[Pas] Probability Abstracts 88
pas at www.economia.unimi.it
pas at www.economia.unimi.it
Thu Sep 1 17:51:54 CEST 2005
September 1, 2005
Letter 88
Probability Abstract Service
( http://www.economia.unimi.it/PAS )
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3516. OPTIMAL LONG TERM INVESTMENT MODEL WITH MEMORY
Akihiko Inoue and Yumiharu Nakano
We consider an investment model with memory in which the prices of n
risky
assets are driven by an n-dimensional Gaussian process with stationary
increments that is different from Brownian motion. The driving
process consists
of n independent components, and each component is characterized by two
parameters describing the memory. For the model, we explicitly solve the
problem of maximizing the expected growth rate as well as that of
maximizing
the probability of overperforming a given benchmark.
http://front.math.ucdavis.edu/math.PR/0506621
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3517. GRADIENT BOUNDS FOR SOLUTIONS OF ELLIPTIC AND PARABOLIC EQUATIONS
Vladimir I. Bogachev and Giuseppe Da Prato and Michael R\"ockner
and Zeev Sobol
Let $L$ be a second order elliptic operator on $R^d$ with a constant
diffusion matrix and a dissipative (in a weak sense) drift $b \in L^p_
{loc}$
with some $p>d$.
We assume that $L$ possesses a Lyapunov function, but no local
boundedness of
$b$ is assumed. It is known that then there exists a unique
probability measure
$\mu$ satisfying the equation $L^*\mu=0$ and that the closure of $L$ in
$L^1(\mu)$ generates a Markov semigroup $\{T_t\}_{t\ge 0}$ with the
resolvent
$\{G_\lambda\}_{\lambda > 0}$.
We prove that, for any Lipschitzian function $f\in L^1(\mu)$ and all
$t,\lambda>0$, the functions $T_tf$ and $G_\lambda f$ are
Lipschitzian and
|\nabla T_tf(x)| \leq T_t|\nabla f|(x) and |\nabla G_\lambda f(x)| \leq
\frac{1}{\lambda} G_\lambda |\nabla f|(x).
An analogous result is proved in the parabolic case.
http://front.math.ucdavis.edu/math.PR/0507079
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3518. THE KINETIC LIMIT OF A SYSTEM OF COAGULATING PLANAR BROWNIAN
PARTICLES
Alan Hammond and Fraydoun Rezakhanlou
We study a model of mass-bearing coagulating planar Brownian particles.
Coagulation is prone to occur when two particles become within a
distance of
order $\epsilon$. We assume that the initial number of particles is
of the
order of $| \log \epsilon |. Under suitable assumptions on the initial
distribution of particles and the microscopic coagulation
propensities, we show
that the macroscopic particle densities satisfy a Smoluchowski-type
equation.
http://front.math.ucdavis.edu/math.PR/0507522
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3519. WEAK CONVERGENCE OF THE SCALED MEDIAN OF INDEPENDENT BROWNIAN
MOTIONS
Jason Swanson
We consider the median of n independent Brownian motions, and show
that this
process, when properly scaled, converges weakly to a centered
Gaussian process.
The chief difficulty is establishing tightness, which is proved
through direct
estimates on the increments of the median process. An explicit
formula is given
for the covariance function of the limit process. The limit process
is also
shown to be Holder continuous with exponent gamma for all gamma < 1/4.
http://front.math.ucdavis.edu/math.PR/0507524
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3520. CONCENTRATION INEQUALITIES WITH EXCHANGEABLE PAIRS (PH.D. THESIS)
Sourav Chatterjee
The purpose of this dissertation is to introduce a version of Stein's
method
of exchangeable pairs to solve problems in measure concentration. We
specifically target systems of dependent random variables, since that
is where
the power of Stein's method is fully realized. Because the theory is
quite
abstract, we have tried to put in as many examples as possible. Some
of the
highlighted applications are as follows: (a) We shall find an easily
verifiable
condition under which a popular heuristic technique originating from
physics,
known as the ``mean field equations'' method, is valid. No such
condition is
currently known. (b) We shall present a way of using couplings to derive
concentration inequalities. Although couplings are routinely used for
proving
decay of correlations, no method for using couplings to derive
concentration
bounds is available in the literature. This will be used to obtain (c)
concentration inequalities with explicit constants under Dobrushin's
condition
of weak dependence. (d) We shall give a method for obtaining
concentration of
Haar measures using convergence rates of related random walks on
groups. Using
this technique and one of the numerous available results about rates of
convergence of random walks, we will then prove (e) a quantitative
version of
Voiculescu's celebrated connection between random matrix theory and free
probability.
http://front.math.ucdavis.edu/math.PR/0507526
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3521. A GENERALIZATION OF STATIONARY AR(1) SCHEMES
S Satheesh and E Sandhya and S Sherly
Here we develop a first order autoregressive model {Xn} that is
marginally
stationary where Xn is the sum/ extreme of k i.i.d observations. We
prove that
stationary solutions to these models are either semi- selfdecomposable/
extreme-semi-selfdecomposable or, sum/ extreme stable with respect to
Harris
distribution.
http://front.math.ucdavis.edu/math.PR/0507535
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3522. ON PATHWISE UNIQUENESS FOR STOCHASTIC HEAT EQUATIONS WITH NON-
LIPSCHITZ COEFFICIENTS
Leonid Mytnik and Edwin Perkins and Anja Sturm
We consider the existence and pathwise uniqueness of the stochastic heat
equation with a multiplicative colored noise term on IR^d for d
greater or
equal to 1. We focus on the case of non-Lipschitz noise coefficients and
singular spatial noise correlations. In the course of the proof a new
result on
Hoelder continuity of the solutions near zero is established.
http://front.math.ucdavis.edu/math.PR/0507545
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3523. THE DISTRIBUTION OF THE MINIMUM HEIGHT AMONG PIVOTAL SITES IN
CRITICAL TWO-DIMENSIONAL PERCOLATION
Gregory J. Morrow and Yu Zhang
Let L_n denote the lowest crossing of the 2n \times 2n square box B(n)
centered at the origin for critical site percolation on Z^2 or
critical site
percolation on the triangular lattice imbedded in Z^2, and denote by
Q_n the
set of pivotal sites along this crossing. On the event that a pivotal
site
exists, denote the minimum height that a pivotal site attains above
the bottom
of B(n) by M_n:= min{m:(x,-n+m)\in Q_n for some -n\le x\le n}. Else,
define M_n
= 2n. We prove that P(M_n < m) \asymp m/n, uniformly for 1\le m\le n.
This
relation extends Theorem 1 of van den Berg and Jarai (2003) who
handle the
corresponding distribution for the lowest crossing in a slightly
different
context. As a corollary we establish the asymptotic distribution of
the minimum
height of the set of cut points of a certain chordal SLE_6 in the
unit square
of C.
http://front.math.ucdavis.edu/math.PR/0507566
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3524. ON COMPLETE CHARACTERIZATION OF COEFFICIENTS OF A.E.
CONVERGING ORTHOGONAL SERIES
Adam Paszkiewicz
We characterize sequences of numbers $(a_n)$ such that $\sum_{n\geq 1}
a_n\Phi_n$ converges a.e. for any orthonormal system $(\Phi_n)$ in any
$L_2$-space. In our criterion, we use the set $B =\{\sum_{m\geq n} |
a_m|^2;
n\geq 1\}$ and its information function $$h_B(t) = -\log_3(\beta-
\alpha)$$ for
$t\in (\alpha, \beta]$, $[\alpha, \beta]\cap B =\{\alpha, \beta\}.$
http://front.math.ucdavis.edu/math.AP/0507568
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3525. LIMIT THEOREMS FOR WEIGHTED SAMPLES WITH APPLICATIONS TO
SEQUENTIAL MONTE CARLO METHODS
R. Douc (\'Ecole Polytechnique and Palaiseau) and France E.
Moulines (\'Ecole Nationale Sup\'erieure des T\'el\'ecommunications,
Paris)
In the last decade, sequential Monte-Carlo methods (SMC) emerged as a
key
tool in computational statistics. These algorithms approximate a
sequence of
distributions by a sequence of weighted empirical measures associated
to a
weighted population of particles. These particles and weights are
generated
recursively according to elementary transformations: mutation and
selection.
Examples of applications include the sequential Monte-Carlo
techniques to solve
optimal non-linear filtering problems in state-space models, molecular
simulation, genetic optimization, etc.
Despite many theoretical advances the asymptotic property of these
approximations remains of course a question of central interest. In
this paper,
we analyze sequential Monte Carlo methods from an asymptotic
perspective, that
is, we establish law of large numbers and invariance principle as the
number of
particles gets large. We introduce the concepts of "weighted sample"
consistency and asymptotic normality, and derive conditions under
which the
mutation and the selection procedure used in the sequential Monte-Carlo
build-up preserve these properties. To illustrate our findings, we
analyze SMC
algorithms to approximate the filtering distribution in state-space
models. We
show how our techniques allow to relax restrictive technical
conditions used in
previously reported works and provide grounds to analyze more
sophisticated
sequential sampling strategies.
http://front.math.ucdavis.edu/math.ST/0507042
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3526. THE CONTACT PROCESS SEEN FROM A TYPICAL INFECTED SITE
J.M. Swart
This paper considers contact processes on general lattices. Assuming
that the
expected number of infected sites grows subexponentially, it is shown
that the
configuration as seen from a typical (`Palmed') infected site at an
exponentially distributed time converges, as time tends to infinity,
to the
upper invariant law conditioned on the origin being infected. The
assumption
that the expected number of infected sites grows subexponentially is
shown to
be satisfied if the lattice has subexponential growth and the
infection rates
satisfy an exponential moment condition.
http://front.math.ucdavis.edu/math.PR/0507578
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3527. ESTIMATES OF POTENTIAL KERNEL AND HARNACK'S INEQUALITY FOR
ANISOTROPIC FRACTIONAL LAPLACIAN
Krzysztof Bogdan and Pawe{\l} Sztonyk
We characterize those homogeneous translation invariant symmetric non-
local
operators with positive maximum principle whose harmonic functions
satisfy
Harnack's inequality. We also estimate the corresponding semigroup
and the
potential kernel.
http://front.math.ucdavis.edu/math.PR/0507579
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3528. INTERNAL DIFFUSION LIMITED AGGREGATION ON DISCRETE GROUPS
HAVING EXPONENTIAL GROWTH
Sebastien Blachere and Sara Brofferio
The Internal Diffusion Limited Aggregation has been introduced by
Diaconis
and Fulton in 1991. It is a growth model defined on an infinite set and
associated to a Markov chain on this set. We focus here on sets which
are
finitely generated groups with exponential growth. We prove a shape
theorem for
the Internal DLA on such groups associated to symmetric random walks.
For that
purpose, we introduce a new distance associated to the Green
function, which
happens to have some interesting properties. In the case of
homogeneous trees,
we also get the right order for the fluctuations of that model around
its
limiting shape.
http://front.math.ucdavis.edu/math.PR/0507582
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3529. GEOMETRIC CHARACTERISATION OF INTERMITTENCY IN THE PARABOLIC
ANDERSON MODEL
J. Gaertner and W. Koenig and S. Molchanov
We consider the parabolic Anderson problem $\partial_t u =\Delta u+\xi
(x) u$
on $\R_+\times \Z^d$ with localized initial condition $u(0,x)=\delta_0
(x)$ and
random i.i.d. potential $\xi$. Under the assumption that the
distribution of
$\xi(0)$ lies in the vicinity of, or beyond, the double-exponential
distribution, we prove the following geometric characterisation of
intermittency: with probability one, as $t\to\infty$, the overwhelming
contribution to the total mass $\sum_x u(t,x)$ comes from a slowly
increasing
number of islands which are located far from each other. These
islands are
local regions of those high exceedances of the field $\xi$ in a box
with radius
$t\log^2t$ for which the (local) principal Dirichlet eigenvalue of
the random
operator $\Delta+\xi$ is close to maximal. We also prove that the
shape of
$\xi$ in these regions is non-random and that $u(t,\cdot)$ is close
to the
corresponding positive eigenfunction. This is the geometric picture
suggested
by localization theory for the Anderson Hamiltonian.
http://front.math.ucdavis.edu/math.PR/0507585
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3530. COAGULATION-FRAGMENTATION DUALITY, POISSON-DIRICHLET
DISTRIBUTIONS AND RANDOM RECURSIVE TREES
Rui Dong and Christina Goldschmidt and James B. Martin
In this paper we give a new example of duality between fragmentation and
coagulation operators. Consider the space of partitions of mass (that
is,
decreasing sequences of non-negative real numbers whose sum is 1) and
the
two-parameter family of Poisson-Dirichlet distributions PD
(alpha,theta), taking
values in this space. We introduce families of random fragmentation and
coagulation operators, Frag_{alpha} and Coag_{alpha,theta}
respectively, with
the following property: if the input to Frag_{alpha} has PD(alpha,theta)
distribution then the output has PD(alpha,theta+1) distribution,
while the
reverse is true for Coag_{alpha,theta}. This result may be proved
using a
subordinator representation, and provides a companion set of
relations to those
of Pitman between PD(alpha,theta) and PD(alpha*beta,theta). Repeated
application of the Frag_{alpha} operators gives rise to a family of
fragmentation chains. We show that these Markov chains can be encoded
natuarally by certain random recursive trees, and use this
representation to
give an alternative and more concrete proof of the coagulation-
fragmentation
duality.
http://front.math.ucdavis.edu/math.PR/0507591
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3531. EXPLICIT INVARIANT MEASURES FOR PRODUCTS OF RANDOM MATRICES
Jens Marklof and Yves Tourigny and Lech Wolowski
We construct explicit invariant measures for a family of infinite
products of
random, independent, identically-distributed elements of SL(2,C). The
matrices
in the product are such that one entry is gamma-distributed along a
ray in the
complex plane. When the ray is the positive real axis, the products
are those
associated with a continued fraction studied by Letac and Seshadri
[Z. Wahr.
Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the
continued fraction is a generalised inverse Gaussian. We extend this
result by
finding the distribution for an arbitrary ray in the complex right-
half plane,
and thus compute the corresponding Lyapunov exponent explicitly. When
the ray
lies on the imaginary axis, the matrices in the infinite product
coincide with
the transfer matrices associated with a one-dimensional discrete
Schroedinger
operator with a random, gamma-distributed potential. Hence, the explicit
knowledge of the Lyapunov exponent may be used to estimate the
(exponential)
rate of localisation of the eigenstates.
http://front.math.ucdavis.edu/math-ph/0507069
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3532. INTER-ARRIVAL TIME DISTRIBUTION FOR THE NON-HOMOGENEOUS POISSON
PROCESS
Gleb Yakovlev and John B. Rundle and Robert Shcherbakov and and
Donald L. Turcotte
We derive an analytical expression of the inter-arrival time
distribution for
a non-homogeneous Poisson process (NHPP). This expression is exact
and is
applicable to any time interval, finite or infinite. As an
illustration, we
present simulation results for three different intensity functions.
http://front.math.ucdavis.edu/cond-mat/0507657
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3533. A FAST ALGORITHM FOR SIMULATING THE CHORDAL SCHRAMM-LOEWNER
EVOLUTION
Tom Kennedy
The Schramm-Loewner evolution (SLE) can be simulated by dividing the
time
interval into N subintervals and approximating the random conformal
map of the
SLE by the composition of N random, but relatively simple, conformal
maps. In
the usual implementation the time required to compute a single point
on the SLE
curve is O(N). We give an algorithm for which the time to compute a
single
point is O(N^p) with p<1. Simulations with kappa=8/3 and kappa=6 both
give a
value of p of approximately 0.4.
http://front.math.ucdavis.edu/math.PR/0508002
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3534. ASYMPTOTIC ANALYSIS OF MULTISCALE APPROXIMATIONS TO REACTION
NETWORKS
Karen Ball and Tom Kurtz and Lea Popovic and and Greg Rempala
A reaction network is a chemical system involving multiple reactions and
chemical species. Stochastic models of such networks treat the system
as a
continuous time Markov chain on the number of molecules of each
species with
reactions as possible transitions of the chain. In many cases of
biological
interest some of the chemical species in the network are present in much
greater abundance than others and reaction rate constants can vary
over several
orders of magnitude. We consider approaches to approximation of such
models
that take the multiscale nature of the system into account. Our
primary example
is a model of a cell's viral infection for which we apply a
combination of
averaging and law of large number arguments to show that the ``slow''
component
of the model can be approximated by a deterministic equation and to
characterize the asymptotic distribution of the ``fast'' components.
The main
goal is to illustrate techniques that can be used to reduce the
dimensionality
of much more complex models.
http://front.math.ucdavis.edu/math.PR/0508015
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3535. A NECESSARY CONDITION FOR THE UNIQUENESS OF THE STATIONARY
STATE OF A MARKOV SYSTEM
Ivan Werner
We continue the study of Markov systems started in \cite{Wer1}. In this
paper, we prove a generalization of Breiman's strong low of large
numbers
\cite{Br} which implies a necessary condition for the uniqueness of the
stationary state of a Markov system.
http://front.math.ucdavis.edu/math.PR/0508054
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3536. QUANTUM FILTERING: A REFERENCE PROBABILITY APPROACH
Luc Bouten and Ramon van Handel
These notes are intended as an introduction to noncommutative (quantum)
filtering theory. An introduction to quantum probability theory is
given,
focusing on the spectral theorem and the conditional expectation as
the least
squares estimate, and culminating in the construction of Wiener and
Poisson
processes on the Fock space. Next we describe the Hudson-
Parthasarathy quantum
Ito calculus and its use in the modelling of physical systems.
Finally, we use
a reference probability method to obtain quantum filtering equations,
in the
Belavkin-Zakai (unnormalized) form, for several system-observation
models from
quantum optics. The normalized (Belavkin-Kushner-Stratonovich) form
is obtained
through a noncommutative analogue of the Kallianpur-Striebel formula.
http://front.math.ucdavis.edu/math-ph/0508006
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3537. POSITION PLAY IN CAROM BILLIARDS AS A MARKOV PROCESS
Mathieu Bouville
Using certain techniques a billiards player can have long series of easy
shots --each shot leading to another easy shot-- and very high
scores. As the
usual model for carom billiards assumes a Bernoulli process which
does not
account for such correlations, it cannot capture this important
feature of the
game. Modelling carom billiards as a Markov process, the probability
to make a
shot can be made to depend on the previous shot. The improved
agreement with
data is an indication that a Markov process indeed captures the
effects of
position play better. Moreover it is possible to quantify how much a
player
plays position. Given two players with the same average, one can tell
the good
shot-maker from the good position player. This can be useful for
players (and
their coaches) to evaluate their strengths and weaknesses.
http://front.math.ucdavis.edu/math.PR/0508089
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3538. GEODESICS IN FIRST PASSAGE PERCOLATION
Christopher Hoffman
We consider a wide class of ergodic first passage percolation
processes on
Z^2 and prove that there exist at least four one-sided geodesics a.s.
We also
show that coexistence is possible with positive probability in a four
color
Richardson's growth model. This improves earlier results of Haggstrom
and
Pemantle, Garet and Marchand, and Hoffman who proved that first passage
percolation has at least two geodesics and that coexistence is
possible in a
two color Richardson's growth model.
http://front.math.ucdavis.edu/math.PR/0508114
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3539. LIMIT SHAPES AND THE COMPLEX BURGERS EQUATION
Richard Kenyon and Andrei Okounkov
In this paper we study surfaces in R^3 that arise as limit shapes in
a class
of random surface models arising from dimer models. The limit shapes are
minimizers of a surface tension functional, that is, they minimize,
for fixed
boundary conditions, the integral of a quantity (the surface tension)
depending
only on the slope of the surface. The surface tension as a function
of the
slope has singularities and is not strictly convex, which leads to
formation of
facets and edges in the limit shapes.
We find a change of variables that reduces the Euler-Lagrange
equation for
the variational problem to the complex inviscid Burgers equation
(complex Hopf
equation). The equation can thus be solved in terms of an arbitrary
holomorphic
function, which is somewhat similar in spirit to Weierstrass
parametrization of
minimal surfaces. We further show that for a natural dense set of
boundary
conditions, the holomorphic function in question is, in fact,
algebraic. The
tools of algebraic geometry can thus be brought in to study the the
minimizers
and, especially, the formation of their singularities. This is
illustrated by
several explicitly computed examples.
http://front.math.ucdavis.edu/math-ph/0507007
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3540. A REFINEMENT OF THE EULERIAN NUMBERS, AND THE JOINT
DISTRIBUTION OF $\PI(1)$ AND DES($\PI$) IN $S_N$
Mark Conger
Given a permutation $\pi$ chosen uniformly from $S_n$, we explore the
joint
distribution of $\pi(1)$ and the number of descents in $\pi$. We
obtain a
formula for the number of permutations with $\des(\pi)=d$ and $\pi(1)
=k$, and
use it to show that if $\des(\pi)$ is fixed at $d$, then the expected
value of
$\pi(1)$ is $d+1$. We go on to derive generating functions for the joint
distribution, show that it is unimodal if viewed correctly, and show
that when
$d$ is small the distribution of $\pi(1)$ among the permutations with
$d$
descents is approximately geometric. Applications to Stein's method
and the
Neggers-Stanley problem are presented.
http://front.math.ucdavis.edu/math.CO/0508112
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3541. COHERENT PERMUTATIONS WITH DESCENT STATISTIC AND THE BOUNDARY
PROBLEM FOR THE GRAPH OF ZIGZAG DIAGRAMS
Alexander Gnedin and Grigori Olshanski
The graph of zigzag diagrams is a close relative of Young's lattice. The
boundary problem for this graph amounts to describing coherent random
permutations with descent-set statistic, and is also related to certain
positive characters on the algebra of quasi-symmetric functions. We
establish
connections to some further relatives of Young's lattice and solve
the boundary
problem by reducing it to the classification of spreadable total
orders on
integers, as recently obtained by Jacka and Warren.
http://front.math.ucdavis.edu/math.CO/0508131
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3542. RAINBOW HAMILTON CYCLES IN RANDOM REGULAR GRAPHS
Svante Janson and Nicholas Wormald
A rainbow subgraph of an edge-coloured graph has all edges of distinct
colours. A random d-regular graph with d even, and having edges coloured
randomly with d/2 of each of n colours, has a rainbow Hamilton cycle
with
probability tending to 1 as n tends to infinity, provided d is at
least 8.
http://front.math.ucdavis.edu/math.CO/0508145
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3543. BOUNDS FOR CRITICAL VALUES OF THE BAK-SNEPPEN MODEL ON
TRANSITIVE GRAPHS
Alexis Gillett and Ronald Meester and Misja Nuyens
We study the Bak-Sneppen model on locally finite transitive graphs $G
$, in
particular on $\mathbb{Z}^d$ and on $T_{\Delta}$, the regular tree
with common
degree $\Delta$. We show that the avalanches of the Bak-Sneppen model
dominate
independent site percolation, in a sense to be made precise. Together
with the
fact that avalanches of the Bak-Sneppen model are dominated by a simple
branching process, this yields upper and lower bounds for the
critical value
$p_c^{BS}(G)$ of the Bak-Sneppen model. Our main results state that
$\frac{1}{\Delta+1} \le p_c^{BS}(T_\Delta) \le \frac{1}{\Delta -1}$,
and that
$\frac{1}{2d+1}\leq p_c^{BS}(\mathbb{Z}^d)\leq \frac{1}{2d}+
\frac{1}{(2d)^2}+O\big(d^{-3}\big)$, as $d\to\infty$.
http://front.math.ucdavis.edu/math.PR/0508167
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3544. ON A SIMPLE STRATEGY WEAKLY FORCING THE STRONG LAW OF LARGE
NUMBERS IN THE BOUNDED FORECASTING GAME
Masayuki Kumon and Akimichi Takemura
In the framework of the game-theoretic probability of Shafer and Vovk
(2001)
it is of basic importance to construct an explicit strategy weakly
forcing the
strong law of large numbers (SLLN) in the bounded forecasting game.
We present
a simple finite-memory strategy based on the past average of
Reality's moves,
which weakly forces the strong law of large numbers with the
convergence rate
of $O(\sqrt{\log n/n})$. We also give a detailed analysis of the
paths of
Skeptic's capital process for the case of the fair-coin game when our
strategy
is used. We show that if Reality violates SLLN, then the exponential
growth
rate of Skeptic's capital process is explicitly described in terms of
the
Kullback divergence between the average of Reality's moves when she
violates
SLLN and the average when she observes SLLN.
http://front.math.ucdavis.edu/math.PR/0508190
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3545. HARMONICITY OF GIBBS MEASURES
Chris Connell and Roman Muchnik
In this paper we extend the construction of random walks with a
prescribed
Poisson boundary to the case of measures in the class of a
generalized Gibbs
state. The price for dropping the $\alpha$-quasiconformal assumptions
is that
we must restrict our attention to CAT($-\kappa$) groups. Apart from
the new
estimates required, we prove a new approximation scheme to provide a
positive
basis for positive functions in a metric measure space.
http://front.math.ucdavis.edu/math.GR/0507033
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3546. LOGICAL STRUCTURE OF PHYSICAL PROBABILITY ASSERTIONS
Joseph F. Johnson
A modification and generalisation of von Plato's fix of the frequency
theory
of probability is presented. It is thermodynamic in nature. Von Plato
already
fixed the logical circle in the frequency theory, we generalise his
results to
not necessarily ergodic systems of classical and quantum mechanics.
This turns
out to be precisely what is needed for the problem of Quantum
Measurement and
the problem of induction.
http://front.math.ucdavis.edu/quant-ph/0508059
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3547. A SIMPLE INVARIANCE THEOREM
Sourav Chatterjee
We present a simple extension of Lindeberg's argument for the Central
Limit
Theorem to get a general invariance result. We apply the technique to
prove
results from random matrix theory, spin glasses, and maxima of random
fields.
http://front.math.ucdavis.edu/math.PR/0508213
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3548. NORMAL APPROXIMATIONS FOR DESCENTS AND INVERSIONS OF
PERMUTATIONS OF MULTISETS
Mark Conger and D. Viswanath
Normal approximations for descents and inversions of permutations of
the set
$\{1,2,...,n\}$ are well known. A number of sequences that occur in
practice,
such as the human genome and other genomes, contain many repeated
elements.
Motivated by such examples, we consider the number of inversions of a
permutation $\pi(1), \pi(2),...,\pi(n)$ of a multiset with $n$
elements, which
is the number of pairs $(i,j)$ with $1\leq i < j \leq n$ and $\pi(i)>
\pi(j)$.
The number of descents is the number of $i$ in the range $1\leq i < n
$ such
that $\pi(i) > \pi(i+1)$. We prove that, appropriately normalized, the
distribution of both inversions and descents of a random permutation
of the
multiset approaches the normal distribution as $n\to\infty$, provided
that the
permutation is equally likely to be any possible permutation of the
multiset
and no element occurs more than $\alpha n$ times in the multiset for
a fixed
$\alpha$ with $0<\alpha < 1$. Both normal approximation theorems are
proved
using the size biased version of Stein's method of auxiliary
randomization and
are accompanied by error bounds.
http://front.math.ucdavis.edu/math.PR/0508242
---------------------------------------------------------------
3549. LAWS OF THE ITERATED LOGARITHM FOR \ALPHA-TIME BROWNIAN MOTION
Erkan Nanw
We introduce a class of iterated processes called $\alpha$-time Brownian
motion for $0<\alpha \leq 2$. These are obtained by taking Brownian
motion and
replacing the time parameter with a symmetric $\alpha$-stable
process. We prove
a Chung-type law of the iterated logarithm (LIL) for these processes
which is a
generalization of LIL proved in \cite{hu} for iterated Brownian
motion. When
$\alpha =1$ it takes the following form $$ \liminf_{T\to\infty}T^
{-1/2}(\log
\log T) \sup_{0\leq t\leq T}|Z_{t}|=\pi^{2}\sqrt{\lambda_{1}} a.s. $$
where
$\lambda_{1}$ is the first eigenvalue for the Cauchy process in the
interval
$[-1,1].$ We also define the local time $L^{*}(x,t)$ and range $R^{*}
(t)=|\{x:
Z(s)=x \text{for some} s\leq t\}|$ for these processes for $1<\alpha
<2$. We
prove that there are universal constants $c_{R},c_{L}\in (0,\infty) $
such that
$$ \limsup_{t\to\infty}\frac{R^{*}(t)}{(t/\log \log t)^{1/2\alpha}
\log \log t}=
c_{R} a.s. $$ $$
\liminf_{t\to\infty} \frac{\sup_{x\in \RR{R}}L^{*}(x,t)}{(t/\log \log
t)^{1-1/2\alpha}}= c_{L} a.s. $$
http://front.math.ucdavis.edu/math.PR/0508261
---------------------------------------------------------------
3550. HIGHER ORDER PDE'S AND ITERATED PROCESSES
Erkan nane
We introduce a class of stochastic processes based on symmetric
$\alpha$-stable processes, for $\alpha \in (0,2]$ rational.
These are obtained by taking Markov processes and replacing the time
parameter
with the modulus of a symmetric $\alpha$-stable process. We call them
$\alpha$-time processes. They generalize Brownian time processes
studied in
\cite{allouba1, allouba2, allouba3}, and they introduce new interesting
examples. We establish the connection of
$\alpha-$time processes to some higher order PDE's. We also study the
exit
problem for $\alpha$-time processes as they exit regular domains and
connect
them to elliptic PDE's. We also obtain the PDE connection of
subordinate killed
Brownian motion in bounded domains of regular boundary.
http://front.math.ucdavis.edu/math.PR/0508262
---------------------------------------------------------------
3551. THE ARITHMETIC OF DISTRIBUTIONS IN FREE PROBABILITY THEORY
G. Chistyakov and F. G\"otze
We give a new approach to the definition of additive and
multiplicative free
convolutions which is based on the theory of Nevanlinna and of Schur
functions.
We consider the set of probability distributions as a semigroup M
equipped with
the operation of free convolution and prove a Khintchine type theorem
for
factorization of elements of this semigroup. Any element of M
contains either
indecomposable factors or it belongs to a class, say I_0, of
distributions
without indecomposable factors. In contrast to the classical convolution
semigroup in the free additive and multiplicative convolution
semigroups the
class I_0 consists of units (i.e. Dirac measures) only. Furthermore
we show
that the set of indecomposable elements is dense in M.
http://front.math.ucdavis.edu/math.OA/0508245
---------------------------------------------------------------
3552. AUTOMATIC FILTERS FOR THE DETECTION OF COHERENT STRUCTURE IN
SPATIOTEMPORAL SYSTEMS
Cosma Rohilla Shalizi and Robert Haslinger and Jean-Baptiste
Rouquier and Kristina Lisa Klinkner, Cristopher Moore
Most current methods for identifying coherent structures in
spatially-extended systems rely on prior information about the form
which those
structures take. Here we present two new approaches to automatically
filter the
changing configurations of spatial dynamical systems and extract
coherent
structures. One, local sensitivity filtering, is a modification of
the local
Lyapunov exponent approach suitable to cellular automata and other
discrete
spatial systems. The other, local statistical complexity filtering,
calculates
the amount of information needed for optimal prediction of the system's
behavior in the vicinity of a given point. By examining the changing
spatiotemporal distributions of these quantities, we can find the
coherent
structures in a variety of pattern-forming cellular automata, without
needing
to guess or postulate the form of that structure. We apply both
filters to
elementary and cyclical cellular automata (ECA and CCA) and find that
they
readily identify particles, domains and other more complicated
structures. We
compare the results from ECA with earlier ones based upon the theory
of formal
languages, and the results from CCA with a more traditional approach
based on
an order parameter and free energy. While sensitivity and statistical
complexity are equally adept at uncovering structure, they are based on
different system properties (dynamical and probabilistic,
respectively), and
provide complementary information.
http://front.math.ucdavis.edu/nlin.CG/0508001
---------------------------------------------------------------
3553. ALMOST SURE RECURRENCE OF THE SIMPLE RANDOM WALK PATH
Itai Benjamini and Ori Gurel-Gurevich
It is shown that the simple random walk path on a bounded degree graph,
consisting of all vertices visited and edges crossed by the walk, is
almost
surely a recurrent subgraph.
http://front.math.ucdavis.edu/math.PR/0508270
---------------------------------------------------------------
3554. CONTINUITY OF THE MIXING OPERATOR
Mikhail Kovtun
Mixed distributions are considered as a results of application of a
linear
operator, which maps mixing measures to mixed measures. The main
result is a
proof of continuity of this mixing operator. Corollaries for parametric
families of distributions (usually considered in literature) are also
discussed.
http://front.math.ucdavis.edu/math.PR/0508296
---------------------------------------------------------------
3555. EVERY DECISION TREE HAS AN INFLUENTIAL VARIABLE
Ryan O'Donnell and Michael Saks and Oded Schramm and Rocco A.
Servedio
We prove that for any decision tree calculating a boolean function
$f:\{-1,1\}^n\to\{-1,1\}$, \[ \Var[f] \le \sum_{i=1}^n \delta_i \Inf_i
(f), \]
where $\delta_i$ is the probability that the $i$th input variable is
read and
$\Inf_i(f)$ is the influence of the $i$th variable on $f$. The variance,
influence and probability are taken with respect to an arbitrary product
measure on $\{-1,1\}^n$. It follows that the minimum depth of a
decision tree
calculating a given balanced function is at least the reciprocal of
the largest
influence of any input variable. Likewise, any balanced boolean
function with a
decision tree of depth $d$ has a variable with influence at least
$\frac{1}{d}$. The only previous nontrivial lower bound known was $
\Omega(d
2^{-d})$. Our inequality has many generalizations, allowing us to prove
influence lower bounds for randomized decision trees, decision trees on
arbitrary product probability spaces, and decision trees with non-
boolean
outputs. As an application of our results we give a very easy proof
that the
randomized query complexity of nontrivial monotone graph properties
is at least
$\Omega(v^{4/3}/p^{1/3})$, where $v$ is the number of vertices and $p
\leq
\half$ is the critical threshold probability. This supersedes the
milestone
$\Omega(v^{4/3})$ bound of Hajnal and is sometimes superior to the
best known
lower bounds of Chakrabarti-Khot and Friedgut-Kahn-Wigderson.
http://front.math.ucdavis.edu/cs.CC/0508071
---------------------------------------------------------------
3556. COMBINATIONS AND MIXTURES OF OPTIMAL POLICIES IN UNICHAIN
MARKOV DECISION PROCESSES ARE OPTIMAL
Ronald Ortner
We show that combinations of optimal (stationary) policies in
unichain Markov
decision processes are optimal. That is, let M be a unichain Markov
decision
process with state space S, action space A and policies \pi_j^*: S ->
A (1\leq
j\leq n) with optimal average infinite horizon reward. Then any
combination \pi
of these policies, where for each state i in S there is a j such that
\pi(i)=\pi_j^*(i), is optimal as well. Furthermore, we prove that any
mixture
of optimal policies, where at each visit in a state i an arbitrary
action
\pi_j^*(i) of an optimal policy is chosen, yields optimal average
reward, too.
http://front.math.ucdavis.edu/math.CO/0508319
---------------------------------------------------------------
3557. CENTRAL LIMIT THEOREMS FOR A CLASS OF IRREDUCIBLE MULTICOLOR
URN MODELS
Gopal K. Basak and Amites Dasgupta
We take a unified approach to central limit theorems for a class of
irreducible urn models with constant replacement matrix. Depending on
the
eigenvalue, we consider appropriate linear combinations of the number
of balls
of different colors. Then under appropriate norming the multivariate
distribution of the weak limits of these linear combinations is
obtained and
independence and dependence issues are investigated.
http://front.math.ucdavis.edu/math.PR/0507084
---------------------------------------------------------------
3558. A LATTICE SCHEME FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
OF ELLIPTIC TYPE IN DIMENSION $D\GE 4$
Teresa Mart\'inez and Marta Sanz-Sol\'e
We study a stochastic boundary value problem on $(0,1)^d$ of elliptic
type in
dimension $d\ge 4$, driven by a coloured noise. An approximation
scheme based
on a suitable discretization of the Laplacian on a lattice of $(0,1)^d
$ is
presented; we also give the rate of convergence to the original SPDE in
$L^p(\Omega;L^{2}(D))$--norm, for some values of $p$.
http://front.math.ucdavis.edu/math.PR/0508339
---------------------------------------------------------------
3559. THE SCALING LIMIT OF LOOP-ERASED RANDOM WALK IN THREE DIMENSIONS
Gady Kozma
We show that the scaling limit exists and is invariant to dilations and
rotations. We give some tools that might be useful to show universality.
http://front.math.ucdavis.edu/math.PR/0508344
---------------------------------------------------------------
3560. HYDRODYNAMIC LIMIT FLUCTUATIONS OF SUPER-BROWNIAN MOTION WITH A
STABLE CATALYST
Klaus Fleischmann and Peter Moerters and and Vitali Wachtel
We consider the behaviour of a continuous super-Brownian motion
catalysed by
a random medium with infinite overall density under the hydrodynamic
scaling of
mass, time, and space. We show that, in supercritical dimensions, the
scaled
process converges to a macroscopic heat flow, and the appropriately
rescaled
random fluctuations around this macroscopic flow are asymptotically
bounded, in
the sense of log-Laplace transforms, by generalised stable Ornstein-
Uhlenbeck
processes. The most interesting new effect we observe is the
occurrence of an
index-jump from a 'Gaussian' situation to stable fluctuations of
index 1+gamma,
where gamma is an index associated to the medium.
http://front.math.ucdavis.edu/math.PR/0508368
---------------------------------------------------------------
3561. RANDOM ORDERINGS OF THE INTEGERS AND CARD SHUFFLING
Saul Jacka and Jon Warren
In this paper we study random orderings of the integers with a certain
invariance property. We describe all such orders in a simple way. We
define and
represent random shuffles of a countable set of labels and then give an
interpretation of these orders in terms of a class of generalized riffle
shuffles.
http://front.math.ucdavis.edu/math.PR/0508369
---------------------------------------------------------------
3562. ALMOST SURE CONVERGENCE OF SOLUTIONS TO NON-HOMOGENEOUS
STOCHASTIC DIFFERENCE EQUATION
Gregory Berkolaiko and Alexandra Rodkina
We consider a non-homogeneous nonlinear stochastic difference equation
X_{n+1} = X_n (1 + f(X_n)\xi_{n+1}) + S_n, and its important
special case
X_{n+1} = X_n (1 + \xi_{n+1}) + S_n, both with initial value X_0,
non-random
decaying free coefficient S_n and independent random variables \xi_n. We
establish results on \as convergence of solutions X_n to zero. The
necessary
conditions we find tie together certain moments of the noise \xi_n
and the rate
of decay of S_n. To ascertain sharpness of our conditions we discuss
some
situations when X_n diverges. We also establish a result concerning
the rate of
decay of X_n to zero.
http://front.math.ucdavis.edu/math.PR/0508371
---------------------------------------------------------------
3563. ON CONVERGENCE TO EQUILIBRIUM DISTRIBUTION, II. THE WAVE
EQUATION IN ODD DIMENSIONS, WITH MIXING
T.V. Dudnikova and A.I. Komech and N.E. Ratanov and Yu.M. Suhov
The paper considers the wave equation, with constant or variable
coefficients
in $\R^n$, with odd $n\geq 3$. We study the asymptotics of the
distribution
$\mu_t$ of the random solution at time $t\in\R$ as $t\to\infty$. It
is assumed
that the initial measure $\mu_0$ has zero mean, translation-invariant
covariance matrices, and finite expected energy density. We also
assume that
$\mu_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type space mixing
condition. The main result is the convergence of $\mu_t$ to a
Gaussian measure
$\mu_\infty$ as $t\to\infty$, which gives a Central Limit Theorem
(CLT) for the
wave equation. The proof for the case of constant coefficients is
based on an
analysis of long-time asymptotics of the solution in the Fourier
representation
and Bernstein's `room-corridor' argument. The case of variable
coefficients is
treated by using a version of the scattering theory for infinite energy
solutions, based on Vainberg's results on local energy decay.
http://front.math.ucdavis.edu/math-ph/0508039
---------------------------------------------------------------
3564. RANK STATISTICS IN BIOLOGICAL EVOLUTION
E. Ben-Naim and P.L. Krapivsky
We present a statistical analysis of biological evolution processes.
Specifically, we study the stochastic replication-mutation-death
model where
the population of a species may grow or shrink by birth or death,
respectively,
and additionally, mutations lead to the creation of new species. We
rank the
various species by the chronological order by which they originate.
The average
population N_k of the kth species decays algebraically with rank, N_k
~ M^{mu}
k^{-mu}, where M is the average total population. The characteristic
exponent
mu=(alpha-gamma)/(alpha+beta-gamma)$ depends on alpha, beta, and
gamma, the
replication, mutation, and death rates. Furthermore, the average
population P_k
of all descendants of the kth species has a universal algebraic
behavior, P_k ~
M/k.
http://front.math.ucdavis.edu/q-bio.PE/0508023
---------------------------------------------------------------
3565. CLASSICAL BI-POISSON PROCESS: AN INVERTIBLE QUADRATIC HARNESS
Wlodzimierz Bryc and Jacek Wesolowski
We give an elementary construction of a time-invertible Markov
process which
is discrete except at one instance. The process is one of the quadratic
harnesses studied in our previous papers and can be regarded as a
random joint
of two independent Poisson processes.
http://front.math.ucdavis.edu/math.PR/0508383
---------------------------------------------------------------
3566. ROUTING IN POISSON SMALL-WORLD NETWORKS
M. Draief and A. Ganesh
In recent work, Jon Kleinberg considered a small-world network model
consisting of a d-dimensional lattice augmented with shortcuts. The
probability
of a shortcut being present between two points decays as a power of the
distance between them. Kleinberg studied the efficiency of greedy
routing
depending on the value of the power. The results were extended to a
continuum
model by Franceschetti and Meester. In our work, we extend the result
to more
realistic models constructed from a Poisson point process, wherein
each point
is connected to all its neighbours within some fixed radius, as well as
possessing random shortcuts to more distant nodes as described above.
http://front.math.ucdavis.edu/math.PR/0508410
---------------------------------------------------------------
3567. BROWNIAN LOCAL MINIMA AND OTHER RANDOM DENSE COUNTABLE SETS
Boris Tsirelson
We compare two examples of random dense countable sets, `Brownian local
minima' and `unordered uniform infinite sample'. They appear to be
identically
distributed. A framework for such notions is proposed. In addition,
random
elements of other singular spaces (especially, reals modulo
rationals) are
considered.
http://front.math.ucdavis.edu/math.PR/0508414
---------------------------------------------------------------
3568. ON THE STRONG CONSISTENCY OF APPROXIMATED M-ESTIMATORS
Djalil Chafai (LSProba and Upte Umr Inra/Envt 181) and Didier
Concordet (LSProba, Upte Umr Inra/Envt 181)
The aim of this article is to provide a strong consistency Theorem for
approximated M-estimators. It contains both Wald and Pfanzagl type
results for
maximum likelihood. The proof relies, in particular, on the existence
of a sort
of contraction of the parameter space which admits the true parameter
as a
fixed point. In a way, it can be seen as a simplification of ideas of
Wang and
Pfanzagl, generalised to approximated M-estimators. Proofs are short and
elementary.
http://front.math.ucdavis.edu/math.PR/0507102
---------------------------------------------------------------
3569. ON CONVERGENCE TO EQUILIBRIUM DISTRIBUTION, I. THE KLEIN -
GORDON EQUATION WITH MIXING
T.V. Dudnikova and A.I. Komech and E.A. Kopylova and Yu.M. Suhov
Consider the Klein-Gordon equation (KGE) in $\R^n$, $n\ge 2$, with
constant
or variable coefficients. We study the distribution $\mu_t$ of the
random
solution at time $t\in\R$. We assume that the initial probability
measure
$\mu_0$ has zero mean, a translation-invariant covariance, and a
finite mean
energy density. We also asume that $\mu_0$ satisfies a Rosenblatt- or
Ibragimov-Linnik-type mixing condition. The main result is the
convergence of
$\mu_t$ to a Gaussian probability measure as $t\to\infty$ which gives
a Central
Limit Theorem for the KGE. The proof for the case of constant
coefficients is
based on an analysis of long time asymptotics of the solution in the
Fourier
representation and Bernstein's `room-corridor' argument. The case of
variable
coefficients is treated by using an `averaged' version of the
scattering theory
for infinite energy solutions, based on Vainberg's results on local
energy
decay.
http://front.math.ucdavis.edu/math-ph/0508042
---------------------------------------------------------------
3570. ON A TWO-TEMPERATURE PROBLEM FOR WAVE EQUATION
T.V. Dudnikova and A.I. Komech and H. Spohn
Consider the wave equation with constant or variable coefficients in $
\R^3$.
The initial datum is a random function with a finite mean density of
energy
that also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing
condition.
The random function converges to different space-homogeneous
processes as
$x_3\to\pm\infty$, with the distributions $\mu_\pm$. We study the
distribution
$\mu_t$ of the random solution at a time $t\in\R$. The main result is
the
convergence of $\mu_t$ to a Gaussian translation-invariant measure as
$t\to\infty$ that means central limit theorem for the wave equation.
The proof
is based on the Bernstein `room-corridor' argument. The application
to the case
of the Gibbs measures $\mu_\pm=g_\pm$ with two different temperatures
$T_{\pm}$
is given. Limiting mean energy current density formally is $-\infty\cdot
(0,0,T_+ -T_-)$ for the Gibbs measures, and it is finite and equals to
$-C(0,0,T_+ -T_-)$ with $C>0$ for the convolution with a nontrivial test
function.
http://front.math.ucdavis.edu/math-ph/0508044
---------------------------------------------------------------
3571. RAMDOM WALKS ON HYPERGROUP OF CIRCLES IN FINITE FIELDS
Le Anh Vinh
In this paper we study random walks on the hypergroup of circles in a
finite
field of prime order p = 4l + 3. We investigating the behavior of
random walks
on this hypergroup, the equilibrium distribution and the mixing
times. We use
two different approaches - comparision of Dirichlet forms (geometric
bound of
eigenvalues), and coupling methods, to show that the mixing time of
random
walks on hypergroup of circles is only linear.
http://front.math.ucdavis.edu/math.CO/0508403
---------------------------------------------------------------
3572. MALLIAVIN CALCULUS FOR LIE GROUP-VALUED WIENER FUNCTIONS
Tai Melcher
Let G be a Lie group equipped with a set of left invariant vector
fields.
These vector fields generate a function \xi on Wiener space into G
via the
stochastic version of Cartan's rolling map. It is shown here that,
for any
smooth function f with compact support, f(\xi) is Malliavin
differentiable to
all orders and these derivatives belong to L^p(\mu) for all p>1,
where \mu is
Wiener measure.
http://front.math.ucdavis.edu/math.PR/0508419
---------------------------------------------------------------
3573. ON A QUESTION OF CHUNG, DIACONIS, AND GRAHAM
Martin Hildebrand
Chung, Diaconis, and Graham considered random processes of the form
X_{n+1}=2X_n+b_n (mod p) where X_0=0, p is odd, and b_n for
n=0,1,2,... are
i.i.d. random variables on {-1,0,1}. If Pr(b_n=-1)= Pr(b_n=1)=\beta and
Pr(b_n=0)=1-2\beta, they asked which value of \beta makes X_n get
close to
uniformly distributed on the integers mod p the slowest. In this
paper, we
extend the results of Chung, Diaconis, and Graham in the case p=2^t-1
to show
that for 0<\beta<=1/2, there is no such value of \beta.
http://front.math.ucdavis.edu/math.PR/0508427
---------------------------------------------------------------
3574. LONG-RANGE PERCOLATION IN R^D
Bela Bollobas and Svante Janson and Oliver Riordan
Let $X$ be either $Z^d$ or the points of a Poisson process in $R^d$ of
intensity 1. Given parameters $r$ and $p$, join each pair of points
of $X$
within distance $r$ independently with probability $p$. This is the
simplest
case of a `spread-out' percolation model studied by Penrose, who
showed that,
as $r\to\infty$, the average degree of the corresponding random graph
at the
percolation threshold tends to 1, i.e., the percolation threshold and
the
threshold for criticality of the naturally associated branching process
approach one another. Here we show that this result follows
immediately from of
a general result of the authors on inhomogeneous random graphs.
http://front.math.ucdavis.edu/math.PR/0508430
---------------------------------------------------------------
3575. HYPOELLIPTIC HEAT KERNEL INEQUALITIES ON LIE GROUPS
Tai Melcher
This paper discusses the existence of gradient estimates for second
order
hypoelliptic heat kernels on manifolds. It is now standard that such
inequalities, in the elliptic case, are equivalent to a lower bound
on the
Ricci tensor of the Riemannian metric. For hypoelliptic operators, the
associated ``Ricci curvature'' takes on the value -\infty at points of
degeneracy of the semi-Riemannian metric associated to the operator.
For this
reason, the standard proofs for the elliptic theory fail in the
hypoelliptic
setting.
This paper presents recent results for hypoelliptic operators.
Malliavin
calculus methods transfer the problem to one of determining certain
infinite
dimensional estimates. Here, the underlying manifold is a Lie group,
and the
hypoelliptic operators are invariant under left translation. In
particular,
``L^p-type'' gradient estimates hold for p\in(1,\infty), and the p=2
gradient
estimate implies a Poincar\'e estimate in this context.
http://front.math.ucdavis.edu/math.AP/0508420
---------------------------------------------------------------
3576. THE KLEE-MINTY RANDOM EDGE CHAIN MOVES WITH LINEAR SPEED
Jozsef Balogh and Robin Pemantle
An infinite sequence of 0's and 1's evolves by flipping each~1 to a~0
exponentially at rate one. When a~1 flips, all bits to its right also
flip.
Starting from any configuration with finitely many 1's to the left of
the
origin, we show that the leftmost~1 moves right with linear speed.
Upper and
lower bounds are given on the speed.
http://front.math.ucdavis.edu/math.PR/0506626
---------------------------------------------------------------
3577. FAST COMPUTATION OF THE ECONOMIC CAPITAL, THE VALUE AT RISK AND
THE GREEKS OF A LOAN PORTFOLIO IN THE GAUSSIAN FACTOR MODEL
P.Okunev
We propose a fast algorithm for computing the economic capital, Value
at Risk
and Greeks in the Gaussian factor model. The algorithm proposed here
is much
faster than brute force Monte Carlo simulations or Fourier transform
based
methods \cite{MD}. While the algorithm of Hull-White \cite{HW} is
comparably
fast, it assumes that all the loans in the portfolio have equal
notionals and
recovery rates. This is a very restrictive assumption which is
unrealistic for
many portfolios encountered in practice. Our algorithm makes no
assumptions
about the homogeneity of the portfolio. Additionally, it is easier to
implement
than the algorithm of Hull-White. We use the implicit function
theorem to
derive analytic expressions for the Greeks.
http://front.math.ucdavis.edu/math.ST/0507082
---------------------------------------------------------------
3578. ON FILTERING OF MARKOV CHAINS IN STRONG NOISE
P.Chigansky
The filtering problem for a finite state Markov chain observed in
white noise
is addressed in continuous time. The low signal to noise asymptotic
is derived
for the performance indices of MAP and MMSE estimates of the signal.
http://front.math.ucdavis.edu/math.PR/0508446
---------------------------------------------------------------
3579. WEAK TYPE ESTIMATES ASSOCIATED TO BURKHOLDER'S MARTINGALE
INEQUALITY
Javier Parcet
Given a probability space $(\Omega, \mathsf{A}, \mu)$, let $\mathsf{A}
_1,
\mathsf{A}_2, ...$ be a filtration of $\sigma$-subalgebras of $\mathsf
{A}$ and
let $\mathsf{E}_1, \mathsf{E}_2, ...$ denote the corresponding family of
conditional expectations. Given a martingale $f = (f_1, f_2, ...)$
adapted to
this filtration and bounded in $L_p(\Omega)$ for some $2 \le p <
\infty$,
Burkholder's inequality claims that $$\|f\|_{L_p(\Omega)} \sim_
{\mathrm{c}_p}
\Big\| \Big(\sum_{k=1}^\infty \mathsf{E}_{k-1}(|df_k|^2) \Big)^{1/2}
\Big\|_{L_{p}(\Omega)} + \Big(\sum_{k=1}^\infty \|df_k\|_p^p \Big)^{1/
p}.$$
Motivated by quantum probability, Junge and Xu recently extended this
result to
the range $1 < p < 2$. In this paper we study Burkholder's inequality
for
$p=1$, for which the techniques (as we shall explain) must be
different. Quite
surprisingly, we obtain two non-equivalent estimates which play the
role of the
weak type $(1,1)$ analog of Burkholder's inequality. As application,
we obtain
new properties of Davis decomposition for martingales.
http://front.math.ucdavis.edu/math.PR/0508447
---------------------------------------------------------------
3580. UTILITY MAXIMIZATION IN INCOMPLETE MARKETS
Ying Hu and Peter Imkeller and Matthias Muller
We consider the problem of utility maximization for small traders on
incomplete financial markets. As opposed to most of the papers
dealing with
this subject, the investors' trading strategies we allow underly
constraints
described by closed, but not necessarily convex, sets. The final wealths
obtained by trading under these constraints are identified as stochastic
processes which usually are supermartingales, and even martingales for
particular strategies. These strategies are seen to be optimal, and the
corresponding value functions determined simply by the initial values
of the
supermartingales. We separately treat the cases of exponential, power
and
logarithmic utility.
http://front.math.ucdavis.edu/math.PR/0508448
---------------------------------------------------------------
3581. EQUIVALENT AND ABSOLUTELY CONTINUOUS MEASURE CHANGES FOR JUMP-
DIFFUSION PROCESSES
Patrick Cheridito and Damir Filipovic and Marc Yor
We provide explicit sufficient conditions for absolute continuity and
equivalence between the distributions of two jump-diffusion processes
that can
explode and be killed by a potential.
http://front.math.ucdavis.edu/math.PR/0508450
---------------------------------------------------------------
3582. ON THE POWER OF TWO CHOICES: BALLS AND BINS IN CONTINUOUS TIME
Malwina J. Luczak and Colin McDiarmid
Suppose that there are n bins, and balls arrive in a Poisson process
at rate
\lambda n, where \lambda >0 is a constant. Upon arrival, each ball
chooses a
fixed number d of random bins, and is placed into one with least
load. Balls
have independent exponential lifetimes with unit mean. We show that
the system
converges rapidly to its equilibrium distribution; and when d\geq 2,
there is
an integer-valued function m_d(n)=\ln \ln n/\ln d+O(1) such that, in the
equilibrium distribution, the maximum load of a bin is concentrated
on the two
values m_d(n) and m_d(n)-1, with probability tending to 1, as n\to
\infty. We
show also that the maximum load usually does not vary by more than a
constant
amount from \ln \ln n/\ln d, even over quite long periods of time.
http://front.math.ucdavis.edu/math.PR/0508451
---------------------------------------------------------------
3583. HYPOELLIPTICITY IN INFINITE DIMENSIONS AND AN APPLICATION IN
INTEREST RATE THEORY
Fabrice Baudoin and Josef Teichmann
We apply methods from Malliavin calculus to prove an infinite-
dimensional
version of Hormander's theorem for stochastic evolution equations in
the spirit
of Da Prato-Zabczyk. This result is used to show that HJM-equations from
interest rate theory, which satisfy the Hormander condition, have the
conceptually undesirable feature that any selection of yields admits
a density
as multi-dimensional random variable.
http://front.math.ucdavis.edu/math.PR/0508452
---------------------------------------------------------------
3584. THE COALESCENT EFFECTIVE SIZE OF AGE-STRUCTURED POPULATIONS
Serik Sagitov and Peter Jagers
We establish convergence to the Kingman coalescent for a class of
age-structured population models with time-constant population size.
Time is
discrete with unit called a year. Offspring numbers in a year may
depend on
mother's age.
http://front.math.ucdavis.edu/math.PR/0508454
---------------------------------------------------------------
3585. REPRESENTATION OF SOLUTIONS TO BSDES ASSOCIATED WITH A
DEGENERATE FSDE
Jianfeng Zhang
In this paper we investigate a class of decoupled forward-backward SDEs,
where the volatility of the FSDE is degenerate and the terminal value
of the
BSDE is a discontinuous function of the FSDE. Such an FBSDE is
associated with
a degenerate parabolic PDE with discontinuous terminal condition. We
first
establish a Feynman-Kac type representation formula for the spatial
derivative
of the solution to the PDE. As a consequence, we show that there
exists a
stopping time \tau such that the martingale integrand of the BSDE is
continuous
before \tau and vanishes after \tau. However, it may blow up at \tau, as
illustrated by an example. Moreover, some estimates for the martingale
integrand before \tau are obtained. These results are potentially
useful for
pricing and hedging discontinuous exotic options (e.g., digital
options) when
the underlying asset's volatility is small, and they are also useful for
studying the rate of convergence of finite-difference approximations for
degenerate parabolic PDEs.
http://front.math.ucdavis.edu/math.PR/0508457
---------------------------------------------------------------
3586. THE SIZES OF THE PIONEERING, LOWEST CROSSING AND PIVOTAL SITES
IN CRITICAL PERCOLATION ON THE TRIANGULAR LATTICE
G. J. Morrow and Y. Zhang
Let L_n denote the lowest crossing of a square 2n\times2n box for
critical
site percolation on the triangular lattice imbedded in Z^2. Denote
also by F_n
the pioneering sites extending below this crossing, and Q_n the
pivotal sites
on this crossing. Combining the recent results of Smirnov and Werner
[Math.
Res. Lett. 8 (2001) 729-744] on asymptotic probabilities of multiple
arm paths
in both the plane and half-plane, Kesten's [Comm. Math. Phys. 109 (1987)
109-156] method for showing that certain restricted multiple arm
paths are
probabilistically equivalent to unrestricted ones, and our own second
and
higher moment upper bounds, we obtain the following results. For each
positive
integer \tau, as n\to\infty: 1. E(|L_n|^{\tau})=n^{4\tau/3+o(1)}. 2.
E(|F_n|^{\tau})=n^{7\tau/4+o(1)}. 3. E(|Q_n|^{\tau})=n^{3\tau/4+o
(1)}. These
results extend to higher moments a discrete analogue of the recent
results of
Lawler, Schramm and Werner [Math. Res. Lett. 8 (2001) 401-411] that the
frontier, pioneering points and cut points of planar Brownian motion
have
Hausdorff dimensions, respectively, 4/3, 7/4 and 3/4.
http://front.math.ucdavis.edu/math.PR/0508459
---------------------------------------------------------------
3587. A LARGE DEVIATIONS APPROACH TO ASYMPTOTICALLY OPTIMAL CONTROL
OF CRISSCROSS NETWORK IN HEAVY TRAFFIC
Amarjit Budhiraja and Arka Prasanna Ghosh
In this work we study the problem of asymptotically optimal control of a
well-known multi-class queuing network, referred to as the ``crisscross
network,'' in heavy traffic. We consider exponential inter-arrival
and service
times, linear holding cost and an infinite horizon discounted cost
criterion.
In a suitable parameter regime, this problem has been studied in
detail by
Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996)
2133-2171] using
viscosity solution methods. In this work, using the pathwise solution
of the
Brownian control problem, we present an elementary and transparent
treatment of
the problem (with the identical parameter regime as in [SIAM J.
Control Optim.
34 (1996) 2133-2171]) using large deviation ideas introduced in [Ann.
Appl.
Probab. 10 (2000) 75-103, Ann. Appl. Probab. 11 (2001) 608-649]. We
obtain an
asymptotically optimal scheduling policy which is of threshold type.
The proof
is of independent interest since it is one of the few results which
gives the
asymptotic optimality of a control policy for a network with a more than
one-dimensional workload process.
http://front.math.ucdavis.edu/math.PR/0508460
---------------------------------------------------------------
3588. THE PROBABILITY OF EXCEEDING A HIGH BOUNDARY ON A RANDOM TIME
INTERVAL FOR A HEAVY-TAILED RANDOM WALK
Serguei Foss and Zbigniew Palmowski and Stan Zachary
We study the asymptotic probability that a random walk with heavy-tailed
increments crosses a high boundary on a random time interval. We use new
techniques to extend results of Asmussen [Ann. Appl. Probab. 8 (1998)
354-374]
to completely general stopping times, uniformity of convergence over all
stopping times and a wide class of nonlinear boundaries. We also give
some
examples and counterexamples.
http://front.math.ucdavis.edu/math.PR/0508461
---------------------------------------------------------------
3589. EQUILIBRIUM FOR FRAGMENTATION WITH IMMIGRATION
Benedicte Haas
This paper introduces stochastic processes that describe the
evolution of
systems of particles in which particles immigrate according to a Poisson
measure and split according to a self-similar fragmentation. Criteria
for
existence and absence of stationary distributions are established and
uniqueness is proved. Also, convergence rates to the stationary
distribution
are given. Linear equations which are the deterministic counterparts of
fragmentation with immigration processes are next considered. As in the
stochastic case, existence and uniqueness of solutions, as well as
existence
and uniqueness of stationary solutions, are investigated.
http://front.math.ucdavis.edu/math.PR/0508462
---------------------------------------------------------------
3590. CONVERGENCE OF RANDOM MEASURES IN GEOMETRIC PROBABILITY
Mathew D. Penrose
Given $n$ independent random marked $d$-vectors $X_i$ with a common
density,
define the measure $\nu_n = \sum_i \xi_i $, where $\xi_i$ is a
measure (not
necessarily a point measure) determined by the (suitably rescaled)
set of
points near $X_i$. Technically, this means here that $\xi_i$
stabilizes with a
suitable power-law decay of the tail of the radius of stabilization. For
bounded test functions $f$ on $R^d$, we give a law of large numbers
and central
limit theorem for $\nu_n(f)$. The latter implies weak convergence of
$\nu_n(\cdot)$, suitably scaled and centred, to a Gaussian field
acting on
bounded test functions. The general result is illustrated with
applications
including the volume and surface measure of germ-grain models with
unbounded
grain sizes.
http://front.math.ucdavis.edu/math.PR/0508464
---------------------------------------------------------------
3591. A SIMPLE SOLUTION TO THE K-CORE PROBLEM
Svante Janson and Malwina Luczak
We study the k-core of a random (multi)graph on n vertices with a given
degree sequence. We let n tend to infinity. Then, under some regularity
conditions on the degree sequences, we give conditions on the
asymptotic shape
of the degree sequence that imply that with high probability the k-
core is
empty, and other conditions that imply that with high probability the
k-core is
non-empty and the sizes of its vertex and edge sets satisfy a law of
large
numbers; under suitable assumptions these are the only two
possibilities. In
particular, we recover the result by Pittel, Spencer and Wormald on the
existence and size of a k-core in G(n,p) and G(n,m).
Our method is based on the properties of empirical distributions of
independent random variables, and leads to simple proofs.
http://front.math.ucdavis.edu/math.CO/0508453
---------------------------------------------------------------
3592. CONCENTRATION OF HAAR MEASURES, WITH AN APPLICATION TO RANDOM
MATRICES
Sourav Chatterjee
In this article, we present a general technique for analyzing the
concentration of Haar measures on compact groups using the properties of
certain kinds of random walks. As an application, we obtain a new
kind of
measure concentration for random unitary matrices, which allows us to
directly
establish the concentration of the empirical distribution of
eigenvalues of a
class of random matrices. The end-result of this application is a
quantitative
version of Voiculescu's celebrated connection between random matrices
and free
probability.
http://front.math.ucdavis.edu/math.PR/0508518
---------------------------------------------------------------
3593. A GENERALIZATION OF THE LINDEBERG PRINCIPLE
Sourav Chatterjee
We present a generalization of Lindeberg's method of proving the central
limit theorem to encompass general smooth functions (instead of just
sums) and
dependent random variables. The technique is then used to obtain an
invariance
result for smooth functions of exchangeable random variables. As an
illustrative application of this theorem, we then establish
``convergence to
Wigner's law'' for eigenspectra of matrices with exchangeable random
entries.
http://front.math.ucdavis.edu/math.PR/0508519
---------------------------------------------------------------
3594. ON THE CASCADE ROLLBACK SYNCHRONIZATION
Anatoli Manita and Francois Simonot
We consider a cascade model of $N$ different processors performing a
distributed parallel simulation. The main goal of the study is to
show that the
long-time dynamics of the system has a cluster behavior. To attack
this problem
we combine two methods: stochastic comparison and Foster-Lyapunov
functions.
http://front.math.ucdavis.edu/math.PR/0508533
---------------------------------------------------------------
3595. A BERNSTEIN-TYPE INEQUALITY FOR VECTOR FUNCTIONS ON FINITE
MARKOV CHAINS
Vladislav Kargin
An analogue of the Bernstein inequality is derived for partial sums of a
vector-valued function on a finite reversible Markov chain. The
inequality
gives an upper bound for the probability of a large deviation of the
partial
sum. The bound depends on the chain's spectral gap, the dimension of
the space
where the function takes values, and the upper bound on the size and the
variance of the function.
http://front.math.ucdavis.edu/math.PR/0508538
---------------------------------------------------------------
3596. ON THE CONVERGENCE TO A STATISTICAL EQUILIBRIUM IN THE CRYSTAL
COUPLED TO A SCALAR FIELD
T.V. Dudnikova and A.I. Komech
We consider the dynamics of a field coupled to a harmonic crystal
with $n$
components in dimension $d$, $d,n\ge 1$. The crystal and the dynamics
are
translation-invariant with respect to the subgroup $\Z^d$ of $\R^d$. The
initial data is a random function with a finite mean density of
energy which
also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition.
Moreover, initial correlation functions are translation-invariant
with respect
to the discrete subgroup $\Z^d$. We study the distribution $\mu_t$ of
the
solution at time $t\in\R$. The main result is the convergence of $
\mu_t$ to a
Gaussian measure as $t\to\infty$, where $\mu_\infty$ is translation-
invariant
with respect to the subgroup $\Z^d$.
http://front.math.ucdavis.edu/math-ph/0508053
---------------------------------------------------------------
3597. CONNECTION BETWEEN DERIVING BRIDGES AND RADIAL PARTS FROM
MULTIDIMENSIONAL ORNSTEIN-UHLENBECK PROCESSES
Matyas Barczy and Gyula Pap
First we give a construction of bridges derived from a general Markov
process
using only its transition densities. We give sufficient conditions
for their
existence and uniqueness (in law). Then we prove that the law of the
radial
part of the bridge with endpoints zero derived from a special
multidimensional
Ornstein-Uhlenbeck process equals the law of the bridge with
endpoints zero
derived from the radial part of the same Ornstein-Uhlenbeck process.
We also
construct bridges derived from general multidimensional Ornstein-
Uhlenbeck
processes.
http://front.math.ucdavis.edu/math.PR/0508542
---------------------------------------------------------------
3598. VALLEYS AND THE MAXIMUM LOCAL TIME FOR RANDOM WALK IN RANDOM
ENVIRONMENT
Amir Dembo and Nina Gantert and Yuval Peres and Zhan Shi
Let $\xi(n, x)$ be the local time at $x$ for a recurrent one-dimensional
random walk in random environment after $n$ steps, and consider the
maximum
$\xi^*(n) = \max_x \xi(n,x)$. It is known that $\limsup \xi^*(n)/n$ is a
positive constant a.s. We prove that $\liminf_n (\log\log\log n)\xi^*
(n)/n$ is
a positive constant a.s.; this answers a question of P. R\'ev\'esz
(1990). The
proof is based on an analysis of the {\em valleys /} in the environment,
defined as the potential wells of record depth. In particular, we
show that
almost surely, at any time $n$ large enough, the random walker has
spent almost
all of its lifetime in the two deepest valleys of the environment it has
encountered. We also prove a uniform exponential tail bound for the
ratio of
the expected total occupation time of a valley and the expected local
time at
its bottom.
http://front.math.ucdavis.edu/math.PR/0508579
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3599. RANDOM-TURN HEX AND OTHER SELECTION GAMES
Yuval Peres and Oded Schramm and Scott Sheffield and David B. Wilson
The game of Hex has two players who take turns placing stones of
their colors
on the hexagons of a rhombus-shaped hexagonal grid. Black wins by
completing a
crossing between two opposite edges, while White wins by completing a
crossing
between the other pair of opposite edges. Although ordinary Hex is
famously
difficult to analyze, random-turn Hex--in which players toss a coin
before each
turn to decide who gets to place the next stone--has a simple optimal
strategy.
It belongs to a general class of random-turn games--called selection
games--in
which the expected payoff when both players play the random-turn game
optimally
is the same as when both players play randomly. We also describe the
optimal
strategy and study the expected length of the game under optimal play
for
random-turn Hex and several other selection games.
http://front.math.ucdavis.edu/math.PR/0508580
---------------------------------------------------------------
3600. NUMERICAL SOLUTIONS TO INTEGRODIFFERENTIAL EQUATIONS WHICH
INTERPOLATE HEAT AND WAVE EQUATIONS
Piotr Rozmej and Anna Karczewska
In the paper we study some numerical solutions to Volterra equations
which
interpolate heat and wave equations. We present a scheme for
construction of
approximate numerical solutions for one and two spatial dimensions. Some
solutions to the stochastic version of such equations (for one spatial
dimension) are presented as well.
http://front.math.ucdavis.edu/math.NA/0508564
---------------------------------------------------------------
3601. DISTRIBUTED ALGORITHMS IN AN ERGODIC MARKOVIAN ENVIRONMENT
Francis Comets (PMA) and Francois Delarue (PMA) and Rene Schott
(IEC and LORIA)
We provide a probabilistic analysis of the banker algorithm when
transition
probabilities may depend on time and space. The transition probabilities
evolve, as time goes by, along the trajectory of an ergodic Markovian
environment, whereas the spatial parameter just acts on long runs.
Our model
appears as a new (small) step towards more general time and space
dependent
protocols. Our analysis relies on well-known results in stochastic
homogenization theory and investigates the asymptotic behaviour of
the rescaled
algorithm as the total amount of resource available for allocation
tends to the
infinity. In the two dimensional setting, we manage to exhibit three
different
possible regimes for the deadlock time of the limit system.
http://front.math.ucdavis.edu/math.PR/0507115
---------------------------------------------------------------
3602. ON MULTIDIMENSIONAL BRANCHING RANDOM WALKS IN RANDOM ENVIRONMENT
Francis Comets (PMA) and Serguei Popov (IME)
We study branching random walks in random i.i.d. environment in $
\Z^d, d \geq
1$. For this model, the population size cannot decrease, and a natural
definition of recurrence is introduced. We prove a dichotomy for
recurrence/transience, depending only on the support of the
environmental law.
We give sufficient conditions for recurrence and for transience. In the
recurrent case, we study the asymptotics of the tail of the
distribution of the
hitting times and prove a shape theorem for the set of lattice sites
which are
visited up to a large time.
http://front.math.ucdavis.edu/math.PR/0507126
---------------------------------------------------------------
3603. COMPETITION BETWEEN GROWTHS GOVERNED BY BERNOULLI PERCOLATION
Olivier Garet (MAPMO) and R\'{e}gine Marchand (IEC)
We study a competition model on $\mathbb{Z}^d$ where the two
infections are
driven by supercritical Bernoulli percolations with distinct
parameters $p$ and
$q$. We prove that, for any $q$, there exist at most countably many
values of
$p<\min(q, \overrightarrow{p\_c})$ such that coexistence can occur.
http://front.math.ucdavis.edu/math.PR/0507133
---------------------------------------------------------------
3604. POLYMER PINNING IN A RANDOM MEDIUM AS INFLUENCE PERCOLATION
Vincent Beffara (UMPA-ENSL) and Vladas Sidoravicius (BR-IMPA) and
Herbert Spohn (D-MUTU-ZM), Eulalia Vares (BR-CBPF)
In this article we discuss a set of geometric ideas which shed some
light on
the question of directed polymer pinning in the presence of bulk
disorder.
Differing from standard methods and techniques, we transform the
problem to a
particular dependent percolative system and relate the pinning
transition to a
percolation transition.
http://front.math.ucdavis.edu/math.PR/0507142
---------------------------------------------------------------
3605. LINEAR STOCHATIC DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH
CONSTANT COEFFICIENTS
Aureli Alabert and Marco Ferrante
We consider linear stochastic differential-algebraic equations with
constant
coefficients and additive white noise. Due to the nature of this
class of
equations, the solution must be defined as a generalised process (in
the sense
of Dawson and Fernique). We provide sufficient conditions for the law
of the
variables of the solution process to be absolutely continuous with
respect to
Lebesgue measure.
http://front.math.ucdavis.edu/math.PR/0507159
---------------------------------------------------------------
3606. LIKELIHOOD INFERENCE FOR INCOMPLETELY OBSERVED STOCHASTIC
PROCESSES: IGNORABILITY CONDITIONS
Daniel Commenges and Anne Gegout-Petit
We define a general coarsening model for stochastic processes. We
decribe
incomplete data by means of sigma-fields and we give conditions of
ignorability
for likelihood inference.
http://front.math.ucdavis.edu/math.ST/0507151
---------------------------------------------------------------
3607. DETERMINISTIC EQUIVALENTS FOR CERTAIN FUNCTIONALS OF LARGE
RANDOM MATRICES
W. Hachem and P. Loubaton and J. Najim
Consider a $N\times n$ random matrix $ Y_n$ where the entries are
independent
but not identically distributed (matrices with a variance profile)
Consider now
a deterministic $N\times n$ matrix $A_n$ whose columns and rows are
uniformly
bounded for the Euclidean norm.
Let $\Sigma_n=Y_n+A_n$. We prove in this article that there exists a
deterministic equivalent to the empirical Stieltjes transform of the
distribution of the eigenvalues of $\Sigma_n \Sigma_n^T$ which is
itself the
Stieltjes transform of a probability measure.
This work is motivated by the context of performance evaluation of
Multiple
Inputs / Multiple Output (MIMO) wireless digital communication
channels. As an
application, we derive a deterministic equivalent to the mutual
information of
a wireless channel.
http://front.math.ucdavis.edu/math.PR/0507172
---------------------------------------------------------------
3608. A RENEWAL THEORY APPROACH TO PERIODIC COPOLYMERS WITH ADSORPTION
Francesco Caravenna and Giambattista Giacomin and Lorenzo Zambotti
We consider a general model of an heterogeneous polymer chain
fluctuating in
the proximity of an interface between two selective solvents. The
heterogeneous
character of the model comes from the fact that monomer units
interact with the
solvents and with the interface according to some charges that they
carry. The
charges repeat themselves along the chain in a periodic fashion. The
main
question on this model is whether the polymer remains tightly close
to the
interface, a phenomenon called localization, or there is a marked
preference
for one of the two solvents yielding thus a delocalization phenomenon.
We propose an approach to this model, based on renewal theory,
that yields
sharp estimates on the partition function of the model in all the
regimes
(localized, delocalized and critical). This in turn allows to get a very
precise description of the polymer measure, both in a local sense
(thermodynamic limit) and in a global sense (scaling limits). A key
point, but
also a byproduct, of our analysis is the closeness of the polymer
measure to
suitable Markov Renewal Processes.
http://front.math.ucdavis.edu/math.PR/0507178
---------------------------------------------------------------
3609. LEVY PROCESSES: HITTING TIME, OVERSHOOT AND UNDERSHOOT II -
ASYMPTOTIC BEHAVIOUR
Bernard Roynette and Pierre Vallois and Agnes Volpi
Let (X_t, t>=0) be a Levy process started at 0, with Levy measure nu
and T_x
the first hitting time of level x>0: T_x:=inf{t>=0; X_t>x}. Let $F
(theta, mu,
rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F
(theta,mu,rho,x)
:=E(e^(-theta T_x - mu K_x \rho L_x) 1_(T_x<+infinity)), where
theta>=0, mu>=0,
rho>=0, x>=0, K_x:=X_(T_x)-x and L_x:=x-X_(T_(x^-)). If we assume
that nu has
finite exponential moments we exhibit an asymptotic expansion for
F(theta,mu,rho,x), as x -> +infinity. A limit theorem involving a
normalization
of the triplet (T_x,K_x,L_x) as x -> +infinity, may be deduced. At
last, if
nu_(|_R_+) has finite moment of fixed order, we prove that the ruin
probability
P(T_x<+infinity) has at most a polynomial decay.
http://front.math.ucdavis.edu/math.PR/0507193
---------------------------------------------------------------
3610. COMPLETENESS WITH RESPECT TO THE PROBABILISTIC POMPEIU-
HAUSDORFF METRIC
Stefan Cobza\c{s}
The aim of the present paper is to prove that the family of all closed
nonempty subsets of a complete probabilistic metric space $L$ is
complete with
respect to the probabilistic Pompeiu-Hausdorff metric $H$. The same
is true for
the families of all closed bounded, respectively compact, nonempty
subsets of
$L$. If $L$ is a complete random normed space in the sense of \v{S}
erstnev,
then the family of all nonempty closed convex subsets of $L$ is also
complete
with respect to $H$.
http://front.math.ucdavis.edu/math.PR/0507207
---------------------------------------------------------------
3611. PERCOLATION THEORY
Vincent Beffara (UMPA-ENSL) and Vladas Sidoravicius (IMPA)
This is a survey article to be part of the Encyclopedia of Mathematical
Physics, to be published by Elsevier in the beginning of 2006.
http://front.math.ucdavis.edu/math.PR/0507220
---------------------------------------------------------------
3612. THE RANDOM AVERAGE PROCESS AND RANDOM WALK IN A SPACE-TIME
RANDOM ENVIRONMENT IN ONE DIMENSION
Marton Balazs and Firas Rassoul-Agha and Timo Seppalainen
We study space-time fluctuations around a characteristic line for a
one-dimensional interacting system known as the random average
process. The
state of this system is a real-valued function on the integers. New
values of
the function are created by averaging previous values with random
weights. The
fluctuations analyzed occur on the scale n^{1/4} where n is the ratio of
macroscopic and microscopic scales in the system. The limits of the
fluctuations are described by a family of Gaussian processes. In
cases of known
product-form equilibria, this limit is a two-parameter process whose
time
marginals are fractional Brownian motions with Hurst parameter 1/4.
Along the
way we study the limits of quenched mean processes for a random walk
in a
space-time random environment. These limits also happen at scale n^
{1/4} and
are described by certain Gaussian processes that we identify. In
particular,
when we look at a backward quenched mean process, the limit process
is the
solution of a stochastic heat equation.
http://front.math.ucdavis.edu/math.PR/0507226
---------------------------------------------------------------
3613. A NEW EFFICIENT ALGORITHM FOR CONSTRUCTION OF LLS MODELS
Mikhail Kovtun and Igor Akushevich and Kenneth G. Manton and H.
Dennis Tolley
We present a new efficient algortithm for construction of linear latent
structure (LLS) models. This algorithm reduces a problem of
estimation of model
parameters to a sequence of problems of linear algebra, which assures
a low
computational complexity and ability to handle on desktop computers
data that
involve up to thousands of variables.
http://front.math.ucdavis.edu/math.PR/0507021
---------------------------------------------------------------
3614. BROWNIAN SHEET AND REFLECTIONLESS POTENTIALS
Setsuo Taniguchi
The bijectivity of the mapping, which is represented as expectation,
from a
family of Gaussian measures parametrized by linear combinations of Dirac
measures to the space of classical reflectionless potentials is
shown. It is
also shown that the bijectivity extends to the space of generalized
reflectionless potentials, which was used by V. Marchenko to study
the Cauchy
problem for the KdV equation. In the extension, the stochastic
calculus based
on the Brownian sheet plays a key role.
http://front.math.ucdavis.edu/math.PR/0507229
---------------------------------------------------------------
3615. ANALYTICITY OF ENTROPY RATE IN FAMILIES OF HIDDEN MARKOV CHAINS
Guangyue Han and Brian Marcus
We prove that under a mild positivity assumption the entropy rate of
a hidden
Markov chain varies analytically as a function of the underlying
Markov chain
parameters. We give examples to show how this can fail in some cases.
And we
study two natural special classes of hidden Markov chains in more
detail:
binary hidden Markov chains with an unambiguous symbol and binary
Markov chains
corrupted by binary symmetric noise. Finally, we show that under the
positivity
assumption the hidden Markov chain {\em itself} varies analytically,
in a
strong sense, as a function of the underlying Markov chain parameters.
http://front.math.ucdavis.edu/math.PR/0507235
---------------------------------------------------------------
3616. LIKELY PATH TO EXTINCTION FOR SIMPLE BRANCHING MODEL (LARGE
DEVIATIONS APPROACH)
F. Klebaner and R. Liptser
We give an explicit formula for the most likely path to extinction
for the
Galton-Watson processes with large initial population. We establish
this result
with the help of the large deviation principle (LDP) which also
recovers the
asymptotics of extinction probability.
Due to the nonnegativity of the Galton-Watson processes, the proof
of LDP
verification at the point of extinction uses a nonstandard argument of
independent interest.
http://front.math.ucdavis.edu/math.PR/0507257
---------------------------------------------------------------
3617. CRAMER'S THEOREM FOR NONNEGATIVE SUMMANDS
F. Klebaner and R. Liptser
We clarify the boundary effect in Cramer's theorem on the Large
Deviations
Principle (LDP) for normed sums of non-negative i.i.d. random
variables $
S_n=\frac{1}{n}\sum_{i=1}^n\xi_i $. We show that the LDP holds true
with the
rate function possibly infinite at the boundary point $x=0$.
We also consider a continuous time version of Cramer's theorem with
nonnegative summands $ S_t=\frac{1}{t}\sum_{i:\tau_i\le t}\xi_i, t \to
\infty, $
where $(\tau_i,\xi_i)_{i\ge 1}$ is a sequence of random variables
such that
$tS_t$ is a random process with independent increments.
http://front.math.ucdavis.edu/math.PR/0507258
---------------------------------------------------------------
3618. STATIONARITY OF SWITCHING VAR AND OTHER RELATED MODELS
Gopal K. Basak and Zhan-Qian Lu
Switching ARMA models greatly enhance the standard linear models to the
extent that different ARMA model is allowed in a different regime,
and the
regime switching is typically assumed a Markov chain on the finite
states of
potential regimes. Although statistical issues have been the subject
of many
recent papers, there is few systematic study of the probabilistic
aspects of
this new class of nonlinear models. This paper discusses some basic
issues
concerning this class of models including strict stationarity,
influence of
initial conditions, and second-order property by studying SVAR
models. A number
of examples are given to illustrate the theory and the variety of
applications.
Extensions to other models such as mean-shifting, and inhomogeneous
transition
probabilities are discussed.
http://front.math.ucdavis.edu/math.ST/0507267
---------------------------------------------------------------
3619. THE FERMAT CUBIC, ELLIPTIC FUNCTIONS, CONTINUED FRACTIONS, AND
A COMBINATORIAL EXCURSION
Eric van Fossen Conrad and Philippe Flajolet
Elliptic functions considered by Dixon in the nineteenth century and
related
to Fermat's cubic, $x^3+y^3=1$, lead to a new set of continued fraction
expansions with sextic numerators and cubic denominators. The
functions and the
fractions are pregnant with interesting combinatorics, including a
special
P\'olya urn, a continuous-time branching process of the Yule type, as
well as
permutations satisfying various constraints that involve either
parity of
levels of elements or a repetitive pattern of order three. The
combinatorial
models are related to but different from models of elliptic functions
earlier
introduced by Viennot, Flajolet, Dumont, and Fran{\c{c}}on.
http://front.math.ucdavis.edu/math.CO/0507268
---------------------------------------------------------------
3620. EARTHQUAKE RECURRENCE AS A RECORD BREAKING PROCESS
Joern Davidsen and Peter Grassberger and Maya Paczuski
We extend the notion of waiting times for a point process to
recurrent events
in space-time. Earthquake $B$ is a recurrence of a previous one, $A$,
if no
intervening earthquake happens after $A$ and before $B$ in the
spatial disc
centered on $A$ with radius $\bar{AB}$. The cascade of recurrent
events, where
each later recurrence to an event is closer in space than all
previous ones,
forms a sequence of records. Representing each record by a directed link
between nodes defines a network of earthquakes. For Southern
California, this
network exhibits robust scaling laws. The rupture length emerges as a
fundamental scale for distance between recurrent events. Also, the
distribution
of relative separations for the next record in space (time) $\sim
r^{-\delta_r}$ ($\sim t^{-\delta_t}$), with $\delta_r \approx
\delta_t \approx
0.6$. While the in-degree distribution agrees with a random network, the
out-degree distribution shows large deviations from Poisson statistics.
Comparison with randomized data and a theory of records for
independent events
occurring on a fractal shows that these statistics capture non-
trivial features
of the complex spatiotemporal organization of seismicity.
http://front.math.ucdavis.edu/physics/0507082
---------------------------------------------------------------
3621. LINEAR LATENT STRUCTURE ANALYSIS: MIXTURE DISTRIBUTION MODELS
WITH LINEAR CONSTRAINTS
Mikhail Kovtun and Igor Akushevich and Kenneth G. Manton and H.
Dennis Tolley
A new method for analyzing high-dimensional categorical data, Linear
Latent
Structure (LLS) analysis, is presented. LLS models belong to the
family of
latent structure models, which are mixture distribution models
constrained to
satisfy the local independence assumption. LLS analysis explicitly
considers a
family of mixed distributions as a linear space and LLS models are
obtained by
imposing linear constraints on the mixing distribution. LLS models are
identifiable under modest conditions and are consistently estimable. A
remarkable feature of LLS analysis is the existence of a high-
performance
numerical algorithm, which reduces parameter estimation to a sequence
of linear
algebra problems. Preliminary simulation experiments with a prototype
of the
algorithm demonstrated a good quality of restoration of model
parameters.
http://front.math.ucdavis.edu/math.PR/0507025
---------------------------------------------------------------
3622. EULER INTEGRALS FOR COMMUTING SLES
Julien Dubedat
Schramm-Loewner Evolutions (SLEs) have proved an efficient way to
describe a
single continuous random conformally invariant interface in a simply
connected
planar domain; the admissible probability distributions are
parameterized by a
single positive parameter $\kappa$. As shown in \cite{D6}, the
coexistence of
$n$ interfaces in such a domain implies algebraic ("commutation")
conditions.
In the most interesting situations, the admissible laws on systems of
$n$
interfaces are parameterized by $\kappa$ and the solution of
particular (finite
rank) holonomic systems. The study of solutions of differential
systems, in
particular their global behaviour, often involves the use of integral
representations. In the present article, we provide Euler integral
representations for solutions of holonomic systems arising from SLE
commutation. Applications to critical percolation (general crossing
formulae),
loop-erased random walks (direct derivation of Fomin's formulae in
the scaling
limit), and uniform spanning trees are discussed. The connection with
conformal
restriction and Poissonized non-intersection for chordal SLEs is also
studied.
http://front.math.ucdavis.edu/math.PR/0507276
---------------------------------------------------------------
3623. BRYC'S RANDOM FIELDS: THE EXISTENCE AND DISTRIBUTIONS ANALYSIS
Wojciech Matysiak and Pawe{\l} J. Szab{\l}owski
We examine problem of existence of stationary random fields with linear
regressions and quadratic conditional variances, introduced by Bryc in
"Stationary random fields with linear regressions" (Annals of
Probability 29,
No. 1, 504-519). Distributions of the fields are identified and
almost complete
description of the possible sets of parameters defining the first two
conditional moments is given. This note almost solves Bryc's problem
concerning
fields undetermined by moments - the only remaining set of parameters
for which
the existence of Bryc's fields is unclear has Lebesgue measure zero.
http://front.math.ucdavis.edu/math.PR/0507296
---------------------------------------------------------------
3624. Q-WIENER AND RELATED PROCESSES. A BRYC PROCESSES CONTINUOUS
TIME GENERALIZATION
Pawe{\l} J. Szab{\l}owski
We define two Markov processes. The finite dimensional distributions
of the
first one (say $\mathbf{X=}(X_{t})_{t\geq0})$ depend on one parameter
$q\in(-1,1>$ and of the second one (say $\mathbf{Y=}(Y_{t})_{t\in
\mathbb{R}})$
on two parameters $(q,\alpha) \in(-1,1>\times(0,\infty).$ The first one
resembles Wiener process in the sense that for $q=1$ it is Wiener
process but
also that for $q<1$ and $\forall n\geq1$ $t^{n/2}H_{n}(X_{t}/\sqrt{t}|
q) ,$
where $(H_{n})_{n\geq0}$ are so called $q-$Hermite polynomials, are
martingales. It does not have however independent increments. The
second one
resemble Orstein-Ulehnbeck processes. For $q=1$ it is a classical OU
process.
For $q<1$ it is stationary with correlation function equal to $\exp
(-\alpha|t-s|).$When defining these processes and proving their
existence we
use properties of discrete time Bryc processes and solve the problem
of their
existence for $q>1.$ On the way we deny Wesolowski's martingale
characterization of Wiener process.
http://front.math.ucdavis.edu/math.PR/0507303
---------------------------------------------------------------
3625. THE MIXING TIME OF THE THORP SHUFFLE
Ben Morris
The Thorp shuffle is defined as follows. Cut the deck into two equal
piles.
Drop the first card from the left pile or the right pile according to
the
outcome of a fair coin flip; then drop from the other pile. Continue
this way
until both piles are empty. We show that the mixing time for the
Thorp shuffle
with $2^d$ cards is polynomial in $d$.
http://front.math.ucdavis.edu/math.PR/0507307
---------------------------------------------------------------
3626. TAIL BOUNDS FOR THE STABLE MARRIAGE OF POISSON AND LEBESGUE
Christopher Hoffman and Alexander E. Holroyd and Yuval Peres
Let \Xi be a discrete set in R^d. Call the elements of \Xi centers. The
well-known Voronoi tessellation partitions R^d into polyhedral
regions (of
varying volumes) by allocating each site of R^d to the closest
center. Here we
study allocations of R^d to \Xi in which each center attempts to
claim a region
of equal volume \alpha.
We focus on the case where \Xi arises from a Poisson process of unit
intensity. It was proved in math.PR/0505668 that there is a unique
allocation
which is stable in the sense of the Gale-Shapley marriage problem. We
study the
distance X from a typical site to its allocated center in the stable
allocation.
The model exhibits a phase transition in the appetite \alpha. In
the critical
case \alpha=1 we prove a power law upper bound on X in dimension d=1.
It is an
open problem to prove any upper bound in d\geq 2. (Power law lower
bounds were
proved in math.PR/0505668 for all d). In the non-critical cases
\alpha<1 and
\alpha>1 we prove exponential upper bounds on X.
http://front.math.ucdavis.edu/math.PR/0507324
---------------------------------------------------------------
3627. NON-MARKOV RANDOM FIELDS WITH LINEAR REGRESSIONS - A TOEPLITZ
OPERATORS APPROACH
Wojciech Matysiak and Pawe{\l} J. Szab{\l}owski
The aim of the paper is to analyze square integrable random sequences
$\mathbf{X}=(X_{k})_{k\in\mathbb{Z}}$ satisfying condition \[
\wwo{X_k}{...,X_{k-2},X_{k-1},X_{k+1},X_{k+2},...}=\sum_{j=1}^n b_j
(X_{k-j}+X_{k+j}) \] with $b_{j}\in\mathbb{R}$ and
$n\in\nat\cup\left\{+\infty\right\}$. The question of existence of such
sequences for all $n\in\mathbb{N}$ including $n=+\infty$ is examined
and some
conditions guaranteeing existence are provided. In order to give these
conditions we analyze general problem of existence of processes
defined by
regression coefficients. The problem is closely related to one
considered by
Kingman and Williams.
One of the results presented in the paper is that one sided
regressions of
$\mathbf{X}$ are also linear: \[
\mathbb{E}(X_{k}|...,X_{k-2},X_{k-1})=\sum_{j=1}^{n}\beta_{j}X_{k -j}
% \] for
some $\beta_{j}\in\mathbb{R}$ and with the same $n$ as before.
http://front.math.ucdavis.edu/math.PR/0507332
---------------------------------------------------------------
3628. RECTANGULAR RANDOM MATRICES, RELATED CONVOLUTION
Florent Benaych-Georges (DMA)
We characterize asymptotic collective behaviour of rectangular random
matrices, the sizes of which tend to infinity at different rates:
when embedded
in a space of larger square matrices, independent rectangular random
matrices
are asymtotically free with amalgamation over a subalgebra. Therefore
we can
define a "rectangular free convolution", linearized by cumulants and
by an
analytic integral transform, called the "rectangular R-transform".
http://front.math.ucdavis.edu/math.PR/0507336
---------------------------------------------------------------
3629. LIMIT SHAPES OF MULTIPLICATIVE MEASURES ASSOCIATED WITH
COAGULATION-FRAGMENTATION PROCESSES AND RANDOM COMBINATORIAL STRUCTURES
Michael Erlihson and Boris Granovsly
We find limit shapes for a family of multiplicative measures on the
set of
partitions, induced by exponential generating functions with expansive
parameters, $a_k\sim k^{p-1}, k\to\infty, p>0$. The measures
considered are
associated with reversible coagulation-fragmentation processes and
certain
combinatorial structures. We prove the functional central limit
theorem for the
fluctuations of a scaled random partition around its limit shape. We
also
demonstrate that when the component size passes beyond the threshold
value, the
independence of numbers of components transforms into their conditional
independence. Among other things, the paper discusses, in a general
setting,
the interplay between limit shapes, threshold and gelation.
http://front.math.ucdavis.edu/math.PR/0507343
---------------------------------------------------------------
3630. CONDITIONAL ASSOCIATION AND SPIN SYSTEMS
Thomas M. Liggett
A 1977 theorem of T. Harris states that an attractive spin system
preserves
the class of associated probability measures. We study analogues of
this result
for measures that satisfy various conditional positive correlations
properties.
In particular, we show that a spin system preserves measures
satisfying the FKG
lattice condition (essentially) if and only if distinct spins flip
independently. The downward FKG property, which has been useful
recently in the
study of the contact process, lies between the properties of lattice
FKG and
association. We prove that this property is preserved by a spin
system if the
death rates are constant and the birth rates are additive (e.g., the
contact
process), and prove a partial converse to this statement. Finally, we
introduce
a new property, which we call downward conditional association, which
lies
between the FKG lattice condition and downward FKG, and find essentially
necessary and sufficient conditions for this property to be preserved
by a spin
system. This suggests that the latter property may be more natural
than the
downward FKG property.
http://front.math.ucdavis.edu/math.PR/0507392
---------------------------------------------------------------
3631. SOME RESULTS CONCERNING MAXIMUM RENYI ENTROPY DISTRIBUTIONS
Oliver Johnson and Christophe Vignat
We consider the Student-t and Student-r distributions, which maximise
Renyi
entropy under a covariance condition. We show that they have
information-theoretic properties which mirror those of the Gaussian
distributions, which maximise Shannon entropy under the same
condition. We
introduce a convolution which preserves the Renyi maximising family,
and show
that the Renyi maximisers are the case of equality in a version of
the Entropy
Power Inequality. Further, we show that the Renyi maximisers satisfy
a version
of the heat equation, motivating the definition of a generalized Fisher
information.
http://front.math.ucdavis.edu/math.PR/0507400
---------------------------------------------------------------
3632. RECURRENCE FOR PERSISTENT RANDOM WALKS IN TWO DIMENSIONS
Marco Lenci
We discuss the question of recurrence for persistent, or Newtonian,
random
walks in Z^2, i.e., random walks whose transition probabilities
depend both on
the walker's position and incoming direction. We use results by Toth and
Schmidt-Conze to prove recurrence for a large class of such processes,
including all "invertible" walks in elliptic random environments.
Furthermore,
rewriting our Newtonian walks as ordinary random walks in a suitable
graph, we
gain a better idea of the geometric features of the problem, and
obtain further
examples of recurrence.
http://front.math.ucdavis.edu/math.PR/0507411
---------------------------------------------------------------
3633. DONSKER THEOREMS FOR DIFFUSIONS: NECESSARY AND SUFFICIENT
CONDITIONS
Aad van der Vaart and Harry van Zanten
We consider the empirical process G_t of a one-dimensional diffusion
with
finite speed measure, indexed by a collection of functions F. By the
central
limit theorem for diffusions, the finite-dimensional distributions of
G_t
converge weakly to those of a zero-mean Gaussian random process G. We
prove
that the weak convergence G_t\Rightarrow G takes place in \ell^
{\infty}(F) if
and only if the limit G exists as a tight, Borel measurable map. The
proof
relies on majorizing measure techniques for continuous martingales.
Applications include the weak convergence of the local time density
estimator
and the empirical distribution function on the full state space.
http://front.math.ucdavis.edu/math.PR/0507412
---------------------------------------------------------------
3634. THE MULTIFRACTAL SPECTRUM OF BROWNIAN INTERSECTION LOCAL TIMES
Achim Klenke and Peter Morters
Let \ell be the projected intersection local time of two independent
Brownian
paths in R^d for d=2,3. We determine the lower tail of the random
variable
\ell(U), where U is the unit ball. The answer is given in terms of
intersection
exponents, which are explicitly known in the case of planar Brownian
motion. We
use this result to obtain the multifractal spectrum, or spectrum of thin
points, for the intersection local times.
http://front.math.ucdavis.edu/math.PR/0507437
---------------------------------------------------------------
3635. VALIDITY OF THE EXPECTED EULER CHARACTERISTIC HEURISTIC
Jonathan Taylor and Akimichi Takemura and Robert J. Adler
We study the accuracy of the expected Euler characteristic
approximation to
the distribution of the maximum of a smooth, centered, unit variance
Gaussian
process f. Using a point process representation of the error, valid for
arbitrary smooth processes, we show that the error is in general
exponentially
smaller than any of the terms in the approximation. We also give a
lower bound
on this exponential rate of decay in terms of the maximal variance of
a family
of Gaussian processes f^x, derived from the original process f.
http://front.math.ucdavis.edu/math.PR/0507442
---------------------------------------------------------------
3636. LEVY PROCESSES: HITTING TIME, OVERSHOOT AND UNDERSHOOT - PART
I: FUNCTIONAL EQUATIONS
Bernard Roynette and Pierre Vallois and Agnes Volpi
Let (X_t, t >=0) be a Levy process started at 0, with Levy measure
nu, and
T_x the first hitting time of level x>0: T_x := inf{t>=0; X_t>x}. Let
F(theta,mu,rho,.) be the joint Laplace transform of (T_x, K_x, L_x):
F(theta,mu,rho,x) := E (e^{-theta T_x - mu K_x - rho L_x} 1_{T_x<
+infinity}),
where theta>=0, mu>=0, rho>=0, x>0, K_x := X_{T_x} - x and L_x := x -
X_{T_{x^-}}. If nu(R) < + \infinity and integral_1^{+\infty} e^{sy}
nu (dy) <
+infinity for some s>0, then we prove that F(theta,mu,rho,.) is the
unique
solution of an integral equation and has a subexponential decay at
infinity
when theta>0 or theta=0 and E(X_1)<0. If nu is not necessarily a
finite measure
but verifies integral_{-infinity}^{-1} e^{-sy} nu (dy) < +infinity
for any s>0,
then the x-Laplace transform of F(theta,mu,rho,.) satisfies some kind of
integral equation. This allows us to prove that F(theta,mu,rho,.) is
a solution
to a second integral equation.
http://front.math.ucdavis.edu/math.PR/0507034
---------------------------------------------------------------
3637. CORNER PERCOLATION ON Z^2 AND THE SQUARE ROOT OF 17
Gabor Pete
We consider a dependent bond percolation model on Z^2, introduced by
Balint
Toth, in which every edge is present with probability 1/2, and each
vertex has
exactly two incident edges, perpendicular to each other. We prove
that all
components are finite cycles almost surely, but the expected diameter
of the
cycle containing the origin is infinite. A more detailed analysis
leads to the
derivation of the following critical exponents: the tail probability
\Pr(diameter of the cycle of the origin > n) \approx n^{-\gamma}, and
the
expectation \E(length of a cycle conditioned on having diameter n)
\approx
n^\delta. We show that \gamma=(5-\sqrt{17})/4=0.219... and
\delta=(\sqrt{17}+1)/4=1.28... The relation \gamma+\delta=3/2
corresponds to
the fact that the scaling limit of the natural height function in the
model is
the Additive Brownian Motion, whose level sets have Hausdorff
dimension 3/2.
http://front.math.ucdavis.edu/math.PR/0507457
---------------------------------------------------------------
3638. BARYCENTERS OF MEASURES TRANSPORTED BY STOCHASTIC FLOWS
Marc Arnaudon and Xue-Mei Li
We investigate the evolution of barycenters of masses transported by
stochastic flows. The state spaces under consideration are smooth affine
manifolds with certain convexity structure. Under suitable conditions
on the
flow and on the initial measure, the barycenter {Z_t} is shown to be a
semimartingale and is described by a stochastic differential
equation. For the
hyperbolic space the barycenter of two independent Brownian particles
is a
martingale and its conditional law converges to that of a Brownian
motion on
the limiting geodesic. On the other hand for a large family of discrete
measures on suitable Cartan-Hadamard manifolds, the barycenter of the
measure
carried by an unstable Brownian flow converges to the Busemann
barycenter of
the limiting measure.
http://front.math.ucdavis.edu/math.PR/0507460
---------------------------------------------------------------
3639. STOCHASTIC EQUIVARIANT COHOMOLOGIES AND CYCLIC COHOMOLOGY
Remi Leandre
We give two stochastic diffeologies on the free loop space which
allow us to
define stochastic equivariant cohomology theories in the Chen-Souriau
sense and
to establish a link with cyclic cohomology. With the second one, we can
establish a stochastic fixed point theorem.
http://front.math.ucdavis.edu/math.PR/0507461
---------------------------------------------------------------
3640. SOME RESULTS ON TWO-SIDED LIL BEHAVIOR
Uwe Einmahl and Deli Li
Let {X,X_n;n\geq 1} be a sequence of i.i.d. mean-zero random
variables, and
let S_n=\sum_{i=1}^nX_i,n\geq 1. We establish necessary and sufficient
conditions for having with probability 1, 0<lim sup_{n\to
\infty}|S_n|/\sqrtnh(n)<\infty, where h is from a suitable subclass
of the
positive, nondecreasing slowly varying functions. Specializing our
result to
h(n)=(\log \log n)^p, where p>1 and to h(n)=(\log n)^r, r>0, we obtain
analogues of the Hartman-Wintner LIL in the infinite variance case.
Our proof
is based on a general result dealing with LIL behavior of the
normalized sums
{S_n/c_n;n\ge 1}, where c_n is a sufficiently regular normalizing
sequence.
http://front.math.ucdavis.edu/math.PR/0507462
---------------------------------------------------------------
3641. RANDOM WALK ATTRACTED BY PERCOLATION CLUSTERS
Serguei Popov and Marina Vachkovskaia
Starting with a percolation model in $\Z^d$ in the subcritical
regime, we
consider a random walk described as follows: the probability of
transition from
$x$ to $y$ is proportional to some function $f$ of the size of the
cluster of
$y$. This function is supposed to be increasing, so that the random
walk is
attracted by bigger clusters. For $f(t)=e^{\beta t}$ we prove that
there is a
phase transition in $\beta$, i.e., the random walk is subdiffusive
for large
$\beta$ and is diffusive for small $\beta$.
http://front.math.ucdavis.edu/math.PR/0507054
---------------------------------------------------------------
3642. LIMIT THEOREMS FOR THE TYPICAL POISSON-VORONOI CELL AND THE
CROFTON CELL WITH A LARGE INRADIUS
Pierre Calka and Tomasz Schreiber
In this paper, we are interested in the behavior of the typical
Poisson-Voronoi cell in the plane when the radius of the largest disk
centered
at the nucleus and contained in the cell goes to infinity. We prove a
law of
large numbers for its number of vertices and the area of the cell
outside the
disk. Moreover, for the latter, we establish a central limit theorem
as well as
moderate deviation type results. The proofs deeply rely on precise
connections
between Poisson-Voronoi tessellations, convex hulls of Poisson
samples and
germ-grain models in the unit ball. Besides, we derive analogous
facts for the
Crofton cell of a stationary Poisson line process in the plane.
http://front.math.ucdavis.edu/math.PR/0507463
---------------------------------------------------------------
3643. STATISTICAL DUALITY OF THE LAPLACE DISTRIBUTION
E.A. Barkova and S.I. Bityukov and V.A. Taperechkina
The statistical duality of distributions is a powerful tool for
statistical
inferences. In the paper the statistical duality of Laplace
distribution is
discussed. As shown the confidence density of the parameter of this
distribution is uniquely determined.
http://front.math.ucdavis.edu/math.ST/0507452
---------------------------------------------------------------
3644. ESTIMATES FOR MOMENTS OF RANDOM MATRICES WITH GAUSSIAN ELEMENTS
O. Khorunzhiy
We describe an elementary method to get non-asymptotic estimates for the
moments of hermitian random matrices whose elements are gaussian
independent
random variables. As the basic example, we consider the GUE matrices.
Immediate
applications include GOE and gaussian skew-symmetric hermitian matrices.
http://front.math.ucdavis.edu/math-ph/0507060
---------------------------------------------------------------
3645. DISTANCES BETWEEN THE WINNING NUMBERS IN LOTTERY
Konstantinos Drakakis
We prove an interesting fact about Lottery: the winning 6 numbers
(out of 49)
in the game of the Lottery contain two consecutive numbers with a
surprisingly
high probability (almost 50%).
http://front.math.ucdavis.edu/math.CO/0507469
---------------------------------------------------------------
3646. A GENERAL LOWER BOUND FOR MIXING OF SINGLE-SITE DYNAMICS ON GRAPHS
Thomas P. Hayes and Alistair Sinclair
We prove that any Markov chain that performs local, reversible
updates on
randomly chosen vertices of a bounded-degree graph necessarily has
mixing time
at least $\Omega(n\log n)$, where $n$ is the number of vertices. Our
bound
applies to the so-called ``Glauber dynamics'' that has been used
extensively in
algorithms for the Ising model, independent sets, graph colorings and
other
structures in computer science and statistical physics, and
demonstrates that
many of these algorithms are optimal up to constant factors within
their class.
Previously no super-linear lower bound for this class of algorithms
was known.
Though widely conjectured, such a bound had been proved previously
only in very
restricted circumstances, such as for the empty graph and the path.
We also
show that the assumption of bounded degree is necessary by giving a
family of
dynamics on graphs of unbounded degree with mixing time $O(n)$.
http://front.math.ucdavis.edu/math.PR/0507517
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