesimal
freeness for a family of unital subalgebras A_1, ..., A_k in an
infinitesimal
noncommutative probability space (A, phi, phi'), and we introduce a
concept of
infinitesimal non-crossing cumulant functionals for (A, phi, phi'),
obtained by
taking a formal derivative in the formula for usual non-crossing
cumulants. We
prove that the infinitesimal freeness of A_1, ... A_k is equivalent to a
vanishing condition for mixed cumulants; this gives the infinitesimal
counterpart for a theorem of Speicher from "usual" free probability.
We show
that the lattices of non-crossing partitions of type B appear in the
combinatorial study of (A, phi, phi'), in the formulas for infinitesimal
cumulants and when describing alternating products of infinitesimally
free
random variables. As an application of alternating free products, we
observe
the infinitesimal analogue for the well-known fact that freeness is
preserved
under compression with a free projection. As another application, we
observe
the infinitesimal analogue for a well-known procedure used to
construct free
families of free Poisson elements. Finally, we discuss situations when
the
freeness of A_1, ..., A_k in (A, phi) can be naturally upgraded to
infinitesimal freeness in (A, phi, phi'), for a suitable choice of a
"companion
functional" phi'.
http://arxiv.org/abs/0906.2017
---------------------------------------------------------------
8630. CONVERGENCE RATES OF THE SPLITTING SCHEME FOR PARABOLIC LINEAR
STOCHASTIC CAUCHY PROBLEMS
Sonja Cox and Jan van Neerven
We study the splitting scheme associated with the linear stochastic
Cauchy
problem dU(t) = AU(t) dt + dW(t), where A is the generator of an
analytic
C_0-semigroup S={S(t)} on a Banach space E and W={W(t)} is a Brownian
motion
with values in a fractional domain space E_\b associated with A. We
prove that
if \a,\b,\g,\th \ge 0 are such that \g + \th < 1 and max[0,(\a-\b+
\th)] + \g <
1/2, then the approximate solutions U_n (where n is the number of time
steps)
converge to the solution U in the Holder space C^\g([0,T];E_\a), both in
L^p-means and almost surely, with rate 1/n^\th.
http://arxiv.org/abs/0906.2129
---------------------------------------------------------------
8631. ASYMPTOTIC RESULTS FOR THE TWO-PARAMETER POISSON-DIRICHLET
DISTRIBUTION
Shui Feng and Fuqing Gao
The two-parameter Poisson-Dirichlet distribution is the law of a
sequence of
decreasing nonnegative random variables with total sum one. It can be
constructed from stable and Gamma subordinators with the two-parameters,
$\alpha$ and $\theta$, corresponding to the stable component and Gamma
component respectively. The moderate deviation principles are
established for
the two-parameter Poisson-Dirichlet distribution and the corresponding
homozygosity when $\theta$ approaches infinity, and the large deviation
principle is established for the two-parameter Poisson-Dirichlet
distribution
when both $\alpha$ and $\theta$ approach zero.
http://arxiv.org/abs/0906.2217
---------------------------------------------------------------
8632. ON THE EXPECTATIONS OF MAXIMA OF SETS OF INDEPENDENT RANDOM
VARIABLES
D. V. Tokarev and K. A. Borovkov
Let $X^1, ..., X^k$ and $Y^1, ..., Y^m$ be jointly independent copies of
random variables $X$ and $Y$, respectively. For a fixed total number $n
$ of
random variables, we aim at maximising $M(k,m):= E \max \{X^1, ...,
X^k, Y^1,
>..., Y^{m} \}$ in $k = n-m\ge 0$, which corresponds to maximising
the expected
lifetime of an $n$-component parallel system whose components can be
chosen
from two different types. We show that the lattice $\{M(k,m): k, m\ge
0\}$ is
concave, give sufficient conditions on $X$ and $Y$ for M(n,0) to be
always or
ultimately maximal and derive a bound on the number of sign changes in
the
sequence $M(n,0)-M(0,n)$, $n\ge 1$. The results are applied to a mixed
population of Bienayme-Galton-Watson processes, with the objective to
derive
the optimal initial composition to maximise the expected time to
extinction.
http://arxiv.org/abs/0906.2270
---------------------------------------------------------------
8633. COEXISTENCE IN STOCHASTIC SPATIAL MODELS
Rick Durrett
In this paper I will review twenty years of work on the question: When
is
there coexistence in stochastic spatial models? The answer, announced in
Durrett and Levin [Theor. Pop. Biol. 46 (1994) 363--394], and that we
explain
in this paper is that this can be determined by examining the mean-
field ODE.
There are a number of rigorous results in support of this picture, but
we will
state nine challenging and important open problems, most of which date
from the
1990's.
http://arxiv.org/abs/0906.2293
---------------------------------------------------------------
8634. CRITICALLY LOADED QUEUEING MODELS THAT ARE THROUGHPUT SUBOPTIMAL
Rami Atar and Gennady Shaikhet
This paper introduces and analyzes the notion of throughput
suboptimality for
many-server queueing systems in heavy traffic. The queueing model under
consideration has multiple customer classes, indexed by a finite set
$\mathcal{I}$, and heterogenous, exponential servers. Servers are
dynamically
chosen to serve customers, and buffers are available for customers
waiting to
be served. The arrival rates and the number of servers are scaled up
in such a
way that the processes representing the number of class-$i$ customers
in the
system, $i\in\mathcal{I}$, fluctuate about a static fluid model, that is
assumed to be critically loaded in a standard sense. At the same time,
the
fluid model is assumed to be throughput suboptimal. Roughly, this
means that
the servers can be allocated so as to achieve a total processing rate
that is
greater than the total arrival rate. We show that there exists a dynamic
control policy for the queueing model that is efficient in the
following strong
sense: Under this policy, for every finite $T$, the measure of the set
of times
prior to $T$, at which at least one customer is in the buffer,
converges to
zero in probability as the arrival rates and number of servers go to
infinity.
On the way to prove our main result, we provide a characterization of
throughput suboptimality in terms of properties of the buffer-station
graph.
http://arxiv.org/abs/0906.2305
---------------------------------------------------------------
8635. NO ARBITRAGE WITHOUT SEMIMARTINGALES
Robert A. Jarrow and Philip Protter and Hasanjan Sayit
We show that with suitable restrictions on allowable trading
strategies, one
has no arbitrage in settings where the traditional theory would admit
arbitrage
possibilities. In particular, price processes that are not
semimartingales are
possible in our setting, for example, fractional Brownian motion.
http://arxiv.org/abs/0906.2318
---------------------------------------------------------------
8636. PORTFOLIO CHOICE WITH JUMPS: A CLOSED-FORM SOLUTION
Yacine A\"it-Sahalia and Julio Cacho-Diaz and T. R. Hurd
We analyze the consumption-portfolio selection problem of an investor
facing
both Brownian and jump risks. We bring new tools, in the form of
orthogonal
decompositions, to bear on the problem in order to determine the optimal
portfolio in closed form. We show that the optimal policy is for the
investor
to focus on controlling his exposure to the jump risk, while exploiting
differences in the Brownian risk of the asset returns that lies in the
orthogonal space.
http://arxiv.org/abs/0906.2324
---------------------------------------------------------------
8637. THE ASYMPTOTIC DISTRIBUTION OF A CLUSTER-INDEX FOR I.I.D. NORMAL
RANDOM VARIABLES
Yannis G. Yatracos
In a sample variance decomposition, with components functions of the
sample's
spacings, the largest component $\tilde{I}_n$ is used in cluster
detection. It
is shown for normal samples that the asymptotic distribution of $
\tilde{I}_n$
is the Gumbel distribution.
http://arxiv.org/abs/0906.2334
---------------------------------------------------------------
8638. CONDITIONS FOR RAPID MIXING OF PARALLEL AND SIMULATED TEMPERING
ON MULTIMODAL DISTRIBUTIONS
Dawn B. Woodard and Scott C. Schmidler and Mark Huber
We give conditions under which a Markov chain constructed via parallel
or
simulated tempering is guaranteed to be rapidly mixing, which are
applicable to
a wide range of multimodal distributions arising in Bayesian statistical
inference and statistical mechanics. We provide lower bounds on the
spectral
gaps of parallel and simulated tempering. These bounds imply a single
set of
sufficient conditions for rapid mixing of both techniques. A direct
consequence
of our results is rapid mixing of parallel and simulated tempering for
several
normal mixture models, and for the mean-field Ising model.
http://arxiv.org/abs/0906.2341
---------------------------------------------------------------
8639. ERROR BOUNDS FOR COMPUTING THE EXPECTATION BY MARKOV CHAIN MONTE
CARLO
Daniel Rudolf
We study the error of reversible Markov chain Monte Carlo methods for
approximating the expectation of a function. Explicit error bounds
with respect
to different norms of the function are proven. By the estimation the
well known
asymptotical limit of the error is attained, i.e. there is no gap
between the
estimate and the asymptotical behavior. We discuss the dependence of
the error
on a burn-in of the Markov chain. Furthermore we suggest and justify a
specific
burn-in for optimizing the algorithm.
http://arxiv.org/abs/0906.2359
---------------------------------------------------------------
8640. OPTIMAL PORTFOLIO LIQUIDATION WITH EXECUTION COST AND RISK
Idris Kharroubi (PMA and Crest) and Huyen Pham (PMA and Crest)
We study the optimal portfolio liquidation problem over a finite
horizon in a
limit order book with bid-ask spread and temporary market price impact
penalizing speedy execution trades. We use a continuous-time modeling
framework, but in contrast with previous related papers (see e.g. [24]
and
[25]), we do not assume continuous-time trading strategies. We
consider instead
real trading that occur in discrete-time, and this is formulated as an
impulse
control problem under a solvency constraint, including the lag variable
tracking the time interval between trades. A first important result of
our
paper is to show that nearly optimal execution strategies in this
context lead
actually to a finite number of trading times, and this holds true
without
assuming ad hoc any fixed transaction fee. Next, we derive the dynamic
programming quasi-variational inequality satisfied by the value
function in the
sense of constrained viscosity solutions. We also introduce a family
of value
functions converging to our value function, and which is characterized
as the
unique constrained viscosity solutions of an approximation of our
dynamic
programming equation. This convergence result is useful for numerical
purpose,
postponed in a further study.
http://arxiv.org/abs/0906.2565
---------------------------------------------------------------
8641. A FLUCTUATION LIMIT THEOREM OF BRANCHING PROCESSES WITH
IMMIGRATION AND STATISTICAL APPLICATIONS
Chunhua Ma
We prove a general fluctuation limit theorem for Galton-Watson branching
processes with immigration. The limit is a time-inhomogeneous OU type
process
driven by a spectrally positive Levy process. As applications of this
result,
we obtain some asymptotic estimates for the conditional least-squares
estimator
of the offspring mean.
http://arxiv.org/abs/0906.2586
---------------------------------------------------------------
8642. CHARACTERISTIC POLYNOMIALS OF SAMPLE COVARIANCE MATRICES
Holger K\"osters
We investigate the second-order correlation function of the
characteristic
polynomial of a sample covariance matrix. Starting from an explicit
formula for
the generating function, we re-obtain several well-known kernels from
random
matrix theory.
http://arxiv.org/abs/0906.2763
---------------------------------------------------------------
8643. HOT SCATTERERS AND TRACERS FOR THE TRANSFER OF HEAT IN
COLLISIONAL DYNAMICS
Raphael Lefevere and Lorenzo Zambotti
We introduce stochastic models for the transport of heat in systems
described
by local collisional dynamics. The dynamics consists of tracer
particles moving
through an array of hot scatterers describing the effect of heat baths
at fixed
temperatures. Those models have the structure of Markov renewal
processes. We
study their ergodic properties in details and provide a useful formula
for the
cumulant generating function of the time integrated energy current. We
observe
that out of thermal equilibrium, the generating function is not
analytic. When
the set of temperatures of the scatterers is fixed by the condition
that in
average no energy is exchanged between the scatterers and the system,
different
behaviours may arise. When the tracer particles are allowed to travel
freely
through the whole array of scatterers, the temperature profile is
linear. If
the particles are locked in between scatterers, the temperature
profile becomes
nonlinear. In both cases, the thermal conductivity is interpreted as a
frequency of collision between tracers and scatterers.
http://arxiv.org/abs/0904.0020
---------------------------------------------------------------
8644. ALGORITHMIC INFORMATION THEORY AND MARTINGALES
Laurent Bienvenu and Alexander Shen
The notion of an individual random sequence goes back to von Mises. We
describe the evolution of this notion, especially the use of martingales
(suggested by Ville), and the development of algorithmic information
theory in
1960s and 1970s (Solomonov, Kolmogorov, Martin-Lof, Levin, Chaitin,
Schnorr and
others). We conclude with some remarks about the use of the algorithmic
information theory in the foundations of probability theory.
http://arxiv.org/abs/0906.2614
---------------------------------------------------------------
8645. RESISTANCE BOUNDARIES OF INFINITE NETWORKS
Palle E. T. Jorgensen and Erin P. J. Pearse
A resistance network is a connected graph $(G,c)$ with edges (and edge
weights) determined by the conductance function $c_{xy}$. The
Dirichlet energy
form $\mathcal E$ produces a Hilbert space structure (which we call
the energy
space ${\mathcal H}_{\mathcal E}$) on the space of functions of finite
energy.
In a previous paper, we constructed a reproducing kernel $\{v_x\}$ for
this
Hilbert space and used it to prove a discrete Gauss-Green identity \
[{\mathcal
E}(u,v) = \sum_{G} u \Delta v + \sum_{\operatorname{bd}G} u
\frac{\partial}{\partial \mathbf{n}} v,\] where the latter sum is
understood in
a limiting sense. Applying this formula to a harmonic function $u \in
{\mathcal
H}_{\mathcal E}$ gives a boundary representation \[u(x) =
\sum_{\operatorname{bd}G} u \frac{\partial}{\partial \mathbf{n}} v +
u(o),\]
where $o$ is a fixed reference vertex.
In this paper, we use techniques from stochastic integration to
make the
boundary $\operatorname{bd}G$ precise as a measure space, and replace
the
latter formula with a boundary integral representation (in a sense
analogous to
that of Poisson or Martin boundary theory). We construct a Gel'fand
triple $S
\ci {\mathcal H}_{\mathcal E} \ci S'$ and obtain a probability measure
$\mathbb{P}$ and an isometric embedding of ${\mathcal H}_{\mathcal E}$
into
$L^2(S',\mathbb{P})$. This gives a concrete representation of the
boundary as a
certain subset of $S'$.
http://arxiv.org/abs/0906.2745
---------------------------------------------------------------
8646. A SOLVABLE MODEL FOR HOMOPOLYMERS AND THE CRITICAL PHENOMENA
M. Cranston and L. Koralov and S. Molchanov and B. Vainberg
We consider a model for the distribution of a long homopolymer with a
zero-range potential at the origin in $\mathbb{R}^3$. The distribution
can be
obtained as a limit of Gibbs distributions corresponding to properly
normalized
potentials concentrated in small neighborhoods of the origin as the
size of the
neighborhoods tends to zero. The distribution depends on the length $T
$ of the
polymer and a parameter $\gamma$ that corresponds, roughly speaking,
to the
difference between the inverse temperature in our model and the
critical value
of the inverse temperature.
At the critical point $\gamma_{cr} = 0$ the transition occurs from
the
globular phase (positive recurrent behavior of the polymer, $\gamma >
0$) to
the extended phase (Brownian type behavior, $\gamma < 0$). The main
result of
the paper is a detailed analysis of the behavior of the polymer when $
\gamma$
is near $\gamma_{cr}$.
Our approach is based on analyzing the semigroups generated by the
self-adjoint extensions $\mathcal{L}_\gamma$ of the Laplacian on
$C_0^\infty(\mathbb{R}^3 \setminus \{0\})$ parametrized by $\gamma$,
which are
related to the distribution of the polymer. The main technical tool of
the
paper is the explicit formula for the resolvent of the operator
$\mathcal{L}_\gamma$.
http://arxiv.org/abs/0906.2816
---------------------------------------------------------------
8647. A THREE-PARAMETER BINOMIAL APPROXIMATION
Vydas \v{C}ekanavi\v{c}ius and Erol A. Pek\"oz and Adrian R\"ollin
and Michael Shwartz
We approximate the distribution of the sum of independent but not
necessarily
identically distributed Bernoulli random variables using a shifted
binomial
distribution where the three parameters (the number of trials, the
probability
of success, and the shift amount) are chosen to match up the first three
moments of the two distributions. We give a bound on the approximation
error in
terms of the total variation metric using Stein's method. A numerical
study is
discussed that shows shifted binomial approximations typically are more
accurate than Poisson or standard binomial approximations. The
application of
the approximation to solving a problem arising in Bayesian hierarchical
modeling is also discussed.
http://arxiv.org/abs/0906.2855
---------------------------------------------------------------
8648. INFINITE VARIANCE STABLE LIMITS FOR SUMS OF DEPENDENT RANDOM
VARIABLES
Katarzyna Bartkiewicz and Adam Jakubowski and Thomas Mikosch and
Olivier Wintenberger (CEREMADE)
The aim of this paper is to provide conditions which ensure that the
affinely
transformed partial sums of a strictly stationary process converge in
distribution to an in?nite variance stable distribution. Conditions
for this
convergence to hold are known in the literature. However, most of
these results
are qualitative in the sense that the parameters of the limit
distribution are
expressed in terms of some limiting point process. In this paper we
will be
able to determine the parameters of the limiting stable distribution
in terms
of some tail characteristics of the underlying stationary sequence. We
will
apply our results to some standard time series models, including the
GARCH(1,
1) process and its squares, the stochastic volatility models and
solutions to
stochastic recurrence equations.
http://www.arxiv.org
---------------------------------------------------------------
8649. MATHEMATICAL ANALYSIS OF STOCHASTIC MODELS FOR TUMOR-IMMUNE
SYSTEMS
O. Chis and D. Opris
In this paper we investigate some stochastic models for tumor-immune
systems.
To describe these models we used a Wiener process, as the noise has a
stabilization effect. Their dynamics are studied in terms of stochastic
stability in the equilibrium points, by constructing the Lyapunov
exponent,
depending on the parameters that describe the model. We have studied and
analyzed a Kuznetsov-Taylor like stochastic model and a Bell
stochastic model
for tumor-immune systems. These stochastic models are studied from
stability
point of view and they were represented using the Euler second order
scheme.
http://arxiv.org/abs/0906.2794
---------------------------------------------------------------
8650. REGULAR SETS AND COUNTING IN FREE GROUPS
Elizaveta Frenkel and Alexei G. Myasnikov and Vladimir N.
Remeslennikov
In this paper we study asymptotic behavior of regular subsets in a
free group
F of finite rank, compare their sizes at infinity, and develop
techniques to
compute the probabilities of sets relative to distributions on F that
come
naturally from no-return random walks on the Cayley graph of F. We
apply these
techniques to study cosets, double cosets, and Schreier
representatives of
finitely generated subgroups of F and also to analyze relative sizes
of regular
prefixed-closed subsets in F.
http://arxiv.org/abs/0906.2850
---------------------------------------------------------------
8651. A FINITE DIFFERENCE APPROACH TO THE INFINITY LAPLACE EQUATION
AND TUG-OF-WAR GAMES
Scott N. Armstrong and Charles K. Smart
We present a modified version of the $\epsilon$-step tug-of-war game
recently
introduced by Peres, Schramm, Sheffield, and Wilson. This new tug-of-
war game
is identical to the original except near the boundary of the domain $
\partial
\Omega$, but its associated value functions are more regular. Using
the dynamic
programming principle, we show that the value functions satisfy a
certain
finite difference equation. By studying solutions of this difference
equation
directly, we are able to adapt techniques from viscosity solution
theory to
prove a number of new results.
The finite difference equation has unique maximal and minimal
solutions,
which are identified as the value functions for the two tug-of-war
players. We
demonstrate uniqueness, and hence the existence of a value for the
game, in the
case that the running payoff function is nonnegative. In the limit $
\epsilon
\to 0$, we obtain the convergence of the value functions to a viscosity
solution of the normalized infinity Laplace equation.
We also obtain several new results for the normalized infinity
Laplace
equation $-\Delta_\infty u = f$. In particular, we demonstrate the
existence of
solutions to the Dirichlet problem for any $f \in C(\Omega) \cap
L^\infty(\Omega)$ and continuous boundary data, as well as the
uniqueness of
solutions to this problem in the generic case. We also give a new
elementary
proof of Jensen's theorem on the uniqueness of infinity harmonic
functions.
http://arxiv.org/abs/0906.2871
---------------------------------------------------------------
8652. UNIQUELY ERGODIC MINIMAL TILING SPACES WITH POSITIVE ENTROPY
Ian Palmer and Jean Bellissard
Strictly ergodic spaces of tilings with positive entropy are constructed
using tools from information and probability theory. Statistical
estimates are
made to create a one-dimensional subshift with these dynamical
properties,
yielding a space of repetitive tilings of R^D wit finite local
complexity that
is also equivalent to a symbolic dynamical system with a Z^D action.
http://arxiv.org/abs/0906.2997
---------------------------------------------------------------
8653. ON THE DENSITY OF THE SUM OF TWO INDEPENDENT STUDENT T-RANDOM
VECTORS
C. Berg and C. Vignat
In this paper, we find an expression for the density of the sum of two
independent $d-$dimensional Student $t-$random vectors $\mathbf{X}$ and
$\mathbf{Y}$ with arbitrary degrees of freedom. As a byproduct we also
obtain
an expression for the density of the sum $\mathbf{N}+\mathbf{X}$, where
$\mathbf{N}$ is normal and $\mathbf{X}$ is an independent Student $t-
$vector.
In both cases the density is given as an infinite series \
[ \sum_{n=0}^{\infty}
c_{n}f_{n} \] where $f_{n}$ is a sequence of probability densities on
$\mathbb{R}^{d}$ and $(c_{n} )$ is a sequence of positive numbers of
sum 1,
i.e. the distribution of a non-negative integer-valued random variable
$C$,
which turns out to be infinitely divisible for $d=1$ and $d=2.$ When
$d=1$ and
the degrees of freedom of the Student variables are equal, we recover
an old
result of Ruben.
http://arxiv.org/abs/0906.3037
---------------------------------------------------------------
8654. FEYNMAN-KAC FORMULA FOR HEAT EQUATION DRIVEN BY FRACTIONAL WHITE
NOISE
Yaozhong Hu and David Nualart and Jian Song
In this paper we establish a version of the Feynman-Kac formula for the
stochastic heat equation with a multiplicative fractional Brownian
sheet. We
prove the smoothness of the density of the solution, and the H\"older
regularity in the space and time variables.
http://arxiv.org/abs/0906.3076
---------------------------------------------------------------
8655. ZERO DISSIPATION LIMIT IN THE ABELIAN SANDPILE MODEL
Antal A.Jarai and F. Redig and E. Saada
We study the abelian avalanche model, an analogue of the abelian
sandpile
model with continuous heights, which allows for arbitrary small values
of
dissipation. We prove that for non-zero dissipation, the infinite
volume limit
of the stationary measures of the abelian avalanche model exists and
can be
obtained via a weighted spanning tree measure. Moreover we obtain
exponential
decay of spatial covariances of local observables in the non-zero
dissipation
regime. We then study the zero dissipation limit and prove that the
self-organized critical model is recovered, both for the stationary
measures
and for the dynamics.
http://arxiv.org/abs/0906.3128
---------------------------------------------------------------
8656. FIXED POINTS OF THE MIN-TRANSFORMATION
Matthias Meiners and Gerold Alsmeyer
In this paper, the stochastic fixed-point equation X \stackrel{d}{=}
\inf_{i
\geq 1: T_i > 0} X_i/T_i is studied, where $\stackrel{d}{=}$ means
equality in
law, $(X_{n})_{n\ge 1}$ and $T = (T_i)_{i \geq 1}$ denote independent
sequences
of non-negative random variables, and $X_1, X_2, ...$ are independent
copies of
the random variable $X$. Under suitable conditions on the given weight
sequence
$T$, most notably that $1<\E N<\infty$ for $N=\sum_{i\ge 1}\1_{\
{T_{i}>0\}}$
and that $T$ has a characteristic exponent, defined as the minimal $
\alpha>0$
such that $\sum_{i\ge 1}\E T_{i}^{\alpha}=1$, we determine the set of
all
solutions to the equation, viz. all distributions of $X$ such that the
distributional identity holds true. These turn out to be certain
mixtures of
Weibull distributions (or periodic variants). This extends earlier
results by
Alsmeyer and R\"osler for the case of deterministic $T$ and by the
present
authors who showed (under weaker assumptions on $T$) that these
distributions
are the only ones within a certain subclass of distributions on
$[0,\infty)$.
It further extends results by Durrett and Liggett and by Liu on the
related
equation $X \stackrel{d}{=} \sum_{i \geq 1} T_i X_i$ after the
observation that
any Laplace transform of a solution to this latter equation may also
be viewed
as the survival function of a solution to the above min-type equation.
http://arxiv.org/abs/0906.3133
---------------------------------------------------------------
8657. FIXATION FOR DISTRIBUTED CLUSTERING PROCESSES
Marcelo R. Hilario and Oren Louidor and Charles M. Newman and
Leonardo T. Rolla, Scott Sheffield, Vladas Sidoravicius
We study a discrete-time resource flow in $Z^d$, where wealthier
vertices
attract the resources of their less rich neighbors. For any
translation-invariant probability distribution of initial resource
quantities,
we prove that the flow at each vertex terminates after finitely many
steps.
This answers (a generalized version of) a question posed by Van den
Berg and
Meester in 1991. The proof uses the mass-transport principle and
extends to
other graphs.
http://arxiv.org/abs/0906.3154
---------------------------------------------------------------
8658. ESTIMATION FOR THE CHANGE POINT OF THE VOLATILITY IN A
STOCHASTIC DIFFERENTIAL EQUATION
Stefano M. Iacus and Nakahiro Yoshida
We consider a multidimensional It\^o process $Y=(Y_t)_{t\in[0,T]}$
with some
unknown drift coefficient process $b_t$ and volatility coefficient
$\sigma(X_t,\theta)$ with covariate process $X=(X_t)_{t\in[0,T]}$, the
function
$\sigma(x,\theta)$ being known up to $\theta\in\Theta$. For this model
we
consider a change point problem for the parameter $\theta$ in the
volatility
component. The change is supposed to occur at some point $t^*\in (0,T)
$. Given
discrete time observations from the process $(X,Y)$, we propose quasi-
maximum
likelihood estimation of the change point. We present the rate of
convergence
of the change point estimator and the limit thereoms of aymptotically
mixed
type.
http://arxiv.org/abs/0906.3108
---------------------------------------------------------------
8659. SPECTRAL PROPERTIES OF THE CAUCHY PROCESS
Tadeusz Kulczycki and Mateusz Kwa\'snicki and Jacek Ma{\l}ecki and
Andrzej Stos
We study the spectral properties of the transition semigroup of the
killed
one-dimensional Cauchy process on the half-line (0,infty) and the
interval
(-1,1). This process is related to the square root of one-dimensional
Laplacian
A = -sqrt(-d^2/dx^2) with a Dirichlet exterior condition (on a
complement of a
domain), and to a mixed Steklov problem in the half-plane. For the
half-line,
an explicit formula for generalized eigenfunctions psi_lambda of A is
derived,
and then used to construct spectral representation of A. Explicit
formulas for
the transition density of the killed Cauchy process in the half-line
(or the
heat kernel of A in (0,infty)), and for the distribution of the first
exit time
from the half-line follow. The formula for psi_lambda is also used to
construct
approximations to eigenfunctions of A in the interval. For the
eigenvalues
lambda_n of A in the interval the asymptotic formula lambda_n = n pi/2
- pi/8 +
O(1/n) is derived, and all eigenvalues lambda_n are proved to be simple.
Finally, efficient numerical methods of estimation of eigenvalues
lambda_n are
applied to obtain lower and upper numerical bounds for the first few
eigenvalues up to 9th decimal point.
http://arxiv.org/abs/0906.3113
---------------------------------------------------------------
8660. AFFINE PROCESSES ARE REGULAR
Martin Keller-Ressel and Walter Schachermayer and Josef Teichmann
We show that stochastically continuous, time-homogeneous affine
processes on
the canonical state space $\mathbb{R}_{\geq 0}^m \times \mathbb{R}^n$
are
always regular. In the paper of Duffie-Filipovi\'c-Schachermayer (2003)
regularity was used as a crucial basic assumption. However, a
counterexample of
a non-regular, stochastically continuous affine process was neither
known nor
expected. We now show that the regularity assumption is indeed
superfluous,
since regularity follows from stochastic continuity and the
exponentially
affine behavior of the characteristic function. For the proof we combine
classic results on the differentiability of transformation semigroups
with the
method of the moving frame which has been recently found to be useful
in the
theory of SPDEs.
http://arxiv.org/abs/0906.3392
---------------------------------------------------------------
8661. AROUND TSIRELSON'S EQUATION, OR: THE EVOLUTION PROCESS MAY NOT
EXPLAIN EVERYTHING
Kouji Yano and Marc Yor
We present a synthesis of a number of developments which have been made
around the celebrated Tsirelson's equation (1975), conveniently
modified in the
framework of a Markov chain taking values in a compact group $ G $,
and indexed
by negative time. To illustrate, we discuss in detail the case of the
one-dimensional torus $ G=\bT $.
http://arxiv.org/abs/0906.3442
---------------------------------------------------------------
8662. RATE OF CONVERGENCE FOR NUMERICAL SOLUTIONS TO SFDES WITH JUMPS
Jianhai Bao and Xuerong Mao and Chenggui Yuan
In this paper, we are interested in the numerical solutions of
stochastic
functional differential equations (SFDEs) with {\it jumps}. Under the
global
Lipschitz condition, we show that the $p$th moment convergence of the
Euler-Maruyama (EM) numerical solutions to SFDEs with jumps has order
$1/p$ for
any $p\ge 2$. This is significantly different from the case of SFDEs
without
jumps where the order is 1/2 for any $p\ge 2$. It is therefore best to
use the
mean-square convergence for SFDEs with jumps. Consequently, under the
local
Lipschitz condition, we reveal that the order of the mean-square
convergence is
close to 1/2, provided that the local Lipschitz constants, valid on
balls of
radius $j$, do not grow faster than $\log j$.
http://arxiv.org/abs/0906.3455
---------------------------------------------------------------
8663. A WEAK TRAPEZOIDAL METHOD FOR A CLASS OF STOCHASTIC
DIFFERENTIAL EQUATIONS
David F. Anderson and Jonathan C. Mattingly
We present a numerical method for the approximation of solutions for the
class of stochastic differential equations driven by Brownian motions
which
induce stochastic variation in fixed directions. This class of
equations arises
naturally in the study of population processes and chemical reaction
kinetics.
We show that the method constructs paths that are second order
accurate in the
weak sense. The method is simpler than many second order methods in
that it
neither requires the construction of iterated Ito integrals nor the
evaluation
of any derivatives. The method consists of two steps. In the first an
explicit
Euler step is used to take a fractional step. This fractional point is
then
combined with the initial point to obtain a higher order, trapezoidal
like,
approximation. The higher order of accuracy stems from the fact that
both the
drift and the quadratic variation of the underlying SDE are
approximated to
second order.
http://arxiv.org/abs/0906.3475
---------------------------------------------------------------
8664. BROWNIAN MOTION IN A BALL IN THE PRESENCE OF SPHERICAL OBSTACLES
Julie O'Donovan
We study the problem of when a Brownian motion in the unit ball has a
positive probability of avoiding a countable collection of spherical
obstacles.
We give a necessary and sufficient integral condition for such a
collection to
be avoidable.
http://arxiv.org/abs/0906.3481
---------------------------------------------------------------
8665. CRITICAL HOMOGENIZATION OF LEVY PROCESS DRIVEN SDES IN RANDOM
MEDIUM
R\'emi Rhodes (CEREMADE) and Bamba A. Sow (LERSTAD)
We are concerned with homogenization of stochastic differential
equations
(SDE) with stationary coefficients driven by Poisson random measures and
Brownian motions in the critical case, that is when the limiting
equation
admits both a Brownian part as well as a pure jump part. We state an
annealed
convergence theorem. This problem is deeply connected with
homogenization of
integral partial differential equations
http://arxiv.org/abs/0906.3569
---------------------------------------------------------------
8666. ON MONOCHROMATIC ARM EXPONENTS FOR 2D CRITICAL PERCOLATION
Vincent Beffara (UMPA-ENSL) and Pierre Nolin (CIMS)
We investigate the so-called monochromatic arm exponents for critical
percolation in two dimensions. These exponents, describing the
probability of
observing j disjoint macroscopic paths, are shown to exist and to form a
different family from the (now well-understood) polychromatic exponents.
http://arxiv.org/abs/0906.3570
---------------------------------------------------------------
8667. CENTRAL LIMIT THEOREM FOR COLOURED HARD-DIMERS
Maria Simonetta Bernabei and Horst Thaler
Using an averaged generating function for coloured hard-dimers, some
random
variables of interest are studied. The main result lies in the fact
that all
their probability distributions obey a central limit theorem.
http://arxiv.org/abs/0906.3652
---------------------------------------------------------------
8668. SHARP THRESHOLD FOR PERCOLATION ON EXPANDERS
Itai Benjamini and Stephane Boucheron and Gabor Lugosi and Raphael
Rossignol
We study the appearance of the giant component in random subgraphs of
a given
finite graph G=(V,E) in which each edge is present independently with
probability p. We show that if G is an expander with vertices of bounded
degree, then for any c in ]0,1[, the property that the random subgraph
contains
a giant component of size c|V| has a sharp threshold.
http://arxiv.org/abs/0906.3657
---------------------------------------------------------------
8669. ZEROS OF AIRY FUNCTION AND RELAXATION PROCESS
Makoto Katori and Hideki Tanemura
One-dimensional system of Brownian motions called Dyson's model is the
particle system with long-range repulsive forces acting between any
pair of
particles, where the strength of force is $\beta/2$ times the inverse of
particle distance. When $\beta=2$, it is realized as the Brownian
motions in
one dimension conditioned never to collide with each other. For any
initial
configuration, it is proved that Dyson's model with $\beta=2$ and $N$
particles, $\X(t)=(X_1(t), ..., X_N(t)), t \in [0,\infty), 2 \leq N <
\infty$,
is determinantal in the sense that any multitime correlation function
is given
by a determinant with a continuous kernel. The Airy function $\Ai(z)$
is an
entire function with zeros all located on the negative part of the
real axis
$\R$. We consider Dyson's model with $\beta=2$ starting from the first
$N$
zeros of $\Ai(z)$, $0 > a_1 > ... > a_N$, $N \geq 2$. In order to
properly
control the effect of such initial confinement of particles in the
negative
region of $\R$, we put the drift term to each Brownian motion, which
increases
in time as a parabolic function :
$Y_j(t)=X_j(t)+t^2/4+\{d_1+\sum_{\ell=1}^{N}(1/a_{\ell})\}t, 1 \leq j
\leq N$,
where $d_1=\Ai'(0)/\Ai(0)$. We show that, as the $N \to \infty$ limit of
$\Y(t)=(Y_1(t), ..., Y_N(t)), t \in [0, \infty)$, we obtain an infinite
particle system, which is the relaxation process from the
configuration, in
which every zero of $\Ai(z)$ on the negative $\R$ is occupied by one
particle,
to the stationary state $\mu_{\Ai}$. The stationary state $\mu_{\Ai}$
is the
determinantal point process with the Airy kernel, which is spatially
inhomogeneous on $\R$ and in which the Tracy-Widom distribution
describes the
rightmost particle position.
http://arxiv.org/abs/0906.3666
---------------------------------------------------------------
8670. THE NATURAL PARAMETRIZATION FOR THE SCHRAMM-LOEWNER EVOLUTION
Gregory F. Lawler and Scott Sheffield
The Schramm-Loewner evolution (SLE_\kappa) is a candidate for the
scaling
limit of random curves arising in two-dimensional critical phenomena.
When
\kappa < 8, an instance of SLE_\kappa is a random planar curve with
almost sure
Hausdorff dimension d = 1 + \kappa/8 < 2. This curve is conventionally
parametrized by its half plane capacity, rather than by any measure of
its
d-dimensional volume. For \kappa < 8, we use a Doob-Meyer
decomposition to
construct the unique (under mild assumptions) Markovian
parametrization of
SLE_\kappa that transforms like a d-dimensional volume measure under
conformal
maps. We prove that this parametrization is non-trivial (i.e., the
curve is not
entirely traversed in zero time) for \kappa < 4(7 - \sqrt{33}) =
5.021 ....
http://arxiv.org/abs/0906.3804
---------------------------------------------------------------
8671. MARKOV CHAINS CONDITIONED NEVER TO WAIT TOO LONG AT THE ORIGIN
Saul Jacka
Motivated by Feller's coin-tossing problem, we consider the problem of
conditioning an irreducible Markov chain never to wait too long at 0.
Denoting
by $\tau$ the first time that the chain, $X$, waits for at least one
unit of
time at the origin, we consider conditioning the chain on the event $
(\tau>T)$.
We show there is a weak limit as $T\to \infty$ in the cases where
either the
statespace is finite or $X$ is transient. We give sufficient
conditions for the
existence of a weak limit in other cases and show that we have vague
convergence to a defective limit if the time to hit zero has a lighter
tail
than $\tau$ and $\tau$ is subexponential.
http://arxiv.org/abs/0906.3876
---------------------------------------------------------------
8672. CONCENTRATION OF MEASURES VIA SIZE BIASED COUPLINGS
Subhankar Ghosh and Larry Goldstein
Let $Y$ be a nonnegative random variable with mean $\mu$ and finite
positive
variance $\sigma^2$, and let $Y^s$, defined on the same space as $Y$,
have the
$Y$ size biased distribution, that is, the distribution characterized by
E[Yf(Y)]=\mu E f(Y^s) for all functions $f$ for which these
expectations exist.
Under a variety of conditions on the coupling of Y and $Y^s$, including
combinations of boundedness and monotonicity, concentration of measure
inequalities hold. Examples include the number of relatively ordered
subsequences of a random permutation, sliding window statistics
including the
number of m-runs in a sequence of coin tosses, the number of local
maximum of a
random function on a lattice, the number of urns containing exactly
one ball in
an urn allocation model, the volume covered by the union of $n$ balls
placed
uniformly over a volume n subset of d diml Euclidean space, the number
of bulbs
switched on at the terminal time in the so called lightbulb process,
the number
of isolated vertices in the Erdos-Renyi random graph model, and the
infinitely
divisible and compound Poisson distributions that satisfy a bounded
moment
generating function condition.
http://arxiv.org/abs/0906.3886
---------------------------------------------------------------
8673. SURVIVAL AND GROWTH OF A BRANCHING RANDOM WALK IN RANDOM
ENVIRONMENT
Christian Bartsch and Nina Gantert and Michael Kochler
We consider a particular Branching Random Walk in Random Environment
(BRWRE)
on $N_0$ started with one particle at the origin. Particles reproduce
according
to an offspring distribution (which depends on the location) and move
either
one step to the right (with a probability in $(0,1]$ which may also
depend on
the location) or stay in the same place. We give criteria for local
and global
survival and show that global survival is equivalent to exponential
growth of
the moments.
http://arxiv.org/abs/0906.4033
---------------------------------------------------------------
8674. ONE-SIDED CAUCHY-STIELTJES KERNEL FAMILIES
Wlodzimierz Bryc and Abdelhamid Hassairi
This paper continues the study of a kernel family which uses the
Cauchy-Stieltjes kernel in place of the celebrated exponential kernel
of the
exponential families theory. We extend the theory to cover generating
measures
with support that is unbounded on one side. We illustrate the need for
such an
extension by showing that cubic pseudo-variance functions correspond to
free-infinitely divisible laws without the first moment. We also
determine the
domain of means, advancing the understanding of
Cauchy-Stieltjes kernel families also for compactly supported
generating
measures.
http://arxiv.org/abs/0906.4073
---------------------------------------------------------------
8675. IMPROVEMENT OF TWO HUNGARIAN BIVARIATE THEOREMS
Nathalie Castelle (LM-Orsay)
We introduce a new technique to establish Hungarian multivariate
theorems. In
this article we apply this technique to the strong approximation
bivariate
theorems of the uniform empirical process. It improves the Komlos,
Major and
Tusn\'ady (1975) result, as well as our own (1998). More precisely, we
show
that the error in the approximation of the uniform bivariate $n$-
empirical
process by a bivariate Brownian bridge is of order $n^{-1/2}(log
(nab))^{3/2}$
on the rectangle $[0,a]x[0,b]$, $0 <a, b <1$, and that the error in the
approximation of the uniform univariate $n$-empirical process by a
Kiefer
process is of order $n^{-1/2}(log (na))^{3/2}$ on the interval $[0,a]
$, $0 < a
< 1$. In both cases, the global error bound is therefore of order
$n^{-1/2}(log
(n))^{3/2}$. Previously, from the 1975 article of Komlos, Major and
Tusn\'ady,
the global error bound was of order $n^{-1/2}(log (n))^{2}$, and from
our 1998
article, the local error bounds were of order $n^{-1/2}(log
(nab))^{2}$ or
$n^{-1/2}(log (na))^{2}$. We think that, in the d-variate case, the
global
error bound between the uniform d-variate $n$-empirical process and the
associated Gaussian process is of order $n^{-1/2}(log (n))^{(d+1)/2}$,
and that
this result is optimal. The new feature of this article is to identify
martingales in the error terms and to apply to them an exponential
inequality.
The idea is to bound of the compensator of the error term, instead of
bounding
of the error term itself.
http://arxiv.org/abs/0906.3952
---------------------------------------------------------------
8676. THE SPECTRAL EDGE OF SOME RANDOM BAND MATRICES
Sasha Sodin
We study the asymptotic distribution of the eigenvalues of random
Hermitian
periodic band matrices, focusing on the spectral edges. The
eigenvalues close
to the edges converge in distribution to the Airy point process if
(and only
if) the band is sufficiently wide (W >> N^{5/6}.) Otherwise, a different
limiting distribution appears.
http://arxiv.org/abs/0906.4047
---------------------------------------------------------------
8677. A CENTRAL LIMIT THEOREM VIA DIFFERENTIAL EQUATIONS
Taral Guldahl Seierstad
In a paper from 1995, Wormald gave general criteria for certain
parameters in
a family of discrete random processes to converge to the solution of a
system
of differential equations. Based on this method, we show that if some
further
conditions are satisfied, the parameters converge to a multivariate
normal
distribution.
http://arxiv.org/abs/0906.4202
---------------------------------------------------------------
8678. POISSON--VORONOI APPROXIMATION
Matthias Heveling and Matthias Reitzner
Let $X$ be a Poisson point process and $K\subset\mathbb{R}^d$ a
measurable
set. Construct the Voronoi cells of all points $x\in X$ with respect
to $X$,
and denote by $v_X(K)$ the union of all Voronoi cells with nucleus in
$K$. For
$K$ a compact convex set the expectation of the volume difference
$V(v_X(K))-V(K)$ and the symmetric difference $V(v_X(K)\triangle K)$ is
computed. Precise estimates for the variance of both quantities are
obtained
which follow from a new jackknife inequality for the variance of
functionals of
a Poisson point process. Concentration inequalities for both
quantities are
proved using Azuma's inequality.
http://arxiv.org/abs/0906.4238
---------------------------------------------------------------
8679. RATES OF CONVERGENCE OF SOME MULTIVARIATE MARKOV CHAINS WITH
POLYNOMIAL EIGENFUNCTIONS
Kshitij Khare and Hua Zhou
We provide a sharp nonasymptotic analysis of the rates of convergence
for
some standard multivariate Markov chains using spectral techniques.
All chains
under consideration have multivariate orthogonal polynomial as
eigenfunctions.
Our examples include the Moran model in population genetics and its
variants in
community ecology, the Dirichlet-multinomial Gibbs sampler, a class of
generalized Bernoulli--Laplace processes, a generalized Ehrenfest urn
model and
the multivariate normal autoregressive process.
http://arxiv.org/abs/0906.4242
---------------------------------------------------------------
8680. TREE BASED FUNCTIONAL EXPANSIONS FOR FEYNMAN--KAC PARTICLE MODELS
Pierre Del Moral and Fr\'ed\'eric Patras and Sylvain Rubenthaler
We design exact polynomial expansions of a class of Feynman--Kac
particle
distributions. These expansions are finite and are parametrized by
coalescent
trees and other related combinatorial quantities. The accuracy of the
expansions at any order is related naturally to the number of
coalescences of
the trees. Our results include an extension of the Wick product
formula to
interacting particle systems. They also provide refined nonasymptotic
propagation of chaos-type properties, as well as sharp $\mathbb{L}_p$-
mean
error bounds, and laws of large numbers for $U$-statistics.
http://arxiv.org/abs/0906.4249
---------------------------------------------------------------
8681. ENERGY MEASURES AND INDICES OF DIRICHLET FORMS, WITH
APPLICATIONS TO DERIVATIVES ON SOME FRACTALS
Masanori Hino
We introduce the concept of index for regular Dirichlet forms by means
of
energy measures, and discuss its properties. In particular, it is
proved that
the index of strong local regular Dirichlet forms is identical with the
martingale dimension of the associated diffusion processes. As an
application,
a class of self-similar fractals is taken up as an underlying space.
We prove
that first-order derivatives can be defined for functions in the
domain of the
Dirichlet forms and their total energies are represented as the square
integrals of the derivatives.
http://arxiv.org/abs/0906.4251
---------------------------------------------------------------
8682. ON LARGE DEVIATION REGIMES FOR RANDOM MEDIA MODELS
M. Cranston and D. Gauthier and T. S. Mountford
The focus of this article is on the different behavior of large
deviations of
random subadditive functionals above the mean versus large deviations
below the
mean in two random media models. We consider the point-to-point first
passage
percolation time $a_n$ on $\mathbb{Z}^d$ and a last passage
percolation time
$Z_n$. For these functionals, we have $\lim_{n\to\infty}\frac{a_n}{n}=
\nu$ and
$\lim_{n\to\infty}\frac{Z_n}{n}=\mu$. Typically, the large deviations
for such
functionals exhibits a strong asymmetry, large deviations above the
limiting
value are radically different from large deviations below this
quantity. We
develop robust techniques to quantify and explain the differences.
http://arxiv.org/abs/0906.4254
---------------------------------------------------------------
8683. ERGODICITY OF THE 3D STOCHASTIC NAVIER-STOKES EQUATIONS DRIVEN
BY MILDLY DEGENERATE NOISE
Lihu Xu and Marco Romito
We prove that the any Markov solution to the 3D stochastic Navier-Stokes
equations driven by a mildly degenerate noise (i.e.all but finitely many
Fourier modes are forced) is uniquely ergodic. This follows by proving
strong
Feller regularity and irreducibility.
http://arxiv.org/abs/0906.4281
---------------------------------------------------------------
8684. PARAMETER ESTIMATION IN DIAGONALIZABLE STOCHASTIC HYPERBOLIC
EQUATIONS
W. Liu and S. V. Lototsky
A parameter estimation problem is considered for a linear stochastic
hyperbolic equation driven by additive space-time Gaussian white
noise. The
damping/amplification operator is allowed to be unbounded.
The estimator is of spectral type and utilizes a finite number of
the spatial
Fourier coefficients of the solution. The asymptotic properties of the
estimator are studied as the number of the Fourier coefficients
increases,
while the observation time and the noise intensity are fixed.
http://arxiv.org/abs/0906.4353
---------------------------------------------------------------
8685. BULK UNIVERSALITY FOR WIGNER HERMITIAN MATRICES WITH
SUBEXPONENTIAL DECAY
Laszlo Erdos and Jose Ramirez and Benjamin Schlein and Terence Tao
and Van Vu and Horng-Tzer Yau
We consider the ensemble of $n \times n$ Wigner hermitian matrices
$H = (h_{\ell k})_{1 \leq \ell,k \leq n}$ that generalize the
Gaussian
unitary ensemble (GUE). The matrix elements $h_{k\ell} = \bar h_{\ell
k}$ are
given by $h_{\ell k} = n^{-1/2} (x_{\ell k} + \sqrt{-1} y_{\ell k})$,
where
$x_{\ell k}, y_{\ell k}$ for $1 \leq \ell < k \leq n$ are i.i.d. random
variables with mean zero and variance 1/2, $y_{\ell\ell}=0$ and
$x_{\ell \ell}$
have mean zero and variance 1. We assume the distribution of $x_{\ell
k},
y_{\ell k}$ to have subexponential decay. In a recent paper, four of the
authors recently established that the gap distribution and averaged $k
$-point
correlation of these matrices were \emph{universal} (and in
particular, agreed
with those for GUE) assuming additional regularity hypotheses on the
$x_{\ell
k}, y_{\ell k}$. In another recent paper, the other two authors, using a
different method, established the same conclusion assuming instead
some moment
and support conditions on the $x_{\ell k}, y_{\ell k}$. In this short
note we
observe that the arguments of these two papers can be combined to
establish
universality of the gap distribution and averaged $k$-point
correlations for
all Wigner matrices (with subexponentially decaying entries), with no
extra
assumptions.
http://arxiv.org/abs/0906.4400
---------------------------------------------------------------
8686. STEIN'S METHOD MEETS MALLIAVIN CALCULUS: A SHORT SURVEY WITH
NEW ESTIMATES
Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X)
We provide an overview of some recent techniques involving the Malliavin
calculus of variations and the so-called ``Stein's method'' for the
Gaussian
approximations of probability distributions. Special attention is
devoted to
establishing explicit connections with the classic method of moments: in
particular, we use interpolation techniques in order to deduce some new
estimates for the moments of random variables belonging to a fixed
Wiener
chaos. As an illustration, a class of central limit theorems
associated with
the quadratic variation of a fractional Brownian motion is studied in
detail.
http://arxiv.org/abs/0906.4419
---------------------------------------------------------------
8687. MOST LIKELY PATHS TO ERROR WHEN ESTIMATING THE MEAN OF A
REFLECTED RANDOM WALK
Ken R. Duffy and Sean P. Meyn
It is known that simulation of the mean position of a reflected random
walk
$\{W_n\}$ exhibits non-standard behavior, even for light-tailed
increment
distributions with negative drift. The Large Deviation Principle (LDP)
holds
for deviations below the mean, but for deviations at the usual speed
above the
mean the rate function is null. This paper takes a deeper look at this
phenomenon. Conditional on a large sample mean, a complete sample path
LDP
analysis is obtained. Let $I$ denote the rate function for the one
dimensional
increment process. If $I$ is coercive, then given a large simulated mean
position, under general conditions our results imply that the most
likely
asymptotic behavior, $\psi$, of the paths $n^{-1} W_{\lfloor tn\rfloor}
$ is to
be zero apart from on an interval $[T_0,T_1]\subset[0,1]$ and to
satisfy the
functional equation \nabla I(\ddt\psi(t))=\lambda^*(T_1-t) \quad
\text{whenever} \psi(t)\neq 0. If $I$ is non-coercive, a similar, but
slightly
more involved, result holds.
http://arxiv.org/abs/0906.4514
---------------------------------------------------------------
8688. ON THE TIME SCHEDULE OF BROWNIAN FLIGHTS
Athanasios Batakis (MAPMO) and Michel Zinsmeister (MAPMO)
We are interested on the statistics of the duration of Brownian
diffusions
started at distance \epsilon from a given boundary and stopped when
they hit
back the interface.
http://arxiv.org/abs/0906.4537
---------------------------------------------------------------
8689. COVARIANCE FUNCTION OF VECTOR SELF-SIMILAR PROCESS
Fr\'ed\'eric Lavancier (LMJL) and Anne Philippe (LMJL) and Donatas
Surgailis
The paper obtains the general form of the cross-covariance function of
vector
fractional Brownian motion with correlated components having different
self-similarity indices.
http://arxiv.org/abs/0906.4541
---------------------------------------------------------------
8690. SUMSET AND INVERSE SUMSET THEOREMS FOR SHANNON ENTROPY
Terence Tao
Let $G = (G,+)$ be an additive group. The sumset theory of Pl\"unnecke
and
Ruzsa gives several relations between the size of sumsets $A+B$ of
finite sets
$A, B$, and related objects such as iterated sumsets $kA$ and
difference sets
$A-B$, while the inverse sumset theory of Freiman, Ruzsa, and others
characterises those finite sets $A$ for which $A+A$ is small. In this
paper we
establish analogous results in which the finite set $A \subset G$ is
replaced
by a discrete random variable $X$ taking values in $G$, and the
cardinality
$|A|$ is replaced by the Shannon entropy $\Ent(X)$. In particular, we
classify
the random variable $X$ which have small doubling in the sense that
$\Ent(X_1+X_2) = \Ent(X)+O(1)$ when $X_1,X_2$ are independent copies
of $X$, by
showing that they factorise as $X = U+Z$ where $U$ is uniformly
distributed on
a coset progression of bounded rank, and $\Ent(Z) = O(1)$.
When $G$ is torsion-free, we also establish the sharp lower bound $
\Ent(X+X)
\geq \Ent(X) + {1/2} \log 2 - o(1)$, where $o(1)$ goes to zero as $
\Ent(X) \to
\infty$.
http://arxiv.org/abs/0906.4387
---------------------------------------------------------------
8691. EXCURSIONS OF DIFFUSION PROCESSES AND CONTINUED FRACTIONS
Alain Comtet and Yves Tourigny
It is well-known that the excursions of a one-dimensional diffusion
process
can be studied by considering a certain Riccati equation associated
with the
process. We show that, in many cases of interest, the Riccati equation
can be
solved in terms of an infinite continued fraction. We examine the
probabilistic
significance of the expansion. To illustrate our results, we discuss
some
examples of diffusions in deterministic and in random environments.
http://arxiv.org/abs/0906.4651
---------------------------------------------------------------
8692. THE INCLUSION PROCESS: DUALITY AND CORRELATION INEQUALITIES
C. Giardina and F. Redig and K. Vafayi
We prove a comparison inequality between a system of independent random
walkers and a system of random walkers which interact by attracting
eachother
-a process which we call here the symmetric inclusion process (SIP).
As an
application, correlation inequalities for the SIP, as well as for a
model of
heat conduction, the so-called Brownian momentum process, are
obtained. These
inequalities are counterparts of the inequalities (in the opposite
direction)
for the symmetric exclusion process, confirming that the SIP is a
natural
bosonic analogue of the symmetric exclusion process (which is
fermionic). We
discuss stationary measures of the SIP, and an asymmetric version that
has the
same stationary probability measures, as well as infinite non-
translation
invariant reversible measures. Finally, we consider a boundary driven
version
of the SIP for which we prove duality and correlation inequalities.
http://arxiv.org/abs/0906.4664
---------------------------------------------------------------
8693. THE ORTHOGONAL WEINGARTEN FORMULA IN COMPACT FORM
T. Banica
We present a compact formulation of the orthogonal Weingarten formula,
with
the traditional quantity
$I(i_1,...,i_{2k}:j_1,...,j_{2k})=\int_{O_n}u_{i_1j_1}...
u_{i_{2k}j_{2k}} du$
replaced by the more advanced quantity $I(a)=\int_{O_n}\Pi
u_{ij}^{a_{ij}} du$,
depending on a matrix of exponents $a\in M_n(\mathbb N)$. Among
consequences,
we establish a number of basic facts regarding the integrals $I(a)$:
vanishing
condition, sign, possible poles, asymptotic behavior.
http://arxiv.org/abs/0906.4694
---------------------------------------------------------------
8694. A CLT FOR THE $L^{2}$ MODULI OF CONTINUITY OF LOCAL TIMES OF
LEVY PROCESSES
Michael B. Marcus and Jay Rosen
Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric L\'evy process with local
time
$\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the L\'evy
exponent
$\psi(\la)$ is regularly varying at infinity with index $1<\beta\leq
2$ and
satisfies some additional regularity conditions && \sqrt{h\psi^{2}(1/
h)} \lc
\int (L^{x+h}_{1}- L^{x}_{1})^{2} dx- E(\int (L^{x+h}_{1}-
L^{x}_{1})^{2}
dx)\rc\nn && {1 in} \stackrel{\mathcal{L}}{\Longrightarrow}
(8c_{\beta,1})^{1/2} \eta (\int (L_{1}^{x})^{2} dx)^{1/2} \nn, as $h
\rar 0$,
where $\eta$ is a normal random variable with mean zero and variance
one that
is independent of $L^{x}_{t}$, and $c_{\beta,1}$ is a known constant.
http://arxiv.org/abs/0906.4770
---------------------------------------------------------------
8695. LEVY FLIGHTS IN CONFINING POTENTIALS
Piotr Garbaczewski and Vladimir Stephanovich
We analyze confining mechanisms for L\'{e}vy flights. When they evolve
in
suitable external potentials their variance may exist and show
signatures of a
superdiffusive transport. Two classes of stochastic jump - type
processes are
considered: those driven by Langevin equation with L\'{e}vy noise and
those,
named by us topological L\'{e}vy processes (occurring in systems with
topological complexity like folded polymers or complex networks and
generically
in inhomogeneous media), whose Langevin representation is unknown and
possibly
nonexistent. Our major finding is that both above classes of processes
stay in
affinity and may share common stationary (eventually asymptotic)
probability
density, even if their detailed dynamical behavior look different. That
generalizes and offers new solutions to a reverse engineering (e.g.
targeted
stochasticity) problem due to I. Eliazar and J. Klafter [J. Stat.
Phys. 111,
739, (2003)]: design a L\'{e}vy process whose target pdf equals a priori
preselected one. Our observations extend to a broad class of L\'{e}vy
noise
driven processes, like e.g. superdiffusion on folded polymers,
geophysical
flows and even climatic changes.
http://arxiv.org/abs/0904.4157
---------------------------------------------------------------
8696. SURVIVAL AND COEXISTENCE FOR A MULTITYPE CONTACT PROCESS
J. Theodore Cox and Rinaldo B. Schinazi
We study the ergodic theory of a multitype contact process with equal
death
rates and unequal birth rates on the $d$-dimensional integer lattice and
regular trees. We prove that for birth rates in a certain interval
there is
coexistence on the tree, which by a result of Neuhauser is not
possible on the
lattice. We also prove a complete convergence result when the larger
birth rate
falls outside of this interval.
http://arxiv.org/abs/0906.4845
---------------------------------------------------------------
8697. A BOUNDED DERIVATION METHOD FOR THE MAXIMUM LIKELIHOOD
ESTIMATION ON THE PARAMETERS OF WEIBULL DISTRIBUTION
DeTao Mao and Wenyuan Li
For the basic maximum likelihood estimating function of the two
parameters
Weibull distribution, a simple proof on its global monotonicity is
given to
ensure the existence and uniqueness of its solution. The boundary of the
function's first-order derivation is defined based on its scale-free
property.
With a bounded derivation, the possible range of the root of this
function can
be determined. A novel root-finding algorithm employing these
established
results is proposed accordingly, its convergence is proved
analytically as
well. Compared with other typical algorithms for this problem, the
efficiency
of the proposed algorithm is also demonstrated by numerical experiments.
http://arxiv.org/abs/0906.4823
---------------------------------------------------------------
8698. A SERIES REPRESENTATION OF MULTISTABLE MULTIFRACTIONAL PROCESSES
AND OTHER LOCALISABLE PROCESSES
Ronan Le Gu\'evel (LMJL) and Jacques L\'evy-V\'ehel (INRIA Saclay -
Ile de France)
The study of non-stationary processes whose local form has controlled
properties is a fruitful and important area of research, both in
theory and
applications. We present here a construction of multifractional
multistable
processes, based on a series representation of stable stochastic
integrals. We
consider various particular cases of interest, including multistable L
\'evy
motion, multistable reverse Ornstein-Uhlenbeck process, log-fractional
multistable motion and linear multistable multifractional motion. We
also
compute the finite dimensional distributions of those processes.
Finally, we
display numerical experiments showing graphs of synthesized paths of
such
processes.
http://arxiv.org/abs/0906.5042
---------------------------------------------------------------
8699. USING DIFFERENTIAL EQUATIONS TO OBTAIN JOINT MOMENTS OF FIRST-
PASSAGE TIMES OF INCREASING LEVY PROCESSES
Mark S. Veillette and Murad S. Taqqu
Let $\{D(s), s \geq 0 \}$ be a L\'evy subordinator, that is, a non-
decreasing
process with stationary and independent increments and suppose that
$D(0) = 0$.
We study the first-hitting time of the process $D$, namely, the
process $E(t) =
\inf \{s: D(s) > t \}$, $t \geq 0$.
The process $E$ is, in general, non-Markovian with non-stationary and
non-independent increments. We derive a partial differential equation
for the
Laplace transform of the $n$-time tail distribution function $P[E(t_1) >
s_1,...,E(t_n) > s_n]$, and show that this PDE has a unique solution
given
natural boundary conditions. This PDE can be used to derive all $n$-time
moments of the process $E$.
http://arxiv.org/abs/0906.5083
---------------------------------------------------------------
8700. STUDENTIZED PROCESSES OF U-STATISTICS
Masoud M. Nasari
A uniform in probability approximation is established for Studentized
processes of non degenerate U-statistics of order m greater or equal
to 2 in
terms of a standard Wiener process. The classical condition that the
second
moment of kernel of the underlying U-statistic exists is relaxed to
having 5/3
moments. Furthermore, the conditional expectation of the kernel is
only assumed
to be in the domain of attraction of the normal law (instead of the
classical
two moment condition).
http://arxiv.org/abs/0906.5101
---------------------------------------------------------------
8701. BOUNDS ON THE CONSTANT IN THE MEAN CENTRAL LIMIT THEOREM
Larry Goldstein
Let $X_1,...,X_n$ be independent with mean zero, finite variances
$\sigma_1^2,...,\sigma_n^2$ and finite absolute third moments, $F_n$ the
distribution function of $(X_1+...+X_n)/\sigma$ where $\sigma^2=
\sum_{i=1}^n
\sigma_i^2$, and $\Phi$ that of the standard normal. Then the $L^1$
distance
between $F_n$ and $\Phi$ satisfies $$ ||F_n-\Phi||_1 \le
\frac{1}{\sigma^3}\sum_{i=1}^n E|X_i|^3. $$ In particular, when
$X_1,...,X_n$
are identically distributed with variance $\sigma^2$, $$ ||F_n-\Phi||
_1 \le
\frac{E|X_1|^3}{\sigma^3 \sqrt{n}} \quad {for all $n \in \mathbb{N}$,}
$$
corresponding to an $L^1$ Berry Esseen constant of 1. A lower bound of
$$
\frac{2 \sqrt{\pi} (2\Phi(1)-1) - (\sqrt{\pi}+\sqrt{2})+ 2
e^{-1/2}\sqrt{2}}{\sqrt{\pi}} =0.535377... $$ on the smallest possible
constant
is provided.
http://arxiv.org/abs/0906.5145
---------------------------------------------------------------
8702. STOCHASTIC HOMOGENIZATION OF HOROSPHERIC TREE PRODUCTS
Vadim A. Kaimanovich and Florian Sobieczky
We construct measures invariant with respect to equivalence relations
which
are graphed by horospheric products of trees. The construction is
based on
using conformal systems of boundary measures on treed equivalence
relations.
The existence of such an invariant measure allows us to establish
amenability
of horospheric products of random trees.
http://arxiv.org/abs/0906.5296
---------------------------------------------------------------
8703. GEOMETRIC ERGODICITY AND THE SPECTRAL GAP OF NON-REVERSIBLE
MARKOV CHAINS
Ioannis Kontoyiannis and Sean P. Meyn
We argue that the spectral theory of non-reversible Markov chains may
often
be more effectively cast within the framework of the naturally
associated
weighted-$L_\infty$ space $L_\infty^V$, instead of the usual Hilbert
space
$L_2=L_2(\pi)$, where $\pi$ is the invariant measure of the chain. This
observation is, in part, based on the following results. A discrete-
time Markov
chain with values in a general state space is geometrically ergodic if
and only
if its transition kernel admits a spectral gap in $L_\infty^V$. If the
chain is
reversible, the same equivalence holds with $L_2$ in place of $L_
\infty^V$, but
in the absence of reversibility it fails: There are (necessarily
non-reversible, geometrically ergodic) chains that admit a spectral
gap in
$L_\infty^V$ but not in $L_2$. Moreover, if a chain admits a spectral
gap in
$L_2$, then for any $h\in L_2$ there exists a Lyapunov function $V_h
\in L_1$
such that $V_h$ dominates $h$ and the chain admits a spectral gap in
$L_\infty^{V_h}$. The relationship between the size of the spectral
gap in
$L_\infty^V$ or $L_2$, and the rate at which the chain converges to
equilibrium
is also briefly discussed.
http://arxiv.org/abs/0906.5322
---------------------------------------------------------------
8704. TESTING FOR WHITE NOISE UNDER UNKNOWN DEPENDENCE AND ITS
APPLICATIONS TO GOODNESS-OF-FIT FOR TIME SERIES MODELS
Xiaofeng Shao
Testing for white noise has been well studied in the literature of
econometrics and statistics. For most of the proposed test statistics,
such as
the well-known Box-Pierce's test statistic with fixed lag truncation
number,
the asymptotic null distributions are obtained under independent and
identically distributed assumptions and may not be valid for the
dependent
white noise. Due to recent popularity of conditional heteroscedastic
models
(e.g., GARCH models), which imply nonlinear dependence with zero
autocorrelation, there is a need to understand the asymptotic
properties of the
existing test statistics under unknown dependence. In this paper, we
showed
that the asymptotic null distribution of Box-Pierce's test statistic
with
general weights still holds under unknown weak dependence so long as
the lag
truncation number grows at an appropriate rate with increasing sample
size.
Further applications to diagnostic checking of the ARMA and FARIMA
models with
dependent white noise errors are also addressed. Our results go beyond
earlier
ones by allowing non-Gaussian and conditional heteroscedastic errors
in the
ARMA and FARIMA models and provide theoretical support for some
empirical
findings reported in the literature.
http://arxiv.org/abs/0906.5179
---------------------------------------------------------------
8705. DISTANCE STATISTICS IN QUADRANGULATIONS WITH A BOUNDARY, OR WITH
A SELF-AVOIDING LOOP
J. Bouttier and E. Guitter
We consider quadrangulations with a boundary and derive explicit
expressions
for the generating functions of these maps with either a marked vertex
at a
prescribed distance from the boundary, or two boundary vertices at a
prescribed
mutual distance in the map. For large maps, this yields explicit
formulas for
the bulk-boundary and boundary-boundary correlators in the various
encountered
scaling regimes: a small boundary, a dense boundary and a critical
boundary
regime. The critical boundary regime is characterized by a one-
parameter family
of scaling functions interpolating between the Brownian map and the
Brownian
Continuum Random Tree. We discuss the cases of both generic and self-
avoiding
boundaries, which are shown to share the same universal scaling limit.
We
finally address the question of the bulk-loop distance statistics in the
context of planar quadrangulations equipped with a self-avoiding loop.
Here
again, a new family of scaling functions describing critical loops is
discovered.
http://arxiv.org/abs/0906.4892
---------------------------------------------------------------
8706. SPREAD OF MISINFORMATION IN SOCIAL NETWORKS
Daron Acemoglu and Asuman Ozdaglar and Ali ParandehGheibi
We provide a model to investigate the tension between information
aggregation
and spread of misinformation in large societies (conceptualized as
networks of
agents communicating with each other). Each individual holds a belief
represented by a scalar. Individuals meet pairwise and exchange
information,
which is modeled as both individuals adopting the average of their pre-
meeting
beliefs. When all individuals engage in this type of information
exchange, the
society will be able to effectively aggregate the initial information
held by
all individuals. There is also the possibility of misinformation,
however,
because some of the individuals are "forceful," meaning that they
influence the
beliefs of (some) of the other individuals they meet, but do not
change their
own opinion. The paper characterizes how the presence of forceful agents
interferes with information aggregation. Under the assumption that even
forceful agents obtain some information (however infrequent) from some
others
(and additional weak regularity conditions), we first show that
beliefs in this
class of societies converge to a consensus among all individuals. This
consensus value is a random variable, however, and we characterize its
behavior. Our main results quantify the extent of misinformation in
the society
by either providing bounds or exact results (in some special cases) on
how far
the consensus value can be from the benchmark without forceful agents
(where
there is efficient information aggregation). The worst outcomes obtain
when
there are several forceful agents and forceful agents themselves
update their
beliefs only on the basis of information they obtain from individuals
most
likely to have received their own information previously.
http://arxiv.org/abs/0906.5007
---------------------------------------------------------------
8707. DERIVATION OF AN EIGENVALUE PROBABILITY DENSITY FUNCTION
RELATING TO THE POINCARE DISK
Peter J. Forrester and Manjunath Krishnapur
A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)]
gives
the eigenvalue probability density function for the top N x N sub-
block of a
Haar distributed matrix from U(N+n). In the case n \ge N, we rederive
this
result, starting from knowledge of the distribution of the sub-blocks,
introducing the Schur decomposition, and integrating over all
variables except
the eigenvalues. The integration is done by identifying a recursive
structure
which reduces the dimension. This approach is inspired by an analogous
approach
which has been recently applied to determine the eigenvalue
probability density
function for random matrices A^{-1} B, where A and B are random
matrices with
entries standard complex normals. We relate the eigenvalue
distribution of the
sub-blocks to a many body quantum state, and to the one-component
plasma, on
the pseudosphere.
http://arxiv.org/abs/0906.5223
---------------------------------------------------------------
8708. STOCHASTIC CALCULUS FOR A TIME-CHANGED SEMIMARTINGALE AND THE
ASSOCIATED STOCHASTIC DIFFERENTIAL EQUATIONS
Kei Kobayashi
It is shown that under a certain condition on a semimartingale and a
time-change, any stochastic integral driven by the time-changed
semimartingale
is a time-changed stochastic integral driven by the original
semimartingale. As
a direct consequence, a specialized form of the Ito formula is
derived. When a
standard Brownian motion is the original semimartingale, classical Ito
stochastic differential equations driven by the Brownian motion with
drift
extend to a larger class of stochastic differential equations
involving a
time-change with continuous paths. A form of the general solution of
linear
equations in this new class is established, followed by consideration
of some
examples analogous to the classical equations. Through these examples,
each
coefficient of the stochastic differential equations in the new class
is given
meaning. The new feature is the coexistence of a usual drift term
along with a
term related to the time-change.
http://arxiv.org/abs/0906.5385
---------------------------------------------------------------
8709. LONGEST CONVEX CHAINS
Gergely Ambrus and Imre Barany
Assume $X_n$ is a random sample of $n$ uniform, independent points
from a
triangle $T$. The longest convex chain, $Y$, of $X_n$ is defined
naturally. The
length $|Y|$ of $Y$ is a random variable, denoted by $L_n$. In this
article, we
determine the order of magnitude of the expectation of $L_n$. We show
further
that $L_n$ is highly concentrated around its mean, and that the
longest convex
chains have a limit shape.
http://arxiv.org/abs/0906.5452
---------------------------------------------------------------
8710. ORTHOGONAL SERIES AND LIMIT THEOREMS FOR CANONICAL U- AND V-
STATISTICS OF STATIONARY CONNECTED OBSERVATIONS
I.S.Borisov and N.Volodko
The limit behavior is studied for the distributions of normalized U- and
V-statistics of an arbitrary order with canonical (degenerate)
kernels, based
on samples of increasing sizes from a stationary sequence of
observations
satisfying classical mixing conditions. The corresponding limit
distributions
are represented as infinite multilinear forms of a centered Gaussian
sequence
with a known covariance matrix.
http://arxiv.org/abs/0906.5465
---------------------------------------------------------------
8711. POISSON BOUNDARY OF $GL_D(\Q)$
Sara Brofferio (LM-Orsay) and Bruno Schapira (LM-Orsay)
We construct the Poisson boundary for a random walk supported by the
general
linear group on the rational numbers as the product of flag manifolds
over the
$p$-adic fields. To this purpose, we prove a law of large numbers
using the
Oseledets' multiplicative ergodic theorem.
http://arxiv.org/abs/0906.5548
---------------------------------------------------------------
8712. ON BUFFON MACHINES AND NUMBERS
Philippe Flajolet and Maryse Pelletier and Michele Soria
Buffon's needle experiment is well-known: take a plane on which parallel
lines at unit distance one from the next have been marked, throw a
needle of
unit length at random, and, finally, declare the experiment a success
if the
needle intersects one of the lines. Basic calculus implies that the
probability
of success is 2/pi~0.63661, and the experiment can be regarded as an
analog
(i.e., continuous) device that stochastically "computes'' 2/pi.
Generalizing
the experiment and simplifying the computational framework, we ask
ourselves
which probability distributions can be produced perfectly, from a
discrete
source of unbiased coin flips. We describe and analyse a few simple
Buffon
machines that can generate geometric, Poisson, and logarithmic-series
distributions (these are in particular required to transform continuous
Boltzmann samplers of classical combinatorial structures into purely
discrete
random generators). Say that a number is Buffon if it is the
probability of
success of a probabilistic experiment based on discrete coin
flippings. We
provide human-accessible Buffon machines, which require a dozen coin
flips or
less, on average, and produce experiments whose probabilities are
expressible
in terms of numbers such as pi, exp(-1), log2, sqrt(3), cos(1/4),
zeta(5). More
generally, we develop a collection of constructions based on simple
probabilistic mechanisms that enable one to create Buffon experiments
involving
compositions of exponentials and logarithms, polylogarithms, direct
and inverse
trigonometric functions, algebraic and hypergeometric functions, as
well as
functions defined by integrals, such as the Gaussian error function.
http://arxiv.org/abs/0906.5560
---------------------------------------------------------------
8713. STRONG TAYLOR APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
AND APPLICATION TO THE L\'EVY LIBOR MODEL
Antonis Papapantoleon and Maria Siopacha
In this article we consider the strong approximation of stochastic
differential equations driven by L\'evy processes or general
semimartingales.
The main ingredients of our method is the perturbation of the SDE and
the
Taylor expansion of the resulting parameterized curve. We apply this
method to
develop strong approximation schemes for LIBOR market models. In
particular, we
derive fast and precise algorithms for the valuation of derivatives in
LIBOR
models which are more tractable than the simulation of the full SDE. A
numerical example for the L\'evy LIBOR model illustrates our method.
http://arxiv.org/abs/0906.5581
---------------------------------------------------------------
8714. RELATIVE DENSITY OF THE RANDOM R-FACTOR PROXIMITY CATCH DIGRAPH
FOR TESTING SPATIAL PATTERNS OF SEGREGATION AND ASSOCIATION
(TECHNICAL REPORT)
Elvan Ceyhan and Carey E. Priebe and John C. Wierman
Statistical pattern classification methods based on data-random graphs
were
introduced recently. In this approach, a random directed graph is
constructed
from the data using the relative positions of the data points from
various
classes. Different random graphs result from different definitions of
the
proximity region associated with each data point and different graph
statistics
can be employed for data reduction. The approach used in this article
is based
on a parameterized family of proximity maps determining an associated
family of
data-random digraphs. The relative arc density of the digraph is used
as the
summary statistic, providing an alternative to the domination number
employed
previously. An important advantage of the relative arc density is that,
properly re-scaled, it is a U-statistic, facilitating analytic study
of its
asymptotic distribution using standard U-statistic central limit
theory. The
approach is illustrated with an application to the testing of spatial
patterns
of segregation and association. Knowledge of the asymptotic
distribution allows
evaluation of the Pitman and Hodges-Lehmann asymptotic efficacy, and
selection
of the proximity map parameter to optimize efficacy. Notice that the
approach
presented here also has the advantage of validity for data in any
dimension.
http://arxiv.org/abs/0906.5436
---------------------------------------------------------------
8715. MARTINGALE DIFFERENCES AND THE METRIC THEORY OF CONTINUED
FRACTIONS
Alan K. Haynes and Jeffrey D. Vaaler
We investigate a collection of orthonormal functions that encodes
information
about the continued fraction expansion of real numbers. When suitably
ordered
these functions form a complete system of martingale differences and
are a
special case of a class of martingale differences considered by R. F.
Gundy. By
applying known results for martingales we obtain corresponding metric
theorems
for the continued fraction expansion of almost all real numbers.
http://arxiv.org/abs/0906.5428
---------------------------------------------------------------
8716. RANDOM $K$-NONCROSSING RNA STRUCTURES
William Y.C. Chen and Hillary S.W. Han and Christian M. Reidys
In this paper we derive polynomial time algorithms that generate random
$k$-noncrossing matchings and $k$-noncrossing RNA structures with
uniform
probability. Our approach employs the bijection between $k$-noncrossing
matchings and oscillating tableaux and the $P$-recursiveness of the
cardinalities of $k$-noncrossing matchings. The main idea is to
consider the
tableaux sequences as paths of stochastic processes over shapes and to
derive
their transition probabilities.
http://arxiv.org/abs/0906.5553
---------------------------------------------------------------
8717. AN EXPERIMENTAL MATHEMATICS PERSPECTIVE ON THE OLD, AND STILL
OPEN, QUESTION OF WHEN TO STOP?
Luis A. Medina and Doron Zeilberger
In a recent article in American Scientist, Theodore Hill described a
coin-tossing game whose pay-off is the number of heads over the total
number of
throws. Suppose that at a given point during the game you have 5 heads
and 3
tails, should you stop and get 5/8, or should you keep playing, hoping
to get a
better score? This is still an open problem. In the present article,
we explore
different strategies to this game from the Experimental Mathematics
perspective.
http://arxiv.org/abs/0907.0032
---------------------------------------------------------------
8718. EXPONENTIAL INEQUALITIES FOR THE DISTRIBUTIONS OF CANONICAL U-
AND V-STATISTICS OF DEPENDENT OBSERVATIONS
I.S.Borisov and N.Volodko
The exponential inequalities are obtained for the distribution tails of
canonical (degenerate) U- and V-statistics of an arbitrary order based
on
samples from uniformly strong mixing stationary sequences.
http://arxiv.org/abs/0907.0058
---------------------------------------------------------------
8719. CONTINUITY ON STOCHASTIC CONTROL PROBLEM WITH STOPPING TIME
Qingshuo Song and Jie Yang
Stochastic control problem with stopping time in finite time horizon is
considered. In the general framework, the value function is
characterized as
the unique viscosity solution of Bellman equation on a bounded domain.
It
requires the continuity of the value function on its bounded domain.
However,
the necessary and sufficient condition on continuity still remains
unclear. In
this paper, compared to the existing literature, a more general
sufficient
condition for the continuity of the value function is achieved by
studying
pathwise local behavior of underlying stochastic processes on the
boundary of
the domain. In addition, a simplified proof for the existence of the
value
function is provided by applying the dynamic programming principle to a
naturally chosen stopping time.
http://arxiv.org/abs/0907.0062
---------------------------------------------------------------
8720. STABILITY FOR RANDOM MEASURES, POINT PROCESSES AND DISCRETE
SEMIGROUPS
Youri Davydov and Ilya Molchanov and Sergei Zuyev
A scaling operation on non-negative integers can be defined in a
randomised
way by transforming an integer into the corresponding binomial
distribution
with success probability being the scaling factor. We explore a similar
(thinning) operation defined on counting measures and characterise the
corresponding discrete stablility property of point processes. It is
shown that
these processes are exactly Cox (doubly stochastic Poisson) processes
with
strictly stable random intensity measures. The paper contains spectral
and
LePage representations for strictly stable measures and characterises
some
special cases, e.g. independently scattered measures. As consequence,
spectral
representations are provided for the probability generating functional
and void
probabilities of discrete stable processes. An alternative cluster
representation for discrete stable processes is also derived using the
so-called Sibuya point processes that constitute a new family of
purely random
point processes. The obtained results are then applied to explore
stable random
elements in discrete semigroups, where the scaling is defined by means
of
thinning of a point process on the basis of the semigroup. Particular
examples
include discrete stable vectors that generalise the one-dimensional
case of
discrete random variables studied by Steutel and van Harn (1979) and
the family
of natural numbers with the multiplication operation, where the primes
form the
basis.
http://arxiv.org/abs/0907.0077
---------------------------------------------------------------
8721. A PURE JUMP MARKOV PROCESS WITH A RANDOM SINGULARITY SPECTRUM
Julien Barral and Nicolas Fournier and Stephane Jaffard and
Stephane Seuret
We construct a non-decreasing pure jump Markov process, whose jump
measure
heavily depends on the values taken by the process. We determine the
singularity spectrum of this process, which turns out to be random and
to
depend locally on the values taken by the process. The result relies
on fine
properties of the distribution of Poisson point processes and on
ubiquity
theorems.
http://arxiv.org/abs/0907.0104
---------------------------------------------------------------
8722. COHERENT FREQUENTISM
David R. Bickel
The certainty distribution, a fiducial-like distribution of a scalar
interest
parameter, combines the logical consistency of Bayesian methods with the
reliability of Neyman-Pearson methods. As a probability distribution
over
parameter space, the certainty distribution is coherent in the sense
that it
satisfies the axioms of the decision-theoretic and logic-theoretic
systems
typically cited in support of the Bayesian posterior distribution.
Since the
probabilities of a certainty distribution by definition are equal to the
coverage rates of the corresponding confidence intervals, the resulting
inferences are uniquely minimax to risk in a betting game designed to
quantify
inferential reliability.
http://arxiv.org/abs/0907.0139
---------------------------------------------------------------
8723. SETS OF FINITE PERIMETER AND THE HAUSDORFF-GAUSS MEASURE ON THE
WIENER SPACE
Masanori Hino
In Euclidean space, the integration by parts formula for a set of finite
perimeter is expressed by the integration with respect to a type of
surface
measure. According to geometric measure theory, this surface measure is
realized by the one-codimensional Hausdorff measure restricted on the
reduced
boundary and/or the measure-theoretic boundary, which may be strictly
smaller
than the topological boundary. In this paper, we discuss the
counterpart of
this measure in the abstract Wiener space, which is a typical
infinite-dimensional space. We introduce the concept of the measure-
theoretic
boundary in the Wiener space and provide the integration by parts
formula for
sets of finite perimeter. The formula is presented in terms of the
integration
with respect to the one-codimensional Hausdorff-Gauss measure
restricted on the
measure-theoretic boundary.
http://arxiv.org/abs/0907.0056
---------------------------------------------------------------
8724. THE METRIC THEORY OF P-ADIC APPROXIMATION
Alan K. Haynes
Metric Diophantine approximation in its classical form is the study of
how
well almost all real numbers can be approximated by rationals. There
is a long
history of results which give partial answers to this problem, but
there are
still questions which remain unknown. The Duffin-Schaeffer Conjecture
is an
attempt to answer all of these questions in full, and it has withstood
more
than fifty years of mathematical investigation. In this paper we
establish a
strong connection between the Duffin-Schaeffer Conjecture and its p-adic
analogue. Our main theorems are transfer principles which allow us to
go back
and forth between these two problems. We prove that if the variance
method from
probability theory can be used to solve the p-adic Duffin-Schaeffer
Conjecture
for even one prime p, then almost the entire classical Duffin-Schaeffer
Conjecture would follow. Conversely if the variance method can be used
to prove
the classical conjecture then the p-adic conjecture is true for all
primes.
Furthermore we are able to unconditionally and completely establish
the higher
dimensional analogue of this conjecture in which we allow simultaneous
approximation in any finite number and combination of real and p-adic
fields,
as long as the total number of fields involved is greater than one.
Finally by
using a mass transference principle for Hausdorff measures we are able
to
extend all of our results to their corresponding analogues with Haar
measures
replaced by the Hausdorff measures associated with arbitrary dimension
functions.
http://arxiv.org/abs/0907.0141
-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
------------------------------------------------------------------------------------
Please don't send me Word or PowerPoint attachments if not
absolutely necessary. See:
http://www.gnu.org/philosophy/no-word-attachments.html
More information about the PAS
mailing list