Probability Abstracts 28
This document contains abstracts 520-570.
They have been mailed on August 4, 1995.
Click here to see the
list of all abstract titles.
520. BIASED RANDOM WALKS ON GALTON-WATSON TREES
Russell Lyons, Robin Pemantle and Yuval Peres
We consider random walks with a bias toward the root on the
family tree $T$ of a supercritical Galton-Watson branching process and
show that the speed is positive whenever the walk is transient. The
corresponding harmonic measures are carried by subsets of the boundary
of dimension smaller than that of the whole boundary. When the bias is
directed away from the root and the extinction probability is positive, the
speed may be zero even though the walk is transient; the critical bias
for positive speed is determined.
rdlyons@silver.ucs.indiana.edu pemantle@math.wisc.edu
peres@stat.berkeley.edu
521. A NEW DUALITY RELATION FOR RANDOM WALKS
Holger Dette and Jim Pitman and William J. Studden
For the random walk on the nonnegative integers with reflecting barrier at 0
and transition probabilities p_i and q_i = 1 - p_i for increments
of +1 and -1, i > 0, it is shown that the right tails of the probability of the
first return from state 0 to state 0 are simple transition probabilities of a
dual random walk obtained by switching p_i and q_i. A combinatorical and
an analytical proof are presented. Extensions and relations to other concepts
of duality in the literature are discussed.
dette@namu01.gwdg.de pitman@stat.Berkeley.edu studden@stat.purdue.edu
522. CYCLICALLY STATIONARY BROWNIAN LOCAL TIME PROCESSES
Jim Pitman
Local time processes parameterized by a circle, defined by the occupation
density up to time T of Brownian motion with constant drift on the circle,
are studied for various random times T. While such processes are typically
non-Markovian, their Laplace functionals are expressed by series formulae
related to similar formulae for the Markovian local time processes subject to
the Ray-Knight theorems for BM on the line,
and for squares of Bessel processes and their bridges.
For T the time that BM on the circle first returns to its starting point after
a complete loop around the circle, the local time process is cyclically
stationary, with same two-dimensional distributions, but not the
same three-dimensional distributions, as the sum of squares of two i.i.d.
cyclically stationary Gaussian processes. This local time process is
the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions.
The corresponding intensity measure on path space,
and similar L\'evy measures derived from squares of Bessel processes,
are described in terms of a 4-dimensional Bessel bridge
by Williams' decomposition of It\^o's law of Brownian excursions.
pitman@stat.Berkeley.EDU
523. HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS:
A MARTINGALE APPROACH
Jaksa Cvitanic and Ioannis Karatzas
We derive a formula for the minimal initial wealth needed to
hedge an arbitrary contingent claim in a continuous-time, diffusion model
with proportional transaction costs; the expression obtained can
be interpreted as the supremum of expected discounted values
of the claim, over all (pairs of) probability
measures under which the ``wealth process" is a supermartingale.
Next, we prove the existence of an optimal solution to
the portfolio optimization problem of maximizing utility from
terminal wealth in the same model; we also characterize this solution
via a transformation to a
hedging problem: the optimal portfolio is the one that hedges
the inverse of marginal utility evaluated at the shadow state-price density
solving the corresponding dual problem, if such exists. We can then use
the optimal shadow state-price density for pricing
contingent claims in this market. The mathematical tools
are those of continuous-time martingales, convex analysis, functional
analysis and duality theory.
cj@stat.columbia.edu (TeX or ps file available on request)
524. RANDOM WALKS ON THE LAMPLIGHTER GROUP
Russell Lyons, Robin Pemantle and Yuval Peres
Kaimanovich and Vershik (1983) described certain finitely generated
groups of exponential growth such that simple random walk on the Cayley
graph escapes from the identity at a sublinear rate, or equivalently,
all bounded harmonic functions on the Cayley graph are constant. Here
we focus on a key example, called $G_1$ by Kaimanovich and Vershik, and
show that {\it inward-biased} random walks on the Cayley graph of this
group escape from the identity at a linear rate provided the bias
parameter is smaller than the growth rate of $G_1$. These walks can be
viewed as random walks interacting with a dynamical environment on
$\Z$. The proof uses potential theory to analyze a stationary
environment as seen from the moving particle.
rdlyons@indiana.edu (PlainTeX and hardcopies available)
525. METASTABILITY OF THE d-DIMENSIONAL CONTACT PROCESS
Adilson Simonis
We prove that the d-dimensional
supercritical contact process exhibits metastable behavior, in the
pathwise sense. This is done by proving the property of thermalization
and using the Mountford's theorem. We also extend some
previous results on the loss
of memory of the process.
asimonis@ime.usp.br
- To see some information provided by the author
click here.
526. BIVARIATE MEASURES VIA A POWER SERIES CONDITIONAL CONDITIONAL
DISTRIBUTION AND A REGRESSION FUNCTION
Jacek Wesolowski
Given a mixture $\mu_{Y|X}$, the problem of identifiability of the join
distribution by the posterior mean $E(X|Y)$ is considered. A wide family
of power series mixtures $\mu{Y|X}=PSD(a,X)$ is studied. Recall that
$PSD(a,\theta)$ is defined by the pmf $p_k=a(k)\theta^k/f(\theta)$,
$k=0,1,...$, where $a>0$ is a coefficient function and $f>0$ is a series
function. It is shown that under the condition $\sum_{k=1}^{\infty}
\sqrt[2k]{a(k)}=\infty$ the mixture is identifiable. A similar result
holds for the mixture $\mu_{Y|X}=PSD(a,1/X)$. Examples involving
geometric, binomial and Poisson mixtures are provided.
wesolo@alpha.im.pw.edu.pl
527. EXPONENTIAL STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN
BY DISCONTINUOUS SEMIMARTINGALES
Xuerong Mao and A.E. Rodkina
Both almost sure exponential stability and in mean square of stochastic
differential equations driven by semimartingales, which in general are
discontinuous, are investigated under various hypotheses. New methods
are introduced to incorporate the discontinuity of semimartingales.
The Dol\'eans-Dale stochastic exponential, the convergence of nonnegative
special semimartingales established by Liptser and Shiryayev as well
as the It\^o formula play a great role in this paper.
xuerong@stams.strath.ac.uk
528. LONG TIME ASYMPTOTICS FOR SOME DYNAMICAL
NOISE FREE NON-LINEAR FILTERING PROBLEMS
Frederic Cerou
A good way to study the accuracy of optimal non-linear filtering is to
look at the concentration of the conditional law on arbitrary
neighborhood of the current (unknown) state as the time goes to
infinity. In this paper we consider a
restricted class of systems without noise in their state equation and
under some more assumptions we show some results concerning the
concentration of the conditional law, and also how estimating
the current state could be different from estimating the initial
condition. Our assumptions could seem quite restrictive, however they
concern only the ``observability'' of the system, without any
reference to compactness or ergodicity as in previous works.
cerou@sophia.inria.fr
- To see some information provided by the author
click here.
529. LONG TIME ASYMPTOTICS FOR SOME DYNAMICAL
NOISE FREE NON-LINEAR FILTERING PROBLEMS: NEW CASES
Frederic Cerou
We are interested here in the long time behaviour of the conditional
law for a special case of filtering problem: there is no noise on the
state equation and the prior law of the state process concentrate fast
in some neighborhood of a limit cycle with strictly negative
characteristic exponents. Then assuming a deterministic observability
property on the cycle we show the concentration of the conditional law
on an arbitrary
neighborhood of the current (unknown) state as the time goes to
infinity. This work can be considered as illustrating
how the tools of dynamical systems
theory can be use to study the long time behavior of the filtering
process.
cerou@sophia.inria.fr
- To see some information provided by the author
click here.
530. PARTICLE AND CELL APPROXIMATIONS FOR NONLINEAR FILTERING
Fabien Campillo, F. Cerou, F. Le Gland and R. Rakotozafy
We consider the nonlinear filtering problem for systems
with noise--free state equation.
First, we study a particle approximation of the a
posteriori probability distribution, and we give an estimate of the
approximation error.
Then we show, and we illustrate with numerical examples, that
this approximation can produce a non consistent estimation of the state
of the system when the measurement noise tends to zero. Hence, we
propose a histogram--like modification of the particle
approximation, which is always consistent. Finally, we present an application to
target motion analysis.
campillo@sophia.inria.fr, cerou@sophia.inria.fr, legland@irisa.fr
- To see some information provided by the author
click here.
531. LYAPUNOV EXPONENTS OF CONTROLLED SDE'S AND STABILIZABILITY PROPERTY
Fabien Campillo and Abdoulaye Traore
We consider a stochastic differential equation with linear feedback
control:
$$ dX_t = (A+B\,K)\,X_t\, dt + \sum_k=1^r(A_k+B_k\,K)\,X_t\,\circ dW_k(t)$$
where $K$ is the feedback gain matrix. For each value of $K$, let
$\lambda_K$ be the Lyapunov exponent associated with the solution of
the SDE. The set of $\lambda_K$, as $K$ describe the set of
matrices, is a connected interval of $\R$. We present some examples
where $-\infty$ is the lower bound of this set. For these cases, we
say that the corresponding EDS is stabilizable.
campillo@sophia.inria.fr, traore@sophia.inria.fr
- To see some information provided by the author
click here.
532. A SPHERICAL BOUND FOR THE SHERRINGTON-KIRKPATRICK MODEL.
Francis Comets
The Sherrington-Kirkpatrick model exhibits a phase transition
at temperature 1 (in the sense of non-analyticity of the
pressure). Making use of the domination by the spherical model,
we derive a bound for the pressure and for the ground state
energy. (Sept. 1994, dedicated to J. Neveu on his 60th
birthday).
comets@mathp7.jussieu.fr
533. A CENTRAL LIMIT THEOREM FOR CONDITIONALLY CENTERED RANDOM
FIELDS WITH AN APPLICATION TO MARKOV FIELDS
Francis Comets and Martin Janzura
We prove a central limit theorem for conditionally centered random
field, under condition of moments and of strict positivity of the
empirical variance per observation. We use a random normalization,
which fits to non-stationary situations. The theorem directly
applies to Markov random fields, including the case of phase
transition and lack of stationarity. One consequence is the
asymptotic normality of the maximum pseudo-likelihood estimator
for Markov fields in complete generality.
comets@mathp7.jussieu.fr janzura@utia.cas.cz
534. DETECTING PHASE TRANSITION FOR GIBBS MEASURES
Francis Comets
We propose a new empirical procedure
for detecting phase transition from a single sample of a
Gibbs-Markovrandom field. The method is based on frequencies
for large deviations when the whole sample is divided in
smaller blocks, and estimates for the rate function. We relate
our approach to some almost sure large deviation principle.
comets@mathp7.jussieu.fr
535. KAC'S MOMENT FORMULA FOR ADDITIVE FUNCTIONALS OF A MARKOV PROCESS
Jim Pitman
Mark Kac introduced a method for calculating the distribution of the integral
$A_v = \int_0^T v(X_t) dt$ for a function $v$ of a Markov process
$(X_t, t \ge 0 )$, and suitable random $T$, which yields the Feynman-Kac
formula for the moment-generating function of $A_v$. This paper reviews Kac's
method, with emphasis on an aspect often overlooked. This is Kac's formula for moments of $A_v$ which may be stated as follows. For any random time $T$ such
that the killed process $(X_t, 0 \le t <T)$ is Markov with some substochastic
semi-group $K_t(x,dy) = P_x(X_t \in dy, T > t)$, and any non-negative measurable $v$, for $X$ with initial distribution $\lambda$ the $n$th moment of
$A_v$ is $P_\lambda A_v^n = n! \lambda (G M_v) ^n 1 $ where
$G = \int_{0}^\infty K_t dt$ is the Green's operator of the killed process,
$M_v$ is the operator of multiplication by $v$,
and $1$ is the function that is identically 1.
pitman@stat.Berkeley.EDU
536. CONSTRAINED VARIATIONAL PROBLEM WITH APPLICATIONS TO THE
ISING MODEL
Roberto H. Schonmann and Senya B. Shlosman
We continue our study of the behavior of the two-dimensional nearest
neighbor ferromagnetic Ising model under an external magnetic field $h$,
initiated in [SS](*). We strengthen further a result previously proven by
Martirosyan at low enough temperature, which roughly states that for
finite systems with $(-)$-boundary conditions under a positive external
field, the boundary effect dominates in the system if the linear size
of the system is of order $B/h$ with $B$ small enough, while if $B$ is large
enough, then the external field dominates in the system. In [SS] this result
was extended to every subcritical value of the temperature.
Here for every subcritical value of the temperature we show the existence
of the critical value $B_0(T)$, which separates the two regimes specified
above. We also find the asymptotic shape of the region occupied by the
$(+)$-phase in the second regime, which turns out to be a ``squeezed Wulff
shape''.
The main step in our study is the solution of the variational
problem of finding the curve minimizing the Wulff functional,
which curve is constrained to the unit square.
Other tools used are the results and techniques developed to study
large deviations for the block magnetization in the absence of the
magnetic field, extended to all temperatures below the
critical one.
(*) [SS] is the paper: "Complete analyticity for 2D Ising completed"
by the same authors. To appear in Commun. Math. Phys.
rhs@math.ucla.edu (Postscript file available)
537. PARTITION STRUCTURES DERIVED FROM BROWNIAN MOTION AND
STABLE SUBORDINATORS
Jim Pitman
Explicit formulae are obtained for the distribution of various random
partitions of a positive integer $n$, both ordered and unordered, derived
from the zero set $M$ of a Brownian motion by the following
scheme: pick $n$ points uniformly at random from $[0,1]$, and classify them
by whether they fall in the same or different component intervals of the
complement of $M$. Corresponding results are obtained for $M$ the range of a
stable subordinator and for bridges defined by conditioning on $1 \in M$.
These formulae are related to discrete renewal theory by a general method of
discretizing a subordinator using the points of an independent homogeneous
Poisson process.
pitman@stat.Berkeley.EDU
538. FINITE WIDTH FOR A RANDOM STATIONARY INTERFACE
Carl Mueller and Roger Tribe
This paper is part of a program to study the behavior
of travelling waves for stochastic partial differential
equations. We study the asymptotic shape of the solution
$u(t,x) \in [0,1]$ to a one-dimensional heat equation
with a multiplicative white noise term:
$u_t=u_{xx}+\sqrt{u(1-u)\dot W(t,x)$, for $x\in R$.
Here, $\dot W(t,x)$ is 2-parameter white noise. At time
zero the solution is an interface, that is $u(0,x)$ is $0$
for all large positive $x$ and $u(0,x)$ is $1$ for all
large negative $x$. The special form of the noise term
preserves this property at all times $t \geq 0$. The main
result is that, in contrast to the deterministic heat
equation, the width of the interface remains stochastically
bounded. Thus, the interface behaves somewhat like a
travelling wave, but with 0 velocity.
cmlr@troi.cc.rochester.edu (AmsLatex file available)
539. REFINEMENTS OF THE GIBBS CONDITIONING PRINCIPLE
Amir Dembo and Ofer Zeitouni
Refinements of Sanov's large deviations theorem lead via
Csisz{\'{a}}r's information theoretic identity to refinements
of the Gibbs conditioning principle which are valid for blocks whose length
increase with the length of the conditioning sequence.
Sharp bounds on the growth of the block length with the length of the
conditioning sequence are derived.
Extensions of Csisz{\'{a}}r's triangle inequality and information
theoretic identity to the Markov chain set--up lead to similar refinements in
the Markov case.
amir@playfair.stanford.edu zeitouni@ee.technion.ac.il
(LaTeX and Postscript files available)
540. LARGE DEVIATIONS AND STRONG MIXING
W{\l}odzimierz Bryc and Amir Dembo
The Large Deviation Principle (LDP) with respect to the $\tau $-topology
holds for the empirical measure of any
$\alpha$-mixing or any $\phi $-mixing stationary process with a
hyper-exponential
mixing rate of at least $\alpha (n)\ll \exp (-n(\log n)^{1+\delta })$, for
some $\delta >0$ or at least $\phi (n)\ll \exp (-n\ell (n))$ with $\ell
(n)\to \infty $. Positive recurrent Doeblin Markov chain examples
for which the LDP
does not hold demonstrate the tightness of these rates and the relationship
between the LDP for the empirical means of all bounded $R^d$-valued
functionals and the LDP for the empirical measure.
amir@playfair.stanford.edu (LaTeX or Postscript file available)
541. LARGE DEVIATIONS FOR QUADRATIC FUNCTIONALS OF GAUSSIAN PROCESSES
W{\l}odzimierz Bryc and Amir Dembo
The Large Deviation Principle is derived for several unbounded additive
functionals of centered stationary Gaussian processes. For example, the rate
function corresponding to
$\frac{1}{T} \int_0^T X_t^2 dt$ is the Fenchel-Legendre transform of
$L(y)=-\frac{1}{4\pi}\int_{-\infty}^\infty \log(1-4\pi y f(s)) ds$, where
$X_t$ is a continuous time process with the bounded spectral density $f(s)$.
The spectral density condition for quadratic functionals is weaker than
the necessary condition for bounded continuous functionals.
Similar results in the discrete-time version are obtained
for the energy of multivariate Gaussian processes and for the sums of $p<2$
powers. Moderate deviations and optimal CLTs are obtained for unbounded
spectral density.
Explicit rate functions are obtained in several instances.
amir@playfair.stanford.edu (LaTeX or Postscript file available)
542. ON LARGE DEVIATIONS OF EMPIRICAL MEASURES FOR STATIONARY
GAUSSIAN PROCESSES
W{\l}odzimierz Bryc and Amir Dembo
We show that the Large Deviation Principle
with respect to the weak topology holds for the empirical
measure of any stationary continuous time Gaussian process with
continuous vanishing at infinity spectral density. We also point out that
Large Deviation Principle might fail in both continuous and
discrete time if the spectral density is bounded but discontinuous.
amir@playfair.stanford.edu (LaTeX or Postscript file available)
543. MODERATE DEVIATIONS FOR MARTINGALES WITH BOUNDED JUMPS
Amir Dembo
We prove that the Moderate Deviation Principle (MDP) holds for the
trajectory of a locally square integrable martingale with bounded jumps
as soon as its quadratic covariation, properly scaled, converges in
probability at an exponential rate.
A consequence of this MDP is the tightness of the method of bounded
martingale differences in the regime of moderate deviations.
amir@playfair.stanford.edu (LaTeX or Postscript file available)
544. FILLING THE HYPERCUBE IN THE SUPERCRITICAL CONTACT PROCESS IN
EQUILIBRIUM
Adilson Simonis
We consider the supercritical $d-$dimensional contact process,
obtaining news results about the asymptotic distribution of the time
of first ocurrence of an anomalous density of particles in a fixed region
of the space, when the supercritical process starts from equilibrium. In
particular, we get uniform sharp bounds for the rates of convergence
in distribution of this time to a mean one exponential random
variable, when suitably rescaled.
asimonis@ime.usp.br
- To see some information provided by the author
click here.
545. ON THE CONNECTED COMPONENTS OF THE SUPPORT OF SUPER BROWNIAN
MOTION AND OF ITS EXIT MEASURE
Romain Abraham
R. Tribe proved in a previous paper that a typical point of the support
of super Brownian motion considered at a fixed time is a.s. disconnected
from the others when the space dimension is greater than equal to 3.
We give here a new proof of the result based on Brownian snake. This
proof can then be adapted in order to obtain an analogous result for
the support of the exit measure of the super Brownian motion from a smooth
domain when the dimension is greater than equal to 4.
abraham@math-info.univ-paris5.fr
546. SPECTRA OF GRAPHS AND FRACTAL DIMENSION III
Andras Telcs
The paper provides local central limit theorem, a result on the exponent of
the spectral density and a large deviation theorem of random walks on a big
class of fractals - i.e. fulfilling smoothness condition. This crucial
assumption on the fractals, the smoothness condition - which ensures
the Einstein relation - is reconsidered and it is shown that in the
transient case ( d_{Omega} >2 ) it is equivalent with the condition on the
Green function G(x,y) and the graph distance d(x,y):
c< G(x,y)/d^{2-d_{Omega}}(x,y) <C
where c and C constants do not depend on x,y vertices.
Similar equivalence is valid for recurrent graphs as well.
This interpretation of the smoothness uses boundary theory.
Illustrating the necessity, a counterexample in the transient case is given.
In the recurrent case the Einstein relation is proven for nonsmooth graphs
of finitely many smooth subgraphs. Results are applied to a branching random
walk problem.
h197tel@ella.hu (hard copy available)
547. ENLARGEMENT OF THE WIENER FILTRATION BY A MANIFOLD VALUED RANDOM ELEMENT
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 and take their values in some
product of Grassmannians or flag manifold.
Continuing the analysis of a previous paper, we therefore consider the
enlargement of the Wiener filtration by a random element $F$ with values
in some compact Riemannian manifold. Applying the tools of Malliavin's
calculus, we derive smoothness conditions on $F$ under which the semimartingale
property with respect to the enlarged filtration is preserved, and a priori
martingale inequalities between the norms of stochastic intrgrals of
processes adapted to the enlarged filtration and their quadratic variations
are valid.
imkeller@vega.univ-fcomte.fr
548. MULTIPLE INTERSECTION LOCAL TIME OF PLANAR BROWNIAN MOTION AS A PARTICULAR
HIDA DISTRIBUTION
Peter Imkeller and Jia-An Yan
Subspaces of the space of Hida distributions are quantified in which the
multiple intersection local time of planar Brownian motion, renormalized by
taking off the term of order zero in its chaos decomposition, lives. This is
done by means of a new formula expressing the expectation of higher products
of multiple Wiener-Ito-integrals by certain functions of the scalar products of
their kernels.
imkeller@vega.univ-fcomte.fr
549. ON THE CONVERGENCE OF MULTI-TYPE BRANCHING PROCESSES WITH VARYING
ENVIRONMENTS
Owen Jones
Using the ergodic theory of non-negative matrices, conditions are obtained
for the ${\cal L}^2$ and almost sure convergence of a supercritical
multi-type branching process with varying environment, normed by its mean.
We also give conditions for the limit to be continuous on $(0,\infty)$ and
for the extinction probability of the limit to equal that of the process.
The theory developed allows for different types to grow at different rates,
and an example of this is given, taken from the construction of a spatially
inhomogeneous diffusion on the Sierpinski gasket.
O.D.Jones@sheffield.ac.uk (LaTeX 2e, postscript or hardcopy available)
550. ON ADDITIVE FUNCTIONALS OF MARKOV CHAINS
Endre Cs\'aki and Mikl\'os Cs\"org\H{o}
Consider a Harris-recurrent Markov chain $S_n$ on a measurable state space.
In this paper strong approximation results are proved for additive functionals
of the form $\sum_{i=0}^N f(S_i)$. Both positive and null recurrent cases are
treated. In the former case the limiting process is a Brownian motion
itself, while in the latter case it is a composition of a Brownian motion
and an inverse of stable subordinator. The proofs are based on an idea of
Athreya-Ney and Nummelin to split the chain into independent blocks.
csaki@novell.math-inst.hu (TeX file available)
551. ON ADDITIVE FUNCTIONALS OF DIFFUSION PROCESSES
Endre Cs\'aki and Paavo Salminen
Consider a recurrent difusion process $X_t$ with its local time $L_t^x$.
In this paper strong approximation results are proved for additive
functionals of the form
$$Z_t=\int_0^t f(X_s)\, ds=\int f(x)L_t^x\, m(dx),$$
where $m$ is the speed measure. Both positive and null recurrent cases are
treated. In the former case the limiting process is a Brownian motion
itself, while in the latter case it is a composition of a Brownian motion
and an inverse of stable subordinator.
csaki@novell.math-inst.hu (TeX file available)
552. ON THE LENGTH OF THE LONGEST MONOTONE BLOCK
Endre Cs\'aki and Ant\'onia F\"oldes
Consider a sequence of i.i.d. random variables $X_1,X_2,\dots$. The
section $(X_{i+1},\dots ,X_{i+j})$ is called a monotone block (MB) of
length $j$ if $X_{i+1}\leq\dots\leq X_{i+j}$. It is called a strictly
monotone block if $X_{i+1}<\dots<X_{i+j}$. Extending results of Pittel and
R\'ev\'esz, asymptotic properties of the length of the longest MB and SMB
up to the $n$-th observation for discrete distributions are studied. In
contrast to the continuous case (when the length of the longest monotone
block is of order $\log n/\log\log n$ it turns out that the length of the
longest MB is of the order $\log n$ for any discrete distribution. On the
other hand, the order of the length of the longest SMB depends on the
distribution of the random variables $X_n$. Several examples, such as
geometric distribution, Poisson distribution, etc. are treated.
csaki@novell.math-inst.hu (TeX file available)
553. GLOBAL STRASSEN-TYPE THEOREMS FOR ITERATED BROWNIAN MOTIONS
Endre Cs\'aki, Mikl\'os Cs\"org\H{o}, Ant\'onia F\"oldes and P\'al
R\'ev\'esz
A class of iterated processes is studied by proving a joint
functional limit theorem for a pair of independent Brownian
motions. This Strassen method is applied to prove global $(t\to
\infty)$, as well as local ($t\to 0$), LIL type results
for various iterated processes. Similar results are also proved for
iterated random walks via invariance.
csaki@novell.math-inst.hu (LaTeX file available)
554. POISSON APPROXIMATIONS FOR FUNCTIONALS OF RANDOM TREES
Robert P. Dobrow and Robert T. Smythe
We use Poisson approximation techniques for sums of
indicator random variables to derive explicit error
bounds and central limit theorems for several functionals
of random trees. In particular, we consider (i) the
number of comparisons for successful and unsuccessful
search in a binary search tree and (ii) internode distances
in increasing trees. The Poisson approximation setting
is shown to be a natural and fairly simple framework
for deriving asymptotic results.
dobrow@cam.nist.gov smythe@gwuvm.gwu.edu
555. GENERALIZED RAY-KNIGHT THEORY AND LIMIT THEOREMS
FOR SELF-INTERACTING RANDOM WALKS ON $\Bbb Z$
Balint Toth
We consider non-Markovian, self-interacting random walks (SIRW)
on the one-dimensional integer lattice $\Bbb Z$. The walk starts
from the origin and at each step jumps to a neighbouring site,
the probability of jumping along a bond being proportional to
$w(\hbox{number of previous jumps along that lattice bond})$,
where $w:\Bbb N\to\Bbb R_+$ is a monotone weight function.
Exponential and subexponential weight functions were considered
in earlier papers. In the present paper we consider weight
functions $w$ with power-law asymptotics. These weight functions
define variants of the so-called `reinforced random walk'.
We prove functional limit theorems for the local time processes
of these random walks and local limit theorems for the position
of the random walker at late times. A generalization of the
Ray-Knight theory of local time arises.
balint@math-inst.hu (AMSTeX or hardcopy available)
556. COUPLING IN BRANCHING -- A TUMOR MODEL WITH POPULATION SIZE DEPENDENCE
Peter Jagers
Think of a tumour where cells at birth either enter a quiescent state
or else pass through the cell cycle, ending by mitotic division. Assume
that the probability of entering the cell cycle is a decreasing function
of tumor size, whose limit is strictly greater than 1/2, as size tends
to infinity. In deterministic structured population dynamics such models
have been studied under the name of non-linear tumour models with quiescence.
Here a coupling argument in a branching process formulation yields a
condition for the tumour to exhibit so called balanced or asynchronous
exponential growth, i.e. exponential growth plus stabilization of composition.
The argument extends to general branching processes which are regulated
in a sense close to that introduced by Sevastyanov: A population size
dependent coin toss determines whether an individual will reproduce
or not. If she does, that occurs like in a general (not population
size dependent) branching population.
jagers@math.chalmers.se
557. THE METASTABLE BEHAVIOR OF THE THREE DIMENSIONAL STOCHASTIC ISING MODEL
Dayue Chen, Jianfeng Feng and Minping Qian
We study the metastable behavior of a ferromagnetic spin system in a
finite three dimensional torus, in the limit as the temperature goes
to zero. It is proved that small clusters of +1 spins are likely to
shrink and disappear, while large clusters are likely to grow. The
critical droplet is identified. All metastable states are charac-
terized and a hierarchic structure is found. For a large class of
initial states, we estimate the logarithmic asymptotics of the hit-
ting time of the state that all spins are +1 or -1.
dayue@stat.pku.edu.cn (AMSTeX file available)
558. OPTIMAL SCALING OF DISCRETE APPROXIMATIONS TO LANGEVIN DIFFUSIONS
Gareth O. Roberts and Jeffrey S. Rosenthal
In recent work of Roberts, Gelman and Gilks (1994), the problem of
optimal scaling of proposal variances for random-walk Metropolis
algorithms was considered. It was proved that, for Gaussian proposals
and certain target distributions, the optimal asymptotic acceptance
probability in high dimensions is approximately $0.234$. Furthermore,
it was shown that the proposal variance should scale as $n^{-1}$ as the
dimension $n \to \infty$.
In this paper, we pursue a similar study for a class of algorithms
given by discrete approximations to Langevin diffusions. We show that,
for certain target distributions $\pi$, the optimal asymptotic
acceptance probability in high dimensions is approximately $0.574$.
Furthermore, the proposal variance should scale as $n^{-1/3}$.
Therefore, Langevin algorithms are considerably more efficient than
random-walk based Metropolis methods; the optimal proposal variances
and acceptance probabilities are both substantially larger.
jeff@utstat.toronto.edu (Postscript file or hardcopy available.)
- To see some information provided by the author
click here.
559. FAITHFUL COUPLINGS OF MARKOV CHAINS: NOW EQUALS FOREVER
Jeffrey S. Rosenthal
This short note considers the usual coupling approach to bounding
convergence of Markov chains. It addresses the question of whether it
suffices to have two chains become equal at a {\it single} time, or
whether it is necessary to have them then {\it remain} equal for all
future times. It provides sufficient conditions (``faithful
couplings'') under which the condition $X_T = Y_T$, for some random
time $T$, implies that $\|L(X_k)-L(Y_k)\| \le P(T \ge k)$.
jeff@utstat.toronto.edu (Postscript file or hardcopy available.)
- To see some information provided by the author
click here.
560. EXISTENCE, UNIQUENESS AND EGODICITY FOR STOCHASTIC QUANTIZATION EQUATION
Dariusz Gatarek and Beniamin Goldys
In this note we discuss the so-called stochastic quantization
equation in two-dimensional finite area $D$:
$$dX=[-{1\over 2}AX-\lambda A^{-2\alpha}:X^3:]dt+A^{-\alpha}dW,$$
where $A$ is a properly chosen power of the operator $I-\Delta$
and $W$ is a cylindrical Wiener process in the space $L^2(D)$.
Nonlinear term in this equation is the so-called Wick power.
This equation is of some importance in quantum field theory.
Existence, uniqueness and ergodicity of weak solutions to
equation of stochastic quantization is obtained
as a simple consequence of the Girsanov theorem.
gatarek@solution.maths.unsw.edu.au (PlainTex available)
561. ON THE SUPPORT OF HARMONIC MEASURE FOR THE RANDOM WALK
Itai Benjamini
Let Q^{d}(n) denote the d-dimensional n x ...x n cube
in the lattice Z^d i.e. Q^{d}(n)= {(x_1,...x_d): x_i \in 0,..,n-1}.
Given a set of vertices A in the cube, let \mu_{v}(S) denote
the harmonic measure supported on A for the SRW starting at v.
That is for S \subset A, \mu_{v}(S) is the probability that the
SRW first visit to A is in S. We prove the following.
Theorem: For any \epsilon > 0 there is N(\epsilon) so that for any
n > N(\epsilon)
and any v and A in Q^{d}(n), there is a set S(v,A) with no more
than \epsilon n^{d} vertices, \mu_{v}(S(v,A)) > 1- \epsilon.
itai@wisdom.weizmann.ac.il (latex file available)
562. THE RETRIEVAL PHASE OF THE HOPFIELD MODEL: A RIGOROUS ANALYSIS OF
THE OVERLAP DISTRIBUTION
Anton Bovier, V\'eronique Gayrard
Standard large deviation estimates
or the use of the Hubbard-Stratonovich transformation reduce the
analysis of the distribution of the overlap parameters essentially to that of
an explicitly known random function $\Phi_{N,\b}$
on $\R^M$. In this article we present a
rather careful study of the structure of the minima of this random function
related to the retrieval of the stored patterns. We denote by $m^*(\b)$ the
modulus of the spontaneous magnetization in the
Curie-Weiss model and by $\a$ the ratio between the number of the
stored patterns and the system size. We show that there exist strictly positive
numbers $0<\g_a<\g_c$ such that
1) If $\sqrt\a\leq \g_a (m^*(\b))^2$, then the
absolute minima of $\Phi$ are located within small balls
around the points $\pm m^*e^\mu$, where $e^\mu$ denotes the $\mu$-th
unit vector while
2) if $\sqrt\a\leq \g_c (m^*(\b))^2$ at least a local minimum
surrounded by extensive energy barriers exists near these points.
The random location of these minima is given within precise bounds.
These are used to prove sharp estimates on the support of the Gibbs measures.
bovier@iaas-berlin.d400.de or gayrard@cpt.univ-mrs.fr
(hardcopy or PS-file available upon request)
563. SYMMETRIZATION AND CONCENTRATION INEQUALITIES FOR MULTILINEAR FORMS WITH
APPLICATIONS TO ZERO-ONE LAWS FOR L\'EVY CHAOS
Jan Rosinski and Gennady Samorodnitsky
We consider stochastic processes ${\bf X} = \{ X(t),\, t\in T \}$
represented as a L\'evy chaos of finite order, i.e. as a finite sum of
multiple stochastic integrals with respect to a symmetric infinitely
divisible random measure. For a measurable subspace $V$ of ${\bf R}^T$
we prove a very general zero-one law $P({\bf X} \in V)=0$ or $1$,
providing a complete analog to the corresponding situation in the case
of symmetric infinitely divisible processes (single integrals with
respect to an infinitely divisible random measure).
Our argument requires developing a new symmetrization technique for
multilinear Rademacher forms, as well as generalizing Kanter's
concentration inequality to multiple sums.
rosinski@novell.math.utk.edu or gennady@orie.cornell.edu
(LaTeX or hardcopy available)
564. CLASSES OF MIXING STABLE PROCESSES
Jan Rosinski and Gennady Samorodnitsky
Every measurable stationary $\alpha$-stable process with $0<\alpha<2$
can be related to a nonsingular flow on a $\sigma$-finite measure space.
We establish the relationship between properties of the flow and mixing
of the stationary stable process. We provide the first example of a
mixing stationary stable process corresponding to a conservative flow.
We show further the connection between the expected return time of the
flow to sets of finite positive measure and the mixing properties of the
process.
rosinski@novell.math.utk.edu or gennady@orie.cornell.edu
(LaTeX or hardcopy available)
565. SIMPLE CONDITIONS FOR MIXING OF INFINITELY DIVISIBLE PROCESSES
Jan Rosinski and Tomasz Zak
Let ${\bf X} = (X_t)_{t\in T}$ be a real-valued, stationary, infinitely
divisible stochastic process, where $T = {\bf R}$ or $\bf Z$. We show
that $\bf X$ is mixing if and only if $X_0 - X_t$ converges in
distribution to the symmetrization of $X_0$. The latter condition can be
further simplified when the location of atoms of the L\'evy measure of
$X_0$ is known.
If $X_t = \int f_t \, dM$, where $f_t$ is a deterministic function and
$M$ is an infinitely divisible random measure without Gaussian
component, then $\bf X$ is mixing if and only if the supports of $f_0$
and $f_t$ are essentially asymptotically disjoint. This condition is
much stronger than the corresponding condition for mixing of Gaussian
processes.
We also give parallel characterizations of the weak mixing property of
infinitely divisible processes.
rosinski@novell.math.utk.edu or zak@graf.im.pwr.wroc.pl
(AMSTeX or hardcopy available)
566. THE EQUIVALENCE OF ERGODICITY AND WEAK MIXING FOR INFINITELY DIVISIBLE
PROCESSES
Jan Rosinski and Tomasz Zak
The equivalence of ergodicity and weak mixing for general infinitely
divisible processes is first proven. Using this result and the authors'
earlier work "Simple conditions for mixing of infinitely divisible
processes", simple conditions for ergodicity of infinitely divisible
processes are derived.
The notion of codifference for infinitely divisible processes is
investigated; it plays the crucial role in the proofs but it may be also
of independent interest.
rosinski@novell.math.utk.edu or zak@graf.im.pwr.wroc.pl
(AMSTeX or hardcopy available)
567. SOME POLAR SETS FOR THE BROWNIAN SHEET
Davar Khoshnevisan
Let W denote d dimensional Brownian sheet with
N parameters. In a recent paper, Chen (1994) has
proven that W has intersection local times for
points of k-multiplicity when (d-2N)k <d.
This is a generalization of the results of
Adler (1981), Ehm (1981) and Rosen (1984). It is
also well known that W has local times at points
when d<2N.
The goal of this paper is to prove two complementary
results, posed in the review of Fristedt (1995): when
(d-2N)k \geq d, W does not have points of k multiplicity,
and when d \geq 2N, W does not hit points. The methods
presented here are elementary and work for a large class
of Gaussian random fields as they do not rely on capacity
theory of Markov processes.
davar@math.utah.edu (Plain TeX file available)
568. NONLINEAR PARABOLIC P.D.E. AND ADDITIVE FUNCTIONALS OF
SUPERDIFFUSIONS
E.B.Dynkin, S.E.Kuznetsov
Suppose that E is an arbitrary domain in R^d, L is a second order
elliptic differential operator in S=R_+\times E and S^e is the extremal
part of the Martin boundary for the corresponding diffusion \xi. Let
1<\alpha\le 2. We investigate a boundary value problem
{\partial u}/{\partial r}+Lu-u^\alpha = -\eta in S,
u = \nu on S^e,
u = 0 on \infty\times E (*)
involving two measures \eta and \nu. For the existence of a solution,
we give sufficient conditions in terms of Martin capacities and necessary
conditions in terms of hitting probabilities for an
(L,\alpha)-superdiffusion X. If a solution exists, then it can be
expressed by an explicit formula through an additive functional A of X.
An (L,\alpha)-superdiffusion is a branching measure-valued process. A
natural linear additive (NLA) functional A of X is determined uniquely by
its potential h defined by the formula:
P_\mu A(0,\infty)=\int h(r,x)\mu(dr,dx)
for all \mu in M^* (the determining set of A). The potential h is an
exit rule for \xi and it has a unique decomposition into extremal
exit rules. If \eta and \nu are measures which appear in this decomposition,
then (*) can be replaced by an integral equation
u(r,x)+\int p(r,x;t,dy)u(t,y)^\alpha ds=h(r,x) (**)
where p(r,x;t,dy) is the transition function of \xi. We prove that
h is the potential of an NLA functional if and only if (**) has a
solution u. Moreover,
u(r,x)=-\log P_{r,x} exp{-A(0,\infty)}.
By applying these results to homogeneous functionals of time-homogeneous
superdiffusions, we get a stronger version of theorems proved in our
earlier publication. The basis for our present investigation is laid by a
general theory developed in an accompanying paper.
sk47@cornell.edu
569. NATURAL LINEAR ADDITIVE FUNCTIONALS OF SUPERPROCESSES
E.B.Dynkin, S.E.Kuznetsov
We investigate natural linear additive (NLA) functionals of a general
critical (\xi,K,\psi)-superprocess X. We prove that all of them have only
fixed discontinuities. All homogeneous NLA functionals of
time-homogeneous superprocesses are continuous [this was known before
only in the case of quadratic branching.]
We introduce an energy operator \E(u) defined in terms of (\xi,K,\psi)
and we prove that the potential h and the log-potential u of an NLA
functional A are connected by the equation u+\E(u)=h. The potential is
always an exit rule for \xi and the condition h+\E(h)<\infty a.e. is
sufficient for an exit rule h to be a potential.
In an accompanying paper, these results are applied to boundary value
problems for partial differential equations involving nonlinear operator
Lu-u^\alpha where L is a second order elliptic differential operator and
1<\alpha\le 2.
sk47@cornell.edu
570. LIMIT PROCESSES FOR AGE-DEPENDENT BRANCHING PARTICLE SYSTEMS
Ingemar Kaj and Serik Sagitov
We consider systems of spatially distributed branching particle systems in
$R^$. The particle lifelengths are of general form, hence the time propagation
of the system is typically not Markov. A natural time-space-mass scaling is
applied to a sequence of particle systems and we derive limit results for the
corresponding sequence of measure-valued processes.
The limit is identified as the projection of a superprocess in $R_+\times R^d$.
The additive functional characterizing the superprocess is the scaling limit
of certain point processes, which count generations along a line of descent for
the branching particles.
ingemar.kaj math.uu.se
