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Probability Abstracts 98
This document contains abstracts 5550-5756 from
May-1-2007 to June-30-2007.
They have been mailed on July 4th, 2007.
Author(s): Davar Khoshnevisan and Yimin Xiao
Abstract: In order to compute the packing dimension of orthogonal projections
Falconer and Howroyd (1997) introduced a family of packing dimension profiles
${\rm Dim}_s$ that are parametrized by real numbers $s>0$. Subsequently,
Howroyd (2001) introduced alternate $s$-dimensional packing dimension profiles
$\hbox{${\rm P}$-$\dim$}_s$ and proved, among many other things, that
$\hbox{${\rm P}$-$\dim$}_s E={\rm Dim}_s E$ for all integers $s>0$ and all
analytic sets $E\subseteq\R^N$. The goal of this article is to prove that
$\hbox{${\rm P}$-$\dim$}_s E={\rm Dim}_s E$ for all real numbers $s>0$ and
analytic sets $E\subseteq\R^N$. This answers a question of Howroyd (2001, p.
159). Our proof hinges on a new property of fractional Brownian motion.
http://arxiv.org/abs/0705.0135
Author(s): Davar Khoshnevisan
Abstract: H\"aggstr\"om, Peres, and Steif (1997) have introduced a dynamical version of
percolation on a graph $G$. When $G$ is a tree they derived a necessary and
sufficient condition for percolation to exist at some time $t$. In the case
that $G$ is a spherically symmetric tree, H\"aggstr\"om, Peres, and Steif
(1997) derived a necessary and sufficient condition for percolation to exist at
some time $t$ in a given target set $D$. The main result of the present paper
is a necessary and sufficient condition for the existence of percolation, at
some time $t\in D$, in the case that the underlying tree is not necessary
spherically symmetric. This answers a question of Yuval Peres (personal
communication). We present also a formula for the Hausdorff dimension of the
set of exceptional times of percolation.
http://arxiv.org/abs/0705.0140
Author(s): Erhan Bayraktar and Virginia R. Young
Abstract: We show that the mutual fund theorems of Merton (1971) extend to the problem
of optimal investment to minimize the probability of lifetime ruin. We obtain
four such theorems by considering a financial market both with and without a
riskless asset and by considering both constant and random consumption.
http://arxiv.org/abs/0705.0053
Author(s): Boris Baeumer and Mark M. Meerschaert and Erkan Nane
Abstract: A Brownian time process is a Markov process subordinated to the absolute
value of an independent one-dimensional Brownian motion. Its transition
densities solve an initial value problem involving the square of the generator
of the original Markov process. An apparently unrelated class of processes,
emerging as the scaling limits of continuous time random walks, involve
subordination to the inverse or hitting time process of a classical stable
subordinator. The resulting densities solve fractional Cauchy problems, an
extension that involves fractional derivatives in time. In this paper, we will
show a close and unexpected connection between these two classes of processes,
and consequently, an equivalence between these two families of partial
differential equations.
http://arxiv.org/abs/0705.0168
Author(s): N. Fountoulakis and D. K\"uhn and D. Osthus
Abstract: Let ccl(G) denote the order of the largest complete minor in a graph G (also
called the contraction clique number) and let G(n,p) denote a random graph on n
vertices with edge probability p. Bollobas, Catlin and Erdos asymptotically
determined ccl(G (n,p)) when p is a constant. Luczak, Pittel and Wierman gave
bounds on ccl(G(n,p)) when p is very close to 1/n, i.e. inside the phase
transition. Extending the results of Bollobas, Catlin and Erdos, we determine
ccl(G(n,p)) quite tightly, for p>C/n where C is a large constant. If p=C/n, for
an arbitrary constant C>1, then we show that asymptotically almost surely ccl(G
(n,p)) is of order square-root of n. This answers a question of Krivelevich and
Sudakov.
http://arxiv.org/abs/0705.0325
Author(s): Vladimir Vovk
Abstract: This paper gives game-theoretic versions of several results on "merging of
opinions" obtained in measure-theoretic probability and algorithmic randomness
theory. An advantage of the game-theoretic versions over the measure-theoretic
results is that they are pointwise, their advantage over the algorithmic
randomness results is that they are non-asymptotic, but the most important
advantage over both is that they are very constructive, giving explicit and
efficient strategies for players in a game of prediction.
http://arxiv.org/abs/0705.0372
Author(s): Zongxia Liang
Abstract: The large deviations principles are established for a class of
multidimensional degenerate stochastic differential equations with reflecting
boundary conditions. The results include two cases where the initial conditions
are adapted and anticipated.
http://arxiv.org/abs/0705.0405
Author(s): Olivier Aj Bardou (GDF-RDD) and Sandrine Bouthemy (GDF-RDD) and Gilles Pag\`es (PMA)
Abstract: In this paper we investigate a class of swing options with firm constraints
in view of the modeling of supply agreements. We show, for a fully general
payoff process, that the premium, solution to a stochastic control problem, is
concave and piecewise affine as a function of the global constraints of the
contract. The existence of bang-bang optimal controls is established for a set
of constraints which generates by affinity the whole premium function. When the
payoff process is driven by an underlying Markov process, we propose a
quantization based recursive backward procedure to price these contracts. A
priori error bounds are established, uniformly with respect to the global
constraints.
http://arxiv.org/abs/0705.0466
Author(s): Noga Alon and Chen Avin and Michal Koucky and Gady Kozma and Zvi Lotker and Mark R. Tuttle
Abstract: We consider a fundamental new question regarding random walks on graphs: How
long does it take for several independent random walks to cover an entire
graph? We study the {\em cover time}, the expected time required to visit every
node in a graph at least once, and we show that for a large collection of
interesting graphs, running many random walks in parallel yields a speed-up in
the cover time that is linear in the number of the parallel walks. We
demonstrate that an exponential speed-up is sometimes possible, but that some
natural graphs allow only a logarithmic speed-up.
http://arxiv.org/abs/0705.0467
Author(s): Geoffrey Grimmett
Abstract: The contact model for the spread of disease may be viewed as a directed
percolation model on $\ZZ \times \RR$ in which the continuum axis is oriented
in the direction of increasing time. Techniques from percolation have enabled a
fairly complete analysis of the contact model at and near its critical point.
The corresponding process when the time-axis is unoriented is an undirected
percolation model to which now standard techniques may be applied. One may
construct in similar vein a random-cluster model on $\ZZ \times \RR$, with
associated continuum Ising and Potts models. These models are of independent
interest, in addition to providing a path-integral representation of the
quantum Ising model with transverse field. This representation may be used to
obtain a bound on the entanglement of a finite set of spins in the quantum
Ising model on $\ZZ$, where this entanglement is measured via the entropy of
the reduced density matrix. The mean-field version of the quantum Ising model
gives rise to a random-cluster model on $K_n \times \RR$, thereby extending the
Erdos-Renyi random graph on the complete graph $K_n$.
http://arxiv.org/abs/0705.0506
Author(s): Nicolas Bouleau (CIRED)
Abstract: This lecture presents recent advances in the theory of errors propagation. We
first explain in which cases the propagation of errors may be performed with a
first order differential calculus or needs a second order differential
calculus. Then we point out the link between error propagation and the concept
of second order vector in differential geometry, emphasizing the existence of a
slight ambiguity concerning the bias operator. The third part in devoted to the
powerful framework of Dirichlet forms whose main feature is to apply easily to
infinite dimensional models including the Wiener space (giving an
interpretation of Malliavin calculus in terms of errors), the Poisson space and
the Monte Carlo space. In the fourth part we show how an error in the usual
mathematical sense, i.e. an approximate quantity, may yield a Dirichlet form
and we introduce the four bias operators. Eventually we connect the Dirichlet
form with statistics by identifying the square of field operator with the
inverse of the Fisher information matrix.
http://arxiv.org/abs/0705.0519
Author(s): Alessandro De Gregorio and Stefano M. Iacus
Abstract: The telegraph process models a random motion with finite velocity and it is
usually proposed as an alternative to diffusion models. The process describes
the position of a particle moving on the real line, alternatively with constant
velocity $+ v$ or $-v$. The changes of direction are governed by an homogeneous
Poisson process with rate $\lambda >0.$ In this paper, we consider a change
point estimation problem for the rate of the underlying Poisson process by
means of least squares method. The consistency and the rate of convergence for
the change point estimator are obtained and its asymptotic distribution is
derived. Applications to real data are also presented.
http://arxiv.org/abs/0705.0503
Author(s): Ivan Nourdin (PMA)
Abstract: This note is devoted to a fine study of the convergence of some weighted
quadratic and cubic variations of a fractional Brownian motion B with Hurst
index H in (0,1/2). With the help of Malliavin calculus, we show that,
correctly renormalized, the weighted quadratic variation of B that we consider
converges in L^2 to an explicit limit when H<1/4, while we conjecture that it
converges in law when H>1/4. In the same spirit, we also show that, correctly
renormalized, the weighted cubic variation of B converges in L^2 to an explicit
limit when H<1/6.
http://arxiv.org/abs/0705.0570
Author(s): Jean B\'erard and Alejandro Ram\'irez
Abstract: We prove that a law of large numbers and a central limit theorem hold for the
excited random walk model in every dimension $d \geq 2$.
http://arxiv.org/abs/0705.0658
Author(s): R. Fernandez and G. Maillard
Abstract: We establish a one-to-one correspondence between one-sided and two-sided
regular systems of conditional probabilities on the half-line that preserves
the associated chains and Gibbs measures. As an application, we determine
uniqueness and non-uniqueness regimes in one-sided versions of ferromagnetic
Ising models with long range interactions. Our study shows that the interplay
between chain and Gibbsian theories yields more information than that contained
within the known theory of each separate framework. In particular: (i) A
Gibbsian construction due to Dyson yields a new family of chains with phase
transitions; (ii) these transitions show that a square summability uniqueness
condition of chains is false in the general non-shift-invariant setting, and
(iii) an uniqueness criterion for chains shows that a Gibbsian conjecture due
to Kac and Thompson is false in this half-line setting.
http://arxiv.org/abs/0705.0808
Author(s): Bo'az Klartag
Abstract: We provide a sharp rate of convergence in the central limit theorem for
random vectors with an unconditional, log-concave density. The argument relies
on analysis of the Neumann laplacian on convex domains and on the theory of
optimal transportation of measures.
http://arxiv.org/abs/0705.0832
Author(s): Ren\'e Blacher (LJK)
Abstract: We prove a central limit theorem with aassumptions which are many weak than
classical conditions
http://arxiv.org/abs/0705.0853
Author(s): Noga Alon and Eyal Lubetzky
Abstract: Random walks on expander graphs were thoroughly studied, with the important
motivation that, under some natural conditions, these walks mix quickly and
provide an efficient method of sampling the vertices of a graph. Alon,
Benjamini, Lubetzky and Sodin studied non-backtracking random walks on regular
graphs, and showed that their mixing rate may be up to twice as fast as that of
the simple random walk. As an application, they showed that the maximal number
of visits to a vertex, made by a non-backtracking random walk of length $n$ on
a high-girth $n$-vertex regular expander, is typically $(1+o(1))\frac{\log
n}{\log\log n}$, as in the case of the balls and bins experiment. They further
asked whether one can establish the precise distribution of the visits such a
walk makes.
In this work, we answer the above question by combining a generalized form of
Brun's sieve with some extensions of the ideas in Alon et al. Let $N_t$ denote
the number of vertices visited precisely $t$ times by a non-backtracking random
walk of length $n$ on a regular $n$-vertex expander of fixed degree and girth
$g$. We prove that if $g=\omega(1)$, then for any fixed $t$, $N_t/n$ is
typically $\frac{1}{\mathrm{e}t!}+o(1)$. Furthermore, if $g=\Omega(\log\log
n)$, then $N_t/n$ is typically $\frac{1+o(1)}{\mathrm{e}t!}$ uniformly on all
$t \leq (1-o(1))\frac{\log n}{\log\log n}$ and 0 for all $t \geq
(1+o(1))\frac{\log n}{\log\log n}$. In particular, we obtain the above result
on the typical maximal number of visits to a single vertex, with an improved
threshold window. The essence of the proof lies in showing that variables
counting the number of visits to a set of sufficiently distant vertices are
asymptotically independent Poisson variables.
http://arxiv.org/abs/0705.0867
Author(s): Claude Dellacherie and Servet Martinez and Jaime San Martin
Abstract: We study infinite tree and ultrametric matrices, and their action on the
boundary of the tree. For each tree matrix we show the existence of a symmetric
random walk associated to it and we study its Green potential. We provide a
representation theorem for harmonic functions that includes simple expressions
for any increasing harmonic function and the Martin kernel. In the boundary, we
construct the Markov kernel whose Green function is the extension of the matrix
and we simulate it by using a cascade of killing independent exponential random
variables and conditionally independent uniform variables. For ultrametric
matrices we supply probabilistic conditions to study its potential properties
when immersed in its minimal tree matrix extension.
http://arxiv.org/abs/0705.0967
Author(s): Mingyu Xu
Abstract: In this paper, we prove the existence and uniqueness result of the reflected
BSDE with two continuous barriers under monotonicity and general increasing
condition on $y$, with Lipschitz condition on $z$.
http://arxiv.org/abs/0705.1026
Author(s): Jonathan Novak
Abstract: We investigate the combinatorics of truncated Haar-distributed random unitary
matrices. Specifically, if $U$ is a random matrix from the unitary group
$U(d),$ let $U_k$ denote its $k \times k$ upper left corner, where $1 \leq k
\leq d.$ We give an explicit formula for the moments of the trace of $U_k$ in
terms of pairs of Standard Young Tableaux on distinct shapes. This formula can
be restated as counting configurations of non-intersecting walkers on the
integer lattice. Our main tool is the Colour-Flavour Transformation of lattice
gauge theory.
http://arxiv.org/abs/0705.0984
Author(s): Sourav Chatterjee
Abstract: Linear statistics of eigenvalues in many familiar classes of random matrices
are known to obey gaussian central limit theorems. The proofs of such results
are usually rather difficult, involving hard computations specific to the model
in question. In this article we attempt to formulate a unified technique for
deriving such results via relatively soft arguments. Our approach is based on a
notion of `extending the Poincare inequality to the second order' via Stein's
method of normal approximation. Just as ordinary Poincare inequalities give
variance bounds, our second order Poincare inequalities (based on second order
partial derivatives) give central limit theorems. A number of examples,
complete with total variation error bounds, are worked out. On the downside, we
require stringent distributional assumptions and our theorems do not provide
information about the variances of the linear statistics, which have to be
computed separately.
http://arxiv.org/abs/0705.1224
Author(s): Fabio Gobbi and Cecilia Mancini
Abstract: In this paper we consider two processes driven by diffusions and jumps. The
jump components are Levy processes and they can both have finite activity and
infinite activity. Given discrete observations we estimate the covariation
between the two diffusion parts and the co-jumps. The detection of the co-jumps
allows to gain insight in the dependence structure of the jump components and
has important applications in finance. Our estimators are based on a threshold
principle allowing to isolate the jumps. This work follows Gobbi and Mancini
(2006) where the asymptotic normality for the estimator of the covariation,
with convergence speed given by the squared root of h, was obtained when the
jump components have finite activity. Here we show that the speed is the
squared root of h only when the activity of the jump components is moderate.
http://arxiv.org/abs/0705.1268
Author(s): Qingyang Guan
Abstract: By constructing super harmonic functions, we give a direct proof for the
existence of the continuous curve of SLE_8. This method can also be applied to
driven function of Brownian motion with variant speeds.
http://arxiv.org/abs/0705.1273
Author(s): Janko Gravner and Alexander E. Holroyd
Abstract: In the bootstrap percolation model, sites in an L by L square are initially
infected independently with probability p. At subsequent steps, a healthy site
becomes infected if it has at least 2 infected neighbours. As
(L,p)->(infinity,0), the probability that the entire square is eventually
infected is known to undergo a phase transition in the parameter p log L,
occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy
between the critical parameter and its limit lambda is at least Omega((log
L)^(-1/2)). In contrast, the critical window has width only Theta((log
L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds
which imply for example that the relative discrepancy is at least 1% even when
L = 10^3000. Our results shed some light on the observed differences between
simulations and rigorous asymptotics.
http://arxiv.org/abs/0705.1347
Author(s): Alan J Bray and Richard Smith
Abstract: We consider a Brownian particle, with diffusion constant D, moving inside an
expanding d-dimensional sphere whose surface is an absorbing boundary for the
particle. The sphere has initial radius L_0 and expands at a constant rate c.
We calculate the joint probability density, p(r,t|r_0), that the particle
survives until time t, and is at a distance r from the centre of the sphere,
given that it started at a distance r_0 from the centre.
http://arxiv.org/abs/0705.0501
Author(s): Elizabeth Meckes
Abstract: The main result of this paper is a bound on the distance between the
distribution of an eigenfunction of the Laplacian on a compact Riemannian
manifold and the Gaussian distribution. If $X$ is a random point on a manifold
$M$ and $f$ is an eigenfunction of the Laplacian with $L^2$-norm one and
eigenvalue $-\mu$, then $$d_{TV}(f(X),Z)\le\frac{2}{\mu}\E\big|\|\nabla
f(X)\|^2-\E\|\nabla f(X) \|^2\big|.$$ This result is applied to construct
specific examples of spherical harmonics of arbitrary (odd) degree which are
close to Gaussian in distribution. A second application is given to random
linear combinations of eigenfunctions on flat tori.
http://arxiv.org/abs/0705.1342
Author(s): P.Collet S.Martinez
Abstract: We consider the asymptotic behaviour of the solution of one dimensional
stochastic differential equations and Langevin equations in periodic
backgrounds with zero average. We prove that in several such models, there is
generically a non vanishing asymptotic velocity, despite of the fact that the
average of the background is zero.
http://arxiv.org/abs/0705.1435
Author(s): Qingyang Guan
Abstract: Let 1<\alpha<2. We prove boundary Harnack inequalities for regional
fractional Laplacian on C^{1,1} open set G in \R^n. This operator is the
generator of the \alpha-stable-like process on G taking \kappa(x,y)I_{G\times
G}/|x-y|^{n+\alpha} as the jumping measure. When \kappa is a constant, this
explicit boundary Harnack inequality was proved in Bogdan, Burdzy and Chen [9]
on C^{1,1} open sets. We prove that it holds also for C^{1,\beta-1} open sets
with \kappa\in C^1(\bar{G}\times\bar{G}) bounded between two positive values,
where 1<\alpha<\beta\leq 2.
http://arxiv.org/abs/0705.1614
Author(s): Sandrine Peche
Abstract: For sample covariance matrices with iid entries with sub-Gaussian tails, when
both the number of samples and the number of variables become large and the
ratio approaches to one, it is a well-known result of A. Soshnikov that the
limiting distribution of the largest eigenvalue is same as the of Gaussian
samples. In this paper, we extend this result to two cases. The first case is
when the ratio approaches to an arbitrary finite value. The second case is when
the ratio becomes infinity or arbitrarily small.
http://arxiv.org/abs/0705.1701
Author(s): Itamar Landau and Lionel Levine
Abstract: The rotor-router model is a deterministic analogue of random walk. It can be
used to define a deterministic growth model analogous to internal DLA. We show
that if the initial rotor configuration is acyclic, then the set of occupied
sites for rotor-router aggregation on an infinite regular tree is a perfect
ball whenever it can be. This is proved by defining the rotor-router group of a
graph, which we show is isomorphic to the sandpile group. We also address the
question of recurrence and transience: We give two rotor configurations on the
infinite ternary tree, one for which chips exactly alternate escaping to
infinity with returning to the origin, and one for which every chip returns to
the origin. We also characterize the possible "escape sequences" for the
ternary tree, that is, binary words $a_1 ... a_n$ for which there exists a
rotor configuration so that the $k$-th chip escapes to infinity if and only if
$a_k=1$.
http://arxiv.org/abs/0705.1562
Author(s): Marc Wouts (PMA)
Abstract: By the mean of a multi-scale analysis we describe the typical geometrical
structure of the clusters under the FK measure in random media. Our result
holds in any dimension greater or equal to 2 provided that slab percolation
occurs under the annealed measure, which should be the case in the whole
supercritical phase. This work extends the one of Pisztora and provides an
essential tool for the analysis of the supercritical regime in disordered FK
models and in the corresponding disordered Ising and Potts models.
http://arxiv.org/abs/0705.1630
Author(s): Yuliya Mishura and Georgiy Shevchenko
Abstract: The paper focuses on discrete-type approximations of solutions to
non-homogeneous stochastic differential equations (SDEs) involving fractional
Brownian motion (fBm). We prove that the rate of convergence for Euler
approximations of solutions of pathwise SDEs driven by fBm with Hurst index
$H>1/2$ can be estimated by $O(\delta^{2H-1})$ ($\delta$ is the diameter of
partition). For discrete-time approximations of Skorohod-type quasilinear
equation driven by fBm we prove that the rate of convergence is $O(\delta^H)$.
We also establish that the rate of weak convergence for the approximations of
solutions of pathwise SDE with bounded smooth coefficients is $O(\delta)$.
http://arxiv.org/abs/0705.1773
Author(s): N. Lazrieva and T. Sharia and T. Toronjadze
Abstract: The semimartingale stochastic approximation procedure, namely, the
Robbins-Monro type SDE is introduced which naturally includes both generalized
stochastic approximation algorithms with martingale noises and recursive
parameter estimation procedures for statistical models associated with
semimartingales. General results concerning the asymptotic behaviour of the
solution are presented. In particular, the conditions ensuring the convergence,
rate of convergence and asymptotic expansion are established. The results
concerning the Polyak weighted averaging procedure are also presented.
http://arxiv.org/abs/0705.1794
Author(s): Shige Peng and Zhe Yang
Abstract: In this paper, we discuss a new type of differential equations which we call
anticipated backward stochastic differential equations (anticipated BSDEs). In
these equations the generator includes not only the values of solutions of the
present but also the future. We show that these anticipated BSDEs have unique
solutions, a comparison theorem for their solutions, and a duality between them
and stochastic differential delay equations.
http://arxiv.org/abs/0705.1822
Author(s): Dapeng Zhan
Abstract: We prove that the chordal SLE$_\kappa$ trace is reversible for
$\kappa\in(0,4]$.
http://arxiv.org/abs/0705.1852
Author(s): K. A. Borovkov and G. Last
Abstract: We consider a piecewise-deterministic Markov process governed by a jump
intensity function, a rate function that determines the behaviour between
jumps, and a stochastic kernel describing the conditional distribution of jump
sizes. We study the point process of upcrossings of a level $b$ by the Markov
process. Our main result shows that, under a suitable scaling $\nu(b)$, the
point process converges, as $b$ tends to infinity, weakly to a geometrically
compound Poisson process. We also prove a version of Rice's formula relating
the stationary density of the process to level crossing intensities. This
formula provides an interpretation of the scaling factor $\nu(b)$. While our
proof of the limit theorem requires additional assumptions, Rice's formula
holds whenever the (stationary) overall intensity of jumps is finite.
http://arxiv.org/abs/0705.1863
Author(s): Sebastian M\"uller
Abstract: We develop a criterion for transience for a general model of branching Markov
chains. In the case of multi-dimensional branching random walk in random
environment (BRWRE) this criterion becomes explicit. In particular, we show
that \emph{Condition L} of Comets and Popov is necessary and sufficient for
transience as conjectured. Furthermore, the criterion applies to two important
classes of branching random walks and implies that the critical branching
random walk is transient resp. dies out locally.
http://arxiv.org/abs/0705.1874
Author(s): Marius Junge and Quanhua Xu
Abstract: We show norm estimates for the sum of independent random variables in
noncommutative $L_p$-spaces for $1
http://arxiv.org/abs/0705.1952
Author(s): Qi Zhong
Abstract: The purpose of this paper is to determine the asymptotic of the average
energy of a configuration of N zeros of system of random polynomials of degree
N as N tends to infinity and more generally the zeros of random holomorphic
sections of a line bundle L over any Riemann surface M. And we compare our
results to the well-known minimum of energies.
http://arxiv.org/abs/0705.2000
Author(s): Olivier Aj Bardou (GDF-RDD) and Sandrine Bouthemy (GDF-RDD) and Gilles Pag\`es (PMA)
Abstract: In this paper, we investigate a numerical algorithm for the pricing of swing
options, relying on the so-called optimal quantization method. The numerical
procedure is described in details and numerous simulations are provided to
assert its efficiency. In particular, we carry out a comparison with the
Longstaff-Schwartz algorithm.
http://arxiv.org/abs/0705.2110
Author(s): Jean Picard
Abstract: We consider a real-valued path; it is possible to associate a tree to this
path, and we explore the relations between the tree, the properties of
$p$-variation of the path, and integration with respect to the path. In
particular, the fractal dimension of the tree is estimated from the variations
of the path, and Young integrals with respect to the path, as well as integrals
from the rough paths theory, are written as integrals on the tree. Examples
include some stochastic paths such as martingales, L\'evy processes and
fractional Brownian motions.
http://arxiv.org/abs/0705.2128
Author(s): Jacob J. H. Simmons and Peter Kleban and and Robert M. Ziff
Abstract: Using conformal field theory, we derive several new crossing formulas at the
two-dimensional percolation point. High-precision simulation confirms these
results. Integrating them gives a unified derivation of Cardy's formula for the
horizontal crossing probability $\Pi_h(r)$, Watts' formula for the
horizontal-vertical crossing probability $\Pi_{hv}(r)$, and Cardy's formula for
the expected number of clusters crossing horizontally $\mathcal{N}_h(r)$. The
main step in our approach implies the identification of the derivative of one
primary operator with another. We present operator identities that support this
idea and suggest the presence of additional symmetry in $c=0$ conformal field
theories.
http://arxiv.org/abs/0705.1933
Author(s): Bertrand Duplantier
Abstract: We describe in detail the history of Brownian motion, as well as the
contributions of Einstein, Sutherland, Smoluchowski, Bachelier, Perrin and
Langevin to its theory. The always topical importance in physics of the theory
of Brownian motion is illustrated by recent biophysical experiments, where it
serves, for instance, for the measurement of the pulling force on a single DNA
molecule.
In a second part, we stress the mathematical importance of the theory of
Brownian motion, illustrated by two chosen examples. The by-now classic
representation of the Newtonian potential by Brownian motion is explained in an
elementary way. We conclude with the description of recent progress seen in the
geometry of the planar Brownian curve. At its heart lie the concepts of
conformal invariance and multifractality, associated with the potential theory
of the Brownian curve itself.
http://arxiv.org/abs/0705.1951
Author(s): Jon. Aaronson and Kyewon Koh Park
Abstract: We show that a certain type of conservative, ergodic measure preserving
transformation always has a maximal zero entropy factor, generated by
predictable sets. We also consider distribution asymptotics of information;
e.g. for Boole's transformation, information is asymptotically mod-normal, a
property shared by certain ergodic, probability preserving transformations with
zero entropy.
http://arxiv.org/abs/0705.2148
Author(s): Roberto I. Oliveira
Abstract: We consider Kac's random walk on n-dimensional rotation matrices, where each
step is a random rotation in the plane generated by two randomly picked
coordinates. We show that this process converges to the uniform (Haar) measure
in the (Wasserstein) transportation cost metric in O(n^2 ln n) steps. This
improves on previous results of Diaconis/Saloff Coste and Pak/Sidenko and is a
ln n factor away from being optimal. Our proof method includes a general result
akin to the path coupling method of Bubley and Dyer. Suppose that P is a Markov
chain on a Polish length space (M,d) and that for all x,y in M with d(x,y)<< 1
there is a coupling (X,Y) of one step from P from x and y (respectively) that
is (c+o(1))-contracting on average. Then the map from a initial distribution m
to the distribution mP after one step is c-contracting in the transportation
cost metric. Other applications of this result are also presented.
http://arxiv.org/abs/0705.2253
Author(s): David \v{S}i\v{s}ka and Istv\'an Gy\"ongy
Abstract: A general result on the method of randomized stopping is proved. It is
applied to optimal stopping of controlled diffusion processes with unbounded
coefficients to reduce it to optimal control problem without stopping. This is
motivated by recent results of Krylov on numerical solutions to the Bellman
equation.
http://arxiv.org/abs/0705.2302
Author(s): Qingyang Guan
Abstract: Schramm Loewner Evolutions (SLE) are random increasing hulls defined through
the Loewner equation driven by Brownian motion. It is known that the increasing
hulls are generated by continuous curves. When the driving process is of the
form \sqrt{\kappa} B+\theta^{1/\alpha}S for a Brownian motion B and a symmetric
\alpha-stable process S with \kappa not equal to 4 and 8, we prove that the
corresponding increasing hulls are generated by Cadlag curves.
http://arxiv.org/abs/0705.2321
Author(s): Jeremy Quastel and Benedek Valko
Abstract: The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on
$\mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$.
Sepp\"{a}l\"ainen and Bal\'azs recently proved this conjecture for the nearest
neighbor case. We extend their results to general finite range exclusion by
proving that the Laplace transform of the diffusivity is of the conjectured
order. We also obtain the correct order pointwise upper bound for $D(t)$.
http://arxiv.org/abs/0705.2416
Author(s): Guilhem Semerjian
Abstract: The set of solutions of random constraint satisfaction problems (zero energy
groundstates of mean-field diluted spin glasses) undergoes several structural
phase transitions as the amount of constraints is increased. This set first
breaks down into a large number of well separated clusters. At the freezing
transition, which is in general distinct from the clustering one, some
variables (spins) take the same value in all solutions of a given cluster. In
this paper we study the critical behavior around the freezing transition, which
appears in the unfrozen phase as the divergence of the sizes of the
rearrangements induced in response to the modification of a variable. The
formalism is developed on generic constraint satisfaction problems and applied
in particular to the random satisfiability of boolean formulas and to the
coloring of random graphs. The computation is first performed in random tree
ensembles, for which we underline a connection with percolation models and with
the reconstruction problem of information theory. The validity of these results
for the original random ensembles is then discussed in the framework of the
cavity method.
http://arxiv.org/abs/0705.2147
Author(s): Makoto Katori and Hideki Tanemura
Abstract: A system of one-dimensional Brownian motions (BMs) conditioned never to
collide with each other is realized as (i) Dyson's BM model, which is a process
of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian
unitary ensemble (GUE), and as (ii) the $h$-transform of absorbing BM in a Weyl
chamber, where the harmonic function $h$ is the product of differences of
variables (the Vandermonde determinant). The Karlin-McGregor formula gives
determinantal expression to the transition probability density of absorbing BM.
We show from the Karlin-McGregor formula, if the initial state is in the
eigenvalue distribution of GUE, the noncolliding BM is a determinantal process,
in the sense that any multitime correlation function is given by a determinant
specified by a matrix-kernel. By taking appropriate scaling limits, spatially
homogeneous and inhomogeneous infinite determinantal processes are derived. We
note that the determinantal processes related with noncolliding diffusion
processes have a feature in common such that the matrix-kernels are expressed
using spectral projections of appropriate effective Hamiltonians. Using the
common properties of matrix-kernels, continuity of processes in time is proved
and Dirichlet forms are provided.
http://arxiv.org/abs/0705.2460
Author(s): Youri Davydov and Vladimir Rotar
Abstract: We consider some general facts concerning convergence P_{n}-Q_{n}\to 0 as
n\to \infty, where P_{n} and Q_{n} are probability measures in a complete
separable metric space. The main point is that the sequences {P_{n}} and
{Q_{n}} are not assumed to be tight. We compare different possible definitions
of the above convergence, and establish some general properties.
http://arxiv.org/abs/0705.2677
Author(s): Miklos Bona
Abstract: We prove that the generating polynomials of partitions of an $n$-element set
into non-singleton blocks, counted by the number of blocks, have real roots
only. We apply this information to find the most likely number of blocks. As
another application of the real zeros result, we prove that the number of
blocks is normally distributed in such partitions. We present a quick way to
prove the corresponding statement for cycles of permutations in which each
cycle is longer than a given integer $r$.
http://arxiv.org/abs/0705.2734
Author(s): Daniel Alpay and David Levanony
Abstract: We present decompositions of various positive kernels as integrals or sums of
positive kernels. Within this framework we study the reproducing kernel Hilbert
spaces associated with the fractional and bi-fractional Brownian motions. As a
tool, we define a new function of two complex variables, which is a natural
generalization of the classical Gamma function for the setting we consider
http://arxiv.org/abs/0705.2863
Author(s): Alexander V. Gnedin
Abstract: For $\tau$ a stopping rule adapted to a sequence of $n$ iid observations, we
define the loss to be $\ex [ q(R_\tau)]$, where $R_j$ is the rank of the $j$th
observation, and $q$ is a nondecreasing function of the rank. This setting
covers both the best choice problem with $q(r)={\bf 1}(r>1)$, and Robbins'
problem with $q(r)=r$. As $n\to\infty$ the stopping problem acquires a limiting
form which is associated with the planar Poisson process. Inspecting the limit
we establish bounds on the stopping value and reveal qualitative features of
the optimal rule. In particular, we show that the complete history dependence
persists in the limit, thus answering a question asked by Bruss in the context
of Robbins' problem.
http://arxiv.org/abs/0705.2976
Author(s): Patrizia Berti and Pietro Rigo
Abstract: Let (\Omega,\mathcal{B},P) be a probability space, \mathcal{A} a
sub-sigma-field of \mathcal{B}, and \mu a regular conditional distribution for
P given \mathcal{A}. For various, classically interesting, choices of
\mathcal{A} (including tail and symmetric) the following 0-1 law is proved:
There is a set A_0 in \mathcal{A} such that P(A_0)=1 and \mu(\omega)(A) is 0 or
1 for all A in \mathcal{A} and \omega in A_0. Provided \mathcal{B} is countably
generated (and certain regular conditional distributions exist), the result
applies whatever P is.
http://arxiv.org/abs/0705.3028
Author(s): Gregory Schehr and Satya N. Majumdar
Abstract: We consider a class of real random polynomials, indexed by an integer d, of
large degree n and focus on the number of real roots of such random
polynomials. For n even, the probability that such polynomials have no real
root decays as a power law n^{-4 \theta(d)} where \theta(d)>0 is the exponent
associated to the decay of the persistence probability for the diffusion
equation with random initial conditions in space dimension d. Considering the
particular case d=1, this connection allows for a physical realization of real
random polynomials. We further show that the probability that such polynomials
have exactly k real roots (n and k having the same parity) has an unusual
scaling form given by n^{-\tilde \phi(k/\log n)} where \tilde \phi(x) a
universal large deviation function.
http://arxiv.org/abs/0705.2648
Author(s): Christian Mastrodonato and Roderich Tumulka
Abstract: We provide an elementary proof for a theorem due to Petz and R\'effy which
states that for a random $n\times n$ unitary matrix with distribution given by
the Haar measure on the unitary group U(n), the upper left (or any other)
$k\times k$ submatrix converges in distribution, after multiplying by a
normalization factor $\sqrt{n}$ and as $n\to\infty$, to a matrix of independent
complex Gaussian random variables with mean 0 and variance 1.
http://arxiv.org/abs/0705.3146
Author(s): Joan-Andreu L\'azaro-Cam\'{\i} and Juan-Pablo Ortega
Abstract: We present reduction and reconstruction procedures for the solutions of
symmetric stochastic differential equations, similar to those available for
ordinary differential equations. The general methods introduced in the first
part of the paper are then adapted to the Hamiltonian case, which is studied
with special care and illustrated with several examples.
http://arxiv.org/abs/0705.3156
Author(s): Valentin Konakov and Enno Mammen
Abstract: We consider triangular arrays of Markov chains that converge weakly to a
diffusion process. Second order Edgeworth type expansions for transition
densities are proved. The paper differs from recent results in two respects. We
allow nonhomogeneous diffusion limits and we treat transition densities with
time lag converging to zero. Small time asymptotics are motivated by
statistical applications and by resulting approximations for the joint density
of diffusion values at an increasing grid of points.
http://arxiv.org/abs/0705.3139
Author(s): Luigi Manca
Abstract: We prove an extension theorem for a small perturbation of the
Ornstein-Uhlenbeck operator $(L,D(L))$ in the space of all uniformly continuous
and bounded functions $f:H\to \Rset$, where $H$ is a separable Hilbert space.
We consider a perturbation of the form $N_0\phi=L\phi+< D\phi,F>$ where $F:H\to
H$ is bounded and Fr\'echet differentiable with uniformly continuous and
bounded differential. Hence, we prove that $N_0$ is $m$-dissipative and its
closure in $C_b(H)$ coincides with the infinitesimal generator of a diffusion
semigroup associated to a stochastic differential equation in $H$.
http://arxiv.org/abs/0705.3126
Author(s): Wouter Kager
Abstract: We derive diffusion constants and martingales for senile random walks with
the help of a time-change. We provide direct computations of the diffusion
constants for the time-changed walks. Alternatively, the values of these
constants can be derived from martingales associated with the time-changed
walks. Using an inverse time-change, the diffusion constants for senile random
walks are then obtained via these martingales. When the walks are diffusive,
weak convergence to Brownian motion can be shown using a martingale central
limit theorem.
http://arxiv.org/abs/0705.3305
Author(s): B. Ferrario
Abstract: For the 1-dimensional Kuramoto-Sivashinsky equation with random forcing term,
existence and uniqueness of solutions is proved. Then, the Markovian semigroup
is well defined; its properties are analyzed, in order to provide sufficient
conditions for existence and uniqueness of invariant measures for this
stochastic equation. Finally, regularity results are obtained by means of
Girsanov theorem.
http://arxiv.org/abs/0705.3321
Author(s): Nadine Guillotin-Plantard (ICJ) and Arnaud Le Ny (LM-Orsay)
Abstract: We prove a non-standard functional limit theorem for a two dimensional simple
random walk on some randomly oriented lattices. This random walk, already known
to be transient, has different horizontal and vertical fluctuations leading to
different normalizations in the functional limit theorem, with a non-Gaussian
horizontal behavior. We also prove that the horizontal and vertical components
are not asymptotically independent.
http://arxiv.org/abs/0705.3342
Author(s): Kenji Handa (Saga University)
Abstract: The two-parameter Poisson-Dirichlet distribution is a probability
distribution on the totality of positive decreasing sequences with sum 1 and
hence considered to govern masses of a random discrete distribution. A
characterization of the associated point process (i.e., the random point
process obtained by regarding the masses as points in the positive real line)
is given in terms of the correlation functions. Relying on this, we apply the
theory of point processes to reveal mathematical structure of the two-parameter
Poisson-Dirichlet distribution. Also, developing the Laplace transform approach
due to Pitman and Yor, we will be able to extend several results previously
known for the one-parameter case, and the Markov-Krein identity for the
generalized Dirichlet process is discussed from a point of view of functional
analysis based on the two-parameter Poisson-Dirichlet distribution.
http://arxiv.org/abs/0705.3496
Author(s): Lincoln Chayes and Pierre Nolin
Abstract: We reinvestigate the 2D problem of the inhomogeneous incipient infinite
cluster where, in an independent percolation model, the density decays to p_c
with an inverse power, \lambda, of the distance to the origin. Assuming the
existence of critical exponents (as is known in the case of the triangular site
lattice) if the power is less than 1/\nu, with \nu the correlation length
exponent, we demonstrate an infinite cluster with scale dimension given by
D_H=2-\beta\lambda. Further, we investigate the critical case \lambda_c=1/\nu
and show that iterated logarithmic corrections will tip the balance between the
possibility and impossibility of an infinite cluster.
http://arxiv.org/abs/0705.3570
Author(s): Kouji Yano
Abstract: An invariance principle is obtained for a Markov process on a half line with
continuous paths on the interior. Investigated are the domains of attraction of
the two different types of self-similar processes introduced by Lamperti. Our
approach is to establish pathwise convergence of excursions, which is based on
It\^o's excursion theory and a recent result of convergence of excursion
measures by Fitzsimmons and the author.
http://arxiv.org/abs/0705.3588
Author(s): Luisa Beghin
Abstract: We study here a heat-type differential equation of order n greater than two,
in the case where the time-derivative is supposed to be fractional. The
corresponding solution can be described as the transition function of a
pseudoprocess (coinciding with the one governed by the standard,
non-fractional, equation) with a time argument T which is itself random. The
distribution of T is presented together with some features of the solution
(such as analytic expressions for its moments).
http://arxiv.org/abs/0705.3598
Author(s): B\'en\'edicte Haas (CEREMADE) and Jim Pitman and Matthias Winkel
Abstract: We develop some theory of spinal decompositions of discrete and continuous
fragmentation trees. Specifically, we consider a coarse and a fine spinal
integer partition derived from spinal tree decompositions. We prove that for a
two-parameter Poisson-Dirichlet family of continuous fragmentation trees,
including the stable trees of Duquesne and Le Gall, the fine partition is
obtained from the coarse one by shattering each of its parts independently,
according to the same law. As a second application of spinal decompositions, we
prove that among the continuous fragmentation trees, stable trees are the only
ones whose distribution is invariant under uniform re-rooting.
http://arxiv.org/abs/0705.3602
Author(s): Giovanni Peccati (LSTA) and Marc Yor (PMA)
Abstract: We provide a simple proof, as well as several generalizations, of a recent
result by Davis and Suh, characterizing a class of continuous submartingales
and supermartingales that can be expressed in terms of a squared Brownian
motion and of some appropriate powers of its maximum. Our techniques involve
elementary stochastic calculus, as well as the Doob-Meyer decomposition of
continuous submartingales. These results can be used to obtain an explicit
expression of the constants appearing in the Burkholder-Davis-Gundy
inequalities. A connection with some balayage formulae is also established.
http://arxiv.org/abs/0705.3633
Author(s): Stefan Ankirchner and Peter Imkeller and Alexandre Popier
Abstract: We consider insurance derivatives depending on an external physical risk
process, for example a temperature in a low dimensional climate model. We
assume that this process is correlated with a tradable financial asset. We
derive optimal strategies for exponential utility from terminal wealth,
determine the indifference prices of the derivatives, and interpret them in
terms of diversification pressure. Moreover we check the optimal investment
strategies for standard admissibility criteria. Finally we compare the static
risk connected with an insurance derivative to the reduced risk due to a
dynamic investment into the correlated asset. We show that dynamic hedging
reduces the risk aversion in terms of entropic risk measures by a factor
related to the correlation.
http://arxiv.org/abs/0705.3760
Author(s): Guangming Pan and Wang Zhou
Abstract: Consider the empirical spectral distribution of complex random $n\times n$
matrix whose entries are independent and identically distributed random
variables with mean zero and variance $1/n$. In this paper, via applying
potential theory in the complex plane and analyzing extreme singular values, we
prove that this distribution converges, with probability one, to the uniform
distribution over the unit disk in the complex plane, i.e. the well known
circular law, under the finite fourth moment assumption on matrix elements.
http://arxiv.org/abs/0705.3773
Author(s): Olivier Couronn\'e (MODAL'X)
Abstract: Using the Chen-Stein method, we show that the spatial distribution of large
finite clusters in the supercritical FK model approximates a Poisson process
when the ratio weak mixing property holds.
http://arxiv.org/abs/0705.3781
Author(s): Stefan Ankirchner and Peter Imkeller and Alexandre Popier
Abstract: We consider backward stochastic differential equations (BSDE) with nonlinear
generators typically of quadratic growth in the control variable. A measure
solution of such a BSDE will be understood as a probability measure under which
the generator is seen as vanishing, so that the classical solution can be
reconstructed by a combination of the operations of conditioning and using
martingale representations. We show that classical solutions entail the
existence of measure solutions. To go the other way, we prove a priori
inequalities providing bounds on exponential moments of the control processes.
Then we give some algorithms based on approximations of singular generators by
smoother ones, or of exponentially integrable terminal variables by bounded
ones, which construct measure solutions from first principles, in particular
without reference to classical solutions. This way we provide an elegant and
efficient method to at least recover classical existence Theorems for BSDE.
http://arxiv.org/abs/0705.3788
Author(s): Remco van der Hofstad and Mark Holmes and Gordon Slade
Abstract: This paper extends the inductive approach to the lace expansion of van der
Hofstad and Slade in order to prove Gaussian asymptotic behaviour for models
with critical dimension other than 4. The results are applied by Holmes to
study sufficiently spread-out lattice trees in dimensions d>8 and may also be
applicable to percolation in dimensions d>6.
http://arxiv.org/abs/0705.3798
Author(s): Deborah H. Glueck and Anis Karimpour-Fard and Jan Mandel and Larry Hunter and Keith E. Muller
Abstract: The joint cumulative distribution function for order statistics arising from
several different populations is given in terms of the distribution function of
the populations. The computational cost of the formula in the case of two
populations is still exponential in the worst case, but it is a dramatic
improvement compared to the general formula by Bapat and Beg. In the case when
only the joint distribution function of a subset of the order statistics of
fixed size is needed, the complexity is polynomial, for the case of two
populations.
http://arxiv.org/abs/0705.3851
Author(s): Firas Rassoul-Agha and Timo Seppalainen
Abstract: We consider a multidimensional random walk in a product random environment
with bounded steps, transience in some spatial direction, and high enough
moments on the regeneration time. We prove an invariance principle, or
functional central limit theorem, under almost every environment for the
diffusively scaled centered walk. The main point behind the invariance
principle is that the quenched mean of the walk behaves subdiffusively.
http://arxiv.org/abs/0705.4116
Author(s): Maria Deijfen and Henri van den Esker and Remco van der Hofstad and Gerard Hooghiemstra
Abstract: In this paper, a random graph process {G(t)}_{t\geq 1}$ is studied and its
degree sequence is analyzed. Let {W_t}_{t\geq 1} be an i.i.d. sequence. The
graph process is defined so that, at each integer time t, a new vertex, with
W_t edges attached to it, is added to the graph. The new edges added at time t
are then preferentially connected to older vertices, i.e., conditionally on
G(t-1), the probability that a given edge is connected to vertex i is
proportional to d_i(t-1)+\delta, where d_i(t-1) is the degree of vertex i at
time t-1, independently of the other edges. The main result is that the
asymptotical degree sequence for this process is a power law with exponent
\tau=\min{\tau_{W}, \tau_{P}}, where \tau_{W} is the power-law exponent of the
initial degrees {W_t}_{t\geq 1} and $\tau_{P} the exponent predicted by pure
preferential attachment. This result extends previous work by Cooper and
Frieze, which is surveyed.
http://arxiv.org/abs/0705.4151
Author(s): Remco van der Hofstad and Gerard Hooghiemstra
Abstract: In this paper, we investigate the diameter in preferential attachment (PA-)
models, thus quantifying the statement that these models are small worlds.
There is a substantial amount of literature proving that, in quite generality,
PA-graphs possess power-law degree sequences with exponent \tau>2. The models
studied here are such that edges are attached to older vertices proportional to
the degree plus a constant, i.e., we consider linear PA-models. We prove that
the diameter is bounded by a constant times \log{t}, where t is the size of the
graph. When the power-law exponent \tau exceeds 3, then we also prove a lower
bound of the form \log{t}/\log\log{t}}, while when \tau\in (2,3), we improve
the upper bound to a constant times \log\log{t}. These bounds are consistent
with predictions by physicists that the distances in PA-graphs are similar to
the ones in other scale-free random graphs, where distances have been shown to
be of order \log\log{t}, when \tau\in (2,3), and of order \log{t} when \tau>3.
http://arxiv.org/abs/0705.4153
Author(s): A. Kurtzmann
Abstract: Self-interacting diffusions are solutions to SDEs with a drift term depending
on the process and its normalized occupation measure $\mu_t$ (via an
interaction potential and a confinement potential). We establish a relation
between the asymptotic behavior of $\mu_t$ and the asymptotic behavior of a
deterministic dynamical flow (defined on the space of the Borel probability
measures). We extend previous results on $\mathbb{R}^d$ or more generally a
smooth complete connected Riemannian manifold without boundary. We will also
give some sufficient conditions for the convergence of $\mu_t$. Finally, we
will illustrate our study with an example on $\mathbb{R}^2$.
http://arxiv.org/abs/0705.4245
Author(s): Ai-Hua Fan (LAMFA) and Joerg Schmeling and Serge Troubetzkoy (CPT and FRUMAM and IML)
Abstract: Let $\mu$ be a Gibbs measure of the doubling map $T$ of the circle. For a
$\mu$-generic point $x$ and a given sequence $\{r_n\} \subset \R^+$, consider
the intervals $(T^nx - r_n \pmod 1, T^nx + r_n \pmod 1)$. In analogy to the
classical Dvoretzky covering of the circle we study the covering properties of
this sequence of intervals. This study is closely related to the local entropy
function of the Gibbs measure and to hitting times for moving targets. A mass
transference principle is obtained for Gibbs measures which are multifractal.
Such a principle was shown by Beresnevich and Velani \cite{BV} only for
monofractal measures. In the symbolic language we completely describe the
combinatorial structure of a typical relatively short sequence, in particular
we can describe the occurrence of ''atypical'' relatively long words. Our
results have a direct and deep number-theoretical interpretation via
inhomogeneous diadic diophantine approximation by numbers belonging to a given
(diadic) diophantine class.
http://arxiv.org/abs/0705.4203
Author(s): Jean-Pierre Kahane (LM-Orsay) and Yitzhak Katznelson (U STANFORD)
Abstract: We study properties of a sequence $\Lambda$ obtained by a randomselection of
integers $n$, where $n\in\Lambda$ with probability $\varpi_{n}$, independently
of the other choices. We distinguish two cases : if
$\limsup_{n\to\infty}n\varpi_{n}<\infty$, $\Lambda$ is a.s. a Sidon set,
non-dense in the Bohr group ; if $\lim_{n\to\infty}n\varpi_{n}=\infty$, then
$\Lambda$ is a.s. a set of analyticity and is dense in the Bohr group.
http://arxiv.org/abs/0705.4261
Author(s): Alberto Piatti and Marco Zaffalon and Fabio Trojani and Marcus Hutter
Abstract: It is well known that complete prior ignorance is not compatible with
learning, at least in a coherent theory of (epistemic) uncertainty. What is
less widely known, is that there is a state similar to full ignorance, that
Walley calls near-ignorance, that permits learning to take place. In this paper
we provide new and substantial evidence that also near-ignorance cannot be
really regarded as a way out of the problem of starting statistical inference
in conditions of very weak beliefs. The key to this result is focusing on a
setting characterized by a variable of interest that is latent. We argue that
such a setting is by far the most common case in practice, and we show, for the
case of categorical latent variables (and general manifest variables) that
there is a sufficient condition that, if satisfied, prevents learning to take
place under prior near-ignorance. This condition is shown to be easily
satisfied in the most common statistical problems.
http://arxiv.org/abs/0705.4312
Author(s): Gordan Zitkovic
Abstract: We introduce a linear space of finitely additive measures to treat the
problem of optimal expected utility from consumption under a stochastic clock
and an unbounded random endowment process. In this way we establish existence
and uniqueness for a large class of utility maximization problems including the
classical ones of terminal wealth or consumption, as well as the problems
depending on a random time-horizon or multiple consumption instances. As an
example we treat explicitly the problem of maximizing the logarithmic utility
of a consumption stream, where the local time of an Ornstein-Uhlenbeck process
acts as a stochastic clock.
http://arxiv.org/abs/0705.4487
Author(s): Lutz Duembgen and Christoph Leuenberger
Abstract: Benford's law states that for many random variables X > 0 the leading digit D
= D(X) satisfies approximately the equation P(D = d) = log_{10}(1 + 1/d) for d
= 1,2,...,9. This phenomenon follows from another, maybe more intuitive fact,
applied to Y := log_{10}(X): For many real random variables Y, the remainder U
= U(Y) := Y - floor(Y) is approximately uniformly distributed on [0,1). The
present paper provides new explicit bounds for the latter approximation in
terms of the total variation of the density of Y or some derivative of it.
These bounds are an interesting alternative to traditional Fourier methods
which yield mostly qualitative results. As a by-product we obtain explicit
bounds for the approximation error in Benford's law.
http://arxiv.org/abs/0705.4488
Author(s): Remco van der Hofstad and Wouter Kager
Abstract: We study occurrences of patterns on clusters of size n in random fields on
Z^d. We prove that for a given pattern, there is a constant a>0 such that the
probability that this pattern occurs at most an times on a cluster of size n is
exponentially small. Moreover, for random fields obeying a certain Markov
property, we show that the ratio between the numbers of occurrences of two
distinct patterns on a cluster is concentrated around a constant value. This
leads to an elegant and simple proof of the ratio limit theorem for these
random fields, which states that the ratio of the probabilities that the
cluster of the origin has sizes n+1 and n converges as n tends to infinity.
Implications for the maximal cluster in a finite box are discussed.
http://arxiv.org/abs/0705.4534
Author(s): Tibor Antal and P. L. Krapivsky and and Kirone Mallick
Abstract: Molecular spiders are synthetic bio-molecular systems which have "legs" made
of short single-stranded segments of DNA. Spiders move on a surface covered
with single-stranded DNA segments complementary to legs. Different mappings are
established between various models of spiders and simple exclusion processes.
For spiders with simple gait and varying number of legs we compute the
diffusion coefficient; when the hopping is biased we also compute their
velocity.
http://arxiv.org/abs/0705.2594
Author(s): Tibor Antal and P. L. Krapivsky
Abstract: Synthetic bio-molecular spiders with "legs" made of single-stranded segments
of DNA can move on a surface which is also covered by single-stranded segments
of DNA complementary to the leg DNA. In experimental realizations, when a leg
detaches from a segment of the surface for the first time it alters that
segment, and legs subsequently bound to these altered segments more weakly.
Inspired by these experiments we investigate spiders moving along a
one-dimensional substrate, whose legs leave newly visited sites at a slower
rate than revisited sites. For a random walk (one-leg spider) the slowdown does
not effect the long time behavior. For a bipedal spider, however, the slowdown
generates an effective bias towards unvisited sites, and the spider behaves
similarly to the excited walk. Surprisingly, the slowing down of the spider at
new sites increases the diffusion coefficient and accelerates the growth of the
number of visited sites.
http://arxiv.org/abs/0705.2596
Author(s): Dmitry Bilyk and Michael Lacey and Armen Vagharshakyan
Abstract: Let h_R denote an L ^{\infty} normalized Haar function adapted to a dyadic
rectangle R contained in the unit cube in dimension d. We establish a
non-trivial lower bound on the L^{\infty} norm of the `hyperbolic' sums $$ \sum
_{|R|=2 ^{-n}} \alpha(R) h_R (x) $$ The lower bound is non-trivial in that we
improve the average case bound by n^{\eta} for some positive \eta, a function
of dimension d. As far as the authors know, this is the first result of this
type in dimension 4 and higher.
This question is related to Conjectures in (1) Irregularity of Distributions,
(2) Approximation Theory and (3) Probability Theory. The method of proof of
this paper gives new results on these conjectures in all dimensions 4 and
higher.
This paper builds upon prior work of Jozef Beck, from 1989, and first two
authors from 2006. These results were of the same nature, but only in dimension
3.
http://arxiv.org/abs/0705.4619
Author(s): Gordan Zitkovic
Abstract: We extend the Bipolar Theorem of Brannath and Schachermayer (1999) to the
space of nonnegative cadlag supermartingales on a filtered probability space.
We formulate the notion of fork-convexity as an analogue to convexity in this
setting. As an intermediate step in the proof of our main result we establish a
conditional version of the Bipolar theorem. In an application to mathematical
finance we describe the structure of the set of dual processes of the utility
maximization problem of Kramkov and Schachermayer (1999) and give a
budget-constraint characterization of admissible consumption processes in an
incomplete semimartingale market.
http://arxiv.org/abs/0706.0049
Author(s): Ioannis Karatzas and Gordan Zitkovic
Abstract: We consider the problem of maximizing expected utility from consumption in a
constrained incomplete semimartingale market with a random endowment process,
and establish a general existence and uniqueness result using techniques from
convex duality. The notion of asymptotic elasticity of Kramkov and
Schachermayer is extended to the time-dependent case. By imposing no smoothness
requirements on the utility function in the temporal argument, we can treat
both pure consumption and combined consumption/terminal wealth problems, in a
common framework. To make the duality approach possible, we provide a detailed
characterization of the enlarged dual domain which is reminiscent of the
enlargement of $L^1$ to its topological bidual $(L^{\infty})^*$, a space of
finitely-additive measures. As an application, we treat the case of a
constrained It\^ o-process market-model.
http://arxiv.org/abs/0706.0051
Author(s): Mireille Capitaine (LSProba) and Catherine Donati-Martin (PMA) and Delphine F\'eral (LSProba)
Abstract: We investigate the asymptotic spectrum of deformed Wigner matrices. The
deformation is deterministic will all but finitely many eigenvalues equal to
zero. We show that, as soon as the first largest or last smallest eigenvalues
of the deformation are sufficiently far from 0, the corresponding eigenvalues
of the deformed Wigner matrix almost surely exit the limiting semicircle
compact support as the size of the matrix becomes large. In the particular case
of a diagonal pertubation of rank 1, we prove that the fluctuations of the
largest eigenvalue are not universal and depend on the particular distribution
of the entries of the Wigner matrix.
http://arxiv.org/abs/0706.0136
Author(s): Walid Hachem (LTCI) and Philippe Loubaton (IGM-LabInfo) and Jamal Najim (LTCI)
Abstract: Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are
given by $$ Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}} X_{ij}^{n} $$ the
$X_{ij}^{n}$ being centered, independent and identically distributed random
variables with unit variance and $(\sigma_{ij}(n); 1\le i\le N, 1\le j\le n)$
being an array of numbers we shall refer to as a variance profile. We study in
this article the fluctuations of the random variable $$ \log\det(Y_n Y_n^* +
\rho I_N) $$ where $Y^*$ is the Hermitian adjoint of $Y$ and $\rho > 0$ is an
additional parameter. We prove that when centered and properly rescaled, this
random variable satisfies a Central Limit Theorem (CLT) and has a Gaussian
limit whose parameters are identified. A complete description of the scaling
parameter is given; in particular it is shown that an additional term appears
in this parameter in the case where the 4$^\textrm{th}$ moment of the
$X_{ij}$'s differs from the 4$^{\textrm{th}}$ moment of a Gaussian random
variable. Such a CLT is of interest in the field of wireless communications.
http://arxiv.org/abs/0706.0166
Author(s): Denis Feyel and Arnaud De La Pradelle (IMJ) and Gabriel Mokobodzki (IMJ)
Abstract: In a preceding paper [E.J.ofProb.34,860-892,(2006)], we proved a sewing lemma
which was a key result for the study of Holder continuous functions. In this
paper we give a non-commutative version of this lemma with some applications.
http://arxiv.org/abs/0706.0202
Author(s): Jean-Fran\c{c}ois Delmas (CERMICS) and Jean-St\'ephane Dhersin (MAP5) and Arno Siri-Jegousse (MAP5)
Abstract: We give the asymptotic distribution of the length of partial coalescent trees
for Beta and related coalescents. This allows us to give the asymptotic
distribution of the number of (neutral) mutations in the partial tree. This is
a first step to study the asymptotic distribution of a natural estimator of DNA
mutation rate for species with large families.
http://arxiv.org/abs/0706.0204
Author(s): Gabriela Ciuperca
Abstract: We consider a multi-phase random regression model, discontinuous in each
change-point, with an arbitrary error $\epsilon$. In the case that the number
of jumps is known, the M-estimator for the locations of the jumps and for the
coefficient parameters are studied. These estimators are consistent and the
distribution for the estimators of the coefficients is Gaussian. The estimators
of the change-points converge, with the rate $n^{-1}$, to the smallest
minimizer of the independent compound Poisson processes.
http://arxiv.org/abs/0706.0153
Author(s): J. van den Berg and Y. Peres and V. Sidoravicius and M.E. Vares
Abstract: We study models of spatial growth processes where initially there are sources
of growth (indicated by the colour green) and sources of a growth-stopping
(paralyzing) substance (indicated by red). The green sources expand and may
merge with others (there is no `inter-green' competition). The red substance
remains passive as long as it is isolated. However, when a green cluster comes
in touch with the red substance, it is immediately invaded by the latter, stops
growing and starts to act as red substance itself. In our main model space is
represented by a graph, of which initially each vertex is randomly green, red
or white (vacant), and the growth of the green clusters is similar to that in
first-passage percolation. The main issues we investigate are whether the model
is well-defined on an infinite graph (e.g. the $d$-dimensional cubic lattice),
and what can be said about the distribution of the size of a green cluster just
before it is paralyzed. We show that, if the initial density of red vertices is
positive, and that of white vertices is sufficiently small, the model is indeed
well-defined and the above distribution has an exponential tail. In fact, we
believe this to be true whenever the initial density of red is positive. This
research also led to a relation between invasion percolation and critical
Bernoulli percolation which seems to be of independent interest.
http://arxiv.org/abs/0706.0219
Author(s): Paul Bourgade and Chris Hughes and Ashkan Nikeghbali and Marc Yor
Abstract: In this paper, we propose a probabilistic approach to the study of the
characteristic polynomial of a random unitary matrix. We recover the Mellin
Fourier transform of such a random polynomial, first obtained by Keating and
Snaith, using a simple recursion formula, and from there we are able to obtain
the joint law of its radial and angular parts in the complex plane. In
particular, we show that the real and imaginary parts of the logarithm of the
characteristic polynomial of a random unitary matrix can be represented in law
as the sum of independent random variables. From such representations, the
celebrated limit theorem obtained by Keating and Snaith is now obtained from
the classical central limit theorems of Probability Theory, as well as some new
estimates for the rate of convergence and law of the iterated logarithm type
results.
http://arxiv.org/abs/0706.0333
Author(s): Giambattista Giacomin
Abstract: A class of discrete renewal processes with super-exponentially decaying
inter-arrival distributions coincides with the infinite volume limit of general
homogeneous pinning models in their localized phase. Pinning models are
statistical mechanics systems to which a lot of attention has been devoted both
for their relevance for applications and because they are solvable models
exhibiting a non-trivial phase transition. The spatial decay of correlations in
these systems is directly mapped to the speed of convergence to equilibrium for
the associated renewal processes. We show that close to criticality, under
general assumptions, the correlation decay rate, or the renewal convergence
rate, coincides with the inter-arrival decay rate. We also show that, in
general, this is false away from criticality. Under a stronger assumption on
the inter-arrival distribution we establish a local limit theorem, capturing
thus the sharp asymptotic behavior of correlations.
http://arxiv.org/abs/0706.0341
Author(s): Russell Lyons and Benjamin J. Morris and Oded Schramm
Abstract: It has hitherto been known that in a transitive unimodular graph, each tree
in the wired spanning forest has only one end a.s. We dispense with the
assumptions of transitivity and unimodularity, replacing them with a much
broader condition on the isoperimetric profile that requires just slightly more
than uniform transience.
http://arxiv.org/abs/0706.0358
Author(s): Soeren Asmussen and Pierre Fiorini and Lester Lipsky and Tomasz Rolski and Robert Sheahan
Abstract: Many processes must complete in the presence of failures. Different systems
respond to task failure in different ways. The system may resume a failed task
from the failure point (or a saved checkpoint shortly before the failure
point), it may give up on the task and select a replacement task from the ready
queue, or it may restart the task. The behavior of systems under the first two
scenarios is well documented, but the third ({\em RESTART}) has resisted
detailed analysis. In this paper we derive tight asymptotic relations between
the distribution of {\em task times} without failures to the {\em total time}
when including failures, for any failure distribution. In particular, we show
that if the task time distribution has an unbounded support then the total time
distribution $H$ is always heavy-tailed. Asymptotic expressions are given for
the tail of $H$ in various scenarios. The key ingredients of the analysis are
the Cram\'er--Lundberg asymptotics for geometric sums and integral asymptotics,
that in some cases are obtained via Tauberian theorems and in some cases by
bare-hand calculations.
http://arxiv.org/abs/0706.0403
Author(s): Pawe{\l} Sztonyk
Abstract: We prove that bounded harmonic functions of anisotropic fractional Laplacians
are H\"older continuous under mild regularity assumptions on the corresponding
L\'evy measure. Under some stronger assumptions the Green function, Poisson
kernel and the harmonic functions are even differentiable of order up to three.
http://arxiv.org/abs/0706.0413
Author(s): Gordan Zitkovic
Abstract: Existence of stochastic financial equilibria giving rise to semimartingale
asset prices is established under a general class of assumptions. These
equilibria are expressed in real terms and span complete markets or markets
with withdrawal constraints.We deal with random endowment density streams which
admit jumps and general time-dependent utility functions on which only
regularity conditions are imposed. As an integral part of the proof of the main
result, we establish a novel characterization of semimartingale functions.
http://arxiv.org/abs/0706.0462
Author(s): Kasper Larsen and Gordan Zitkovic
Abstract: This paper provides a new version of the condition of Di Nunno et al. (2003),
Ankirchner and Imkeller (2005) and Biagini and \{O}ksendal (2005) ensuring the
semimartingale property for a large class of continuous stochastic processes.
Unlike our predecessors, we base our modeling framework on the concept of
portfolio proportions which yields a short self-contained proof of the main
theorem, as well as a counterexample, showing that analogues of our results do
not hold in the discontinuous setting.
http://arxiv.org/abs/0706.0468
Author(s): Kasper Larsen and Gordan Zitkovic
Abstract: The effectiveness of utility-maximization techniques for portfolio management
relies on our ability to estimate correctly the parameters of the dynamics of
the underlying financial assets. In the setting of complete or incomplete
financial markets, we investigate whether small perturbations of the market
coefficient processes lead to small changes in the agent's optimal behavior
derived from the solution of the related utility-maximization problems.
Specifically, we identify the topologies on the parameter process space and the
solution space under which utility-maximization is a continuous operation, and
we provide a counterexample showing that our results are best possible, in a
certain sense. A novel result about the structure of the solution of the
utility-maximization problem where prices are modeled by continuous
semimartingales is established as an offshoot of the proof of our central
theorem.
http://arxiv.org/abs/0706.0474
Author(s): Mark Owen and Gordan Zitkovic
Abstract: This paper studies the problem of maximizing the expected utility of terminal
wealth for a financial agent with an unbounded random endowment, and with a
utility function which supports both positive and negative wealth. We prove the
existence of an optimal trading strategy within a class of permissible
strategies -- those strategies whose wealth process is a supermartingale under
all pricing measures with finite relative entropy. We give necessary and
sufficient conditions for the absence of utility-based arbitrage, and for the
existence of a solution to the primal problem.
We consider two utility based methods which can be used to price contingent
claims. Firstly we investigate marginal utility-based price processes
(MUBPP's). We show that such processes can be characterized as local
martingales under the normalized optimal dual measure for the utility
maximizing investor. Finally, we present some new results on utility
indifference prices, including continuity properties and volume asymptotics for
the case of a general utility function, unbounded endowment and unbounded
contingent claims.
http://arxiv.org/abs/0706.0478
Author(s): Traian A. Pirvu and Gordan Zitkovic
Abstract: We investigate the ergodic problem of growth-rate maximization under a class
of risk constraints in the context of incomplete, It\^{o}-process models of
financial markets with random ergodic coefficients. Including {\em
value-at-risk} (VaR), {\em tail-value-at-risk} (TVaR), and {\em limited
expected loss} (LEL), these constraints can be both wealth-dependent(relative)
and wealth-independent (absolute). The optimal policy is shown to exist in an
appropriate admissibility class, and can be obtained explicitly by uniform,
state-dependent scaling down of the unconstrained (Merton) optimal portfolio.
This implies that the risk-constrained wealth-growth optimizer locally behaves
like a CRRA-investor, with the relative risk-aversion coefficient depending on
the current values of the market coefficients.
http://arxiv.org/abs/0706.0480
Author(s): Constantinos Kardaras and Gordan Zitkovic
Abstract: We perform a stability analysis for the utility maximization problem in a
general semimartingale model where both liquid and illiquid assets (random
endowments) are present. Small misspecifications of preferences (as modeled via
expected utility), as well as views of the world or the market model (as
modeled via subjective probabilities) are considered. Simple sufficient
conditions are given for the problem to be well-posed, in the sense that
optimal wealths and marginal utility-based prices are continuous functionals of
the inputs.
http://arxiv.org/abs/0706.0482
Author(s): Michael Behrisch and Amin Coja-Oghlan and Mihyun Kang
Abstract: We establish central and local limit theorems for the number of vertices in
the largest component of a random $d$-uniform hypergraph $\hnp$ with edge
probability $p=c/\binnd$, where $(d-1)^{-1}+\eps
http://arxiv.org/abs/0706.0496
Author(s): Michael Behrisch and Amin Coja-Oghlan and Mihyun Kang
Abstract: Let $\hnp$ signify a random $d$-uniform hypergraph with $n$ vertices in which
each of the $\bink{n}d$ possible edges is present with probability $p=p(n)$
independently, and let $\hnm$ denote a uniformly distributed with $n$ vertices
and $m$ edges. We derive local limit theorems for the joint distribution of the
number of vertices and the number of edges in the largest component of $\hnp$
and $\hnm$ for the regime $\bink{n-1}{d-1}p,dm/n>(d-1)^{-1}+\eps$. As an
application, we obtain an asymptotic formula for the probability that $\hnp$ or
$\hnm$ is connected. In addition, we infer a local limit theorem for the
conditional distribution of the number of edges in $\hnp$ given connectivity.
While most prior work on this subject relies on techniques from enumerative
combinatorics, we present a new, purely probabilistic approach.
http://arxiv.org/abs/0706.0497
Author(s): Fabricio Bandeira Cabral
Abstract: In this paper, we consider systems that can be modelled by $M \mid M \mid n$
queues with heterogeneous servers and non informed customers. Considering any
two servers: we show that the probability that the fastest server is busy is
smaller than the probability that the slowest server is busy. Moreover, we show
that the effective rate of service done by the fastest server is larger than
effective rate of service done by the slowest server.
http://arxiv.org/abs/0706.0560
Author(s): Attila Andai
Abstract: In this paper we consider the space of those probability distributions which
maximize the $q$-R\'enyi entropy. These distributions have the same parameter
space for every $q$, and in the $q=1$ case these are the normal distributions.
Some methods to endow this parameter space with Riemannian metric is presented:
the second derivative of the $q$-R\'enyi entropy, Tsallis-entropy and the
relative entropy give rise to a Riemannian metric, the Fisher-information
matrix is a natural Riemannian metric, and there are some geometrically
motivated metrics which were studied by Siegel, Calvo and Oller, Lovri\'c,
Min-Oo and Ruh. These metrics are different therefore our differential
geometrical calculations based on a unified metric, which covers all the above
mentioned metrics among others. We also compute the geometrical properties of
this metric, the equation of the geodesic line with some special solutions, the
Riemann and Ricci curvature tensors and scalar curvature. Using the
correspondence between the volume of the geodesic ball and the scalar curvature
we show how the parameter $q$ modulates the statistical distinguishability of
close points. We show that some frequently used metric in quantum information
geometry can be easily recovered from classical metrics.
http://arxiv.org/abs/0706.0606
Author(s): Remco van der Hofstad and Mark Holmes and Gordon Slade
Abstract: We extend the inductive approach to the lace expansion, previously developed
to study models with critical dimension 4, to be applicable more generally. In
particular, the result of this note has recently been used to prove Gaussian
asymptotic behaviour for the Fourier transform of the two-point function for
sufficiently spread-out lattice trees in dimensions d>8, and it is potentially
also applicable to percolation in dimensions d>6.
http://arxiv.org/abs/0706.0611
Author(s): Remco van der Hofstad and Mark Holmes
Abstract: We derive a perturbation expansion for general interacting random walks,
where steps are made on the basis of the history of the path. Examples of
models where this expansion applies are reinforced random walk, excited random
walk, the true (weakly) self-avoiding walk and loop-erased random walk. We use
the expansion to prove a law of large numbers and central limit theorem for two
models: (i) A directed version of once-reinforced random walk on \Z^d for
sufficiently small reinforcement parameters. This model is such that if the
reinforcement parameter is set to zero, then the resulting random walk has
independent increments with a non-zero drift; and (ii) Excited random walk in
dimension d>8 when the excitement parameter is sufficiently small.
http://arxiv.org/abs/0706.0614
Author(s): Abdelhamid Hassairi and Sallouha Lajmi and Raoudha Zine
Abstract: Bobecka and Wesolowski (2002) have shown that, in the Olkin and Rubin
characterization of the Wishart distribution (See Casalis and Letac (1996)),
when we use the division algorithm defined by the quadratic representation and
replace the property of invariance by the existence of twice differentiable
densities, we still have a characterization of the Wishart distribution. In the
present work, we show that, when we use the division algorithm defined by the
Cholesky decomposition, we get a characterization of the Riesz distribution.
http://arxiv.org/abs/0706.0679
Author(s): Manuel Lladser
Abstract: Let P(z) and Q(y) be polynomials of the same degree k>=1 in the complex
variables z and y, respectively. In this extended abstract we study the
non-linear functional equation P(z)=Q(y(z)), where y(z) is restricted to be
analytic in a neighborhood of z=0. We provide sufficient conditions to ensure
that all the roots of Q(y) are contained within the range of y(z) as well as to
have y(z)=z as the unique analytic solution of the non-linear equation. Our
results are motivated from uniqueness considerations of polynomial canonical
representations of the phase or amplitude terms of oscillatory integrals
encountered in the asymptotic analysis of the coefficients of mixed powers and
multivariable generating functions via saddle-point methods. Uniqueness shall
prove important for developing algorithms to determine the Taylor coefficients
of the terms appearing in these representations. The uniqueness of Levinson's
polynomial canonical representations of analytic functions in several variables
follows as a corollary of our one-complex variables results.
http://arxiv.org/abs/0705.2345
Author(s): N. Raj Rao and Roland Speicher
Abstract: This note extends Voiculescu's S-transform based analytical machinery for
free multiplicative convolution to the case where the mean of the probability
measures vanishes. We show that with the right interpretation of the
S-transform in the case of vanishing mean, the usual formula makes perfectly
good sense.
http://arxiv.org/abs/0706.0323
Author(s): Marius Junge and Javier Parcet
Abstract: Given any $1 < q \le 2$, we use new free probability techniques to construct
a completely isomorphic embedding of $\ell_q$ (equipped with its natural
operator space structure) into the predual of a sufficiently large QWEP von
Neumann algebra.
http://arxiv.org/abs/0706.0550
Author(s): Martin P.W. Zerner
Abstract: In this note we present a simplified proof of the zero-one law by Merkl and
Zerner (2001) for directional transience of random walks in i.i.d. random
environments (RWRE) on the square lattice. Also, we indicate how to construct a
two-dimensional counterexample in a non-uniformly elliptic and stationary
environment which has better ergodic properties than the example given by Merkl
and Zerner.
http://arxiv.org/abs/0706.0745
Author(s): Sandrine Peche and Alexander Soshnikov
Abstract: We show that the spectral radius of an $N\times N$ random symmetric matrix
with i.i.d. bounded centered but non-symmetrically distributed entries is
bounded from below by $ 2 \*\sigma - o(N^{-6/11+\epsilon}), $ where $\sigma^2 $
is the variance of the matrix entries and $\epsilon $ is an arbitrary small
positive number. Combining with our previous result from [6], this proves that
for any $\epsilon >0, $ one has $$ \|A_N\| =2 \*\sigma + o(N^{-6/11+\epsilon})
$$ with probability going to 1 as $N \to \infty. $
http://arxiv.org/abs/0706.0748
Author(s): Srikanth K. Iyer and D. Manjunath and D. Yogeshwaran
Abstract: Let P := {X_i}_{i >= 1} be a stationary point process in R^d. {C_i}_{i>= 1}
be a sequence of i.i.d random sets in R^d. and {Y^t_i}_{t >= 0, i >= 1} be
i.i.d. {0,1}-valued continuous time stationary Markov chains. We define the
Markov-Boolean model C_t := {Y_t^i(Xi + Ci)}_{i>=1}. C_t represents the
coverage process at time t.
We first obtain limit laws for k-coverage of an area at an arbitrary instant.
We then derive limit laws for the k-coverage induced on a one-dimensional path
at an arbitrary instant. Finally, we obtain the limit laws for the k-coverage
seen by a particle as it moves along a one-dimensional path
http://arxiv.org/abs/0706.0789
Author(s): Pascal Moyal
Abstract: In this paper we present a stability criterion for finite measure-valued
stochastic recursions, generalizing Loynes's Theorem to spaces of measures.
This result provides conditions for the reach of a "total stationary state" for
the queue with an infinity of servers and the single-server SRPT queue. Indeed,
we give in both cases a condition of existence of a stationary measure-valued
recursive sequence characterizing the queueing system exhaustively.
http://arxiv.org/abs/0706.0817
Author(s): J\'er\^ome Lelong (CERMICS)
Abstract: We study the almost sure convergence of randomly truncated stochastic
algorithms. We present a new convergence theorem which extends the already
known results by making vanish the classical condition on the noise terms. The
aim of this work is to prove an almost sure convergence result of randomly
truncated stochastic algorithms under easily verifiable conditions
http://arxiv.org/abs/0706.0841
Author(s): Lutz Mattner and Bero Roos
Abstract: We prove optimal constant over root $n$ upper bounds for the maximal
probabilities of $n$th convolution powers of discrete uniform distributions.
http://arxiv.org/abs/0706.0843
Author(s): Elizabeth Meckes
Abstract: In this paper, explicit error bounds are derived in the approximation of rank
$k$ projections of certain $n$-dimensional random vectors by standard
$k$-dimensional Gaussian random vectors. The bounds are given in terms of $k$,
$n$, and a basis of the $k$-dimensional space onto which we project. The random
vectors considered are two generalizations of the case of a vector with
independent, identically distributed components. In the first case, the random
vector has components which are independent but need not have the same
distribution. The second case deals with finite exchangeable sequences of
random variables.
http://arxiv.org/abs/0706.0844
Author(s): Adrian R\"ollin
Abstract: The application of Stein's method for distributional approximation often
involves so called magic factors in the bound of the solutions to Stein
equations. However, these factors sometimes contain additional terms such as a
logarithmic term for Poisson point process approximation, leading to
unsatisfactory estimates. Despite the fact that is has been shown for many of
these magic factors that the known bounds are sharp and thus that the
additional terms cannot be avoided in general, no probabilistic examples have
been presented in the literature, which justify these magic factors. In this
article we close this gap by constructing such examples more or less
explicitly. As a side effect, a new interpretation of the solutions to Stein
equations is given.
http://arxiv.org/abs/0706.0879
Author(s): Brian Rider and Xin Zhou
Abstract: For a broad class of unitary ensembles of random matrices we demonstrate the
universal nature of the Janossy densities of eigenvalues near the spectral
edge, providing a different formulation of the probability distributions of the
limiting second, third, etc. largest eigenvalues of the ensembles in question.
The approach is based on a representation of the Janossy densities in terms of
a system of orthogonal polynomials, plus the steepest descent method of Deift
and Zhou for the asymptotic analysis of the associated Riemann-Hilbert problem.
http://arxiv.org/abs/0706.0921
Author(s): Alexei Borodin and Grigori Olshanski
Abstract: The present paper originated from our previous study of the problem of
harmonic analysis on the infinite symmetric group. This problem leads to a
family {P_z} of probability measures, the z-measures, which depend on the
complex parameter z. The z-measures live on the Thoma simplex, an
infinite-dimensional compact space which is a kind of dual object to the
infinite symmetric group. The aim of the paper is to introduce stochastic
dynamics related to the z-measures. Namely, we construct a family of diffusion
processes in the Toma simplex indexed by the same parameter z. Our diffusions
are obtained from certain Markov chains on partitions of natural numbers n in a
scaling limit as n goes to infinity. These Markov chains arise in a natural
way, due to the approximation of the infinite symmetric group by the increasing
chain of the finite symmetric groups. Each z-measure P_z serves as a unique
invariant distribution for the corresponding diffusion process, and the process
is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so
that the process is reversible. We describe the spectrum of its generator and
compute the associated (pre)Dirichlet form.
http://arxiv.org/abs/0706.1034
Author(s): A. D. Neate and A. Truman
Abstract: The inviscid limit of the stochastic Burgers equation is discussed in terms
of the level surfaces of the minimising Hamilton-Jacobi function, the classical
mechanical caustic and the Maxwell set and their algebraic pre-images under the
classical mechanical flow map. The problem is analysed in terms of a reduced
(one dimensional) action function. We demonstrate that the geometry of the
caustic, level surfaces and Maxwell set can change infinitely rapidly causing
turbulent behaviour which is stochastic in nature. The intermittence of this
turbulence is demonstrated in terms of the recurrence of two processes.
http://arxiv.org/abs/0706.1159
Author(s): J. Gaertner and F. den Hollander and G. Maillard
Abstract: The present paper provides an overview of results obtained in four recent
papers by the authors. These papers address the problem of intermittency for
the Parabolic Anderson Model in a \emph{time-dependent random medium},
describing the evolution of a ``reactant'' in the presence of a ``catalyst''.
Three examples of catalysts are considered: (1) independent simple random
walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is
on the annealed Lyapunov exponents, i.e., the exponential growth rates of the
successive moments of the reactant. It turns out that these exponents exhibit
an interesting dependence on the dimension and on the diffusion constant.
http://arxiv.org/abs/0706.1171
Author(s): Dominic Schuhmacher
Abstract: Based on Stein's method, we derive upper bounds for Poisson process
approximation in the L_1-Wasserstein metric d_2^(p), which is based on a
slightly adapted L_p-Wasserstein metric between point measures. For the case
p=1, this construction yields the metric d_2 introduced in [Barbour, A.D. and
Brown, T.C. (1992), Stochastic Process. Appl. 43(1), pp. 9--31], for which
Poisson process approximation is well studied in the literature. We demonstrate
the usefulness of the extension to general p by showing that d_2^(p)-bounds
control differences between expectations of certain p-th order average
statistics of point processes.
http://arxiv.org/abs/0706.1172
Author(s): A. D. Neate and A. Truman
Abstract: The inviscid limit of the stochastic Burgers equation, with body forces white
noise in time, is discussed in terms of the level surfaces of the minimising
Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set
and their algebraic pre-images under the classical mechanical flow map. The
problem is analysed in terms of a reduced (one dimensional) action function. We
give an explicit expression for an algebraic surface containing the Maxwell set
and caustic in the polynomial case. Those parts of the caustic and Maxwell set
which are singular are characterised. We demonstrate how the geometry of the
caustic, level surfaces and Maxwell set can change infinitely rapidly causing
turbulent behaviour which is stochastic in nature, and we determine its
intermittence in terms of the recurrent behaviour of two processes.
http://arxiv.org/abs/0706.1173
Author(s): Tomasz Grzywny and Micha{\l}Ryznar
Abstract: The purpose of this paper is to find optimal estimates for the Green function
of a half-space of {\it the relativistic $\alpha$-stable process} with
parameter $m$ on $\Rd$ space. This process has an infinitesimal generator of
the form $mI-(m^{2/\alpha}I-\Delta)^{\alpha/2},$ where $0<\alpha<2$, $m>0$, and
reduces to the isotropic $\alpha$-stable process for $m=0$. Its potential
theory for open bounded sets has been well developed throughout the recent
years however almost nothing was known about the behaviour of the process on
unbounded sets. The present paper is intended to fill this gap and we provide
two-sided sharp estimates for the Green function for a half-space. As a
byproduct we obtain some improvements of the estimates known for bounded sets
specially for balls. The advantage of these estimates is a clarification of the
relationship between the diameter of the ball and the parameter $m$ of the
process.
The main result states that the Green function is comparable with the Green
function for the Brownian motion if the points are away from the boundary of a
half-space and their distance is greater than one. On the other hand for the
remaining points the Green function is somehow related the Green function for
the isotropic $\alpha$-stable process. For example, for $d\ge3$, it is
comparable with the Green function for the isotropic $\alpha$-stable process,
provided that the points are close enough.
http://arxiv.org/abs/0706.1175
Author(s): Mohammed Debbarh and Bertrand Maillot
Abstract: In the setting of additive regression model for continuous time process, we
establish the optimal uniform convergence rates and optimal asymptotic
quadratic error of additive regression. To build our estimate, we use the
marginal integration method.
http://arxiv.org/abs/0706.1154
Author(s): Mohammed Debbarh
Abstract: We establish some uniform limit results in the setting of additive regression
model estimation. Our results allow to give an asymptotic 100% confidence bands
for these components. These results are stated in the framework of i.i.d random
vectors when the marginal integration estimation method is used.
http://arxiv.org/abs/0706.1161
Author(s): Pawel Pralat and Jacques Verstraete and Nicholas Wormald
Abstract: The $k$-core of a graph is the largest subgraph of minimum degree at least
$k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random
graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular
subgraph. Thus the threshold for the appearance of a $k$-regular subgraph of a
random graph is at most the threshold for the $(k+2)$-core. In particular, this
pins down the point of appearance of a $k$-regular subgraph in $\G(n,p)$ to a
window for $p$ of width roughly $2/n$ for large $n$ and moderately large $k$.
http://arxiv.org/abs/0706.1103
Author(s): Istvan Gy\"ongy and Annie Millet (PMA and Ces and Matisse and Samos)
Abstract: Stochastic evolution equations in Banach spaces with unbounded nonlinear
drift and diffusion operators are considered. Under some regularity condition
assumed for the solution, the rate of convergence of various numerical
approximations are estimated under strong monotonicity and Lipschitz
conditions. The abstract setting involves general consistency conditions and is
then applied to a class of quasilinear stochastic PDEs of parabolic type.
http://arxiv.org/abs/0706.1404
Author(s): Serban Belinschi and Florent Benaych-Georges (PMA) and Alice Guionnet (UMPA-ENSL)
Abstract: The free convolution (resp. its rectangular analogue) is the binary operation
on the set of probability measures on the real line which allows to deduce,
from the individual spectral (resp. singular) distributions, the spectral
(resp. singular) distribution of a sum of independent unitarily invariant
square (resp. rectangular) random matrices. In this paper, we consider these
free convolutions, and study the possibility to find probability measures close
to the Dirac mass at zero with regularization properties on the whole real
line. More specifically, we try to find continuous semigroups $(\mu_t)$ of
probability measures such that $\mu_0$ is the Dirac mass at zero and such that
for all positive $t$ and all probability measure $\nu$, the free convolution of
$\mu_t$ with $\nu$ (or, in the rectangular context, the rectangular free
convolution of $\mu_t$ with $\nu$) is absolutely continuous with respect to the
Lebesgue measure, with a positive analytic density on the whole real line. In
the square case, we prove that in semigroups satisfying this property, no
measure can have a finite second moment, and we give a sufficient condition on
semigroups to satisfy this property, with examples. In the rectangular case, we
prove that in most cases, for $\mu$ in a continuous
rectangular-convolution-semigroup, the rectangular convolution of $\mu$ with
$\nu$ either has an atom at the origin or doesn't put any mass in a
neighborhood of the origin, thus the expected property does not hold. However,
we give sufficient conditions for analyticity of the density of the rectangular
convolution of $\mu$ with $\nu$ except on a negligible set of points, as well
as existence and continuity of a density everywhere.
http://arxiv.org/abs/0706.1419
Author(s): Violetta Bernyk and Robert C. Dalang and Goran Peskir
Abstract: Let $X=(X_t)_{t \ge 0}$ be a stable L\'evy process of index $\alpha \in
(1,2)$ with no negative jumps, and let $S_t = \sup_{0 \le s \le t} X_s$ denote
its running supremum for $t>0$. We show that the probability density function
$f_t$ of $S_t$ can be characterized as the unique solution to a weakly singular
Volterra integral equation of the first kind, or equivalently, as the unique
solution to a first-order Riemann-Liouville fractional differential equation
satisfying a boundary condition at zero. This yields an explicit series
representation for $f_t$. Recalling the familiar relation between $S_t$ and the
first entry time $\tau_x$ of $X$ into $[x,\infty)$, this further translates
into an explicit series representation for the probability density function of
$\tau_x$.
http://arxiv.org/abs/0706.1503
Author(s): Davar Khoshnevisan and David A. Levin and Pedro J. Mendez-Hernandez
Abstract: Let X1, X2, . . . denote i.i.d. random bits, each taking the values 1 and 0
with respective probabilities p and 1-p. A well-known theorem of Erdos and
Renyi (1970) describes the length of the longest contiguous stretch, or "run",
of ones in X1, . . ., Xn for large values of n. Benjamini, Haggstrom, Peres,
and Steif (2003, Theorem 1.4) demonstrated the existence of unusual times,
provided that the bits undergo equilibrium dynamics in time. The first of the
two main results of this paper describes what happens if we allow for a fixed
and finite number of "impurities" [or zeros] in the longest run of ones. This
resolves a recent conjecture of Revesz (2005, p. 61). We also compute the
Hausdorff dimension of the collection of all unusual times at which this
long-run-with-impurities occur.
The second main contribution of this paper describes a sharp capacity
criterion for a parity test of Benjamini, Haggstrom, Peres, and Steif (2003)
that was initially motivated by problems in complexity theory. This refines the
existing sufficient condition and necessary condition of Benjamini, Haggstrom,
Peres, and Steif (2003, Theorem 3.4) to a necessary and sufficient condition
which is potential-theoretic in nature. The proof hinges on a combinatorial
argument which does not appear to have an obvious connection to the Markov
property. This is worth mentioning because probabilistic potential theory is
often associated strongly with the Markov, or even strong Markov, property.
http://arxiv.org/abs/0706.1520
Author(s): Thomas P. Hayes and Juan C. Vera and and Eric Vigoda
Abstract: We study Markov chains for randomly sampling $k$-colorings of a graph with
maximum degree $\Delta$. Our main result is a polynomial upper bound on the
mixing time of the single-site update chain known as the Glauber dynamics for
planar graphs when $k=\Omega(\Delta/\log{\Delta})$. Our results can be
partially extended to the more general case where the maximum eigenvalue of the
adjacency matrix of the graph is at most $\Delta^{1-\eps}$, for fixed $\eps >
0$.
The main challenge when $k \le \Delta + 1$ is the possibility of ``frozen''
vertices, that is, vertices for which only one color is possible, conditioned
on the colors of its neighbors. Indeed, when $\Delta = O(1)$, even a typical
coloring can have a constant fraction of the vertices frozen. Our proofs rely
on recent advances in techniques for bounding mixing time using ``local
uniformity'' properties.
http://arxiv.org/abs/0706.1530
Author(s): Harald Andres Helfgott
Abstract: Let f be a cubic polynomial. Then there are infinitely many primes p such
that f(p) is square-free.
http://arxiv.org/abs/0706.1497
Author(s): Yvan Velenik
Abstract: We consider the wetting transition in the framework of an effective interface
model of gradient type, in dimension 2 and higher. We prove pathwise estimates
showing that the interface is localized in the whole thermodynamically-defined
partial wetting regime considered in earlier works. Moreover, we study how the
interface delocalizes as the wetting transition is approached. Our main tool is
reflection positivity in the form of the chessboard estimate.
http://arxiv.org/abs/0706.1632
Author(s): Ronen Gradwohl and Amir Yehudayoff
Abstract: In this note we prove a large deviation bound on the sum of random variables
with the following dependency structure: there is a dependency graph $G$ with a
bounded chromatic number, in which each vertex represents a random variable.
Variables that are represented by neighboring vertices may be arbitrarily
dependent, but collections of variables that form an independent set in $G$ are
$t$-wise independent.
http://arxiv.org/abs/0706.1637
Author(s): Wing Yan Yip and David Stephens and Sofia Olhede
Abstract: An explicit formula for the chaotic representation of the powers of
increments, (X_{t+t_0}-X_{t_0})^n, of a Levy process is presented. There are
two different chaos expansions of a square integrable functional of a Levy
process: one with respect to the compensated Poisson random measure and the
other with respect to the orthogonal compensated powers of the jumps of the
Levy process. Computationally explicit formulae for both of these chaos
expansions of (X_{t+t_0}-X_{t_0})^n are given in this paper. Simulation results
verify that the representation is satisfactory. The CRP of a number of
financial derivatives can be found by expressing them in terms of
(X_{t+t_0}-X_{t_0})^n using Taylor's expansion.
http://arxiv.org/abs/0706.1698
Author(s): Dimitris Achlioptas and Assaf Naor
Abstract: Given d \in (0,infty) let k_d be the smallest integer k such that d < 2k\log
k. We prove that the chromatic number of a random graph G(n,d/n) is either k_d
or k_d+1 almost surely.
http://arxiv.org/abs/0706.1725
Author(s): Christian Houdr\'e and Hua Xu
Abstract: We derive concentration inequalities for functions of the empirical measure
of large random matrices with infinitely divisible entries and, in particular,
stable ones. We also give concentration results for some other functionals of
these random matrices, such as the largest eigenvalue or the largest singular
value.
http://arxiv.org/abs/0706.1753
Author(s): Zakhar Kabluchko
Abstract: Let $X_i$ be i.i.d. standard gaussian variables. Let $S_n=X_1+...+X_n$ and
$$L_n=\max_{0\leq i
http://arxiv.org/abs/0706.1849
Author(s): F. den Hollander and N. Petrelis
Abstract: In this paper we review some recent results, obtained jointly with Stu
Whittington, for a mathematical model describing a copolymer in an emulsion.
The copolymer consists of hydrophobic and hydrophilic monomers, concatenated
randomly with equal density. The emulsion consists of large blocks of oil and
water, arranged in a percolation-type fashion. To make the model mathematically
tractable, the copolymer is allowed to enter and exit a neighboring pair of
blocks only at diagonally opposite corners. The energy of the copolymer in the
emulsion is minus $\alpha$ times the number of hydrophobic monomers in oil
minus $\beta$ times the number of hydrophilic monomers in water. Without loss
of generality we may assume that the interaction parameters are restricted to
the cone $\{(\alpha,\beta)\in \mathbb{R}^2\colon |\beta|\leq\alpha\}$.
We show that the phase diagram has two regimes: (1) in the supercritical
regime where the oil blocks percolate, there is a single critical curve in the
cone separating a localized and a delocalized phase; (2) in the subcritical
regime where the oil blocks do not percolate, there are three critical curves
in the cone separating two localized phases and two delocalized phases, and
meeting at two tricritical points. The different phases are characterized by
different behavior of the copolymer inside the four neighboring pairs of
blocks.
http://arxiv.org/abs/0706.1876
Author(s): Ljuben Mutafchiev
Abstract: The family tree of a Galton-Watson branching process may contain N-ary
subtrees, i.e. subtrees whose vertices have at least N>0 children. For family
trees without infinite N-ary subtrees, we study how fast N-ary subtrees of
height t disappear as t goes to infinity.
http://arxiv.org/abs/0706.1904
Author(s): Vassili N. Kolokoltsov
Abstract: Functional limit theorem for continuous-time random walks (CTRW) are found in
general case of dependent waiting times and jump sizes that are also position
dependent. The limiting anomalous diffusion is described in terms of fractional
dynamics. Probabilistic interpretation of generalized fractional evolution is
given in terms of the random time change (subordination) by means of hitting
times processes.
http://arxiv.org/abs/0706.1928
Author(s): Richard F. Bass and Krzysztof Burdzy
Abstract: Pathwise uniqueness holds for the Skorokhod stochastic differential equation
in $C^{1+\gamma}$-domains in $\R^d$ for $\gamma>1/2$ and $d\geq3$.
http://arxiv.org/abs/0706.1993
Author(s): Nicholas James and Russell Lyons and Yuval Peres
Abstract: We give an example of a transient reversible Markov chain that a.s. has only
a finite number of cutpoints. We explain how this is relevant to a conjecture
of Diaconis and Freedman and a question of Kaimanovich. We also answer
Kaimanovich's question when the Markov chain is a nearest-neighbor random walk
on a tree.
http://arxiv.org/abs/0706.2013
Author(s): Nicolas Fournier and Philippe Laurencot
Abstract: The Marcus-Lushnikov process is a finite stochastic particle system in which
each particle is entirely characterized by its mass. Each pair of particles
with masses $x$ and $y$ merges into a single particle at a given rate $K(x,y)$.
We consider a {\it strongly gelling} kernel behaving as $K(x,y)=x^\alpha y + x
y^\alpha$ for some $\alpha\in (0,1]$. In such a case, it is well-known that
{\it gelation} occurs, that is, giant particles emerge. Then two possible
models for hydrodynamic limits of the Marcus-Lushnikov process arise: the
Smoluchowski equation, in which the giant particles are inert, and the Flory
equation, in which the giant particles interact with finite ones. We show that,
when using a suitable cut-off coagulation kernel in the Marcus-Lushnikov
process and letting the number of particles increase to infinity, the possible
limits solve either the Smoluchowski equation or the Flory equation. We also
study the asymptotic behaviour of the largest particle in the Marcus-Lushnikov
process without cut-off and show that there is only one giant particle. This
single giant particle represents, asymptotically, the lost mass of the solution
to the Flory equation.
http://arxiv.org/abs/0706.2057
Author(s): T. S. Turova
Abstract: We study the "rank 1 case" of the inhomogeneous random graph model. In the
subcritical case we derive an exact formula for the asymptotic size of the
largest connected component scaled to log n. This result is new, it completes
the corresponding known result in the supercritical case. We provide some
examples of application of a new formula.
http://arxiv.org/abs/0706.2106
Author(s): Gerard Ben Arous and Anton Bovier and Jiri Cerny
Abstract: We consider a version of a Glauber dynamics for a p-spin
Sherrington--Kirkpatrick model of a spin glass that can be seen as a time
change of simple random walk on the N-dimensional hypercube. We show that, for
any p>2 and any inverse temperature \beta>0, there exist constants g>0, such
that for all exponential time scales, $\exp(\gamma N)$, with $\gamma< g$, the
properly rescaled clock process (time-change process), converges to an
\alpha-stable subordinator where \alpha=\gamma/\beta^2<1. Moreover, the
dynamics exhibits aging at these time scales with time-time correlation
function converging to the arcsine law of this \alpha-stable subordinator. In
other words, up to rescaling, on these time scales (that are shorter than the
equilibration time of the system), the dynamics of p-spin models ages in the
same way as the REM, and by extension Bouchaud's REM-like trap model,
confirming the latter as a universal aging mechanism for a wide range of
systems. The SK model (the case p=2) seems to belong to a different
universality class.
http://arxiv.org/abs/0706.2135
Author(s): Po-Shen Loh and Benny Sudakov
Abstract: Let G be a graph with n vertices, and let k be an integer dividing n. G is
said to be strongly k-colorable if for every partition of V(G) into disjoint
sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex
k-coloring of G with each color appearing exactly once in each V_i. In the case
when k does not divide n, G is defined to be strongly k-colorable if the graph
obtained by adding k \lceil n/k \rceil - n isolated vertices is strongly
k-colorable. The strong chromatic number of G is the minimum k for which G is
strongly k-colorable. In this paper, we study the behavior of this parameter
for the random graph G(n, p). In the dense case when p >> n^{-1/3}, we prove
that the strong chromatic number is a.s. concentrated on one value \Delta+1,
where \Delta is the maximum degree of the graph. We also obtain several weaker
results for sparse random graphs.
http://arxiv.org/abs/0706.2110
Author(s): Michael J. Kearney and Satya N. Majumdar and and Richard J. Martin
Abstract: An exact expression for the distribution of the area swept out by a drifted
Brownian motion till its first-passage time is derived. A study of the
asymptotic behaviour confirms earlier conjectures and clarifies their range of
validity. The analysis also leads to a simple closed-form solution for the
moments of the Airy distribution.
http://arxiv.org/abs/0706.2038
Author(s): Marcos Kiwi and Martin Loebl
Abstract: We address the following question: When a randomly chosen regular bipartite
multi--graph is drawn in the plane in the ``standard way'', what is the
distribution of its maximum size planar matching (set of non--crossing disjoint
edges) and maximum size planar subgraph (set of non--crossing edges which may
share endpoints)? The problem is a generalization of the Longest Increasing
Sequence (LIS) problem (also called Ulam's problem). We present combinatorial
identities which relate the number of r-regular bipartite multi--graphs with
maximum planar matching (maximum planar subgraph) of at most d edges to a
signed sum of restricted lattice walks in Z^d, and to the number of pairs of
standard Young tableaux of the same shape and with a ``descend--type''
property. Our results are obtained via generalizations of two combinatorial
proofs through which Gessel's identity can be obtained (an identity that is
crucial in the derivation of a bivariate generating function associated to the
distribution of LISs, and key to the analytic attack on Ulam's problem).
We also initiate the study of pattern avoidance in bipartite multigraphs and
derive a generalized Gessel identity for the number of bipartite 2-regular
multigraphs avoiding a specific (monotone) pattern.
http://arxiv.org/abs/0706.2223
Author(s): S. V. Lototsky and B. L. Rozovskii
Abstract: A procedure is described for defining a generalized solution for stochastic
differential equations using the Cameron-Martin version of the Wiener Chaos
expansion. Existence and uniqueness of this Wiener Chaos solution is
established for parabolic stochastic PDEs such that both the drift and the
diffusion operators are of the second order.
http://arxiv.org/abs/0706.2390
Author(s): S. V. Lototsky and K. Stemmann
Abstract: The paper studies stochastic integration with respect to Gaussian processes
and fields. It is more convenient to work with a field than a process: by
definition, a field is a collection of stochastic integrals for a class of
deterministic integrands. The problem is then to extend the definition to
random integrands. An orthogonal decomposition of chaos space of the random
field leads to two such extensions, corresponding to the \Ito-Skorokhod and the
Stratononovich integrals, and provides an efficient tool to study these
integrals, both analytically and numerically. For a Gaussian process, a natural
definition of the integral follows from a canonical correspondence between
random processes and a special class of random fields.
http://arxiv.org/abs/0706.2391
Author(s): James Allen Fill and Take Nakama
Abstract: When algorithms for sorting and searching are applied to keys that are
represented as bit strings, we can quantify the performance of the algorithms
not only in terms of the number of key comparisons required by the algorithms
but also in terms of the number of bit comparisons. Some of the standard
sorting and searching algorithms have been analyzed with respect to key
comparisons but not with respect to bit comparisons. In this paper, we
investigate the expected number of bit comparisons required by Quickselect
(also known as Find). We develop exact and asymptotic formulae for the expected
number of bit comparisons required to find the smallest or largest key by
Quickselect and show that the expectation is asymptotically linear with respect
to the number of keys. Similar results are obtained for the average case. For
finding keys of arbitrary rank, we derive an exact formula for the expected
number of bit comparisons that (using rational arithmetic) requires only finite
summation (rather than such operations as numerical integration) and use it to
compute the expectation for each target rank.
http://arxiv.org/abs/0706.2437
Author(s): Loic Chaumont (LAREMA) and Juan Carlos Pardo Millan (PMA)
Abstract: We give a realization of the stable L\'evy forest of a given size conditioned
by its mass from the path of the unconditioned forest. Then, we prove an
invariance principle for this conditioned forest by considering $k$ independent
Galton-Watson trees whose offspring distribution is in the domain of attraction
of any stable law conditioned on their total progeny to be equal to $n$. We
prove that when $n$ and $k$ tend towards $+\infty$, under suitable rescaling,
the associated coding random walk, the contour and height processes converge in
law on the Skorokhod space respectively towards the "first passage bridge" of a
stable L\'evy process with no negative jumps and its height process.
http://arxiv.org/abs/0706.2605
Author(s): Andreas Neuenkirch
Abstract: We study the approximation of stochastic differential equations driven by a
fractional Brownian motion with Hurst parameter $H>1/2$. For the mean-square
error at a single point we derive the optimal rate of convergence that can be
achieved by any approximation method using an equidistant discretization of the
driving fractional Brownian motion. We find that there are mainly two cases:
either the solution can be approximated perfectly or the best possible rate of
convergence is $n^{-H-1/2},$ where $n$ denotes the number of evaluations of the
fractional Brownian motion. In addition, we present an implementable
approximation scheme that obtains the optimal rate of convergence in the latter
case.
http://arxiv.org/abs/0706.2636
Author(s): Wolfgang P. Angerer
Abstract: We introduce a Mellin transform of functions which live on all of $\bR$ and
discuss its applications to the limiting theory of Bellman-Harris processes,
and specifically Luria-Delbr\"uck processes. More precisely, we calculate the
life-time distribution of particles in a Bellman-Harris process from their
first-generation offspring and limiting distributions, and prove a formula for
the Laplace transform of the distribution of types in a Luria-Delbr\"uck
process in the Mittag-Leffler limit. As a by-product, we show how to easily
derive the (classical) Mellin transforms of certain stable probability
distributions from their Fourier transform.
http://arxiv.org/abs/0706.2638
Author(s): Fedor Nazarov and Mikhail Sodin
Abstract: Let N(f) be a number of nodal domains of a random Gaussian spherical harmonic
f of degree n. We prove that as n grows to infinity, the mean of N(f)/n^2 tends
to a positive constant, and that N(f)/n^2 exponentially concentrates around
that constant.
This result is consistent with predictions made by Bogomolny and Schmit using
a percolation-like model for nodal domains of random Gaussian plane waves.
http://arxiv.org/abs/0706.2409
Author(s): Omer Friedland and Sasha Sodin
Abstract: Let || . || be a norm on R^n. Averaging || (\eps_1 x_1, ..., \eps_n x_n) ||
over all the 2^n choices of \eps = (\eps_1, ..., \eps_n) in {-1, +1}^n, we
obtain an expression ||| . ||| which is an unconditional norm on R^n.
Bourgain, Lindenstrauss and Milman showed that, for a certain (large)
constant \eta > 1, one may average over (\eta n) (random) choices of \eps and
obtain a norm that is isomorphic to ||| . |||. We show that this is the case
for any \eta > 1.
http://arxiv.org/abs/0706.2483
Author(s): Omer Friedland and Sasha Sodin
Abstract: We demonstrate a simple analytic argument that may be used to bound the Levy
concentration function of a sum of independent random variables. The main
application is a version of a recent inequality due to Rudelson and Vershynin.
http://arxiv.org/abs/0706.2679
Author(s): Frank Aurzada and Mikhail Lifshits
Abstract: We obtain several extensions of Talagrand's lower bound for the small
deviation probability using metric entropy. For Gaussian processes, our
investigations are focused on processes with sub-polynomial and, respectively,
exponential behaviour of covering numbers. The corresponding results are also
proved for non-Gaussian symmetric stable processes, both for the cases of
critically small and critically large entropy. The results extensively use the
classical chaining technique; at the same time they are meant to explore the
limits of this method.
http://arxiv.org/abs/0706.2720
Author(s): Antonio Di Crescenzo and Elvira Di Nardo and Luigi M. Ricciardi
Abstract: We consider the first-crossing-time problem through a constant boundary for a
Wiener process perturbed by random jumps driven by a counting process. On the
base of a sample-path analysis of the jump-diffusion process we obtain explicit
lower bounds for the first-crossing-time density and for the
first-crossing-time distribution function. In the case of the distribution
function, the bound is improved by use of processes comparison based on the
usual stochastic order. The special case of constant jumps driven by a Poisson
process is thoroughly discussed.
http://arxiv.org/abs/0706.2755
Author(s): A.Vershik and A.Gorbulsky
Abstract: We study the notion of the scaled entropy of a filtration of $\sigma$-fields
(= decreasing sequence of $\sigma$-fields) introduced by the first author
({V4}). We suggest a method for computing this entropy for the sequence of
$\sigma$-fields of pasts of a Markov process determined by a random walk over
the trajectories of a Bernoulli action of a commutative or nilpotent countable
group (Theorems~5,~6). Since the scaled entropy is a metric invariant of the
filtration, it follows that the sequences of $\sigma$-fields of pasts of random
walks over the trajectories of Bernoulli actions of lattices (groups ${\Bbb
Z}^d$) are metrically nonisomorphic for different dimensions $d$, and for the
same $d$ but different values of the entropy of the Bernoulli scheme. We give a
brief survey of the metric theory of filtrations, in particular, formulate the
standardness criterion and describe its connections with the scaled entropy and
the notion of a tower of measures.
http://arxiv.org/abs/0706.2758
Author(s): Anne-Laure Basdevant and Christina Goldschmidt
Abstract: We work in the context of the infinitely many alleles model. The allelic
partition associated with a coalescent process started from n individuals is
obtained by placing mutations along the skeleton of the coalescent tree; for
each individual, we trace back to the most recent mutation affecting it and
group together individuals whose most recent mutations are the same. The number
of blocks of each of the different possible sizes in this partition is the
allele frequency spectrum. The celebrated Ewens sampling formula gives precise
probabilities for the allele frequency spectrum associated with Kingman's
coalescent. This (and the degenerate star-shaped coalescent) are the only
Lambda coalescents for which explicit probabilities are known, although they
are known to satisfy a recursion due to Moehle. Recently, Berestycki,
Berestycki and Schweinsberg have proved asymptotic results for the allele
frequency spectra of the Beta(2-alpha,alpha) coalescents with alpha in (1,2).
In this paper, we prove full asymptotics for the case of the
Bolthausen-Sznitman coalescent.
http://arxiv.org/abs/0706.2808
Author(s): Domenico Marinucci (DIPMAT) and Giovanni Peccati (LSTA)
Abstract: We study the weak convergence (in the high-frequency limit) of the frequency
components associated with Gaussian-subordinated, spherical and isotropic
random fields. In particular, we provide conditions for asymptotic Gaussianity
and we establish a new connection with random walks on the the dual of SO(3),
which mirrors analogous results previously established for fields defined on
Abelian groups (see Marinucci and Peccati (2007)). Our work is motivated by
applications to cosmological data analysis, and specifically by the
probabilistic modelling and the statistical analysis of the Cosmic Microwave
Background radiation, which is currently at the frontier of physical research.
To obtain our main results, we prove several fine estimates involving
convolutions of the so-called Clebsch-Gordan coefficients (which are elements
of unitary matrices connecting reducible representations of SO(3); this allows
to intepret most of our asymptotic conditions in terms of coupling of angular
momenta in a quantum mechanical system. Part of the proofs are based on
recently established criteria for the weak convergence of multiple Wiener-It\^o
integrals. In particular, an Appendix contains some useful new results
concerning the asymptotic Gaussianity of sequences of vector-valued multiple
integrals.
http://arxiv.org/abs/0706.2851
Author(s): Max-K. Von Renesse and Marc Yor (PMA) and Lorenzo Zambotti (PMA)
Abstract: We study absolute-continuity properties of a class of stochastic processes,
including the gamma and the Dirichlet processes. We prove that the laws of a
general class of non-linear transformations of such processes are locally
equivalent to the law of the original process and we compute explicitly the
associated Radon-Nikodym densities. This work unifies and generalizes to random
non-linear transformations several previous results on quasi-invariance of
gamma and Dirichlet processes.
http://arxiv.org/abs/0706.3010
Author(s): Kerry Michael Soileau
Abstract: If two random variables X and A are functionally related via f(X)=A for some
strictly monotone continuously differentiable function f:R->R, the distribution
of X may easily be computed from the distribution of A.
http://arxiv.org/abs/0706.3037
Author(s): Paul Bourgade and Ashkan Nikeghbali and Alain Rouault
Abstract: This note presents some equalities in law for $Z_N:=\det(\Id-G)$, where $G$
is an element of a subgroup of the set of unitary matrices of size $N$, endowed
with its unique probability Haar measure. Indeed, under some general
conditions, $Z_N$ can be decomposed as a product of independent random
variables, whose laws are explicitly known. Our results can be obtained in two
ways : either by a recursive decomposition of the Haar measure or by previous
results by Killip and Nenciu on orthogonal polynomials with respect to some
measure on the unit circle. This latter method leads naturally to a study of
determinants of a class of principal submatrices.
http://arxiv.org/abs/0706.3057
Author(s): Matteo Ortisi
Abstract: In this paper we study the long time behavior of the continuum time version
of the Minority Game in terms of existence of an invariant measure for the
stochastic differential equation governing it and convergence to such a
measure. We consider the special case of a game where the number of possible
choices for each individual is S=2 and agents play ``mixed'' strategies
$(\Gamma<\infty)$. Our approach is based on polynomial mixing bounds for
stochastic differential equations.
http://arxiv.org/abs/0706.3114
Author(s): Kouji Yano and Yuko Yano
Abstract: Smoothness and asymptotic behaviors are studied for the densities of the law
of the occupation time on the positive line for Bessel bridges and the
normalized excursion of strictly stable processes. The key role is played by
these properties for functions defined by Riemann--Liouville fractional
integrals.
http://arxiv.org/abs/0706.3162
Author(s): D. Arinkin and A. Borodin
Abstract: We introduce the tau-function of a rational d-connection and its isomonodromy
transformations. We show that in a continuous limit our tau-function agrees
with the Jimbo-Miwa-Ueno tau-function, compute the tau-function for the
isomonodromy transformations leading to difference Painleve V and difference
Painleve VI equations, and prove that the gap probability for a wide class of
discrete random matrix type models can be viewed as the tau-function for an
associated d-connection.
http://arxiv.org/abs/0706.3073
Author(s): Yunfen Bai (1 and 2) and Xinhua Hu (1) and Zhongxing Ye (1) ((1)Department of Mathematics, Shanghai Jiaotong University; (2)Department of Mathematics,
Shijiazhuang College)
Abstract: In this paper, a geometric function is introduced to reflect the attenuation
speed of impact of one firm's default to its partner. If two firms are
competitions (copartners), the default intensity of one firm will decrease
(increase) abruptly when the other firm defaults. As time goes on, the impact
will decrease gradually until extinct. In this model, the joint distribution
and marginal distributions of default times are derived by employing the change
of measure, so can we value the fair swap premium of a CDS.
http://arxiv.org/abs/0706.3331
Author(s): Gr\'egory Miermont (LM-Orsay) and Mathilde Weill (DMA)
Abstract: We prove some asymptotic results for the radius and the profile of large
random rooted planar maps with faces of arbitrary degrees. Using a bijection
due to Bouttier, Di Francesco and Guitter between rooted planar maps and
certain four-type trees with positive labels, we derive our results from a
conditional limit theorem for four-type spatial Galton-Watson trees.
http://arxiv.org/abs/0706.3334
Author(s): B. Rajeev and S. Thangavelu
Abstract: In this paper we prove a stochastic representation for solutions of the
evolution equation $ \partial_t \psi_t = {1/2}L^*\psi_t $ where $ L^* $ is the
formal adjoint of an elliptic second order differential operator with smooth
coefficients corresponding to the infinitesimal generator of a finite
dimensional diffusion $ (X_t).$ Given $ \psi_0 = \psi $, a distribution with
compact support, this representation has the form $ \psi_t = E(Y_t(\psi))$
where the process $ (Y_t(\psi))$ is the solution of a stochastic partial
differential equation connected with the stochastic differential equation for $
(X_t) $ via Ito's formula.
http://arxiv.org/abs/0706.3352
Author(s): S. V. Lototsky and K. Stemmann
Abstract: An Ito-Skorokhod bi-linear equation driven by infinitely many independent
colored noises is considered in a normal triple of Hilbert spaces. The special
feature of the equation is the appearance of the Wick product in the definition
of the Ito-Skorokhod integral, requiring innovative approaches to computing the
solution. A chaos expansion of the solution is derived and several truncations
of this expansion are studied. A recursive approximation of the solution is
suggested and the corresponding approximation error bound is computed.
http://arxiv.org/abs/0706.3392
Author(s): Felipe Cucker and Ernesto Mordecki
Abstract: We consider a perturbed version of the dynamics of a flock introduced by
Cucker and Smale ("Emergent behaviour in flocks") and prove, under similar
conditions, that nearly-alignment (a concept that is precised in the text) is
achieved with a certain probability, bounded from below.
http://arxiv.org/abs/0706.3343
Author(s): Erick Herbin and Ely Merzbach
Abstract: The set-indexed fractional Brownian motion (sifBm) has been defined by
Herbin-Merzbach (2006) for indices that are subsets of a metric measure space.
In this paper, the sifBm is proved to be the only set-indexed process whose
projection on any increasing path is a one-dimensional fractional Brownian
motion. The limitation of its definition for a set-similarity parameter 0
http://arxiv.org/abs/0706.3472
Author(s): Sourav Chatterjee
Abstract: We introduce some applications of Stein's method in the high temperature
analysis of spin glasses. One of the main advantages of Stein's method is that
it gives limit theorems with total variation error bounds, which is not
possible by the method of moments (the only method used to prove such results
in spin glasses till now). Considering the Sherrington-Kirkpatrick model as a
specific example, we obtain error bounds in quenched central limit theorems for
(i) the cavity field, (ii) the hamiltonian in the high temperature phase under
zero external field, and (iii) the local fields in the absence of a cavity. The
last result deserves special mention because the limit distributions are
asymmetric mixtures of pairs of gaussians rather than pure gaussians. A
surprising byproduct of this result is a simple and transparent proof of the
famous Thouless-Anderson-Palmer (TAP) equations that solve the high temperature
phase of the SK model.
http://arxiv.org/abs/0706.3500
Author(s): Sourav Chatterjee and Soumik Pal
Abstract: Consider a time-varying collection of n points on the positive real axis,
modeled as exponentials of n Brownian motions whose drift vector at every time
point is determined by the relative ranks of the coordinate processes at that
time. If at each time point we divide the points by their sum, under suitable
assumptions the rescaled point process converges to a stationary distribution
(depending on n and the vector of drifts) as time goes to infinity. This
stationary distribution can be exactly computed using a recent result of Pal
and Pitman. The model and the rescaled point process are both central objects
of study in models of equity markets introduced by Banner, Fernholz, and
Karatzas. In this paper, we look at the behavior of this point process under
the stationary measure as $n$ tends to infinity. Under a certain `continuity at
the edge' condition on the drifts, we show that one of the following must
happen: either (i) all points converge to zero, or (ii) the maximum goes to one
and the rest go to zero, or (iii) the processes converge in law to a
non-trivial Poisson-Dirichlet distribution. The proof employs, among other
things, techniques from Talagrand's analysis of the low temperature phase of
Derrida's Random Energy Model of spin glasses. The main result establishes a
universality property for the BFK models and aids in explicit asymptotic
computations using known results about the Poisson-Dirichlet law.
http://arxiv.org/abs/0706.3558
Author(s): Harry Kesten and Vladas Sidoravicius
Abstract: Let $\{X(v), v \in \Bbb Z^d \times \Bbb Z_+\}$ be an i.i.d. family of random
variables such that $P\{X(v)= e^b\}=1-P\{X(v)= 1\} = p$ for some $b>0$. We
consider paths $\pi \subset \Bbb Z^d \times \Bbb Z_+$ starting at the origin
and with the last coordinate increasing along the path, and of length $n$.
Define for such paths $W(\pi) = \text{number of vertices $\pi_i, 1 \le i \le
n$, with}X(\pi_i) = e^b$. Finally let $N_n(\al) = \text{number of paths $\pi$
of length $n$ starting at $\pi_0 = \bold 0$ and with $W(\pi) \ge \al n$.}$ We
establish several properties of $\lim_{n \to \infty} [N_n]^{1/n}$.
http://arxiv.org/abs/0706.3626
Author(s): Thomas Richthammer
Abstract: The conservation of translation as a symmetry in two-dimensional systems with
interaction is a classical subject of statistical mechanics. Here we establish
such a result for Gibbsian particle systems with two-body interaction, where
the interesting cases of singular, hard-core and discontinuous interaction are
included. We start with the special case of pure hard core repulsion in order
to show how to treat hard cores in general.
http://arxiv.org/abs/0706.3637
Author(s): Pablo A. Ferrari and Sebastian P. Grynberg
Abstract: Let a
http://arxiv.org/abs/0706.3714
Author(s): Kasper Pedersen
Abstract: This paper is concerned with sampling from the uniform distribution on
H-colourings of the n-vertex path using systematic scan Markov chains. An
H-colouring of the n-vertex path is a homomorphism from the n-vertex path to
some fixed graph H. We show that systematic scan for H-colourings of the
n-vertex path mixes in O(log n) scans for any fixed H. This is a significant
improvement over the previous bound on the mixing time which was O(n^5) scans.
Furthermore we show that for a slightly more restricted family of H (where any
two vertices are connected by a 2-edge path) systematic scan also mixes in
O(log n) scans for any scan order. Finally, for completeness, we show that a
random update Markov chain mixes in O(n log n) updates for any fixed H,
improving the previous bound on the mixing time from O(n^5) updates.
http://arxiv.org/abs/0706.3794
Author(s): Karlheinz Gr\"ochenig and Richard F. Bass
Abstract: We consider the problem of random sampling for band-limited functions. When
can a band-limited function $f$ be recovered from randomly chosen samples
$f(x_j), j\in \mathbb{N}$? We estimate the probability that a sampling
inequality of the form
A\|f\|_2^2 \leq \sum_{j\in \mathbb{N}} |f(x_j)|^2 \leq B \|f\|_2^2 hold
uniformly all functions $f\in L^2(\mathbb{R}^d)$ with supp $\hat{f} \subseteq
[-1/2,1/2]^d$ or some subset of \bdl functions. In contrast to discrete models,
the space of band-limited functions is infinite-dimensional and its functions
``live'' on the unbounded set $\mathbb{R}^d$. This fact raises new problems and
leads to both negative and positive results. (a) With probability one, the
sampling inequality fails for any reasonable definition of a random set on
$\mathbb{R}^d$, e.g., for spatial Poisson processes or uniform distribution
over disjoint cubes. (b) With overwhelming probability, the sampling inequality
holds for certain compact subsets of the space of band-limited functions and
for sufficiently large sampling size.
http://arxiv.org/abs/0706.3818
Author(s): Roland Alkemper and Martin Hutzenthaler
Abstract: We derive a unified stochastic picture for the duality of a
resampling-selection model with a branching-coalescing particle process (cf.
http://www.ams.org/mathscinet-getitem?mr=MR2123250) and for the self-duality of
Feller's branching diffusion with logistic growth (cf. math/0509612). The two
dual processes are approximated by particle processes which are forward and
backward processes in a graphical representation. We identify duality relations
between the basic building blocks of the particle processes which lead to the
two dualities mentioned above.
http://arxiv.org/abs/0706.3852
Author(s): Deborah H. Glueck and Anis Karimpour-Fard and Jan Mandel and Keith E. Muller
Abstract: Consider a set of order statistics that arise from sorting samples from two
different populations, each with their own, possibly different distribution
function. The probability that these order statistics fall in disjoint, ordered
intervals, and that of the smallest statistics, a certain number come from the
first populations, are given in terms of the two distribution functions. The
result is applied to computing the joint probability of the number of
rejections and the number of false rejections for the Benjamini-Hochberg false
discovery rate procedure.
http://arxiv.org/abs/0706.3520
Author(s): Federico Ardila and Elitza Maneva
Abstract: In analyzing the survey propagation algorithm, Maneva, Mossel, and Wainwright
discovered a polynomial identity that holds for a Boolean formula F and a
satisfying assignment a. We show that F and a give rise to a convex geometry,
and that convex geometries are precisely the combinatorial objects satisfying
(the multivariate analog of) that polynomial identity.
http://arxiv.org/abs/0706.3750
Author(s): Rainer Buckdahn and Jin Ma and Catherine Rainer
Abstract: In this paper we study a class of stochastic control problems in which the
control of the jump size is essential. Such a model is a generalized version
for various applied problems ranging from optimal reinsurance selections for
general insurance models to queueing theory. The main novel point of such a
control problem is that by changing the jump size of the system, one
essentially changes the type of the driving martingale. Such a feature does not
seem to have been investigated in any existing stochastic control literature.
We shall first provide a rigorous theoretical foundation for the control
problem by establishing an existence result for the multidimensional structure
equation on a Wiener-Poisson space, given an arbitrary bounded jump size
control process; and by providing an auxiliary counterexample showing the
non-uniqueness for such solutions. Based on these theoretical results we then
formulate the control problem and prove the Bellman Principle, and derive the
corresponding Hamilton-Jacobi-Bellman (HJB) equation, which in this case is a
mixed second-order partial differential/difference equation. Finally we prove a
uniqueness result for the viscosity solution of such an HJB equation.
http://arxiv.org/abs/0706.4018
Author(s): Emmanuel Gobet (LJK) and St\'ephane Menozzi (PMA)
Abstract: For a stopped diffusion process in a time dependent domain, we obtain the
asymptotics of the triplet exit time/exit position/overshoot for the discretely
stopped Euler scheme. Here, the overshoot means the distance to the boundary of
the process when it exits the domain. As a first consequence of this result, we
obtain an expansion for the weak error. From the expansion and the sensitivity
of the underlying Dirichlet problem with respect to the domain, we finally
derive a procedure to improve the convergence by suitably restraining the
domain.
http://arxiv.org/abs/0706.4042
Author(s): Laurent Goergen
Abstract: In the setting of multi-dimensional diffusions in random environment, we
carry on the investigation of condition (T'), introduced by Sznitman in [18]
and by Schmitz in [11] respectively in the discrete and continuous setting, and
which implies a law of large numbers with non vanishing limiting velocity
(ballistic behaviour) as well as a central limit theorem. Specifically, we show
that when $d \geq 2$, (T') is equivalent to an effective condition that can be
checked by local inspection of the environment. When d=1, we prove that
condition (T') is merely equivalent to almost sure transience. As an
application of the effective criterion, we show that when $d \geq 4$ a
perturbation of Brownian motion by a random drift of size at most $\epsilon>0$
whose projection on some direction has expectation bigger than
$\epsilon^{2-\eta}, \eta > 0$, satisfies condition (T') when $\epsilon$ is
small and hence exhibits ballistic behaviour. This class of diffusions contains
new examples of ballistic behaviour which in particular do not fulfill the
condition in [11], (5.4) therein, related to Kalikow's condition, see [21].
http://arxiv.org/abs/0706.4069
Author(s): A.E. Patrick
Abstract: We show that the Euler walk on a Cayley tree exhibits two regimes (dynamic
phases): a condensed phase and a low-density phase. In the condensed phase the
self-organized area grows as a compact domain. In the low-density phase the
proportion of self-organized (visited) nodes decreases rapidly from one
generation of the tree to the next. We describe in detail returns of the Euler
walk to the root and growth of the self-organized domain in the condensed
phase. We also investigate the critical behaviour of the Euler walk at the
point separating the two regimes.
http://arxiv.org/abs/0706.3161
Author(s): Evelina Shamarova
Abstract: A generalized version of Chernoff's theorem has been obtained. Namely, the
version of Chernoff's theorem for semigroups obtained in a paper by Smolyanov,
Weizsaecker, and Wittich is generalized for a time-inhomogeneous case. The main
theorem obtained in the current paper, Chernoff's theorem for evolution
families, deals with a family of time-dependent generators of semigroups $A_t$
on a Banach space, a two-parameter family of operators $Q_{t,t+\Delta t}$
satisfying the relation: $\frac{\partial}{\partial \Delta t}Q_{t,t+\Delta
t}|_{\Delta t = 0}=A_t$, whose products $Q_{t_i,t_{i+1}}... Q_{t_{k-1},t_k}$
are uniformly bounded for all subpartitions $s = t_0 < t_1 < >... < t_n = t$.
The theorem states that $Q_{t_0,t_1}... Q_{t_{n-1},t_n}$ converges to an
evolution family $U(s,t)$ solving a non-autonomous Cauchy problem. Furthermore,
the theorem is formulated for a particular case when the generators $A_t$ are
time dependent second order differential operators. Finally, an example of
application of this theorem to a construction of time-inhomogeneous diffusions
on a compact Riemannian manifold is given.
Keywords: Chernoff's theorem, evolution family, strongly continuous
semigroup, evolution families generated by manifold valued stochastic
processes.
http://arxiv.org/abs/0706.4079
Author(s): Nicolas Champagnat and Amaury Lambert (FESE)
Abstract: We consider a trait-structured population subject to mutation, birth and
competition of logistic type, where the number of coexisting types may
fluctuate. Applying a limit of rare mutations to this population while keeping
the population size finite leads to a jump process, the so-called `trait
substitution sequence', where evolution proceeds by successive invasions and
fixations of mutant types. The probability of fixation of a mutant is
interpreted as a fitness landscape that depends on the current state of the
population. It was in adaptive dynamics that this kind of model was first
invented and studied, under the additional assumption of large population.
Assuming also small mutation steps, adaptive dynamics' theory provides a
deterministic ODE approximating the evolutionary dynamics of the dominant trait
of the population, called `canonical equation of adaptive dynamics'. In this
work, we want to include genetic drift in this models by keeping the population
finite. Rescaling mutation steps (weak selection) yields in this case a
diffusion on the trait space that we call `canonical diffusion of adaptive
dynamics', in which genetic drift (diffusive term) is combined with directional
selection (deterministic term) driven by the fitness gradient. Finally, in
order to compute the coefficients of this diffusion, we seek explicit
first-order formulae for the probability of fixation of a nearly neutral mutant
appearing in a resident population. These formulae are expressed in terms of
`invasibility coefficients' associated with fertility, defense, aggressiveness
and isolation, which measure the robustness (stability w.r.t. selective
strengths) of the resident type. Some numerical results on the canonical
diffusion are also given.
http://arxiv.org/abs/0706.4157
Author(s): Davar Khoshnevisan and Yimin Xiao
Abstract: Let $X_1,...,X_N$ denote $N$ independent $d$-dimensional L\'evy processes,
and consider the $N$-parameter random field \[\X(\bm{t}):=
X_1(t_1)+...+X_N(t_N).\] First we demonstrate that for all nonrandom Borel sets
$F\subseteq\R^d$, the Minkowski sum $\X(\R^N_+)\oplus F$, of the range
$\X(\R^N_+)$ of $\X$ with $F$, can have positive $d$-dimensional Lebesgue
measure if and only if a certain capacity of $F$ is positive. This improves our
earlier joint effort with Yuquan Zhong \ycite{KXZ:03} by removing a
symmetry-type condition there. Moreover, we show that under mild regularity
conditions, our necessary and sufficient condition can be recast in terms of
one-potential densities. This rests on developing results in classical
[non-probabilistic] harmonic analysis that might be of independent interest. As
was shown in \fullocite{KXZ:03}, the potential theory of the type studied here
has a large number of consequences in the theory of L\'evy processes. We
present a few new consequences here.
http://arxiv.org/abs/0706.4164
Author(s): Arnaud Guillin (LATP) and Christian Leonard (CMAP and MODAL'X) and Liming Wu and Nian Yao
Abstract: In this paper, one investigates the following type of
transportation-information $T_cI$ inequalities: $\alpha(T_c(\nu,\mu))\le
I(\nu|\mu)$ for all probability measures $\nu$ on some metric space $(\XX, d)$,
where $\mu$ is a given probability measure, $T_c(\nu,\mu)$ is the
transportation cost from $\nu$ to $\mu$ with respect to some cost function
$c(x,y)$ on $\XX^2$, $I(\nu|\mu)$ is the Fisher-Donsker-Varadhan information of
$\nu$ with respect to $\mu$ and $\alpha: [0,\infty)\to [0,\infty]$ is some left
continuous increasing function. Using large deviation techniques, it is shown
that $T_cI$ is equivalent to some concentration inequality for the occupation
measure of a $\mu$-reversible ergodic Markov process related to $I(\cdot|\mu)$,
a counterpart of the characterizations of transportation-entropy inequalities,
recently obtained by Gozlan and L\'eonard in the i.i.d. case . Tensorization
properties of $T_cI$ are also derived.
http://arxiv.org/abs/0706.4193
Author(s): Paolo Baldi and Gerard Kerkyacharian and Domenico Marinucci and Dominique Picard
Abstract: In a recent paper, we analyzed the properties of a new kind of spherical
wavelets (so-called needlets) for statistical inference procedures on spherical
random fields; the results were mainly motivated by applications to
cosmological data. In the present work, we exploit the asymptotic uncorrelation
of random needlet coefficients at fixed angular distances to construct
subsampling statistics evaluated on Voronoi cells on the sphere. We illustrate
how such statistics can be used for isotropy tests and for bootstrap estimation
of nuisance parameters, even when a single realization of the spherical random
field is observed. The asymptotic theory is developed in details, in the high
resolution sense.
http://arxiv.org/abs/0706.4169
Author(s): Benny Sudakov and Van Vu
Abstract: In this paper, we initiate a systematic study of graph resilience. The
(local) resilience of a graph G with respect to a property P measures how much
one has to change G (locally) in order to destroy P. Estimating the resilience
leads to many new and challenging problems. Here we focus on random and
pseudo-random graphs and prove several sharp results.
http://arxiv.org/abs/0706.4104
Author(s): Ying Hu (IRMAR) and Shanjian Tang (School of Mathematical Sciences)
Abstract: In this paper, we study a multi-dimensional backward stochastic differential
equation (BSDE) with oblique reflection, which is a BSDE reflected on the
boundary of a special unbounded convex domain along an oblique direction, and
which arises naturally in the study of optimal switching problem. The existence
of the adapted solution is obtained by the penalization method, the monotone
convergence, and the a priori estimations. The uniqueness is obtained by a
verification method (the first component of any adapted solution is shown to be
the vector value of a switching problem for BSDEs). As applications, we apply
the above results to solve the optimal switching problem for stochastic
differential equations of functional type, and we give also a probabilistic
interpretation of a system of variational inequalities.
http://arxiv.org/abs/0706.4365
Author(s): Anja Sturm and Jan M. Swart
Abstract: Consider a long-range, one-dimensional voter model started with all zeros on
the negative integers and all ones on the positive integers. If the process
obtained by identifying states that are translations of each other is
positively recurrent, then it is said that the voter model exhibits interface
tightness. In 1995, Cox and Durrett proved that one-dimensional voter models
exhibit interface tightness if their infection rates have a finite third
moment. Recently, Belhaouari, Mountford, and Valle have improved this by
showing that a finite second moment suffices. The present paper gives a new
short proof of this fact. We also prove interface tightness for a long range
swapping voter model, which has a mixture of long range voter model and
exclusion process dynamics.
http://arxiv.org/abs/0706.4405
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