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Probability Abstracts 107
This document contains abstracts 7696-7953
from November-1-2008 to December-31-2008.
They have been mailed on Jan 4, 2009.
Author(s): Benedek Valk\'o and B\'alint Vir\'ag
Abstract: We show that in the point process limit of the bulk eigenvalues of
$\beta$-ensembles of random matrices, the probability of having no eigenvalue
in a fixed interval of size $\lambda$ is given by $$
(\kappa_\beta+o(1))\lambda^{\gamma_\beta} \exp(-
\frac{\beta}{64}\lambda^2+(\frac{\beta}{4}-\frac18)\lambda) $$ as
$\lambda\to\infty$, where $$ \gamma_\beta={1/4}(\frac\beta{2}+\frac{2}{\beta}-
3). $$ This is a slightly corrected version of a prediction by Dyson. Our proof
uses the new Brownian carousel representation of the limit process, as well as
the Cameron-Martin-Girsanov transformation in stochastic calculus.
http://arxiv.org/abs/0811.0007
Author(s): Dan Goreac
Abstract: In this paper we study a criterion for the viability of stochastic semilinear
control systems on a real, separable Hilbert space. The necessary and
sufficient condition is given using the notion of stochastic quasi-tangency. As
a consequence, we prove that approximate viability and the viability property
coincide for stochastic linear control systems. The paper generalizes recent
results from the deterministic framework.
http://arxiv.org/abs/0811.0098
Author(s): Patrick Cattiaux (IMT) and Sylvie M\'el\'eard (CMAP)
Abstract: We are interested in the long time behavior of a two-type density-dependent
biological population conditioned to non-extinction, in both cases of
competition or weak cooperation between the two species. This population is
described by a stochastic Lotka-Volterra system, obtained as limit of
renormalized interacting birth and death processes. The weak cooperation
assumption allows the system not to blow up. We study the existence and
uniqueness of a quasi-stationary distribution, that is convergence to
equilibrium conditioned to non extinction. To this aim we generalize in
two-dimensions spectral tools developed for one-dimensional generalized Feller
diffusion processes. The existence proof of a quasi-stationary distribution is
reduced to the one for a $d$-dimensional Kolmogorov diffusion process under a
symmetry assumption. The symmetry we need is satisfied under a local balance
condition relying the ecological rates. A novelty is the outlined relation
between the uniqueness of the quasi-stationary distribution and the
ultracontractivity of the killed semi-group. By a comparison between the
killing rates for the populations of each type and the one of the global
population, we show that the quasi-stationary distribution can be either
supported by individuals of one (the strongest one) type or supported by
individuals of the two types. We thus highlight two different long time
behaviors depending on the parameters of the model: either the model exhibits
an intermediary time scale for which only one type (the dominant trait) is
surviving, or there is a positive probability to have coexistence of the two
species.
http://arxiv.org/abs/0811.0240
Author(s): Nina Gantert and Yueyun Hu and Zhan Shi
Abstract: Consider a discrete-time one-dimensional supercritical branching random walk.
We study the probability that there exists an infinite ray in the branching
random walk that always lies above the line of slope $\gamma-\epsilon$, where
$\gamma$ denotes the asymptotic speed of the right-most position in the
branching random walk. Under mild general assumptions upon the distribution of
the branching random walk, we prove that when $\epsilon\to 0$, the probability
in question decays like $\exp\{- {\beta + o(1)\over \epsilon^{1/2}}\}$, where
$\beta$ is a positive constant depending on the distribution of the branching
random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables
(with $0
http://arxiv.org/abs/0811.0262
Author(s): Georgi Dimitroff and Michael Scheutzow
Abstract: We study transport properties of isotropic Brownian flows. Under a transience
condition for the two-point motion, we show asymptotic normality of the image
of a finite measure under the flow and -- under slightly stronger assumptions
-- asymptotic normality of the distribution of the volume of the image of a set
under the flow. Finally, we show that for a class of isotropic flows, the
volume of the image of a nonempty open set (which is a martingale) converges to
a random variable which is almost surely strictly positive.
http://arxiv.org/abs/0811.0276
Author(s): Leandro P. R. Pimentel and Raphael Rossignol
Abstract: Let N be distributed as a Poisson random set on R^d with intensity comparable
to the Lebesgue measure. Consider the Voronoi tiling of R^d, (C_v)_{v\in N},
where C_v is composed by points x in R^d that are closer to v than to any other
v' in N. A polyomino P of size n is a connected union (in the usual R^d
topological sense) of n tiles, and we denote by Pi_n the collection of all
polyominos P of size n containing the origin. Assume that the weight of a
Voronoi tile C_v is given by F(C_v), where F is a nonnegative functional on
Voronoi tiles. In this paper we investigate the tail behavior of the maximal
weight among polyominoes in Pi_n for some functionals F, mainly when F(C_v) is
the number of faces of C_v. Next we apply our results to study self-avoiding
paths, first-passage percolation models and the stabbing number on the dual
graph, named the Delaunay triangulation. As the main application we show that
first passage percolation has at most linear variance.
http://arxiv.org/abs/0811.0308
Author(s): Francis Comets and Serguei Popov and Gunter M. Sch\"utz and Marina Vachkovskaia
Abstract: We consider a stochastic billiard in a random tube which stretches to
infinity in the direction of the first coordinate. This random tube is
stationary and ergodic, and also it is supposed to be in some sense
well-behaved. The stochastic billiard can be described as follows: when
strictly inside the tube, the particle moves straight with constant speed. Upon
hitting the boundary, it is reflected randomly, according to the cosine law:
the density of the outgoing direction is proportional to the cosine of the
angle between this direction and the normal vector. We also consider the
discrete-time random walk formed by the particle's positions at the moments of
hitting the boundary. Under the condition of existence of the second moment of
the projected jump length with respect to the stationary measure for the
environment seen from the particle, we prove the quenched invariance principles
for the projected trajectories of the random walk and the stochastic billiard.
http://arxiv.org/abs/0811.0366
Author(s): Yuval Peres and G\'abor Pete and Stephanie Somersille
Abstract: We prove that if X\subset \R^n is compact in the path metric, Y\subset X is
closed, and F is a Lipschitz function on Y, then for each \beta \in \R there
exists a unique viscosity solution to the \beta-biased infinity Laplacian
equation
\beta |\nabla u| + \Delta_\infty u=0 on X\setminus Y, where \Delta_\infty u=
|\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j} . In the proof, we extend
the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the
\beta-biased \eps-game as follows. The starting position is x_0 \in X\backslash
Y. At the k-th step the two players toss a suitably biased coin (in our key
example, player I wins with odds of \exp(\beta\eps) to 1), and the winner
chooses x_k with d(x_k,x_{k-1}) < \eps. The game ends when x_k \in Y, and
player II pays the amount F(x_k) to player I. We prove that the value
u^{\eps}(x_0) of this game exists, and that \|u^\eps - u\|_\infty \to 0 as \eps
\to 0, where u is the unique extension of F to X that satisfies comparison with
\beta-exponential cones. Comparison with exponential cones is a notion that we
introduce here, and generalizing a theorem of Crandall, Evans and Gariepy
regarding comparison with linear cones, we show that a continuous function
satisfies comparison with \beta-exponential cones if and only if it is a
viscosity solution to the \beta-biased infinity Laplacian equation.
http://arxiv.org/abs/0811.0208
Author(s): Volodymyr Nekrashevych and G\'abor Pete
Abstract: Motivated by the renormalization method in statistical physics, Itai
Benjamini defined a finitely generated infinite group G to be scale-invariant
if there is a nested sequence of finite index subgroups G_n that are all
isomorphic to G and whose intersection is a finite group. He conjectured that
every scale-invariant group has polynomial growth, hence is virtually
nilpotent. We disprove his conjecture by showing that the following
self-similar groups of finite automata are also scale-invariant: the
lamplighter groups \F \wr \Z, where \F is any finite Abelian group; the
solvable Baumslag-Solitar groups BS(1,m); the affine groups GL(\Z,d) \ltimes
\Z^d. However, the conjecture remains open with some natural stronger notions
of scale-invariance for groups and transitive graphs. We construct
scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg
group, whose existence is not immediate just from the scale-invariance of the
group.
http://arxiv.org/abs/0811.0220
Author(s): Ernie Croot and Andrew Granville and Robin Pemantle and Prasad Tetali
Abstract: In many integer factoring algorithms, one produces a sequence of integers
(created in a pseudo-random way), and wishes to rapidly determine a subsequence
whose product is a square (which we call a square product). In his lecture at
the 1994 International Congress of Mathematicians, Pomerance observed that the
following problem encapsulates all of the key issues: Select integers a_1, a_2,
>... at random from the interval [1,x], until some (non-empty) subsequence has
product equal to a square. Find good estimate for the expected stopping time of
this process. A good solution to this problem should help one to determine the
optimal choice of parameters for one's factoring algorithm, and therefore this
is a central question.
Pomerance (1994), using an idea of Schroeppel (1985), showed that with
probability 1-o(1) the first subsequence whose product equals a square occurs
after at least J_0^{1-o(1)} integers have been selected, but no more than J_0,
for an appropriate (explicitly determined) J_0=J_0(x). Herein we determine this
expected stopping time up to a constant factor, tightening Pomerance's interval
to $$[ (\pi/4)(e^{-\gamma} - o(1))J_0, (e^{-\gamma} + o(1)) J_0],$$ where
$\gamma = 0.577...$ is the Euler-Mascheroni constant. We will also confirm the
well established belief that, typically, none of the integers in the square
product have large prime factors. We believe the upper of the two bounds to be
asymptotically sharp.
http://arxiv.org/abs/0811.0372
Author(s): Oleg Yu. Vorobyev and Lavrentiy S. Golovkov
Abstract: This article brings in two new discrete distributions: multidimensional
Binomial distribution and multidimensional Poisson distribution. Those
distributions were created in eventology as more correct generalizations of
Binomial and Poisson distributions. Accordingly to eventology new laws take
into account full distribution of events. Also, in article its characteristics
and properties are described
http://arxiv.org/abs/0811.0406
Author(s): Oleg Yu. Vorobyev and Joe J. Goldblatt and Rebecca Finkel
Abstract: The eventological theory of decision-making, the theory of eventfull
decision-making is a theory of decision-making based on eventological
principles and using results of mathematical eventology; a theoretical basis of
the practical eventology. The beginnings of this theory which have arisen from
eventfull representation of the reasonable subject and his decisions in the
form of eventological distributions (E-distributions) of sets of events and
which are based on the eventological H-theorem are offered. The illustrative
example of the eventological decision-making by the reasonable subject on his
own eventfull behaviour in the financial or share market is considered.
http://arxiv.org/abs/0811.0420
Author(s): Elie Aidekon (PMA)
Abstract: Consider a random walk in random environment on a supercritical
Galton--Watson tree, and let $\tau_n$ be the hitting time of generation $n$.
The paper presents a large deviation principle for $\tau_n/n$, both in quenched
and annealed cases. Then we investigate the subexponential situation, revealing
a polynomial regime similar to the one encountered in one dimension. The paper
heavily relies on estimates on the tail distribution of the first regeneration
time.
http://arxiv.org/abs/0811.0438
Author(s): Ivan Kojadinovic and Jean-Luc Marichal
Abstract: We study the moments and the distribution of the discrete Choquet integral
when regarded as a real function of a random sample drawn from a continuous
distribution. Since the discrete Choquet integral includes weighted arithmetic
means, ordered weighted averaging functions, and lattice polynomial functions
as particular cases, our results encompass the corresponding results for these
aggregation functions. After detailing the results obtained in [1] in the
uniform case, we present results for the standard exponential case, show how
approximations of the moments can be obtained for other continuous
distributions such as the standard normal, and elaborate on the asymptotic
distribution of the Choquet integral. The results presented in this work can be
used to improve the interpretation of discrete Choquet integrals when employed
as aggregation functions.
http://arxiv.org/abs/0811.0468
Author(s): He-Jing Hong and Ze-Chun Hu
Abstract: Gaussian correlation conjecture states that the Gaussian measure of the
intersection of two symmetric convex sets is greater or equal to the product of
the measures. In this paper, firstly we prove that the inequality holds when
one of the two convex sets is the intersection of finite centered ellipsoids
and the other one is simply symmetric. Then we prove that any symmetric convex
set can be approximated by the intersection of finite centered ellipsoids, and
thus the inequality holds for any two symmetric convex sets in any dimensional
$\mathbb{R}^n$, i.e. Gaussian correlation conjecture is true.
http://arxiv.org/abs/0811.0488
Author(s): Nizar Demni
Abstract: We provide two equivalent approaches for computing the tail distribution of
the first hitting time of the boundary of the Weyl chamber by a radial Dunkl
process. The first approach is based on a spectral problem with initial value.
The second one expresses the tail distribution by means of the $W$-invariant
Dunkl-Hermite polynomials. Illustrative examples are given by the irreducible
root systems of types $A$, $B$, $D$. The paper ends with an interest in the
case of Brownian motions for which our formulae take determinantal forms.
http://arxiv.org/abs/0811.0504
Author(s): Nizar Demni
Abstract: We write down the generalized Bessel function associated with the root system
of type $D$ by means of multivariate hypergeometric series. Our hint comes from
the particular case of the Brownian motion in the Weyl chamber of type $D$.
http://arxiv.org/abs/0811.0507
Author(s): F. Michel Dekking and Bram Kuijvenhoven
Abstract: We investigate the question under which conditions the algebraic difference
between two independent random Cantor sets $C_1$ and $C_2$ almost surely
contains an interval, and when not. The natural condition is whether the sum
$d_1+d_2$ of the Hausdorff dimensions of the sets is smaller (no interval) or
larger (an interval) than 1. Palis conjectured that \emph{generically} it
should be true that $d_1+d_2>1$ should imply that $C_1-C_2$ contains an
interval. We prove that for 2-adic random Cantor sets generated by a vector of
probabilities $(p_0,p_1)$ the interior of the region where the Palis conjecture
does not hold is given by those $p_0,p_1$ which satisfy $p_0+p_1>\sqrt{2}$ and
$p_0p_1(1+p_0^2+p_1^2)<1$. We furthermore prove a general result which
characterizes the interval/no interval property in terms of the lower spectral
radius of a set of $2\times 2$ matrices.
http://arxiv.org/abs/0811.0525
Author(s): Gabriel Istrate and Madhav V. Marathe and S.S.Ravi
Abstract: In (Istrate, Marathe, Ravi SODA 2001) we advocated the investigation of
robustness of results in the theory of learning in games under adversarial
scheduling models. We provide evidence that such an analysis is feasible and
can lead to nontrivial results by investigating, in an adversarial scheduling
setting, Peyton Young's model of diffusion of norms. In particular, our main
result incorporates into Peyton Young's model.
http://arxiv.org/abs/0803.2495
Author(s): Gabriel Istrate
Abstract: We study nondeterministic and probabilistic versions of a discrete dynamical
system (due to T. Antal, P. L. Krapivsky, and S. Redner) inspired by Heider's
social balance theory. We investigate the convergence time of this dynamics on
several classes of graphs. Our contributions include:
1. We point out the connection between the triad dynamics and a
generalization of annihilating walks to hypergraphs. In particular, this
connection allows us to completely characterize the recurrent states in graphs
where each edge belongs to at most two triangles.
2. We also solve the case of hypergraphs that do not contain edges consisting
of one or two vertices.
3. We show that on the so-called "triadic cycle" graph, the convergence time
is linear.
4. We obtain a cubic upper bound on the convergence time on 2-regular triadic
simplexes G. This bound can be further improved to a quantity that depends on
the Cheeger constant of G. In particular this provides some rigorous
counterparts to previous experimental observations.
We also point out an application to the analysis of the random walk algorithm
on certain instances of the 3-XOR-SAT problem.
http://arxiv.org/abs/0811.0381
Author(s): Ludovic Gouden\`ege (IRMAR)
Abstract: We consider a stochastic partial differential equation with logarithmic (or
negative power) nonlinearity, with one reflection at 0 and with a constraint of
conservation of the space average. The equation, driven by the derivative in
space of a space-time white noise, contains a bi-Laplacian in the drift. The
lack of the maximum principle for the bi-Laplacian generates difficulties for
the classical penalization method, which uses a crucial monotonicity property.
Being inspired by the works of Debussche and Zambotti, we use a method based on
infinite dimensional equations, approximation by regular equations and
convergence of the approximated semi-group. We obtain existence and uniqueness
of solution for nonnegative intial conditions, results on the invariant
measures, and on the reflection measures.
http://arxiv.org/abs/0811.0580
Author(s): Patrick Cattiaux (IMT) and Djalil Chafai (IMT and UPTE) and S\'ebastien Motsch (IMT)
Abstract: The Persistent Turning Walker Model (PTWM) was introduced by Gautrais et al
in Mathematical Biology for the modelling of fish motion. It involves a
nonlinear pathwise functional of a non-elliptic hypo-elliptic diffusion. This
diffusion solves a kinetic Fokker-Planck equation based on an
Ornstein-Uhlenbeck Gaussian process. The long time "diffusive" behavior of this
model was recently studied by Degond & Motsch using partial differential
equations techniques. This model is however intrinsically probabilistic. In the
present paper, we show how the long time diffusive behavior of this model can
be essentially recovered and extended by using appropriate tools from
stochastic analysis. The approach can be adapted to many other kinetic
"probabilistic" models. Beyond the mathematical results, the aim of this short
paper is also to contribute to the diffusion of stochastic techniques in the
domain of partial differential equations. Also, the text aims to be very
accessible for non probabilists.
http://arxiv.org/abs/0811.0600
Author(s): Bero Roos
Abstract: We derive new explicit bounds for the total variation distance between two
convolution products of $n$ probability distributions, one of which having
identical convolution factors. Approximations by finite signed measures of
arbitrary order are considered as well. We are interested in bounds with magic
factors, i.e. roughly speaking $n$ also appears in the denominator. Special
emphasis is given to the approximation by the $n$-fold convolution of the
arithmetic mean of the distributions under consideration. As an application, we
consider the multinomial approximation of the generalized multinomial
distribution. It turns out that here the order of some bounds given in Roos
(2001) and Loh (1992) can significantly be improved. In particular, it follows
that a dimension factor can be dropped. Moreover, better accuracy is achieved
in the context of symmetric distributions with finite support. In the course of
proof, we use a basic Banach algebra technique for measures on a measurable
Abelian group. Though this method was already used by Le Cam (1960), our
central arguments seem to be new. We also derive new smoothness bounds for
convolutions of probability distributions, which might be of independent
interest.
http://arxiv.org/abs/0811.0622
Author(s): Mohammud Foondun and Davar Khoshnevisan
Abstract: We study discrete nonlinear parabolic stochastic heat equations of the form,
$u_{n+1}(x) - u_n(x) = (\sL u_n)(x) + \sigma(u_n(x))\xi_n(x)$, for $n\in \Z_+$
and $x\in \Z^d$, where $\bm\xi:=\{\xi_n(x)\}_{n\ge 0,x\in\Z^d}$ denotes random
forcing and $\sL$ the generator of a random walk on $\Z^d$. Under mild
conditions, we prove that the preceding stochastic PDE has a unique solution
that grows at most exponentially in time. And that, under natural conditions,
it is "weakly intermittent." Along the way, we establish a comparison principle
as well as a finite-support property.
http://arxiv.org/abs/0811.0643
Author(s): Enkelejd Hashorva
Abstract: In this paper we derive the tail asymptotics of a Kotz Type III elliptical
random vector. As an application of our asymptotic expansion we derive an
approximation for the conditional excess distribution. Furthermore, we discuss
the asymptotic dependence of Kotz Type III triangular arrays and provide some
details on the estimation of conditional excess distribution and survivor
function.
http://arxiv.org/abs/0811.0662
Author(s): Ole E. Barndorff-Nielsen and Friedrich Hubalek
Abstract: We investigate the relation of the semigroup probability density of an
infinite activity L\'{e}vy process to the corresponding L\'{e}vy density. For
subordinators, we provide three methods to compute the former from the latter.
The first method is based on approximating compound Poisson distributions, the
second method uses convolution integrals of the upper tail integral of the
L\'{e}vy measure and the third method uses the analytic continuation of the
L\'{e}vy density to a complex cone and contour integration. As a by-product, we
investigate the smoothness of the semigroup density in time. Several concrete
examples illustrate the three methods and our results.
http://arxiv.org/abs/0811.0678
Author(s): Yves Elskens (PIIM) and Etienne Pardoux (LATP)
Abstract: The one-dimensional motion of any number $\cN$ of particles in the field of
many independent waves (with strong spatial correlation) is formulated as a
second-order system of stochastic differential equations, driven by two Wiener
processes. In the limit of vanishing particle mass ${\mathfrak{m}} \to 0$, or
equivalently of large noise intensity, we show that the momenta of all $N$
particles converge weakly to $N$ independent Brownian motions, and this
convergence holds even if the noise is periodic. This justifies the usual
application of the diffusion equation to a family of particles in a unique
stochastic force field. The proof rests on the ergodic properties of the
relative velocity of two particles in the scaling limit.
http://arxiv.org/abs/0811.0801
Author(s): Philippe Barbe (CNRS)
Abstract: This note presents a rather intuitive approach to extreme value theory. This
approach was devised mostly for pedagogical reason.
http://arxiv.org/abs/0811.0753
Author(s): J. Bouttier and E. Guitter
Abstract: We consider planar quadrangulations with three marked vertices and discuss
the geometry of triangles made of three geodesic paths joining them. We also
study the geometry of minimal separating loops, i.e. paths of minimal length
among all closed paths passing by one of the three vertices and separating the
two others in the quadrangulation. We concentrate on the universal scaling
limit of large quadrangulations, also known as the Brownian map, where pairs of
geodesic paths or minimal separating loops have common parts of non-zero
macroscopic length. This is the phenomenon of confluence, which distinguishes
the geometry of random quadrangulations from that of smooth surfaces. We
characterize the universal probability distribution for the lengths of these
common parts.
http://arxiv.org/abs/0811.0509
Author(s): Giambattista Giacomin and Hubert Lacoin and Fabio Lucio Toninelli
Abstract: The effect of disorder on pinning and wetting models has attracted much
attention in theoretical physics. In particular, it has been predicted on the
basis of the Harris criterion that disorder is relevant (annealed and quenched
model have different critical points and critical exponents) if the return
probability exponent alpha, a positive number that characterizes the model, is
larger than 1/2. Weak disorder has been predicted to be irrelevant (i.e.
coinciding critical points and exponents) if alpha < 1/2. Recent mathematical
work has put these predictions on firm grounds. In renormalization group terms,
the case alpha = 1/2 is a 'marginal case' and there is no agreement in the
literature as to whether one should expect disorder relevance or irrelevance at
marginality. The question is particularly intriguing also because the case
alpha = 1/2 includes the classical models of two-dimensional wetting of a rough
substrate, of pinning of directed polymers on a defect line in dimension (3+1)
or (1+1) and of pinning of an heteropolymer by a point potential in
three-dimensional space. Here we prove disorder relevance both for the general
alpha = 1/2 pinning model and for the hierarchical version of the model
proposed by B. Derrida, V. Hakim and J. Vannimenus (JSP, 1992), in the sense
that we prove a shift of the quenched critical point with respect to the
annealed one. In both cases we work with Gaussian disorder and we show that the
shift is at least of order exp(-1/\beta^4) for beta small, if beta is the
standard deviation of the disorder.
http://arxiv.org/abs/0811.0723
Author(s): Romain Couillet and Merouane Debbah
Abstract: In this paper, a practical power detection scheme for OFDM terminals, based
on recent free probability tools, is proposed. The objective is for the
receiving terminal to determine the transmission power and the number of the
surrounding base stations in the network. However, thesystem dimensions of the
network model turn energy detection into an under-determined problem. The focus
of this paper is then twofold: (i) discuss the maximum amount of information
that an OFDM terminal can gather from the surrounding base stations in the
network, (ii) propose a practical solution for blind cell detection using the
free deconvolution tool. The efficiency of this solution is measured through
simulations, which show better performance than the classical power detection
methods.
http://arxiv.org/abs/0811.0731
Author(s): Romain Couillet and Merouane Debbah
Abstract: A Bayesian inference learning process for cognitive receivers is provided in
this paper. We focus on the particular case of signal detectionas an
explanatory example to the learning framework. Under any prior state of
knowledge on the communication channel, an information theoretic criterion is
presented to decide on the presence of informative data in a noisy wireless
MIMO communication. We detail the particular cases of knowledge, or absence of
knowledge at the receiver, of the number of transmit antennas and noise power.
The provided method is instrumental to provide intelligence to the receiver and
gives birth to a novel Bayesian signal detector. The detector is compared to
the classical power detector and provides detection performance upper bounds.
Simulations corroborate the theoretical results and quantify the gain achieved
using the proposed Bayesian framework.
http://arxiv.org/abs/0811.0764
Author(s): Romain Couillet and Merouane Debbah
Abstract: In this work, a new Bayesian framework for OFDM channel estimation is
proposed. Using Jaynes' maximum entropy principle to derive prior information,
we successively tackle the situations when only the channel delay spread is a
priori known, then when it is not known. Exploitation of the time-frequency
dimensions are also considered in this framework, to derive the optimal channel
estimation associated to some performance measure under any state of knowledge.
Simulations corroborate the optimality claim and always prove as good or better
in performance than classical estimators.
http://arxiv.org/abs/0811.0778
Author(s): P.Chigansky and F.C.Klebaner
Abstract: For the one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$, started at $x>0$,
and the first hitting time $\tau=\inf\{t\ge 0:B_t=0\}$, we find the probability
density of $B_{u\tau}$ for a $u\in(0,1)$, i.e. of the Brownian motion on its
way to hitting zero.
http://arxiv.org/abs/0811.0909
Author(s): P. Bianchi and M. Debbah and J. Najim
Abstract: Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE).
Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\
1\leq i\leq p)$, properly rescaled, and eventually included in any
neighbourhood of the support of Wigner's semi-circle law, we prove that the
related counting measures $({\mathcal N}_n(\Delta_{i,n}), 1\leq i\leq p)$,
where ${\mathcal N}_n(\Delta)$ represents the number of eigenvalues within
$\Delta$, are asymptotically independent as the size $n$ goes to infinity, $p$
being fixed.
As a consequence, we prove that the largest and smallest eigenvalues,
properly centered and rescaled, are asymptotically independent; we finally
describe the fluctuations of the condition number of a matrix from the GUE.
http://arxiv.org/abs/0811.0979
Author(s): George Kesidis and Takis Konstantopoulos and Perla Sousi
Abstract: In this paper, we propose a stochastic model for a file-sharing peer-to-peer
network which resembles the popular BitTorrent system: large files are split
into chunks and a peer can download or swap from another peer only one chunk at
a time. We prove that the fluid limits of a scaled Markov model of this system
are of the coagulation form, special cases of which are well-known
epidemiological (SIR) models. In addition, Lyapunov stability and settling-time
results are explored. We derive conditions under which the BitTorrent
incentives under consideration result in shorter mean file-acquisition times
for peers compared to client-server (single chunk) systems. Finally, a
diffusion approximation is given and some open questions are discussed.
http://arxiv.org/abs/0811.1003
Author(s): Svante Linusson
Abstract: We study a problem on percolation on product graphs G x K_2. Here G is any
finite graph and K_2 consists of two vertices {0,1} connected by an edge. In
edge percolation every edge in G x K_2 is present with probability p. In [3]
Olle H\"aggstr\"om stated a conjecture (which he claimed to be folklore) that
for all G and p the probability that (u,0) is in the same component as (v,0) is
greater than the probability that (u,0) is in the same component as (v,1) for
every pair of vertices u,v in G.
We generalize this conjecture and formulate and prove similar statements for
randomly directed graphs. The methods lead to a proof of the original
conjecture for special classes of graphs $G$, in particular outerplanar graphs.
http://arxiv.org/abs/0811.0949
Author(s): Charles Bordenave (IMT) and Pietro Caputo and Djalil Chafai (IMT and UPTE)
Abstract: In this work, we adopt a Random Matrix Theory point of view to study the
spectrum of large reversible Markov chains in random environment. As the number
of states tends to infinity, we consider both the almost sure global behavior
of the spectrum, and the local behavior at the edge including the so called
spectral gap. We study presently two simple models. The first one is on the
complete graph while the second is on the chain graph (birth-and-death
dynamics). These two models exhibit different scalings and limiting objects.
The first model is related to the semi--circle law and Wigner's theorem. It
contains as a special case a natural reversible Dirichlet Markov Ensemble. The
second model is related to homogenization and also to asymptotics for the roots
of random orthogonal polynomials. A special case gives rise to the arc--sine
law as in a theorem by Erdos & Turan. This work raises several open problems.
http://arxiv.org/abs/0811.1097
Author(s): Georgi Dimitroff and Holger van Bargen
Abstract: Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic
flows which has been studied extensively by various authors. Their rich
structure allows for explicit calculations in several situations and makes them
a natural object to start with if one wants to study more general stochastic
flows. Often the intuition gained by understanding the problem in the context
of IBFs transfers to more general situations. However, the obvious link between
stochastic flows, random dynamical systems and ergodic theory cannot be
exploited in its full strength as the IBF does not have an invariant
probability measure but rather an infinite one. Isotropic Ornstein-Uhlenbeck
flows are in a sense localized IBFs and do have an invariant probability
measure. The imposed linear drift destroys the translation invariance of the
IBF, but many other important structure properties like the Markov property of
the distance process remain valid and allow for explicit calculations in
certain situations. The fact that isotropic Ornstein-Uhlenbeck flows have
invariant probability measures allows one to apply techniques from random
dynamical systems theory. We demonstrate this by applying the results of
Ledrappier and Young to calculate the Hausdorff dimension of the statistical
equilibrium of an isotropic Ornstein-Uhlenbeck flow.
http://arxiv.org/abs/0811.1107
Author(s): Allan Sly
Abstract: The reconstruction problem on the tree has been studied in numerous contexts
including statistical physics, information theory and computational biology.
However, rigorous reconstruction thresholds have only been established in a
small number of models. We prove the first exact reconstruction threshold in a
non-binary model establishing the Kesten-Stigum bound for the 3-state Potts
model on regular trees of large degree. We further establish that the
Kesten-Stigum bound is not tight for the $q$-state Potts model when $q \geq 5$.
Moreover, we determine asymptotics for the reconstruction thresholds.
http://arxiv.org/abs/0811.1208
Author(s): Lamia Belhadji and Daniela Bertacchi and Fabio Zucca
Abstract: We consider an interacting particle process on a graph which, from a
macroscopic point of view, looks like $\Z^d$ and, at a microscopic level, is a
complete graph of degree $N$ (called a patch). There are two birth rates: an
inter-patch one $\lambda$ and an intra-patch one $\phi$. Once a site is
occupied, there is no breeding from outside the patch and the probability
$c(i)$ of success of an intra-patch breeding decreases with the size $i$ of the
population in the site. We prove the existence of a critical value
$\lambda_{cr}(\phi, c, N)$ and a critical value $\phi_{cr}(\lambda, c, N)$. We
consider a sequence of processes generated by the families of control functions
$\{c_i\}_{i \in \N}$ and degrees $\{N_i\}_{i \in \N}$; we prove, under mild
assumptions, the existence of a critical value $i_{cr}$. Roughly speaking we
show that, in the limit, these processes behave as the branching random walk on
$\Z^d$ with external birth rate $\lambda$ and internal birth rate $\phi$. Some
examples of models that can be seen as particular cases are given.
http://arxiv.org/abs/0811.1279
Author(s): Philippe Biane (IGM)
Abstract: We give a geometric description of the motion of eigenvalues of a Brownian
motion with values in some matrix spaces. In the second part we consider a
paper by Polya where he introduced a function close to the Riemann zeta
function, which satisfies Riemann hypothesis. We show that each of these two
functions can be related to Brownian motion on a symmetric space.
http://arxiv.org/abs/0811.1490
Author(s): Benjamin Jourdain and Jerome Lelong
Abstract: Adaptive Monte Carlo methods are very efficient techniques designed to tune
simulation estimators on-line. In this work, we present an alternative to
stochastic approximation to tune the optimal change of measure in the context
of importance sampling for normal random vectors. Unlike stochastic
approximation, which requires very fine tuning in practice, we propose to use
sample average approximation and deterministic optimization techniques to
devise a robust and fully automatic variance reduction methodology. The same
samples are used in the sample optimization of the importance sampling
parameter and in the Monte Carlo computation of the expectation of interest
with the optimal measure computed in the previous step. We prove that this
highly non independent Monte Carlo estimator is convergent and satisfies a
central limit theorem with the optimal limiting variance. Numerical experiments
confirm the performance of this estimator : in comparison with the crude Monte
Carlo method, the computation time needed to achieve a given precision is
divided by a factor going from 2 to 10.
http://arxiv.org/abs/0811.1496
Author(s): Andrey Novikov
Abstract: This work deals with a general problem of testing multiple hypotheses about
the distribution of a discrete-time stochastic process. Both the Bayesian and
the conditional settings are considered. The structure of optimal sequential
tests is characterized.
http://arxiv.org/abs/0811.1297
Author(s): Eric Babson and Christopher Hoffman and Matthew Kahle
Abstract: The random 2-complex Y=Y(n,p) is the probability space of all simplicial
complexes on vertex set [n] and edge set [n] \choose 2, with each 2-dimensional
face included with probability p independently. Nathan Linial and Roy Meshulam
showed that if p >> 2\log{n}/n then the probability that H_{1}(Y,F_2) is
trivial goes to 1 as n approaches infinity. This is an analogue of the phase
transition for connectivity of the Erd\H{o}s-R\'enyi random graph G(n,p).
We show here that if p >> n^{-1/2}, then the probability that Y is simply
connected goes to 1 as n approaches infinity, but if p << n^{-1/2} then the
probability that Y is simply connected goes to 0. This implies in particular
that vanishing of H_{1}(Y,F_2) and \pi_1(Y) have distinct thresholds. Finding
the threshold for vanishing of H_{1}(Y,Z}) is still an open problem.
http://arxiv.org/abs/0711.2704
Author(s): Benjamin Steinhurst
Abstract: In this paper we explore the metric-measure spaces introduced by Laakso in
2000. Building upon the work of Barlow and Evans we are able to show the
existence of a large supply of Dirichlet forms, or alternatively Markov
Processes, on these spaces. The construction of Barlow and Evans allows us to
justify the use of a quantum graph perspective to identify and describe a
Laplacian operator generated by minimal generalized upper gradients on any of
the Laakso spaces
http://arxiv.org/abs/0811.1378
Author(s): Serban Belinschi and Amir Dembo and Alice Guionnet
Abstract: We study the asymptotic behavior of the appropriately scaled and possibly
perturbed spectral measure $\mu$ of large random real symmetric matrices with
heavy tailed entries. Specifically, consider the N by N symmetric matrix
$Y_N^\sigma$ whose (i,j) entry is $\sigma(i/N,j/N)X_{ij}$ where $(X_{ij},
0
http://arxiv.org/abs/0811.1587
Author(s): Simon Harris and Matthew Roberts
Abstract: We consider a change of measure by a martingale $Z_t$ and clarify that in
general $1/Z_t$ is only a supermartingale under the changed measure. We then
give a necessary and sufficient condition for the event that the limit of the
martingale is zero to coincide with the event that the martingale hits zero in
finite time (up to a set of zero probability).
http://arxiv.org/abs/0811.1696
Author(s): Simon Harris and Matthew Roberts
Abstract: We give new results on the growth of the number of particles in a dyadic
branching Brownian motion which follow within a fixed distance of a path
$f:[0,\infty)\to \mathbb{R}$. We show that it is possible to count the number
of particles without rescaling the paths. Our results reveal that the number of
particles along certain paths can oscillate dramatically. The methods used are
entirely probabilistic, taking advantage of the spine technique developed by,
amongst others, Lyons et al, Kyprianou, and Hardy & Harris.
http://arxiv.org/abs/0811.1704
Author(s): Noam Berger
Abstract: We consider random walk in an elliptic i.i.d. random environment in dimension
greater than or equal to 4, satisfying the ballisticity condition (T'). We show
that for every $\alpha< d$ and $n$ large enough, the probability of linear
slowdown is bounded above by $\exp(-(\log n)^\alpha)$. This is almost matching
the known lower bound of $\exp(-C(\log n)^d)$, and significantly improves
previously known upper bounds. As a tool for the main result, we show an almost
local version of the quenched central limit theorem under the assumption of
condition (T').
http://arxiv.org/abs/0811.1710
Author(s): Giovanni Peccati (LSTA and Modal'x) and Murad S. Taqqu (BOSTON University)
Abstract: This survey provides a unified discussion of multiple integrals, moments,
cumulants and diagram formulae associated with functionals of completely random
measures. Our approach is combinatorial, as it is based on the algebraic
formalism of partition lattices and M\"obius functions. Gaussian and Poisson
measures are treated in great detail. We also present several combinatorial
interpretations of some recent CLTs involving sequences of random variables
belonging to a fixed Wiener chaos.
http://arxiv.org/abs/0811.1726
Author(s): Nina Gantert and Sebastian M\"uller and Serguei Popov and Marina Vachkovskaia
Abstract: We study survival of nearest-neighbour branching random walks in random
environment (BRWRE) on $\mathbb{Z}$. A priori there are three different regimes
of survival: global survival, local survival, and strong local survival. We
show that local and strong local survival regimes coincide for BRWRE and that
they can be characterized with the spectral radius of the first moment matrix
of the process. These results are generalizations of the classification of
BRWRE in recurrent and transient regimes. Our main result is a characterization
of global survival that is given in terms of Lyapunov exponents of an infinite
product of i.i.d. $2\times 2$ random matrices.
http://arxiv.org/abs/0811.1748
Author(s): Sivan Leviyang
Abstract: We examine genetic statistics used in the study of structured populations. In
a 1999 paper, Wakeley observed that the coalescent process associated with the
finite island model can be decomposed into a scattering phase and a collecting
phase. In this paper, we introduce a class of population structure models,
which we refer to as G/KC models, that obey such a decomposition. In a large
population, large sample limit we derive the distribution of the statistic Fst
for all G/KC models under the assumptions of strong or weak mutation. We show
that in the large population, large sample limit the island and two dimensional
stepping stone models are members of the G/KC class of models, thereby deriving
the distributions of Fst for these two well known models as a special case of a
general formula. We show that our analysis of Fst can be extended to an entire
class of genetic statistics, and we use our approach to examine homozygosity
measures. Our analysis uses coalescent based methods.
http://arxiv.org/abs/0811.1553
Author(s): A.Yu. Olshanskii and M. V. Sapir
Abstract: A $k$-free like group is a $k$-generated group $G$ with a sequence of
$k$-element generating sets $Z_n$ such that the girth of $G$ relative to $Z_n$
is unbounded and the Cheeger constant of $G$ relative to $Z_n$ is bounded away
from 0. By a recent result of Benjamini-Nachmias-Peres, this implies that the
critical bond percolation probability of the Cayley graph of $G$ relative to
$Z_n$ tends to $1/(2k-1)$ as $n\to \infty$. Answering a question of Benjamini,
we construct many non-free groups that are $k$-free like for all sufficiently
large $k$.
http://arxiv.org/abs/0811.1607
Author(s): Richard W. Kenyon and David B. Wilson
Abstract: A grove is a spanning forest of a planar graph in which every component tree
contains at least one of a special subset of vertices on the outer face called
nodes. For the natural probability measure on groves, we compute various
connection probabilities for the nodes in a random grove. In particular, for
``tripartite'' pairings of the nodes, the probability can be computed as a
Pfaffian in the entries of the Dirichlet-to-Neumann matrix (discrete Hilbert
transform) of the graph. These formulas generalize the determinant formulas
given by Curtis, Ingerman, and Morrow, and by Fomin, for parallel pairings.
These Pfaffian formulas are used to give exact expressions for reconstruction:
reconstructing the conductances of a planar graph from boundary measurements.
We prove similar theorems for the double-dimer model on bipartite planar
graphs.
http://arxiv.org/abs/0811.1766
Author(s): Mikko S. Pakkanen
Abstract: We show that any continuous stochastic process having the form Z_t = H_t +
\int_0^t K_s dW_s, t \in [0,T], where (H_t) and (K_t) are continuous processes,
independent of the driving Brownian motion (W_t), has the conditional full
support property, introduced by Guasoni, R\'asonyi, and Schachermayer [Ann.
Appl. Probab. 18 (2) (2008) 491-520] in connection pricing models with
transaction costs. Using this result, we show that several stochastic
volatility (SV) models (e.g. Heston, Hull-White, log-OU, fractional SV) have
the conditional full support property.
http://arxiv.org/abs/0811.1847
Author(s): R. L. Loeffen
Abstract: We consider the classical optimal dividend control problem which was proposed
by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957) 433--443].
Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007) 156--180]
studied the case when the risk process is modeled by a general spectrally
negative L\'{e}vy process. We draw upon their results and give sufficient
conditions under which the optimal strategy is of barrier type, thereby helping
to explain the fact that this particular strategy is not optimal in general. As
a consequence, we are able to extend considerably the class of processes for
which the barrier strategy proves to be optimal.
http://arxiv.org/abs/0811.1862
Author(s): Mark Adler and Patrik L. Ferrari and Pierre van Moerbeke
Abstract: Consider n+m non-intersecting Brownian bridges, with n of them leaving from 0
at time t=-1 and returning to 0 at time t=1, while the m remaining ones
(wanderers) go from m points a_i to m points b_i. First we keep m fixed and we
scale a_i,b_i appropriately with n. In the large-n limit we obtain a new Airy
process with wanderers, in the neighborhood of (2n)^(1/2), the approximate
location of the rightmost particle in the absence of wanderers. This new
process is governed by an Airy-type kernel, with a rational perturbation.
Letting the number m of wanderers tend to infinity as well, leads to two
Pearcey processes about two cusps, a closing and an opening cusp, the location
of the tips being related by an elliptic curve. Upon tuning the starting and
target points, one can let the two tips of the cusps grow very close; this
leads to a new process, which we conjecture to be governed by a kernel,
represented as a double integral involving the exponential of a quintic
polynomial in the integration variables.
http://arxiv.org/abs/0811.1863
Author(s): Herold Dehling and Martin Wendler
Abstract: The asymptotic normality of U-statistics has so far been proved for iid data
and under various mixing conditions such as absolute regularity, but not for
strong mixing. We use a coupling technique introduced 1983 by Bradley to prove
a new generalized covariance inequality similar to Yoshihara's. It follows from
the Hoeffding-decomposition and this inequality, that U-statistics of strong
mixing observations converge to a normal limit if the kernel of the U-statistic
fulfills some moment and continuity conditions.
The validity of the bootstrap for U-statistics has until now only been
established in the case of iid data (see Bickel and Freedman). For mixing data,
Politis and Romano proposed the circular block bootstrap, which leads to a
consistent estimation of the sample mean's distribution. We extend these
results to U-statistics of weakly dependent data and prove a CLT for the
circular block bootstrap version of U-statistics under absolute regularity and
strong mixing. We also calculate a rate of convergence for the bootstrap
variance estimator of a U-statistic.
http://arxiv.org/abs/0811.1888
Author(s): Yan Dolinsky and Yuri Kifer
Abstract: We show that the shortfall risk of binomial approximations of game (Israeli)
options converges to the shortfall risk in the corresponding Black--Scholes
market considering Lipschitz continuous path-dependent payoffs for both
discrete- and continuous-time cases. These results are new also for usual
American style options. The paper continues and extends the study of Kifer
[Ann. Appl. Probab. 16 (2006) 984--1033] where estimates for binomial
approximations of prices of game options were obtained. Our arguments rely, in
particular, on strong invariance principle type approximations via the
Skorokhod embedding, estimates from Kifer [Ann. Appl. Probab. 16 (2006)
984--1033] and the existence of optimal shortfall hedging in the discrete time
established by Dolinsky and Kifer [Stochastics 79 (2007) 169--195].
http://arxiv.org/abs/0811.1896
Author(s): Kenneth S. Alexander and Nikos Zygouras
Abstract: We consider a polymer with configuration modeled by the trajectory of a
Markov chain, interacting with a potential of form $u+V_n$ when it visits a
particular state 0 at time $n$, with $\{V_n\}$ representing i.i.d. quenched
disorder. There is a critical value of $u$ above which the polymer is pinned by
the potential. A particular case not covered in a number of previous studies is
that of loop exponent one, in which the probability of an excursion of length
$n$ takes the form $\varphi(n)/n$ for some slowly varying $\varphi$; this
includes simple random walk in two dimensions. We show that in this case, at
all temperatures, the critical values of $u$ in the quenched and annealed
models are equal, in contrast to all other loop exponents, for which these
critical values are known to differ at least at low temperatures.
http://arxiv.org/abs/0811.1902
Author(s): Thomas Duquesne
Abstract: In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function
for the boundary of supercritical Galton-Watson trees under a certain
assumption on the tail at the infinity of the total mass of the branching
measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as
other other precise results on fractal properties of the boundary of
Galton-Watson trees. The goal of this paper is to provide an elementary proof
of Hawkes's conjecture under a less restrictive assumption than in T.
Watanabe's paper, by use of size-biased Galton-Watson trees introduced by
Lyons, Pemantle and Peres in 1995.
http://arxiv.org/abs/0811.1935
Author(s): Didier Dacunha-Castelle (1) and Lisandro J. Ferm\'in (1 and 2) ((1) Universit\'e Paris Sud, (2) Universit\'e Paris Descartes)
Abstract: We study the aggregation of AR processes and generalized Ornstein-Uhlenbeck
(OU) processes. Mixture of spectral densities with random poles are the main
tool. In this context, we apply our results for the aggregation of doubly
stochastic interactives processes, see Dacunha-Castelle and Fermin (2006).
Thus, we study the relationship between aggregation of autoregressive processes
and long memory considering complex interaction structures. We precise a very
interesting qualitative phenomena: how the long memory creation depends on the
poles concentration near to the boundary of stability (measured in the
Prokhorov sense). Our results extends the results given by Oppenheim and Viano
(2004), and highlight the importance of the angular dispersion measure of poles
in the appearance of the long memory.
http://arxiv.org/abs/0811.1917
Author(s): Alexander Cherny
Abstract: We prove that any Brownian moving average
\[X_t=\int_{-\infty}^t\bigl(f(s-t)-f(s)\bigr) dB_s,\qquad t\ge0,\] satisfies
the conditional full support condition introduced by Guasoni, R\'{a}sonyi and
Schachermayer [Ann. Appl. Probab. 18 (2008) 491--520].
http://arxiv.org/abs/0811.2040
Author(s): Kouji Yano and Yuko Yano and Marc Yor
Abstract: Several aspects of the laws of first hitting times of points are investigated
for one-dimensional symmetric stable L\'evy processes. It\^o's excursion theory
plays a key role in this study.
http://arxiv.org/abs/0811.2046
Author(s): Fabien Panloup (LSProba)
Abstract: We consider a sequence $(\xi_n)_{n\ge1}$ of $i.i.d.$ random values living in
the domain of attraction of an extreme value distribution. For such sequence,
there exists $(a_n)$ and $(b_n)$, with $a_n>0$ and $b_n\in\ER$ for every $n\ge
1$, such that the sequence $(X_n)$ defined by
$X_n=(\max(\xi_1,...,\xi_n)-b_n)/a_n$ converges in distribution to a non
degenerated distribution. In this paper, we show that $(X_n)$ can be viewed as
an Euler scheme with decreasing step of an ergodic Markov process solution to a
SDE with jumps and we derive a functional limit theorem for the sequence
$(X_n)$ from some methods used in the long time numerical approximation of
ergodic SDE's.
http://arxiv.org/abs/0811.2052
Author(s): Giuseppe Da Prato and Michael R\"ockner and Feng-Yu Wang
Abstract: We consider stochastic equations in Hilbert spaces with singular drift in the
framework of [Da Prato, R\"ockner, PTRF 2002]. We prove a Harnack inequality
(in the sense of [Wang, PTRF 1997]) for its transition semigroup and exploit
its consequences. In particular, we prove regularizing and ultraboundedness
properties of the transition semigroup as well as that the corresponding
Kolmogorov operator has at most one infinitesimally invariant measure $\mu$
(satisfying some mild integrability conditions). Finally, we prove existence of
such a measure $\mu$ for non-continuous drifts.
http://arxiv.org/abs/0811.2061
Author(s): Viorel Barbu and Giuseppe Da Prato and Michael R\"ockner
Abstract: We prove that the solutions to fast diffusion stochastic porous media
equations have finite time extinction with strictly positive probability.
http://arxiv.org/abs/0811.2064
Author(s): Guszt\'av Morvai and Benjamin Weiss
Abstract: A binary renewal process is a stochastic process $\{X_n\}$ taking values in
$\{0,1\}$ where the lengths of the runs of 1's between successive zeros are
independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the
future behavior, and the problem of universal estimators is to do so without
any prior knowledge of the distribution. We prove a variety of results of this
type, including universal estimates for the expected time to renewal as well as
estimates for the conditional distribution of the time to renewal. Some of our
results require a moment condition on the time to renewal and we show by an
explicit construction how some moment condition is necessary.
http://arxiv.org/abs/0811.2076
Author(s): Viorel Barbu and Philippe Blanchard and Giuseppe Da Prato and Michael R\"ockner
Abstract: Models of self-organized criticality, which can be described as singular
diffusions with or without (multiplicative) Wiener forcing term (as e.g. the
Bak/Tang/Wiesenfeld- and Zhang-models), are analyzed. Existence and uniqueness
of nonnegative strong solutions are proved. Previously numerically predicted
transition to the critical state in 1-D is confirmed by a rigorous proof that
this indeed happens in finite time with high probability.
http://arxiv.org/abs/0811.2093
Author(s): G. Aletti and C. May and and P. Secchi
Abstract: We prove a Central Limit Theorem for the sequence of random compositions of a
two-color randomly reinforced urn. As a consequence, we are able to show that
the distribution of the urn limit composition has no point masses.
http://arxiv.org/abs/0811.2097
Author(s): Mustapha Mourragui and Enza Orlandi
Abstract: We consider a lattice gas interacting by the exclusion rule in the presence
of a random field given by i.i.d. bounded random variables in a bounded domain
in contact with particles reservoir at different densities. We show, in
dimensions $d \ge 3$, that the rescaled empirical density field almost surely,
with respect to the random field, converges to the unique weak solution of a
non linear parabolic equation having the diffusion matrix determined by the
statistical properties of the external random field and boundary conditions
determined by the density of the reservoir. Further we show that the rescaled
empirical density field, in the stationary regime, almost surely with respect
to the random field, converges to the solution of the associated stationary
transport equation.
http://arxiv.org/abs/0811.2121
Author(s): Chunrong Feng and Huaizhong Zhao
Abstract: In this paper, we will prove that the local time of a L\'evy process is of
finite $p$-variation in the space variable in the classical sense, a.s. for any
$p>2$, $t\geq 0$, and is a rough path of roughness $p$ a.s. for any $2
http://arxiv.org/abs/0811.2179
Author(s): Djalil Chafai (IMT and UPTE) and Florent Malrieu (IRMAR) and Katy Paroux (LM-Besan\c{c}on, IRISA)
Abstract: The TCP window size process appears in the modeling of the famous
Transmission Control Protocol used over the Internet. This continuous time
Markov process takes its values in $[0,\infty)$, is ergodic and irreversible.
It belongs to the Additive Increase Multiplicative Decrease class of processes.
The sample paths are piecewise linear deterministic and the whole randomness of
the dynamics comes from the jump mechanism. Several aspects of this process
have been already investigated in the literature. In the present paper, we
mainly get quantitative estimates for the exponential convergence to
equilibrium, in terms of the $W_1$ Wasserstein coupling distance, for the
process and also for its embedded chain.
http://arxiv.org/abs/0811.2180
Author(s): Hiroki Sumi and Mariusz Urbanski
Abstract: We consider the dynamics of semi-hyperbolic semigroups generated by finitely
many rational maps on the Riemann sphere. Assuming that the nice open set
condition holds it is proved that there exists a geometric measure on the Julia
set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both
$h$-dimensional Hausdorff and packing measures are finite and positive on the
Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly
separated from zero and infinity. All three fractal dimensions, Hausdorff,
packing and box counting are equal. It is also proved that for the canonically
associated skew-product map there exists a unique $h$-conformal measure.
Furthermore, it is shown that this conformal measure admits a unique Borel
probability absolutely continuous invariant (under the skew-product map)
measure. In fact these two measures are equivalent, and the invariant measure
is metrically exact, hence ergodic.
http://arxiv.org/abs/0811.1809
Author(s): St\'ephane Cr\'epey and Anis Matoussi
Abstract: It is now established that under quite general circumstances, including in
models with jumps, the existence of a solution to a reflected BSDE is
guaranteed under mild conditions, whereas the existence of a solution to a
doubly reflected BSDE is essentially equivalent to the so-called Mokobodski
condition. As for uniqueness of solutions, this holds under mild integrability
conditions. However, for practical purposes, existence and uniqueness are not
enough. In order to further develop these results in Markovian set-ups, one
also needs a (simply or doubly) reflected BSDE to be well posed, in the sense
that the solution satisfies suitable bound and error estimates, and one further
needs a suitable comparison theorem. In this paper, we derive such estimates
and comparison results. In the last section, applicability of the results is
illustrated with a pricing problem in finance.
http://arxiv.org/abs/0811.2276
Author(s): Florent Benaych-Georges (PMA and CMAP) and Thierry L\'evy (DMA)
Abstract: In this paper, we investigate a continuous family of notions of independence
which interpolates between the classical and free ones for non-commutative
random variables. These notions are related to the liberation process
introduced by D. Voiculescu. To each notion of independence correspond new
convolutions of probability measures, for which we establish formulae and of
which we compute simple examples. We prove that there exists no reasonable
analogue of classical and free cumulants associated to these notions of
independence.
http://arxiv.org/abs/0811.2335
Author(s): Elena Shmileva
Abstract: Consider a symmetric $\alpha$-stable L\'evy process with $\alpha\in (1,2)$.
We study shifted small ball probabilities for these processes in the uniform
topology, when the shift function is an arbitrary continuous function which
starts at 0. We obtain the exact rate of decrease for these probabilities
including constants.
Using these small ball estimates, we obtain a functional LIL for
$\alpha$-stable L\'evy process with attracting functions that are continuous.
It occurs that the limit set for the family of renormalized $\alpha$-stable
L\'evy processes is equal to the set of all continuous functions on $[0,1]$
which start at 0, under certain choice of normalizing functions.
http://arxiv.org/abs/0811.2583
Author(s): Konstantin A. Borovkov and Andrew N. Downes
Abstract: In this paper, we establish a relationship between the asymptotic form of
conditional boundary crossing probabilities and first passage time densities
for diffusion processes. Namely, we show that, under broad assumptions, the
first crossing time density of a general curvilinear boundary by a general
time-homogeneous diffusion process has a product-form, the factors being the
transition density of the process and the coefficient of the leading term in
the asymptotic representation of the non-crossing probability of the boundary
by the respective diffusion bridge (as the end-point of the bridge approaches
the boundary). Using a similar technique, we also demonstrate that the boundary
crossing probability is a Gateaux differentiable function of the boundary and
give an explicit representation of its derivative.
http://arxiv.org/abs/0811.2629
Author(s): Cyril Roberto (LAMA)
Abstract: We consider Glauber dynamics reversible with respect to Gibbs measures with
heavy tails. Spins are unbounded. The interactions are bounded and finite
range. The self potential enters into two classes of measures, $\kappa$-concave
probability measure and sub-exponential laws, for which it is known that no
exponential decay can occur. We prove, using coercive inequalities, that the
associated infinite volume semi-group decay to equilibrium polynomially and
stretched exponentially, respectively. Thus improving and extending previous
results by Bobkov and Zegarlinski.
http://arxiv.org/abs/0811.2733
Author(s): Daniel Gandolfo (CPT) and Jean Ruiz (CPT) and Marc Wouts (LAGA)
Abstract: The Curie-Weiss Potts model is a mean field version of the well-known Potts
model. In this model, the critical line $\beta = \beta_c (h)$ is explicitly
known and corresponds to a first order transition when $q > 2$. In the present
paper we describe the fluctuations of the density vector in the whole domain
$\beta \geqslant 0$ and $h \geqslant 0$, including the conditional fluctuations
on the critical line and the non-Gaussian fluctuations at the extremity of the
critical line. The probabilities of each of the two thermodynamically stable
states on the critical line are also computed. Similar results are inferred for
the Random-Cluster model on the complete graph.
http://arxiv.org/abs/0811.2735
Author(s): Elizabeth Meckes
Abstract: There is a result of Diaconis and Freedman which says that, in a limiting
sense, for large collections of high-dimensional data most one-dimensional
projections of the data are approximately Gaussian. This paper gives
quantitative versions of that result. For a set of deterministic vectors
$\{x_i\}_{i=1}^n$ in $\R^d$ with $n$ and $d$ fixed, let $\theta\in\s^{d-1}$ be
a random point of the sphere and let $\mu_n^\theta$ denote the random measure
which puts mass $\frac{1}{n}$ at each of the points
$\inprod{x_1}{\theta},\ldots,\inprod{x_n}{\theta}$. For a fixed bounded
Lipschitz test function $f$, $Z$ a standard Gaussian random variable and
$\sigma^2$ a suitable constant, an explicit bound is derived for the quantity
$\ds\P\left[\left|\int f d\mu_n^\theta-\E f( \sigma Z)\right|>\epsilon\right]$.
A bound is also given for $\ds\P\left[d_{BL}(\mu_n^\theta,
N(0,\sigma^2))>\epsilon\right]$, where $d_{BL}$ denotes the bounded-Lipschitz
distance.
http://arxiv.org/abs/0811.2769
Author(s): Jean B\'erard (ICJ) and Jean-Baptiste Gou\'er\'e (MAPMO)
Abstract: The term Brunet-Derrida behavior refers to the 1997 paper by E. Brunet and B.
Derrida "Shift in the velocity of a front due to a cutoff" (see the reference
within the paper), where it is shown, based on numerical simulations and
heuristic arguments, that a certain branching-selection particle system on the
line exhibits the following behavior: as N goes to infinity, the asymptotic
velocity of the system with N particles converges to a limiting value at the
surprisingly slow rate $(\log N)^{-2}$. In this paper, we consider a class of
branching-selection particle systems on $\R$ with N particles, defined through
iterated branching-selection steps of the following type. During a branching
step, each particle is replaced by two new particles, whose positions are
shifted from that of the original particle by independently performing two
random walk steps, according to some distribution $p$. During the selection
step that follows, only the N rightmost particles are kept among the 2N
particles obtained at the branching step, to form a new population of N
particles. Under generic assumptions on $p$, it is shown that Brunet-Derrida
behavior holds for the corresponding particle system. The proofs are based on
ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and rely
on a comparison of the particle system with a family of N independent branching
random walks killed below a linear space-time barrier. The results presented
here both improve and generalize upon previous work by the first author of this
paper, which was completed just before the results by Gantert, Hu and Shi
became publicly available.
http://arxiv.org/abs/0811.2782
Author(s): Jean-Marc Bardet (CES and Matisse and Samos) and Ciprian Tudor (CES and Matisse and Samos)
Abstract: The purpose of this paper is to make a wavelet analysis of self-similar
stochastic processes by using the techniques of the Malliavin calculus and the
chaos expansion into multiple stochastic integrals. Our examples are the
fractional Brownian motion and the Rosenblatt process. We study the asymptotic
behavior of the statistics based on the wavelet coefficients of these
processes. We find that, in the case when driven process is the Rosenblatt
process, this statistics satisfy a non-central limit theorem although a part of
it converges to a Gaussian limit. We also construct estimators for the
self-similarity index and we illustrate our results by numerical simulations.
http://arxiv.org/abs/0811.2664
Author(s): Morris L. Eaton and Robb J. Muirhead
Abstract: This paper is motivated by the following observation. Take a 3 x 3 random
(Haar distributed) orthogonal matrix $\Gamma$, and use it to "rotate" the north
pole, $x_0$ say, on the unit sphere in $R^3$. This then gives a point $u=\Gamma
x_0$ that is uniformly distributed on the unit sphere. Now use the same
orthogonal matrix to transform u, giving $v=\Gamma u=\Gamma^2 x_0$. Simulations
reported in Marzetta et al (2002) suggest that v is more likely to be in the
northern hemisphere than in the southern hemisphere, and, morever, that
$w=\Gamma^3 x_0$ has higher probability of being closer to the poles $\pm x_0$
than the uniformly distributed point u. In this paper we prove these results,
in the general setting of dimension $p\ge 3$, by deriving the exact
distributions of the relevant components of u and v. The essential questions
answered are the following. Let x be any fixed point on the unit sphere in
$R^p$, where $p\ge 3$. What are the distributions of $U_2=x'\Gamma^2 x$ and
$U_3=x'\Gamma^3 x$? It is clear by orthogonal invariance that these
distribution do not depend on x, so that we can, without loss of generality,
take x to be $x_0=(1,0,...,0)'\in R^p$. Call this the "north pole". Then
$x_0'\Gamma^ k x_0$ is the first component of the vector $\Gamma^k x_0$. We
derive stochastic representations for the exact distributions of $U_2$ and
$U_3$ in terms of random variables with known distributions.
http://arxiv.org/abs/0811.2678
Author(s): Nicolas Fournier
Abstract: We consider a class of nonlinear partial-differential equations, including
the spatially homogeneous Fokker-Planck-Landau equation for Maxwell (or
pseudo-Maxwell) molecules. Continuing the work of
Fontbona-Gu\'erin-M\'el\'eard, we propose a probabilistic interpretation of
such a P.D.E. in terms of a nonlinear stochastic differential equation driven
by a standard Brownian motion. We derive a numerical scheme, based on a system
of $n$ particles driven by $n$ Brownian motions, and study its rate of
convergence. We finally deal with the possible extension of our numerical
scheme to the case of the Landau equation for soft potentials, and give some
numerical results.
http://arxiv.org/abs/0811.2688
Author(s): Eric Shellef
Abstract: We show the critical density for activated random walks on Euclidean lattices
is at most one.
http://arxiv.org/abs/0811.2892
Author(s): Russell Lyons
Abstract: Uniform spanning trees on finite graphs and their analogues on infinite
graphs are a well-studied area. On a Cayley graph of a group, we show that they
are related to the first $\ell^2$-Betti number of the group. Our main aim,
however, is to present the basic elements of a higher-dimensional analogue on
finite and infinite CW-complexes, which relate to the higher $\ell^2$-Betti
numbers. One consequence is a uniform isoperimetric inequality extending work
of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent
work of Duval, Klivans, and Martin.
http://arxiv.org/abs/0811.2933
Author(s): Jean Bertoin (DMA and PMA) and Vladas Sidoravicius (UCI and CWI)
Abstract: The initial purpose of this work is to provide a probabilistic explanation of
a recent result on a version of Smoluchowski's coagulation equations in which
the number of aggregations is limited. The latter models the deterministic
evolution of concentrations of particles in a medium where particles coalesce
pairwise as time passes and each particle can only perform a given number of
aggregations. Under appropriate assumptions, the concentrations of particles
converge as time tends to infinity to some measure which bears a striking
resemblance with the distribution of the total population of a Galton-Watson
process started from two ancestors. Roughly speaking, the configuration model
is a stochastic construction which aims at producing a typical graph on a set
of vertices with pre-described degrees. Specifically, one attaches to each
vertex a certain number of stubs, and then join pairwise the stubs uniformly at
random to create edges between vertices. In this work, we use the configuration
model as the stochastic counterpart of Smoluchowski's coagulation equations
with limited aggregations. We establish a hydrodynamical type limit theorem for
the empirical measure of the shapes of clusters in the configuration model when
the number of vertices tends to $\infty$. The limit is given in terms of the
distribution of a Galton-Watson process started with two ancestors.
http://arxiv.org/abs/0811.2988
Author(s): Zbigniew J. Jurek
Abstract: It is shown that operator-selfdecomposable measures, or more precisely their
Urbanik decomposability semigroups, induce generalized Mehler semigroups of
bounded linear operators. Moreover, those semigroups can be represented as
random integrals of operator valued functions with respect to stochastic L\'evy
processes. Our Banach space setting is in the contrast with the Hilbert spaces
on which so far and most often the generalized Mehler semigroups were studied.
Furthermore, we give new proofs of the random integral representation.
http://arxiv.org/abs/0811.2989
Author(s): Darrell Duffie and Semyon Malamud and Gustavo Manso
Abstract: We solve for the equilibrium dynamics of information sharing in a large
population. Each agent is endowed with signals regarding the likely outcome of
a random variable of common concern. Individuals choose the effort with which
they search for others from whom they can gather additional information. When
two agents meet, they share their information. The information gathered is
further shared at subsequent meetings, and so on. Equilibria exist in which
agents search maximally until they acquire sufficient information precision,
and then minimally. A tax whose proceeds are used to subsidize the costs of
search improves information sharing and can in some cases increase welfare. On
the other hand, endowing agents with public signals reduces information sharing
and can in some cases decrease welfare.
http://arxiv.org/abs/0811.3023
Author(s): Darrell Duffie and Gaston Giroux and Gustavo Manso
Abstract: For a setting in which a large number of asymmetrically informed agents are
randomly matched into groups over time, exchanging their information with each
other when matched, we provide an explicit solution for the dynamics of the
cross-sectional distribution of posterior beliefs. We also show that
convergence of the cross-sectional distribution of beliefs to a common
posterior is exponential and that the rate of convergence does not depend on
the size of the groups of agents that meet. The rate of convergence is merely
the mean rate at which an individual agent is matched.
http://arxiv.org/abs/0811.3024
Author(s): A. Kyprianou and J.C. Pardo and V. Rivero
Abstract: Understanding the space-time features of how a L\'evy process crosses a
constant barrier for the first time, and indeed the last time, is a problem
which is central to many models in applied probability such as queueing theory,
financial and actuarial mathematics, optimal stopping problems, the theory of
branching processes to name but a few. In \cite{KD} a new quintuple law was
established for a general L\'evy process at first passage above a fixed level.
In this article we use the quintuple law to establish a family of related joint
laws, which we call $n$-tuple laws, for L\'evy processes, L\'evy processes
conditioned to stay positive and positive self-similar Markov processes at both
first and last passage over a fixed level. Here the integer $n$ typically
ranges from three to seven. Moreover, we look at asymptotic overshoot and
undershoot distributions and relate them to overshoot and undershoot
distributions of positive self-similar Markov processes issued from the origin.
Although the relation between the $n$-tuple laws for L\'evy processes and
positive self-similar Markov processes are straightforward thanks to the
Lamperti transformation, by inter-playing the role of a (conditioned) stable
processes as both a (conditioned) L\'evy processes and a positive self-similar
Markov processes, we obtain a suite of completely explicit first and last
passage identities for so-called Lamperti-stable L\'evy processes. This leads
further to the introduction of a more general family of L\'evy processes which
we call hypergeometric L\'evy processes, for which similar explicit identities
may be considered.
http://arxiv.org/abs/0811.3075
Author(s): Francois David and Mark Dukes and Thordur Jonsson and Sigurdur Orn Stefansson
Abstract: We study a model of growing planar tree graphs where in each time step we
separate the tree into two components by splitting a vertex and then connect
the two pieces by inserting a new link between the daughter vertices. We
develop a mean field theory for the vertex degree distribution, prove that the
mean field theory is exact in some special cases and check that it agrees with
numerical simulations in general. We calculate various correlation functions
and show that the intrinsic Hausdorff dimension can vary from one to infinity,
depending on the parameters of the model.
http://arxiv.org/abs/0811.3183
Author(s): Jacky Cresson (IMCCE and LMA-PAU) and S\'ebastien Darses (BU)
Abstract: We prove that the Navier-Stokes, the Euler and the Stokes equations admit a
Lagrangian structure using the stochastic embedding of Lagrangian systems.
These equations coincide with extremals of an explicit stochastic Lagrangian
functional, i.e. they are stochastic Lagrangian systems in the sense of
[Cresson-Darses, J. Math. Phys. 48, 072703 (2007]
http://arxiv.org/abs/0811.3286
Author(s): Bartolo Luque and Lucas Lacasa
Abstract: Prime numbers seem to distribute among the natural numbers with no other law
than that of chance, however its global distribution presents a quite
remarkable smoothness. Such interplay between randomness and regularity has
motivated sci- entists of all ages to search for local and global patterns in
this distribution that eventually could shed light into the ultimate nature of
primes. In this work we show that a generalization of the well known
first-digit Benford's law, which addresses the rate of appearance of a given
leading digit d in data sets, describes with astonishing precision the
statistical distribution of leading digits in the prime numbers sequence.
Moreover, a reciprocal version of this pattern also takes place in the sequence
of the nontrivial Riemann zeta zeros. We prove that the prime number theorem
is, in the last analysis, the responsible of these patterns. Some new relations
concerning the prime numbers distribution are also deduced, including a new
approximation to the counting function pi(n). Furthermore, some relations
concerning the statistical conformance to this generalized Benford's law are
derived. Some applications are finally discussed.
http://arxiv.org/abs/0811.3302
Author(s): Weihong Yao
Abstract: This paper is concerned with the distribution of normalized zero-sets of
random entire functions. The normalization of the zero-set is performed in the
same way as that of the counting function for an entire function in Nevanlinna
theory. The result generalizes the Shiffman and Zelditch theory on the
distribution of the zeroes of random holomorphic sections of powers for
positive Hermitian holomorphic line bundles from polynomial functions to entire
functions. Our result can also be viewed as the analogy of Nevanlinna's First
Main Theorem in the theory of the distribution of zero-sets of random entire
functions.
http://arxiv.org/abs/0811.3365
Author(s): Ph. Barbe (CNRS) and W.P. McCormick (UGA)
Abstract: Cramer's theorem provides an estimate for the tail probability of the maximum
of a random walk with negative drift and increments having a moment generating
function finite in a neighborhood of the origin. The class of (g,F)-processes
generalizes in a natural way random walks and fractional ARIMA models used in
time series analysis. For those (g,F)-processes with negative drift, we obtain
a logarithmic estimate of the tail probability of their maximum, under
conditions comparable to Cramer's. Furthermore, we exhibit the most likely
paths as well as the most likely behavior of the innovations leading to a large
maximum.
http://arxiv.org/abs/0811.3460
Author(s): Enkelejd Hashorva
Abstract: The residual dependence index of bivariate Gaussian distributions is
determined by the correlation coefficient. This tail index is of certain
statistical importance when extremes and related rare events of bivariate
samples with asymptotic independent components are being modeled. In this paper
we calculate the partial residual dependence indices of a multivariate
elliptical random vector assuming that the associated random radius is in the
Gumbel max-domain of attraction. Furthermore, we discuss the estimation of
these indices when the associated random radius possesses a Weibull-tail
distribution.
http://arxiv.org/abs/0811.3552
Author(s): Theo van Uem
Abstract: We obtain expected number of arrivals, absorption probabilities and expected
time before absorption for an asymmetric discrete random walk on a locally
infinite graph in the presence of multiple function barriers
http://arxiv.org/abs/0811.3682
Author(s): Gideon Amir and Omer Angel and Benedek Valko
Abstract: In a multi-type totally asymmetric simple exclusion process (TASEP) on the
line, each site of Z is occupied by a particle labeled with a number and two
neighboring particles are interchanged at rate one if their labels are in
increasing order. Consider the process with the initial configuration where
each particle is labeled by its position. It is known that in this case a.s.
each particle has an asymptotic speed which is distributed uniformly on [-1,1].
We study the joint distribution of these speeds: the TASEP speed process.
We prove that the TASEP speed process is stationary with respect to the
multi-type TASEP dynamics. Consequently, every ergodic stationary measure is
given as a projection of the speed process measure.
By relating this form to the known stationary measures for multi-type TASEPs
with finitely many types we compute several marginals of the speed process,
including the joint density of two and three consecutive speeds. One striking
property of the distribution is that two speeds are equal with positive
probability and for any given particle there are infinitely many others with
the same speed.
We also study the (partially) asymmetric simple exclusion process (ASEP). We
prove that the ASEP with the above initial configuration has a certain
symmetry. This allows us to extend some of our results, including the
stationarity and description of all ergodic stationary measures, also to the
ASEP.
http://arxiv.org/abs/0811.3706
Author(s): Przemyslaw Klusik and Zbigniew Palmowski and Jakub Zwierz
Abstract: In this paper we consider the problem of the quantile hedging from the point
of view of a better informed agent acting on the market. The additional
knowledge of the agent is modelled by a filtration initially enlarged by some
random variable. By using equivalent martingale measures introduced in
Amendinger (2000) and Amendinger, Imkeller and Schweizer (1998) we solve the
problem for the complete case, by extending the results obtained in F{\"o}llmer
and Leukert (1999) to the insider context. Finally, we consider the examples
with the explicit calculations within the standard Black-Scholes model.
http://arxiv.org/abs/0811.3749
Author(s): Agnieszka Czyzewska-Jankowska and Zbigniew J. Jurek
Abstract: A method of random integral representation, that is, a method of representing
a given probability measure as the probability distribution of some random
integral, was quite successful in the past few decades. In this note we show
that a composition of two random integral mappings $\J^\be$ is again a random
integral mapping. We illustrate our results on some examples.
http://arxiv.org/abs/0811.3750
Author(s): Zbigniew J. Jurek
Abstract: In infinite dimensional Banach spaces there is no complete characterization
of the L\'evy exponents of infinitely divisible probability measures. Here we
propose \emph{a calculus on L\'evy exponents} that is derived from some random
integrals. As a consequence we prove that \emph{each} selfdecomposable measure
can by factorized as another selfdecomposable measure and its background
driving measure that is s-selfdecomposable. This complements a result from the
paper of Iksanov-Jurek-Schreiber in the Annals of Probability \textbf{32},
2004.}
http://arxiv.org/abs/0811.3752
Author(s): C. Hein and P. Imkeller and I. Pavlyukevich
Abstract: In this paper we study the asymptotic properties of the power variations of
stochastic processes of the type X=Y+L, where L is an alpha-stable Levy
process, and Y a perturbation which satisfies some mild Lipschitz continuity
assumptions. We establish local functional limit theorems for the power
variation processes of X. In case X is a solution of a stochastic differential
equation driven by L, these limit theorems provide estimators of the stability
index alpha. They are applicable for instance to model fitting problems for
paleo-climatic temperature time series taken from the Greenland ice core.
http://arxiv.org/abs/0811.3769
Author(s): Gautam Iyer and Alexei Novikov
Abstract: We study the dissipation mechanism of a stochastic particle system for the
Burgers' equation. The velocity field of the viscous Burgers' and Navier-Stokes
equations can be expressed as an expected value of a stochastic process based
on noisy particle trajectories (Constantin, Iyer, Comm. Pure Appl. Math, 2008).
In this paper we study a particle system for the viscous Burgers' equations
using a Monte-Carlo version of the above; we consider $N$ copies of the above
stochastic flow, each driven by independent Wiener processes, and replace the
expected value with $\frac{1}{N}$ times the sum over these copies. A similar
construction for the Navier-Stokes equations was studied by J. Mattingly and
the first author (\texttt{arXiv:0803.1222}, to appear in Nonlinearity).
Surprisingly, for any finite $N$, the particle system for the Burgers'
equations shocks almost surely in finite time. In contrast to the full expected
value, the empirical mean $\frac{1}{N} \sum_1^N$ does not regularize the system
enough to ensure a time global solution. To avoid these shocks, we consider a
resetting procedure, which at first sight should have no regularizing effect at
all. We however prove that this procedure prevents the formation of shocks for
any $N \geq 2$, and consequently as $N \to \infty$ we get convergence to the
solution of the viscous Burgers' equations on long time intervals.
http://arxiv.org/abs/0811.3799
Author(s): Luciano Campi and Mark P. Owen
Abstract: We present an optimal investment theorem for a currency exchange model with
random and possibly discontinuous proportional transaction costs. The
investor's preferences are represented by a multivariate utility function,
allowing for simultaneous consumption of any prescribed selection of the
currencies at a given terminal date. We prove the existence of an optimal
portfolio process under the assumption of asymptotic satiability of the value
function. Sufficient conditions for asymptotic satiability of the value
function include reasonable asymptotic elasticity of the utility function, or a
growth condition on its dual function. We show that the portfolio optimization
problem can be reformulated in terms of maximization of a terminal liquidation
utility function, and that both problems have a common optimizer.
http://arxiv.org/abs/0811.3889
Author(s): Jinqiao Duan
Abstract: Nonlinear systems with model uncertainty are often described by stochastic
differential equations. Some techniques from random dynamical systems are
discussed. They are relevant to better understanding of solution processes of
stochastic differential equations and thus may shed lights on predictability in
nonlinear systems with model uncertainty.
http://arxiv.org/abs/0811.3697
Author(s): Kenric P. Nelson and Sabir Umarov
Abstract: Tsallis statistics (or q-statistics) in nonextensive statistical mechanics is
a one-parameter description of correlated states. In this paper we use a
translated entropic index: $1 - q \to q$ . The essence of this translation is
to improve the mathematical symmetry of the q-algebra and make q directly
proportional to the nonlinear coupling. A conjugate transformation is defined
$\hat q \equiv \frac{{- 2q}}{{2 + q}}$ which provides a dual mapping between
the heavy-tail q-Gaussian distributions, whose translated q parameter is
between $ - 2 < q < 0$, and the compact-support q-Gaussians, between $0 < q <
\infty $ . This conjugate transformation is used to extend the definition of
the q-Fourier transform to the domain of compact support. A conjugate q-Fourier
transform is proposed which transforms a q-Gaussian into a conjugate $\hat q$
-Gaussian, which has the same exponential decay as the Fourier transform of a
power-law function. The nonlinear statistical coupling is defined such that the
conjugate pair of q-Gaussians have equal strength but either couple
(compact-support) or decouple (heavy-tail) the statistical states. Many of the
nonextensive entropy applications can be shown to have physical parameters
proportional to the nonlinear statistical coupling.
http://arxiv.org/abs/0811.3777
Author(s): Fredrik Johansson and Alan Sola
Abstract: We consider radial Loewner evolution driven by unimodular L\'evy processes.
We rescale the hulls of the evolution by capacity, and prove that the weak
limit of the rescaled hulls exists. We then study a random growth model
obtained by driving the Loewner equation with a compound Poisson process. The
process involves two real parameters: the intensity of the underlying Poisson
process and a localization parameter of the Poisson kernel which determines the
jumps. A particular choice of parameters yields a growth process similar to the
Hastings-Levitov $\rm{HL}(0)$ model. We describe the asymptotic behavior of the
hulls with respect to the parameters, showing that growth tends to become
localized as the jump parameter increases. We obtain deterministic evolutions
in one limiting case, and Loewner evolution driven by a unimodular Cauchy
process in another. We show that the Hausdorff dimension of the limiting
rescaled hulls is equal to 1. Using a different type of compound Poisson
process, where the Poisson kernel is replaced by the heat kernel, as driving
function, we recover one case of the aforementioned model and
$\rm{SLE}(\kappa)$ as limits.
http://arxiv.org/abs/0811.3857
Author(s): Christophette Blanchet-Scalliet (ICJ) and Anne Eyraud-Loisel (SAF - EA2429), Manuela Royer-Carenzi (LATP)
Abstract: This article focuses on the mathematical problem of existence and uniqueness
of BSDE with a random terminal time which is a general random variable but not
a stopping time, as it has been usually the case in the previous literature of
BSDE with random terminal time. The main motivation of this work is a financial
or actuarial problem of hedging of defaultable contingent claims or life
insurance contracts, for which the terminal time is a default time or a death
time, which are not stopping times. We have to use progressive enlargement of
the Brownian filtration, and to solve the obtained BSDE under this enlarged
filtration. This work gives a solution to the mathematical problem and proves
the existence and uniqueness of solutions of such BSDE under certain general
conditions. This approach is applied to the financial problem of hedging of
defaultable contingent claims, and an expression of the hedging strategy is
given for a defaultable contingent claim or a life insurance contract.
http://arxiv.org/abs/0811.4039
Author(s): Rodolphe Garbit (LMJL)
Abstract: A result of R. Durrett, D. Iglehart and D. Miller states that Brownian
meander is Brownian motion conditioned to stay positive for a unit of time, in
the sense that it is the weak limit, as $x$ goes to 0, of Brownian motion
started at $x>0$ and conditioned to stay positive for a unit of time. We extend
this limit theorem to the case of multidimensional Brownian motion conditioned
to stay in a smooth convex cone. Properties of the limit process are obtained
and applications to random walks are given.
http://arxiv.org/abs/0811.4079
Author(s): B\'en\'edicte Haas (CEREMADE)
Abstract: The subject of this paper is a fragmentation equation with non-conservative
solutions, some mass being lost to a dust of zero-mass particles as a
consequence of an intensive splitting. Under some assumptions of regular
variation on the fragmentation rate, we describe the large-time behavior of
solutions. Our approach is based on probabilistic tools: the solutions to the
fragmentation equation are constructed via non-increasing self-similar Markov
processes that reach continuously 0 in finite time. Our main probabilistic
result describes the asymptotic behavior of these processes conditioned on
non-extinction and is then used for the solutions to the fragmentation
equation. We notice that two parameters influence significantly these
large-time behaviors: the rate of formation of "nearly-1 relative masses" (this
rate is related to the behavior near 0 of the L\'evy measure associated to the
corresponding self-similar Markov process) and the distribution of large
initial particles. Correctly rescaled, the solutions then converge to a
non-trivial limit which is related to the quasi-stationary solutions to the
equation. Besides, these quasi-stationary solutions, or equivalently the
quasi-stationary distributions of the self-similar Markov processes, are
entirely described.
http://arxiv.org/abs/0811.4267
Author(s): Christophe Sabot (ICJ)
Abstract: We consider random walks in random Dirichlet environment (RWDE) which is a
special type of random walks in random environment where the exit probabilities
at each site are i.i.d. Dirichlet random variables. On $Z^d$, RWDE are
parameterized by a 2d-uplet of positive reals. We prove that for all values of
the parameters, RWDE are transient in dimension $d\ge 3$. We also prove that
the Green function has some finite moments and, on $Z^d$, $d\ge 3$, we
explicitly compute the critical integrability exponent. Our result is more
general and applies forexample to finitely generated transient Cayley graphs.
In terms of reinforced random walks it implies that linearly edge-oriented
reinforced random walks are transient for $d\ge 3$.
http://arxiv.org/abs/0811.4285
Author(s): Daniel Alpay and David Levanony
Abstract: Using the white noise setting, in particular the Wick product, the Hermite
transform, and the Kondratiev space, we present a new approach to study linear
stochastic systems, where randomness is also included in the transfer function.
We prove BIBO type stability theorems for these systems, both in the discrete
and continuous time cases. We also consider the case of dissipative systems for
both discrete and continuous time systems. We further study $\ell_1$-$\ell_2$
stability in the discrete time case, and ${\mathbf L}_2$-${\mathbf L}_\infty$
stability in the continuous time case.
http://arxiv.org/abs/0811.4321
Author(s): Johanna Garz\'on
Abstract: We define weighted fractional Brownian sheets, which are a class of Gaussian
random fields with four parameters that include fractional Brownian sheets as
special cases, and we give some of their properties. We show that for certain
values of the parameters the weighted fractional Brownian sheets are obtained
as limits in law of occupation time fluctuations of a stochastic particle
model. In contrast with some known approximations of fractional Brownian sheets
which use a kernel in a Volterra type integral representation of fractional
Brownian motion with respect to ordinary Brownian motion, our approximation
does not make use of a kernel.
http://arxiv.org/abs/0811.4455
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X) and Gesine Reinert
Abstract: We prove infinite-dimensional second order Poincar\'e inequalities on Wiener
space, thus closing a circle of ideas linking limit theorems for functionals of
Gaussian fields, Stein's method and Malliavin calculus. We provide two
applications: (i) to a new "second order" characterization of CLTs on a fixed
Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields.
http://arxiv.org/abs/0811.4485
Author(s): Raphael Cerf and Reda Messikh
Abstract: We study the 2d-Ising model defined on finite boxes at temperatures that are
below but very close from the critical point. When the temperature approaches
the critical point and the size of the box grows fast enough, we establish
large deviations estimates on FK-percolation events that concern the phenomenon
of phase coexistence.
http://arxiv.org/abs/0811.4507
Author(s): Mohammad Reza Yaghouti and Fraydoun Rezakhanlou and Alan Hammond
Abstract: The Smoluchowski equation is a system of partial differential equations
modelling the diffusion and binary coagulation of a large collection of tiny
particles. The mass parameter may be indexed either by positive integers, or by
positive reals, these corresponding to the discrete or the continuous form of
the equations. In dimension at least 3, we derive the continuous Smoluchowski
PDE as a kinetic limit of a microscopic model of Brownian particles liable to
coalesce, using a similar method to that used to derive the discrete form of
the equations in Hammond and Rezakhanlou [4]. The principal innovation is a
correlation-type bound on particle locations that permits the derivation in the
continuous context while simplifying the arguments of [4]. We also comment on
the scaling satisfied by the continuous Smoluchowski PDE, and its potential
implications for blow-up of solutions of the equations.
http://arxiv.org/abs/0811.4601
Author(s): P. Caputo and A. Faggionato and A. Gaudilliere
Abstract: We consider reversible random walks in random environment obtained from
symmetric long--range jump rates on a random point process. We prove almost
sure transience and recurrence results under suitable assumptions on the point
process and the jump rate function. For recurrent models we obtain almost sure
estimates on effective resistances in finite boxes. For transient models we
construct explicit fluxes with finite energy on the associated electrical
network.
http://arxiv.org/abs/0811.4623
Author(s): Ger\'onimo Uribe Bravo
Abstract: We present a further analysis of the fragmentation at heights of the
normalized Brownian excursion. Specifically we study a representation for the
mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable
subordinator and use it to study its jumps; this accounts for a description of
how a typical fragment falls apart. These results carry over to the height
fragmentation of the stable tree. Additionally, the sizes of the fragments in
the Brownian fragmentation when it is about to reduce to dust are described in
a limit theorem.
http://arxiv.org/abs/0811.4754
Author(s): Yang Xing
Abstract: The classical condition on the existence of uniformly exponentially
consistent tests for testing the true density against the complement of its
arbitrary neighborhood has been widely adopted in study of asymptotics of
Bayesian nonparametric procedures. Because we follow a Bayesian approach, it
seems to be more natural to explore alternative and appropriate conditions
which incorporate the prior distribution. In this paper we supply a new
prior-dependent integration condition to establish general posterior
convergence rate theorems for observations which may not be independent and
identically distributed. The posterior convergence rates for such observations
have recently studied by Ghosal and van der Vaart \cite{ghv1}. We moreover
adopt the Hausdorff $\alpha$-entropy given by Xing and Ranneby
\cite{xir1}\cite{xi1}, which is also prior-dependent and smaller than the
widely used metric entropies. These lead to extensions of several existing
theorems. In particular, we establish a posterior convergence rate theorem for
general Markov processes and as its application we improve on the currently
known posterior rate of convergence for a nonlinear autoregressive model.
http://arxiv.org/abs/0811.4677
Author(s): Eduard Rotenstein
Abstract: We shall study backward stochastic differential equations and we will present
a new approach for the existence of the solution. This type of equation appears
very often in the valuation of financial derivatives in complete markets.
Therefore, the identification of the solution as the unique element in a
certain Banach space where a suitably chosen functional attains its minimum
becomes interesting for numerical computations.
http://arxiv.org/abs/0811.4613
Author(s): Thomas Lim (PMA) and Marie-Claire Quenez (PMA)
Abstract: We adress the maximization problem of expected utility from terminal wealth.
The special feature of this paper is that we consider a financial market where
the price process of risky assets can have a default time. Using dynamic
programming, we characterize the value function with a backward stochastic
differential equation and the optimal portfolio policies. We separately treat
the cases of exponential, power and logarithmic utility.
http://arxiv.org/abs/0811.4715
Author(s): Jon Warren and Peter Windridge
Abstract: We give examples of stochastic processes in the Gelfand Tsetlin cone in which
each component evolves independently apart from a blocking and pushing
interaction. The processes give couplings to certain conditioned Markov
processes, last passage times and asymetric exclusion processes. An example of
a cone valued process whose components cannot escape past a wall at the origin
is also considered.
http://arxiv.org/abs/0812.0022
Author(s): Eyal Lubetzky and Allan Sly
Abstract: The cutoff phenomenon describes a sharp transition in the convergence of a
family of ergodic finite Markov chains to equilibrium. Many natural families of
chains are believed to exhibit cutoff, and yet establishing this fact is often
extremely challenging. An important such family of chains is the random walk on
$\G(n,d)$, a random $d$-regular graph on $n$ vertices. It is well known that
the spectral gap of this class of chains for $d \geq 3$ fixed is constant,
implying a mixing-time of $O(\log n)$. According to a conjecture of Peres, the
simple random walk on $\G(n,d)$ for such $d$ should then exhibit cutoff whp. As
a special case of this, Durrett conjectured that the mixing time of the lazy
random walk on a random 3-regular graph is whp $(6+o(1))\log_2 n$.
In this work we confirm the above conjectures, and establish cutoff in
total-variation, its location and its optimal window, both for simple and for
non-backtracking random walks on $\G(n,d)$. Namely, for any fixed $d \geq 3$,
the simple random walk on $\G(n,d)$ whp has cutoff at $\frac{d}{d-2}\log_{d-1}
n$ with window order $\sqrt{\log n}$. Surprisingly, the non-backtracking random
walk on $\G(n,d)$ whp has cutoff already at $\log_{d-1} n$ with constant window
order. We further extend these results to $\G(n,d)$ for any $d=n^{o(1)}$
(beyond which the mixing time is O(1)), provide efficient algorithms for
testing cutoff, as well as give explicit constructions where cutoff occurs.
http://arxiv.org/abs/0812.0060
Author(s): Luigi Accardi and Ameur Dhahri and Michael Skeide
Abstract: We extend the square of white noise algebra over the step functions on R to
the test function space of bounded square-integrable functions on R^d, and we
show that in the Fock representation the exponential vectors exist for all test
functions bounded by 1/2.
http://arxiv.org/abs/0812.0089
Author(s): Feng Yu and Alison Etheridge and Charles Cuthbertson
Abstract: We consider a biological population in which a beneficial mutation is
undergoing a selective sweep when a second beneficial mutation arises at a
linked locus and we investigate the probability that both mutations will
eventually fix in the population. Previous work has dealt with the case where
the second mutation to arise confers a smaller benefit than the first. In that
case population size plays almost no role. Here we consider the opposite case
and observe that, by contrast, the probability of both mutations fixing can be
heavily dependent on population size. Indeed the key parameter is $\rho N$, the
product of the population size and the recombination rate between the two
selected loci. If $\rho N$ is small, the probability that both mutations fix
can be reduced through interference to almost zero while for large $\rho N$ the
mutations barely influence one another. The main rigorous result is a method
for calculating the fixation probability of a double mutant in the large
population limit.
http://arxiv.org/abs/0812.0104
Author(s): Florian Sobieczky
Abstract: By an eigenvalue comparison-technique, the expected return probability of the
delayed random walk on the finite clusters of critical Bernoulli bond
percolation on the two-dimensional Euclidean lattice is estimated. The results
are generalised to invariant percolations on unimodular graphs with almost
surely finite clusters. A similar method has been used elsewhere to derive
bounds for invariant percolation of finite clusters on unimodular transitive
graphs. It is adapted here to match the special situation of criticality. The
approach followed here involves using the special property of Cartesian
Products of finite graphs with cycles of a certain minimal size to be
Hamiltonian.
http://arxiv.org/abs/0812.0117
Author(s): Achim Klenke and Peter M\"orters
Abstract: Let $p\ge2$, $n_1\le...\le n_p$ be positive integers and $B_1^1, ...,
B_{n_1}^1; ...; B_1^p, ..., B_{n_p}^{p}$ be independent planar Brownian motions
started uniformly on the boundary of the unit circle. We define a $p$-fold
intersection exponent $\varsigma(n_1,..., n_p)$, as the exponential rate of
decay of the probability that the packets $\bigcup_{j=1}^{n_i} B_j^i[0,t^2]$,
$i=1,...,p$, have no joint intersection. The case $p=2$ is well-known and,
following two decades of numerical and mathematical activity, Lawler, Schramm
and Werner (2001) rigorously identified precise values for these exponents. The
exponents have not been investigated so far for $p>2$. We present an extensive
mathematical and numerical study, leading to an exact formula in the case
$n_1=1$, $n_2=2$, and several interesting conjectures for other cases.
http://arxiv.org/abs/0812.0131
Author(s): Uriel Feige and Elchanan Mossel and Dan Vilenchik
Abstract: Experimental results show that certain message passing algorithms, namely,
Survey Propagation, are very effective in finding satisfying assignments for
random satisfiable 3CNF formulas which are considered hard for other SAT
heuristics. Unfortunately, rigorous understanding of this phenomena is still
lacking. In this paper we make a modest step towards providing rigorous
explanation for the effectiveness of message passing algorithms. We analyze the
performance of Warning Propagation, a popular message passing algorithm that is
simpler than Survey Propagation. We show that for 3CNF formulas drawn from a
certain distribution over random satisfiable 3CNF formulas, commonly referred
to as the planted-assignment distribution, running Warning Propagation in the
standard way (run message passing until convergence, simplify the formula
according to the resulting assignment, and satisfy the remaining subformula, if
necessary, using a simple "off the shelf" heuristic) works when the
clause-variable ratio is a sufficiently large constant. We are not aware of
previous rigorous analysis of message passing algorithms for satisfiability
instances, though such analysis was performed for decoding of Low Density
Parity Check (LDPC) Codes. We discuss some of the differences between results
for the LDPC setting and our results.
http://arxiv.org/abs/0812.0147
Author(s): Ramon van Handel
Abstract: We prove that bootstrap type Monte Carlo particle filters approximate the
optimal nonlinear filter in a time average sense uniformly with respect to the
time horizon when the signal is ergodic and the particle system satisfies a
tightness property. The latter is satisfied without further assumptions when
the signal state space is compact, as well as in the noncompact setting when
the signal is geometrically ergodic and the observations satisfy additional
regularity assumptions.
http://arxiv.org/abs/0812.0350
Author(s): Andrey Novikov
Abstract: In this article, a general problem of sequential statistical inference for
general discrete-time stochastic processes is considered. The problem is to
minimize an average sample number given that Bayesian risk due to incorrect
decision does not exceed some given bound. We characterize the form of optimal
sequential stopping rules in this problem. In particular, we have a
characterization of the form of optimal sequential decision procedures when the
Bayesian risk includes both the loss due to incorrect decision and the cost of
observations.
http://arxiv.org/abs/0812.0159
Author(s): Hiroki Sumi
Abstract: We investigate the dynamics of polynomial semigroups (semigroups generated by
a family of polynomial maps on the Riemann sphere) and the random dynamics of
polynomials on the Riemann sphere. Combining the dynamics of semigroups and the
fiberwise (random) dynamics, we give a classification of polynomial semigroups
$G$ such that $G$ is generated by a compact family $\Gamma $, the planar
postcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In one of
the classes, we have that for almost every sequence $\gamma \in \Gamma
^{\Bbb{N}}$, the Julia set $J_{\gamma}$ of $\gamma $ is a Jordan curve but not
a quasicircle, the unbounded component of the Fatou set $F_{\gamma}$ of
$\gamma$ is a John domain, and the bounded component of $F_{\gamma}$ is not a
John domain. Note that this phenomenon does not hold in the usual iteration of
a single polynomial. Moreover, we consider the dynamics of polynomial
semigroups $G$ such that the planar postcritical set of $G$ is bounded and the
Julia set is disconnected. Those phenomena of polynomial semigroups and random
dynamics of polynomials that do not occur in the usual dynamics of polynomials
are systematically investigated.
http://arxiv.org/abs/0811.4536
Author(s): Constantinos Kardaras and Eckhard Platen
Abstract: A financial market model with general semimartingale asset-price processes
and where agents can only trade using no-short-sale strategies is considered.
We show that wealth processes using continuous trading can be approximated very
closely by wealth processes using simple combinations of buy-and-hold trading.
This approximation is based on controlling the proportions of wealth invested
in the assets. As an application, the utility maximization problem is
considered and it is shown that optimal utilities and wealth processes
resulting from continuous trading can be approximated arbitrarily well by the
use of simple combinations of buy-and-hold strategies.
http://arxiv.org/abs/0812.0033
Author(s): Amine Asselah
Abstract: We consider the energy of a randomly charged random walk. We assume that only
charges on the same site interact. We study the upper and lower tails of the
energy, when averaged over both randomness, in dimension three or more.
http://arxiv.org/abs/0812.0443
Author(s): Rami Atar and Amarjit Budhiraja
Abstract: A two-player stochastic differential game representation has recently been
obtained for solutions of the equation -\Delta_\infty u=h in a \calC^2 domain
with Dirichlet boundary condition, where h is continuous and takes values in
\RR\setminus\{0\}. Under appropriate assumptions, including smoothness of u,
the vanishing \delta limit law of the state process, when both players play
\delta-optimally, is identified as a diffusion process with coefficients given
explicitly in terms of derivatives of the function u.
http://arxiv.org/abs/0812.0496
Author(s): Volker Betz and Daniel Ueltschi
Abstract: We consider the distribution of cycles in two models of random permutations,
that are related to one another. In the first model, cycles receive a weight
that depends on their length. The second model deals with permutations of
points in the space and there is an additional weight that involves the length
of permutation jumps. We prove the occurrence of infinite macroscopic cycles
above a certain critical density.
http://arxiv.org/abs/0812.0569
Author(s): Krzysztof Czarkowski
Abstract: We consider the discrete "fast" penalization scheme for SDE's driven by
general semimartingale on orthant $\mathbb{R}_{+}^{d}$ with oblique reflection.
http://arxiv.org/abs/0812.0619
Author(s): Jian Ding and Eyal Lubetzky and Yuval Peres
Abstract: We study Glauber dynamics for the Ising model on the complete graph on $n$
vertices, known as the Curie-Weiss Model. It is well known that at high
temperature ($\beta < 1$) the mixing time is $\Theta(n\log n)$, whereas at low
temperature ($\beta > 1$) it is $\exp(\Theta(n))$. Recently, Levin, Luczak and
Peres considered a censored version of this dynamics, which is restricted to
non-negative magnetization. They proved that for fixed $\beta > 1$, the
mixing-time of this model is $\Theta(n\log n)$, analogous to the
high-temperature regime of the original dynamics. Furthermore, they showed
\emph{cutoff} for the original dynamics for fixed $\beta<1$. The question
whether the censored dynamics also exhibits cutoff remained unsettled.
In a companion paper, we extended the results of Levin et al. into a complete
characterization of the mixing-time for the Currie-Weiss model. Namely, we
found a scaling window of order $1/\sqrt{n}$ around the critical temperature
$\beta_c=1$, beyond which there is cutoff at high temperature. However,
determining the behavior of the censored dynamics outside this critical window
seemed significantly more challenging.
In this work we answer the above question in the affirmative, and establish
the cutoff point and its window for the censored dynamics beyond the critical
window, thus completing its analogy to the original dynamics at high
temperature. Namely, if $\beta = 1 + \delta$ for some $\delta > 0$ with
$\delta^2 n \to \infty$, then the mixing-time has order $(n /
\delta)\log(\delta^2 n)$. The cutoff constant is $(1/2+[2(\zeta^2 \beta /
\delta - 1)]^{-1})$, where $\zeta$ is the unique positive root of
$g(x)=\tanh(\beta x)-x$, and the cutoff window has order $n / \delta$.
http://arxiv.org/abs/0812.0633
Author(s): Li Wang
Abstract: We establish an almost sure scaling limit theorem for super-Brownian motion
on $\mathbb{R}^d$ associated with the semi-linear equation $u_t = {1/2}\Delta u
+\beta u-\alpha u^2$, where $\alpha$ and $\beta$ are positive constants. In
this case, the spectral theoretical assumptions that required in Chen et al
(2008) are not satisfied. An example is given to show that the main results
also hold for some sub-domains in $\mathbb{R}^d$.
http://arxiv.org/abs/0812.0642
Author(s): Wlodek Bryc and Jacek Wesolowski
Abstract: We use orthogonality measures of Askey-Wilson polynomials to construct Markov
processes with linear regressions and quadratic conditional variances.
Askey-Wilson polynomials are orthogonal martingale polynomials for these
processes.
http://arxiv.org/abs/0812.0657
Author(s): Ravi Montenegro and Prasad Tetali
Abstract: The discrete logarithm problem asks to solve for the exponent $x$, given the
generator $g$ of a cyclic group $G$ and an element $h\in G$ such that $g^x=h$.
We give the first rigorous proof that Pollard's Kangaroo method finds the
discrete logarithm in expected time $(3+o(1))\sqrt{b-a}$ when the logarithm
$x\in[a,b]$, and $(2+o(1))\sqrt{b-a}$ when $x\in_{uar}[a,b]$. This matches the
conjectured time complexity and, rare among the analysis of algorithms based on
Markov chains, even the lead constants 2 and 3 are correct.
http://arxiv.org/abs/0812.0789
Author(s): Fran\c{c}ois Bolley (CEREMADE) and Ivan Gentil (CEREMADE)
Abstract: We obtain and study new $\Phi$-entropy inequalities for diffusion semigroups,
with Poincar\'e or logarithmic Sobolev inequalities as particular cases. From
this study we derive the asymptotic behaviour of a large class of linear
Fokker-Plank type equations under simple conditions, widely extending previous
results. Nonlinear diffusion equations are also studied by means of these
inequalities. The $\Gamma_2$ criterion of D. Bakry and M. Emery appears as a
main tool in the analysis, in local or integral forms.
http://arxiv.org/abs/0812.0800
Author(s): Dirk Blomker and Wei Wang
Abstract: The qualitative properties of local random invariant manifolds for stochastic
partial differential equations with quadratic nonlinearities and multiplicative
noise is studied by a cut off technique. By a detail estimates on the Perron
fixed point equation describing the local random invariant manifold, the
structure near a bifurcation is given.
http://arxiv.org/abs/0812.0390
Author(s): Stefaan Vaes and Nikolas Vander Vennet
Abstract: We identify the Poisson boundary of the dual of the universal compact quantum
group A_u(F) with a measurable field of ITPFI factors.
http://arxiv.org/abs/0812.0804
Author(s): O.L.V. Costa and F. Dufour
Abstract: The main goal of this paper is to derive sufficient conditions for the
existence of an optimal control strategy for the long run average continuous
control problem of piecewise deterministic Markov processes (PDMP's) taking
values in a general Borel space and with compact action space depending on the
state variable. In order to do that we apply the so-called vanishing discount
approach to obtain a solution to an average cost optimality inequality
associated to the long run average cost problem. Our main assumptions are
written in terms of some integro-differential inequalities related to the
so-called expected growth condition, and geometric convergence of the post-jump
location kernel associated to the PDMP.
http://arxiv.org/abs/0812.0820
Author(s): Xicheng Zhang
Abstract: In this paper, we first study the existence-uniqueness and large deviation
estimate of solutions for stochastic Volterra integral equations with singular
kernels in 2-smooth Banach spaces. Then, we apply them to a large class of
semilinear stochastic partial differential equations (SPDE) driven by Brownian
motions as well as by fractional Brownian motions, and obtain the existence of
unique maximal strong solutions (in the sense of SDE and PDE) under local
Lipschitz conditions. Lastly, high order SPDEs in a bounded domain of Euclidean
space, second order SPDEs on complete Riemannian manifolds, as well as
stochastic Navier-Stokes equations are investigated.
http://arxiv.org/abs/0812.0834
Author(s): Guodong Pang and Ward Whitt
Abstract: In order to obtain Markov heavy-traffic approximations for infinite-server
queues with general non-exponential service-time distributions and general
arrival processes, possibly with time-varying arrival rates, we establish
heavy-traffic limits for two-parameter stochastic processes. We consider the
random variables $Q^e(t,y)$ and $Q^r(t,y)$ representing the number of customers
in the system at time $t$ that have elapsed service times less than or equal to
time $y$, or residual service times strictly greater than $y$. We also consider
$W^r(t,y)$ representing the total amount of work in service time remaining to
be done at time $t+y$ for customers in the system at time $t$. The
two-parameter stochastic-process limits in the space $D([0,\infty),D)$ of
$D$-valued functions in $D$ draw on, and extend, previous heavy-traffic limits
by Glynn and Whitt (1991), where the case of discrete service-time
distributions was treated, and Krichagina and Puhalskii (1997), where it was
shown that the variability of service times is captured by the Kiefer process
with second argument set equal to the service-time c.d.f.
http://arxiv.org/abs/0812.0877
Author(s): Enkelejd Hashorva
Abstract: Let X,Y,B be three independent random variables such that $X$ has the same
distribution function as Y B. Assume that B is a Beta random variable with
positive parameters a,b and Y has distribution function H. Pakes and Navarro
(2007) show under some mild conditions that the distribution function H_{a,b}
of X determines H. Based on that result we derive in this paper a recursive
formula for calculation of H, if H_{a,b} is known. Furthermore, we investigate
the relation between the tail asymptotic behaviour of X and Y. We present three
applications of our asymptotic results concerning the extremes of two random
samples with underlying distribution functions H and H_{a,b}, respectively, and
the conditional limiting distribution of bivariate elliptical distributions.
http://arxiv.org/abs/0812.0881
Author(s): Marek Bozejko and Eugene Lytvynov
Abstract: Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it (e.g.
$T=\mathbb R^d$ and $\sigma$ is the Lebesgue measure). We introduce and study a
class of non-commutative generalized stochastic processes, indexed by points of
$T$, with freely independent values. Such a process (field),
$\omega=\omega(t)$, $t\in T$, is given a rigorous meaning through smearing out
with test functions on $T$, with $\int_T \sigma(dt)f(t)\omega(t)$ being a
(bounded) linear operator in a full Fock space. We define a set $\mathbf{CP}$
of all continuous polynomials of $\omega$, and then define a con-commutative
$L^2$-space $L^2(\tau)$ by taking the closure of $\mathbf{CP}$ in the norm
$\|P\|_{L^2(\tau)}:=\|P\Omega\|$, where $\Omega$ is the vacuum in the Fock
space. Through procedure of orthogonalization of polynomials, we construct a
unitary isomorphism between $L^2(\tau)$ and a (Fock-space-type) Hilbert space
$\mathbb F=\mathbb R\oplus\bigoplus_{n=1}^\infty L^2(T^n,\gamma_n)$, with
explicitly given measures $\gamma_n$. We identify the Meixner class as those
processes for which the procedure of orthogonalization leaves the set $\mathbf
{CP}$ invariant. (Note that, in the general case, the projection of a
continuous monomial of oder $n$ onto the $n$-th chaos need not remain a
continuous polynomial.) Each element of the Meixner class is characterized by
two continuous functions $\lambda$ and $\eta\ge0$ on $T$, such that, in the
$\mathbb F$ space, $\omega$ has representation
$\omega(t)=\di_t^\dag+\lambda(t)\di_t^\dag\di_t+\di_t+\eta(t)\di_t^\dag\di^2_t$,
where $\di_t^\dag$ and $\di_t$ are the usual creation and annihilation
operators at point $t$.
http://arxiv.org/abs/0812.0895
Author(s): Eugene Lytvynov and Irina Rodionova
Abstract: We compare some properties of the lowering and raising operators for the
classical and free classes of Meixner polynomials on the real line.
http://arxiv.org/abs/0812.0896
Author(s): Laurent Menard
Abstract: We prove that the uniform infinite random quadrangulations defined
respectively by Chassaing-Durhuus and Krikun have the same distribution.
http://arxiv.org/abs/0812.0965
Author(s): Miguel Abadi Nicolas Vergne
Abstract: We describe the statistics of repetition times of a string of symbols in a
stochastic process. Denote by T(A) the time elapsed until the process spells
the finite string A and by S(A) the number of consecutive repetitions of A. We
prove that, if the length of the string grows unbondedly, (1) the distribution
of T(A), when the process starts with A, is well aproximated by a certain
mixture of the point measure at the origin and an exponential law, and (2) S(A)
is approximately geometrically distributed. We provide sharp error terms for
each of these approximations. The errors we obtain are point-wise and allow to
get also approximations for all the moments of T(A) and S(A). To obtain (1) we
assume that the process is phi-mixing while to obtain (2) we assume the
convergence of certain contidional probabilities.
http://arxiv.org/abs/0812.1016
Author(s): Richard F. Bass
Abstract: For $\alpha\in [1,2)$ we consider operators of the form $$L f(x)=\int_{R^d}
[f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h]
\frac{A(x,h)}{|h|^{d+\alpha}}$$ and for $\alpha\in (0,1)$ we consider the same
operator but where the $\nabla f$ term is omitted. We prove, under appropriate
conditions on $A(x,h)$, that the solution $u$ to $L u=f$ will be in
$C^{\alpha+\beta}$ if $f\in C^\beta$.
http://arxiv.org/abs/0812.0982
Author(s): Xavier Bressaud and Nicolas Fournier
Abstract: We consider the so-called one-dimensional forest-fire process. At each site
of $\mathbb{Z}$, a tree appears at rate 1. At each site of $\mathbb{Z}$ a fire
starts at rate $\lambda>0$, destroying immediately the whole corresponding
connected component of trees. We show that when making $\lambda$ tend to 0,
with a correct normalization, the forest-fire process tends to an uniquely
defined process, of which we describe precisely the dynamics. The normalization
consists of accelerating time by a factor $\log (1/\lambda)$ and of compressing
space by a factor $\lambda \log(1/\lambda)$. The limit process is quite simple:
it can be built using a graphical construction, and can be perfectly simulated.
Finally, we derive some asymptotic estimates (when $\lambda\to 0$) for the
cluster-size distribution of the forest-fire process.
http://arxiv.org/abs/0812.1099
Author(s): Steven N. Evans and Peter L. Ralph
Abstract: If we follow an asexually reproducing population through time, then the
amount of time that has passed since the most recent common ancestor (MRCA) of
all current individuals lived will change as time progresses. The resulting
stochastic process has been studied previously when the population has a
constant large size and evolves via the diffusion limit of standard
Wright-Fisher dynamics. We investigate cases in which the population varies in
size and evolves according to a class of models that includes suitably
conditioned $(1+\beta)$-stable continuous state branching processes (in
particular, it includes the conditioned Feller continuous state branching
process). We also consider the discrete time Markov chain that tracks the MRCA
age just before and after its successive jumps. We find transition
probabilities for both the continuous and discrete time processes, determine
when these processes are transient and recurrent, and compute stationary
distributions when they exist. We also introduce a new family of Markov
processes that stand in a relation with respect to the $(1+\beta)$-stable
continuous state branching process that is similar to the one between the
Bessel-squared diffusions and the Feller continuous state branching process.
http://arxiv.org/abs/0812.1302
Author(s): Yu Zhang
Abstract: Kesten showed the exponential decay of percolation probability in the
subcritical phase for the two-dimensional percolation model. This result
implies his celebrated computation that $p_c=0.5$ for bond percolation in the
square lattice, and site percolation in the triangular lattice, respectively.
In this paper, we present a simpler proof for Kesten's theorem.
http://arxiv.org/abs/0812.1384
Author(s): A.B. Dieker and J. Warren
Abstract: Using a change-of-measure argument, we prove an equality in law between the
process of largest eigenvalues in a generalized Wishart random-matrix process
and a last-passage percolation process. This equality in law was conjectured by
Borodin and Peche.
http://arxiv.org/abs/0812.1504
Author(s): Andrey Novikov
Abstract: Suppose that at any stage of a statistical experiment a control variable $X$
that affects the distribution of the observed data $Y$ can be used. The
distribution of $Y$ depends on some unknown parameter $\theta$, and we consider
the classical problem of testing a simple hypothesis $H_0: \theta=\theta_0$
against a simple alternative $H_1: \theta=\theta_1$ allowing the data to be
controlled by $X$, in the following sequential context. The experiment starts
with assigning a value $X_1$ to the control variable and observing $Y_1$ as a
response. After some analysis, we choose another value $X_2$ for the control
variable, and observe $Y_2$ as a response, etc. It is supposed that the
experiment eventually stops, and at that moment a final decision in favour of
$H_0$ or $H_1$ is to be taken.
In this article, our aim is to characterize the structure of optimal
sequential procedures, based on this type of data, for testing a simple
hypothesis against a simple alternative.
http://arxiv.org/abs/0812.1395
Author(s): Elizabeth Beer and James Allen Fill and Svante Janson and and Edward R. Scheinerman
Abstract: We consider three classes of random graphs: edge random graphs, vertex random
graphs, and vertex-edge random graphs. Edge random graphs are Erdos-Renyi
random graphs, vertex random graphs are generalizations of geometric random
graphs, and vertex-edge random graphs generalize both. The names of these three
types of random graphs describe where the randomness in the models lies: in the
edges, in the vertices, or in both. We show that vertex-edge random graphs,
ostensibly the most general of the three models, can be approximated
arbitrarily closely by vertex random graphs, but that the two categories are
distinct.
http://arxiv.org/abs/0812.1410
Author(s): Castell Fabienne
Abstract: We prove a large deviations principle for the q-fold (q>1) self-intersection
local time of a continuous time simple random walk on the d-dimensional
lattice, in the critical dimension d=(2q)/(q-1). When q is integer, we obtain
similar results for the intersection local times of q independent simple random
walks.
http://arxiv.org/abs/0812.1639
Author(s): Nicolas Champagnat (INRIA Sophia Antipolis / INRIA Lorraine / IECN) and Sylvie M\'el\'eard (CMAP)
Abstract: We are interested in the study of models describing the evolution of a
polymorphic population with mutation and selection in the specific scales of
the biological framework of adaptive dynamics. The population size is assumed
to be large and the mutation rate small. We prove that under a good combination
of these two scales, the population process is approximated in the long time
scale of mutations by a Markov pure jump process describing the successive
trait equilibria of the population. This process, which generalizes the
so-called trait substitution sequence, is called polymorphic evolution
sequence. Then we introduce a scaling of the size of mutations and we study the
polymorphic evolution sequence in the limit of small mutations. From this study
in the neighborhood of evolutionary singularities, we obtain a full
mathematical justification of a heuristic criterion for the phenomenon of
evolutionary branching. To this end we finely analyze the asymptotic behavior
of 3-dimensional competitive Lotka-Volterra systems.
http://arxiv.org/abs/0812.1655
Author(s): Francis Comets and Serguei Popov and Gunter M. Sch\"utz and Marina Vachkovskaia
Abstract: We consider transport diffusion in a stochastic billiard in a random tube
which is elongated in the direction of the first coordinate (the tube axis).
Inside the random tube, which is stationary and ergodic, non-interacting
particles move straight with constant speed. Upon hitting the tube walls, they
are reflected randomly, according to the cosine law: the density of the
outgoing direction is proportional to the cosine of the angle between this
direction and the normal vector. Steady state transport is studied by
introducing an open tube segment as follows: We cut out a large finite segment
of the tube with segment boundaries perpendicular to the tube axis. Particles
which leave this piece through the segment boundaries disappear from the
system. Through stationary injection of particles at one boundary of the
segment a steady state with non-vanishing stationary particle current is
maintained. We prove (i) that in the thermodynamic limit of an infinite open
piece the coarse-grained density profile inside the segment is linear, and (ii)
that the transport diffusion coefficient obtained from the ratio of stationary
current and effective boundary density gradient equals the diffusion
coefficient of a tagged particle in an infinite tube. Thus we prove equality of
transport diffusion and self-diffusion coefficients for quite generic rough
(random) tubes.
http://arxiv.org/abs/0812.1659
Author(s): Gianluca Casseses
Abstract: We prove results on the existence of Dol\'{e}ans-Dade measures and of the
Doob-Meyer decomposition for supermartingales indexed by a general index set
http://arxiv.org/abs/0812.1664
Author(s): Quansheng Liu (LMAM) and Fr\'ed\'erique Watbled (LMAM)
Abstract: The objective of the present paper is to establish exponential large
deviation inequalities, and to use them to show exponential concentration
inequalities for the free energy of a polymer in general random environment,
its rate of convergence, and an expression of its limit value in terms of those
of some multiplicative cascades.
http://arxiv.org/abs/0812.1719
Author(s): Max-K. von Renesse and Michael Scheutzow
Abstract: We provide sufficient conditions on the coefficients of a stochastic
functional differential equation with bounded memory driven by Brownian motion
which guarantee existence and uniqueness of a maximal local and global strong
solution for each initial condition. Our results extend those of previous
works. For local existence and uniqueness, we only require the coefficients to
be continuous and to satisfy a one-sided local Lipschitz (or monotonicity)
condition. In an appendix we formulate and prove four lemmas which may be of
independent interest: three of them can be viewed as rather general stochastic
versions of Gronwall's Lemma, the final one - which we call Dereich-Lemma -
provides tail bounds for Hoelder norms of stochastic integrals.
http://arxiv.org/abs/0812.1726
Author(s): J. Brian Conrey and David W. Farmer and and \"Ozlem Imamoglu
Abstract: We show that if a real trigonometric polynomial has few real roots, then the
trigonometric polynomial obtained by writing the coefficients in reverse order
must have many real roots. This is used to show that a class of random
trigonometric polynomials has, on average, many real roots. In the case that
the coefficients of a real trigonometric polynomial are independently and
identically distributed, but with no other assumptions on the distribution, the
expected fraction of real zeros is at least one-half. This result is best
possible.
http://arxiv.org/abs/0812.1752
Author(s): Yevgeniy Kovchegov and Nick Meredith and Eyal Nir
Abstract: This study of occupation time densities for continuous-time Markov processes
was inspired by the work of E.Nir et al (2006) in the field of Single Molecule
FRET spectroscopy. There, a single molecule fluctuates between two or more
states, and the experimental observable depends on the state's occupation time
distribution. To mathematically describe the observable there was a need to
calculate a single state occupation time distribution.
In this paper, we consider a Markov process with countably many states. In
order to find a one-stete occupation time density, we use a combination of
Fourier and Laplace transforms in the way that allows for inversion of the
Fourier transform. We derive an explicit expression for an occupation time
density in the case of a simple continuous time random walk on Z. Also we
examine the spectral measures in Karlin-McGregor diagonalization in an attempt
to represent occupation time densities via modified Bessel functions.
http://arxiv.org/abs/0812.1775
Author(s): Yevgeniy Kovchegov
Abstract: In this article, we will explore why Karlin-McGregor method of using
orthogonal polynomials in the study of Markov processes was so successful for
one dimensional nearest neighbor processes, but failed beyond nearest neighbor
transitions. We will proceed by suggesting and testing possible fixtures.
http://arxiv.org/abs/0812.1779
Author(s): Michal Baran and Jerzy Zabczyk
Abstract: The completeness problem of the bond market model with noise given by the
independent Wiener process and Poisson random measure is studied. Hedging
portfolios are assumed to have maturities in a countable, dense subset of a
finite time interval. It is shown that under some assumptions the market is not
complete unless the support of the Levy measure consists of a finite number of
points. Explicit constructions of contingent claims which can not be replicated
are provided.
http://arxiv.org/abs/0812.1796
Author(s): Martin T. Barlow and Richard F. Bass and Takashi Kumagai and and Alexander Teplyaev
Abstract: We prove that, up to scalar multiples, there exists only one local regular
Dirichlet form on a generalized Sierpinski carpet that is invariant with
respectto the local symmetries of the carpet. Consequently for each suchfractal
the law of Brownian motion is uniquely determined and theLaplacian is well
defined.
http://arxiv.org/abs/0812.1802
Author(s): Lorenzo Finesso and Peter Spreij
Abstract: In this paper we attempt at understanding how to build an optimal normal
factor analysis model. The criterion we have chosen to evaluate the distance
between different models is the I-divergence between the corresponding normal
laws. The algorithm that we propose for the construction of the best
approximation is of an the alternating minimization kind.
http://arxiv.org/abs/0812.1804
Author(s): Raluca Balan and Ciprian Tudor (CES and SAMOS)
Abstract: We consider the stochastic heat equation with multiplicative noise
$u_t={1/2}\Delta u+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where
$\diamond$ denotes the Wick product, and the solution is interpreted in the
mild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq
1/2$), and colored in space (with spatial covariance kernel $f$). We prove that
if $f$ is the Riesz kernel of order $\alpha$, or the Bessel kernel of order
$\alpha1/2$), respectively $d<2+\alpha$ (if $H=1/2$), whereas
if $f$ is the heat kernel or the Poisson kernel, then the equation has a
solution for any $d$. We give a representation of the $k$-th order moment of
the solution, in terms of an exponential moment of the "convoluted weighted"
intersection local time of $k$ independent $d$-dimensional Brownian motions.
http://arxiv.org/abs/0812.1913
Author(s): Gabriel Faraud
Abstract: We study a very general model of random walk in random environment on trees,
for which we present a recurrence criterion and a functional central limit
theorem. This last result is a generalization of a result of Y. Peres and O.
Zeitouni (2006).
http://arxiv.org/abs/0812.1948
Author(s): Mikael Falconnet (IF)
Abstract: We consider models of nucleotidic substitution processes where the rate of
substitution at a given site depends on the state of its neighbours. For a wide
class of such nonreversible models, we show how to compute consistent,
mathematically exact, estimators of the time elapsed between aligned sequences,
for an ancestral sequence and a present one, and also for two present
sequences. In both cases, we provide asymptotic confidence intervals, valid for
nucleotidic sequences of finite length. We compute explicit formulas for the
estimators and for their confidence intervals in the simplest nontrivial case,
the Jukes-Cantor model with CpG influence.
http://arxiv.org/abs/0812.1962
Author(s): Dong Wang
Abstract: In this paper we prove that the integral of the Wishart ensemble is, in a
certain sense, a KP $\tau$ function, and generalize the result to other random
matrix models, especially the Hermitian matrix model with external source.
Besides potential application in multivariate statistics, we obtain some
interesting combinatorial results.
http://arxiv.org/abs/0810.0280
Author(s): A. C. D. van Enter and C. Kuelske and A. A. Opoku and W. M. Ruszel
Abstract: We review some recent developments in the study of Gibbs and non-Gibbs
properties of transformed n-vector lattice and mean-field models under various
transformations. Also, some new results for the loss and recovery of the Gibbs
property of planar rotor models during stochastic time evolution are presented.
http://arxiv.org/abs/0812.1751
Author(s): Wei Wang and A. J. Roberts
Abstract: The macroscopic behavior of dissipative stochastic partial differential
equations usually can be described by a finite dimensional system. This article
proves that a macroscopic reduced model may be constructed for stochastic
reaction-diffusion equations with cubic nonlinearity by artificial separating
the system into two distinct slow-fast time parts. An averaging method and a
deviation estimate show that the macroscopic reduced model should be a
stochastic ordinary equation which includes the random effect transmitted from
the microscopic timescale due to the nonlinear interaction. Numerical
simulations of an example stochastic heat equation confirms the predictions of
this stochastic modelling theory. This theory empowers us to better model the
long time dynamics of complex stochastic systems.
http://arxiv.org/abs/0812.1837
Author(s): Ohad N. Feldheim and Sasha Sodin
Abstract: After proper rescaling and under some technical assumptions, the smallest
eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1
converges to the Tracy--Widom distribution. This complements the results on the
largest eigenvalue, due to Soshnikov and Peche.
http://arxiv.org/abs/0812.1961
Author(s): Mohammud Foondun
Abstract: We consider the operator $\sL$ defined on $C^2(\bR^d)$ functions by \sL
f(x)&=&{1/2}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial^2f(x)}{\partial x_i\partial
x_j}+\sum_{i=1}^d b_i(x)\frac{\partial f(x)}{\partial x_i}
&+&\int_{\bR^d\backslash\{0\}}[f(x+h)-f(x)-1_{(|h|\leq1)}h\cdot \grad
f(x)]n(x,h)dh. Under the assumption that the local part of the operator is
uniformly elliptic and with suitable conditions on $n(x,h)$, we establish a
Harnack inequality for functions that are nonnegative in $\bR^d$ and harmonic
in a domain. We also show that the Harnack inequality can fail without suitable
conditions on $n(x,h)$. A regularity theorem for those nonnegative harmonic
functions is also proved
http://arxiv.org/abs/0812.2082
Author(s): Agnieszka Czyzewska-Jankowska and Zbigniew J. Jurek
Abstract: The method of \emph{random integral representation}, that is, the method of
representing given probability measure as the probability distribution of some
random integral, was quite successful in the past few decades. In this note we
will find such a representation for the class $L^f$ of selfdecomposable
distributions that posses the factorization property. The class $L^f$ was
introduced in the paper of Iksanov, Jurek and Schreiber, \textbf{Ann. Probab.}
vol. 32, 2004. In addition, we also study composition of some random integral
mappings.
http://arxiv.org/abs/0812.2129
Author(s): Richard Durrett and Daniel Remenik
Abstract: We investigate an interacting particle system inspired by the gypsy moth,
whose populations grow until they become sufficiently dense so that an epidemic
reduces them to a low level. We consider this process on a random 3-regular
graph and on the $d$-dimensional lattice and torus, with $d\geq2$. On the
finite graphs with global dispersal or with a dispersal radius that grows with
the number of sites, we prove convergence to a dynamical system that is chaotic
for some parameter values. We conjecture that on the infinite lattice with a
fixed finite dispersal distance, distant parts of the lattice oscillate out of
phase so there is a unique non-trivial stationary distribution.
http://arxiv.org/abs/0812.2248
Author(s): Shankar Bhamidi and Guy Bresler and and Allan Sly
Abstract: Exponential random graphs are used extensively in the sociology literature.
This model seeks to incorporate in random graphs the notion of reciprocity,
that is, the larger than expected number of triangles and other small
subgraphs. Sampling from these distributions is crucial for parameter
estimation hypothesis testing, and more generally for understanding basic
features of the network model itself. In practice sampling is typically carried
out using Markov chain Monte Carlo, in particular either the Glauber dynamics
or the Metropolis-Hasting procedure.
In this paper we characterize the high and low temperature regimes of the
exponential random graph model. We establish that in the high temperature
regime the mixing time of the Glauber dynamics is $\Theta(n^2 \log n)$, where
$n$ is the number of vertices in the graph; in contrast, we show that in the
low temperature regime the mixing is exponentially slow for any local Markov
chain. Our results, moreover, give a rigorous basis for criticisms made of such
models. In the high temperature regime, where sampling with MCMC is possible,
we show that any finite collection of edges are asymptotically independent;
thus, the model does not possess the desired reciprocity property, and is not
appreciably different from the Erd\H{o}s-R\'enyi random graph.
http://arxiv.org/abs/0812.2265
Author(s): Andras Balint
Abstract: For parameters p and q such that the random-cluster measure \phi for Z^d with
parameters p and q is unique, the q-divide and colour (DaC(q)) model on Z^d is
defined as follows. First we draw a bond configuration distributed according to
\phi. Then to each FK cluster (i.e., to every vertex in the FK cluster),
independently for different FK clusters, we assign a spin from the set
{1,2,...,s} in such a way that spin i has probability a_i.
In this paper we prove that the resulting measure on the spin configurations
is a Gibbs measure for small values of p, and it is not a Gibbs measure for
large p, except in the special case of a_1=a_2=...=a_s=1/q, when the DaC(q)
model coincides with the random-cluster representation of the q-state Potts
model.
Our analysis is based on Haggstrom's methods developed for the fuzzy Potts
model.
http://arxiv.org/abs/0812.2399
Author(s): Roman Vershynin
Abstract: We study the spectral norm of matrices M that can be factored as M=BA, where
A is a random matrix with independent mean zero entries, and B is a fixed
matrix. Under the (4+epsilon)-th moment assumption on the entries of A, we show
that the spectral norm of such an m by n matrix M is bounded by \sqrt{m} +
\sqrt{n}, which is sharp. In other words, in regard to the spectral norm,
products of random and deterministic matrices behave similarly to random
matrices with independent entries. This result along with the previous work of
M. Rudelson and the author implies that the smallest singular value of a random
m times n matrix with i.i.d. mean zero entries and bounded (4+epsilon)-th
moment is bounded below by \sqrt{m} - \sqrt{n-1} with high probability.
http://arxiv.org/abs/0812.2432
Author(s): Alexandre F. Roch
Abstract: We extend the model of liquidity risk of Cetin et al. [5] to allow for price
impacts. Starting from simple principles, we show that the impact of a trade on
prices is directly proportional to the size of the transaction and the amount
of liquidity of the asset. This leads to a new characterization of
self-financing trading strategies and a sufficient condition for no arbitrage.
We show that, with the use of volatility swaps, contingent claims whose payoffs
depend on the value of the asset can be approximately replicated. The
replicating costs of such payoffs are obtained from the solutions of BSDEs with
http://arxiv.org/abs/0812.2440
Author(s): Alexandre F. Roch
Abstract: In this paper, we study the valuation of American type derivatives in the
stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We
characterize the value of such derivatives as the unique viscosity solution of
an integral-partial differential equation when the payoff function satisfies a
Lipschitz condition.
http://arxiv.org/abs/0812.2444
Author(s): Leonardo T. Rolla
Abstract: * ACTIVATED RANDOM WALK MODEL * This is a conservative particle system on the
lattice, with a Markovian continuous-time evolution. Active particles perform
random walks without interaction, and they may as well change their state to
passive, then stopping to jump. When particles of both types occupy the same
site, they all become active. This model exhibits phase transition in the sense
that for low initial densities the system locally fixates and for high
densities it keeps active. Though extensively studied in the physics
literature, the matter of giving a mathematical proof of such phase transition
remained as an open problem for several years. In this work we identify some
variables that are sufficient to characterize fixation and at the same time are
stochastically monotone in the model's parameters. We employ an explicit
graphical representation in order to obtain the monotonicity. With this method
we prove that there is a unique phase transition for the one-dimensional
finite-range random walk. Joint with V. Sidoravicius.
* BROKEN LINE PROCESS * We introduce the broken line process and derive some
of its properties. Its discrete version is presented first and a natural
generalization to the continuum is then proposed and studied. The broken lines
are related to the Young diagram and the Hammersley process and are useful for
computing last passage percolation values and finding maximal oriented paths.
For a class of passage time distributions there is a family of boundary
conditions that make the process stationary and reversible. One application is
a simple proof of the explicit law of large numbers for last passage
percolation with exponential and geometric distributions. Joint with V.
Sidoravicius, D. Surgailis, and M. E. Vares.
http://arxiv.org/abs/0812.2473
Author(s): Loic Chaumont and Andreas Eos Kyprianou and Juan Carlos Pardo and Victor Rivero
Abstract: For a positive self-similar Markov process, $X$, we construct a local time
for the random set, $\Theta,$ of times where the process reaches its past
supremum. Using this local time we describe an exit system for the excursions
of $X$ out of its past supremum. Next, we define and study the ladder process
$(R,H)$ associated to a positive self-similar Markov process $X$, viz. a
bivariate Markov process with a scaling property whose coordinates are the
right inverse of the local time of the random set $\Theta$ and the process $X$
sampled on the local time scale. The process $(R,H)$ is described in terms of
ladder process associated to the Levy process associated to $X$ via Lamperti's
transformation. In the case where $X$ never hits 0 and the upward ladder height
process is not arithmetic and has finite mean we prove the finite dimensional
convergence of $(R,H)$ as the starting point of $X$ tends to $0.$ Finally, we
use these results to provide an alternative proof to the weak convergence of
$X$ as the starting point tends to $0.$ Our approach allows us to address two
issues that remained open in \cite{CCh}, namely to remove a redundant
hypothesis and to provide a formula for the entrance law of $X$ in the case
where the underlying Levy process oscillates.
http://arxiv.org/abs/0812.2506
Author(s): Alexander Fribergh (ICJ)
Abstract: We study the speed of a biased random walk on a percolation cluster on $\Z^d$
in function of the percolation parameter $p$. We obtain a first order expansion
of the speed at $p=1$ which proves that percolating slows down the random walk
at least in the case where the drift is along a component of the lattice.
http://arxiv.org/abs/0812.2532
Author(s): Andreas Nordvall Lager{\aa}s
Abstract: Copulas have been popular to model dependence for multivariate distributions,
but have not been used much in modelling temporal dependence of univariate time
series. This paper shows some difficulties with using copulas even for Markov
processes: some tractable copulas such as mixtures between copulas of complete
co- and countermonotonicity and independence (Fr{\'e}chet copulas) are shown to
imply quite a restricted type of Markov process, and Archimedean copulas are
shown to be incompatible with Markov chains. We also investigate Markov chains
that are spreadable, or equivalently, conditionally i.i.d.
http://arxiv.org/abs/0812.2548
Author(s): Bela Bollobas and Oliver Riordan
Abstract: Recently, Bollob\'as, Janson and Riordan have introduced a family of random
graph models producing inhomogeneous graphs with $n$ vertices and $\Theta(n)$
edges whose distribution is characterized by a kernel, i.e., a symmetric
measurable function $\ka:[0,1]^2 \to [0,\infty)$. To understand these models,
we should like to know when different kernels $\ka$ give rise to `similar'
graphs, and, given a real-world network, how `similar' is it to a typical graph
$G(n,\ka)$ derived from a given kernel $\ka$.
The analogous questions for dense graphs, with $\Theta(n^2)$ edges, are
answered by recent results of Borgs, Chayes, Lov\'asz, S\'os, Szegedy and
Vesztergombi, who showed that several natural metrics on graphs are equivalent,
and moreover that any sequence of graphs converges in each metric to a graphon,
i.e., a kernel taking values in $[0,1]$.
Possible generalizations of these results to graphs with $o(n^2)$ but
$\omega(n)$ edges are discussed in a companion paper [arXiv:0708.1919]; here we
focus only on graphs with $\Theta(n)$ edges, which turn out to be much harder
to handle. Many new phenomena occur, and there are a host of plausible metrics
to consider; many of these metrics suggest new random graph models, and vice
versa.
http://arxiv.org/abs/0812.2656
Author(s): Omar Boukhadra
Abstract: We consider the nearest-neighbor simple random walk on $\Z^{d}$, $d\geq 4$,
driven by a field of i.i.d. random nearest-neighbor conductances
$\omega_{xy}\in[0,1]$. Our aim is to derive estimates of the heat-kernel decay
in a case where ellipticity assumption is absent. We consider the case of
independant conductances with polynomial tail near 0 and obtain for almost
every environment an anomalous lower bound on the heat-kernel.
http://arxiv.org/abs/0812.2669
Author(s): Steven N. Evans and Bernd Sturmfels and Caroline Uhler
Abstract: We use methods from combinatorics and algebraic statistics to study analogues
of birth-and-death processes that have as their state space a finite subset of
the $m$-dimensional lattice and for which the $m$ matrices that record the
transition probabilities in each of the lattice directions commute pairwise.
One reason such processes are of interest is that the transition matrix is
straightforward to diagonalize, and hence it is easy to compute $n$ step
transition probabilities. The set of commuting birth-and-death processes
decomposes as a union of toric varieties, with the main component being the
closure of all processes whose nearest neighbor transition probabilities are
positive. We exhibit an explicit monomial parametrization for this main
component, and we explore the boundary components using primary decomposition.
http://arxiv.org/abs/0812.2724
Author(s): Balazs Rath
Abstract: We define a modification of the Erdos-Renyi random graph process which can be
regarded as the mean field frozen percolation process. We describe the behavior
of the process using differential equations and investigate their solutions in
order to show the self-organized critical and extremum properties of the
critical frozen percolation model. We prove two limit theorems about the
distribution of the size of the component of a typical frozen vertex.
http://arxiv.org/abs/0812.2750
Author(s): Sophie Dede (PMA)
Abstract: In this paper, we derive asymptotic results for L^1-Wasserstein distance
between the distribution function and the corresponding empirical distribution
function of a stationary sequence. Next, we give some applications to dynamical
systems and causal linear processes. To prove our main result, we give a
Central Limit Theorem for ergodic stationary sequences of random variables with
values in L^1. The conditions obtained are expressed in terms of
projective-type conditions. The main tools are martingale approximations.
http://arxiv.org/abs/0812.2839
Author(s): M. Y. Mo
Abstract: We studied universality of Wishart ensembles whose covariance matrix has 2
distinct eigenvalues and the number of each of these eigenvalue goes to
infinity in the asymptotic limit. In this case, the limiting eigenvalue
distribution can be supported on 1 or 2 disjoint intervals. In our previous
work the case when the support consists of 2 intervals was studied. This paper
complements our previous analysis and studied the case when the support
consists of a single interval. By using Riemann-Hilbert analysis, we have shown
that under proper rescaling of the eigenvalues, the limiting correlation kernel
is given by the sine kernel and the Airy kernel in the bulk and the edge of the
spectrum respectively. As a consequence, the behavior of the largest eigenvalue
in this model is described by the Tracy-Widom distribution.
http://arxiv.org/abs/0812.2863
Author(s): Andrey Novikov
Abstract: Suppose that at any stage of a statistical experiment a control variable $X$
that affects the distribution of the observed data $Y$ at this stage can be
used. The distribution of $Y$ depends on some unknown parameter $\theta$, and
we consider the problem of testing multiple hypotheses $H_1: \theta=\theta_1$,
$H_2: \theta=\theta_2, ...$, $H_k: \theta=\theta_k$ allowing the data to be
controlled by $X$, in the following sequential context.
The experiment starts with assigning a value $X_1$ to the control variable
and observing $Y_1$ as a response. After some analysis, another value $X_2$ for
the control variable is chosen, and $Y_2$ as a response is observed, etc. It is
supposed that the experiment eventually stops, and at that moment a final
decision in favor of one of the hypotheses $H_1,...$, $H_k$ is to be taken. In
this article, our aim is to characterize the structure of optimal sequential
testing procedures based on data obtained from an experiment of this type in
the case when the observations $Y_1, Y_2,..., Y_n$ are independent, given
controls $X_1,X_2,..., X_n$, $n=1,2,...$.
http://arxiv.org/abs/0812.2712
Author(s): Delphine F\'eral (IMB) and Sandrine P\'ech\'e (IF)
Abstract: We consider large complex random sample covariance matrices obtained from
"spiked populations", that is when the true covariance matrix is diagonal with
all but finitely many eigenvalues equal to one. We investigate the limiting
behavior of the largest eigenvalues when the population and the sample sizes
both become large. Under some conditions on moments of the sample distribution,
we prove that the asymptotic fluctuations of the largest eigenvalues are the
same as for a complex Gaussian sample with the same true covariance. The real
setting is also considered.
http://arxiv.org/abs/0812.2320
Author(s): Shamgar Gurevich (University of California Berkeley) and Ronny Hadani (University of Chicago)
Abstract: In this article we present a statistical version of the Candes-Tao restricted
isometry property (SRIP for short) which holds in general for any incoherent
dictionary which is a disjoint union of orthonormal bases. In addition, we show
that, under appropriate normalization, the eigenvalues of the associated Gram
matrix fluctuate around 1 according to the Wigner semicircle distribution. The
result is then applied to various dictionaries that arise naturally in the
setting of finite harmonic analysis, giving, in particular, a better
understanding on a remark of Applebaum-Howard-Searle-Calderbank concerning RIP
for the Heisenberg dictionary of chirp like functions.
http://arxiv.org/abs/0812.2602
Author(s): Wenlian Lu and Fatihcan M. Atay and J\"urgen Jost
Abstract: We study the local complete synchronization of discrete-time dynamical
networks with time-varying couplings. Our conditions for the temporal variation
of the couplings are rather general and include both variations in the network
structure and in the reaction dynamics; the reactions could, for example, be
driven by a random dynamical system. A basic tool is the concept of Hajnal
diameter which we extend to infinite Jacobian matrix sequences. The Hajnal
diameter can be used to verify synchronization and we show that it is
equivalent to other quantities which have been extended to time-varying cases,
such as the projection radius, projection Lyapunov exponents, and transverse
Lyapunov exponents. Furthermore, these results are used to investigate the
synchronization problem in coupled map networks with time-varying topologies
and possibly directed and weighted edges. In this case, the Hajnal diameter of
the infinite coupling matrices can be used to measure the synchronizability of
the network process. As we show, the network is capable of synchronizing some
chaotic map if and only if there exists an integer T>0 such that for any time
interval of length T, there exists a vertex which can access other vertices by
directed paths in that time interval.
http://arxiv.org/abs/0812.2706
Author(s): Asuka Takatsu
Abstract: In this paper, we prove that if a base space has a cone structure, then so
does its $L^2$-Wasserstein space. Furthermore, we investigate relations between
the base spaces of the both cones. Conversely, we show when an
$L^2$-Wasserstein space has a cone structure satisfying certain conditions,
then its underlying space is also a cone.
http://arxiv.org/abs/0812.2752
Author(s): Tatyana S. Turova
Abstract: We study the inhomogeneous random graphs in the subcritical case. We derive
an exact formula for the size of the largest connected component scaled to
$\log n$ where $n$ is the size of the graph. This generalizes the recent result
for the "rank 1 case". Here we discover that the same well-known equation for
the survival probability, whose positive solution determines the asymptotics of
the size of the largest component in the supercritical case, plays the crucial
role in the subcritical case as well. But now these are the negative solutions
which come into play.
http://arxiv.org/abs/0812.3007
Author(s): Blandine Berard Bergery (IECN) and Christophe Profeta (IECN) and Etienne Tanr\'e (INRIA Sophia Antipolis / INRIA Lorraine / IECN)
Abstract: We propose a mathematical model for one pattern of charts studied in
technical analysis: in a phase of consolidation, the price of a risky asset
goes down $\xi$ times after hitting a resistance level. We construct a
mathematical strategy and we calculate the expectation of the wealth for the
logaritmic utility function. Via simulations, we compare the strategy with the
standard one.
http://arxiv.org/abs/0812.3027
Author(s): Larry Goldstein and Mathew D. Penrose
Abstract: We give error bounds which demonstrate optimal rates of convergence in the
CLT for the total covered volume and the number of isolated shapes, for
germ-grain models with fixed grain radius over a binomial point process of $n$
points in a toroidal spatial region of volume $n$. The proof is based on
Stein's method via size-biased couplings.
http://arxiv.org/abs/0812.3084
Author(s): Anastasia Papavasiliou and Christophe Ladroue
Abstract: We construct an estimator based on "signature matching" for differential
equations driven by rough paths and we prove its consistency and asymptotic
normality. Note that the the Moment Matching estimator is a special case of
this estimator.
http://arxiv.org/abs/0812.3102
Author(s): Jakob E. Bj\"ornberg
Abstract: We present a rigorous determination of the critical value of the ground-state
quantum Ising model in a transverse field, on a class of planar graphs which we
call star-like. These include the star graph, which is a junction of several
copies of Z at a single point. Our approach is to use the graphical, or FK-,
representation of the model, and the probabilistic and geometric tools
associated with it.
http://arxiv.org/abs/0812.3113
Author(s): Vadim Gorin
Abstract: We introduce a stochastic dynamics related to the measures that arise in
harmonic analysis on the infinite-dimensional unitary group. Our dynamics is
obtained as a limit of a sequence of natural Markov chains on Gelfand-Tsetlin
graph. We compute finite-dimensional distributions of the limit Markov process,
the generator and eigenfunctions of the semigroup related to this process. The
limit process can be identified with Doob h-transform of a family of
independent diffusions. Space-time correlation functions of the limit process
have a determinantal form.
http://arxiv.org/abs/0812.3146
Author(s): Emilio Porcu and Rene L. Schilling
Abstract: We show that a large subclass of variograms is closed under Schur products
and that some desirable stability properties, like the Schur product of
\emph{ad hoc} compositions, can be obtained under the proposed setting. We
introduce new classes of kernels of Schoenberg-L\'{e}vy type and show some
important properties of eventually constant, radially symmetric variograms. In
particular, we characterize eventually constant variograms in terms of their
permissibility in Euclidean spaces of arbitrary high dimension.
http://arxiv.org/abs/0812.2936
Author(s): Laurent Decreusefond (LTCI) and Ald\'eric Joulin and Nicolas Savy
Abstract: In this paper, we provide upper bounds on several Rubinstein-type distances
on the configuration space equipped with the Poisson measure. Our inequalities
involve the two well-known gradients, in the sense of Malliavin calculus, which
can be defined on this space. Actually, we show that depending on the distance
between configurations which is considered, it is one gradient or the other
which is the most effective. Some applications to distance estimates between
Poisson and other more sophisticated processes are also provided, and an
investigation of our results to functional inequalities completes this work.
http://arxiv.org/abs/0812.3221
Author(s): J. Gaertner and F. den Hollander and G. Maillard
Abstract: We continue our study of intermittency for the parabolic Anderson model
$\partial u/\partial t = \kappa\Delta u + \xi u$ in a space-time random medium
$\xi$, where $\kappa$ is a positive diffusion constant, $\Delta$ is the lattice
Laplacian on $\Z^d$, $d \geq 1$, and $\xi$ is a simple symmetric exclusion
process on $\Z^d$ in Bernoulli equilibrium. This model describes the evolution
of a \emph{reactant} $u$ under the influence of a \emph{catalyst} $\xi$.
In G\"artner, den Hollander and Maillard (2007) we investigated the behavior
of the annealed Lyapunov exponents, i.e., the exponential growth rates as
$t\to\infty$ of the successive moments of the solution $u$. This led to an
almost complete picture of intermittency as a function of $d$ and $\kappa$. In
the present paper we finish our study by focussing on the asymptotics of the
Lyaponov exponents as $\kappa\to\infty$ in the \emph{critical} dimension $d=3$,
which was left open in G\"artner, den Hollander and Maillard (2007) and which
is the most challenging. We show that, interestingly, this asymptotics is
characterized not only by a \emph{Green} term, as in $d\geq 4$, but also by a
\emph{polaron} term. The presence of the latter implies intermittency of
\emph{all} orders above a finite threshold for $\kappa$.
http://arxiv.org/abs/0812.3311
Author(s): Luc Devroye and Svante Janson
Abstract: We consider a conditioned Galton-Watson tree and prove an estimate of the
number of pairs of vertices with a given distance, or, equivalently, the number
of paths of a given length.
We give two proofs of this result, one probabilistic and the other using
generating functions and singularity analysis.
Moreover, the second proof yields a more general estimate for generating
functions, which is used to prove a conjecture by Bousquet-Melou and Janson
saying that the vertical profile of a randomly labelled conditioned
Galton-Watson tree converges in distribution, after suitable normalization, to
the density of ISE (Integrated Superbrownian Excursion).
http://arxiv.org/abs/0812.3326
Author(s): Olivier Aj Bardou (PMA and GDF-RDD) and Noufel Frikha (PMA and GDF-RDD) and G. Pag\`es (PMA)
Abstract: Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are two risk
measures which are widely used in the practice of risk management. This paper
deals with the problem of computing both VaR and CVaR using stochastic
approximation (with decreasing steps): we propose a first Robbins-Monro
procedure based on Rockaffelar-Uryasev's identity for the CVaR. The convergence
rate of this algorithm to its target satisfies a Gaussian Central Limit
Theorem. As a second step, in order to speed up the initial procedure, we
propose a recursive importance sampling (I.S.) procedure which induces a
significant variance reduction of both VaR and CVaR procedures. This idea,
which goes back to the seminal paper of B. Arouna, follows a new approach
introduced by V. Lemaire and G. Pag\`es. Finally, we consider a deterministic
moving risk level to speed up the initialization phase of the algorithm. We
prove that the convergence rate of the resulting procedure is ruled by a
Central Limit Theorem with minimal variance and its efficiency is illustrated
by considering several typical energy portfolios.
http://arxiv.org/abs/0812.3381
Author(s): Felipe Cucker and Teresa Krick and Gregorio Malajovich and Mario Wschebor
Abstract: In a recent paper [A numerical algorithm for zero counting I: complexity and
accuracy . J. of Complexity 24, 5-6, pp 582-605 (Oct-Dec 2008)] we analyzed a
numerical algorithm for computing the number of real zeros of a polynomial
system. The analysis relied on a condition number k(f) for the input system f.
In this paper we continue this analysis by looking at k(f) as a random variable
derived from imposing a probability measure on the space of polynomial systems.
We give bounds for both the tail P{k(f) > a} and the expected value E(log
k(f)).
http://arxiv.org/abs/0812.3281
Author(s): Nicolas Dirr and Federica Dragoni and Max von Renesse
Abstract: We study the phenomenon of evolution by horizontal mean curvature flow in
sub-Riemannian geometries. We use a stochastic approach to prove the existence
of a generalized evolution in these spaces. In particular we show that the
value function of suitable family of stochastic control problems solves in the
viscosity sense the level set equation for the evolution by horizontal mean
curvature flow.
http://arxiv.org/abs/0812.3288
Author(s): Tom Britton and David Lindenstrand
Abstract: Epidemic models are always simplifications of real world epidemics. Which
real world features to include, and which simplifications to make, depend both
on the disease of interest and on the purpose of the modelling. In the present
paper we discuss some such purposes for which a \emph{stochastic} model is
preferable to a \emph{deterministic} counterpart. The two main examples
illustrate the importance of allowing the infectious and latent periods to be
random when focus lies on the \emph{probability} of a large epidemic outbreak
and/or on the initial \emph{speed}, or growth rate, of the epidemic. A
consequence of the latter is that estimation of the basic reproduction number
$R_0$ is sensitive to assumptions about the distributions of the infectious and
latent periods when using the data from the early stages of an outbreak, which
we illustrate with data from the SARS outbreak. Some further examples are also
discussed as are some practical consequences related to these stochastic
aspects.
http://arxiv.org/abs/0812.3505
Author(s): Bernard Bercu (IMB and INRIA Bordeaux - Sud-Ouest) and Peggy C\'enac (IMB) and Guy Fayolle (INRIA Rocquencourt)
Abstract: We investigate the almost sure asymptotic properties of vector martingale
transforms. Assuming some appropriate regularity conditions both on the
increasing process and on the moments of the martingale, we prove that
normalized moments of any even order converge in the almost sure cental limit
theorem for martingales. A conjecture about almost sure upper bounds under
wider hypotheses is formulated. The theoretical results are supported by
examples borrowed from statistical applications, including linear
autoregressive models and branching processes with immigration, for which new
asymptotic properties are established on estimation and prediction errors.
http://arxiv.org/abs/0812.3528
Author(s): A. Alvarez and F. Panloup and M. Pontier and N. Savy
Abstract: Concerning price processes, the fact that the volatility is not constant has
been observed for a long time. So we deal with models as
$dX_t=\mu_tdt+\sigma_tdW_t$ where $\sigma$ is a stochastic process. Recent
works on volatility modeling suggest that we should incorporate jumps in the
volatility process. Empirical observations suggest that simultaneous jumps on
the price \underline{and} the volatility \cite{BarShep1,ConTan} exist. The
hypothesis that jumps occur simultaneously makes the problem of volatility jump
detection reduced to the prices jump detection. But in case of this hypothesis
failure, we try to work in this direction.
Among others, we use Jacod and Ait-Sahalia' recent work \cite{jac1} giving
estimators of cumulated volatility $\int_0^t|\sigma_s|^pds$ for any $p\geq 2.$
This tool allows us to deliver an estimator of instantaneous volatility.
Moreover we prove a central limit theorem for it. Obviously, such a theorem
provides a confidence interval for the instantaneous volatility and leads us to
a test of the jump existence hypothesis. For instance, we consider a simplest
model having volatility jumps, when volatility is piecewise constant:
$\sigma_t=\sum_{i=0}^{N_t-1} \sigma_i \1_{[\tau_i,\tau_{i+1}[}(t).$ The jump
times are $\tau_i, i\geq 1,$ and $\sigma_i$ is a $\F_{\tau_{i}}$-measurable
random variable. Another example is studied: $\sigma_t=|Y_t|$ where $(Y_t)$ is
a solution to a L\'evy driven SDE, with suitable coefficients.
http://arxiv.org/abs/0812.3538
Author(s): Jonathon Peterson and Ofer Zeitouni
Abstract: We derive properties of the rate function in Varadhan's (annealed) large
deviation principle for multidimensional, ballistic random walk in random
environment, in a certain neighborhood of the zero set of the rate function.
Our approach relates the LDP to that of regeneration times and distances. The
analysis of the latter is possible due to the i.i.d. structure of
regenerations.
http://arxiv.org/abs/0812.3619
Author(s): Wojciech Sarnowski and Krzysztof Szajowski
Abstract: We register a Markov process. At random moment $\theta$ the distribution of
observed sequence changes. Using probability maximizing approach the optimal
stopping rule is identified. For the particular case of disorder the explicit
solution is obtained.
http://arxiv.org/abs/0812.3632
Author(s): Anna Karpowicz and Krzysztof Szajowski
Abstract: The aim of the paper is to extend the model of "fishing problem". The simple
formulation is following. The angler goes to fishing. He buys fishing ticket
for a fixed time. There are two places for fishing at the lake. The fishes are
caught according to renewal processes which are different at both places. The
fishes' weights and the inter-arrival times are given by the sequences of
i.i.d. random variables with known distribution functions. These distributions
are different for the first and second fishing place. The angler's satisfaction
measure is given by difference between the utility function dependent on size
of the caught fishes and the cost function connected with time. On each place
the angler has another utility functions and another cost functions. In this
way, the angler's relative opinion about these two places is modeled. For
example, on the one place better sort of fish can be caught with bigger
probability or one of the places is more comfortable. Obviously our angler
wants to have as much satisfaction as possible and additionally he have to
leave the lake before the fixed moment. Therefore his goal is to find two
optimal stopping times in order to maximize his satisfaction. The first time
corresponds to the moment, when he eventually should change the place and the
second time, when he should stop fishing. These stopping times have to be less
than the fixed time of fishing. The value of the problem and the optimal
stopping times are derived.
http://arxiv.org/abs/0812.3651
Author(s): John H. J. Einmahl and Johan Segers
Abstract: Consider a random sample from a bivariate distribution function $F$ in the
max-domain of attraction of an extreme value distribution function $G$. This
$G$ is characterized by two extreme value indices and a spectral measure, the
latter determining the tail dependence structure of $F$. A major issue in
multivariate extreme value theory is the estimation of the spectral measure
$\Phi_p$ with respect to the $L_p$ norm. For every $p \in [1, \infty]$, a
nonparametric maximum empirical likelihood estimator is proposed for $\Phi_p$.
The main novelty is that these estimators are guaranteed to satisfy the moment
constraints by which spectral measures are characterized. Asymptotic normality
of the estimators is proved under conditions that allow for tail independence.
Moreover, the conditions are easily verifiable as we demonstrate through a
number of theoretical examples. A simulation study shows substantially improved
performance of the new estimators. Two case studies illustrate how to implement
the methods in practice.
http://arxiv.org/abs/0812.3485
Author(s): Krzysztof Szajowski
Abstract: A version of the secretary problem is considered. The ranks of items, whose
values are independent, identically distributed random variables
$X_1,X_2,...,X_n$ from a uniform distribution on $[0; 1]$, are observed
sequentially by the grader. He has to select exactly one item, when it appears,
and receives a payoff which is a function of the unobserved realization of
random variable assigned to the item diminished by some cost. The methods of
analysis are based on the existence of an embedded Markov chain and use the
technique of backward induction. The result is a generalization of the
selection model considered by Bearden(2006). The asymptotic behaviour of the
solution is also investigated.
http://arxiv.org/abs/0812.3483
Author(s): Nizar Demni
Abstract: We characterize probability distributions of all order finite moments gaving
ultraspherical type generating functions for orthogonal polynomials.
http://arxiv.org/abs/0812.3666
Author(s): Jean-Christophe Breton and Christian Houdr\'e
Abstract: Given a random word of size n whose letters are drawn independently from an
ordered alphabet of size m, the fluctuations of the shape of the associated
random Young tableaux are investigated, when both n and m converge together to
infinity. If m does not grow too fast and if the draws are uniform, the
limiting shape is the same as the limiting spectrum of the GUE. In the
non-uniform case, a control of both highest probabilities will ensure the
convergence of the first row of the tableau towards the Tracy-Widom
distribution.
http://arxiv.org/abs/0812.3672
Author(s): Li-Xin Zhang and Feifang Hu
Abstract: The Friedman's urn model is a popular urn model which is widely used in many
disciplines. In particular, it is extensively used in treatment allocation
schemes in clinical trials. In this paper, we prove that both the urn
composition process and the allocation proportion process can be approximated
by a multi-dimensional Gaussian process almost surely for a multi-color
generalized Friedman's urn model with non-homogeneous generating matrices. The
Gaussian process is a solution of a stochastic differential equation. This
Gaussian approximation together with the properties of the Gaussian process is
important for the understanding of the behavior of the urn process and is also
useful for statistical inferences. As an application, we obtain the asymptotic
properties including the asymptotic normality and the law of the iterated
logarithm for a multi-color generalized Friedman's urn model as well as the
randomized-play-the-winner rule as a special case.
http://arxiv.org/abs/0812.3697
Author(s): Charles E.M. Pearce and Krzysztof Szajowski and Mitsushi Tamaki
Abstract: We treat a version of the multiple-choice secretary problem called the
multiple-choice duration problem, in which the objective is to maximize the
time of possession of relatively best objects. It is shown that, for the
$m$--choice duration problem, there exists a sequence (s1,s2,...,sm) of
critical numbers such that, whenever there remain k choices yet to be made,
then the optimal strategy immediately selects a relatively best object if it
appears at or after time $s_k$ ($1\leq k\leq m$). We also exhibit an
equivalence between the duration problem and the classical best-choice
secretary problem. A simple recursive formula is given for calculating the
critical numbers when the number of objects tends to infinity. Extensions are
made to models involving an acquisition or replacement cost.
http://arxiv.org/abs/0812.3765
Author(s): C\'edric Boutillier and B\'eatrice de Tili\`ere
Abstract: We study a large class of critical two-dimensional Ising models namely
critical Z-invariant Ising models on periodic graphs, example of which are the
classical square, triangular and honeycomb lattice at the critical temperature.
Fisher introduced a correspondence between the Ising model and the dimer model
on a decorated graph, thus setting dimer techniques as a powerful tool for
understanding the Ising model. In this paper, we give a full description of the
dimer model corresponding to the critical Z-invariant Ising model. We prove
that the dimer characteristic polynomial is equal (up to a constant) to the
critical Laplacian characteristic polynomial, and defines a Harnack curve of
genus 0. We prove an explicit expression for the free energy, and for the Gibbs
measure obtained as weak limit of Boltzmann measures.
http://arxiv.org/abs/0812.3848
Author(s): Li-Xin Zhang and Feifang Hu and Siu Hung Cheung and Wei Sum Chan
Abstract: Urn models have been widely studied and applied in both scientific and social
disciplines. In clinical studies, the adoption of urn models in treatment
allocation schemes has been proved to be beneficial to both researchers, by
providing more efficient clinical trials, and patients, by increasing the
probability of receiving the better treatment. In this paper, we endeavor to
derive a very general class of immigrated urn models that incorporates the
immigration mechanism into the urn process. Important asymptotic properties are
developed and illustrative examples are provided to demonstrate the
applicability of our proposed class of urn models. In general, the immigrated
urn model has smaller variability than the corresponding urn model. Therefore,
it is more powerful when used in clinical trials.
http://arxiv.org/abs/0812.3698
Author(s): Sam Payne and Elina Robeva
Abstract: We present a Monte Carlo algorithm for efficiently finding near optimal moves
and bids in the game of Bidding Hex. The algorithm is based on the recent
solution of Random-Turn Hex by Peres, Schramm, Sheffield, and Wilson together
with Richman's work connecting random-turn games to bidding games.
http://arxiv.org/abs/0812.3677
Author(s): Jinqiao Duan
Abstract: Macroscopic models for spatially extended systems under random influences are
often described by stochastic partial differential equations (SPDEs).
Some techniques for understanding solutions of such equations, such as
estimating correlations, Liapunov exponents and impact of noises, are
discussed. They are relevant for understanding predictability in spatially
extended systems with model uncertainty, for example, in physics, geophysics
and biological sciences. The presentation is for a wide audience.
http://arxiv.org/abs/0812.3679
Author(s): Li-Xin Zhang and Feifang Hu
Abstract: It is often important to incorporating covariate information in the design of
clinical trials. In literature, there are many designs of using stratification
and covariate-adaptive randomization to balance on certain known covariate.
Recently Zhang, Hu, Cheung and Chan (2007) have proposed a family of
covariate-adjusted response-adaptive (CARA) designs and studied their
asymptotic properties. However, these CARA designs often have high
variabilities. In this paper, we propose a new family of covariate-adjusted
response-adaptive (CARA) designs. We show that the new designs have smaller
variabilities and therefore more efficient.
http://arxiv.org/abs/0812.3691
Author(s): Li-Xin Zhang and Feifang Hu and Siu Hung Cheung and Wei Sum Chan
Abstract: The response-adaptive design driven by randomly reinforced urn model is
optimal in the sense that it allocate patients to the best treatment with
probability converging to one. This paper illustrates asymptotic properties for
multi-color reinforced urn models. Results on the rate of convergence of the
number of patients assigned to each treatment are obtained under minimum
requirement of conditions and the distributions of the limits are found.
Asymptotic distributions of the Wald test statistic for testing mean
differences are obtained both under the null hypothesis and alternate
hypothesis. The asymptotic behavior for the non-homogenous is also studied.
http://arxiv.org/abs/0812.3699
Author(s): Michael R. Hoare and Mizan Rahman
Abstract: We present here a probabilistic approach to the generation of new polynomials
in two discrete variables. This extends our earlier work on the 'classical'
orthogonal polynomials in a previously unexplored direction, resulting in the
discovery of an exactly soluble eigenvalue problem corresponding to a bivariate
Markov chain with a transition kernel formed by a convolution of simple
binomial and trinomial distributions. The solution of the relevant
eigenfunction problem, giving the spectral resolution of the kernel, leads to
what we believe to be a new class of orthogonal polynomials in two discrete
variables. Possibilities for the extension of this approach are discussed.
http://arxiv.org/abs/0812.3879
Author(s): Remi Rhodes; Vincent Vargas
Abstract: We are concerned with scaling limits of the solutions to stochastic
differential equations with stationary coefficients driven by Poisson random
measures and Brownian motions. We state an annealed convergence theorem, in
which the limit exhibits a diffusive or superdiffusive behavior, depending on
the integrability properties of the Poisson random measure
http://arxiv.org/abs/0812.3904
Author(s): Bogdan K. Muciek and Krzysztof J. Szajowski
Abstract: The optimal stopping problem for the risk process with interests rates and
when claims are covered immediately is considered. An insurance company
receives premiums and pays out claims which have occured according to a renewal
process and which have been recognized by them. The capital of the company is
invested at interest rate $\alpha\in\Re^{+}$, the size of claims increase at
rate $\beta\in\Re^{+}$ according to inflation process. The immediate payment of
claims decreases the company investment by rate $\alpha_1$. The aim is to find
the stopping time which maximizes the capital of the company. The improvement
to the known models by taking into account different scheme of claims payment
and the possibility of rejection of the request by the insurance company is
made. It leads to essentially new risk process and the solution of optimal
stopping problem is different.
http://arxiv.org/abs/0812.3925
Author(s): S.Hamadene and Y.Ouknine
Abstract: In the first part of this paper we give a solution for the one-dimensional
reflected backward stochastic differential equation (BSDE for short) when the
noise is driven by a Brownian motion and an independent Poisson point process.
The reflecting process is right continuous with left limits (rcll for short)
whose jumps are arbitrary. We first prove existence and uniqueness of the
solution for a specific coefficient in using a method based on a combination of
penalization and the Snell envelope theory. To show the result in the general
framework we use a fixed point argument in an appropriate space. The second
part of the paper is related to BSDEs with two reflecting barriers. Once more
we prove the existence and uniqueness of the solution of the BSDE.
http://arxiv.org/abs/0812.3965
Author(s): Nizar Demni
Abstract: We stduy radial Dunkl processes associated with dihedral systems: we derive
the semi group, the generalized Bessel function, the Dunkl-Hermite polynomials.
Then we give a skew product decomposition by means of independent Bessel
processes and we compute the tail distribution of the first hitting time of the
boundary of Weyl chamber.
http://arxiv.org/abs/0812.4002
Author(s): Federico Camia and Charles M. Newman
Abstract: We provide a representation for the scaling limit of the d=2 critical Ising
magnetization field as a (conformal) random field using SLE (Schramm-Loewner
Evolution) clusters and associated renormalized area measures. The renormalized
areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn)
clusters and the random field is a convergent sum of the area measures with
random signs. Extensions to off-critical scaling limits, to d=3 and to Potts
models are also considered.
http://arxiv.org/abs/0812.4030
Author(s): Wenbo V. Li and Natesh S. Pillai and Robert L. Wolpert
Abstract: We consider a family of stochastic processes $\{X_t^\epsilon, t \in T\}$ on a
metric space $T$, with a parameter $\epsilon \downarrow 0$. We study the
conditions under which \lim_{\e \to 0} \P \Big(\sup_{t \in T} |X_t^\e| < \delta
\Big) =1 when one has the \textit{a priori} estimate on the modulus of
continuity and the value at one point. We compare our problem to the celebrated
Kolmogorov continuity criteria for stochastic processes, and finally give an
application of our main result for stochastic intergrals with respect to
compound Poisson random measures with infinite intensity measures.
http://arxiv.org/abs/0812.4062
Author(s): Delia Coculescu and Monique Jeanblanc and Ashkan Nikeghbali
Abstract: In this paper we give a financial justification, based on non arbitrage
conditions, of the $(H)$ hypothesis in default time modelling. We also show how
the $(H)$ hypothesis is affected by an equivalent change of probability
measure. The main technique used here is the theory of progressive enlargements
of filtrations.
http://arxiv.org/abs/0812.4064
Author(s): Jason Swanson
Abstract: We consider $n$ independent, identically distributed one-dimensional Brownian
motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth density
function $f$. The empirical quantiles, or pointwise order statistics, are
denoted by $B_{j:n}(t)$, and we are interested in a sequence of quantiles
$Q_n(t) = B_{j(n):n}(t)$, where $j(n)/n \to \alpha \in (0,1)$. This sequence
converges in probability in $C[0,\infty)$ to $q(t)$, the $\alpha$-quantile of
the law of $B_j(t)$. Our main result establishes the convergence in law in
$C[0,\infty)$ of the fluctuation processes $F_n = n^{1/2}(Q_n - q)$. The limit
process $F$ is a centered Gaussian process and we derive an explicit formula
for its covariance function. We also show that $F$ has many of the same local
properties as $B^{1/4}$, the fractional Brownian motion with Hurst parameter $H
= 1/4$. For example, it is a quartic variation process, it has H\"older
continuous paths with any exponent $\gamma < 1/4$, and (at least locally) it
has increments whose correlation is negative and of the same order of magnitude
as those of $B^{1/4}$.
http://arxiv.org/abs/0812.4102
Author(s): Enkelejd Hashorva
Abstract: In this paper we consider elliptical random vectors X in R^d,d>1 with
stochastic representation A R U where R is a positive random radius independent
of the random vector U which is uniformly distributed on the unit sphere of R^d
and A is a given matrix. The main result of this paper is an asymptotic
expansion of the tail probability of the norm of X derived under the assumption
that R has distribution function is in the Gumbel or the Weibull max-domain of
attraction.
http://arxiv.org/abs/0812.4105
Author(s): Makoto Katori and Hideki Tanemura
Abstract: Dyson's model is a one-dimensional system of Brownian motions with long-range
repulsive forces acting between any pair of particles with strength
proportional to the inverse of distances. We give sufficient conditions for
initial configurations so that Dyson's model with infinite number of particles
is well defined in the sense that any multitime correlation function is given
by a determinant with a locally integrable kernel. The class of
infinite-dimensional configurations satisfying our conditions is large enough
to study non-equilibrium dynamics. For example, a relaxation process starting
from a configuration, in which each lattice point of $\Z$ is occupied by one
particle, to the stationary state, which is the determinantal point process
with the sine kernel $\mu_{\sin}$, is determined. The invariant measure
$\mu_{\sin}$ also satisfies our conditions and Dyson's model starting from
$\mu_{\sin}$, which is a reversible process, is identified with the infinite
particle system, which is determinantal with the extended sine kernel studied
in the random matrix theory. We also show that this infinite-dimensional
reversible process is Markovian.
http://arxiv.org/abs/0812.4108
Author(s): Frank Ball and David Sirl and Pieter Trapman
Abstract: This paper considers a stochastic SIR (susceptible$\to$infective$\to$removed)
epidemic model in which individuals may make infectious contacts in two ways,
both within `households' (which for ease of exposition are assumed to have
equal size) and along the edges of a random graph describing additional social
contacts. Heuristically-motivated branching process approximations are
described, which lead to a threshold parameter for the model and methods for
calculating the probability of a major outbreak, given few initial infectives,
and the expected proportion of the population who are ultimately infected by
such a major outbreak. These approximate results are shown to be exact as the
number of households tends to infinity by proving associated limit theorems.
Moreover, simulation studies indicate that these asymptotic results provide
good approximations for modestly sized finite populations. The extension to
unequal sized households is discussed briefly.
http://arxiv.org/abs/0812.4110
Author(s): Pieter Trapman and Martin Bootsma
Abstract: In this paper we establish a relation between the spread of infectious
diseases and the dynamics of so called M/G/1 queues with processor sharing. The
in epidemiology well known relation between the spread of epidemics and
branching processes and the in queueing theory well known relation between
M/G/1 queues and birth death processes will be combined to provide a framework
in which results from queueing theory can be used in epidemiology and vice
versa.
In particular, we consider the number of infectious individuals in a standard
SIR epidemic model at the moment of the first detection of the epidemic, where
infectious individuals are detected at a constant per capita rate. We use a
result from the literature on queueing processes to show that this number of
infectious individuals is geometrically distributed.
http://arxiv.org/abs/0812.4135
Author(s): Nizar Demni
Abstract: This note encloses relatively short proofs of the following known results:
the radial Dunkl process associated with a reduced system and a strictly
positive multiplicity function is the unique strong solution for all time t of
a stochastic differential equation of a singular drift (see [11] for the
original proof and [4] for a proof under additional restrictions), the first
hitting time of the Weyl chamber by a radial Dunkl process is finite almost
surely for small values of the multiplicity function. Our proof of the second
mentioned result gives more information than the original one.
http://arxiv.org/abs/0812.4269
Author(s): Jean-Baptiste Bardet (IRMAR and LMRS) and Gerhard Keller and Roland Zweim\"uller
Abstract: We study systems of globally coupled interval maps, where the identical
individual maps have two expanding, fractional linear, onto branches, and where
the coupling is introduced via a parameter - common to all individual maps -
that depends in an analytic way on the mean field of the system. We show: 1)
For the range of coupling parameters we consider, finite-size coupled systems
always have a unique invariant probability density which is strictly positive
and analytic, and all finite-size systems exhibit exponential decay of
correlations. 2) For the same range of parameters, the self-consistent
Perron-Frobenius operator which captures essential aspects of the corresponding
infinite-size system (arising as the limit of the above when the system size
tends to infinity), undergoes a supercritical pitchfork bifurcation from a
unique stable equilibrium to the coexistence of two stable and one unstable
equilibrium.
http://arxiv.org/abs/0812.4040
Author(s): Ronald Meester and Pieter Trapman
Abstract: We introduce a new percolation model to describe and analyze the spread of an
epidemic on a general directed and locally finite graph. We assign a
two-dimensional random weight vector to each vertex of the graph in such a way
that the weights of different vertices are i.i.d., but the two entries of the
vector assigned to a vertex need not be independent. The probability for an
edge to be open depends on the weights of its end vertices, but conditionally
on the weights, the states of the edges are independent of each other. In an
epidemiological setting, the vertices of a graph represent the individuals in a
(social) network and the edges represent the connections in the network. The
weights assigned to an individual denote its (random) infectivity and
susceptibility, respectively. We show that one can bound the percolation
probability and the expected size of the cluster of vertices that can be
reached by an open path starting at a given vertex from above and below by the
corresponding quantities for respectively independent bond and site percolation
with certain densities; this generalizes a result of Kuulasmaa. Many models in
the literature are special cases of our general model.
http://arxiv.org/abs/0812.4353
Author(s): Mikhail Gordin
Abstract: A stationary random sequence admits under some assumptions a representation
as the sum of two others: one of them is a martingale difference sequence, and
another is a so-called coboundary. Such a representation can be used for
proving some limit theorems by means of the martingale approximation. A
multivariate version of such a decomposition is presented in the paper for a
class of random fields generated by several commuting non-invertible
probability preserving transformations. In this representation summands of
mixed type appear which behave with respect to some groupof directions of the
parameter space as reversed multiparameter martingale differences (in the sense
of one of several known definitions) while they look as coboundaries relative
to the other directions. Applications to limit theorems will be published
elsewhere.
http://arxiv.org/abs/0812.4414
Author(s): Julien Barral and Xiong Jin and Benoit Mandelbrot
Abstract: The theory of positive $T$-martingales was developed in order to set up a
general framework including the positive measure-valued martingales initially
considered for intermittent turbulence modelling. We consider the natural
extension consisting in allowing the martingale to take complex values. We
focus on martingales constructed on the line: $T$ is the interval $[0,1]$.
Then, random measures are replaced by random functions. We specify a large
class of such martingales, which contains the complex extension of $b$-adic
canonical cascades, compound Poisson cascades, and more generally infinitely
divisible cascades. For the elements of this class, we find a sufficient
condition for their almost sure uniform convergence to a non-trivial limit.
Such limit provide new examples of multifractal processes.
http://arxiv.org/abs/0812.4556
Author(s): Julien Barral and Xiong Jin and Benoit Mandelbrot
Abstract: This paper extends the familiar sequences of random measures obtained on
$[0,1]$ via $b$-adic independent cascades by allowing the random weights
invoked in the cascades to take real, or complex values. This yields sequences
of random functions. The asymptotic behavior of these sequences is
investigated. We obtain a sufficient condition for the almost sure convergence
of these signed cascades to non-trivial statistically self-similar limit. Under
suitable assumptions, the limit function can be represented almost surely as a
monofractal function in multifractal time. When the sufficient condition for
convergence does not hold, in most of the cases we show that either the limit
is 0 or the sequence diverges almost surely. In the later case, under some
condition we prove a functional central limit theorem, which claims that there
is a natural normalization making the sequence convergent in law to a standard
Brownian motion in multifractal time.
http://arxiv.org/abs/0812.4557
Author(s): Christa Cuchiero and Martin Keller-Ressel and Josef Teichmann
Abstract: We introduce a class of Markov stochastic processes called $m$-polynomial,
for which the calculation of (mixed) moments up to order $m$ only requires the
computation of matrix exponentials. This class contains affine processes,
Feller processes with quadratic squared diffusion coefficient, as well as
L\'evy-driven SDEs with affine vector fields. Thus, many popular models such as
the classical Black-Scholes, exponential L\'evy or affine models are covered by
this setting. The applications range from statistical GMM estimation to option
pricing. For instance, the efficient and easy computation of moments can
successfully be used for variance reduction techniques in Monte Carlo
simulations.
http://arxiv.org/abs/0812.4740
Author(s): Aihua Xia and Fuxi Zhang
Abstract: The polynomial birth-death distribution (abbr. as PBD) on $\ci=\{0,1,2,
>...\}$ or $\ci=\{0,1,2, ..., m\}$ for some finite $m$ introduced in Brown &
Xia (2001) is the equilibrium distribution of the birth-death process with
birth rates $\{\alpha_i\}$ and death rates $\{\beta_i\}$, where $\a_i\ge0$ and
$\b_i\ge0$ are polynomial functions of $i\in\ci$. The family includes Poisson,
negative binomial, binomial and hypergeometric distributions. In this paper, we
give probabilistic proofs of various Stein's factors for the PBD approximation
with $\a_i=a$ and $\b_i=i+bi(i-1)$ in terms of the Wasserstein distance. The
paper complements the work of Brown & Xia (2001) and generalizes the work of
Barbour & Xia (2006) where Poisson approximation ($b=0$) in the Wasserstein
distance is investigated. As an application, we establish an upper bound for
the Wasserstein distance between the PBD and Poisson binomial distribution and
show that the PBD approximation to the Poisson binomial distribution is much
more precise than the approximation by the Poisson or shifted Poisson
distributions.
http://arxiv.org/abs/0812.4847
Author(s): V. P. Maslov (1 and 2) and V. E. Nazaikinskii (2) ((1) Moscow State University, (2) Institute for Problems in Mechanics, RAS, Moscow)
Abstract: For a system of identical Bose particles sitting on integer energy levels, we
give sharp estimates for the convergence of the sequence of occupation numbers
to the Bose-Einstein distribution and for the Bose condensation effect.
http://arxiv.org/abs/0812.4885
Author(s): Ahmed Guellil (USTHB) and Tewfik Kernane (USTHB)
Abstract: We propose a new approach for estimating the parameters of a probability
distribution. It consists on combining two new methods of estimation. The first
is based on the definition of a new distance measuring the difference between
variations of two distributions on a finite number of points from their support
and on using this measure for estimation purposes by the method of minimum
distance. For the second method, given an empirical discrete distribution, we
build up an auxiliary discrete theoretical distribution having the same support
of the first and depending on the same parameters of the parent distribution of
the data from which the empirical distribution emanated. We estimate then the
parameters from the empirical distribution by the usual statistical methods. In
practice, we propose to compute the two estimations, the second based on
maximum likelihood principle of known theoretical properties, and the first
being as a control of the effectiveness of the obtained estimation, and for
which we prove the convergence in probability, so we have also a criterion on
the quality of the information contained in the observations. We apply the
approach to truncated or grouped and censored data situations to give the
flavour on the effectiveness of the approach. We give also some interesting
perspectives of the approach including model selection from truncated data,
estimation of the initial trial value in the celebrate EM algorithm in the case
of truncation and merged normal populations, a test of goodness of fit based on
the new distance, quality of estimations and data.
http://arxiv.org/abs/0802.2155
Author(s): Hiroki Sumi
Abstract: We investigate the random dynamics of rational maps and the dynamics of
semigroups of rational maps on the Riemann sphere. We see that the both fields
are related to each other very deeply. We investigate spectral properties of
transition operators and the dynamics of associated semigroups of rational
maps. We define several kinds of Julia sets of the associated Markov processes
and we study the properties and the dimension of them. Moreover, we investigate
"singular functions on the complex plane". In particular, we consider the
functions $T$ which represent the probability of tending to infinity with
respect to the random dynamics of polynomials. Under certain conditions these
functions $T$ are complex analogues of the devil's staircase and Lebesgue's
singular functions. More precisely, we show that these functions $T$ are
continuous on the Riemann sphere and vary only on the Julia sets of associated
semigroups. Furthermore, by using ergodic theory and potential theory, we
investigate the non-differentiability and regularity of these functions. We
find many phenomena which can hold in the random complex dynamics and the
dynamics of semigroups of rational maps, but cannot hold in the usual iteration
dynamics of a single holomorphic map. We carry out a systematic study of these
phenomena and their mechanisms.
http://arxiv.org/abs/0812.4483
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