Probability Abstracts 9

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153. NEW CLASSES OF H-SELF-SIMILAR STATIONARY INCREMENTS ALPHA STABLE PROCESSES

Piotr S. Kokoszka  and Murad S. Taqqu

We investigate two new classes of  symmetric stable
self-similar random fields with stationary increments, 
one  of the moving average type, the other of the harmonizable type. 
We then define related `projection processes' whose parameter space 
is $\Bbb R^1$, compare these processes with each other and other 
well-known self-similar processes and study their asymptotic dependence 
structure.
 
murad@math.bu.edu

154. ASYMPTOTIC DEPENDENCE OF STABLE SELF-SIMILAR PROCESSES OF CHENTSOV TYPE

Piotr S. Kokoszka  and Murad S. Taqqu

This paper studies the asymptotic dependence structure of self-similar
symmetric $\alpha$-stable random processes of Chentsov type with 
$0<\alpha<2$. The results allow us to easily verify that Chentsov type
processes are different from many other self-similar symmetric 
$\alpha$-stable processes with stationary increments.

murad@math.bu.edu

155. ASYMPTOTIC DEPENDENCE OF MOVING AVERAGE TYPE SELF-SIMILAR STABLE RANDOM FIELDS

Piotr S. Kokoszka  and Murad S. Taqqu

We investigate the asymptotic dependence structure of
a large class of self-similar stable random fields which are
extensions of the linear fractional L\'evy motion to the parameter
space $\Bbb R^n$. The dependence is measured through the difference of
the joint characteristic function and the product of the marginal
characteristic functions.  We show that the  intensity of the dependence
decreases to zero  like a power function as the lag
tends to infinity  and we obtain the exact expression for the exponent 
in the  power function. The exponent depends on both the stability 
parameter and  the self-similarity parameter.

murad@math.bu.edu

156. DOES ASYMPTOTIC LINEARITY OF THE REGRESSION EXTEND TO STABLE DOMAINS OF ATTRACTION?

Renata Cioczek-Georges and Murad S. Taqqu

Hardin, Samorodnitsky, Taqqu (1991) have
shown that the regression $E[Y|X=x]$ is typically asymptotically linear when
$(X,Y)$ is an $\alpha$-stable random vector with $\alpha <2$. We provide
necessary and sufficient conditions for asymptotic linearity of 
$E[Y|X+{\cal E}=z]$ where $(X,Y)$ is an $\alpha$-stable random vector 
and ${\cal E}$ is a random variable, independent of
$(X,Y)$, such that $X+{\cal E}$ is in the domain of normal attraction
of $X$.  Asymptotic linearity does not always hold even when $E[Y|X=x]$ is
linear.  For some distributions of ${\cal E}$, the asymptotic rate of 
$ E[Y|X+{\cal E}=z]$ fluctuates.

murad@math.bu.edu

157. EXTINCTION PROBABILITY OF A MUTANT GENE

Fred M. Hoppe

We present the identity 
$$1-q_\epsilon = \frac {2} {I_\epsilon} (m_\epsilon -1) \eqno (1)$$
relating the asymptotic behaviour of the extinction probability $q_\epsilon$
to $m_\epsilon$, the mean number of offspring per individual, 
for a family (indexed by the parameter $\epsilon$) slightly supercritical 
Bienayme-Galton-Watson  processes as $m_\epsilon$ approaches 1. 
Here ${I_\epsilon}$ is given explicitly by
$$ {I_\epsilon} = 2\int_{u=0}^{1}(1-u)f''(1-u(1-q_\epsilon),\epsilon)du.  $$ 
where $f(x,\epsilon)$ is the offspring p.g.f. of the approximating
process with parameter $\epsilon$.  This gives the most general branching 
analogue of an old result of Haldane for a
Wright-Fisher model of a slightly advantageous mutant gene.
Previously, Ewens had shown that $1-q_\epsilon \sim 2\epsilon$
for Poisson-distributed offspring; then Eshel, under certain conditions,  
obtained $1-q_\epsilon \sim \frac {2} {\sigma^{2}} (m_\epsilon -1)$
where $\sigma^{2} > 0$ is the offspring variance of
a limit process.  Sometimes, the quantity $I_\epsilon$ approaches the 
variance of a limit process, which then recovers
Eshel's result.  But, in general, we must replace the
variance by the second factorial moment  $b_\epsilon$ (these are not always 
asymptotically equal) and, moreover, these should come from the
approximating processes not from some, apriori assumed, limit 
process, in order not to exclude the possibility that they tend
to zero or infinity.
From (1) we can obtain necessary and sufficient conditions for 
$$1-q_\epsilon \sim \frac {2} {\sigma_\epsilon ^{2}} (m_\epsilon -1)$$
as well as a variety of easily verifiable sufficient conditions (and 
corresponding results with $\sigma_\epsilon ^{2}$ replaced by $b_\epsilon$). 
Perhaps more importantly, the identity shows that completely 
unexpected behaviour can occur which has no analogue in the original setting, 
for instance with constants other than 2 in the numerator on the right 
side of (2), and non-linear behaviour in $m_\epsilon-1$.  Surprisingly,
slight changes in the underlying family of offspring probability
distributions can result in dramatically different rates of growth.  As
illustration we consider examples having identical means and variances to
first order but whose rates are algebraic with different powers.
These results have been extended to the multitype setting.


hoppe@sscvax.cis.McMaster.ca

158. A NOTE ON YAMADA'S THEOREM OF SUCCESSIVE APPROXIMATE SOLUTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS

Xuerong Mao

Yamada in 1981 proved a celebrated theorem on the successive 
approximation of the solution for a stochastic differential
equation where the coefficients were assumed to be weaker than
Lipschitz continuity. The aim of this paper is to establish
another successive approximation for the solution under the 
same condition. The proof of the convergence of our approximations
represents an alternative to the procedure for establishing
the existence and uniqueness of the solution to the equation.
Moreover, our proof is much shorter than Yamada's.

xm@uk.ac.warwick.maths

159. RATES OF CONVERGENCE TO BROWNIAN LOCAL TIME

Richard F. Bass and Davar Khoshnevisan

Let $S_n$ be the partial sums of i.i.d. mean 0, variance 1
random variables. Let $\eta(x,n)$ be the number of times $S_j$
is within a distance 1/2 of x before time n, the ``local time''
of $S_n$. If the summands are lattice valued and have more than
5 moments or if they are nonlattice valued, have more than 6 moments,
and for some j, $S_j$ has a nonzero absolutely continuous part,
then we can give a Skorokhod embedding of the $S_n$ in a 
Brownian motion such that $\sup_{x} |\eta(x,n)-L(x,n)|=O(r_n)$, a.s.,
where $L(x,t)$ is the local time of the Brownian motion and
$r_n=n^{1/4} (\log n)^{1/2}(\log \log n)^{1/4}$.

bass@math.washington.edu

160. BLOWUP FOR THE HEAT EQUATION WITH A NOISE TERM.

Carl Mueller and Richard Sowers

In this paper we study blowup of the equation $u_t = u_{xx} + u^\gamma W_{tx}$,
where $W_{tx}$ is a two-dimensional white noise field and where Dirichlet
boundary conditions are enforced.  It is known that if $\gamma<3/2$, then
the solution exists for all time; in this paper we show that
if $\gamma$ is much larger than $3/2$, then the solution 
blows up in finite time with
positive probability.  We prove this by considering
how peaks in the solution propagate.  If a peak of high mass forms, we
rescale the equation and divide the mass of the peak into a collection
of peaks of smaller mass, and these peaks evolve independently.  In this
way we compare the evolution of $u$ to
a branching process.  Large peaks are regarded
as particles in this branching process.  Offspring are peaks which
are higher by some factor.  We show that the expected number of
offspring is greater than one when $\gamma$ is much larger than $3/2$, and thus
the branching process survives with positive probability, corresponding
to blowup in finite time.

sowers@math.usc.edu

161. INTERSECTION LOCAL TIME FOR POINTS OF INFINITE MULTIPLICITY

R. Bass, K. Burdzy and D. Khoshnevisan

Let $X$ be a 2-dimensional Brownian motion starting from 0
and killed on exiting of the unit disc. Let $N^x_\epsilon$ be
the number of excursions of $X$ from $x$ which hit the circle
with center $x$ and radius $\epsilon$ (for "most" points $x$
$N^x_\epsilon = 0$). Let $D_a$ be the set of all points $x$
such that
$$\lim_{\epsilon\to 0} N^x_\epsilon / |\log \epsilon| = a.$$
Theorem. (i) For every $a\in (0,1/2)$ there exists a measure 
$\beta_a$ which is supported on $D_a$ and such that the Hausdorff
dimension of its support is equal to $2-a$.
(ii) For every $a > 0$ the Hausdorff dimension of $D_a$
is less than or equal to $2-a/e$.
(iii) $\beta_a$ is supported on points of infinite multiplicity.

burdzy@math.washington.edu (AmS-TeX file available)

162. ERGODIC PROPERTIES OF RANDOM MEASURES ON STATIONARY SEQUENCES OF SETS

Aaron Gross and James Robertson

We study a class of stationary sequences having spectral
representation $(M(\tau^n A))_{n\in {\bf Z}}$, where $A$ is
a set in a measure space $(E, {\cal E}, \mu)$, $\tau$ is an invertible
measure-preserving transformation on $(E, {\cal E}, \mu)$,
and $M$ is a random measure on $(E, {\cal E}, \mu)$.  We explore
the relationship between the ergodic properties of the sequence
and the properties of $\tau$, and construct examples 
with various mixing properties using a
stacking method on the half-line $[0,\infty)$.

gross@bernoulli.ucsb.edu (Latex file available)

163. GREEDY LATTICE ANIMALS

Alberto Gandolfi and Harry Kesten

Let {X(v):v a vertex of Z^d} be an i.i.d. family of positive random
variables, and for any set C of vertices let S(C) be the sum of the
X(v) over v in C. We are interested in the behavior for large n of
the maximum of S(C) over various sets C of cardinality n, which contain
the origin. We write M(n) for the maximum of S(C) when C ranges over
all selfavoiding paths of n vertices, which start at the origin; N(n)
denotes the maximum of S(C) when C ranges over all connected sets of
n vertices containing the origin. We show that if X(0) has a (d+a)-th
moment for some a > 0, then lim M(n)/n and lim N(n)/n exist and are
finite w.p. 1. We believe the two limits to be different in general. If
the d-th moment of X(0) is infinite, then limsup M(n)/n = limsup N(n)/n
= infinity w.p. 1.

HAK@CORNELLA.cit.cornell.edu

164. BOUNDS FOR LEAST RELATIVE VACANCY IN A SIMPLE MOSAIC PROCESS

Peter March and Timo Seppalainen

Let 0<d<m<1 and consider the mosaic process formed by centering 
$d \times d$ squares on the points of a Poisson process of intensity
$\lambda$ in the unit square $D \subseteq R^2.$ If $G$ denotes the union 
of these squares then least relative vacancy is the infimum of the
quantity $m^{-2} \times |S \cap G^c| taken over all $m \times m$ squares
$S$ such that $S \subseteq D.$ We prove two-sided bounds for the
distribution of least relative vacancy and show that the bounds are
asymptotically sharp, in the logarithmic sense, as d tends to zero.

march@function.mps.ohio-state.edu

165. DEGENERATE STOCHASTIC DIFFERENTIAL EQUATIONS AND HYPOELLIPTIC PARABOLIC OPERATORS

Denis R. Bell and Salah-Eldin A. Mohammed

This paper establishes the existence of smooth densities for a class
of stochastic differential equations of the form

     dx(t) = g(t, x(t - r))dW(t) + H(t, x)dt,

where r is a positive time delay and H denotes a non-anticipating
function defined on the space of paths.  It is shown that if g has
any finite number of points of degeneracy of polynomial order, then
x(t) admits a smooth density for all positive times t.  Similar
results are also obtained for time-dependent diffusion processes
whose diffusion coefficients have isolated degenerate points of
exponential order.  The latter results are shown to imply the
hypoellipticity of a large class of exponentially degenerate second
order differential operators L for which Hormander's condition
fails.  In the case where the coefficients of L are space-independent,
the same condition is established under a very weak mean ellip-
ticity condition.

(This paper is a modification of the paper announced in the
April 30 abstracts mailing, with stronger versions of the
theorems relating to diffusion processes and PDEs).

dbell@unf1vm.bitnet

166. INVERSE SUBORDINATION OF EXCESSIVE FUNCTIONS

R. K. Getoor and M. J. Sharpe

Let $M$ be a multiplicative functional of a right process $X$, and let
$v$ be excessive for $(X,M)$. We characterize those positive functions $u$
satisfying $u=v+P_Mu$, where $P_Mu(x):=u(x)$ if ${\bold P}^x(M_0=1)=0$,
$={\bold P}^x\int_0^\infty u(X_t)(-dM_t)$ if not. Under sufficiently strong
conditions, it turns out that $u$ is necessarily excessive for $X$, but
in general, $u$ is excessive only for a subprocess $(X,T)$, where the
terminal
time $T$ is determined by $M$. Part of the method of proof involves an
explicit
formula for the $n$th power of the operator $P_M$. We also give a version
of the
results within the framework of (super)martingale theory, one part of which
provides a characterization of the positive solutions $u$ of $u=P_Mu$ as
functions that are locally harmonic up to $S:=\inf\{t:M_t=0\}$.

msharpe@euclid.ucsd.edu

167. MULTIPLICITIES OF A RANDOM SAUSAGE

Steven N. Evans

Consider a particle that executes a transient random walk
 or a transient L\'evy process on some group. Attach a set
 to the particle and trace out a sausage.  Each point in 
the sausage  that has been traced out over the inverval 
$[0,t]$ has an associated multiplicity - the amount of 
time in $[0,t]$ that the point has been covered by the 
moving set.  Using potential theory, we investigate the 
asymptotics as $t \rightarrow \infty$ of the ensemble of 
multiplicities.  Our results involve some interesting 
connections with the theory of Fredholm integral equations.

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

168. LEVY PROCESSES THAT CAN CREEP DOWNWARDS NEVER INCREASE

Jean Bertoin

A Levy process can creep downwards if the probability that it
does not jump at the first instant when it passes below a given
negative level is positive. This notion was introduced by Millar,
and also studied by Rogers. We show that in this case, the Levy
process never increases, where increase is taken in the sense of 
Dvoretzky, Erdos and Kakutani. In particular, a Levy process with
no negative jumps, or with non-zero Gaussian component never increases.

bertoin@frcirp81.bitnet

169. COMPUTER ALGEBRA IN PROBABILITY AND STATISTICS

Wilfrid S. Kendall

This paper discusses the uses of computer algebra within statistics and
probability. A distinction is drawn between the use of computer algebra
packages to {\sl support} investigations, by performing calculations,
and their use {\sl to implement structure}; to build in elements of a
theory (such as stochastic calculus or the Taylor string theory of
Barndorff Nielsen and others) as a preliminary to research
investigations.  Brief surveys are given of instances in the literature
of use of computer algebra in probability and statistics. Two examples
of implementations of structure are discussed, both drawn from the
author's own work with the computer algebra package REDUCE. One is a
simple demonstration using moments of the Poisson distribution. The
other is {\sl itovsn3}, an implementation of the semimartingale
stochastic calculus. It is described how {\sl itovsn3} may be used to
derive the characteristic function of the L\'evy stochastic area,
following a proof due to S.~Janson. Prospects for future work and for
work in progress are discussed.

w.s.kendall@warwick.ac.uk

170. ON THE EMPTY CELLS OF POISSON HISTOGRAMS

Wilfrid S. Kendall

This paper considers the histogram of unit cell size built up from $m$
independent observations on a Poisson$(\mu)$ distribution.  The
following question is addressed: what is the limiting probability of
the event that there are no unoccupied cells lying to the left of
occupied cells of the histogram?  It is shown that the probability of
there being no such isolated empty cells (or isolated finite groups of
empty cells) tends to unity as $m$ tends to infinity, but that the
corresponding almost--sure convergence fails. Moreover this probability
does {\sl not} tend to unity when the Poisson distribution is replaced
by the Negative Binomial distribution arising when $\mu$ is randomized
by a Gamma distribution. The relevance to empirical Bayes statistical
methods is discussed.

w.s.kendall@warwick.ac.uk

171. HEAT KERNEL ON A LIPSCHITZ MANIFOLD COMPOSED BY TWO HALFSPACES WITH DIFFERENT METRICS

Weian Zheng

When we consider a Lipschitz Riemannian manifold $M$
there are many difficulties due to the non-smoothness of the
geometric structure. In this paper, we will just consider the simplest
case where there is a global coordinate $(x_1,...,x_n)$ system on
$M$  and the Lipschitz Riemannian metirc
under this global coordinate system is given as
\begin{eqnarray}
g_{ij}(x)=I_{\lbrack x_1\leq 0\rbrack }A+I_{\lbrack x_1> 0\rbrack }B.
\end{eqnarray}
In other words, the Riemannian matrix as a function of point
$x\in M$ may only assume
two positive definite matrices as its possible values, which
is equal to A on the lower half-space and equal to B on the
upper half-space. In this paper, we give an precise
expression of the heat kernel on this manifold. As a direct consequence,
we found that the heat kernel is directionally
differentiable although the metric is discontinuous, which extends the 
known results about Holder continuity of general heat kernels.

This study also has a clear physical meaning. We can
use it to discuss the propagation of heat on a body
made by parts with materials of different thermal conductivities.

wzheng@math.uci.edu

172. ON A LIMIT THEOREM FOR NON-STATIONARY BRANCHING PROCESSES

Tetsuya Hattori,  Hiroshi Watanabe

The purpose of this paper is to give a limit theorem for a certain class
of discrete-time multi-type non-stationary branching processes.
    We consider the case where the generating functions
$$      f_N (z) = \sum_{k} z^k Prob [X_N=k | X_{N-1}=1]       $$
has a simple generation N dependece (non-stationarity) of 
$$         f_N (z) = D_{N-1}^{-1} F (D_N z) ,                 $$
where $D_N$'s are diagonal matrices.  Our results cover those cases 
where the function F has a singularity at $\lim_{N \to \infty} D_N \vec{1}$, 
so that a consideration of non-stationarity becomes essential.  
Our limit theorem is of a kind that reduces to the limit theorem
for supercritical branching processes in the stationary case.
    This work grew out of (and is related to) our attempt 
with K. Hattori in constructing a non-self-similar diffusion 
on Sierpinski gasket and related fractals.

hattori@tansei.cc.u-tokyo.ac.jp   or   watmath@tansei.cc.u-tokyo.ac.jp
     (Plain Tex file available)