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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.
Author(s): Tomasz Schreiber
Abstract: 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
Author(s): Glenn Shafer and Vladimir Vovk and and Akimichi Takemura
Abstract: 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
Author(s): Alexander Gnedin and Grigori Olshanski
Abstract: 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
Author(s): Svante Janson and Oliver Riordan
Abstract: 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
Author(s): Pierre Tarr\`es and Pierre Vandekerkhove
Abstract: 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
Author(s): Martin Hutzenthaler and Arnulf Jentzen
Abstract: 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
Author(s): E. Kowalski and A. Nikeghbali
Abstract: 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
Author(s): Svante Janson and Oliver Riordan
Abstract: 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
Author(s): Nicole El Karoui (PMA and CMAP) and Monique Jeanblanc (DP) and Ying Jiao (PMA)
Abstract: 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
Author(s): Svante Janson and Tomasz {\L}uczak and Ilkka Norros
Abstract: 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
Author(s): Robin Pemantle and Herbert S. Wilf
Abstract: 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
Author(s): Kay Kirkpatrick
Abstract: 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
Author(s): Alexei Borodin and Vadim Gorin and Eric M. Rains
Abstract: 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
Author(s): J\'er\'emie Unterberger (IECN)
Abstract: 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
Author(s): Peter Imkeller and Goncalo dos Reis
Abstract: 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
Author(s): Theo van Uem
Abstract: 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
Author(s): Vadim Shcherbakov and Stanislav Volkov
Abstract: 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
Author(s): Svante Janson and Andrzej Rucinski
Abstract: 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
Author(s): Lijun Bo (XIDIAN) and Kehua Shi (NANKAI) and Yongjin Wang (NANKAI)
Abstract: 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
Author(s): Frank Aurzada and Leif D\"oring
Abstract: 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
Author(s): Rapha\"el Lachi\`eze-Rey
Abstract: 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
Author(s): Subhankar Ghosh
Abstract: 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
Author(s): Vitali Wachtel
Abstract: 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
Author(s): Bela Bollobas and Oliver Riordan
Abstract: 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
Author(s): Michael Bjorklund
Abstract: 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
Author(s): Wouter Kager and Lionel Levine
Abstract: 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
Author(s): Damir Filipovic and Stefan Tappe and Josef Teichmann
Abstract: 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
Author(s): Es-Sebaiy Khalifa (SAMOS) and Idir Ouassou and Youssef Ouknine
Abstract: 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
Author(s): Dmitry Panchenko
Abstract: 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
Author(s): Yunjiang Jiang
Abstract: 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
Author(s): Vygantas Paulauskas
Abstract: 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
Author(s): Andrey Novikov and Petr Novikov
Abstract: 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
Author(s): Predrag R. Jelenkovic and Mariana Olvera-Cravioto
Abstract: 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
Author(s): Hakima Bessaih and Annie Millet (CES and SAMOS and PMA)
Abstract: 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
Author(s): Mathieu Faure (UNINE) and Roth Gregory (UNINE)
Abstract: 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
Author(s): Arash Fahim and Nizar Touzi and Xavier Warin
Abstract: 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
Author(s): Kevin P. Costello and Van Vu
Abstract: 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
Author(s): W. Hebisch and B. Zegarlinski
Abstract: 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
Author(s): Gerardo Hernandez-del-Valle
Abstract: 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
Author(s): Gerardo Hernandez-del-Valle
Abstract: In this paper we derive the density $\varphi$ of the first time $T$ that a
continuous martingale $M$ with non-random quadratic variation
$_\cdot:=\int_0^\cdot h^2(u)du$ 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
Author(s): Mariusz Bieniek and Krzysztof Burdzy and Sam Finch
Abstract: 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
Author(s): Raluca Balan
Abstract: 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
Author(s): Lo\"ic Chaumont and Ger\'onimo Uribe Bravo
Abstract: 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
Author(s): Milton Jara and Tomasz Komorowski
Abstract: 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
Author(s): Gideon Amir and Omer Angel and Balint Virag
Abstract: 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
Author(s): Joseph Najnudel and Bernard Roynette and Marc Yor
Abstract: 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
Author(s): Philippe Biane G\'erard Letac
Abstract: 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
Author(s): Nelly Litvak and Philippe Robert (INRIA)
Abstract: 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
Author(s): Nizar Demni
Abstract: 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
Author(s): Kilian Raschel
Abstract: 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
Author(s): T. M. Liggett and R. B. Schinazi
Abstract: 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
Author(s): Jeremy J. Becnel
Abstract: 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
Author(s): Massimiliano Gubinelli (CEREMADE) and Antoine Lejay (IECN and INRIA Sophia Antipolis / INRIA Lorraine / IECN)
Abstract: 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
Author(s): V'\ictor Rivero
Abstract: 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
Author(s): Baruch Fischer and Moshe Zakai
Abstract: 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
Author(s): Omer Angel and Itai Benjamini and Ori Gurel-Gurevich and Tom Meyerovitch and Ron Peled
Abstract: 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
Author(s): Beno\^it Collins (ICJ) and Ion Nechita (ICJ)
Abstract: 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
Author(s): J. Martin Lindsay and Adam G. Skalski
Abstract: 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
Author(s): Krzysztof Bogdan and Tomasz Grzywny and Michal Ryznar
Abstract: 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
Author(s): Raluca Balan
Abstract: 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
Author(s): Yves F. Atchade
Abstract: 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
Author(s): Qi Zhang
Abstract: 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
Author(s): David Applebaum and Markus Riedle
Abstract: 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
Author(s): Svante Linusson
Abstract: 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
Author(s): A. Faggionato
Abstract: 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
Author(s): Iosif Pinelis
Abstract: 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
Author(s): Michael J. Kozdron (University of Regina)
Abstract: 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
Author(s): G. Ben Arous and I. Corwin
Abstract: 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
Author(s): Xin Chen and Xue-Mei Li and Bo Wu
Abstract: 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
Author(s): F. Martinelli (Matematica and Roma 3) and F. Toninelli (CNRS and ENS Lyon)
Abstract: 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
Author(s): Thierry L\'evy (DMA) and Myl\`ene Ma\"ida (LM-Orsay)
Abstract: 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
Author(s): Giacomo Aletti and Caterina May and Piercesare Secchi
Abstract: 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
Author(s): Frank Aurzada
Abstract: 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
Author(s): Franz Lehner
Abstract: 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
Author(s): A. Mijatovi\'c and P. Schneider
Abstract: 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
Author(s): Laurent Duvernet
Abstract: 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
Author(s): Takahiro Hasebe
Abstract: 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
Author(s): Adrien Joseph (PMA)
Abstract: 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
Author(s): Federica Masiero
Abstract: 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
Author(s): Denis Villemonais (CMAP)
Abstract: 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
Author(s): Qiang Zhen and Johan S. H. van Leeuwaarden and Charles Knessl
Abstract: 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
Author(s): Aleksandar Mijatovic and Mikhail Urusov
Abstract: 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
Author(s): Erhan Bayraktar and Song Yao
Abstract: 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
Author(s): Rodolphe Garbit (LMJL)
Abstract: 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
Author(s): Christophe Sabot (ICJ) and Laurent Tournier (ICJ)
Abstract: 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
Author(s): Jin Ma and Song Yao
Abstract: 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
Author(s): N. Modarresi and S. Rezakhah
Abstract: 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
Author(s): Hirofumi Osada
Abstract: 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
Author(s): Alexei Borodin and Patrik L. Ferrari and Michael Praehofer and Tomohiro Sasamoto, Jon Warren
Abstract: 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
Author(s): C. Borgs and J. Chayes and L. Lov\'asz and V.T. S\'os and K. Vesztergombi
Abstract: 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
Author(s): Zakhar Kabluchko and Martin Schlather
Abstract: 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
Author(s): George P. Yanev and M. Ahsanullah
Abstract: 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
Author(s): Saidbek S.Mirakhmedov and Sherzod M.Mirakhmedov
Abstract: 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
Author(s): Da-Sheng Zhou and Dang-Zheng Liu and Tao Qian
Abstract: 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
Author(s): Camille Petit (IF)
Abstract: 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
Author(s): Nathan Keller
Abstract: 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
Author(s): R. Liptser
Abstract: 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
Author(s): Xavier Bardina and David Bascompte
Abstract: 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
Author(s): Ryoki Fukushima
Abstract: 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
Author(s): Shankar Bhamidi and Remco van der Hofstad and Gerard Hooghiemstra
Abstract: 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
Author(s): Joshua N. Cooper
Abstract: 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
Author(s): Andreas Eberle and Carlo Marinelli
Abstract: 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
Author(s): Luigi Accardi and Andreas Boukas
Abstract: 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
Author(s): Sunil Abraham and Greg Brockman and Stephanie Sapp and Anant P. Godbole
Abstract: 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
Author(s): Bernardo N. B. de Lima and Remy Sanchis and Roger W. C. Silva
Abstract: 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
Author(s): Jozsef Balogh and Bela Bollobas and Mark Walters
Abstract: 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
Author(s): Laszlo Erdos and Jose A. Ramirez and Benjamin Schlein and Horng-Tzer Yau
Abstract: 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
Author(s): Daniil Ryabko (INRIA Futurs and LIFL and INRIA Lille - Nord Europe)
Abstract: 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
Author(s): Sami Assaf and Persi Diaconis and K. Soundararajan
Abstract: 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
Author(s): Pierre Calka (MAP5) and Julien Michel (UMPA-ENSL) and Sylvain Porret-Blanc (UMPA-ENSL)
Abstract: 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
Author(s): Daniil Ryabko (INRIA Futurs and LIFL and INRIA Lille - Nord Europe)
Abstract: 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
Author(s): Luc Devroye and Svante Janson
Abstract: 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
Author(s): Lasse Leskel\"a (Helsinki University of Technology)
Abstract: 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
Author(s): Gianluca Cassese
Abstract: We prove a version of Rao decomposition for quasi-martingales indexed by a
linearly ordered set.
http://arxiv.org/abs/0906.0183
Author(s): Sven Erick Alm and Svante Linusson
Abstract: 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
Author(s): Ioannis Kontoyiannis and Sean P. Meyn
Abstract: 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
Author(s): Simon Harris and Matthew Roberts
Abstract: 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
Author(s): Iosif Pinelis and Raymond Molzon
Abstract: 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
Author(s): Michael Krivelevich and Tobias Muller
Abstract: 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
Author(s): Imre B\'ar\'any and Ferenc Fodor and Viktor V\'igh
Abstract: 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
Author(s): Yakov Nikitin
Abstract: 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
Author(s): Terence Tao and Van Vu
Abstract: 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
Author(s): Kei Funano
Abstract: 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
Author(s): Sven Erick Alm and Svante Linusson
Abstract: 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
Author(s): Freddy Delbaen (Department of Mathematics) and Ying Hu (IRMAR) and Adrien Richou (IRMAR)
Abstract: 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
Author(s): Kei Funano
Abstract: 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
Author(s): Peter Harremoes and Oliver Johnson and Ioannis Kontoyiannis
Abstract: 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
Author(s): S. N. Ethier and Jiyeon Lee
Abstract: 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
Author(s): Asaf Cohen and Eilon Solan
Abstract: 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
Author(s): David C. Brydges and John Z. Imbrie and Gordon Slade
Abstract: 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
Author(s): Hubert Lacoin (PMA) and Gregorio Moreno Flores (PMA)
Abstract: 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
Author(s): Sourav Chatterjee and Partha S. Dey
Abstract: 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
Author(s): Alexander Nazarov and Natalia Stepanova
Abstract: 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
Author(s): Eduardo Sontag and Doron Zeilberger
Abstract: 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
Author(s): Yu Zhang
Abstract: 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
Author(s): Przemyslaw Klusik and Zbigniew Palmowski
Abstract: 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
Author(s): Zhen-Qing Chen and Joshua Tokle
Abstract: 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
Author(s): Pietro Caputo and Thomas M. Liggett and Thomas Richthammer
Abstract: 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
Author(s): L. Utkin and S. Destercke
Abstract: 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
Author(s): Jeremie Unterberger (IECN)
Abstract: 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
Author(s): Francois Bolley (CEREMADE) and Arnaud Guillin and Florent Malrieu (IRMAR)
Abstract: 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
Author(s): S. N. Ethier and Fred M. Hoppe
Abstract: 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
Author(s): Anders Bj\"orner
Abstract: 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
Author(s): Kei Fukuda and Akihiko Inoue and Yumiharu Nakano
Abstract: 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
Author(s): Sergey G. Bobkov and Michel Ledoux
Abstract: 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
Author(s): Matteo Novaga and Pietro Majer
Abstract: We determine the probability tresholds for the existence of infinite paths in
random shift graphs.
http://arxiv.org/abs/0906.1689
Author(s): Olga Aryasova and Andrey Pilipenko
Abstract: 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
Author(s): Reinhard Hoepfner
Abstract: 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
Author(s): A. B. Dieker
Abstract: 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
Author(s): Xicheng Zhang
Abstract: 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
Author(s): Joseph Najnudel and Ashkan Nikeghbali
Abstract: 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
Author(s): Erik Alfsen and Fred Shultz
Abstract: 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
Author(s): Raoul Normand
Abstract: 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
Author(s): Reinhard Hoepfner
Abstract: 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
Author(s): Beno\^it Collins (ICJ) and Ion Nechita (ICJ)
Abstract: 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
Author(s): Imen Boutouria and Abdelhamid Hassairi
Abstract: 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
Author(s): Jeffrey Matayoshi
Abstract: 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
Author(s): M. A. Rajabpour
Abstract: 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
Author(s): Jian Ding and Jeong Han Kim and Eyal Lubetzky and Yuval Peres
Abstract: 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
Author(s): Jian Ding and Jeong Han Kim and Eyal Lubetzky and Yuval Peres
Abstract: 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
Author(s): Jan van Neerven and Lutz Weis
Abstract: 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
Author(s): Michel Bonnefont
Abstract: 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
Author(s): G\"unter Last and Hermann Thorisson
Abstract: 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
Author(s): Panayotis Mertikopoulos and Aris L. Moustakas
Abstract: 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
Author(s): Irmina Czarna and Zbigniew Palmowski
Abstract: 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
Author(s): David Dereudre and Hans-Otto Georgii
Abstract: 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
Author(s): Maxime Fevrier and Alexandru Nica
Abstract: 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
Author(s): Sonja Cox and Jan van Neerven
Abstract: 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
Author(s): Shui Feng and Fuqing Gao
Abstract: 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
Author(s): D. V. Tokarev and K. A. Borovkov
Abstract: 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
Author(s): Rick Durrett
Abstract: 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
Author(s): Rami Atar and Gennady Shaikhet
Abstract: 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
Author(s): Robert A. Jarrow and Philip Protter and Hasanjan Sayit
Abstract: 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
Author(s): Yacine A\"it-Sahalia and Julio Cacho-Diaz and T. R. Hurd
Abstract: 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
Author(s): Yannis G. Yatracos
Abstract: 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
Author(s): Dawn B. Woodard and Scott C. Schmidler and Mark Huber
Abstract: 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
Author(s): Daniel Rudolf
Abstract: 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
Author(s): Idris Kharroubi (PMA and Crest) and Huyen Pham (PMA and Crest)
Abstract: 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
Author(s): Chunhua Ma
Abstract: 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
Author(s): Holger K\"osters
Abstract: 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
Author(s): Raphael Lefevere and Lorenzo Zambotti
Abstract: 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
Author(s): Laurent Bienvenu and Alexander Shen
Abstract: 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
Author(s): Palle E. T. Jorgensen and Erin P. J. Pearse
Abstract: 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
Author(s): M. Cranston and L. Koralov and S. Molchanov and B. Vainberg
Abstract: 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
Author(s): Vydas \v{C}ekanavi\v{c}ius and Erol A. Pek\"oz and Adrian R\"ollin and Michael Shwartz
Abstract: 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
Author(s): Katarzyna Bartkiewicz and Adam Jakubowski and Thomas Mikosch and Olivier Wintenberger (CEREMADE)
Abstract: 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
Author(s): O. Chis and D. Opris
Abstract: 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
Author(s): Elizaveta Frenkel and Alexei G. Myasnikov and Vladimir N. Remeslennikov
Abstract: 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
Author(s): Scott N. Armstrong and Charles K. Smart
Abstract: 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
Author(s): Ian Palmer and Jean Bellissard
Abstract: 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
Author(s): C. Berg and C. Vignat
Abstract: 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
Author(s): Yaozhong Hu and David Nualart and Jian Song
Abstract: 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
Author(s): Antal A.Jarai and F. Redig and E. Saada
Abstract: 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
Author(s): Matthias Meiners and Gerold Alsmeyer
Abstract: 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
Author(s): Marcelo R. Hilario and Oren Louidor and Charles M. Newman and Leonardo T. Rolla, Scott Sheffield, Vladas Sidoravicius
Abstract: 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
Author(s): Stefano M. Iacus and Nakahiro Yoshida
Abstract: 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
Author(s): Tadeusz Kulczycki and Mateusz Kwa\'snicki and Jacek Ma{\l}ecki and Andrzej Stos
Abstract: 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
Author(s): Martin Keller-Ressel and Walter Schachermayer and Josef Teichmann
Abstract: 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
Author(s): Kouji Yano and Marc Yor
Abstract: 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
Author(s): Jianhai Bao and Xuerong Mao and Chenggui Yuan
Abstract: 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
Author(s): David F. Anderson and Jonathan C. Mattingly
Abstract: 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
Author(s): Julie O'Donovan
Abstract: 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
Author(s): R\'emi Rhodes (CEREMADE) and Bamba A. Sow (LERSTAD)
Abstract: 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
Author(s): Vincent Beffara (UMPA-ENSL) and Pierre Nolin (CIMS)
Abstract: 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
Author(s): Maria Simonetta Bernabei and Horst Thaler
Abstract: 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
Author(s): Itai Benjamini and Stephane Boucheron and Gabor Lugosi and Raphael Rossignol
Abstract: 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
Author(s): Makoto Katori and Hideki Tanemura
Abstract: 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
Author(s): Gregory F. Lawler and Scott Sheffield
Abstract: 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
Author(s): Saul Jacka
Abstract: 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
Author(s): Subhankar Ghosh and Larry Goldstein
Abstract: 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
Author(s): Christian Bartsch and Nina Gantert and Michael Kochler
Abstract: 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
Author(s): Wlodzimierz Bryc and Abdelhamid Hassairi
Abstract: 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
Author(s): Nathalie Castelle (LM-Orsay)
Abstract: 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
http://arxiv.org/abs/0906.3952
Author(s): Sasha Sodin
Abstract: 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
Author(s): Taral Guldahl Seierstad
Abstract: 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
Author(s): Matthias Heveling and Matthias Reitzner
Abstract: 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
Author(s): Kshitij Khare and Hua Zhou
Abstract: 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
Author(s): Pierre Del Moral and Fr\'ed\'eric Patras and Sylvain Rubenthaler
Abstract: 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
Author(s): Masanori Hino
Abstract: 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
Author(s): M. Cranston and D. Gauthier and T. S. Mountford
Abstract: 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
Author(s): Lihu Xu and Marco Romito
Abstract: 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
Author(s): W. Liu and S. V. Lototsky
Abstract: 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
Author(s): Laszlo Erdos and Jose Ramirez and Benjamin Schlein and Terence Tao and Van Vu and Horng-Tzer Yau
Abstract: 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
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X)
Abstract: 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
Author(s): Ken R. Duffy and Sean P. Meyn
Abstract: 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
Author(s): Athanasios Batakis (MAPMO) and Michel Zinsmeister (MAPMO)
Abstract: 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
Author(s): Fr\'ed\'eric Lavancier (LMJL) and Anne Philippe (LMJL) and Donatas Surgailis
Abstract: 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
Author(s): Terence Tao
Abstract: 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
Author(s): Alain Comtet and Yves Tourigny
Abstract: 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
Author(s): C. Giardina and F. Redig and K. Vafayi
Abstract: 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
Author(s): T. Banica
Abstract: 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
Author(s): Michael B. Marcus and Jay Rosen
Abstract: 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
Author(s): Piotr Garbaczewski and Vladimir Stephanovich
Abstract: 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
Author(s): J. Theodore Cox and Rinaldo B. Schinazi
Abstract: 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
Author(s): DeTao Mao and Wenyuan Li
Abstract: 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
Author(s): Ronan Le Gu\'evel (LMJL) and Jacques L\'evy-V\'ehel (INRIA Saclay - Ile de France)
Abstract: 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
Author(s): Mark S. Veillette and Murad S. Taqqu
Abstract: 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
Author(s): Masoud M. Nasari
Abstract: 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
Author(s): Larry Goldstein
Abstract: 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
Author(s): Vadim A. Kaimanovich and Florian Sobieczky
Abstract: 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
Author(s): Ioannis Kontoyiannis and Sean P. Meyn
Abstract: 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
Author(s): Xiaofeng Shao
Abstract: 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
Author(s): J. Bouttier and E. Guitter
Abstract: 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
Author(s): Daron Acemoglu and Asuman Ozdaglar and Ali ParandehGheibi
Abstract: 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
Author(s): Peter J. Forrester and Manjunath Krishnapur
Abstract: 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
Author(s): Kei Kobayashi
Abstract: 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
Author(s): Gergely Ambrus and Imre Barany
Abstract: 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
Author(s): I.S.Borisov and N.Volodko
Abstract: 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
Author(s): Sara Brofferio (LM-Orsay) and Bruno Schapira (LM-Orsay)
Abstract: 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
Author(s): Philippe Flajolet and Maryse Pelletier and Michele Soria
Abstract: 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
Author(s): Antonis Papapantoleon and Maria Siopacha
Abstract: 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
Author(s): Elvan Ceyhan and Carey E. Priebe and John C. Wierman
Abstract: 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
Author(s): Alan K. Haynes and Jeffrey D. Vaaler
Abstract: 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
Author(s): William Y.C. Chen and Hillary S.W. Han and Christian M. Reidys
Abstract: 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
Author(s): Luis A. Medina and Doron Zeilberger
Abstract: 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
Author(s): I.S.Borisov and N.Volodko
Abstract: 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
Author(s): Qingshuo Song and Jie Yang
Abstract: 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
Author(s): Youri Davydov and Ilya Molchanov and Sergei Zuyev
Abstract: 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
Author(s): Julien Barral and Nicolas Fournier and Stephane Jaffard and Stephane Seuret
Abstract: 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
Author(s): David R. Bickel
Abstract: 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
Author(s): Masanori Hino
Abstract: 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
Author(s): Alan K. Haynes
Abstract: 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
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