Probability Abstracts 62
This document contains abstracts 1691-1725.
They have been mailed on May 1, 2001.
1691. DISCREPANCY CONVERGENCE FOR THE DRUNKARD'S WALK ON THE SPHERE
Francis Edward Su
We analyze the drunkard's walk on the unit sphere with step size theta and
show that the walk converges in order constant/sin^2(theta) steps in the
discrepancy metric. This is an application of techniques we develop for
bounding the discrepancy of random walks on Gelfand pairs generated by
bi-invariant measures. In such cases, Fourier analysis on the acting group
admits tractable computations involving spherical functions. We advocate the
use of discrepancy as a metric on probabilities for state spaces with isometric
group actions.
su@math.hmc.edu
- This article is available
from the xxx mathematics archive as
math.PR/0102205.
1692. RANDOM WALKS WITH BADLY APPROXIMABLE NUMBERS
Doug Hensley, Francis Edward Su
Using the discrepancy metric, we analyze the rate of convergence of a random
walk on the circle generated by d rotations, and establish sharp rates that
show that badly approximable d-tuples in R^d give rise to walks with the
fastest convergence. We use the discrepancy metric because the walk does not
converge in total variation. For badly approximable d-tuples, the discrepancy
is bounded above and below by (constant)k^(-d/2), where k is the number of
steps in the random walk. We show how the constants depend on the d-tuple.
su@math.hmc.edu
- This article is available
from the xxx mathematics archive as
math.PR/0102206.
1693. SOURCE CODING, LARGE DEVIATIONS, AND APPROXIMATE PATTERN MATCHING
A. Dembo, I. Kontoyiannis
We present a development of parts of rate-distortion theory and pattern-
matching algorithms for lossy data compression, centered around a lossy version
of the Asymptotic Equipartition Property (AEP). This treatment closely
parallels the corresponding development in lossless compression, a point of
view that was advanced in an important paper of Wyner and Ziv in 1989. In the
lossless case we review how the AEP underlies the analysis of the Lempel-Ziv
algorithm by viewing it as a random code and reducing it to the idealized
Shannon code. This also provides information about the redundancy of the
Lempel-Ziv algorithm and about the asymptotic behavior of several relevant
quantities. In the lossy case we give various versions of the statement of the
generalized AEP and we outline the general methodology of its proof via large
deviations. Its relationship with Barron's generalized AEP is also discussed.
The lossy AEP is applied to: (i) prove strengthened versions of Shannon's
source coding theorem and universal coding theorems; (ii) characterize the
performance of mismatched codebooks; (iii) analyze the performance of pattern-
matching algorithms for lossy compression; (iv) determine the first order
asymptotics of waiting times (with distortion) between stationary processes;
(v) characterize the best achievable rate of weighted codebooks as an optimal
sphere-covering exponent. We then present a refinement to the lossy AEP and use
it to: (i) prove second order coding theorems; (ii) characterize which sources
are easier to compress; (iii) determine the second order asymptotics of waiting
times; (iv) determine the precise asymptotic behavior of longest match-lengths.
Extensions to random fields are also given.
yiannis@stat.purdue.edu
- This article is available
from the xxx mathematics archive as
math.PR/0103007.
1694. LIMIT THEOREMS FOR THE PAINTING OF GRAPHS BY CLUSTERS
Olivier Garet
We consider a generalization of the so-called divide and color model recently
introduced by Haggstrom. We investigate the behaviour of the magnetization in
large boxes and its fluctuations. Thus, laws of large numbers and Central Limit
theorems are proved, both quenched and annealed. We show that the properties of
the underlying percolation process roughly influence the behaviour of the
colorying model. In the subcritical case, the limit magnetization is
deterministic and the Central Limit Theorem admits a Gaussian limit. A
contrario, the limit magnetization is not deterministic in the supercritical
case and the limit of the Central Limit Theorem is not Gaussian, except in the
particular model with exactly two colors which are equally probable.
olivier.garet@labomath.univ-orleans.fr
- This article is available
from the xxx mathematics archive as
math.PR/0103027.
1695. STOCHASTIC VERSION OF THE ERDOS-RENYI LIMIT THEOREM
A. Khorunzhy
We generalise the Erdos-Renyi limit theorem on the maximum of the partial
sums of random variables to the case when the number of terms in these sums is
randomly distributed. Certain relations between the limiting theorems of this
type and the spectral theory of random graphs and random matrices are
discussed.
khorunjy@math.jussieu.fr
- This article is available
from the xxx mathematics archive as
math.PR/0103043.
1696. A DYNAMIC ONE-DIMENSIONAL INTERFACE INTERACTING WITH A WALL
F. M. Dunlop, P. A. Ferrari, L. R. G. Fontes
We study a symmetric randomly moving line interacting by exclusion with a
wall. We show that the expectation of the position of the line at the origin
when it starts attached to the wall satisfies the following bounds: c_1t^{1/4}
\le\E\xi_t(0) \le c_2 t^{1/4}\log t The result is obtained by comparison with a
``free'' process, a random line that has the same behavior but does not see the
wall. The free process is isomorphic to the symmetric nearest neigbor
one-dimensional simple exclusion process. The height at the origin in the
interface model corresponds to the integrated flux of particles through the
origin in the simple exclusion process. We compute explicitly the asymptotic
variance of the flux and show that the probability that this flux exceeds
Kt^{1/4}\log t is bounded above by const. t^{2-K}. We have also performed
numerical simulations, which indicate \E\xi_t(0)^2 \sim t^{1/2}\log t as
t\to\infty.
pablo@ime.usp.br
- This article is available
from the xxx mathematics archive as
math.PR/0103049.
1697. CLUSTERS AND RECURRENCE IN THE TWO-DIMENSIONAL ZERO-TEMPERATURE
STOCHASTIC ISING MODEL
F. Camia, E. De Santis, C. M. Newman
We analyze clustering and (local) recurrence of a standard Markov process
model of spatial domain coarsening. The continuous time process, whose state
space consists of assignments of +1 or -1 to each site in ${\bf Z}^2$, is the
zero-temperature limit of the stochastic homogeneous Ising ferromagnet (with
Glauber dynamics): the initial state is chosen uniformly at random and then
each site, at rate one, polls its 4 neighbors and makes sure it agrees with the
majority, or tosses a fair coin in case of a tie. Among the main results
(almost sure, with respect to both the process and initial state) are: clusters
(maximal domains of constant sign) are finite for times $t< \infty$, but the
cluster of a fixed site diverges (in diameter) as $t \to \infty$; each of the
two constant states is (positive) recurrent. We also present other results and
conjectures concerning positive and null recurrence and the role of absorbing
states.
fc276@scires.nyu.edu
- This article is available
from the xxx mathematics archive as
math.PR/0103050.
1698. ON SOME PROPERTIES OF TRANSITIONS OPERATORS
Fabio Zucca
We study a general transition operator, generated by a random walk on a graph
$X$; in particular we give necessary and sufficient condition on the matrix
coefficient (1-step transition probablilities) to be a bounded operator from
$l^\infty(X)$ into itself. Moreover we characterize compact operators and we
relate this property to the behaviour of the associated random walk. We give a
necessary and sufficient condition for the pre-adjoint of the discrete Laplace
operator to be an injective map.
zucca@mat.unimi.it
- This article is available
from the xxx mathematics archive as
math.PR/0103072.
1699. DETECTION OF SPATIAL PATTERN THROUGH INDEPENDENCE OF THINNED PROCESSES
Renato M. Assuncao and Pablo A. Ferrari
Let N, N' and N'' be point processes such that N' is obtained from N by
homogeneous independent thinning and N''= N- N'. We give a new elementary proof
that N' and N'' are independent if and only if N is a Poisson point process. We
present some applications of this result to test if a homogeneous point process
is a Poisson point process.
pablo@ime.usp.br
- This article is available
from the xxx mathematics archive as
math.PR/0103104.
1700. A PARADOX OF DIFFUSION MARKET MODEL RELATED WITH EXISTENCE OF WINNING
COMBINATIONS OF OPTIONS
Nikolai Dokuchaev
We consider strategies of investments into options and diffusion market
model. It is shown that there exists a correct proportion between "put" and
"call" in the portfolio such that the average gain is almost always positive
for a generic Black and Scholes model. This gain is zero if and only if the
market price of risk is zero. It is discussed a paradox related to the
corresponding loss of option's seller.
dokuchaev@pobox.spbu.ru
- This article is available
from the xxx mathematics archive as
math.PR/0103118.
1701. GENERATING A RANDOM SINK-FREE ORIENTATION IN QUADRATIC TIME
Henry Cohn, Robin Pemantle, James Propp
A sink-free orientation of a finite undirected graph is a choice of
orientation for each edge such that every vertex has out-degree at least 1.
Bubley and Dyer (1997) use Markov Chain Monte Carlo to sample approximately
from the uniform distribution on sink-free orientations in time O(m^3 log
(1/epsilon)), where m is the number of edges and epsilon the degree of
approximation. Huber (1998) uses coupling from the past to obtain an exact
sample in time O(m^4). We present a simple randomized algorithm inspired by
Wilson's cycle popping method which obtains an exact sample in mean time at
most O(nm), where n is the number of vertices.
cohn@microsoft.com
- This article is available
from the xxx mathematics archive as
math.PR/0103189.
1702. STRONG POINTWISE ESTIMATES FOR THE WEAKLY SELF-AVOIDING WALK
Erwin Bolthausen and Christine Ritzmann
This paper presents asymptotics for the weakly self-avoiding walk on the
hypercubic lattice in five or more dimensions. Our main result is a general
central limit theorem for distributions defined by certain renewal type
equations. In addition, we give good error estimates and Gaussian tail
estimates which have not been obtained by other methods. We use the lace
expansion and at the same time develop a new perspective on this method: While
all earlier approaches to the lace expansion use Fourier transformation, we
work here with a fixed point argument directly in the x-space.
chritz@amath.unizh.ch
- This article is available
from the xxx mathematics archive as
math.PR/0103218.
1703. FLUX FLUCTUATIONS IN THE ONE DIMENSIONAL NEAREST NEIGHBORS SYMMETRIC
SIMPLE EXCLUSION PROCESS
A. De Masi, P. A. Ferrari
Let $J(t)$ be the the integrated flux of particles in the symmetric simple
exclusion process starting with the product invariant measure $\nu_\rho$ with
density $\rho$. We compute its rescaled asymptotic variance: \[
\lim_{t\to\infty} t^{-1/2} \V J(t) = \sqrt{2/\pi} (1-\rho)\rho \] Furthermore
we show that $t^{-1/4}J(t)$ converges weakly to a centered normal random
variable with this variance. From these results we compute the asymptotic
variance of a tagged particle in the nearest neighbor case and show the
corresponding central limit theorem, results previously proven by Arratia.
pablo@ime.usp.br
- This article is available
from the xxx mathematics archive as
math.PR/0103233.
1704. SPECTRAL DENSITIES DESCRIBING OFF-WHITE NOISES
Boris Tsirelson
For the white noise, the spectral density is constant, and the past
(restriction to (-\infty,0)) is independent from the future (restriction to
(0,+\infty)). If the spectral density is not too far from being constant, then
dependence between the past and the future can be eliminated by an equivalent
measure change; that is called an off-white noise. I derive from well-known
results a necessary and sufficient condition for a spectral density to describe
an off-white noise.
tsirel@math.tau.ac.il
- This article is available
from the xxx mathematics archive as
math.PR/0104027.
1705. ASYMPTOTIC BEHAVIOR OF A STATIONARY SILO WITH ABSORBING WALLS
S. R. M. Barros, P. A. Ferrari, N. L. Garcia, S. Martinez
We study the nearest neighbors one dimensional uniform q-model of force
fluctuations in bead packs [Coppersmith et al (1996)], a stochastic model to
simulate the stress of granular media in two dimensional silos. Interpreting
the vertical coordinate as time and the horizontal one as space the system is a
discrete time Markov process with state space $\R^{\{1,...,N\}}$. At each layer
(time), the weight supported by each grain is a random variable of mean one
(its own weight) plus the sum of random fractions of the weights supported by
the nearest neighboring grains at the previous layer. The fraction of the
weight given to the right neighbor of the successive layer is an independent
uniform random variable in $[0,1]$. The remaining weight is given to the left
neighbor. A uniform fraction of the weight at the extreme sites leans on the
walls of the silo; this corresponds to \emph{absorbing boundary conditions}. We
show that there exists a unique invariant measure. The mean weight at site $i$
under the invariant measure is $i(N+1-i)$; we prove that its variance is
$\frac12(i(N+1-i))^2 + O(N^3)$ and the covariances between grains $i\neq j$ are
of order $O(N^3)$. Moreover, as $N\to\infty$, the law under the invariant
measure of the weights divided by $N^2$ around site (integer part of) $rN$,
$r\in (0,1)$, converges to a product of gamma distributions with parameters 2
and $2(r(1-r))^{-1}$ (sum of two exponentials of mean $r(1-r)/2$). This shows
the mean field conjecture of Liu {\it et al} (1995) for this model.
pablo@ime.usp.br
- This article is available
from the xxx mathematics archive as
math.PR/0104043.
1706. PHASE TRANSITION FOR THE FROG MODEL
O.S.M.Alves, F.P.Machado, S.Yu.Popov
We study a system of simple random walks on graphs, known as frog model. This
model can be described as follows: There are active and sleeping particles
living on some graph G. Each active particle performs a simple random walk with
discrete time and at each moment it may disappear with probability 1-p. When an
active particle hits a sleeping particle, the latter becomes active. Phase
transition results and asymptotic values for critical parameters are presented
for Z^d and regular trees.
fmachado@ime.usp.br
- This article is available
from the xxx mathematics archive as
math.PR/0104044.
1707. A NOTE ON UNIVERSALITY OF THE DISTRIBUTION OF THE LARGEST EIGENVALUES IN
CERTAIN SAMPLE COVARIANCE MATRICES
Alexander Soshnikov
Recently Johansson and Johnstone proved that the distribution of the
(properly rescaled) largest principal component of the complex (real) Wishart
matrix $ X^* \* X (X^t \*X) $ converges to the Tracy-Widom law as $ n, p $ (the
dimensions of $ X $) tend to $ \infty $ in some ratio $ n/p \to \gamma>0. $ We
extend these results in two directions. First of all, we prove that the joint
distribution of the first, second, third, etc. eigenvalues of a Wishart matrix
converges (after a proper rescaling) to the Tracy-Widom distribution. Second of
all, we explain how the combinatorial machinery developed for Wigner matrices
allows to extend the results by Johansson and Johnstone to the case of $ X $
with non-Gaussian entries, provided $ n-p =O(p^{1/3}) . $ We also prove that $
\lambda_{max} \leq (n^{1/2}+p^{1/2})^2 +O(p^{1/2}\*\log(p)) $ (a.e.) for
general $ \gamma >0.$
soshniko@math.ucdavis.edu
- This article is available
from the xxx mathematics archive as
math.PR/0104113.
1708. COUPLING AND BERNOULLICITY IN RANDOM-CLUSTER AND POTTS MODELS
Olle Haggstrom, Johan Jonasson, Russell Lyons
An explicit coupling construction of random-cluster measures is presented. As
one of the applications of the construction, the Potts model on amenable Cayley
graphs is shown to exhibit at every temperature the mixing property known as
Bernoullicity.
rdlyons@indiana.edu
- This article is available
from the xxx mathematics archive as
math.PR/0104174.
1709. CONDITIONAL EXPECTATION AS QUANTILE DERIVATIVE
Dirk Tasche
For a linear combination of random variables, fix some confidence level and
consider the quantile of the combination at this level. We are interested in
the partial derivatives of the quantile with respect to the weights of the
random variables in the combination. It turns out that under suitable
conditions on the joint distribution of the random variables the derivatives
exist and coincide with the conditional expectations of the variables given
that their combination just equals the quantile. Moreover, using this result,
we deduce formulas for the derivatives with respect to the weights of the
variables for the so-called expected shortfall (first or higher moments) of the
combination. Finally, we study in some more detail the coherence properties of
the expected shortfall in case it is defined as a first conditional moment. Key
words: quantile; value-at-risk; quantile derivative; conditional expectation;
expected shortfall; conditional value-at-risk; coherent risk measure.
tasche@math.ethz.ch
- This article is available
from the xxx mathematics archive as
math.PR/0104190.
1710. CLASSIFICATION ON THE AVERAGE OF RANDOM WALKS
Daniela Bertacchi, Fabio Zucca
We introduce a new method for studying large scale properties of random
walks. The new concepts of transience and recurrence on the average are
compared with the ones introduced by R.Burioni, D.Cassi and A.Vezzani and with
the usual ones; their relationships are analyzed and various examples are
provided.
zucca@mat.unimi.it
- This article is available
from the xxx mathematics archive as
math.PR/0104252.
1711. THE QUANTILE OF A DIFFUSION. PRICING A QUANTILE LOOKBACK OPTION
Emilson Andrianjakaherivola and Francesco Russo
The quantile of a general diffusion $\left( X_{t}\right) $
is deeply studied. When $\left(X_{t}\right) $ is a Brownian motion with drift,
we explicitly calculate the joint distribution of the triple constituted by the quantile,
$\left( X_{t}\right) $ and its local time.
We introduce a new path dependent option,
baptized {\bf quantile lookback} which
generalizes the classical lookback option.
As a significant application of the joint
distribution above, we evaluate the price
of quantile lookback options.
herivola@math.univ-paris13.fr russo@math.univ-paris13.fr
- To see a preprint or other
information provided by the author
click here.
1712. COALESCENCE OF SYNCHRONOUS COUPLINGS
Krzysztof Burdzy and Zhenqing Chen
We consider a synchronous coupling (X_t,Y_t) of reflected
Brownian motions in a bounded planar domain D. The processes
X_t and Y_t are reflected Brownian motions driven
by the same 2-dimensional Brownian motion B_t.
We prove that |X_t - Y_t| converges to 0 a.s.
if the boundary of D is a polygon or if D is a ``lip''
domain, i.e., a domain sandwiched between graphs
of two functions which are Lipschitz with constant 1.
burdzy@math.washington.edu zchen@math.washington.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
1713. DISCRETIZING A BACKWARD SDE
Yinnan Zhang and Weian Zheng
In this paper, we show a simple method to discretize Pardoux-Peng's
non-linear backward stochatic differential equation.
This discretization scheme also gives a numerical method to solve
a class of semi-linear PDEs.
wzheng@uci.edu
1714. THE FLUID LIMIT OF A HEAVILY LOADED PROCESSOR SHARING QUEUE
H. Christian Gromoll, Amber L. Puha and Ruth J. Williams
Consider a single server queue with renewal arrivals and i.i.d. service
times in which the server operates under a processor sharing service
discipline. To describe the evolution of this system, we use a measure
valued process that keeps track of the residual service times of all jobs
in the system at any given time. From this measure valued process,
one can recover the traditional performance processes, including queue
length and workload. We propose and study a critical fluid model
(or formal law of large numbers approximation) for a heavily loaded
processor sharing queue. The fluid model state descriptor is a measure
valued function whose dynamics are governed by a nonlinear integral
equation. Under mild assumptions, we prove existence and uniqueness of
fluid model solutions. Furthermore, we justify the critical fluid model as a
first order approximation of a heavily loaded processor sharing queue by
showing that, when appropriately rescaled, the measure valued processes
corresponding to a sequence of heavily loaded processor sharing queues
converge in distribution to a limit that is almost surely a fluid model
solution.
williams@math.ucsd.edu
1715. A UNIVERSAL RESULT IN ALMOST SURE CENTRAL LIMIT THEORY
Istvan Berkes and Endre Csaki
The discovery of the almost sure central limit theorem (Brosamler, 1988;
Schatte, 1988) revealed a new phenomenon in classical central limit theory
and has led to an extensive literature in the past decade. In particular,
a.s. central limit theorems and various related `logarithmic' limit theorems
have been obtained for several classes of independent and dependent random
variables. In this paper we extend this theory and show that not only the
central limit theorem, but every weak limit theorem for independent
random variables, subject to minor technical conditions, has an analogous
almost sure version. For many classical limit theorems this involves
logarithmic averaging, as in the case of the CLT, but we need radically
different averaging processes for `more sensitive' limit theorems. Several
examples of such a.s. limit theorems are discussed.
berkes@renyi.hu csaki@renyi.hu
- To see a preprint or other
information provided by the author
click here.
1716. ALMOST SURE LIMIT THEOREMS FOR SUMS AND MAXIMA FROM THE
DOMAIN OF GEOMETRIC PARTIAL ATTRACTION OF SEMISTABLE LAWS
Istvan Berkes, Endre Csaki, Sandor Csorgo, Zoltan Megyesi
The possible limiting distributions of sums of independent
identically distributed random variables along subsequences
$\{k_n\}\subset\N$ satisfying $k_{n+1}/k_n\to c\ge 1$ are
the semistable laws and the domain of geometric partial
attraction of a semistable law consists of distributions
attracted to it along such a subsequence. The aim of this
paper is to show that sums and maxima from the domain of
geometric partial attraction of a semistable law satisfy
almost sure limit theorems along the whole sequence
$\{n\}=\N$ of natural numbers, despite the fact that
ordinary convergence in distribution typically takes place
in both cases only along $\{k_n\}$ and related subsequences.
We describe the class of all possible almost sure asymptotic
distributions both for sums and maxima.
berkes@renyi.hu csaki@renyi.hu csorgo@math.u-szeged.hu megyesiz@math.u-szeged.hu
- To see a preprint or other
information provided by the author
click here.
1717. ESTIMATING SOME FEATURES OF NK FITNESS LANDSCAPES
Steven N. Evans and David Steinsaltz
Some time ago, Stuart Kauffman introduced a class of models
for the evolution of hereditary systems which he called
``NK fitness landscapes''. Inspired by spinglasses, these
models have the attractive feature of being tunable, with
regard to both overall size (through the parameter $N$) and
connectivity (through $K$). There are $N$ genes, each of
which exists in two possible alleles (leading to
a system indexed by $\{0,1\}^N$); the fitness score of an
allele at a given site is determined by the alleles of $K$
neighboring sites. Otherwise the fitnesses are as simple as
possible, namely i.i.d., and the fitnesses of different
sites are simply averaged.
Much attention has been focused on these fitness landscapes
as paradigms for investigating the interaction between size
and complexity in making evolution possible.
In this paper, some asymptotic features of NK fitness
landscapes are reduced to questions about eigenvalues and
Lyapunov exponents. When $K$ is fixed, the expected number
of local maxima grows exponentially with $N$ at a rate
depending on the top eigenvalue of a kernel derived from
the distribution of the fitnesses, and the average height
of a local maximum converges to a value determined by the
corresponding eigenfunction.
The global maximum converges in probability as
$N \to \infty$ to a constant given by the top Lyapunov
exponent for a system of i.i.d. max--plus random matrices,
and this constant is non-decreasing with $K$.
Various such quantities are computed for certain special
cases when $K$ is small, and these calculations can,
in principle, be extended to larger $K$.
evans@stat.berkeley.edu dstein@stat.berkeley.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
1718. MESOSCOPIC LIMIT FOR NON-ISOTHERMAL PHASE TRANSITION
Nicolas Dirr and Stephan Luckhaus
Motivated by the problem of modelling nucleation in non-
isothermal systems,we consider the stochastic evolution of
a coupled system of a lattice spin variable $\sigma$ and a
continuous variable e (corresponding to the phase and the
energy density of a continuum system). The spin variables
flip with rates depending both on a Kac-potential type
interaction with the spins and on an interaction with the
e-field, which plays the role of the external field in
ferromagnetics but evolves by a diffusion equation with a
forcing depending on the spins.
We analyse the mesoscopic limit, where space scales like
the diverging interaction range of the Kac potential,
$\gamma^{-1},$ while time is not rescaled. By writing
the spins as random time change of a family of independent
spins, we show that as $\gamma\to 0$ the average of the
spins over small cubes and the field e converge in
probability to the solution of a system of nonlocal
evolution equations which is similar to the phase field
equations.
ndirr@mis.mpg.de
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information provided by the author
click here.
1719. A STOCHASTIC SELECTION PRINCIPLE IN CASE OF FATTENING FOR CURVATURE FLOW
Nicolas Dirr, Stephan Luckhaus and Matteo Novaga
Consider two disjoint circles moving by mean curvature plus
a forcing term which makes them touch with zero velocity.
It is known that the generalized solution in the viscosity
sense ceases to be a curve after the touching (the so-called
fattening phenomenon). We show that after adding a small
noise in time, the measure selects in the limit two evolving
curves, the upper and lower barrier in the sense of
De Giorgi. Further we show partial results for finite noise.
ndirr@mis.mpg.de
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information provided by the author
click here.
1720. DEGENERATE STOCHASTIC DIFFERENTIAL EQUATIONS
AND SUPER-MARKOV CHAINS
Siva Athreya, Martin Barlow, Richard Bass, and Ed Perkins
We consider diffusions corresponding to the generator
$$ L f(x) = \sum_{i=1}^d x_i \gamma_i(x)\frac{\partial^2}
{\partial{x_i}^2}f(x) +
b_i(x) \frac{\partial}{\partial{x_i}}f(x),$$
$x \in [0,\infty)^d,$ for continuous $\gamma_i, b_i :
[0,\infty)^d \rightarrow \bR$ with $\gamma_i$ nonnegative.
We show uniqueness for the corresponding martingale
problem under certain non-degeneracy conditions on
$b_i, \gamma_i$ and present a counter-example when
these conditions are not satisfied. As a special case,
we establish uniqueness in law for some classes of
super-Markov chains with state dependent branching
rates and spatial motions.
athreya@math.ubc.ca barlow@math.ubc.ca bass@math.uconn.edu perkins@math.ubc.ca
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information provided by the author
click here.
1721. MEAN-FIELD CRITICALITY FOR PERCOLATION ON PLANAR NON-AMENBLE GRAPHS
Roberto H. Schonmann
The critical exponents $\beta$, $\gamma$, $\delta$ and $\Delta$
are proved to exist and to take their mean-field values for independent
percolation on the following classes of
infinite, locally finite, connected transitive graphs:
1) Non-amenable planar with one end.
2) Unimodular with infinitely many ends.
rhs@math.ucla.edu
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information provided by the author
click here.
1722. INFORMATION INEQUALITIES AND A DEPENDENT CENTRAL LIMIT THEOREM
Oliver Johnson
We adapt arguments concerning information-theoretic
convergence in the Central Limit Theorem to the case of
dependent random variables under Rosenblatt mixing
conditions. The key is to work with random variables
perturbed by the addition of a normal random variable,
giving us good control of the joint density and the
mixing coefficient. We strengthen results of Takano and of
Carlen and Soffer to provide entropy-theoretic, not weak
convergence.
otj1000@cam.ac.uk
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information provided by the author
click here.
1723. LIMIT THEOREMS FOR MONOTONIC PARTICLE SYSTEMS
AND SEQUENTIAL DEPOSITION
Mathew D. Penrose
We prove spatial laws of large numbers and central
limit theorems for the ultimate number of adsorbed
particles in a large class of multidimensional random
and cooperative seuqential adsorption schemes on
the lattice, and also for the Johnson-Mehl model
of birth, linear growth and spatial exclusion in the
continuum. The lattice result is also applicable to
certain telecommunications networks. The proofs are based
on a general law of large numbers and central limit
theorem for sums of random variables determined
by the restriction of a white noise process to large
spatial regions.
mathew.penrose@durham.ac.uk
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here.
1724. A DIFFERENT CONSTRUCTION OF GAUSSIAN FIELDS
FROM MARKOV CHAINS: DIRICHLET COVARIANCES
Persi Diaconis and Steven N. Evans
We study a class of Gaussian random fields with negative
correlations. These fields are easy to simulate. They are
defined in a natural way from a Markov chain that has the
index space of the Gaussian field as its state space. In
parallel with Dynkin's investigation of Gaussian fields
having covariance given by the Green's function of a Markov
process, we develop connections between the occupation times
of the Markov chain and the prediction properties of the
Gaussian field. Our interest in such fields was initiated
by their appearance in random matrix theory.
evans@stat.berkeley.edu diaconis@math.stanford.edu
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1725. EIGENVALUES OF THE LAGUERRE PROCESS
AS NON-COLLIDING SQUARED BESSEL PROCESSES
Wolfgang Koenig and Neil O'Connell
Let $A(t)$ be a $n\times p$ matrix with independent
standard complex Brownian entries and set
$M(t)=A(t)^{\rm *}A(t)$. This is a process version of
the Laguerre ensemble and as such we shall refer to it
as the {\em Laguerre process}. The purpose of this note
is to remark that, assuming $n> p-1$, the eigenvalues
of $M(t)$ evolve like $p$ independent squared Bessel
processes of dimension $2(n-p+1)$, conditioned (in the
sense of Doob) never to collide. More precisely, the
function $h(x)=\prod_{i&<;j}(x_i-x_j)$ is harmonic
with respect to $p$ independent squared Bessel processes
of dimension $2(n-p+1)$, and the eigenvalue process
has the same law as the corresponding Doob $h$-transform.
In the case where the entries of $A(t)$ are {\em real\/}
Brownian motions, $(M(t))_{t\geq0}$
is the Wishart process considered by Bru~\cite{Br91}.
There it is shown that the eigenvalues of $M(t)$
evolve according to a certain diffusion process, the
generator of which is given explicitly. An interpretation
in terms of non-colliding processes does not seem to
be possible in this case.
We also identify a class of processes (including Brownian
motion, squared Bessel processes and generalised
Ornstein-Uhlenbeck processes) which are all
amenable to the same $h$-transform, and compute the
corresponding transition densities and upper tail
asymptotics for the first collision time.
koenig@math.tu-berlin.de
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