Probability Abstracts 37

804. HIGH TEMPERATURE REGIME FOR A MULTIDIMENSIONAL SHERRINGTON-KIRKPATRICK MODEL OF SPIN GLASS

Alain Toubol

Comets and Neveu have initiated in 1995 a method to
prove convergence of the partition function of disordered systems to a
log-normal random variable in the high temperature regime by means of 
stochastic calculus. We
generalize their approach to a multidimensional
Sherrington-Kirkpatrick model with an application to the Heisenberg
model of uniform spins on a sphere of $\rr^d$.
The main tool that we use is a truncation of
the partition function outside a small neighbourhood of the typical
energy path. 


toubol@cermics.enpc.fr

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805. EQUIVALENT CONDITIONS ON THE CENTRAL LIMIT THEOREM FOR A SEQUENCE OF PROBABILITY MEASURES ON R

Toshio Mikami

In this note we give equivalent conditions on the central limit  theorem in
total variation norm
for a sequence of probability measures on $R$ by those in weak convergence.
This generalizes Cacoullos, Papathanasiou and Utev's central limit  theorem
in $L^1$-norm for a sequence of probability density functions on $R$, where
they used Stein's method.
Stein's method can be used only to probability density functions.

mikami@math.hokudai.ac.jp 

806. NO DIRECTED FRACTAL PERCOLATION IN ZERO AREA

L. Chayes,  Robin Pemantle and  Yuval Peres

We consider the fractal percolation process on the unit square, with fixed 
decimation parameter N  and level dependent  retention parameters  {p_k};
that is, for all k>0, at the k`th stage every retained square of side-length 
N^{1-k} is partitioned into N^2 congruent subsquares, and each of these is 
retained with probability p_k, independently of all others. We show that if 
the product of all the p_k vanishes (i.e., if the area  of the limiting set 
vanishes a.s.) then a.s. the limiting set contains no directed crossings of 
the unit square (a directed crossing is a path that  crosses the unit square 
from left to right, and moves only up, down and to the right).

peres@math.huji.ac.il         

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807. DIFFUSIONS ON THE SIERPINSKI CARPET

Richard F. Bass

The Sierpinski carpet is the fractal formed by taking the unit
square, dividing it into nine equal pieces, removing the 
central square, dividing each of the remaining eight squares
into nine, and continuing. If $F_n$ is the $n$th stage of
the construction and $W_n(t)$ is Brownian motion with reflection
on the boundaries of $F_n$, then suitable time changes of $W_n(t)$
have subsequential limit points with respect to weak convergence.
The limit points are continuous strong Markov processes whose
state space is the Sierpinski carpet and which are invariant under
the local isometries of the carpet.

This paper is a survey paper on research done in the past ten
year with M. Barlow on the construction of such processes,  
their properties, and estimates for their transition densities.

bass@math.washington.edu

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808. QUENCHED SUB-EXPONENTIAL TAIL ESTIMATES FOR ONE-DIMENSIONAL RANDOM WALK IN RANDOM ENVIRONMENT

Nina Gantert and  Ofer Zeitouni

Suppose that the integers are assigned i.i.d. random variables
$\{\omega_x\}$ (taking values in the unit interval), which serve as 
an environment.  This environment defines a random walk $\{X_k\}$ 
(called a RWRE) which, when at $x$, moves one step to the right 
with probability $\omega_x$, and one step to the left with probability 
$1-\omega_x$.  Solomon (1975) determined the almost-sure asymptotic 
speed (=rate of escape) of a RWRE.  Greven and den Hollander
(1994) have proved a large deviation principle for $X_k/k$, conditional 
upon the environment, with deterministic rate function.
For certain environment distributions where the drifts $2
\omega_x-1$ can take both positive and negative values,  their
rate function vanishes on an interval $(0,v_\alpha)$.
We find the rate of decay on  this interval and prove it is
a stretched exponential of appropriate exponent, that is
the absolute value of the log of the probability that the
empirical mean is smaller than $v$, $v\in (0,v_\alpha)$,
behaves roughly like a fractional power of $n$.
The annealed estimates of Dembo, Peres and Zeitouni (1996)
play a crucial role in the proof.\hfil\break
We also deal with the case of positive and zero drifts, and prove
there a quenched decay of the form $\exp (-cn/(\log n)^2)$.

nina@ee.technion.ac.il (postscript file available) 

809. SMALL DEVIATION PROBABILITIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

T. Dunker, M. A. Lifshits and W. Linde 

Let $Z_1,Z_2,\ldots$ be an i.i.d. sequence of non--negative 
random variables and let $(\phi(j))_{j=1}^\infty$ be a summable 
sequence of positive real numbers. Recently, M.A.Lifshits (to
appear Ann.Probab.) gave a description of the asymptotic
behavior of
  $$ P(\sum_{j=1}^\infty \phi(j) Z_j < r ) $$
as $r\to 0$ provided that the distribution function of
$Z_1$ behaves regularly near zero. This description is very
implicite, thus our aim is to prove an explicite formula for
this behavior in the case of log--convex functions $\phi$. It is
applied to concrete examples as $\phi(t)=t^{-A}$ and 
$\phi(t)=\exp(1-t)$. Furthermore, we show that a former result
by V.M.Zolotarev (1986) for $Z_j={\xi_j}^2$ with $\xi_j$ standard
normal is not valid without further assumptions on $\phi$.
We construct an example where a surprising extra oscillating
term appears.

Our results heavily depend on an assumption on the logarithm of the 
Laplace transform of $Z_j$. Using a concept of higher order slow
variation we prove direct conditions on $Z_j$ such that it's Laplace
transform possesses the required property.

dunker@minet.uni-jena.de   (Postscript and tex-files available)

810. LARGE DEVIATIONS FOR A HEAVY-TAILED MIXING SEQUENCE

Nina Gantert

Let $Y_1, Y_2, ... $ be a stationary, ergodic sequence of non-negative random   
variables with heavy tails. Under mixing conditions, we prove large deviation 
principles for the distribution of the arithmetic mean in a finer scale. 
As in the i.i.d. case, the rate of convergence in the law of large numbers 
is determined by the tail of the maximum of the sequence.

nina@ee.technion.ac.il    (postscript file available)

811. CONVERGENCE TIME TO EQUILIBRIUM FOR LARGE FINITE MARKOV CHAINS

Anatoli D. Manita                           

For a sequence of finite Markov chains ${\cal L}(N)$
we introduce the notion of ``convergence time to equilibrium'' $T(N)$.
For sequences that are obtained by  truncating some countable
Markov chain ${\cal L}$ we find the convergence time to equilibrium
in terms of Lyapunov function of Markov chain ${\cal L}$.
We apply this result to queueing systems with a limited number 
of customers: a priority system with several customer types  
and the  Jackson network.

manita@mech.math.msu.su

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• Or here.

812. MODERATE DEVIATIONS FOR THE BLOCKWISE BOOTSTRAP

Amir Dembo and Magda Peligrad

We study the moderate deviations for the empirical process of the blockwise 
bootstrap estimator for stationary sequences.  In this context we show 
that the Moderate Deviation Principle (MDP) holds for essentially the 
same class of weakly dependent processes for which the C.L.T. applies. 
In particular, for many dependent sequences the bootstrapped estimators of 
the mean satisfy the MDP while the empirical means do not.

amir@playfair.stanford.edu  (LaTeX file)

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813. REFLECTED FORWARD-BACKWARD SDE's AND OBSTACLE PROBLEMS WITH BOUNDARY CONDITIONS

Jin Ma and Jaksa Cvitanic

In this paper we study a class of forward-backward stochastic differential
equations with reflecting boundary conditions (FBSDER for short). More 
precisely, we consider the case in which the forward component of the 
FBSDER is restricted to a fixed, convex region, and the backward
component will stay, at each fixed time, in a convex region that may
depend on time and is possibly random. The solvability of such FBSDER 
is studied in a fairly general way.  We also prove that if the coefficients 
are all deterministic and the backward equation is one dimensional, then the 
adapted solution of such FBSDER will give the viscosity solution of a 
quasilinear variational inequality (obstacle problem) with a Neumann 
boundary condition.

In an application, we prove that the solvability
of such FBSDER will lead to the solvability of the minimum hedging price
of an  American game option, in which both the  buyer and the seller
of the option have the right to exercise.  Another possible
interpretation is that of a guided missile - moving target dynamics. 

cj@stat.columbia.edu

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814. DIFFUSIONS WITH A NONLINEAR IRREGULAR DRIFT COEFFICIENT AND PROBABILISTIC INTERPRETATION OF GENERALIZED BURGERS EQUATIONS

Benjamin Jourdain

In this paper, we prove existence and uniqueness for two classes of diffusions
which are the sum of a Brownian motion and a nonlinear irregular drift. Within
each class we construct processes linked with generalized Burgers' equations.
Finally, we extend the existence and uniqueness result to diffusions with a
similar drift and a linear, uniformly elliptic and Lipschitz continuous 
diffusion coefficient.

jourdain@cermics.enpc.fr
http://cermics.enpc.fr/~jourdain/home.html

815. FUNCTIONAL ERD\"{O}S-RENYI-LAWS FOR SEMIEXPONENTIAL RANDOM VARIABLES

Nina Gantert
 
For an i.i.d. sequence of random variables with a semiexponential
distribution, we give a functional form of the Erd\"{o}s-Renyi law
for partial sums.  In contrast to the classical case, i.e. the case 
where the random variables have exponential moments of all orders, the 
set of limit points is not a subset of the continuous functions.
This reflects the bigger influence of extreme values.
The proof is based on a large deviation principle for the
sample path of the corresponding random walk.  Here, the scale of 
the large deviation principle differs from the usual scale and
depends on the tail of the distribution.
We also get a functional limit law for moving averages.
 
nina@ee.technion.ac.il  (postscript file available) 

816. OPTIMALITIES FOR RANDOM FUNCTIONS LEE - WIENER'S NETWORK AND NON-CANONICAL REPRESENTATION OF STATIONARY GAUSSIAN PROCESSES

Win Win Htay

This paper illustrates a significance of a non-canonical representation of a 
Gaussian process in contrast with the canonical representation,the
importance of which is very much appreciated. It is shown that, in a case
of a non-canonical representation, how much extra information is contatined
in the white noise input {B'(t)} compared to the observed process {X(t)}
when the input and the output are evaluated up to the same fixed time t. At
the same time our results give an actual explanation to the non-canonical
representations, from the view point of the optimality, when the theory is
applied to communication system. Moreover it is shown that the effect of
non-canonical can be realized by the so-called Lee-Wiener's networks in
electrical engineering.

d95002x@math.nagoya-u.ac.jp

817. BOSON FOCK REPRESENTATIONS OF STOCHASTIC PROCESSES

Luigi Accardi, Takeyuki Hida and Win Win Htay

We begin a classification theory of quantum stationary processes in analogy
with the corresponding theory for classical stationary processes which
admit a joint Boson Fock canonical representation. 

thida@meijo-u.ac.jp

818. ON THE "HOT SPOTS" CONJECTURE OF J. RAUCH

Rodrigo Ba\~nuelos and Krzysztof Burdzy

The "hot spots" conjecture asserts that the second
Neumann eigenfunction in Euclidean domains
attains its maximum on the boundary of the domain.
The conjecture is proved for obtuse triangles
and related domains. It is also proved for a large
class of planar convex domains with a line
of symmetry. The proof uses a coupling technique.

burdzy@math.washington.edu  (Amstex file or hardcopy available)