Probability Abstracts 21

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331. RANDOM WALKS AND THE GROWTH OF GROUPS

Russell Lyons

The critical value separating transience from recurrence for the amount
of radial drift of a random walk on a Cayley graph of any finitely
generated group is shown to equal the exponential growth rate of the group.

rdlyons@silver.ucs.indiana.edu    (plain-Tex file or hardcopy available)

332. A PROBABILISTIC FORMULA FOR THE CONCAVE HULL OF A FUNCTION

Robert J. Vanderbei

Let $D$
be a domain in $d$-dimensional Euclidean space and let $f$ be a positive        real-valued function defined on $D$.  The classical optimal stopping
problem is to find a stopping time $\tau^*$ that attains the
following supremum:
$$ v(x) = \sup_\tau E_x f(B(\tau)).  $$
Here, $B$ is a $d$-dimensional Brownian motion with absorbtion on the
boundary of $D$ and the supremum is over all stopping times.
It is well-known that
$v$ is characterized as the smallest superharmonic majorant of $f$.

In this paper, we modify this problem by allowing
$B$ to be an arbitrary drift-free
diffusion (with absorbtion, as before, on the boundary of $D$).
For example, it could be
a Brownian motion diffusing on some lower dimensional affine set.
In addition, one is allowed to switch
among these diffusions at any time.  The problem is to find a
stopping time and a switching strategy that together attain the
supremum over all stopping times and all switching strategies.
For this problem, we show that $v$ is characterized
as the smallest concave majorant of $f$.
The domain $D$ can be decomposed into a disjoint union of open
convex sets on each of
which the function $v$ is affine.  Furthermore,
the union of the $0$-dimensional convex sets is contained in
the set on which $v = f$.
An optimal switching strategy is any strategy that at all times
diffuses in the affine hull of the
current convex set.  When the diffusion reaches the boundary of the
current convex set it will lie on a lower dimensional convex set and
must then diffuse on the affine hull of this new set.  This
process continues until the set on which $v = f$ is reached, which is
the optimal stopping time.

rvdb@Princeton.EDU

333. LADDER HEIGHT DISTRIBUTIONS WITH MARKS

Soren Asmussen and Volker Schmidt

For risk processes with a general stationary input,
a representation formula of ladder height
distributions is proved which
includes some additional information on process behaviour at the ladder epoch.
The proof is short and probabilistic, and utilizes time reversal, occupation
measures and Campbell's formula.
The results are applied to stochastic fluid models driven by
a general stationary process and the probability is determined that  
ruin occurs in a given state of the
environment.

asmus@iesd.auc.dk

334. RATE MODUALTION IN DAMS AND RUIN PROBLEMS

Soren Asmussen and Offer Kella

We consider a dam in which the release rate depends both on the
state and some modulating process. Conditions for the existence of a limiting
distribution are established in terms of an associated
risk process.
The  case where the release rate
is a product of the
state and the modulating process 
is given special
attention, and in particular explicit formulas are obtained for
a finite state space Markov
modulation.

asmus@iesd.auc.dk

335. RUIN PROBABILITIES VIA LOCAL ADJUSTMENT COEFFICIENTS

Soren Asmussen and Hanne Mandrup Nielsen

Let $\psi(u)$ be the ruin probability in a
risk process with initial reserve $u$, Poisson arrival rate $\beta$,
claim size distribution $B$ and premium rate $p(x)$ at level $x$ of the reserve.
Let $\gamma(x)$ be the non--zero solution of the local Lundberg equation
$\beta(\hat{B}[\gamma(x)]-1)-\gamma(x) p(x)$ $=0$ and
$I(u)=$ $\int_0^u\gamma(x)dx$. It is shown that
$\psi(u)\le e^{-I(u)}$ provided $p(x)$ is non--decreasing
and that $\log\psi(u)$ $\approx -I(u)$ in a slow Markov walk limit.
Though the results and conditions are of large deviations type,
the proofs are elementary and utilize piecewise comparisons
with standard risk processes with a constant $p$.
Also simulation via importance sampling using local exponential
change of measure defined in terms of the $\gamma(x)$ is discussed
and some numerical results are presented.


asmus@iesd.auc.dk

336. STRONG APPROXIMATIONS IN ALMOST SURE LOCAL CENTRAL LIMIT THEOREMS

Lajos Horvath and Davar Khoshnevisan

Let S_n denote the usual random walk. Under 
almost minimal conditions, we show that 
	\sum_{k\leq n} k^{-1/2} I\{ S_k =x_0 \}
can be approximated with a suitably chosen Brownian 
motion. This, in turn, answers a question of Csaki
et al.

davar@math.utah.edu

337. HAMMERSLEY'S INTERACTING PARTICLE PROCESS AND LONGEST INCREASING SUBSEQUENCES

David Aldous and Persi Diaconis

In a famous paper Hammersley (1972)
investigated the length $L_n$ of the longest increasing subsequence
of a random $n$-permutation.
Implicit in that paper is a certain
one-dimensional continuous-space interacting particle process.
By studying a hydrodynamical limit for Hammersley's process we
show by fairly ``soft" arguments that 
$c \equiv \lim n^{-1/2} EL_n$
satisfies $c \geq 2$.  The argument is analogous to that in Rost's (1981)
seminal study of hydrodynamics for the asymmetric exclusion process.
Previous proofs of $c \geq 2$ 
relied on hard analysis of combinatorial asymptotics.
By including an elementary combinatorial proof (due to Vershik - Kerov)
that $EL_n \leq 2 n^{1/2}$ for all $n$,
we provide a self-contained proof that $c = 2$.

aldous@stat.berkeley.edu

338. STABLE NON-GAUSSIAN RANDOM PROCESSES: STOCHASTIC MODELS WITH INFINITE VARIANCE

Gennady Samorodnitsky and Murad S. Taqqu

The familiar Gaussian models do not allow for large deviations and  are thus
often inadequate for modelling high variability. Non-Gaussian stable
models do not possess such limitations. They all share a familiar
feature which differentiates them from the Gaussian ones.
Their marginal distributions possess heavy "probability tails," 
always with infinite variance, and in some cases with infinite first moment.

The aim of this book is to make this exciting material easily accessible to
graduate students and practitioners. Assuming only a first-year graduate
course in probability, it includes material which has appeared 
only recently in journals and material not published anywhere.  

Each chapter begins with a brief overview and concludes with a range of
exercises at varying levels of difficulty. Proofs are spelled out in
detail. The exercises at varying levels of difficulty.
Proofs are spelled out in detail.

The book also includes a discussion of self-similar processes,
ARMA, and fractional ARIMA time series with stable innovations.

murad@math.bu.edu       

339. LARGE DEVIATIONS IN QUEUEING NETWORKS

Neil O'Connell

The main result of this paper describes how the
large deviation properties of a traffic stream
are altered when the traffic passes through a
buffer, possibly sharing that buffer with
cross traffic.  We also consider the effect of
priority service policies.  The heuristics behind
the approach suggest a general method for determining
the large deviation properties of traffic (and hence
overflow probabilities) at any point in an arbitrary
network, which we illustrate with an example.

oconnell@stp.dias.ie

340. WEAK CONVERGENCE OF CONDITIONED BIRTH AND DEATH PROCESSES

Gareth O. Roberts and Saul D. Jacka


We consider the problem of conditioning a non-explosive
birth and death process to remain positive until time $T$, and
consider weak convergence of this conditional process as 
$T\to\infty$.
By a suitable a.s. construction we prove weak convergence.
The a.s. construction used is of independent interest
but relies heavily on the strong monotonic properties of birth and 
death
processes.

s.d.jacka@warwick.ac.uk

341. WEAK CONVERGENCE OF CONDITIONED PROCESSES ON A COUNTABLE STATE SPACE

We consider the problem of conditioning a continuous time
Markov chain (on a countably infinite state space) not to  
hit an absorbing barrier before time $T$; and the 
weak convergence of this conditional process as $T\to\infty$.
We prove a characterization of convergence in terms of the 
distribution
of the process at some arbitrary positive time, $t$, introduce a
decay parameter for the time to absorption, give an example where 
weak
convergence fails, and give sufficient conditions for weak 
convergence in 
terms of the existence of a quasi-stationary
limit, and a recurrence property of the original process.   

s.d.jacka@warwick.ac.uk

342. NON-EXPLOSIVITY OF LIMITS OF CONDITIONED BIRTH AND DEATH PROCESSES

Gareth O Roberts, Saul D. Jacka and Phillip K Pollett

Let $X$ be a birth and death process on $\Z^+$ with absorption at 
zero
and suppose that $X$ is suitably recurrent, irreducible, and
non-explosive. In a recent paper, Roberts and Jacka (1993) showed 
that
as $T\to\infty$ the process conditioned to non-absorption until time 
$T$
convergences weakly to a time-homogeneous Markov limit, 
$X^\infty$,
which is itself a birth and death process. However the question of the
possibility of explosiveness of $X^\infty$ remains open. The major
result of this paper establishes that $X^\infty$ is always
non-explosive.



s.d.jacka@warwick.ac.uk

343. KEEPING A SATELLITE ALOFT: TWO FINITE FUEL STOCHASTIC CONTROL MODELS

Saul D Jacka

We consider two models for the control of
a satellite---in the first, fuel is expended in a linear fashion to move
a satellite following a diffusion---the aim is to keep the satellite
above a critical level for as long as possible (or indeed to reach a
higher, \lq safe' level). Under suitable assumptions on the drift and
diffusion co-efficients, it is shown that the stochastic maximum of the
time to fall below the critical level is attained by the policy which
imposes a reflecting boundary at the critical level until the fuel is
exhausted and jumps the satellite directly to the safe level if this is
ever possible. In the second model, there is a \lq non-linear' 
response
to the expenditure of fuel, and no safe level. It is shown that the
optimal policy for maximising the expected discounted time for the
satellite to ` crash' is similar in that equal packets of fuel are 
used to jump the satellite upwards each time it reaches the critical 
level.

s.d.jacka@warwick.ac.uk

344. OPTIMAL INVESTMENT OF A LIFE INTEREST

We consider the problem of a trustee faced
with investing a sum of money, the interest from which will be 
received
by one party (the life-tenant) during his lifetime whilst the capital
will go to another party (the survivor) on the death of the life-tenant.
Under assumptions as to the utility functions of the two parties and as
to the stochastic behaviour of available investments, we find the
collection of Pareto-optimal investment strategies for the trustee
together with the corresponding payoffs. We do this by optimising the
payoff of the Lagrangian for the problem. We go on to present the
Nash-optimal solution for the trustee.

s.d.jacka@warwick.ac.uk

345. BISMUT_NUALART_PARDOUX COHOMOLOGY AND ENTIRE HOCHSCHILD COHOMOLOGY

R. Leandre

Over the stochastic free loop space of a riemannian compact manifold, we
define a stochastic exterior derivative: it is related to the definition of
an anticipative Stratonovitch integral. We establish the commutative
diagramm, which in the case of smooth loop allows to show that the
Hochschild cohomology is equals to the cohomology of the loop space, and we
give a map between the entire Hochschild cohomology (in the manner of Alain
Connes's cyclic cohomology) and the cohomology of the stochastic loop
space. 

a18631@frccsc21.bitnet

346. INVARIANCE OF THE LYAPUNOV EXPONENT UNDER NONLINEAR PERTURBATION

Mark A. Pinsky

Consider a system of Ito SDE in the form $dX = A(X) dt + \sum_r B_r( X) dw^r  $
with Lyapunov exponent 
$$\lambda(A,B) = \lim_{t\rightarrow \infty} t^{-1}\log |X(t)|  <0$$
Under a mild condition of non-degeneracy we prove that $\lambda(A,B) =
\lambda({\bar A},{\bar B}) $, where $\bar A,\bar B$ denote the 
drift and diffusion coefficients of the associated linearized system at $X=0$.
A corresponding result is obtained when $\lambda >0$

pinsky@math.nwu.edu 

347. BISEXUAL BRANCHING DIFFUSIONS

Leonid Mytnik and  Robert J. Adler

We study the limiting behaviour of large systems of two types of Brownian 
particles undergoing bisexual branching. Particles of each type generate 
individuals of both types, and the respective branching law is 
asymptotically critical for the two-dimensional system, while being 
subcritical for each individual population.

The main result of the paper is that the limiting behaviour of  suitably 
scaled sums and differences of the two populations is given  by a pair 
of measure and distribution valued processes which, together, determine 
the limit behaviours of the individual populations.

Our proofs are based on the martingale problem approach to general state 
space processes.  The fact that our limit involves both measure and 
distribution valued processes requires the development of some new 
methodologies of independent interest.

robert@ieadler.technion.ac.il

348. SUPER FRACTIONAL BROWNIAN MOTION, FRACTIONAL SUPER BROWNIAN MOTION AND RELATED SELF-SIMILAR (SUPER) PROCESSES

Robert J. Adler and Gennady Samorodnitsky

We consider the full weak convergence, in appropriate function spaces,
of systems of noninteracting particles undergoing critical branching
and following a self similar spatial motion with stationary increments.
The limit processes are measure valued, and are of the super and
historical process type. In the case in which the underlying motion is
that of a fractional Brownian motion, we obtain a characterisation of the
limit process as a kind of stochastic integral against the historical
process of a Brownian motion defined on  the full real line.

robert@ieadler.technion.ac.il

349. A SUPERPROCESS WITH A DISAPPEARING SELF-INTERACTION

Robert J. Adler and Lidia Ivanitskaya

We start with a system of independent branching Brownian motions which,
properly organised and normalised, generate a super Brownian motion in 
the high density limit. We introduce a weak interaction between the  
particles, that affects the diffusion but not the branching. The  
interaction is chosen in such a way that the infinite density limit is 
absolutely continuous with respect to the non-interacting system.

We find that, despite the fact that the interaction mechanism never
completely disappears, the limiting superprocesses are {\it identical}. 
We study what actually happens to the interaction mechanism,  or 
``ghost-process".

robert@ieadler.technion.ac.il

350. LAWS OF LARGE NUMBERS FOR QUADRATIC FORMS, MAXIMA OF PRODUCTS AND TRUNCATED SUMS OF I.I.D. RANDOM VARIABLES

Jack Cuzick, Evarist Gin\'e and Joel Zinn

Let ${X,\ X_i}$ be i.i.d. real random variables with $E  
X^2=\infty$. Necessary and sufficient conditions in terms of  
the law of $X$ are given for ${1\over{\gamma_n}}\max_{1\le  
i<j\le n} |X_iX_j|\to 0$ a.s. in general and for  
${1\over{\gamma_n}}\sum_{1\le i\ne j\le n} X_iX_j\to 0$ a.s.  
when the
variables ${X_i}$ are symmetric or regular and the normalizing  
sequence $\{\gamma_n\}$ is (mildly) regular. The rates of a.s.  
convergence of sums and maxima of products turn out to be  
different in general but to coincide under mild regularity  
conditions on both the law of $X$ and the sequence
$\{\gamma_n\}$. Strong laws are also established for  
$X_{1:n}X_{k:n}$ where $X_{j:n}$ is the $j$-th  largest in  
absolute value among $X_1,\dots,X_n$, and it is found  that,  
under some regularity, the rate is the same for all $k\ge3$.
Sharp asymptotic bounds for  
$b_n^{-1}\sum_{i=1}^nX_iI_{|X_i|<b_n}$, for
$b_n$ relatively small, are also obtained.

jzinn@plevy.math.tamu.edu (TeX file available)

351. BRANCHING RANDOM WALK WITH A CRITICAL BRANCHING PART

Harry Kesten

We consider the branching tree $T(n)$ of the first $(n+1)$
generations of a critical branching process, conditioned on survival till
time $\beta n$ for some fixed $\beta > 0$ or on extinction occurring
at time $k_n$ with $k_n/n \rightarrow \beta$.  We attach to each 
vertex $v$ of this tree a random variable $X(v)$ and define
$S(v) = \ds\sum_{w\in\pi(0,v)} X(w)$, where $\pi(0,v)$ is the
unique path  in the family tree from its root to $v$.
Finally $M_n$ is the maximal displacement of the branching
random walk $S(\cdot)$, that is
$M_n = \max\{S(v):v \in T(n)\}$.  We show that if the $X(v)$,
$v \in T(n)$, are i.i.d\. with mean $0$, then under some further
moment condition $n^{-1/2}M_n$ converges in distribution.
In particular $\{n^{-1/2}M_n\}_{n\ge 1}$ is a tight family.
This is closely related to recent results about Aldous'
continuum tree and Le Gall's Brownian snake.

kesten@math.cornell.edu 

352. ASYMPTOTICALLY ONE-DIMENSIONAL DIFFUSIONS ON SCALE-IRREGULAR GASKETS

Tetsuya Hattori 

A simple class of fractals which lack exact self-similarity is introduced, 
and the asymptotically one-dimensional diffusion process is constructed.
The process moves mostly horizontally for very small scales, while for large 
scales it diffuses almost isotropically, in the sense of the off-horizontal 
relative jump rate for the decimated random walks of the process.
An essential step in the construction of diffusion is to prove the existence 
of appropriate time-scaling factors.  For this purpose, a limit theorem for 
a discrete-time multi-type supercritical branching processes with 
singular and irregular (varying) environment, is developed.

hattori@tansei.cc.u-tokyo.ac.jp     (LaTeX file available)

353. EXPLICIT STOCHASTIC INTEGRAL REPRESENTATIONS FOR HISTORICAL FUNCTIONALS

Steven N. Evans and Edwin A. Perkins

It is known from previous work of the authors that any square - integrable 
functional of a superprocess may be represented as a constant plus a stochastic integral against the associated orthogonal martingale measure.  Here we give 
for a large class of such functionals an explicit description of the integrand
that is analogous to Clark's formula for the representation of certain Brownian
functionals.  As a consequence, we develop a partial analogue of the Wiener 
chaos expansion in the superprocess setting.  Rather than work with 
superprocesses {\it per se} our results are stated and proved in the richer 
and more natural context of the associated historical process.

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

354. THE DOOB-MEYER DECOMPOSITION REVISITED

Richard F. Bass

A new proof is given of the Doob-Meyer decomposition of
a supermartingale into martingale and decreasing parts.
Although not the most concise proof, the proof is elementary
in the sense that nothing more sophisticated than Doob's inequality
is used. If the supermartingale is bounded and the jump times are totally
inaccessible, then it is shown that discrete time approximations
converge to the decreasing part in $L^2$. The general case is
handled by reduction to the above special case.

bass@math.washington.edu

355. MARKOV SNAKES AND SUPERPROCESSES

E.B.Dynkin and S.E.Kuznetsov

We suggest the name Markov snakes for a class of path-valued Markov
processes introduced recently by J.-F. Le Gall in connection with the
theory of branching measure-valued processes. Le Gall applied this
class to investigate path properties of superdiffusions and to
approach probabilistically partial differential equations involving a
nonlinear operator $\Delta v-v^2$.

We establish an isomorphism theorem which allows
to translate results on continuous superprocesses into the language
of Markov snakes and vice versa. By using this theorem, we get limit theorems
for discrete Markov snakes.

sek@math.cornell.edu

356. SHORT-TIME GEOMETRY OF RANDOM HEAT KERNELS

Richard B. Sowers

We study the short-time
behavior of the random Green's function associated with the
stochastic partial differential equation $du = {1\over 2} \Delta u dt + (\sigma,\grad u)\circ dW_t$, on some Riemannian manifold $M$.  Here
$\Delta$ is the Laplace-Beltrami operator, $\sigma$ is some vector field on $M$,
and $\grad$ is the gradient operator.
Also, $W$ is a standard
Wiener process and $\circ$ denotes Stratonovich integration.  Our
interest here is to show how classical short-time
estimates of deterministic heat kernels must be corrected to account
for the noise term.  We find that the dominant exponential term is classical and
depends only on the Riemannian distance function.  The second exponential
term is a work term, and also has classical meaning.  The third exponential
term blows up, and thus provides an interesting object of study.
We find an expression for this third exponential term which involves
a random translation of the index form and the equations of Jacobi fields.

rbs@math.umd.edu

357. ON THE OCCUPATION TIMES OF CONES BY BROWNIAN MOTION

Thierry Meyre and Wendelin Werner

We study some features concerning the occupation time $A_t$ 
of a d-dimensional cone $C$ by Brownian motion. In particular, we give the
precise integral test, which decides whether $\liminf_{t \rightarrow \infty}
A_t/(t f (t))= 0 $ or $\infty$ almost surely (for a decreasing function
$f$ and a convex cone $C$). We also investigate the asymptotic behaviour of 
$P(A_1 < u)$ as $u \rightarrow 0$, when the Brownian motion starts at the vertex
of $C$, in the case where $C$ is convex. More precisely, we prove that
for all $u \in [0,1]$,
$k_1 u^{1/ \xi} \leq P(A_1 < u) \leq k_2 u^{1/ \xi}$, 
for some deterministic constants $k_1$, $k_2$, $\xi$
depending on $C$.

W.J.J.Werner@statslab.cam.ac.uk