Probability Abstracts 25

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430. HARMONIC FUNCTIONS ON PLANAR AND ALMOST PLANAR GRAPHS AND MANIFOLDS, VIA CIRCLE PACKINGS

Itai Benjamini and Oded Schramm 

The circle packing theorem is used to show that
on any transient planar graph there exists
a non constant, harmonic, bounded, Dirichlet function.
If $P$ is a bounded circle packing in $\R^2$ whose
contacts graph is a bounded valence triangulation of a disk, then,
with probability $1$, the simple
random walk on $P$ converges to a limit point.
Moreover, in this situation any continuous function on the
limit set of $P$ extends to a continuous harmonic
function on the closure of the contacts graph of $P$;
that is, this Dirichlet problem is solvable.

We define the notions of almost planar graphs and manifolds,
and show that under the assumptions of transience and
bounded local geometry these possess 
non constant, harmonic, bounded, Dirichlet functions.
Let us stress that an almost planar graph is not necessarily
roughly isometric to a planar graph.

itai@math.huji.ac.il or schramm@wisdom.weizmann.ac.il
                       (AMSTeX or hardcopy available)

• To see a preprint or other information provided by the author click here.
• Or here.

431. DIFFUSIONS AND RANDOM SHADOWS IN NEGATIVELY-CURVED MANIFOLDS

Russell Lyons

Let $M$ be a $d$-dimensional complete simply-connected negatively-
curved manifold. There is a natural notion of Hausdorff dimension for
its boundary at infinity. This is shown to provide a notion of global
curvature or average rate of growth in two probabilistic senses: First,
on surfaces ($d = 2$), it is twice the critical drift separating
transience from recurrence for Brownian motion with constant-length
radial drift. Equivalently, it is twice the critical $\beta$ for the
existence of a Green function for the operator $\Delta/2 - \beta
\partial_r$.  Second, for any $d$, it is the critical intensity for
almost sure coverage of the boundary by random shadows cast by balls,
appropriately scaled, produced from a constant-intensity Poisson point
process. These two theorems are analogues of some earlier results of
the author about random walks and percolation on trees.

rdlyons@silver.ucs.indiana.edu     (hard copy available)

432. THE TRACE OF SPATIAL BROWNIAN MOTION IS CAPACITY-EQUIVALENT TO THE UNIT SQUARE

Robin Pemantle, Yuval Peres and Jonathan W. Shapiro
 
   We show that with probability 1, the trace $B[0,1]$ of 
Brownian motion in space, has positive capacity with respect
to exactly the same kernels as the unit square. (Here all
kernels that are decreasing functions of distance are allowed). 
   More precisely, the energy of occupation measure on $B[0,1]$  
in the kernel $f(|x-y|)$, is bounded above and below by constant
multiples of the energy of Lebesgue measure on the unit square.
(The constants are random, but do not depend on the kernel.)
   The  upper bound on energy is based on a  strong law
for the approximate self-intersections of the Brownian path:
an  asymptotic estimate for the probability, conditional on
the path, that two points chosen independently according to
occupation measure on $B[0,1]$, will be close to each other.
  We also prove analogous capacity estimates for planar 
Brownian motion and for the zero-set of one-dimensional
Brownian motion, which rely on Varadhan's renormalization
and on Kingman's intrinsic construction of local time, respectively.

peres@gandalf.Berkeley.EDU

433. NON-INCREASE OF BROWNIAN MOTION VIA BRANCHING PROCESS ARGUMENT

Krzysztof Burdzy

This proof of non-increase of Brownian motion is not
the shortest of all existing proofs but it is elementary.
One can define a branching structure related to
the Brownian motion. Non-existence of points of increase
is equivalent to dying out of the branching process.
The idea is related to a recent article of Yuval Peres (PAS 426).

burdzy@math.washington.edu

• To see some information provided by the author click here.

434. HAUSDORFF DIMENSION OF CUT POINTS FOR BROWNIAN MOTION

Gregory F. Lawler

Let $B$ be a Brownian motion in $R^d, $d=2,3$. A time $t \in [0,1]$
is called a cut time for $B[0,1]$ if 
\[   $B[0,t) \cap $B(t,1] = \emptyset . \]
We show that the Hausdorff dimension of the set of cut times equals
$1 - \zeta$ where $\zeta = \zeta_d$ is the intersection exponent.
The theorem, combined with known estimates on $\zeta_3$,
shows that the percolation dimension of Brownian motion (the
minimal Hausdorff dimension of a subpath of a Brownian path) is
strictly greater than $1$ in $R^3$.

jose@math.duke.edu

• To see some information provided by the author click here.
• Or here.

435. PARAMETER ESTIMATION FOR MOVING AVERAGES WITH POSITIVE INNOVATIONS

Paul D. Feigin, Marie F. Kratz and Sidney I. Resnick

This paper continues the study of time series models generated by
non-negative innovations which was begun in Feigin and Resnick (1992,1994).
We concentrate on moving average processes. Estimators for moving average
coefficients are proposed and consistency and asymptotic distributions
established for the case of an order one moving average assuming either the
right or left tail of the innovation distribution is regularly varying.  The
rate of convergence can be superior to that of the Yule--Walker or maximum
likelihood estimators.

paulf@ie.technion.ac.il, kratz@math-info.univ-paris5.fr, sid@orie.cornell.edu

436. SECOND--ORDER REGULAR VARIATION AND RATES OF CONVERGENCE IN EXTREME--VALUE THEORY

Laurens de Haan and Sidney I. Resnick

Rates of convergence of the distribution of the extreme order
statistic to its limit distribution are given in the uniform metric and the
total variation metric. A second--order regular variation condition is
imposed by supposing a Von Mises type condition which allows a unified
treatment. Rates are constructed from the parameters of the second--order
regular variation condition. Some connections with Poisson processes are
discussed.

dehaan@cs.few.eur.nl, sid@orie.cornell.edu

437. LARGE DEVIATIONS FROM THE ALMOST EVERYWHERE CENTRAL LIMIT THEOREM

Peter March, Timo Seppalainen

We prove large deviation principles
for the almost everywhere central limit theorem, assuming that
the i.i.d. summands have finite moments of all orders. 
The level 3 rate function is a specific entropy relative to
Wiener measure and the level 2 rate the Donsker-Varadhan entropy 
of the Ornstein-Uhlenbeck process. In particular, the rate  functions
are independent of the particular distribution of the i.i.d. process
under study. We deduce these results from a large deviation
theory for Brownian motion via  Skorokhod's representation of 
random walk as Brownian motion evaluated at random
times. The results for Brownian motion come from the well-known large
deviation theory of the Ornstein-Uhlenbeck process,
by a mapping between the two processes. 

seppalai@ml.kva.se (hardcopy and amstex file available)

438. REFLECTING DIFFUSION PROCESSES AND ITS APPLICATION TO ATM NETWORK WITH FEEDBACK CONTROL

Tetsuya Suda, Char-Lin Wu and Weian Zheng

We discuss the diffusion approximation problems in Asynchronous Transfer Model
(ATM) networks with feedback control. We obtain the time averages of
queueing size and sell loss. Those values will enable us to optimize 
the control.
 The arriving ATM cells will be allowed through a control device
to enter into the node and to be stored in the buffer of finite size
$M$ before they are sent to their destinations. After certain peiord of
time the queue size may reach a prefixed threshold value $\beta .$ 
As soon as it reaches the threshold value a BECN (Backward Explicit Control 
Notification) cell will be sent back to the
control device and the transmission rate will be reduced in order to relieve
the congestion in the node. After an another time period, the queue
size may drop to a prefixed recovery value $\alpha .$ As soon as it reaches the
recovery value, an another BECN cell will be sent to the control device and
the transmission rate will be resumed. ......

wzheng@math.uci.edu  (LaTeX file available) 

439. MINIMIZING TRANSACTION COSTS OF OPTION HEDGING STRATEGIES

Eric R. Grannan and Glen H. Swindle

This paper introduces a method for constructing option hedging
strategies in the presence of transaction costs.  The approach
begins with the prescription of a large, but tractable class of
strategies.  A variational problem is constructed in which the
expected square replication error is minimized subject to a fixed
initial portfolio value from among the class of strategies.  Solution
of this variational problem results in a replicating strategy which
simulations show outperforms strategies previously considered.  We
illustrate this method in a particular class of strategies which
contains Leland's discrete time replication scheme.  We show that a
strategy which uses varying time intervals between hedging can
significantly reduce replication error for a given initial wealth.
We will also construct and assess strategies obtained by optimizing
a mean-variance criterion.  This methodology extends to other
optimization problems involving initial portfolio value and expected square
replication error, as well as to other classes of strategies

swindle@pstat.ucsb.edu   (Postscript version available)

440. SHIFT-COUPLING AND CONVERGENCE RATES OF ERGODIC AVERAGES

Gareth O. Roberts  and  Jeffrey S. Rosenthal

Recent work of various authors has used minorization conditions and
drift conditions to obtain quantitative bounds on convergence to
stationarity of certain Markov chains.  Such bounds are particularly
important in the applied area of Markov chain Monte Carlo (MCMC).  Much
of this analysis has used the method of {\it coupling}.  One of the
difficulties in applying such results is that the two coupled Markov
chains must become equal {\it at the same time}.  

In this paper, we avoid such difficulties by examining convergence of
ergodic averages ${1 \over n} \sum_{k=1}^n P(X_k \in \cdot)$ of the
Markov chain distributions rather than of the individual distributions
$P(X_n \in \cdot)$ themselves.  This allows us to replace the method of
coupling with the related method of shift-coupling, in which two Markov
chains are allowed to become equal at {\it different} times.  We
present general bounds concerning such convergence, which are often
much sharper than previous related bounds.  Various examples of MCMC
are also presented.

jeff@utstat.toronto.edu

• To see some information provided by the author click here.

441. MARKOV CHAIN CONVERGENCE: FROM FINITE TO INFINITE

Jeffrey S. Rosenthal

For Markov chains on \it finite \rm state spaces, previous authors have
obtained a number of very useful quantitative bounds on convergence
rate, including those which involve choices of paths.  In this paper,
we consider the extent to which bounds for finite Markov chains can be
extended to infinite chains.

Our results take two forms.  For countably-infinite state spaces $\X$,
we consider the process of \it enlargements \rm of Markov chains,
namely considering Markov chains on finite state spaces
$\X_1,\X_2,\ldots$ whose union is $\X$.  Bounds for the Markov chains
restricted to $\X_d$, if uniform in $d$, immediately imply bounds on
$\X$.  Results for finite Markov chains, involving choices of paths,
can then be applied to countable chains.  We develop these ideas and
apply them to several examples of the Metropolis-Hastings algorithm on
countable state spaces.

For continuous state spaces, we consider the process of \it refinements
\rm of Markov chains.  Namely, we break the original state space $\X$
into countable numbers of smaller and smaller pieces, and define a
Markov chain on these finite pieces which approximates the original
chain.  Under certain continuity assumptions, bounds on the countable
Markov chains, including those related to choices of paths, will imply
bounds on the original chain.  We develop these ideas and apply them to
an example of a continuous state space Metropolis algorithm.

jeff@utstat.toronto.edu

• To see some information provided by the author click here.

442. EXPONENTIAL STABILITY AND INSTABILITY OF STOCHASTIC NEURAL NETWORKS

X.X. Liao and X. Mao

In this paper we shall discuss stochastic effects to the stability property
of a neural network $\dot u(t)=-Bu(t)+Ag(u(t))$. Suppose the stochastically 
perturbed neural network is described by an It\^o equation 
$dx(t)=[-Bx(t) +Ag(x(t))]dt + \sigma (x(t)) dw(t)$. The general theory on 
the almost sure exponential stability and instability of the stochastically 
perturbed neural network is first established. The theory is then applied to 
investigate the stochastic stabilization and destabilization of the neural 
network. Several interesting examples are also given for illustration.

xuerong@stams.strath.ac.uk

443. PROBABILISTIC TECHNIQUES IN ANALYSIS

Richard F. Bass

This book, published by Springer-Verlag,  deals with those areas in 
analysis to which probability has contributed something, either in 
new ideas, new proofs, or new insights. The table of contents follows.

 I PROBABILITY 
   Preliminaries 
   Brownian motion 
   Markov properties 
   Martingales 
   Stochastic integrals 
   Stochastic calculus 
   Weak convergence 
 II POTENTIAL THEORY 
   The Dirichlet problem 
   Choquet capacities 
   Newtonian potentials and Green functions 
   Transition densities 
   Newtonian capacity 
   Excessive functions 
   Martin boundary 
 III LIPSCHITZ DOMAINS 
   The boundary Harnack principle 
   The Martin boundary for Lipschitz domains 
   The conditional lifetime problem 
   Fatou theorems 
   Support of harmonic measure 
 IV SINGULAR INTEGRALS 
   Maximal functions 
   Hilbert transforms 
   Riesz transforms 
   Littlewood-Paley functions 
   Singular integral operators and Fourier multipliers 
   The space $H^1$ 
   Bounded mean oscillation 
 V ANALYTIC FUNCTIONS 
   Conformal invariance 
   Range of analytic functions 
   Boundary behavior of analytic functions 
   Harmonic measure 
   The angular derivative problem 
   The corona problem 

To see a more detailed table of contents, an up-to-date list of errata,
or ordering information, see the "url" below.

bass@math.washington.edu

• To see some information provided by the author click here.

444. ON ASYMPTOTIC PROPERTIES OF MAXIMUM LIKELIHOOD ESTIMATORS FOR PARABOLIC SPDE'S

M. Huebner and B. Rozovskii

We investigate asymptotic properties of the maximum likelihood estimators 
for parameters occuring in parabolic SPDE's of the form
$$ du(t,x) = (A_0 + \theta A_1) u(t,x) dt + dW(t,x).  $$
where $A_0$ and $A_1$ are partial differential operators and $W$ is
a cylindrical Brownian motion.
We introduce a spectral method for  computing MLE's based on 
finite dimensional approximations to solutions of such systems, and 
establish criteria for consistency, asymptotic normality and asymptotic 
efficiency as the dimension of the approximation goes to 
infinity.  We derive the asymptotic properties of the MLE from a
condition on the order of the operators. In particular,
the MLE is consistent if and only if
$\hbox{ord} (A_1) \ge \frac12 ( \hbox{ord} (A_0 + \theta A_1) -d) .$
(to appear in Prob.Th.Rel.Fields)

mhuebner@huebner.stt.msu.edu    (hardcopy available)

445. ON THE EXISTENCE OF SMOOTH DENSITIES FOR JUMP PROCESSES

J. Picard

We consider a L\'evy process $X_t$ and the solution
$Y_t$ of a stochastic differential equation driven by $X_t$; we
suppose that $X_t$ has infinitely many small jumps, but its L\'evy
measure may be very singular (for instance it may have a countable
support). We obtain sufficient conditions ensuring
the existence of a smooth density for $Y_t$; these conditions are
similar to those of the classical Malliavin calculus for continuous diffusions.
More generally, we study the smoothness of the law of variables $F$
defined on a Poisson probability space; the basic tool is a duality
formula from which we estimate the characteristic function of $F$.

picard@ucfma.univ-bpclermont.fr

• To see a preprint or other information provided by the author click here.

446. REFINED LARGE DEVIATION LIMIT THEOREMS

Vladimir Vinogradov

The monograph is mainly devoted to studying the asymptotic behavior of 
the probabilities of large deviations in the case when the famous 
Cramer's condition is failed. In the three initial chapters, we 
systematically construct asymptotic expansions of the probabilities of 
large deviations for the classical scheme of summation of independent 
and identically distributed random variables. To this end, we develop the 
direct probabilistic method that enables one to construct expansions of 
increasing accuracy.
In Chapter 4, we modify this direct probabilistic method in order to 
investigate the asymptotics of the probabilities of large deviations of 
certain order statistics and their sums. Such results in turn provide a 
better understanding of the nature of formation of large deviations in 
the case when Cramer's condition fails.
In the conclusive Chapter 5, we treat the case when Cramer's condition is 
fulfilled only on a finite interval, which often takes place in various 
applied models. Combining a modification of the classical Cramer's 
techniques with some results and methods developed in the previous 
chapters enabled the author to establish a number of new results and also 
to suggest their probabilistic interpretation.

vinograd@unbc.edu

447. POISSONIAN APPROXIMATION FOR THE TAGGED PARTICLE IN ASYMMETRIC SIMPLE EXCLUSION

P.A. Ferrari,  L. R. G. Fontes

We consider the position of a tagged particle in the one dimensional
asymmetric nearest neighbors simple exclusion process. Each particle attempts
to jump to the site to its right at rate $p$ and to the site to its left at
rate $q$. The jump is realized if the destination site is empty. We assume
$p>q$. The initial distribution is the product measure with density $\lambda$,
conditioned to have a particle at the origin.  We call $X_t$ the position
at time $t$ of this particle. Using a result recently proved by the authors
for a semi-infinite zero range process, it is shown that for all $t\ge 0$,
$X_t= N_t -B_t+B_0$, where $\{N_t\}$ is a Poisson process of parameter
$(p-q)(1-\lambda)$ and $\{B_t\}$ is a stationary process satisfying $E\exp
(\theta \vert B_t\vert)<\infty$ for some $\theta>0$. As a corollary we
obtain that ---properly centered and rescaled--- the process $\{X_t\}$
converges to Brownian motion. A previous result says that in the scale
$t^{1/2}$, the position $X_t$ is given by the initial number of empty sites
in the interval $(0,\lambda t)$ divided by $\lambda$.  We use this to compute 
the asymptotic covariance at time $t$ of two tagged particles initially at sites
$0$ and $rt$. The results also hold for the net flux between two queues in a
system of infinitely many queues in series.

pablo@ime.usp.br

448. ASYMMETRIC CONSERVATIVE PROCESSES WITH RANDOM RATES

I. Benjamini, P. A. Ferrari, C. Landim

We study a one dimensional nearest neighbor simple
exclusion process for which the rates of jump are chosen randomly at time zero
and fixed for the rest of the evolution. The $i$-th particle's right and left
jump rates are denoted $p_i$ and $q_i$ respectively; $p_i+q_i=1$.  We fix
$c\in (1/2,1)$ and assume that $p_i\in[c,1]$ is a stationary ergodic process.
We show that there exists a critical density $\rho^*$ depending only on
the distribution of $\{p_i\}$ such that for almost all choices of the 
rates and a (fixed) density $\rho^*< \rho \le 1$ 
there exists an invariant distribution for the process as seen from a
tagged particle with asymptotic density $\rho$. Under this measure, the
distribution of the distances between particles are independent random
variables. We also show that under the invariant distribution, the position
$X_t$ of the tagged particle at time $t$ can be sharply approximated 
by a Poisson process. Finally, we prove the hydrodynamical limit 
for zero range processes with random rate jumps.

pablo@ime.usp.br

449. p-ADIC WHITE NOISE, CHAOS EXPANSIONS AND STOCHASTIC INTEGRATION

Steven N. Evans

Previous work of the author has studied a notion of
``Gaussianity'' for random variables taking values in a vector
space over the $p$-adic numbers.  Here we develop some of the
theory for the associated analogue of white noise.
We derive the counterpart of the Wiener chaos decomposition
and use it to define a Skorohod-type stochastic integral.  We also
define an It\^o-type stochastic integral, and examine the connection between
these two integrals.

evans@stat.berkeley.edu (plain TeX file available)

450. THE CENTRAL LIMIT THEOREM FOR WEIGHTED MINIMAL SPANNING TREES ON RANDOM POINTS

Harry Kesten and Sungchul Lee

Let $\{X_i,1 \le i < \infty\}$ be i.i.d.\ with uniform distribution
on $[0,1]^d$ and let $M(X_1,\ldots,X_n;\alpha)$ be 
$\min\Big\{\ds\sum_{e\in T'} |e|^\alpha;T'$ a spanning tree on
$\{X_1,\ldots,X_n\}\Big\}$.  Then we show that for $\alpha > 0$
$$
\frac{M(X_1,\ldots,X_n;\alpha) - EM(X_1,\ldots,X_n;\alpha)}
        {n^{(d-2\alpha)/2d}} 
   \rightarrow N(0,\sigma^2_{\alpha,d})
$$
in distribution for some $\sigma^2_{\alpha,d} > 0$.

kesten@math.cornell.edu

451. ASYMPTOTICS FOR OCCUPATION TIMES OF HALF-LINES BY STABLE PROCESSES AND REFLECTED BROWNIAN MOTION

Zhan Shi, Wendelin Werner

We derive the integral tests for the sample path behaviour
of the occupation times of half-lines by reflecting Brownian 
motion perturbed by its local time at 0 (i.e. 
$X_t=|B_t| - \mu \ell_t$, where $B$ is a linear Brownian 
motion, $\ell$ its local time at 0 and $\mu>0$) and stable processes,
which correspond to the generalized Arcsine laws which have been
derived for such processes.

wwerner@dmi.ens.fr

452. SOME REMARKS ON PERTURBED REFLECTING BROWNIAN MOTION

Wendelin Werner

In this short note, we give short elementary proofs of 
a remarkable hitting time property of perturbed reflecting Brownian
motion, and of some of its consequences such as the generalized
Ray-Knight theorems for the local times of perturbed reflecting 
Brownian motion.

wwerner@dmi.ens.fr

453. ON WINDINGS OF MULTIDIMENSIONAL BROWNIAN MOTION

Ellen H. Toby, Wendelin Werner

We investigate the aymptotic behaviour of winding numbers of
two-dimensional reflected Brownian motion (around points) and
three-dimensional transient reflected Brownian motion in
a cone (around lines). We also derive some other properties
of the latter process.

toby@math.tamu.edu, wwerner@dmi.ens.fr

454. NON-COLLIDING BROWNIAN MOTIONS ON THE CIRCLE

David G. Hobson, Wendelin Werner

We compute the probability transition density
for $n$ Brownian motions on the unit circle
killed when any two of them collide, and we investigate some
related questions.

D.G.Hobson@maths.bath.ac.uk, wwerner@dmi.ens.fr

455. GENERAL ASYMPTOTIC ESTIMATES FOR THE COUPON COLLECTOR PROBLEM

Shahar Boneh and Vassilis G. Papanicolaou 

We develop techniques of computing the asymptotics of the 
expected number of coupons that a collector has to buy 
in order to find all $N$ existing
different coupons, as $N\rightarrow \infty $. The probabilities 
(occurring frequencies) of the coupons can be quite arbitrary.

papanico@cs.twsu.edu

456. DIFFUSION LIMITED AGGREGATION ON A TREE

Martin T. Barlow, Robin Pemantle, Edwin A. Perkins

We study DLA on a regular $d$-ary tree.  Points are added 
to existing clusters according to harmonic measure on the
boundary for the tree-valued random walk with transition 
probabilities determined by assigning conductance 
$\alpha^{-n}$ to edges from generation $n-1$ to generation 
$n$.  The heights of these clusters are shown to increase
linearly with their total size if $\alpha <1$.  
 This complements the known results for $\alpha \geq 1$ 
where the cluster heights increase logarithmically.

barlow@math.ubc.ca  (Tex or postscript source file available).