Probability Abstracts 24
This document contains abstracts 405-429.
They have been mailed on December 27, 1994.
Click here to see the
list of all abstract titles.
405. A STRASSEN LAW OF THE ITERATED LOGARITHM
FOR PROCESSES WITH INDEPENDENT INCREMENTS
Jia-Gang Wang
Let $X=\{X(t),\, t\ge 0\}$ be a process with independent increments (PII)
such that $E[X(t)]=0$, $Var[X(t)]=E[X(t)]^2<\infty$,
$\lim_{t\to\infty}\frac{D_X(t)}t=1$,
and there exists a majoring measure $G$ for the jump $\Delta X$ of $X$. Under
these assumptions, using rather a direct method, a Strassen's law of the
iterated logarithm (Strassen LIL) is established. As some special cases,
the Strassen LIL for homogeneous PII and for partial sum process of i.i.d.
random variables are comprised.
jgwang@fudan.ihep.ac.cn
406. SELF-AFFINE CARPETS ON THE SQUARE LATTICE
Irene Hueter and Yuval Peres
We explore the ``Hausdorff dimension at infinity" for self-affine
carpets defined on the square lattice. This notion of dimension
(due to Barlow and Taylor), which is the correct notion from a
probabilistic perspective, differs for these sets from more "naive"
indices of fractal dimension.
hueter@stat.berkeley.edu or peres@stat.berkeley.edu (LaTeX file available)
407. EQUILIBRIUM STATISTICAL MECHANICS OF FRUSTRATED SPIN GLASSES:
A SURVEY OF MATHEMATICAL RESULTS
Dimitri Petritis
After a rapid introduction to the physical motivations and a succint
presentation of heuristic results, this survey summarises the main
mathematical results known on the Edwards-Anderson and
the Sherrington-Kirkpatrick models of spin glasses. Although
not complete proofs but rather sketches of the relevant
steps and important ideas are given, only results for which
complete proofs are known --- and for which the author has been able to
reproduce all the intermediate logical steps ---
are presented in the sections entitled `mathematical results'.
This paper is intended to
both physicists, interested to know which articles among the multitude
of papers published on the subject go beyond the heuristic arguments
to obtain rigorous irrefutable results, but also to the mathematicians,
interested in finding out how rich is the physical intuitive
way of thinking and in being inspired by the heuristic results
in view of a mathematical rigorisation. An extended, but not
exhaustive, bibliography is included
petritis@levy.univ-rennes1.fr (LaTeX or postscript file available).
408. ON THE DISTRIBUTION OF DISTANCES IN RECURSIVE TREES
Robert P. Dobrow
Recursive trees have been used to model such things as the
spread of epidemics, family trees of ancient manuscripts, and
pyramid schemes. A tree $T_n$ with $n$ labeled nodes is a
recursive tree if $n=1$, or $n>1$ and $T_n$ can be
constructed by joining node $n$ to a node of some recursive
tree $T_{n-1}$. For arbitrary nodes $i<n$ in a random
recursive tree we give the exact distribution of $D_{i,n}$,
the distance between nodes $i$ and $n$. We characterize this
distribution as the convolution of the law of $D_{i,i+1}$ and
$n-i-1$ Bernoulli distributions. We further characterize the
law of $D_{i,i+1}$ as a mixture of sums of Bernoullis. For $i=i_n$
growing as a function of $n$, we show that $D_{i_n,n} is
asymptotically normal in several settings.
dobrow@cam.nist.gov
409. THE MOVE-TO-ROOT RULE FOR SELF-ORGANIZING TREES WITH
MARKOV DEPENDENT REQUESTS
Robert P. Dobrow
The move-to-root (MTR) heuristic is a self-organizing rule
which attempts to keep a binary search tree in near-optimal
form. It is a tree analogue of the well-studied move-to-front
(MTF) scheme. We study a Markov move-to-root (MMTR) model,
where the sequence of record requests is a Markov chain, and
analyze several characteristics of the tree chain, including
stationary distribution, eigenvalues, and stationary expected
search cost.
dobrow@cam.nist.gov
410. STEPPING STONE MODELS WITH EXTINCTION AND RECOLONIZATION
Hyun-Chung Kang, Stephen M. Krone, and Claudia Neuhauser
The stepping stone model is widely used in population genetics to describe
the evolution of a population with mating and geographical structure.
It is typically formulated on a countable set, where each
element of the set corresponds to a colony. Each colony consists of
a population of a fixed number of haploid individuals. Individuals
undergo random mating within each colony and migrate to neighboring
colonies. In the model considered here, we are interested in
the changes that may occur at a single locus on a single chromosome
at which two alleles are possible. Changes in the gene frequencies are caused
by migration between colonies, in addition to mutation, selection and
random mating within each colony. It is frequently assumed that
local populations persist indefinitely. This, however, is a
reasonable assumption only if the average time until extinction
of a local population is much larger than the time interval over
which the whole population is studied. In this paper, using the framework
of interacting particle systems, we address the
question of how extinction and recolonization affect the spatial distribution
of gene frequencies in both homogeneous and inhomogeneous environments.
It turns out that extinction does not change the qualitative behavior of
the model in a homogeneous environment; that is, in one and two dimensions,
the population clusters in the absence of mutation (i.e., it
consolidates into larger and larger blocks), whereas in higher
dimensions, coexistence is possible. However, there are quantitative
differences. For example, we find that extinction speeds up clustering in
one dimension. In inhomogeneous environments, extinction can drastically
change the behavior and new phenomena, such as coexistence in $d=2$, appear.
krone@cms.wisc.edu (Tex or hard copy available)
411. PERPETUITIES WITH THIN TAILS
Charles M. Goldie and Rudolf Gr\"ubel
We investigate the behaviour of $P(R\ge r)$ and $P(R\le-r)$ as $r\to\infty$
for the random variable $R:=\sum_{n=1}^\infty Q_n\prod_{k=1}^{n-1} M_k$, where
$\bigl((Q_k,M_k)\bigr)_{k\in{\bf N}}$ is an independent, identically
distributed sequence with $P(-1\le M\le1)=1$. Random variables of this type
appear in insurance mathematics, as solutions of stochastic difference
equations, in the analysis of probabilistic algorithms and
elsewhere. Exponential and Poissonian tail behaviour can arise.
C.M.Goldie@qmw.ac.uk (LamsTeX file available)
412. DYNAMIC SCHEDULING WITH CONVEX DELAY COSTS: THE GENERALIZED $c\mu$ RULE
Jan A. Van Mieghem
We consider a general single-server multiclass queueing system that incurs
a delay cost $C_k(\tau_k)$ for each class $k$ job
that resides $\tau_k$ units of time in the system. This paper
derives a scheduling policy that minimizes the total cumulative delay cost
when the system operates during a finite time horizon.
Denote the marginal delay cost and (instantaneous) service rate functions of
class $k$ by $c_k= C_k'$ and $\mu_k$, and let $a_k(t)$ be the ``age'' or time
that the oldest class $k$ job has been waiting at time $t$.
We call the scheduling policy that at time $t$ serves the oldest waiting job
of that class $k$ with the highest index $\mu_k(t) c_k(a_k(t))$,
the {\em Generalized $c\,\mu$ Rule}. As a dynamic priority rule that depends
on very little data, the Generalized $c\,\mu$ Rule is attractive to
implement.
We show that with non-decreasing convex delay costs, the
Generalized $c \, \mu$ Rule is asymptotically optimal if the system
operates in heavy traffic, and give explicit expressions for the
associated performance characteristics: the delay (throughput time) process
and the minimum cumulative delay cost.
The optimality result is robust in that it holds for a countable number of
classes and several homogeneous servers in a non-stationary, deterministic or
stochastic environment where arrival and service processes can be general and
interdependent.
janvm@leland.stanford.edu (postscript file available)
413. SOME PROBLEMS CONCERNING RANDOM WALKS ON TREES
Russell Lyons, Robin Pemantle, and Yuval Peres
We state some unsolved problems and describe relevant examples
concerning random walks on trees. Most of the problems involve the
behavior of random walks with drift: e.g., is the speed on
Galton-Watson trees monotonic in the drift parameter? These random
walks have been used in Monte-Carlo algorithms for sampling from the
vertices of a tree; in general, their behavior reflects the size and
regularity of the underlying tree. Random walks are related to
conductance. The distribution function for the conductance of
Galton-Watson trees satisfies an interesting functional equation; is
this distribution function absolutely continuous?
rdlyons@silver.ucs.indiana.edu (hard copy available)
414. LYAPUNOV EXPONENTS OF LINEAR STOCHASTIC FUNCTIONAL DIFFERENTIAL
EQUATIONS DRIVEN BY SEMIMARTINGALES
Salah-Eldin A . Mohammed and Michael K. R. Scheutzow
We consider a class of stochastic linear functional differential systems driven
by semimartingales with stationary ergodic increments. Stochastic functional
differential systems are SDE's whose coefficients depend on the past evolution
of the state. We allow smooth convolution-type dependence of the noise term on
the history of the state. Using a stochastic variational technique, we construct
a compactifying stochastic semiflow on an infinite-dimensional state space. As
a necessary ingredient of this construction, we prove a general perfection
theorem for crude cocycles with values in a metrizable second countable
topological group. Our theorem is an extension of a previous result of
de Sam Lazaro and Meyer. Using a similar argument, the above perfection
theorem was further generalized in Arnold and Scheutzow's paper
{\it Perfect cocycles through stochastic differential equations
(Probability Abstract 389)\/}, in order to cover the case of crude cocycles
parametrized by locally compact topological groups.
In the paper we also prove the existence of a countable {\it deterministic\/}
Lyapunov spectrum for the stochastic functional differential system using
a Ruelle-Oseledec multiplicative ergodic theorem.
salah@c-math1.siu.edu (Hard copy available)
415. CONTROLE DE LA NORME Hp D'UNE MARTINGALE PAR DES MAXIMUMS DE
TEMPS LOCAUX.
Christophe Leuridan
Let B be a brownian motion starting at 0. We note L(t,*) = max{L(t,x),x real}
the maximum of local times at time t. In the well-known Barlow-Yor inequali-
ties, can we replace L(t,*) with L(t,F) = max{L(t,x),x in F}, where F is a gi-
ven closed set? In this paper, we give a condtition on F that is necessary and
sufficient: roughly speaking, if p>=1, F must not be far from containing two
opposite geometric sequences. We prove various related results too.
leuridan@fourier.grenet.fr (teX file avalaible)
416. LES THEOREMES DE RAY-KNIGHT ET LA MESURE D'ITO POUR LE
MOUVEMENT BROWNIEN SUR LE TORE R/Z.
Christophe Leuridan
Let B be a brownian motion on the torus R/Z, starting at 0. In this paper ,we
give two theorems giving the law of local-times-process of B at certain stop-
ping times, and a description of characteristic measure of the excursions of
B from the origin. These results are the equivalents (related to B) of the
well-known Ray-Knight theorems and Williams' description of Ito's measure for
a brownian motion in R. We give a few applications of these results.
leuridan@fourier.grenet.fr (teX file avalaible)
417. ON THE RATE OF CONVERGENCE IN THE FUNCTIONAL LAW OF
THE ITERATED LOGARITHM FOR NONSTANDARD NORMALIZING FACTORS
A.V. Bulinskii, M.A.Lifshits
Let $W(.)$ be a Wiener process and $ Y_t(s)= t^{-1/2} W(ts)/ L_t $.
Let $K$ be a unit ball of RKHS corresponding to Wiener measure. In 1964
V.Strassen proved that for $ L_t = (2loglogt)^{1/2} $ the distance
between $Y_t$ and $K$ tends to zero for large $t$. Recently K.Grill
and M.Talagrand proved independently that under this normalization the
order of the mentioned distance is $ (loglogt)^{-2/3} $ . We solve this
problem for arbitrary nondecreasing normalizer L_t. It turns out, in par-
ticular, that the answer of Grill and Talagrand is very unstable:
for L_t = (2loglogt)^{1/2} +1 approximation rate is (loglogt)^{-3/4} ;
for L_t = (2loglogt)^{1/2} -1 approximation rate is (loglogt)^{-1/2} ;
Our results relate also to some striking phenomenas encountered by D.Mason
and U.Einmal in the similar problem for the process generated by sums of
i.i.d. random variables.
mikhail@lifshits.spb.su
418. ON THE BEST POSSIBLE CONVERGENCE RATE
IN THE STRASSEN LAW FOR RANDOM POLIGONS
A.V. Bulinskii, M.A.Lifshits
Let $Z(.)$ be a random poligonal function generated in a usual way by
sums of i.i.d. random variables with zero mean and unit variance. Let
$ Y_n(s)= (n 2loglogn)^{-1/2} Z(ns) $.
Let $K$ be a unit ball of RKHS corresponding to Wiener measure. In 1964
V.Strassen proved that the distance between $Y_n$ and $K$ tends to zero
for large $n$. Recently K.Grill and M.Talagrand proved independently that
for Gaussian summands the order of the mentioned distance is
$(loglogn)^{-2/3}$. Later on, U.Einmal and D.Mason investigated the gene-
ral case. In particular,they discovered that for specific summands appro-
ximation rate can be faster (!) than the mentioned Gaussian rate. We find
out the exact range of possible rates showing that the rate
$(loglogn)^{-3/4} G(loglogn)$ can be attained by sums of i.i.d. summands
iff the integral
$ \int^{\infty} dx / (x G(x)^4) $
is finite.
For example, the rate $ (loglogn)^{-3/4} (logloglogn)^A $ is avaliable
whenever $A>1/4$.
mikhail@lifshits.spb.su
419. SCANNING BROWNIAN PROCESSES
Robert J. Adler and Ron Pyke
The `scanning process' $Z(t),\ t\in\R^k$, of the title is a Gaussian random
field obtained by associating with $Z(t)$ the value of a set-indexed Brownian
motion on the translate $t+A_0$ of some `scanning set' $A_0$.
We study the basic properties of the random field $Z$, relating, for
example, its continuity and other sample path properties
to the geometrical properties of $A_0$.
We ask if the set $A_0$ determines the scanning process, and
investigate when, and how, it is possible to recover the structure of $A_0$
from realisations of the sample paths of the random field $Z$.
pyke@math.washington.edu (TeX file available)
420. THE RATE OF CONVERGENCE IN THE RATIO ERGODIC THEOREM FOR
MARKOV PROCESSES
Davar Khoshnevisan
Let A(t) and B(t) denote the continuous additive functionals
of a recurrent standard Markov process possessing local times.
Let mu_A and mu_B denote their respective Revuz measures.
If mu_A and mu_B are finite measures, then it is well
known that a.s., A(t)/B(t) converges to mu_A(R)/mu_B(R).
Under minimal conditions, this article describes the precise
rate at which this convergence takes place. This extends the
recent work of Marcus and Rosen as well as Bertoin; it also
complements the Kakutani-Peterson theorem of ergodic theory.
davar@math.utah.edu
421. SOLUTIONS OF Lu=u^a DOMINATED BY L-HARMONIC FUNCTIONS
E.B.Dynkin and S.E.Kuznetsov
Let L be a second order elliptic differential operator on a
differentiable manifold M and let a be a number in the interval (1,2]. We
investigate connections between class U of all positive solutions of the
equation Lu=u^a and class H of all positive L-harmonic functions (i.e.,
solutions of the equation Lh = 0). Define U_0 as a collection of all
functions u from U such that u does not exceed an L-harmonic function h.
To every element u of U_0 there corresponds a minimal L-harmonic function
h_u which dominates u and this is a 1-1 mapping from U_0 onto a subset H_0
of H. The inverse mapping associates with every element h of H_0 the
maximal element of U dominated by h.
Denote by g(x,dy) Green's kernel, by k(x,y) the Martin kernel and
by M' the Martin boundary associated with L. A subset B of M' is called
R-polar if it is not hit by the range R of the (L,a)-superdiffusion. It is
called M-polar if
\int_M g(c,dx) [\int_B k(x,y)\nu(dy)]^a
is equal to 0 or \infty for every point c of M and every measure \nu on B.
Every L-harmonic function h has a unique representation
h(x)=\int_M' k(x,y) \nu(dy)
where \nu is a measure concentrated on the minimal part M^* of M'.
We show that, the condition:
(a) \nu(B)=0 for all R-polar sets B --
is necessary and the condition:
(b) \nu(B)=0 for all M-polar sets B --
is sufficient for h to belong to H_0. If M is a bounded domain of class
C^{2,\lambda} in R^d, then conditions (a) and (b) are equivalent and
therefore each of them characterizes H_0. This has been conjectured by
Dynkin a few years ago and proved recently by Le Gall for the case
L= the Laplacian, a=2 and M is a domain of class C^5 in R^d.
ebd1@cornell.edu
422. HEDGING OPTIONS FOR A LARGE INVESTOR AND FORWARD-BACKWARD SDE'S
Jaksa Cvitanic and Jin Ma
In the classical continuous-time financial market model, the stock prices
have been understood as solutions to linear stochastic differential
equations; and an important problem to solve is the problem of hedging
options (functions of the stock price values at the expiration date). In
this paper we consider the hedging problem with a price model that is
not only nonlinear, but the coefficients of the price equations can also
depend on the portfolio startegy and the wealth process of the hedger.
In mathematical terminology, the problem translates to solving a Forward-
Backward Stochastic Differential Equation with the forward diffusion part
being degenerate. We show that, under some conditions, the Four Step
Scheme of Ma, Protter and Yong for solving forward-backward SDEs still
works in this case, and we extend the classical results of hedging
contingent claims to this new model. Included in the examples is the
case of the stock volatility increase caused by overpricing the option,
as well as the case of different interest rates for borrowing and lending.
majin@math.purdue.edu/cj@harold.stat.columbia.edu
423. THE RSW THEOREM FOR CONTINUUM PERCOLATION AND THE CLT FOR
EUCLIDEAN MINIMAL SPANNING TREES.
Kenneth S. Alexander
We prove a central limit theorem for the length of the minimal spanning tree
of the set of sites of a Poisson process of intensity lambda in [0,1]^2 as
lambda approaches infinity. As observed previously by Ramey (Ph.D. thesis,
Yale, 1982), the main difficulty is the dependency between the contributions
to this length from different regions of [0,1]^2, and a percolation-theoretic
result on circuits surrounding a fixed site can be used to control this
dependency. We prove such a result on circuits via a continuum-percolation
version of the Russo-Seymour Welsh Theorem on occupied crossings of a
rectangle. This RSW Theorem also yields a variety of results for
two-dimensional fixed-radius continuum percolation already well-known for
lattice models, including a finite-box criterion for percolation, and absense
of percolation at the critical point.
alexandr@math.usc.edu (hard copy available)
424. STOCHASTIC DYNAMICS MACROSCOPICALLY
GOVERNED BY THE POROUS MEDIUM EQUATION
Michael Ekhaus and Timo Seppalainen
We introduce a class of interacting lattice models
on the torus whose special feature is that the macroscopic equation
of the empirical density is a degenerate parabolic equation.
The models come in two versions: One with continuous variables and
one with particles on the sites. In the particle model a degenerate
equation is obtained only if the size of the particle vanishes in the
limit, otherwise the limiting equation is a nondegenerate equation that
also governs the densities of certain exclusion processes with speed
change. We establish basic properties of these models such as
attractiveness and reversibility, and prove the hydrodynamic scaling
limits for the empirical densities. As a special case of the degenerate
equation we get the equation of an ideal gas flowing isothermally
through a porous medium.
seppalai@ml.kva.se (hardcopy and AmSTeX file available)
425. NON-INCREASE OF BROWNIAN MOTION AND RANDOM COVERINGS
Krzysztof Burdzy
An old result of Dvoretzky, Erd\"os and Kakutani says
that Brownian motion has no points of increase.
We give a new proof based on excursion theory
and a theorem of Shepp on random coverings.
burdzy@math.washington.edu (AmS-TeX file available)
426. INTERSECTION-EQUIVALENCE OF BROWNIAN PATHS
AND CERTAIN BRANCHING PROCESSES
Yuval Peres
We show that sample paths of Brownian motion
(and other stable processes) intersect the same sets as certain
random Cantor sets constructed by a branching process.
With this approach, the classical result that two
independent Brownian paths in four dimensions do not
intersect reduces to the dying out of a critical
branching process, and a rough form of
estimates due to Lawler~(1982) for the
long-range intersection probability of several random walk paths,
reduces to Kolmogorov's 1938 law for the lifetime of a
critical branching process.
Extensions to random walks with long jumps and applications to
Hausdorff dimension are also derived.
peres@stat.berkeley.edu
427. SOME MARKOV PROCESSES WITH BROWNIAN EXIT DISTRIBUTIONS
Zoran Vondracek
Let $D$ be a bounded domain in $R^d$ with regular boundary. Let $X=(X_t, P^x)$
be a standard Markov process in $D$ with continuous paths up to its lifetime.
If $X$ satisfies some weak conditions, then it is possible to add a non-local
part to its generator, and construct the corresponding standard Markov process
in $D$ with Brownian exit distributions from $D$.
vondra@math.hr (latex file available)
428. A CHARACTERIZATION OF CERTAIN DIFFUSIONS ON THE SIERPINSKI GASKET
BY THE EXIT DISTRIBUTIONS
Zoran Vondracek
Let $K$ be the Sierpinski gasket with vertices $a_1, a_2, a_3$
forming an equilateral triangle. Suppose that $(X_t,P^x)$ and $(Y_t, Q^x)$
are diffusions on $K$ with the same hitting probabilities to vertices
$a_1, a_2, a_3$. We show that if $X$ is an asymptotically one-dimensional
diffusion or a $p$-stream diffusion, then $Y$ is a time-change of $X$.
vondra@math.hr (latex file available)
429. SOME ANALYTICAL SEMIGROUPS OCCURING IN PROBABILITY THEORY
K.H.Hofmann and Z.J.Jurek
In the theory of operator--limit distributions (see:
Z.J.Jurek, J.D.Mason, Operator -- limit distributions in probability
theory, J.Wiley, New York, 1993) one needs to deal with some semigroups.
In the present paper we define semigroup ${\bold S}$ as the semidirect
product of probability measures ${\Cal P}$ (on a Banach space $E$) and
endomorphisms ${\bold E}nd$ of $E$, with the operation $"\qed"$ given by
$(\mu,A)\qed(\nu,B):=(\mu\ast A\nu,AB)$. We prove that each one--parameter
subsemigroup $T$ of ${\bold S}$ is of the form
$$
T(t)=({\Cal L}(\int_{(0,t]}e^{sQ}dY(s)),e^{tQ}),\ \ t\ge0\ ,
$$
for uniquely determined operator $Q$ and L\'evy process $Y$; [${\Cal
L}(\cdot)$ denotes the probability distribution of stochastic integral].
Observe that the above formula, as a particular case
corresponding to the degenerate L'evy processes $Y$, contains the
well--known characterization of one--parameter subgroups of the Lie group
of invertible affine self--maps of $E$. Furthermore relations between the
semigroup ${\bold S}$ and the Urbanik decomposability semigroups are
studied.
ZJJUREK@math.uni.wroc.pl (TeX file available)
