Probability Abstracts 3

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45. CONNECTING INTERNALLY BALANCED QUASIREVERSIBLE MARKOV PROCESSES

W. Henderson, C.E.M. Pearce, P.K. Pollett and P.G. Taylor

60J

We shall provide a general framework for interconnecting a collection of 
quasireversible nodes in such a way that the resulting process exhibits a 
product-form invariant measure. The individual nodes can be quite general,
although some degree of internal balance will be assumed. Any of the nodes may 
possess a feedback mechanism. Indeed, we shall give particular attention to a 
class of feedback queues, characterized by the fact that their state description
allows one to maintain a record of the order in which events occur. We shall 
also examine in some detail the problem of determining for which values of the 
arrival rates a node does exhibit quasireversibility.

pkp@axiom.maths.uq.oz.au (AmSTeX (Ver. 3) file available)

46. LEVY BANDITS: MULTI-ARMED BANDITS DRIVEN BY LEVY PROCESSES

Haya Kaspi and  Avi Mandelbaum

Multi-armed bandits are used to model dynamic allocation of a scarce
resource
among competing projects.  It is customary to interpret the resource
as ``time", which is dynamically allocated among several independent
stochastic processes, each of which represents the evolution of an
arm.  The goal is then to find an optimal allocation strategy which
maximizes, for example, cumulative reward discounted over an
infinite horizon.

In this work we focus on Levy Bandits, which are multi-armed bandits
whose arms evolve like Levy processes.  We assume that the reward
depends on the state of the arm being operated and is determined by a
continuous non-decreasing function. As might be anticipated
from previous research, such bandits are optimally controlled by an
index strategy: one can associate with each arm an index function of
its state; the index function of an arm is independent of the other
arms; it is optimal to allocate time to those arms whose states have
the largest index, and the optimal reward can be expressed in terms
of all the indices.

Somewhat less anticipated, however, is the fact that the index
function of an arm driven by a Levy process has a representation in
terms of the decreasing
ladder sets and the exit systems of its driver.
Further, one may use the Wiener Hopf factorization of the Levy
exponents
of the arms to obtain the characteristic function of the excursion laws
through which the indices are defined.

We use this representation to explicitly calculate index functions of
some interesting Levy arms.  We then calculate optimal rewards,
rediscovering along the way that local time naturally quantifies
continuous-time switching.


ierkh01@technion.bitnet

47. A NECESSARY AND SUFFICIENT CONDITION FOR THE MARKOV PROPERTY OF THE LOCAL TIME PROCESS OF A SYMMETRIC MARKOV PROCSS

Nathalie Eisenbaum  and Haya Kaspi

Let $X$ be a symmetric Markov process on an interval $E \subset
\real,$ with a finite potential density $g(x,y).$  Let $(\phi_x;  x \in E)$
be a zero mean Gaussian process with $g$ as the covariance function.
Dynkin and Atkinson have proved that $\phi$ is a Markov process if, and
only if, $X$ has continuous paths.

Assuming that $g$ is continuous, we have shown that the local time
process $(L_{\zeta}^x: x \in E)$ of $X$ conditioned to start at a fixed
point and die at another fixed point is a Markov process if, and only if,
$(\phi_x: x \in E)$ is a Markov process.  The sufficiency of this
condition follows immediately from Atkinson's results and Walsh's
study of the local time process of a real valued diffusion.  The
necessity is proved using arguments based on excursion theory.

ierkh01@technion.bitnet

48. LOCAL TIMES ON CURVES AND UNIFORM INVARIANCE PRINCIPLES

Richard Bass and Davar Khoshnevisan

60J

Sufficient conditions are given for a family of local times
$L_t^\mu$ of d--dimensional Brownian motion 
to be jointly continuous as a function of $t$ and
$\mu$. ($L_t^\mu$ is the additive functional corresponding to the
measure $\mu$.) Then invariance principles are given for the weak convergence
of local times of lattice valued random walks to the local 
times of Brownian motion, uniformly over a large family of
measures. Applications include some new results for
intersection local times for Brownian motions on $R^2$ and $R^3$.

bass@math.washington.edu or davar@math.washington.edu
     (Plain Tex file available)

49. U-STATISTICS OF RANDOM-SIZE SAMPLES AND LIMIT THEOREMS FOR SYSTEMS OF MARKOVIAN PARTICLES WITH NON-POISSON INITIAL DISTRIBUTION

Raisa Feldman and Svetlozar Rachev 

60G, 60F

Study of the limit distributions of square-integrable U-statistics was
started by Dynkin and Mandelbaum (1983) and Mandelbaum and Taqqu (1984). 
We extend their results to the case of non-Poisson random sample size. 
Multiple integrals of non-Gaussian generalized fields are constructed
to identify the limiting distributions. An invariance principle is also 
established. 

We use these results to study the limiting distribution of the amount of 
charge left in some set by an infinite system of signed Markovian particles 
when the initial particle density goes to infinity. 
By selecting the initial particle distribution, we determine the limiting
distribution of charge, constructing different non-Gaussian generalized 
random fields, including Laplace, $\alpha$-stable, and their multiple 
integrals. 

epstein@bernoulli.ucsb.edu

50. EXIT DISTRIBUTIONS FOR SYMMETRIC MARKOV PROCESSES VIA GAUSSIAN TECHNIQUES

Raisa Feldman

60G, 60J

We derive the distribution of the first exit value for a class of symmetric
real-valued Markov processes with finite Green's functions  using prediction
theory for Gaussian processes and Dynkin's theory which relates Markov and 
Gaussian processes.
For L\'{e}vy processes with exponential lifetime this method allows us to 
easily rederive Rogozin's infinitely divisible factorization 
and to obtain the Fourier transform of the distribution of the first
exit value. 

epstein@bernoulli.ucsb.edu

51. WEAK CONVERGENCE TO SELF-AFFINE PROCESSES IN DYNAMICAL SYSTEMS

Michael T. Lacey

This is an abstract for four recent papers devoted to different aspects of 
this situation.   Let f be an intergrable functions on a discrete dynamical 
system  (X,m,T).   Here, (X,m)  is  a probability space, and  T maps  X  into 
X, preserving  m  measure.  Then the sequence  {f(T^k(x))  :  k > 0}  is 
stationary,  and we study the weak convergence of the normalized partial 
sums to a stochastic process  Z(t).   The limiting process  Z(t)  is
necessarily  self-similar, with stationary increments, a condition which
is also refered to self-affine.     Can  Z(t)  be  a Brownian motion?  The
answer  is yes,  under the sole assumption that   the dynamical system be
aperiodic,  which is strictly weaker than being ergodic.  For a  particular
example,  take the system to be an irrational rotation.  Then there is a 
continuous function  f  for which the partial sums, normalized by the
usual square root factor, converge to a Brownian motion.  This case is 
studied in more detail in [4] below.  The same is true for a wide variety of 
self affine processes, including the fractional Brownian motions. See
papers  [2] and [3]  below.

1.   "Weak convergence to self-afine processes in dynamical systems."  A 
 brief survey paper.  Availible as a plainTeX  file.

2.  "Weak convergence to self-similar processes with spectral 
representation in dynamical systems."  To appear in Trans. AMS. Hard copy 
only.

3.  "Weak convergence to self-similar processes with moving average 
representation in dynamical systems."  Much more technical than [2].  Hard
 copy only.

4.   "Central limit theorems,  Holder continuity, and Diophantine type for 
certain irrational rotations."  For irrational rotations, we link the 
metric structure of the dynamical system to information about the kinds 
of functions which give rise to a fractional Brownian motion.  
Available as a plainTeX file.

mlacey@ucs.indiana.edu

52. LIMIT THEOREMS FOR TRIMMED SUMS

Yuji Kasahara and Makoto Maejima

60F

This paper studies the heavily trimmed sums (*)$\sum_{[ns]+1}^{[nt]}X_j^{(n)}$,
where $\{ X_j^{(n)}\} _{j=1}^n$ are the order statistics from independent 
random variables $\{ X_1, \cdots , X_n\}$ having a common distribution $F$.
The main theorem gives the limiting process of (*) as a process of $t$ under 
mild conditions on $F$.  A more smoothly trimmed sums like
$\sum_{j=1}^{[nt]}J(\frac{j}{n})X_j^{(n)}$ are also discussed.

maejima@math.keio.ac.jp (AMSTex file available)

53. FORMS OF INCLUSIONS BETWEEN PROCESSES

Frank B. Knight

Let $X_t$ and $Y_t$ be r.c.l.l. processes on $(\Omega,F,P)$, 
with past and future $\sigma$--fields $M_t^{X(Y)}$, 
$N_t^{X(Y)}$ respectively, and prediction processes $Z^X_t$, 
$Z^Y_t$:
$$
Z^X_t(S)=P\{X_{t+(\cdot)}\in S|M^X_{t+}\}, \quad S\in \times_{s\geq 
0} \b.
$$
Then $\sigma(Z^X_t)$ is the minimal splitting field of 
$M^X_{t+}$ and $N^X_t$ in $M^X_{t+}$. A principle result is 
that if $M^X_{t+}\equiv M^Y_{t+}$ for all $t$, then (a)\,\, 
$Z^Y_t$ is linearly dependent on $Z^X_t$, $\forall t$, if and 
only if $N^X_t\supset N^Y_t$ (up to $P$--nullsets) and (b)\,\, 
$Z^Y_t$ is nonlinearly dependent on $Z^X_t$, $\forall t$, if and 
only if $N^{Z^X}_t \supset N^{Z^Y}_t$ (up to $P$--nullsets). 
Here linear dependence means that, for all $S$, $Z^Y_t(S)\equiv 
Z^X_t(\psi_tS)$, for a $\sigma$--homomorphism $\psi_t$.

Contact Hilda Britt at britt@symcom.math.uiuc.edu

54. THE BOUNDARY HARNACK PRINCIPLE FOR NON-DIVERGENCE FORM ELLIPTIC OPERATORS

R. Bass and K. Burdzy

60J

If L is a uniformly elliptic operator in non-divergence form, the 
boundary Harnack principle for the ratio of positive L-harmonic 
functions holds in Holder domains of order alpha if alpha>1/2. A 
counterexample shows that 1/2 is sharp. For Holder domains of order 
alpha with alpha in (0,1], the boundary Harnack principle holds provided 
the domain also satisfies a strong uniform regularity condition.

bass@math.washington.edu or burdzy@math.washington.edu  (plain Tex)

55. ON QUASI-SUPPORTS OF SMOOTH MEASURES AND CLOSABILITY OF PRE-DIRICHLET FORMS

M.Fukushima and Y.LeJan

60J

In the context of a regular Dirichlet form, the notion of the quasi-support
of a smooth measure and its characterizations in terms of classes of
quasi-continuous functions are investigated and applied to closability
criteria of perturbed pre-Dirichlet forms.

e64025@jpnkudpc.bitnet

56. LIMIT DISTRIBUTIONS FOR MINIMAL DISPLACEMENT OF BRANCHING RANDOM WALKS

F.M.Dekking and B.Host

60J

We study the minimal displacement X(n) of a branching random walk with
non-negative steps. It is shown that (X(n)-EX(N)) is tight under a mild
moment condition on the displacements. For supercritical BRW (X(n))
converges almost surely. For critical BRW we determine the possible
limit points of (X(n)-EX(n)), and we prove a generalization of
Kolmogorov's theorem on the extinction probability of a critical 
branching process. Finally we generalize Bramson's results on the
almost sure convergence of X(n)log(2)/loglog(n).

dekking@dutiosa.tudelft.nl

57. LIMITING ANGLE OF BROWNIAN MOTION IN CERTAIN TWO--DIMENSIONAL CARTAN-HADAMARD MANIFOLDS.

Pei Hsu and Wilfrid S. Kendall.

60J

Suppose that {\bf H} is a two--dimensional Cartan-Hadamard manifold
with sectional curvatures satisfying a weak negative upper bound and no
lower bound.  Then the angular part of Brownian motion on {\bf H} tends
to a limit as time tends to infinity. Moreover this limit angle has a
distribution which is dense on the circle at infinity.

elton@math.nwu.edu
w.s.kendall@cu.warwick.ac.uk

58. SYMBOLIC IT\^O CALCULUS: AN INTRODUCTION.

Wilfrid S. Kendall.

60H

The ito procedures are an implementation of stochastic calculus for the
computer algebra package REDUCE. In this paper it is explained how the
implementation of ito grows naturally out of the formulation of
stochastic calculus using modules of stochastic differentials. Two
examples are given of ito in action: a simple example concerning
various exponential martingales and a more involved example concerning
the escape rate of the Bessel process of dimension exceeding $2$.  A
basic subset of the ito procedures is listed in six appendices; details
are given of how to obtain the full set from the author.

w.s.kendall@cu.warwick.ac.uk

59. CLOSING VALUES OF MARTINGALES WITH FINITE LIFETIMES

M. Sharpe

60G

A process $M_t$ defined only for $t<\zeta$, where $\zeta$ is a given
stopping time, is called a martingale with lifetime $\zeta$ provided it 
has a martingale extension $\bar M_t$ defined for all $t\ge 0$. It can 
be proved that there is a unique extension $\bar M$ satisfying 
(i) $\bar M_t$ is constant for $t\ge\zeta$;
(ii) $\bar M_\zeta\in{\Cal F}_{\zeta-}$. 
The following (closely related) questions are examined. 
(a) Knowing $M_t$ only for $t<\zeta$, how may one detect that it is a 
martingale in the above sense? (b) How may one compute the
closing value $\bar M_\zeta$ from $M$ in this case? 

msharpe@euclid.ucsd.edu  (AMSTeX source file available)