Probability Abstracts 35
This document contains abstracts 736-770.
They have been mailed on October 5, 1996.
736. DOES PLANAR BROWNIAN MOTION HAVE CUT LINES?
Richard F. Bass and Krzysztof Burdzy
Let $Z_t$ be two-dimensional Brownian motion. We
say that a straight line $L$ is a cut line if there
exists a time $t\in (0,1)$ such that
the trace of $\{Z_s: 0\leq s<t\}$ lies on one side of $L$ and
the trace of $\{Z_s: t<s<1\}$ lies on the other side of $L$.
In this paper we prove that with probability one cut lines do not exist.
This provides a solution to Problem 8 in Taylor (1986).
A compressed postscript file can be found at the url below.
bass@math.washington.edu, burdzy@math.washington.edu
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737. ON REGULARITY PROPERTIES OF NONSYMMETRIC
ORNSTEIN-UHLENBECK SEMIGROUP IN $L^p$ SPACES
Anna Chojnowska-Michalik and Beniamin Goldys
We study properties of the transition semigroup $R_t$ corresponding to the
Hilbert space valued nonsymmetric Ornstein-Uhlenbeck process possessing an
invariant measure $\mu$. Necessary and sufficient condition is given for
$R_t\phi$ to be infinitely smooth in the direction of the Reproducing Kernel
of $\mu$ for every bounded Borel $\phi$. Estimates on the derivatives of
$R_t\phi$ are obtained and the same estimates are obtained for the adjoint
semigroup. We give also necessary and sufficient conditions for $R_t$ to be
an integral operator on $L^p(H,\mu )$ extending earlier results by
DaPrato-Zabczyk and Fuhrman. It is shown also that the integral kernel
possesses strong integrability properties. The transition semigroup is also
investigated in the scale of Sobolev spaces. The transition semigroup
turns out to be strongly continuous in those spaces and compact if it is
integral in $L^p(H,\mu )$. Finally, an application to some parabolic PDE's on
Hilbert space is given.
beng@alpha.maths.unsw.edu.au
738. A PREDATOR-PREY AND A HOST-PARASITE SPATIAL STOCHASTIC MODELS
Rinaldo B. Schinazi
We consider two interacting particle systems on Z^d
to model predator-prey and host-parasite interactions.
In both models we have two types of particles (1 and 2)
and each site in Z^d can be in one of four states:
empty, occupied by a type 1 particle, occupied by a type 2 particle
or occupied by two particles (one of each type).
Each type gives birth to particles of the same type on nearest
neighbor sites. The interaction between the two types of particles occurs
only when a site is occupied by one particle of each type.
For both models we show that coexistence and non-coexistence are possible
in any dimension.
schinazi@vision.uccs.edu (plain Tex file available)
739. A LARGE DEVIATION PRINCIPLE FOR THE ORDER OF A RANDOM PERMUTATION
Neil O'Connell
We obtain a large deviation principle for the scaled logarithm
of the order of a random permutation of a large number of objects,
and give an explicit expression for the convex dual of the rate function.
noc@hplb.hpl.hp.com (Postscript version available)
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740. STRONGER TOPOLOGIES FOR SAMPLE PATH LARGE DEVIATIONS IN EUCLIDEAN SPACE
Neil O'Connell
In this paper we present sufficient conditions for
sample path large deviation principles to be extended
to finer topologies. We consider extensions of the
uniform topology by Orlicz functionals and we consider
Lipschitz spaces: the former are concerned with cumulative
path behaviour while the latter are more sensitive to
extremes in local variation. We also consider sample paths
indexed by the half line, where the usual projective limit
topologies are not strong enough for many applications,
particularly in queueing theory.
noc@hplb.hpl.hp.com (Postcript version available.)
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741. RENORMALIZED SELF-INTERSECTION LOCAL TIMES AND WICK
POWER CHAOS PROCESSES
Michael Marcus and Jay Rosen
Sufficient conditions are obtained for the continuity of
renormalized self-intersection local times for the
multiple intersections of a large class of strongly
symmetric L\'{e}vy processes in R^m, m=1,2. In R^2 these
include Brownian motion and stable processes of index greater than 3/2, as well
as many processes in their domains of attraction. In R^1 these include stable
processes of index 3/4<\beta\leq 1 and many processes in their domains of
attraction.
Let (\Omega,\FF(t),X(t), P^{x}) be one of these radially
symmetric L\'evy processes with 1-potential density u^1(x,y).
Let G^{2n}_F denote the class of positive
finite measures \mu on R^m for which
\int \int (u^1(x,y))^{2n}\,d\mu(x)\,d\mu(y)<\infty. \] For \mu\in G^{2n}_F, let
\begin{eqnaarray} &&\alpha_{n,\epsilon}(\mu,\lambda)
\stackrel{def}{=}\nn
\int \int_{\{0\leq t_1\leq \cdots \leq t_n\leq \lambda\}}\\&&\qquad
f_{\epsilon}(X(t_1)-x)\prod_{j=2}^n f_{\epsilon}(X(t_j)-
X(t_{j-1}))\,dt_1\cdots\,dt_n \,d\mu(x)\nn
\end{eqnaarray} where f_{\epsilon} is an approximate delta-function at zero and
\lambda is an random exponential time, with mean one, independent
of X (with probability measure P_\lambda). The renormalized
self-intersection local time of X with respect
to the measure \mu is defined as
\[
\gamma_{n}(\mu)=\lim_{\epsilon\rightarrow
0}\,\sum_{k=0}^{n-1}(-1)^{k}\(\stackrel{n-1}{k}\)(u^1_{\epsilon}(0))^{k}
\alpha_{n-k,\epsilon}(\mu,\la)
\] where, u^1(x)\st u^1(x+z,z), for all z\in R^m, and
u^1_{\epsilon}(x)\st\int f_{\epsilon}(x-y)u^1(y)\,dy. Conditons are
obtained under which
this limit exists in L^2(\Omega\times R^+,P^y_\la) for all y\in R^m, where
P^y_\la\st P^y\times P_\lambda.
Let \{\mu_x,x\in R^m\} denote the set of translates of the measure \mu. The
main result in this paper is a sufficient condition for the continuity of
\{\gamma_{n}(\mu_x),\,x\in R^m\}, namely that this process is continuous
P^y_\lambda almost surely for all y\in R^m, if the corresponding
2n-th Wick power chaos process, \{:G^{2n}\mu_x:,\,x\in R^m\} is continuous
almost surely. This chaos process is obtained in the following way.
A Gaussian process G_{x,\delta} is defined
which has covariance u^1_\delta(x,y), where \lim_{\delta\to
0}u_\delta^1(x,y)=u^1(x,y). Then \[ :G^{2n}\mu_x:\st\lim_{\delta\to
0}\int :G_{y,\de}^{2n}:\,d\mu_x(y) \] where the limit is taken in
L^2. (:G_{y,\delta}^{2n}: is the 2n-th Wick power of G_{y,\delta}, that is, a
normalized Hermite polynomial of degree 2n in G_{y,\delta}).
This process has a natural metric \begin{eqnarray}&&
d(x,y)\st\frac1{(2n)!}\(E(:G^{2n}\mu_x:-:G^{2n}\mu_y:)^2\)^{1/2}\nn\\
&&\quad=\(\int\!\! \int
\(u^1(u,v)\)^{2n} \left( d(\mu_x(u)-\mu_y(u)) \right) \left(
d(\mu_x(v)-\mu_y(v)) \right)\)^{1/2}\nn.
\end{eqnarray} A well known metric entropy condition with
respect to d gives a sufficient condition for the continuity of
\{:G^{2n}\mu_x:,\,x\in R^m\} and hence for
\{\gamma_{n}(\mu_x),\,x\in R^m\}.
jrisi@cunyvm.cuny.edu (DVI file available)
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742. GAUSSIAN CHAOS AND SAMPLE PATH PROPERTIES OF
ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES
Michael Marcus and Jay Rosen
Let X be a strongly symmetric Hunt process with \alpha-potential density
u^\alpha(x,y). Let
\[G_{\alpha}^2={ \mu\,|\,\,\int\!\! \int \(u^{\alpha}(x,y)\)^{2}
\,d\mu(x)\,d\mu(y) <\infty},\]
and let L^{\mu}_{t} denote the continuous additive functional with Revuz
measure \mu.
For a set of positive measures M\subset G_{\alpha}^2, subject to some
additional regularity
conditions, we consider families of continuous (in time),
additive functionals L={L^{\mu}_{t}, (t,\mu)\in R^{+}\times M} of X and a
second order Gaussian chaos H_{\alpha}={H_{\alpha}(\mu), \mu\in M} which is
associated with L by an isomorphism
theorem of Dynkin.
A general theorem is obtained which shows that, with some additional
regularity conditions depending on X and M, if H_{\alpha} has a
continuous version on M almost surely then so does
L and furthermore, that moduli of continuity for H_{\alpha} are also moduli
of continuity for L.
Special attention is given to L\'evy processes in R^n and T^n, the
n-dimensional torus, with M taken to be the set of translates of
a fixed measure. Many concrete examples are
given, especially when X is Brownian motion in R^n and T^n for n=2 and 3.
For certain radially symetric smooth measures and processes, including
Brownian motion in T^3, we show
that if the 1-potential of a given finite measure \mu, U^1\mu(0) is finite then
{L^{\mu}_{t}, (t,\mu)\in R^{+}\times M} is continuous almost surely, where
M is the set of all translates of \mu on T^n.
jrisi@cunyvm.cuny.edu (DVI file available)
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743. RANDOM FOURIER SERIES AND CONTINUOUS ADDITIVE
FUNCTIONALS OF LEVYS PROCESSES ON THE TORUS
Michael Marcus and Jay Rosen
Let X be an exponentially killed Levy process on T^n, the
n-dimensional torus, that satisfies a sector condition. (This
includes symmetric Levy processes). Let F_e denote the
extended Dirichlet space of X. Let h\in F_e and let
{h_y, y\in T^n} denote the set of translates of h. That is,
h_y(x)=h(x-y). We consider the family of 0-energy continuous additive
functionals {N^{[h_y]}_t, (y,t)\in T^n\times R^+} as defined by Fukushima.
For a very large class of random functions h we show that<BR>
\[ J_\rho(T^n)=\int (\log N_\rho(T^n,\epsilon))^{1/2}\,d\epsilon<\infty \]
is a necessary and sufficient conditions
for the family {N^{[h_y]}_t, (y,t)\in T^n\times R^+} to have a
continuous version almost surely. Here
N_\rho(T^n,\epsilon) is the minimum number of balls of radius \epsilon in the
metric \rho that covers T^n where the metric \rho is the energy
metric. We argue that this condition is the natural extension of the necessary
and sufficient condition for continuity of local times of Levy
processes of Barlow and Hawkes.
Results on the bounded variation and
p-variation (in t) of N^{[h_y]}_t, for y fixed, are also
obtained for a large class of random functions h.
jrisi@cunyvm.cuny.edu (DVI file available)
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744. JOINT CONTINUITY OF RENORMALIZED INTERSECTION LOCAL
TIMES
Jay Rosen
We study k-fold self-intersection local times \alpha(x_2,\ldots,x_k;t), and
k-fold renormalized self-intersection local times
\gamma(x_2,\ldots,x_k;t) for Levy processes. Our main result says that the
k-fold renormalized self-intersection local time
\gamma(x_2,\ldots,x_k;t) for the symmetric stable process of order \beta in
R^2 is jointly continuous almost surely if and only if (2k-1)(2-\bb)<2.
jrisi@cunyvm.cuny.edu (DVI and postscript file available)
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745. NORMAL FORMS FOR STOCHASTIC DIFFERENTIAL EQUATIONS
L. Arnold and P. Imkeller
We address the following problem from the intersection of dynamical systems
and stochastic analysis: two sde $dx_t = \sum_{j=0}^m f_j(x_t) \circ dW_t^j$
and $dx_t = \sum_{j=0}^m g_j(x_t) \circ dW_t^j$ in ${\bf R}^d$ with smooth
coefficients satisfying $f_j(0) = g_j(0) = 0$ are said to be smoothly equivalent
if there is a smooth random diffeomorphism (coordinate transformation)
$h(\omega)$ with $h(\omega, 0) = 0$ and $D h(\omega, 0) = id$ which conjugates
the corresponding local flows,
$$\phi(t, \omega) \circ h(\omega) = h(\theta_t \omega) \circ \psi(t, \omega),$$
where $\theta_t \omega(s) = \omega(t+s) - \omega(t)$ is the (ergodic) shift
on the canonical Wiener space. The normal form problem for sde consists in
finding the ''simplest possible'' member in the equivalence class of a given
sde, in particular in giving conditions under which it can be linearized, i.e.
$g_j(x) = D f_j(0) x$.
We develop a mathematically rigorous normal form theory for sde which justifies
the engineering and physics literature on that problem. It is based on the
multiplicative ergodic theorem and uses a uniform (with respect to a spatial parameter)
Stratonovitch calculus which allows the handling of non-adapted initial values
and coefficients in the stochastic version of the cohomological equation.
As a by-product, we prove a general theorem on the existence of a stationary
solution of a possibly anticipative affine sde.
arnold@mathematik.uni-bremen.de, imkeller@mathematik.hu-berlin.de
746. METASTATES IN DISORDERED MEAN FIELD MODELS:
RANDOM FIELD AND HOPFIELD MODELS
Christof Kuelske
We rigorously investigate the size dependence of disordered mean field models
with finite local spin space in terms of metastates. Thereby we provide an
illustration of the framework of metastates for systems of randomly competing
Gibbs measures. In particular we consider the thermodynamic limit of
the empirical metastate $1/N\sum_{n=1}^N \d_{\mu_n(\eta)}$ where
$\mu_n(\eta)$ is the Gibbs measure in the finite volume $\{1,\dots,n\}$
and the frozen disorder variable $\eta$ is fixed. We treat explicitly
the Hopfield model with finitely many patterns and the Curie Weiss Random
Field Ising model. In both examples in the phase transition regime the
empirical metastate is dispersed for large $N$. Moreover it does not
converge for a.e. $\eta$ but rather in distribution for whose limits we
give explicit expressions. We also discuss another notion of metastates, due to
Aizenman and Wehr.
(WIAS-preprint 260)
kuelske@wias-berlin.de
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747. ON THE RELATIVE LENGTHS OF EXCURSIONS DERIVED FROM A STABLE SUBORDINATOR
Jim Pitman and Marc Yor
Results are obtained concerning the distribution of ranked relative lengths of
excursions of a recurrent Markov process from a point in its state space whose
inverse local time process is a stable subordinator. It is shown that for a
large class of random times $T$ the distribution of relative excursion lengths
prior to $T$ is the same as if $T$ were a fixed time. It follows that the
generalized arc-sine laws of Lamperti extend to such random times $T$.
For some other random times $T$, absolute continuity relations are obtained
which relate the law of the relative lengths at time $T$ to the law at
a fixed time.
Technical Report No. 469, Department of Statistics, U.C. Berkeley
To appear in S\'em. de Probabilit\'es XXXI
pitman@stat.Berkeley.EDU
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748. ON THE LENGTHS OF EXCURSIONS OF SOME MARKOV PROCESSES
Jim Pitman and Marc Yor
Results are obtained regarding the distribution of the ranked lengths of
component intervals in the complement of the random set of times when a
recurrent Markov process returns to its starting point. Various martingales
are described in terms of the L\'evy measure of the Poisson point process of
interval lengths on the local time scale. The martingales derived from the zero set of a one-dimensional diffusion are related to martingales studied by Az\'ema and Rainer. Formulae are obtained which show how the distribution
of interval lengths is affected when the underlying process is subjected to a
Girsanov transformation. In particular, results for the zero set of an
Ornstein-Uhlenbeck process or a Cox-Ingersoll-Ross process are derived from
results for a Brownian motion or recurrent Bessel process, when the zero set is the range of a stable subordinator.
Technical Report No. 470, Department of Statistics, U.C. Berkeley
To appear in S\'em. de Probabilit\'es XXXI
pitman@stat.Berkeley.EDU
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here.
749. LAPLACE TRANSFORMS RELATED TO EXCURSIONS OF A ONE-DIMENSIONAL DIFFUSION
Jim Pitman and Marc Yor
Various known expressions in terms of hyperbolic functions for the Laplace
transforms of random times related to one-dimensional Brownian motion are
derived in a unified way by excursion theory and extended to one-dimensional
diffusions.
Technical Report No. 467, Department of Statistics, U.C. Berkeley
pitman@stat.Berkeley.EDU
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here.
750. FIRST PASSAGE PERCOLATION AND A MODEL FOR COMPETING SPATIAL GROWTH
Olle H\"aggstr\"om and Robin Pemantle
An interacting particle system modelling competing growth on
the ${\bf Z}^2$ lattice is defined as follows. Each $x\in{\bf Z}^2$
is in one of the states $\{0,1,2\}$. $1$'s and $2$'s remain in their
states forever, while a $0$ flips to a $1$ (resp.\ a $2$) at a rate
equal to the number of its neighbours which are in state $1$ (resp.\ $2$).
This is a generalization of the well known Richardson model. $1$'s and
$2$'s may be thought of as two types of infection, and $0$'s as uninfected
sites. We prove that if we start with a single site in state $1$ and a
single site in state $2$, then there is positive probability for the event
both types of infection reach infinitely many sites. This result implies
that the spanning tree of time-minimizing paths from the origin in first
passage percolation with exponential passage times has at least two
topological ends with positive probability.
olleh@math.chalmers.se, pemantle@math.wisc.edu
751. A NOTE ON (NON-)MONOTONICITY IN TEMPERATURE FOR THE ISING MODEL
Olle H\"aggstr\"om
We consider the Ising model on a finite graph $G$ at reciprocal
temperature $\beta$, and in particular we consider the event
$C_{x\leftrightarrow y}$ that two vertices $x$ and $y$ are in
the same spin-cluster, and the random variable $C_x$ which is
the size of the spin-cluster containing $x$. By means of a simple
counterexample, we show that the probability of $C_{x\leftrightarrow y}$
need not be increasing $\beta$, and that $C_x$ need not be stochastically
increasing in $\beta$. On the other hand, a sufficient condition on $G$
for these monotonicities to hold is established.
olleh@math.chalmers.se
752. ON THE ATTRACTING ORBIT OF A NON-LINEAR TRANSFORMATION ARISING FROM
RENORMALIZATION OF HIERARCHICALLY INTERACTING DIFFUSIONS
PART II: THE NON-COMPACT CASE
J.-B. Baillon, Ph. Cl\'ement, A. Greven, and F. den Hollander
This paper analyzes the $n$-fold composition of a non-linear integral
operator acting on a class of functions on $[0,\infty)$. Attracting
orbits and attracting fixed points are identified. Various results of
convergence to these orbits and to these fixed points are derived. The
proofs are based on order-preserving properties and comparison techniques.
A key role is played by the eigenfunctions of the operator, which are used
as comparison objects. The results imply that the space-time scaling limit
of an infinite system of interacting diffusions has universal behavior
independent of model parameters. The paper can be read independently of
Part I.
denholla@sci.kun.nl (ps-file available)
753. SELF-SIMILAR MEASURES AND INTERSECTIONS OF CANTOR SETS
Yuval Peres and Boris Solomyak
It is natural to expect that the arithmetic sum of two Cantor sets
should have positive Lebesgue measure if the sum of their dimensions exceeds 1,
but there are many known counterexamples, e.g. when both sets are the
middle-$\alpha$ Cantor set and $\alpha$ is in $(1/3,1/2)$. We show that
for any compact set $K$ and for a.e. $\alpha \in (0,1)$, the arithmetic
sum of $K$ and the middle-$\alpha$ Cantor set has positive Lebesgue measure,
provided that the sum of their Hausdorff dimensions exceeds 1.
In this case we also determine the essential supremum, as the translation
parameter $a$ varies, of the dimension of the intersection of $(K+a)$ with
the middle-$\alpha$ Cantor set. The same method yields an interesting
property of infinite Bernoulli convolutions $\nu_u^p$ (the distributions of
random series $\sum s_n u^n$ where the signs $s_n$ are chosen independently
in ${1,-1}$ with probabilities $(p,1-p)$). Let $1 \leq q_1<q_2 \leq 2$.
For $p$ near $1/2$ (but distinct from $1/2$) and for a.e. $u$ in some
nonempty interval, $\nu_u^p$ is absolutely continuous and its density is
in $L^{q_1}$ but not in $L^{q_2}$.
peres@math.huji.ac.il
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information provided by the author
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754. STOCHASTIC CALCULUS FOR BROWNIAN MOTION ON A BROWNIAN FRACTURE
Davar Khoshnevisan and Thomas M. Lewis
The impetus behind this work is a pathwise
development of stochastic integrals with
respect to iterated Brownian motion. Our
results belong to a body of recent work
indicating that the latter process is the
canonical process on a Brownian crack.
davar@math.utah.edu
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information provided by the author
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755. ON WEAK MIXING IN LATTICE MODELS.
Kenneth S. Alexander
For lattice models on Z^d, weak mixing is the property that the influence of
the boundary condition on a finite region decays exponentially with distance
from that region. For a wide class of models on Z^2, including all finite
range models, we show that weak mixing is a consequence of Gibbs uniqueness,
exponential decay of an appropriate form of connectivity, and a natural
coupling property. In particular, the 2D Fortuin-Kastelyn random cluster
model is weak mixing whenever uniqueness holds and the connectivity decays
exponentially, and the 2D q-state Potts model above the critical temperature
is weak mixing whenever correlations decay exponentially, a hypothesis
satisfied if q is sufficiently large. Ratio weak mixing is the property that
uniformly over events A and B occuring on subsets U and V, respectively, of
the lattice, |P(AB)/P(A)P(B) - 1| decreases exponentially in the distance
between U and V. We show that under mild hypotheses, for example finite
range, weak mixing implies ratio weak mixing.
alexandr@math.usc.edu
756. CONFORMAL INVARIANCE OF VORONOI PERCOLATION
Itai Benjamini and Oded Schramm
It is proved that in the Voronoi model for percolation in dimension $2$
and $3$, the crossing probabilities are asymptotically invariant under
conformal change of metric.
To define Voronoi percolation on a manifold $M$, you need a measure
$\mu$, and a Riemannian metric $ds$. Points are scattered according to a
Poisson point process on $(M,\mu)$, with some density $\lambda$. Each
cell in the Voronoi tessellation determined by the chosen points is
declared {\it open\/} with some fixed probability $p$, and {\it closed\/}
with probability $1-p$, independently of the other cells. The above
conformal invariance statement means that under certain conditions, the
probability for an open crossing between two sets is asymptotically
unchanged, as $\lambda\to\infty$, if the metric $ds$ is replaced by any
(smoothly) conformal metric $ds'$. It is also conjectured that if $\mu$
and $\mu'$ are two measures comparable to the Riemannian volume
measure, then replacing $\mu$ by $\mu'$ does not effect the limiting
crossing probabilities.
itai@wisdom.weizmann.ac.il, schramm@wisdom.weizmann.ac.il
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information provided by the author
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here.
757. A DIFFUSION MODEL FOR BOOKSTEIN TRIANGLE SHAPE
Wilfrid S. Kendall
A stochastic dynamical context is developed for Bookstein's shape
theory. It is shown how Bookstein's shape space for triangles arises
naturally when the landmarks are moved around by a special Brownian
motion on the general linear group of invertible
${\oldstyle(2\times2)}$ real matrices. Asymptotics for the Brownian
transition density are used to suggest an exponential family of
distributions analogous to the Fisher-von Mises spherical distribution.
The computer algebra implementation Itovsn3 of stochastic
calculus is used to perform the calculations (some of which actually
date back to work by Dyson on eigenvalues of random matrices and by
Dynkin on Brownian motion on ellipsoids). An interesting feature of
these calculations is that they include the first application (to the
author's knowledge) of the Gr\"obner basis algorithm in a stochastic
calculus context.
wsk@stats.warwick.ac.uk
758. ON SOME WEIGHTED BOOLEAN MODELS
Wilfrid S. Kendall
An overview is given of some recent work (joint with Adrian Baddeley
of Perth and Colette van Lieshout of Warwick) on a new class of random
point and set processes, obtained using a rather natural weighting
procedure employing quermass integrals. The concept of exact (or
perfect) simulation of point processes is then introduced, and a
discussion is given of possibilities for perfect simulation of
quermass weighted processes.
wsk@stats.warwick.ac.uk
759. QUERMASS-INTERACTION PROCESSES
Adrian J. Baddeley, Wilfrid S. Kendall, Marie-Colette van Lieshout
This paper commences an investigation into a new class of random point
and set processes, obtained using a rather natural weighting
procedure. Given a Poisson point process, on each point one places a
{\em grain}, a (possibly random) compact convex set. Let $\Xi$ be the
union of all grains. One can now weight the process using the
exponential of a quermass functional of $\Xi$. If the functional is the
area functional then we recover the area-interaction point process. New
point processes arise if we take the perimeter length functional, or
the Euler functional (number of components minus number of holes). The
main question addressed by the paper is that of when the resulting
point process is well-defined: geometric arguments are used to
establish conditions for the point process to be stable in the sense of
Ruelle.
wsk@stats.warwick.ac.uk
760. RATES OF CONVERGENCE FOR LAMPLIGHTER PROCESSES
Olle H\"aggstr\"om and Johan Jonasson
Consider a graph, $G$, for which the vertices can have two modes, 0 or 1.
Suppose that a particle moves around on $G$ according to a discrete time
Markov chain with the following rules. With (strictly positive) probabilities
$p_{m}$, $p_{c}$ and $p_{r}$ it moves to a randomly chosen neighbour, changes
the mode of the vertex it is at or just stands still respectively. We call
such a random process a $(p_{m},p_{c},p_{r})$-lamplighter process on $G$.
Assume that the process starts with the particle in a fixed position and
with all vertices having mode 0. The convergence rate to stationarity in
terms of the total variation norm is studied for the special cases when
$G={\mathcal K}_N$, the complete graph with $N$ vertices, and when
$G = {\bf Z}$ mod $N$. In the former case we prove that as
$N \rightarrow \infty$, $\frac{2p_{c}+p_{m}}{4p_{c}p_{m}}N\log N$ is a
threshold for the convergence rate. In the latter case we show that the
convergence rate is asymptotically determined by the covering time $C^N$ in
that the total variation norm after $aN^2$ steps is given by
$P(C^N>ap_{m}N^2)$. The limit of this probability can in turn be calculated
by considering a Brownian motion with two absorbing barriers. In particular
this means that there is no threshold for this case.
olleh@math.chalmers.se
761. TAIL INDEX ESTIMATION FOR DEPENDENT DATA
Sidney Resnick and C\u{a}t\u{a}lin St\u{a}ric\u{a}
A popular estimator of the index of regular variation in
heavy tailed models is Hill's estimator. We discuss
consistency of Hill's estimator when it is applied to certain classes of
heavy tailed stationary processes. One class of processes discussed consists
of processes which can be appropriately approximated by
sequences of $m$-dependent random variables
and special cases of our results show the consistency of Hill's estimator
for (i) infinite moving averages with heavy tail innovations,
(ii) a simple stationary bilinear model driven by heavy tail noise
variables, (iii) solutions of stochastic difference equations of the form
$$Y_t=A_t Y_{t-1}+Z_t, \quad -\infty <t<\infty$$ where
$\{(A_n,Z_n), -\infty <n<\infty\}$ are iid and the $Z$'s have regularly
varying tail probabilities. Another class of problems where our methods work
successfully are solutions of stochastic difference equations such as
the ARCH process where the process cannot be successfully approximated
by $m$-dependent random variables. A final class of models where Hill
estimator consistency is proven by our tail empirical process methods is the
class of hidden semi-Markov models.
Available as TR1174.ps.Z at url below.
sid@orie.cornell.edu
- To see a preprint or other
information provided by the author
click here.
762. ASYMPTOTIC BEHAVIOR OF JILL'S ESTIMATOR FOR AUTOREGRESSIVE DATA
Sidney Resnick and C\u{a}t\u{a}lin St\u{a}ric\u{a}
Consider a stationary, $p$th order autoregression $\{X_n\}$ satisfying
$$X_n=\sum_{i=1}^p \phi_i X_{n-i} +Z_n,\quad n=0,\pm 1,\pm
2,\ldots $$
whose innovation sequence $\{Z_n\}$ is iid with regularly varying
tail probabilities of index $-\alpha$. From observations
$X_1,\ldots,X_n$, one may estimate $\alpha^{-1}$ by applying Hill's
estimator to $X_1,\ldots,X_n$. Alternatively, a second procedure is to
use $X_1,\ldots , X_n$ to get estimates $\hat \phi_1,\ldots,\hat
\phi_p$ of the autoregressive coefficients and then to estimate the
residuals by
$$\hat Z_t(n)=X_t-\sum_{i=1}^p \hat \phi_i X_{t-i},\quad t=p+1,\ldots,n,$$
and then to apply Hill's estimator
to the estimated residuals. We show that from the point of asymptotic
variance, the second procedure is superior.
Available as TR1165.ps.Z at url below.
sid@orie.cornell.edu
- To see a preprint or other
information provided by the author
click here.
763. PATTERNS OF BUFFER OVERFLOW IN A CLASS OF QUEUES WITH LONG
MEMORY IN THE INPUT STREAM
David Heath, Sidney Resnick and Gennady Samorodnitsky
We study the time it takes until a a fluid queue with a finite, but
large, holding capacity reaches the overflow point. The queue is
fed by an {\it on/off\/} process, with a heavy tailed {\it on}
distribution, which is known to have long memory. It
turns out that the expected time until overflow, as a function of
capacity $L$, increases only polynomially fast, and so overflows
happen much more often than in the ``classical'' light tailed case,
where the expected overflow time increases as an exponential function
of $L$. Moreover, we show that in the heavy tailed case, overflows are
basically caused by single huge jobs. An implication is that the usual
$GI/G/1$ queue with finite but large holding capacity and heavy tailed
service times, will overflow about equally often {\it no matter how
much we increase the service rate}. We also study the time until
overflow for queues fed by a superposition of $k$ iid. {\it on/off\/}
processes with a heavy tailed {\it on} distribution, and show the benefit of
pooling the system resources as far as time until overflow is
concerned.
Available as TR1169.ps.Z at the url below.
sid@orie.cornell.edu, gennady@orie.cornell.edu, davidh@orie.cornell.edu.
- To see a preprint or other
information provided by the author
click here.
764. ASYMPTOTIC LENGTH OF LONGEST INCREASING SEQUENCES
ON THE PLANAR LATTICE
Timo Seppalainen
In 1977 Vershik and Kerov deduced the asymptotic normalized
length of the longest increasing sequence among independent
points uniformly distributed on the unit square. We solve the
analogous problem for points on the planar square lattice that
are present independently of each other.
seppalai@iastate.edu (Hardcopy and AMS-TeX file available)
765. EVERY GRAPH WITH A POSITIVE CHEEGER CONSTANT
CONTAINS A TREE WITH A POSITIVE CHEEGER CONSTANT
Itai Benjamini and Oded Schramm
It is shown that every (infinite) graph with a positive Cheeger
constant contains a tree with a positive Cheeger constant.
Moreover, for every nonnegative integer $k$ there is a unique
connected graph $T(k)$ that has Cheeger constant $k$, but removing
any edge from it reduces the Cheeger constant. This minimal graph,
$T(k)$, is a tree, and every graph $G$ with Cheeger constant
$h(G)\geq k$ has a spanning forest in which each component is
isomorphic to $T(k)$. In the case $h(G)<1$, the proof is
probabilistic, and uses tree indexed random walks.
itai@wisdom.weizmann.ac.il or schramm@wisdom.weizmann.ac.il
- To see a preprint or other
information provided by the author
click here.
766. THE SCALING LIMIT OF LATTICE TREES IN HIGH DIMENSIONS
Eric Derbez and Gordon Slade.
We prove that above eight dimensions the scaling limit
of sufficiently spread-out
lattice trees is the variant of super-Brownian motion known as
integrated super-Brownian excursion (ISE),
as conjectured by Aldous.
The same is true for nearest-neighbour lattice trees in sufficiently
high dimensions. The proof uses the lace expansion.
slade@mcmaster.ca
767. CONVERGENCE OF MODERATELY INTERACTING PARTICLE SYSTEMS TO A
DIFFUSION-CONVECTION EQUATION
Benjamin Jourdain
We give a probabilistic interpretation of the solution of a
diffusion-convection equation. To do so, we define a martingale problem
in which the drift coefficient is nonlinear and unbounded for small
times whereas the diffusion coefficient is constant. We check that the
time marginals of any solution are given by the solution of the
diffusion-convection equation. Then we prove existence and uniqueness for
the martingale problem and obtain the solution as the propagation of
chaos limit of a sequence of moderately interacting particle systems.
Keywords: nonlinear martingale problem, propagation of chaos, particle
systems, moderate interaction, diffusion-convection equation
jourdain@cermics.enpc.fr
http://cermics.enpc.fr/~jourdain/home.html
768. ASYMPTOTIC PROPERTIES OF AN APPROXIMATE MAXIMUM LILEIHOOD ESTIMATOR
FOR STOCHASTIC PDE'S
M. Huebner, S. Lototsky and B. Rozovskii
A maximum likelihood estimate of a scalar parameter is constructed
for a stochastic evolution system using the Galerkin
approximation of the original equation. Conditions are established
to guarantee the consistency and asymptotic normality of the estimator
as the dimension of the approximation tends to infinity. Examples are
given to illustrate the results.
huebner@stt.msu.edu
769. A CHARACTERIZATION OF ASYMPTOTIC BEHAVIOUR OF MAXIMUM LIKELIHOOD
ESTIMATORS FOR STOCHASTIC PDE'S
M. Huebner
The maximum likelihood method is employed to estimate
multidimensional parameters occuring in stochastic partial
differential equations. The observations of the process are assumed to be
continuous in time for any fixed time interval, but can be discrete
in the space coordinates. Criteria for the asymptotic behaviour of the
maximum likelihood estimator are derived for parameters occuring
in general stochastic partial differential equations involving
possibly noncommuting operators.
Expressions for the rates of convergence of the components of
consistent estimators are derived. Efficient estimation of parameters is
characterized by the order of the operators in the equations
when the operators have a common system of eigenfunctions.
huebner@stt.msu.edu
770. ON THE RECURRENCE OF SOME TIME INHOMOGENEOUS MARKOV PROCESSES
Yoichi Oshima
Given a time inhomogeneous Markov process ${\bf M} = (X_t,P_{(s,x)})$ associated
with a family of Dirichlet forms $(E^{(t)},F)$ on $L^2(X;m)$, we give a general
criterion for recurrence of {\bf M}. In particular, suppose that
$E^{(t)}(\phi,\psi)= \sum_{i,j=1}^d \int_{R^d} a_{ij}(t,x) {\partial \phi \over
\partial x_i} {\partial \psi \over \partial x_j} dx$, $\phi, \psi \in
C^\infty_0(R^d)$ for $a_{ij}(t,x) = \Gamma(t) a_{ij}(x)$ with symmetric
uniformly elliptic functions $(a_{ij})$ on $R^d$ and a positive increasing
function $\Gamma(t)$ on $[0,\infty)$. Then we can use the criterion to assert
that {\bf M} is recurrent if $d=1$ and $\lim_{T \to \infty} {1 \over
\Lambda(T)^2} \int_0^T \lambda(s) ds = \infty$ or $d=2$ and $\lim_{T \to \infty}
\log(\int_0^T \lambda(s) ds) = \infty$, where $\lambda(s) = \inf_{t\leq s}
\Gamma(t)$ and $\Lambda(s) = \sup{t\leq s} \Gamma(t)$.
oshima@sonata.tech.kumamoto-u.ac.jp
