[PAS] Probability Abstracts 98
Probability Abstract Service
pas at lists.imstat.org
Wed Jul 4 16:50:53 CDT 2007
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.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_98.shtml
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5550. PACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION
Davar Khoshnevisan and Yimin Xiao
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
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5551. DYNAMICAL PERCOLATION ON GENERAL TREES
Davar Khoshnevisan
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
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5552. MUTUAL FUND THEOREMS WHEN MINIMIZING THE PROBABILITY OF
LIFETIME RUIN
Erhan Bayraktar and Virginia R. Young
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
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5553. BROWNIAN SUBORDINATORS AND FRACTIONAL CAUCHY PROBLEMS
Boris Baeumer and Mark M. Meerschaert and Erkan Nane
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
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5554. THE ORDER OF THE LARGEST COMPLETE MINOR IN A RANDOM GRAPH
N. Fountoulakis and D. K\"uhn and D. Osthus
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
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5555. MERGING OF OPINIONS IN GAME-THEORETIC PROBABILITY
Vladimir Vovk
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
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5556. LARGE DEVIATIONS FOR MULTIDIMENSIONAL SDES WITH REFLECTION
Zongxia Liang
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
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5557. WHEN ARE SWING OPTIONS BANG-BANG AND HOW TO USE IT
Olivier Aj Bardou (GDF-RDD) and Sandrine Bouthemy (GDF-RDD) and
Gilles Pag\`es (PMA)
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
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5558. MANY RANDOM WALKS ARE FASTER THAN ONE
Noga Alon and Chen Avin and Michal Koucky and Gady Kozma and Zvi
Lotker and Mark R. Tuttle
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
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5559. SPACE-TIME PERCOLATION
Geoffrey Grimmett
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
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5560. BRINGING ERRORS INTO FOCUS
Nicolas Bouleau (CIRED)
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
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5561. CHANGE POINT ESTIMATION FOR THE TELEGRAPH PROCESS OBSERVED AT
DISCRETE TIMES
Alessandro De Gregorio and Stefano M. Iacus
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
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5562. ASYMPTOTIC BEHAVIOR OF SOME WEIGHTED QUADRATIC AND CUBIC
VARIATIONS OF THE FRACTIONAL BROWNIAN MOTION
Ivan Nourdin (PMA)
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
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5563. CENTRAL LIMIT THEOREM FOR THE EXCITED RANDOM WALK IN DIMENSION
$D \GEQ 2$
Jean B\'erard and Alejandro Ram\'irez
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
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5564. UNIQUENESS AND NON-UNIQUENESS OF CHAINS ON HALF LINES
R. Fernandez and G. Maillard
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
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5565. A BERRY-ESSEEN TYPE INEQUALITY FOR CONVEX BODIES WITH AN
UNCONDITIONAL BASIS
Bo'az Klartag
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
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5566. UNE NOUVELLE CONDITION D'INDEPENDANCE POUR LE THEOREME DE LA
LIMITE CENTRALE
Ren\'e Blacher (LJK)
We prove a central limit theorem with aassumptions which are many
weak than
classical conditions
http://arxiv.org/abs/0705.0853
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5567. POISSON APPROXIMATION FOR NON-BACKTRACKING RANDOM WALKS
Noga Alon and Eyal Lubetzky
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
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5568. ULTRAMETRIC AND TREE POTENTIAL
Claude Dellacherie and Servet Martinez and Jaime San Martin
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
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5569. REFLECTED BACKWARD SDES WITH TWO BARRIERS UNDER MONOTONICITY
AND GENERAL INCREASING CONDITIONS
Mingyu Xu
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
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5570. COMBINATORICS OF TRUNCATED RANDOM UNITARY MATRICES
Jonathan Novak
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
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5571. FLUCTUATIONS OF EIGENVALUES AND SECOND ORDER POINCARE INEQUALITIES
Sourav Chatterjee
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
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5572. DIFFUSION COVARIATION AND CO-JUMPS IN BIDIMENSIONAL ASSET
PRICE PROCESSES WITH STOCHASTIC VOLATILITY AND INFINITE ACTIVITY
LEVY JUMPS
Fabio Gobbi and Cecilia Mancini
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
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5573. A NOTE ON SLE CURVES
Qingyang Guan
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
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5574. SLOW CONVERGENCE IN BOOTSTRAP PERCOLATION
Janko Gravner and Alexander E. Holroyd
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
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5575. SURVIVAL OF A DIFFUSING PARTICLE IN AN EXPANDING CAGE
Alan J Bray and Richard Smith
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
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5576. ON THE APPROXIMATE NORMALITY OF EIGENFUNCTIONS OF THE LAPLACIAN
Elizabeth Meckes
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
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5577. ASYMPTOTIC VELOCITY OF ONE DIMENSIONAL DIFFUSIONS WITH PERIODIC
DRIFT
P.Collet S.Martinez
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
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5578. BOUNDARY HARNACK INEQUALITIES FOR REGIONAL FRACTIONAL LAPLACIAN
Qingyang Guan
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
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5579. UNIVERSALITY AT THE SOFT EDGE FOR SOME WHITE SAMPLE COVARIANCE
MATRICES ENSEMBLES
Sandrine Peche
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
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5580. THE ROTOR-ROUTER MODEL ON REGULAR TREES
Itamar Landau and Lionel Levine
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
---------------------------------------------------------------
5581. A COARSE GRAINING FOR THE FORTUIN-KASTELEYN MEASURE IN RANDOM
MEDIA
Marc Wouts (PMA)
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
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5582. THE RATE OF CONVERGENCE OF EULER APPROXIMATIONS FOR SOLUTIONS
OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN
MOTION
Yuliya Mishura and Georgiy Shevchenko
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
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5583. SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURES AND
RECURSIVE ESTIMATION
N. Lazrieva and T. Sharia and T. Toronjadze
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
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5584. ANTICIPATED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
Shige Peng and Zhe Yang
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
---------------------------------------------------------------
5585. REVESIBILITY OF CHORDAL SLE
Dapeng Zhan
We prove that the chordal SLE$_\kappa$ trace is reversible for
$\kappa\in(0,4]$.
http://arxiv.org/abs/0705.1852
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5586. ON LEVEL CROSSINGS FOR A GENERAL CLASS OF PIECEWISE-
DETERMINISTIC MARKOV PROCESSES
K. A. Borovkov and G. Last
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
---------------------------------------------------------------
5587. A CRITERION FOR TRANSIENCE OF MULTIDIMENSIONAL BRANCHING RANDOM
WALK IN RANDOM ENVIRONMENT
Sebastian M\"uller
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
---------------------------------------------------------------
5588. NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS
Marius Junge and Quanhua Xu
We show norm estimates for the sum of independent random variables in
noncommutative $L_p$-spaces for $1<p<\infty$ following our previous
work. These
estimates generalize the classical Rosenthal inequality in the
commutative
case. Among applications, we derive an equivalence for the $p$-norm
of the
singular values of a random matrix with independent entries, and
characterize
those symmetric subspaces and unitary ideals which can be realized as
subspaces
of a noncommutative $L_p$ for $2<p<\infty$.
http://arxiv.org/abs/0705.1952
---------------------------------------------------------------
5589. ENERGY OF ZEROS OF RANDOM SECTIONS ON RIEMANN SURFACE
Qi Zhong
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
---------------------------------------------------------------
5590. OPTIMAL QUANTIZATION FOR THE PRICING OF SWING OPTIONS
Olivier Aj Bardou (GDF-RDD) and Sandrine Bouthemy (GDF-RDD) and
Gilles Pag\`es (PMA)
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
---------------------------------------------------------------
5591. A TREE APPROACH TO $P$-VARIATION AND TO INTEGRATION
Jean Picard
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
---------------------------------------------------------------
5592. PERCOLATION CROSSING FORMULAS AND CONFORMAL FIELD THEORY
Jacob J. H. Simmons and Peter Kleban and and Robert M. Ziff
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
---------------------------------------------------------------
5593. BROWNIAN MOTION, "DIVERSE AND UNDULATING"
Bertrand Duplantier
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
---------------------------------------------------------------
5594. PREDICTABILITY, ENTROPY AND INFORMATION OF INFINITE
TRANSFORMATIONS
Jon. Aaronson and Kyewon Koh Park
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
---------------------------------------------------------------
5595. ON THE CONVERGENCE TO EQUILIBRIUM OF KAC'S RANDOM WALK ON MATRICES
Roberto I. Oliveira
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
---------------------------------------------------------------
5596. ON RANDOMIZED STOPPING
David \v{S}i\v{s}ka and Istv\'an Gy\"ongy
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
---------------------------------------------------------------
5597. CADLAG CURVES OF SLE DRIVEN BY LEVY PROCESSES
Qingyang Guan
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
---------------------------------------------------------------
5598. A NOTE ON THE DIFFUSIVITY OF FINITE-RANGE ASYMMETRIC EXCLUSION
PROCESSES ON Z
Jeremy Quastel and Benedek Valko
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
---------------------------------------------------------------
5599. ON THE FREEZING OF VARIABLES IN RANDOM CONSTRAINT SATISFACTION
PROBLEMS
Guilhem Semerjian
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
---------------------------------------------------------------
5600. NONCOLLIDING BROWNIAN MOTION AND DETERMINANTAL PROCESSES
Makoto Katori and Hideki Tanemura
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
---------------------------------------------------------------
5601. ON ASYMPTOTIC PROXIMITY OF DISTRIBUTIONS
Youri Davydov and Vladimir Rotar
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
---------------------------------------------------------------
5602. REAL ZEROS AND PARTITIONS WITHOUT SINGLETON BLOCKS
Miklos Bona
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
---------------------------------------------------------------
5603. ON THE REPRODUCING KERNEL HILBERT SPACES ASSOCIATED WITH THE
FRACTIONAL AND BI-FRACTIONAL BROWNIAN MOTIONS
Daniel Alpay and David Levanony
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
---------------------------------------------------------------
5604. OPTIMAL STOPPING WITH RANK-DEPENDENT LOSS
Alexander V. Gnedin
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
---------------------------------------------------------------
5605. A CONDITIONAL 0-1 LAW FOR THE SYMMETRIC SIGMA-FIELD
Patrizia Berti and Pietro Rigo
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
---------------------------------------------------------------
5606. STATISTICS OF THE NUMBER OF ZERO CROSSINGS : FROM RANDOM
POLYNOMIALS TO DIFFUSION EQUATION
Gregory Schehr and Satya N. Majumdar
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
---------------------------------------------------------------
5607. ELEMENTARY PROOF FOR ASYMPTOTICS OF LARGE HAAR-DISTRIBUTED
UNITARY MATRICES
Christian Mastrodonato and Roderich Tumulka
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
---------------------------------------------------------------
5608. REDUCTION AND RECONSTRUCTION OF SYMMETRIC STOCHASTIC
DIFFERENTIAL EQUATIONS
Joan-Andreu L\'azaro-Cam\'{\i} and Juan-Pablo Ortega
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
---------------------------------------------------------------
5609. SMALL TIME EDGEWORTH-TYPE EXPANSIONS FOR WEAKLY CONVERGENT
NONHOMOGENEOUS MARKOV CHAINS
Valentin Konakov and Enno Mammen
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
---------------------------------------------------------------
5610. DIFFERENTIABLE PERTURBATIONS OF ORNSTEIN-UHLENBECK OPERATORS
Luigi Manca
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
---------------------------------------------------------------
5611. DIFFUSION CONSTANTS AND MARTINGALES FOR SENILE RANDOM WALKS
Wouter Kager
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
---------------------------------------------------------------
5612. INVARIANT MEASURES FOR A STOCHASTIC KURAMOTO-SIVASHINKY EQUATION
B. Ferrario
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
---------------------------------------------------------------
5613. A FUNCTIONAL LIMIT THEOREM FOR A 2D-RANDOM WALK WITH DEPENDENT
MARGINALS
Nadine Guillotin-Plantard (ICJ) and Arnaud Le Ny (LM-Orsay)
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
---------------------------------------------------------------
5614. THE TWO-PARAMETER POISSON-DIRICHLET POINT PROCESS
Kenji Handa (Saga University)
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
---------------------------------------------------------------
5615. LARGE SCALE PROPERTIES OF THE IIIC FOR 2D PERCOLATION
Lincoln Chayes and Pierre Nolin
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
---------------------------------------------------------------
5616. FUNCTIONAL LIMIT THEOREMS OF MARKOV PROCESSES ON A HALF LINE
VIA PATHWISE CONVERGENCE OF EXCURSIONS
Kouji Yano
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
---------------------------------------------------------------
5617. PSEUDOPROCESSES GOVERNED BY HIGHER-ORDER FRACTIONAL
DIFFERENTIAL EQUATIONS
Luisa Beghin
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
---------------------------------------------------------------
5618. SPINAL PARTITIONS AND INVARIANCE UNDER RE-ROOTING OF CONTINUUM
RANDOM TREES
B\'en\'edicte Haas (CEREMADE) and Jim Pitman and Matthias Winkel
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
---------------------------------------------------------------
5619. BURKHOLDER'S SUBMARTINGALES FROM A STOCHASTIC CALCULUS PERSPECTIVE
Giovanni Peccati (LSTA) and Marc Yor (PMA)
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
---------------------------------------------------------------
5620. OPTIMAL CROSS HEDGING FOR INSURANCE DERIVATIVES
Stefan Ankirchner and Peter Imkeller and Alexandre Popier
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
---------------------------------------------------------------
5621. CIRCULAR LAW, EXTREME SINGULAR VALUES AND POTENTIAL THEORY
Guangming Pan and Wang Zhou
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
---------------------------------------------------------------
5622. POISSON APPROXIMATION FOR LARGE CLUSTERS IN THE SUPERCRITICAL
FK MODEL
Olivier Couronn\'e (MODAL'X)
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
---------------------------------------------------------------
5623. ON MEASURE SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
Stefan Ankirchner and Peter Imkeller and Alexandre Popier
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
---------------------------------------------------------------
5624. EXTENSION OF THE GENERALISED INDUCTIVE APPROACH TO THE LACE
EXPANSION: FULL PROOF
Remco van der Hofstad and Mark Holmes and Gordon Slade
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
---------------------------------------------------------------
5625. FAST COMPUTATION BY BLOCK PERMANENTS OF CUMULATIVE
DISTRIBUTION FUNCTIONS OF ORDER STATISTICS FROM SEVERAL POPULATIONS
Deborah H. Glueck and Anis Karimpour-Fard and Jan Mandel and Larry
Hunter and Keith E. Muller
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
---------------------------------------------------------------
5626. ALMOST SURE FUNCTIONAL CENTRAL LIMIT THEOREM FOR BALLISTIC
RANDOM WALK IN RANDOM ENVIRONMENT
Firas Rassoul-Agha and Timo Seppalainen
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
---------------------------------------------------------------
5627. A PREFERENTIAL ATTACHMENT MODEL WITH RANDOM INITIAL DEGREES
Maria Deijfen and Henri van den Esker and Remco van der Hofstad
and Gerard Hooghiemstra
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
---------------------------------------------------------------
5628. DIAMETERS IN PREFERENTIAL ATTACHMENT MODELS
Remco van der Hofstad and Gerard Hooghiemstra
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
---------------------------------------------------------------
5629. THE ODE METHOD FOR SOME SELF-INTERACTING DIFFUSIONS ON NON-
COMPACT SPACES
A. Kurtzmann
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
---------------------------------------------------------------
5630. DYNAMICAL DIOPHANTINE APPROXIMATION
Ai-Hua Fan (LAMFA) and Joerg Schmeling and Serge Troubetzkoy (CPT
and FRUMAM and IML)
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
---------------------------------------------------------------
5631. ENTIERS AL\'EATOIRES, ENSEMBLES DE SIDON, DENSIT\'E DANS LE
GROUPE DE BOHR ET ENSEMBLES D'ANALYTICIT\'E
Jean-Pierre Kahane (LM-Orsay) and Yitzhak Katznelson (U STANFORD)
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
---------------------------------------------------------------
5632. LEARNING ABOUT A CATEGORICAL LATENT VARIABLE UNDER PRIOR NEAR-
IGNORANCE
Alberto Piatti and Marco Zaffalon and Fabio Trojani and Marcus Hutter
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
---------------------------------------------------------------
5633. UTILITY MAXIMIZATION WITH A STOCHASTIC CLOCK AND AN UNBOUNDED
RANDOM ENDOWMENT
Gordan Zitkovic
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
---------------------------------------------------------------
5634. EXPLICIT BOUNDS FOR THE APPROXIMATION ERROR IN BENFORD'S LAW
Lutz Duembgen and Christoph Leuenberger
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
---------------------------------------------------------------
5635. PATTERN THEOREMS, RATIO LIMIT THEOREMS AND GUMBEL MAXIMAL
CLUSTERS FOR RANDOM FIELDS
Remco van der Hofstad and Wouter Kager
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
---------------------------------------------------------------
5636. MOLECULAR SPIDERS IN ONE DIMENSION
Tibor Antal and P. L. Krapivsky and and Kirone Mallick
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
---------------------------------------------------------------
5637. MOLECULAR SPIDERS WITH MEMORY
Tibor Antal and P. L. Krapivsky
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
---------------------------------------------------------------
5638. ON THE SMALL BALL INEQUALITY IN ALL DIMENSIONS
Dmitry Bilyk and Michael Lacey and Armen Vagharshakyan
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
---------------------------------------------------------------
5639. A FILTERED VERSION OF THE BIPOLAR THEOREM OF BRANNATH AND
SCHACHERMAYER
Gordan Zitkovic
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
---------------------------------------------------------------
5640. OPTIMAL CONSUMPTION FROM INVESTMENT AND RANDOM ENDOWMENT IN
INCOMPLETE SEMIMARTINGALE MARKETS
Ioannis Karatzas and Gordan Zitkovic
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
---------------------------------------------------------------
5641. THE LARGEST EIGENVALUE OF FINITE RANK DEFORMATION OF LARGE WIGNER
Mireille Capitaine (LSProba) and Catherine Donati-Martin (PMA) and
Delphine F\'eral (LSProba)
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
---------------------------------------------------------------
5642. A CLT FOR INFORMATION-THEORETIC STATISTICS OF GRAM RANDOM
MATRICES WITH A GIVEN VARIANCE PROFILE
Walid Hachem (LTCI) and Philippe Loubaton (IGM-LabInfo) and Jamal
Najim (LTCI)
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
---------------------------------------------------------------
5643. A NON COMMUTATIVE SEWING LEMMA
Denis Feyel and Arnaud De La Pradelle (IMJ) and Gabriel Mokobodzki
(IMJ)
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
---------------------------------------------------------------
5644. ASYMPTOTIC RESULTS ON THE LENGTH OF COALESCENT TREES
Jean-Fran\c{c}ois Delmas (CERMICS) and Jean-St\'ephane Dhersin
(MAP5) and Arno Siri-Jegousse (MAP5)
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
---------------------------------------------------------------
5645. THE M-ESTIMATOR IN A MULTI-PHASE RANDOM NONLINEAR MODEL
Gabriela Ciuperca
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
---------------------------------------------------------------
5646. RANDOM SPATIAL GROWTH WITH PARALYZING OBSTACLES
J. van den Berg and Y. Peres and V. Sidoravicius and M.E. Vares
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
---------------------------------------------------------------
5647. THE CHARACTERISTIC POLYNOMIAL OF A RANDOM UNITARY MATRIX: A
PROBABILISTIC APPROACH
Paul Bourgade and Chris Hughes and Ashkan Nikeghbali and Marc Yor
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
---------------------------------------------------------------
5648. RENEWAL CONVERGENCE RATES AND CORRELATION DECAY FOR HOMOGENEOUS
PINNING MODELS
Giambattista Giacomin
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
---------------------------------------------------------------
5649. ENDS IN UNIFORM SPANNING FORESTS
Russell Lyons and Benjamin J. Morris and Oded Schramm
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
---------------------------------------------------------------
5650. ASYMPTOTIC BEHAVIOR OF TOTAL TIMES FOR JOBS THAT MUST START
OVER IF A FAILURE OCCURS
Soeren Asmussen and Pierre Fiorini and Lester Lipsky and Tomasz
Rolski and Robert Sheahan
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
---------------------------------------------------------------
5651. REGULARITY OF HARMONIC FUNCTIONS FOR ANISOTROPIC FRACTIONAL
LAPLACIAN
Pawe{\l} Sztonyk
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
---------------------------------------------------------------
5652. FINANCIAL EQUILIBRIA IN THE SEMIMARTINGALE SETTING: COMPLETE
MARKETS AND MARKETS WITH WITHDRAWAL CONSTRAINTS
Gordan Zitkovic
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
---------------------------------------------------------------
5653. ON THE SEMIMARTINGALE PROPERTY VIA BOUNDED LOGARITHMIC UTILITY
Kasper Larsen and Gordan Zitkovic
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
---------------------------------------------------------------
5654. STABILITY OF UTILITY-MAXIMIZATION IN INCOMPLETE MARKETS
Kasper Larsen and Gordan Zitkovic
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
---------------------------------------------------------------
5655. OPTIMAL INVESTMENT WITH AN UNBOUNDED RANDOM ENDOWMENT WHEN THE
WEALTH CAN BECOME NEGATIVE
Mark Owen and Gordan Zitkovic
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
---------------------------------------------------------------
5656. MAXIMIZING THE GROWTH RATE UNDER RISK CONSTRAINTS
Traian A. Pirvu and Gordan Zitkovic
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
---------------------------------------------------------------
5657. STABILITY OF THE UTILITY MAXIMIZATION PROBLEM WITH RANDOM
ENDOWMENT IN INCOMPLETE MARKETS
Constantinos Kardaras and Gordan Zitkovic
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
---------------------------------------------------------------
5658. THE ORDER OF THE GIANT COMPONENT OF RANDOM HYPERGRAPHS
Michael Behrisch and Amin Coja-Oghlan and Mihyun Kang
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 $(d-1)^{-1}+\eps<c<\infty$. The proof
relies on
a new, purely probabilistic approach, and is based on Stein's method
as well as
exposing the edges of $H_d(n,p)$ in several rounds.
http://arxiv.org/abs/0706.0496
---------------------------------------------------------------
5659. LOCAL LIMIT THEOREMS AND NUMBER OF CONNECTED HYPERGRAPHS
Michael Behrisch and Amin Coja-Oghlan and Mihyun Kang
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
---------------------------------------------------------------
5660. QUEUES WITH HETEROGENEOUS SERVERS AND UNINFORMED CUSTOMERS: WHO
WORKS THE MOST?
Fabricio Bandeira Cabral
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
---------------------------------------------------------------
5661. ON THE GEOMETRY OF GENERALIZED GAUSSIAN DISTRIBUTIONS
Attila Andai
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
---------------------------------------------------------------
5662. AN EXTENSION OF THE INDUCTIVE APPROACH TO THE LACE EXPANSION
Remco van der Hofstad and Mark Holmes and Gordon Slade
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
---------------------------------------------------------------
5663. AN EXPANSION FOR SELF-INTERACTING RANDOM WALKS
Remco van der Hofstad and Mark Holmes
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
---------------------------------------------------------------
5664. A CHARACTERIZATION OF THE RIESZ DISTRIBUTION
Abdelhamid Hassairi and Sallouha Lajmi and Raoudha Zine
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
---------------------------------------------------------------
5665. UNIQUENESS OF POLYNOMIAL CANONICAL REPRESENTATIONS
Manuel Lladser
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
---------------------------------------------------------------
5666. MULTIPLICATION OF FREE RANDOM VARIABLES AND THE S-TRANSFORM:
THE CASE OF VANISHING MEAN
N. Raj Rao and Roland Speicher
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
---------------------------------------------------------------
5667. OPERATOR SPACE LP EMBEDDING THEORY I
Marius Junge and Javier Parcet
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
---------------------------------------------------------------
5668. THE ZERO-ONE LAW FOR PLANAR RANDOM WALKS IN I.I.D. RANDOM
ENVIRONMENTS REVISITED
Martin P.W. Zerner
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
---------------------------------------------------------------
5669. ON THE LOWER BOUND OF THE SPECTRAL NORM OF SYMMETRIC RANDOM
MATRICES WITH INDEPENDENT ENTRIES
Sandrine Peche and Alexander Soshnikov
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
---------------------------------------------------------------
5670. LIMIT LAWS FOR K-COVERAGE OF PATHS BY A MARKOV-BOOLEAN MODEL
Srikanth K. Iyer and D. Manjunath and D. Yogeshwaran
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
---------------------------------------------------------------
5671. MEASURE-VALUED STOCHASTIC RECURRENCES AND THE STABILITY OF QUEUES
Pascal Moyal
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
---------------------------------------------------------------
5672. ALMOST SURE CONVERGENCE OF RANDOMLY TRUNCATED STOCHASTIC
ALGORITHMS UNDER VERIFIABLE CONDITIONS
J\'er\^ome Lelong (CERMICS)
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
---------------------------------------------------------------
5673. MAXIMAL PROBABILITIES OF CONVOLUTION POWERS OF DISCRETE
UNIFORM DISTRIBUTIONS
Lutz Mattner and Bero Roos
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
---------------------------------------------------------------
5674. TWO MULTIVARIATE CENTRAL LIMIT THEOREMS
Elizabeth Meckes
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
---------------------------------------------------------------
5675. ON MAGIC FACTORS AND THE CONSTRUCTION OF EXAMPLES WITH SHARP
RATES IN STEIN'S METHOD
Adrian R\"ollin
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
---------------------------------------------------------------
5676. JANOSSY DENSITIES FOR UNITARY ENSEMBLES AT THE SPECTRAL EDGE
Brian Rider and Xin Zhou
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
---------------------------------------------------------------
5677. INFINITE-DIMENSIONAL DIFFUSIONS AS LIMITS OF RANDOM WALKS ON
PARTITIONS
Alexei Borodin and Grigori Olshanski
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
---------------------------------------------------------------
5678. A ONE DIMENSIONAL ANALYSIS OF TURBULENCE AND ITS INTERMITTENCE
FOR THE D-DIMENSIONAL STOCHASTIC BURGERS EQUATION
A. D. Neate and A. Truman
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
---------------------------------------------------------------
5679. INTERMITTENCY ON CATALYSTS
J. Gaertner and F. den Hollander and G. Maillard
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
---------------------------------------------------------------
5680. STEIN'S METHOD AND POISSON PROCESS APPROXIMATION FOR A CLASS
OF WASSERSTEIN METRICS
Dominic Schuhmacher
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
---------------------------------------------------------------
5681. A ONE DIMENSIONAL ANALYSIS OF SINGULARITIES AND TURBULENCE FOR
THE STOCHASTIC BURGERS EQUATION IN D-DIMENSIONS
A. D. Neate and A. Truman
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
---------------------------------------------------------------
5682. TWO-SIDED OPTIMAL BOUNDS FOR GREEN FUNCTION OF HALF-SPACES FOR
RELATIVISTIC $\ALPHA$-STABLE PROCESS
Tomasz Grzywny and Micha{\l}Ryznar
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
---------------------------------------------------------------
5683. ADDITIVE REGRESSION MODEL FOR CONTINUOUS TIME PROCESSES
Mohammed Debbarh and Bertrand Maillot
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
---------------------------------------------------------------
5684. SOME UNIFORM LIMIT RESULTS IN ADDITIVE REGRESSION MODEL
Mohammed Debbarh
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
---------------------------------------------------------------
5685. ON THE THRESHOLD FOR K-REGULAR SUBGRAPHS OF RANDOM GRAPHS
Pawel Pralat and Jacques Verstraete and Nicholas Wormald
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
---------------------------------------------------------------
5686. RATE OF CONVERGENCE OF SPACE TIME APPROXIMATIONS FOR
STOCHASTIC EVOLUTION EQUATIONS
Istvan Gy\"ongy and Annie Millet (PMA and Ces and Matisse and Samos)
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
---------------------------------------------------------------
5687. REGULARIZATION BY FREE ADDITIVE CONVOLUTION, SQUARE AND
RECTANGULAR CASES
Serban Belinschi and Florent Benaych-Georges (PMA) and Alice
Guionnet (UMPA-ENSL)
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
---------------------------------------------------------------
5688. THE LAW OF THE SUPREMUM OF A STABLE L\'EVY PROCESS WITH NO
NEGATIVE JUMPS
Violetta Bernyk and Robert C. Dalang and Goran Peskir
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
---------------------------------------------------------------
5689. A DYNAMICAL LAW OF LARGE NUMBERS
Davar Khoshnevisan and David A. Levin and Pedro J. Mendez-Hernandez
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
---------------------------------------------------------------
5690. RANDOMLY COLORING PLANAR GRAPHS WITH FEWER COLORS THAN THE
MAXIMUM DEGREE
Thomas P. Hayes and Juan C. Vera and and Eric Vigoda
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
---------------------------------------------------------------
5691. POWER-FREE VALUES, REPULSION BETWEEN POINTS, DIFFERING BELIEFS
AND THE EXISTENCE OF ERROR
Harald Andres Helfgott
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
---------------------------------------------------------------
5692. WETTING OF GRADIENT FIELDS: PATHWISE ESTIMATES
Yvan Velenik
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
---------------------------------------------------------------
5693. T-WISE INDEPENDENCE WITH LOCAL DEPENDENCIES
Ronen Gradwohl and Amir Yehudayoff
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
---------------------------------------------------------------
5694. THE EXPLICIT CHAOTIC REPRESENTATION OF THE POWERS OF INCREMENTS
OF LEVY PROCESSES
Wing Yan Yip and David Stephens and Sofia Olhede
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
---------------------------------------------------------------
5695. THE TWO POSSIBLE VALUES OF THE CHROMATIC NUMBER OF A RANDOM GRAPH
Dimitris Achlioptas and Assaf Naor
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
---------------------------------------------------------------
5696. CONCENTRATION OF THE SPECTRAL MEASURE FOR LARGE RANDOM MATRICES
WITH STABLE ENTRIES
Christian Houdr\'e and Hua Xu
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
---------------------------------------------------------------
5697. EXTREME-VALUE ANALYSIS OF STANDARDIZED GAUSSIAN INCREMENTS
Zakhar Kabluchko
Let $X_i$ be i.i.d. standard gaussian variables. Let $S_n=X_1+...+X_n
$ and
$$L_n=\max_{0\leq i<j\leq n}\frac{S_j-S_i}{\sqrt{j-i}}.$$ We show
that the
distribution of $L_n$, appropriately normalized, converges as $n\to
\infty$ to
the Gumbel distribution. We also prove a version of the above result
for the
Brownian motion.
http://arxiv.org/abs/0706.1849
---------------------------------------------------------------
5698. A MATHEMATICAL MODEL FOR A COPOLYMER IN AN EMULSION
F. den Hollander and N. Petrelis
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
---------------------------------------------------------------
5699. SURVIVAL PROBABILITIES FOR N-ARY SUBTREES ON A GALTON-WATSON
FAMILY TREE
Ljuben Mutafchiev
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
---------------------------------------------------------------
5700. GENERALIZED CONTINUOUS-TIME RANDOM WALKS (CTRW), SUBORDINATION
BY HITTING TIMES AND FRACTIONAL DYNAMICS
Vassili N. Kolokoltsov
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
---------------------------------------------------------------
5701. ON PATHWISE UNIQUENESS FOR REFLECTING BROWNIAN MOTION IN $C^{1+
\GAMMA}$ DOMAINS
Richard F. Bass and Krzysztof Burdzy
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
---------------------------------------------------------------
5702. A TRANSIENT MARKOV CHAIN WITH FINITELY MANY CUTPOINTS
Nicholas James and Russell Lyons and Yuval Peres
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
---------------------------------------------------------------
5703. MARCUS-LUSHNIKOV PROCESSES, SMOLUCHOWSKI'S AND FLORY'S MODELS
Nicolas Fournier and Philippe Laurencot
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
---------------------------------------------------------------
5704. THE SIZE OF THE LARGEST COMPONENT BELOW PHASE TRANSITION IN
INHOMOGENEOUS RANDOM GRAPHS
T. S. Turova
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
---------------------------------------------------------------
5705. UNIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES
Gerard Ben Arous and Anton Bovier and Jiri Cerny
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
---------------------------------------------------------------
5706. ON THE STRONG CHROMATIC NUMBER OF RANDOM GRAPHS
Po-Shen Loh and Benny Sudakov
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
---------------------------------------------------------------
5707. THE FIRST-PASSAGE AREA FOR DRIFTED BROWNIAN MOTION AND THE
MOMENTS OF THE AIRY DISTRIBUTION
Michael J. Kearney and Satya N. Majumdar and and Richard J. Martin
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
---------------------------------------------------------------
5708. TOWARDS THE DISTRIBUTION OF THE SIZE OF A LARGEST PLANAR
MATCHING AND LARGEST PLANAR SUBGRAPH IN RANDOM BIPARTITE GRAPHS
Marcos Kiwi and Martin Loebl
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
---------------------------------------------------------------
5709. STOCHASTIC PARABOLIC EQUATIONS OF FULL SECOND ORDER
S. V. Lototsky and B. L. Rozovskii
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
---------------------------------------------------------------
5710. FROM RANDOM PROCESSES TO GENERALIZED FIELDS: A UNIFIED APPROACH
TO STOCHASTIC INTEGRATION
S. V. Lototsky and K. Stemmann
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
---------------------------------------------------------------
5711. ANALYSIS OF THE EXPECTED NUMBER OF BIT COMPARISONS REQUIRED BY
QUICKSELECT
James Allen Fill and Take Nakama
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
---------------------------------------------------------------
5712. ON THE GENEALOGY ON CONDITIONED STABLE L\'EVY FOREST
Loic Chaumont (LAREMA) and Juan Carlos Pardo Millan (PMA)
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
---------------------------------------------------------------
5713. OPTIMAL POINTWISE APPROXIMATION OF STOCHASTIC DIFFERENTIAL
EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION
Andreas Neuenkirch
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
---------------------------------------------------------------
5714. SOME APPLICATIONS OF THE MELLIN TRANSFORM TO BRANCHING PROCESSES
Wolfgang P. Angerer
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
---------------------------------------------------------------
5715. ON THE NUMBER OF NODAL DOMAINS OF RANDOM SPHERICAL HARMONICS
Fedor Nazarov and Mikhail Sodin
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
---------------------------------------------------------------
5716. AN EXTENSION OF A BOURGAIN--LINDENSTRAUSS--MILMAN INEQUALITY
Omer Friedland and Sasha Sodin
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
---------------------------------------------------------------
5717. DIOPHANTINE BOUNDS ON THE CONCENTRATION FUNCTION
Omer Friedland and Sasha Sodin
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
---------------------------------------------------------------
5718. SMALL DEVIATION PROBABILITY VIA CHAINING
Frank Aurzada and Mikhail Lifshits
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
---------------------------------------------------------------
5719. ON CERTAIN BOUNDS FOR FIRST-CROSSING-TIME PROBABILITIES OF A
JUMP-DIFFUSION PROCESS
Antonio Di Crescenzo and Elvira Di Nardo and Luigi M. Ricciardi
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
---------------------------------------------------------------
5720. SCALED ENTROPY OF FILTRATIONS OF $\SIGMA$-FIELDS
A.Vershik and A.Gorbulsky
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
---------------------------------------------------------------
5721. ASYMPTOTICS OF THE ALLELE FREQUENCY SPECTRUM ASSOCIATED WITH
THE BOLTHAUSEN-SZNITMAN COALESCENT
Anne-Laure Basdevant and Christina Goldschmidt
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
---------------------------------------------------------------
5722. GROUP REPRESENTATIONS AND HIGH-RESOLUTION CENTRAL LIMIT
THEOREMS FOR SUBORDINATED SPHERICAL RANDOM FIELDS
Domenico Marinucci (DIPMAT) and Giovanni Peccati (LSTA)
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
---------------------------------------------------------------
5723. QUASI-INVARIANCE PROPERTIES OF A CLASS OF SUBORDINATORS
Max-K. Von Renesse and Marc Yor (PMA) and Lorenzo Zambotti (PMA)
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
---------------------------------------------------------------
5724. IMPLICIT FUNCTION DENSITY COMPUTATION
Kerry Michael Soileau
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
---------------------------------------------------------------
5725. THE CHARACTERISTIC POLYNOMIAL ON COMPACT GROUPS WITH HAAR
MEASURE : SOME EQUALITIES IN LAW
Paul Bourgade and Ashkan Nikeghbali and Alain Rouault
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
---------------------------------------------------------------
5726. POLYNOMIAL RATE CONVERGENCE TO AN INVARIANT MEASURE FOR THE
CONTINUUM TIME LIMIT OF THE MINORITY GAME
Matteo Ortisi
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
---------------------------------------------------------------
5727. REMARKS ON THE DENSITY OF THE LAW OF THE OCCUPATION TIME FOR
BESSEL BRIDGES AND STABLE EXCURSIONS
Kouji Yano and Yuko Yano
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
---------------------------------------------------------------
5728. TAU-FUNCTION OF DISCRETE ISOMONODROMY TRANSFORMATIONS AND
PROBABILITY
D. Arinkin and A. Borodin
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
---------------------------------------------------------------
5729. A MODEL FOR COUNTERPARTY RISK WITH GEOMETRIC ATTENUATION EFFECT
AND THE VALUATION OF CDS
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)
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
---------------------------------------------------------------
5730. RADIUS AND PROFILE OF RANDOM PLANAR MAPS WITH FACES OF
ARBITRARY DEGREES
Gr\'egory Miermont (LM-Orsay) and Mathilde Weill (DMA)
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
---------------------------------------------------------------
5731. PROBABILISTIC REPRESENTATIONS OF SOLUTIONS OF THE FORWARD
EQUATIONS
B. Rajeev and S. Thangavelu
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
---------------------------------------------------------------
5732. SOLVING SPDES DRIVEN BY COLORED NOISE: A CHAOS APPROACH
S. V. Lototsky and K. Stemmann
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
---------------------------------------------------------------
5733. FLOCKING IN NOISY ENVIRONMENTS
Felipe Cucker and Ernesto Mordecki
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
---------------------------------------------------------------
5734. FRACTAL AND INCREASING PATHS CHARACTERIZATIONS OF THE SET-
INDEXED FRACTIONAL BROWNIAN MOTION
Erick Herbin and Ely Merzbach
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<H<1/2
is studied. When the indexing collection is totally ordered, the
sifBm can be
defined for 0<H<1.
The set-indexed fractional Brownian motion is proved to statisfy a
strenghtened definition of increment stationarity. This new
definition for
stationarity property allows to get a complete characterization of
this process
by its fractal properties. The sifBm is the only set-indexed Gaussian
process
which is self-similar and has stationary increments.
http://arxiv.org/abs/0706.3472
---------------------------------------------------------------
5735. SPIN GLASSES AND STEIN'S METHOD
Sourav Chatterjee
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
---------------------------------------------------------------
5736. A PHASE TRANSITION BEHAVIOR FOR BROWNIAN MOTIONS INTERACTING
THROUGH THEIR RANKS
Sourav Chatterjee and Soumik Pal
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
---------------------------------------------------------------
5737. A PROBLEM IN LAST-PASSAGE PERCOLATION
Harry Kesten and Vladas Sidoravicius
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
---------------------------------------------------------------
5738. TRANSLATION-INVARIANCE OF TWO-DIMENSIONAL GIBBSIAN POINT PROCESSES
Thomas Richthammer
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
---------------------------------------------------------------
5739. NO PHASE TRANSITION FOR GAUSSIAN FIELDS WITH BOUNDED SPINS
Pablo A. Ferrari and Sebastian P. Grynberg
Let a<b, \Omega=[a,b]^{\Z^d} and H be the (formal) Hamiltonian
defined on
\Omega by
H(\eta) = \frac12 \sum_{x,y\in\Z^d} J(x-y) (\eta(x)-\eta(y))^2 where
J:\Z^d\to\R is any summable non-negative symmetric function (J(x)\ge
0 for all
x\in\Z^d, \sum_x J(x)<\infty and J(x)=J(-x)). We prove that there is
a unique
Gibbs measure on \Omega associated to H. The result is a consequence
of the
fact that the corresponding Gibbs sampler is attractive and has a unique
invariant measure.
http://arxiv.org/abs/0706.3714
---------------------------------------------------------------
5740. ON SYSTEMATIC SCAN FOR SAMPLING H-COLOURINGS OF THE PATH
Kasper Pedersen
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
---------------------------------------------------------------
5741. RANDOM SAMPLING OF ENTIRE FUNCTIONS OF EXPONENTIAL TYPE IN
SEVERAL VARIABLES
Karlheinz Gr\"ochenig and Richard F. Bass
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
---------------------------------------------------------------
5742. GRAPHICAL REPRESENTATION OF SOME DUALITY RELATIONS IN
STOCHASTIC POPULATION MODELS
Roland Alkemper and Martin Hutzenthaler
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
---------------------------------------------------------------
5743. ON PROBABILITIES FOR SEPARATING SETS OF ORDER STATISTICS
Deborah H. Glueck and Anis Karimpour-Fard and Jan Mandel and Keith
E. Muller
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
---------------------------------------------------------------
5744. CONVEX GEOMETRIES IN K-SAT PROBLEMS
Federico Ardila and Elitza Maneva
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
---------------------------------------------------------------
5745. STOCHASTIC CONTROL PROBLEMS FOR SYSTEMS DRIVEN BY NORMAL
MARTINGALES
Rainer Buckdahn and Jin Ma and Catherine Rainer
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
---------------------------------------------------------------
5746. STOPPED DIFFUSION PROCESSES: OVERSHOOTS AND BOUNDARY CORRECTION
Emmanuel Gobet (LJK) and St\'ephane Menozzi (PMA)
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
---------------------------------------------------------------
5747. AN EFFECTIVE CRITERION AND A NEW EXAMPLE FOR BALLISTIC
DIFFUSIONS IN RANDOM ENVIRONMENT
Laurent Goergen
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
---------------------------------------------------------------
5748. EULER WALK ON A CAYLEY TREE
A.E. Patrick
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
---------------------------------------------------------------
5749. CHERNOFF'S THEOREM FOR EVOLUTION FAMILIES
Evelina Shamarova
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
---------------------------------------------------------------
5750. ADAPTIVE DYNAMICS IN LOGISTIC BRANCHING POPULATIONS
Nicolas Champagnat and Amaury Lambert (FESE)
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
---------------------------------------------------------------
5751. HARMONIC ANALYSIS OF ADDITIVE LEVY PROCESSES
Davar Khoshnevisan and Yimin Xiao
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
---------------------------------------------------------------
5752. TRANSPORTATION-INFORMATION INEQUALITIES FOR MARKOV PROCESSES
Arnaud Guillin (LATP) and Christian Leonard (CMAP and MODAL'X) and
Liming Wu and Nian Yao
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
---------------------------------------------------------------
5753. SUBSAMPLING NEEDLET COEFFICIENTS ON THE SPHERE
Paolo Baldi and Gerard Kerkyacharian and Domenico Marinucci and
Dominique Picard
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
---------------------------------------------------------------
5754. RESILIENCE OF GRAPHS
Benny Sudakov and Van Vu
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
---------------------------------------------------------------
5755. MULTI-DIMENSIONAL BSDE WITH OBLIQUE REFLECTION AND OPTIMAL
SWITCHING
Ying Hu (IRMAR) and Shanjian Tang (School of Mathematical Sciences)
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
---------------------------------------------------------------
5756. TIGHTNESS OF VOTER MODEL INTERFACES
Anja Sturm and Jan M. Swart
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
-----------------------------
Stefano Iacus
IMS Groups Editor
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