Probability Abstracts 26
This document contains abstracts 457-488.
They have been mailed on April 26, 1995.
Click here to see the
list of all abstract titles.
457. FIRST-PASSAGE TIME MOMENTS OF STRONGLY MIXING STATIONARY SEQUENCES
Dirk Tasche
Consider any stationary sequence of random variables taking only the
values 0 and 1. Then the variance of the first-recurrence
time to 1 conditioned on starting in 1 is finite if and only if the
unconditional expectation of the first-passage time to 1 is finite.
We show that this is the case if the series of the strong mixing
coefficients of the sequence converges. Concerning higher
moments we state similar sufficient conditions. In the case of polynomial
moments these conditions are almost optimal.
dt@math.tu-berlin.de (LaTeX- and Postscript-files available)
458. DYNAMIC AND EQUILIBRIUM BEHAVIOUR OF CONTROLLED LOSS NETWORKS
Nigel Bean, Richard Gibbens and Stan Zachary
We study the behaviour of large loss networks in which the offered
traffic is subject to acceptance controls. In recent work Hunt and
Kurtz proved a functional law of large numbers for the dynamics of such
networks, as capacity and offered traffic are allowed to increase in
proportion. However, limiting dynamics were not in general uniquely
identified. We establish further results identifying these dynamics.
We also prove rigorous results relating their fixed points to limiting
equilibrium behaviour, permitting the investigation of common modelling
assumptions. We give an example of bistability in a single resource
network, and study in detail a two resource network.
stan@ma.hw.ac.uk (postscript or hard copy available)
459. SMOOTHING THE HILL ESTIMATOR
Sidney Resnick and Catalin Starica
For sequences of iid random variables whose common tail 1-F is regularly
varying at
infinity with an unknown index $-\alpha < 0$, it is well known that Hill's
estimator is consistent for $\alpha^{-1}$ and usually asymptotically
normally distributed. However, because the Hill estimator is a function of
$k=k(n)$, the number of upper order statistics used and which is only subject
to the conditions $k\to\infty,\:k/n\to 0$, its use in practice is
problematic since there are few reliable guidelines about how to choose k.
The purpose of this paper is to make the use of Hill's estimator more
reliable through an averaging technique which reduces the asymptotic
variance. As a direct result the range in which the smoothed estimator
varies as a function of $k$ decreases and the successful use of the estimator
is made less dependent on the choice of k. A tail empirical process approach
is used to prove the weak convergence of a process closely related to Hill's
estimator. The smoothed version of Hill's estimator is a functional of
the tail empirical process.
sid@orie.cornell.edu starica@orie.cornell.edu
460. DISTINGUISHING SCENERIES BY OBSERVING THE SCENERY ALONG A
RANDOM WALK PATH
Itai Benjamini and Harry Kesten
Let $\Cal G$ be an infinite connected graph with vertex set
$\Cal V$. A {\it scenery} on $\Cal G$ is a map $\xi:\Cal V
\rightarrow \{0,1\}$ (equivalently, an assignment of zeroes and
ones to the vertices of $\Cal G$). Let
$\{S_n\}_{n\ge 0}$ be a simple random walk on $\Cal G$, starting
at some distinguished vertex $v_0$. Now let $\xi$ and $\eta$ be
two {\it known} sceneries and assume that we observe one of the
two sequences $\{\xi(S_n)\}_{n\ge 0}$ or $\{\eta(S_n)\}_{n\ge
0}$, but we do not know which of the two sequences is observed.
Can we decide, with a zero probability of error, which of the
two sequences is observed? We show that if $\Cal G = \Bbb Z$ or
$\Cal G = \Bbb Z^2$, then the answer is ``yes'' for each fixed
$\xi$ and ``almost all'' $\eta$. We also give some examples of
graphs $\Cal G$ for which almost all pairs $(\xi,\eta)$ are not
distinguishable, and discuss some variants of this problem.
itai@stat.berkeley.edu, kesten@math.cornell.edu
461. LINEAR ADDITIVE FUNCTIONALS OF SUPERDIFFUSIONS AND RELATED
NONLINEAR P.D.E.
E.B.Dynkin and S.E.Kuznetsov
Let L be a second order elliptic differential operator in a bounded
smooth domain D in R^d and let a belong to the interval (1,2]. We get
necessary and sufficient conditions on measures m, n
under which there exists a positive solution of the boundary value problem
-Lv+v^a=m in D,
v=n on the boundary of D.
The conditions are stated, both, analytically (in terms of capacities
related to the Green's and Poisson kernels) and, probabilistically,
in terms of branching measure-valued processes called
(L,a)-superdiffusions.
We also investigate a closely related subject --- linear additive
functionals of superdiffusions. For a superdiffusion in an arbitrary
domain E, we establish a 1-1 correspondence between a class of such
functionals and a class of L-excessive functions h (which we describe in
terms of their Martin integral representation). The Laplace transform of A
satisfies an integral equation which can be considered as a
substitute for the boundary value problem with measures states above.
ebd1@cornell.edu
462. RANDOM WALKS ON RANDOM SIMPLE GRAPHS
Martin Hildebrand
This paper looks at random regular simple graphs and considers nearest
neighbor random walks on such graphs. This paper considers walks where
the degree $d$ of each vertex is around $(\log n)^a$ where $a$ is a
constant which is at least $2$ and where $n$ is the number of vertices. By
extending techniques of Dou, this paper shows that for most such graphs,
the position of the random walk becomes close to uniformly distributed
after slightly more than $\log n/\log d$ steps.
This paper also gets similar results for the random graph $G(n,p)$
where $p=d/(n-1)$.
mvh@math.utexas.edu (TeX and hard copy available)
463. BOUNDEDNESS OF LEVEL LINES FOR TWO-DIMENSIONAL RANDOM FIELDS
Kenneth S. Alexander
Every two-dimensional imcompressible flow follows the level lines of
some scalar function Y on R^2; transport properties of the flow depend in
part on whether all level lines of Y are bounded. We study the structure of
the level lines when Y is a stationary random field. We show that under mild
hypotheses there is only one possible alternative to bounded level lines: the
"treelike" random fields, which, for some interval of values of a, have a
unique unbounded level line at each level a, with this line "winding through
every region of the plane." If the random field has the FKG property then
only bounded level lines are possible. For stationary C^2 Gaussian random
fields with covariance function decaying to 0 at infinity, the treelike
property is the only alternative to bounded level lines provided the density
of the absolutely continuous part of the spectral measure decays at infinity
"slower than exponentially," and only bounded level lines are possible if the
covariance function is nonnegative.
alexandr@math.usc.edu Hard copy available.
464. STOCHASTIC NETWORKS
F. P. Kelly and R. J. Williams (editors)
Research on stochastic networks has applications in the modelling of
manufacturing, telecommunications, and computer systems.
These various applications raise common mathematical issues of some subtlety,
explored in the forthcoming IMA Volume 71 on Stochastic Networks.
The volume edited by Frank Kelly and Ruth Williams contains
articles by experts in operations research, systems science and engineering, and
applied mathematics.
To see the table of contents, a copy of the preface, or ordering information,
see the "url" below.
williams@math.ucsd.edu
- To see some information provided by the author
click here.
465. PERSISTENT SURVIVAL OF ONE-DIMENSIONAL CONTACT PROCESSES
IN RANDOM ENVIRONMENTS
Charles M. Newman and Sergio B. Volchan
Consider an inhomogeneous contact process on ${\bf Z}^{1}$ in
which the recovery rates $\delta(x)$ at site $x$ are i.i.d. random variables
(bounded above) while the infection rate is a constant $\epsilon$. The
following condition implies the survival of the process for {\em every}
$\epsilon > 0$:
$$ u \, {\bf P}(-\log \delta(x) > u) \rightarrow +\infty as u \rightarrow
+\infty $$
newman@cims19.cims.nyu.edu (LaTex or hard copy available)
466. THE DIMENSION OF THE BROWNIAN FRONTIER IS GREATER THAN 1.
Chris Bishop, Peter Jones, Robin Pemantle and Yuval Peres
Run a planar Brownian motion for finite time. The FRONTIER or
"outer boundary" is the boundary of the unbounded component of
the complement. Burdzy (1989) showed that the frontier has
infinite length. We improve this by showing that the Hausdorff
dimension of the frontier is strictly greater than 1. The
proof makes integral use of Jones's Traveling Salesman Theorem
and artifactual use of a self-similar tiling of the plane
by fractal tiles known as "Gosper Islands".
pemantle@stat.berkeley.edu
467. EXPONENTIAL STABILITY OF NEUTRAL STOCHASTIC DIFFERENTIAL
FUNCTIONAL EQUATIONS
X. Mao
Deterministic neutral differential functional equations and their stability
have been studied by many authors e.g. Hale, Meyer. Motivated by
the chemical engineering systems as well as the theory of aeroelasticity,
Kolmanovskii and Nosov introduced the neutral stochastic differential
functional equations of the form
$$ d[x(t)-G(x_t)]=[f(t,x(t))+g(t,x_t)]dt + \sigma (t, x_t)dw(t). $$
Kolmanovskii amd Nosov not only established the theory of existence and
uniqueness of the solution to the equation but also investigated the stability
and asymptotic stability of the equations. However, so far little is known
about the exponential stability of such neutral stochastic differential
functional equations and the aim of this paper is to close this gap.
In this paper the theory of existence and uniqueness of the solutions is
first introduced very briefly. The main results of this paper consists of
several useful criteria on the exponential stability in mean square for the
trivial solution of the equation. Moreover, a number of interesting examples
are given to illustrate the theory.
xuerong@stams.strath.ac.uk
468. ENLARGEMENT OF THE WIENER FILTRATION BY AN ABSOLUTELY CONTINUOUS RANDOM
VARIABLE VIA MALLIAVIN'S CALCULUS
Peter Imkeller
The analytic treatment of problems related to the asymptotic behaviour of
random dynamical systems generated by stochastic differential equations
suffers from the presence of non-adapted random invariant measures.
Semimartingale theory becomes accessible if the underlying Wiener filtration
is enlarged by the information carried by the orthogonal projectors on the
Oseledets spaces of the (linearized) system.
We study the corresponding problem of preservation of the semimartingale
property and the validity of a priori inequalities between the norms of
stochastic integrals in the enlarged filtration and norms of their quadratic
variations in case the random element $F$ enlarging the filtration is real
valued and possesses an absolutely continuous law. Applying the tools of
Malliavin's calculus, we give smoothness conditions on $F$ under which the
semimartingale property is preserved and a priori martingale inequalities are
valid.
imkeller@vega.univ-fcomte.fr (A latex file is available.)
469. ON K-AUTOMORPHISMS, BERNOULLI SHIFTS AND MARKOV RANDOM FIELDS
Frank den Hollander and Jeffrey E. Steif
We show that for translation invariant Markov Random Fields:
(1) the K--property implies a Trivial Full Tail;
(2) the Bernoulli property implies F\o lner Independence.
The existence of bilaterally deterministic Bernoulli Shifts tells us that
neither result is true without
the Markov assumption (even in one dimension). We also show that for general
translation invariant random fields:
(3) F\o lner Independence implies a Trivial Full Tail.
denholla@sci.kun.nl steif@math.chalmers.se
470. REPRESENTATION OF PROBABILITY MEASURE VALUED PROCESSES
Peter March
Let $X$ be a compact metric space, let $X^{\Bbb N}$ be the
countable cartesian product and let $\Cal M_1(X)$ be the space of
probability measures on $X$. We show by construction that any Markov
process on $\Cal M_1(X)$ can be represented as the limiting empirical
distribution of a Markov process on $X^{\Bbb N}$. We also give a
sufficient condition for a process on $X^{\Bbb N}$ to represent some
Markov process on $\Cal M_1(X)$. The special case that $X$ is a finite
set, and so $\Cal M_1(X)$ is the standard unit simplex, is examined
further. We show that every semimartingale diffusion on the simplex
has a characterization similar to that of Fleming-Viot processes.
We also show that, subject to a mild technical condition, each such
diffusion can be represented as the limiting occupancy fractions of
a simple urn scheme in which a large number of balls jump, singly and
in pairs, at random between urns labelled by points in $X$.
march@math.ohio-state.edu
471. RESOLVENT ESTIMATES FOR FLEMING-VIOT OPERATORS WITH
BROWNIAN DRIFT
Peter March
We define a two parameter scale of Banach spaces of functions
defined on $\Cal M_1(\Bbb R^d)$, the space of probability measures
on d dimensional euclidean space, using weighted sums of the classical
Sobolev norms. We prove that the resolvent of the Fleming-Viot operator
with constant diffusion coefficient and brownian drift acts boundedly
between certain members of the scale. These estimates guage the degree
of smoothing performed by the resolvent and separate the contribution
due to the diffusion coefficient and that due to the drift coefficient.
march@math.ohio-state.edu
472. AN ALMOST SURE LARGE DEVIATION PRINCIPLE
FOR THE HOPFIELD MODEL
Anton Bovier and V\'eronique Gayrard
We prove a large deviation principle for the finite dimensional
marginals of the Gibbs distribution of the macroscopic `overlap'-parameters
in the Hopfield model in the case where the number of random patterns, $M$,
as a function of the system size $N$ satisfies $\limsup M(N)/N=0$.
In this case the rate function (or free energy as a function of the overlap
parameters) is independent of the disorder for almost all realization
of the patterns and given by an explicit variational formula.
bovier@iaas-berlin.d400.de or gayrard@cpt.univ-mrs.fr
(hardcopy or TeX-file available upon request)
473. PACKING DIMENSION AND CARTESIAN PRODUCTS
Christopher J. Bishop and Yuval Peres
We show that for any analytic set $A$ in $\R^d$,
its packing dimension $\dim_P(A)$ can be represented as
$
\sup_B \{ \dim_H(A \times B) -\dim_H(B) \}
$
where the supremum is over all compact sets $B$ in $\R^d$,
and $\dim_H$ denotes Hausdorff dimension.
(The lower bound on packing dimension was proved by Tricot in 1982).
Moreover, the supremum above is attained, at least if $\dim_P(A) < d$.
In contrast, we show that the dual quantity
$
\inf_B \{ \dim_P(A \times B) -\dim_P(B) \}
$
is at least the ``lower packing dimension'' of $A$,
but can be strictly greater.
(The lower packing dimension is greater or equal than
the Hausdorff dimension.)
bishop@math.sunysb.edu peres@stat.berkeley.edu
474. SHRINKAGE ESTIMATORS, SKOROKHOD'S PROBLEM, AND STOCHASTIC
INTEGRATION BY PARTS
Steven N. Evans and Philip B. Stark
For a broad class of error distributions that includes the
spherically symmetric ones, we give a short proof that the usual
estimator of the mean in a $d$ - dimensional shift model is inadmissible under
quadratic loss when $d \ge 3$. Our proof involves representing the error
distribution as that of a stopped Brownian motion, and using elementary
stochastic analysis to obtain a generalisation of an integration by parts lemma
due to Stein in the Gaussian case.
evans@stat.berkeley.edu or stark@stat.berkeley.edu (Plain TeX file available)
475. SUPERPROCESSES AND MCKEAN-VLASOV EQUATIONS WITH CREATION OF
MASS
L. Overbeck
Weak solutions of McKean-Vlasov equations with creation of mass are given in
terms of superprocesses. The solutions can be approximated by a sequence of
non-interacting superprocesses or by the mean-field of multitype
superprocesses with mean-field interaction. The latter approximation is
associated with a propagation of chaos statement for weakly interacting
multitype superprocesses.
overbeck@stat.Berkeley.EDU
476. LARGE DEVIATIONS FROM THE MCKEAN-VLASOV LIMIT FOR SUPER-BROWNIAN
MOTION WITH MEAN-FIELD INTERACTION
L. Overbeck
A large deviation principle is proved for the mean-field measure of a
sequence of n-type super-Brownian motions with a mean-field dependent
immigration. The rate-function is identified as a pertubation of the rate
function for a non-interacting case.
overbeck@stat.Berkeley.EDU
477. NON-LINEAR SUPERPROCESSES
L. Overbeck
Non-linear martingale problems in the McKean-Vlasov sense for superprocesses
are studied. The stochastic calculus on historical trees is used in order to
show that there is a unique solution of the non-linear martingale problems
under Lipschitz conditions on the coefficients.
overbeck@stat.Berkeley.EDU
478. THE `TRUE' SELF-AVOIDING WALK WITH BOND REPULSION ON $\Bbb Z$:
LIMIT THEOREMS
Balint Toth
The `true' self-avoiding walk with bond repulsion is a nearest
neighbour random walk on $\Bbb Z$, for which the probability of
jumping along a bond of the lattice is proportional to
$\exp(-g\cdot\text{number of previous jumps along that bond})$.
First we prove a limit theorem for the distribution of the local
time process of this walk. Using this result, later we prove a
local limit theorem, as $A\to\infty$, for the distribution of
$A^{-2/3}X_{\theta_{s/A}}$, where $\theta_{s/A}$ is a random time
of geometric distribution with mean
$\text{e}^{-s/A}\left(1-\text{e}^{-s/A}\right)^{-1}=
\frac As+O(1)$. In more physical terms: an anomalous superdiffusive
behaviuor is established. As a by-product we also obtain an
apparently new identity related to Brownian excursions and Bessel
bridges.
balint@math-inst.hu (to appear in The Annals of Probability,
AMSTeX or hardcopy available)
479. THE `TRUE' SELF-AVOIDING WALK WITH GENERALIZED BOND REPULSION
ON $\Bbb Z$
Balint Toth
We consider a nearest neighbour random walk on $\Bbb Z$,
for which the probability of jumping along a bond of the lattice
is proportional to\newline
$\exp\left(-g\cdot{\text{(number of previous jumps along that
bond)}}^\kappa\right)$,
with $g>0$, $\kappa\in(0,1]$. After a review of earlier results
obtained for the case $\kappa=1$ we outline the generalizations
for $\kappa\in(0,1)$, obtaining a whole range of anomalous
diffusion limits.
balint@math-inst.hu (published in Journ Stat. Phys. v. 77 pp. 17-33 (1994),
AMSTeX or hardcopy available)
480. MULTIPLE COVERING OF THE RANGE OF A RANDOM WALK ON Z
(ON A QUESTION OF P. ERDOS AND P. REVESZ)
Balint Toth
With probability one, for any positive integer $r$ there are
infinitely many instants when all the sites in the range of a
simple symmetric random walk on $\Bbb Z$ had been visited at
least $r$ times. This result disproves a conjecture of
Erdos and Revesz.
balint@math-inst.hu (to appear in Studia Sci. Math. Hung.
AMSTeX or hardcopy available)
481. CONJUGATE DIFFUSIONS ON $\Bbb R_+$ AND GENERALIZED RAY-KNIGHT PROCESSES
Balint Toth
We define the notion of {\it conjugacy\/} of a pair of
diffusions on ${\Bbb R}_+$. Typical examples of conjugate
diffusions are pairs of squared Bessel processes of generalized
dimension $\delta$, respectively $2-\delta$ (the latter one
being stopped at first hitting of $0$). Using pairs of conjugate
diffusions we construct processes with a structure similar to
the processes arising in the Ray-Knight description of Brownian
local time. These {\it generalized Ray-Knight processes\/} arise
naturally as scaling limits of the local time of non-Markovian
random walks. The main technical result of the paper sheds light
on the intimate probabilistic content of conjugacy.
balint@math-inst.hu (AMSTeX or hardcopy available)
482. SLOW RELAXATION OF 2-D STOCHASTIC ISING MODELS WITH
RANDOM AND NON-RANDOM BOUNDARY CONDITIONS
Yasunari Higuchi and Nobuo Yoshida
We consider finite volume stochastic Ising models corresponding to usual
Ising ferromagnet at sufficiently low temperature with zero external
magnetic field. It is shown that for a certain class of boundary
conditions in which neither (+) nor (--) predominates the other, the
spectral gap on a cube decays exponentially fast in the sidelength.
Moreover, we show the same exponential decay of the spectral gap holds
almost surely for random(1/2-Bernoulli, for example) bondary conditions.
nobuo@kusm.kyoto-u.ac.jp
483. STABLE WINDINGS
Jean Bertoin and Wendelin Werner
We derive the asymptotic laws of winding numbers of planar
isotropic stable L\'evy processes and walks.
jbe@ccr.jussieu.fr, wwerner@dmi.ens.fr (LaTeX file available)
484. LE POINT D' UN FERME LE PLUS VISITE PAR LE MOUVEMENT BROWNIEN.
Christophe Leuridan
Let (B(t), t>=O) b e a brownian motion in R, starting at 0 and (L(t,x),
t>=0, x in R) a continuous version of its local times. Given a closed
subset F of R, that contains 0, we study the right-continuous and left-
limited process (A(t,F), t>=0), where A(t,F) is "the" point in F the
most visited at time t, that is "the" point a in F such that L(t,a)=
L(t,F), where L(t,F) denotes the maximum of L(t,x), x in F.
We prove that the procss ((A(t,F)) has a locally bounded vatiation, and
that the increasing process ((L(t,F)) overestimates the symmetric local
time at 0 of the semimartingale (B(t)-A(t,F)).
When F=R, we prove that this overestimation is in fact an equality,
that the process (A(t,F)) is a purely jump process, and that its jumps
are neither isolated, nor "compelled".
leuridan@fourier.grenet.fr (teX file avalaible)
485. THE RANDOM-CLUSTER MODEL ON A HOMOGENEOUS TREE
Olle H\"aggstr\"om
The random-cluster model on a homogeneous tree is defined and studied. It is
shown that for $1\leq q \leq2$, the percolation probability in the maximal
random-cluster measure is continuous in $p$, while for $q>2$ it has a
discontinuity at the critical value $p=p_c(q)$. It is also shown that for
$q>2$, there is nonuniqueness of random-cluster measures for an entire interval
of values of $p$. The latter result is in sharp contrast to what happens on
the integer lattice ${\bf Z}^d$.
olleh@math.chalmers.se
486. NEAREST NEIGHBOUR AND HARD SPHERE MODELS IN CONTINUUM PERCOLATION
Olle H\"aggstr\"om and Ronald Meester
Consider a Poisson process $X$ in ${\bf R}^d$ with density 1. We connect each
point of $X$ to its $k$ nearest neighbours by undirected edges. The number
$k$ is the parameter in this model. We show that for $k=1$, no percolation
occurs in any dimension, while for $k=2$, percolation occurs when the dimension
is sufficiently large. We also show that if percolation occurs, then there is
exactly one infinite cluster. Another percolation model is obtained by putting
balls of radius zero around each point of $X$ and let the radii grow linearly
in time until they hit another ball. We show that this model exists and that
there is no percolation in the limiting configuration. Finally we discuss some
general properties of percolation models where balls placed at Poisson points
are not allowed to overlap (but are allowed to be tangent).
olleh@math.chalmers.se, meester@math.ruu.nl
487. CUT TIMES FOR SIMPLE RANDOM WALK
Gregory F. Lawler
Let $S(n)$ be a simple random walk taking values in $Z^d$. A
time $n$ is called a cut time if
$$ S[0,n] \cap S[n+1,\infty) = \emptyset . $$
We show that in three dimensions the number of cut times less than
$n$ grows like $n^{1 - \zeta}$ where $\zeta = \zeta_d$ is the intersection
exponent. As part of the proof we show that in two
or three dimensions
$$ P\{S[0,n] \cap S[n+1,2n] = \emptyset \} \asymp n^{-\zeta}, $$
where $\asymp$ denotes that each side is bounded by a constant times
the other side.
jose@math.duke.edu
488. RANDOM WALKS AND HARMONIC FUNCTIONS ON INFINITE PLANAR
GRAPHS, USING SQUARE TILINGS
I. Benjamini and O. Schramm
We study a wide class of transient planar graphs, through a geometric
model given by a square tiling of a cylinder.
For many graphs the geometric boundery of the tiling is a circle,
and is easy to describe in general.
The simple random walk on the graph converges (with probability 1)
to a point in the geometric boundary.
We obtain information on harmonic measure, and estimates on the rate of
convergence.
This allows us to extend results we previously proved for triangulations
of a disc.
itai@sunset.huji.ac.il, schramm@wisdom.weizmann.ac.il
(AMSTeX or hardcopy available)
- To see some information provided by the author
click here.
