Probability Abstracts 31
This document contains abstracts 620-645.
They have been mailed on January 28, 1996.
Click here to see the
list of all abstract titles.
620. EXPONENTIAL RELAXATION OF GLAUBER DYNAMICS WITH
SOME SPECIAL BOUNDARY CONDITIONS
Roberto H. Schonmann and Nobuo Yoshida
We consider attractive finite-range Glauber dynamics and show that
if a certain mixing condition is satisfied, then
the system evolving on arbitrary subsets of the lattice,
with appropriate boundary conditions, converges to
equilibrium exponentially fast, in the uniform sense, uniformly
over the subsets of the lattice. This result applies, for instance,
to the ferromagnetic nearest neighbor Ising model in the so called
``Bassuev region'', where complete analyticity is expected to fail.
Technically the result in this paper is an extension of a result
of Martinelli and Olivieri, who proved that under a weaker form of
mixing the infinite system approaches equilibrium exponentially
fast.
Conceptually this paper may be seen as a step towards
developing and exploiting a restricted notion of complete analyticity
in which the boundary conditions, rather than the shapes of
the regions under consideration are being restricted.
rhs@math.ucla.edu nobuo@math.mit.edu
(latex file available)
621. ABSOLUTE CONTINUITY OF BERNOULLI CONVOLUTIONS, A SIMPLE PROOF
Yuval Peres and Boris Solomyak
The distribution $\nu_\lambda$ of the random series $\sum \pm \lambda^n$
has been studied by many authors since the two seminal papers by Erd\H{o}s
in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urba\'{n}ski,
and Ledrappier showed the importance of these distributions in several
problems in Hausdorff dimension estimation. Recently the second author
proved a conjecture made by Garsia in 1962, that $\nu_\lambda$ is absolutely
continuous for a.e.\ $\lambda \in (1/2,1)$. Here we give a considerably
simplified proof of this theorem, using differentiation of measures instead
of Fourier transform methods. This technique is better suited to analyze
more general random power series.
peres@math.huji.ac.il solomyak@math.washington.edu (Amstex file available)
622. EQUIVALENCES OF THE LARGE DEVIATION PRICIPLE FOR GIBBS MEASURES AND
CRITICAL BALANCE IN THE ISING MODEL
Priscilla E. Greenwood and Jiaming Sun
we obtain equivalence of the large deviation principle for Gibbs measures
with and without an external field. For the Ising model, the equivalence
allows us to study the result of competing influences of a positive
external field $h$ and a negative boundary condition in the cube
$\La(B/h)$ as $h$ tends to zero for various $B$. We find a critical
balance at a value $B_0$ of $B$. We also strengthen some results of
Schonmann and Shlosman (1995b) for the 2-d Ising model.
sunj@math.ubc.ca
623. HOW MANY RANDOM WALKS CORRESPOND TO A GIVEN SET
OF RETURN PROBABILITES TO THE ORIGIN ?
Holger Dette and William J. Studden:
We consider the class of simple random walks or birth and death
chains on the nonnegative integers. The set of return
probabilities $P^n_{00}$, $n \geq 0$, uniquely determines the
spectral measure of the process. We characterize the class of
simple random walks with the same spectral measure or same return
probabilities to the origin. The analysis is based on the
spectral theory developed by Karlin and McGregor (1959),
continued fractions and canonical moments.
holger.dette@rz-ruhr-uni-bochum.de
624. STOCHASTIC PARALLEL TRANSPORT ALONG L\'EVY FLOWS OF DIFFEOMORPHISMS
David Applebaum and Serge Cohen
Intrinsic stochastic calculus on manifolds for processes with jumps is
used to prove global existence, uniqueness and isometry for parallel transport
of tangent vectors along the paths induced by a stochastic flow of
diffeomorphisms driven by a L\'evy process.
sc@louise.enpc.fr
625. STOPPED MARKOV CHAINS WITH STATIONARY OCCUPATION TIMES
Steven N. Evans and Jim Pitman
Let $E$ be a finite set equipped with a group $G$
of bijective transformations and suppose that $X$ is an irreducible Markov
chain on $E$ that is equivariant under the action of
$G$. In particular, if $E = G$ with the corresponding transformations
being left or right multiplication, then $X$ is a random walk on $G$.
We show that when $X$ is started at a fixed point
there is a stopping time $U$ such that the distribution of the random
vector of pre-$U$ occupation times is invariant under the action
of $G$. When $G$ acts transitively (that is, $E$ is a homogeneous space),
any non-zero, finite expectation stopping
time with this property can occur no earlier than
the time $S$ of the first return to the
starting point after all states have been visited.
We obtain an expression for the joint Laplace transform
of the pre-$S$ occupation times for an arbitrary finite chain and
show that even for random walk on the group of integers mod $r$
the pre-$S$ occupation times do not generally have a group invariant
distribution. This appears to contrast with the
Brownian analog, as there is considerable support for the conjecture
that the field of local times for Brownian motion on the circle prior to
the counterpart of $S$ is stationary under circular shifts.
evans@stat.berkeley.edu (Plain TeX file available)
Compressed PostScript available from:
626. LYAPUNOV EXPONENTS FOR FINITE STATE NONLINEAR FILTERING
Rami Atar and Ofer Zeitouni
Consider the Wonham optimal filtering problem for a finite state ergodic Markov
process in both discrete and continuous time, and let $\sigma$ be the noise
intensity for the observation. We examine the sensitivity of the solution with
respect to the filter's initial conditions in terms of the gap between the
first two Lyapunov exponents of the Zakai equation for the unnormalized
conditional probability. This gap is studied in the limit as $\sigma\to 0$ by
techniques involving considerations of nonlinear filtering and the stochastic
Feynman-Kac formula. Conditions are given for the limit to be either negative
or $-\infty$. Asymptotic bounds are derived in the latter case.
rami@aluf.technion.ac.il zeitouni@ee.technion.ac.il
627. MARTINGALES AND THE $n$-DIMENSIONAL BEURLING-AHLFORS TRANSFORM
Arthur Lindeman II
Recent progress in the study of quasiregular mappings has involved
a generalization of the Beurling-Ahlfors transform (complex Hilbert
transform). This $n$-dimensional version, $S$, is a differential
geometric operator also known as the signature operator and can be
thought of as a matrix of second-order Riesz transforms. T. Iwaniec
and G. Martin have shown that precise statements about the regularity
of quasiregular mappings can be made in terms of the $L^p$ operator
norm of $S$.
This paper offers a probabilistic approach to identify these norms
following the work of D. Burkholder, R. Ba\~nuelos and G. Wang.
$S$ corresponds to a matrix martingale transform of a vector-valued
martingale on the background radiation filtration. The norm is
estimated by a martingale inequality which requires the notion
of differential subordination. In addition, the non-uniqueness of
the representation of operators as martingale transforms is studied.
We prove that the norm of $S$ restricted to $k$-forms is independent
of $n$:
$$
\|S\|_{L^p(\bold R^n,\Lambda^k)}\leq 2\max(3,2k-1)(p^*-1),
$$
where $p^*=\max(p,{p\over{p-1}})$. It follows that the norm of the
full operator is bounded above by $2(n-1)(p^*-1)$ if $n\geq 4$ and
even, and $2(n-2)(p^*-1)$ if $n\geq 5$ and odd. While it is
conjectured that the norm is simply $p^*-1$, these results are all
new and are currently the best known.
lindaj@math.purdue.edu
- To see some information provided by the author
click here.
628. THE MARTINGALE STRUCTURE OF THE BEURLING-AHLFORS TRANSFORM
Rodrigo Banuelos and Arthur Lindeman II
This paper is a continuation of the probabilistic study of the
complex and $n$-dimensional Beurling-Ahlfors transforms initiated by
R. Ba\~nuelos, G. Wang and A. Lindeman. The problem at hand is to
identify the $L^p$ norms of a family of singular integral operators of
even kernel. These operators are used to study properties of
quasiregular mappings.
There are two focal points of this investigation. First we study
various properties of the operators by carefully examining the
representation of the operator as a martingale transform. By
``optimizing'' the differential subordination constant (done in a
general setting), we are able to prove inequalities of the type:
$$
\| (|Bf|^2+|\tau B\tau f|^2+|2f|^2)^{1/2} \|_p
\leq 4(p^*-1)\|f\|_p,
$$
where $B$ is the complex Beurling-Ahlfors transform,
$\tau f=\overline{f}$ and $p^*=\max(p,{p\over {p-1}})$.
Analogues of the above inequality are proved for the cases
$n\geq 3$. We call these inequalities Ess\'en-type in view of a
similar inequality for the real Hilbert transform. Thus, the
probabilistic technique not only produces the best known constants,
but also yields qualitatively new types of inequalities for these
operators.
Secondly, we point out a natural way to proceed analytically
suggested by this probabilistic approach. The connection is
provided by a function of D. Burkholder used to prove his martingale
inequalities under the hypothesis of differential subordination.
banuelos@math.purdue.edu, lindaj@math.purdue.edu
- To see some information provided by the author
click here.
629. TRANSPORTATION APPROACH TO SOME CONCENTRATION INEQUALITIES
IN PRODUCT SPACES
Amir Dembo and Ofer Zeitouni
For vectors x,y,z in Omega^N let f(x,y,z) be
the number of coordinates k for which x_k neither equals y_k nor z_k.
Using Marton's transportation approach we prove that
for every probability measures P,Q,R on Omega^N with
P a product measure there exist r.c.p.d. S,T such that
int S(.|x) dP(x) = Q(.) ; int T(.|x) dP(x) = R(.)
and for every 0<b<1/2,
b b f(x,y,z)
int dP/dQ(y) dP/dR(z) (1+b (1-2 b)) dS(y|x) dT(z|x) dP(x) =< 1
For Q(.)=R(.)=P(.|A) we essentially recover some of
Talagrand's concentration inequalities in product spaces.
amir@playfair.stanford.edu (LaTeX or Postscript file available)
630. LIMIT THEORY FOR BILINEAR PROCESSES WITH HEAVY TAILED NOISE
Richard A. Davis and Sidney I. Resnick
We consider a simple stationary bilinear model
$$X_t=cX_{t-1}Z_{t-1} +Z_t, \quad t=0,\pm 1,\pm 2,\dots $$
generated by heavy tailed noise variables $\{Z_t\}$. A complete analysis of
weak limit behavior is given by means of a point process analysis. A
striking feature of this analysis is that the sample correlation converges
in distribution to a non-degenerate limit. A warning is sounded about trying
to detect non-linearities in heavy tailed models by means of the sample
correlation function.
rdavis@stat.colostate.edu or sid@orie.cornell.edu
- To see some information provided by the author
click here.
631. TIED FAVOURITE EDGES FOR SIMPLE RANDOM WALK
B\'alint T\'oth and Wendelin Werner
We show that there are almost surely only finitely many times at
which there are at least 4 `tied' favourite edges for a simple
random walk. This (partially) answers a question of P.~Erd\H os
and P.~R\'ev\'esz.
wwerner@dmi.ens.fr, balint@math-inst.hu
- To see some information provided by the author
click here.
632. WEAK CONVERGENCE OF SYMMETRIC DIFFUSION PROCESSES
Kazuhiro Kuwae and Toshihiro Uemura
In this paper, we show the convergence of forms in the sense of
Mosco associated with the part form on relatively compact open set
of Dirichlet forms with locally uniform ellipticity and the locally
uniform boundedness of ground states under regular Dirichlet space setting.
We also get the same assertion under Dirichlet space in infinite dimensional
setting. As a result of this, we get the weak convergence under some
comditions on initial distributions and the growth order of the volume of
the balls defined by (modified) pseudo metric used in K.Th.Sturm.
kuwae@gauss.ma.is.saga-u.ac.jp
633. WEAK CONVERGENCE OF SYMMETRIC DIFFUSION PROCESSES II
Kazuhiro Kuwae and Toshihiro Uemura
In this note, we develop the results of "Weak convergence of symmetric
diffusion processes". We consider the Dirichlet forms associated with
(Girsanov)transformed processes by (super)martingale multiplicative
functionals defined by (singular)drift. In this situation, we can not
apply the results of "Weak convergence of symmetric
diffusion processes", because the drifts are not bounded on compact open
balls. So we can not show the convergence in finite dimensional
distribution globally from that locally as used in "Weak conv,,,,,,",
which use the martingale method.
We show the the convergence of forms in the sense of
Mosco not locally but globally. The key point of this proof is that
every closed forms $\Gamma$-converges for some subsequence. The notion
of $\Cal E$-nest on $X-F$ plays a fundamental role.
kuwae@gauss.ma.is.saga-u.ac.jp
634. ON PSEUDO METRICS OF DIRICHLET FORMS ON SEPARABLE METRIC SPACES
Kazuhiro Kuwae
In this note, we show the pseudo metrics of Dirichlet forms locally
belongs to the domains if the metric function is continuous including
the infinite dimensional setting and the finite dimensional setting.
kuwae@gauss.ma.is.saga-u.ac.jp
635. RANDOM POLYMERS
Frank den Hollander
This paper is a mini-review of some recent developments on probabilistic
models for polymer chains.
denholla@sci.kun.nl (TeX-file available)
636. PROBABILISTIC ASPECTS OF PHYSICS
Frank den Hollander
This paper describes 5 examples from physics where probability is used
to identify an underlying variational principle.
denholla@sci.kun.nl (TeX-file available)
637. BROWNIAN EXCURSIONS, CRITICAL RANDOM GRAPHS
AND THE MULTIPLICATIVE COALESCENT
David Aldous
We introduce the multiplicative coalescent process, defined as follows.
The states are vectors ${\bf x}$ of nonnegative real cluster sizes
$(x_i)$, and clusters with sizes $x_i$ and $x_j$ merge at rate $x_ix_j$.
It turns out that this gives a Feller process on the space $l_2$.
There is a connection with the classical theory of random graphs.
Consider the random graph $\GG(n,n^{-1} + tn^{-4/3})$, whose
largest components have size of order $n^{2/3}$.
Normalizing by $n^{-2/3}$, there is an asymptotic joint distribution
$X(t) = (X_1(t),X_2(t),\ldots)$ of component sizes, and this must
evolve as the multiplicative coalescent, defined for all
$- \infty < t < \infty$.
Informally, we can start the multiplicative coalescent at time $- \infty$
with infinitesimally small clusters.
A remarkable fact is that the distribution of $X(t)$ is the same as the
joint distribution of excursion lengths of a Brownian-type process
$(B^t(s),0 \leq s < \infty)$
defined to be reflecting inhomogeneous Brownian motion with drift $t-s$ at
time $s$,
started with $B^t(0) = 0$.
aldous@stat.berkeley.edu
Compressed Postscript version of paper available via author's homepage
- To see some information provided by the author
click here.
638. MIXING TIMES FOR UNIFORMLY ERGODIC MARKOV CHAINS
David Aldous and Laszlo Lovasz and Peter Winkler
Consider the class of discrete time, general state space
Markov chains which satisfy a ``uniform ergodicity under sampling" condition.
There are many ways to quantify the notion of ``mixing time", that
is time to approach stationarity from a worst initial state.
We prove results asserting equivalence (up to universal
constants) of different quantifications of mixing time.
This work combines three areas of Markov theory which are rarely connected:
the potential-theoretical characterization of optimal stopping times,
the theory of stability and convergence to stationarity for general-state
chains, and the theory surrounding mixing times for finite-state chains.
aldous@stat.berkeley.edu
Compressed Postscript version of paper available via author's homepage
- To see some information provided by the author
click here.
639. A NOTE ON RATIONAL STRUCTURES AND THEIR ASYMPTOTICS
Mark M. Maxwell
There are many combinatorial structures which can be regarded as complexes of
certain basic blocks. Previous results on asymptotic enumeration have yeilded
implicit formulae for the mean and variance of the counts of blocks, together
with Gaussian approximations with these parameters. We will provide explicit
formulae for these basic quantities, together with large deviation estimates
for decay rates from the mean. As a result we obtain laws of large numbers
and, under certain convexity conditions, a new approach to Gaussian
approximations previously obtained by more cumbersome analysis. The results of
this paper are part of the author's Ph.D. thesis under the direction of Ed
Waymire at Oregon State University.
maxwell@math.orst.edu
640. HEAT KERNEL ESTIMATES AND HOMOGENIZATION FOR ASYMPTOTICALLY
LOWER DIMENSIONAL PROCESSES ON SOME NESTED FRACTALS
B.M. Hambly and T. Kumagai
We consider the class of diffusions on fractals first constructed
by Hattori, Hattori and Watanabe on the Sierpinski and $abc$ gaskets.
We give an alternative construction of the diffusion process using
Dirichlet forms and extend the class of fractals considered to some
nested fractals. We use the Dirichlet form to deduce Nash inequalities
which give upper bounds on the short and long time behaviour of the
transition density of the diffusion process. For short times, even
though the diffusion lives mainly on a lower dimensional subset of
the fractal, the heat flows slowly. For the long time scales the diffusion
has a homogenization property in that rescalings converge to the
Brownian motion on the fractal.
bmh@maths.ed.ac.uk
641. WEAK LUMPABILITY OF GENERAL FINITE MARKOV CHAINS AND POSITIVE
INVARIANCE OF CONES
James Ledoux
We consider weak lumpability of general finite homogeneous
Markov chains evolving in discrete time, that is when a lumped Markov
chain with respect to a partition of the initial state space is also a
homogeneous Markov chain. We show that weak lumpability is equivalent to
the existence of a decomposable polyhedral cone which is positively
invariant by the transition probability matrix of the original chain. It
allows us, in a unified way, to derive new results on lumpability of
reducible Markov chains and to obtain spectral properties associated
with lumpability.
James.Ledoux@univ-rennes1.fr
642. ON THE DISTRIBUTION OF BROWNIAN AREAS
Mihael Perman and Jon A. Wellner
We study the distributions of the areas under the positive
parts of a Brownian motion process $B$ and a Brownian bridge
process $U$: with $A^+ = \int_0^1 B^+ (t)dt$
and $A_0^+ = \int_0^1 U^+ (t)dt$ we use excursion theory to show
that the Laplace transforms $\Psi^+ (s) = E \exp (-s A^+)$
and $\Psi_0^+ (s) = E \exp (-s A_0^+)$ of $A^+$ and $A_0^+$
satisfy
$$
\int_0^{\infty} e^{ -\lambda s} \Psi^+ ( \sqrt{2} s^{3/2} ) \;ds
=
{{\lambda^{-1/2} Ai(\lambda) + \left (1/3 - \int_0^\lambda Ai(t) dt \right )}
\over{\sqrt{\lambda} Ai(\lambda) - Ai'(\lambda)}}$$
and
$$
\int_0^{\infty} {{e^{ -\lambda s}}\over {\sqrt{s}}} \Psi_0^+
(\sqrt{2} s^{3/2} ) \;ds
=
2 \sqrt{ \pi } {{ Ai(\lambda )}\over {\sqrt{\lambda} Ai(\lambda) - Ai'(\lambda)}}
$$
where $Ai$ is Airy's function. At the same time our approach via
excursion theory unifies previous calculations of this type due to
Kac, Groeneboom, Louchard, Shepp, and Tak\'{a}cs for other Brownian
areas. Similarly, we use excursion theory to obtain recursion formulas
for the moments of the ``positive part'' areas. We have not yet succeeded
in inverting the double Laplace transforms because of the structure of
the function appearing in the denominators, namely
${\sqrt{\lambda} Ai(\lambda) - Ai'(\lambda)}$.
mihael.perman@fmf.uni-lj.si
jaw@stat.washington.edu
PostScript file available from:
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click here.
643. AN EXCURSION APPROACH TO RAY-KNIGHT THEOREMS FOR PERTURBED BM
Mihael Perman
Perturbed Brownian motion in this paper is defined as
$X_t=\vert B_t\vert-\mu\ell_t$ where $B$ is standard
Brownian motion, $(\ell_t:t\ge 0)$ is its local time
at 0 and $\mu$ is a positive constant. Carmona, Petit
and Yor (1994) have extended the classical second Ray-
Knight theorem about the local time processes in the
space variable taken at an inverse local time to perturbed
Brownian motion with the resulting Bessel square processes
having dimensions depending on $\mu$. In this paper a proof
based on splitting the path of perturbed Brownian motion at
its minimum is presented. The derivation relies mostly on
excursion theory arguments.
mihael.perman@fmf.uni-lj.si
PostScript file available from:
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click here.
644. RANDOM RANDOM WALKS ON THE INTEGERS MOD $n$ WHEN $n$ IS NOT PRIME
Jack J. Dai and Martin V. Hildebrand
This paper considers typical random walks on the integers mod $n$
such that the random walk is supported on a constant $k$ values.
This paper extends a result of Hildebrand to show that for any
integer $n$, roughly $n^{2/(k-1)}$ steps usually suffice to get
the random walk close to uniformly distributed if the $k$ values
satisfy some conditions needed for the random walk to get close
to uniformly distributed.
JDAI@jcvaxa.jcu.edu, mvh@math.utexas.edu (TeX or hard-copy version available)
645. APPROXIMATION OF KILLED BROWNIAN LOCAL TIME
Frank B. Knight
Let $B(t)$ be standard Brownian motion killed at $T(-1)$, let
$R_n(k 2^{-2 n})=B(T_k)$; $T_o=O$,
$T_{k+1}= \inf \{t > T_k:|B(t)- B(T_k)|=2^{-n} \}$,
and let $N_n(k)=\#$ \{steps of $R_n$,
$k2^{-n} \to (k+1)2^{-n}\}$. Then each $N_n$
is a (Galton-Watson)Markov chain and $\lim\limits_{n\to \infty}$
$S_n(x)=L(x)$ uniformly, where $S_n(x)= 2^{-n}
N_n(k),$ $k 2^{-n}\leq x < (k+1)2^{-n}$, and
$L(x)$ is local time of $B, -1 \leq x$.
Theorem. $E(\int^\infty_0(E(L(x)/\sigma(N_n))-L(x))^2 dx|
\sigma(N_n)| = \frac {14}{15} 2^{- n}E (T(-1)|\sigma(N_n))=
\frac{14}{15} 2^{-3 n} M_n$, where
$M_n$:=total number of steps of $R_n$.
Corollary. The conditional mean squared error times
$2^n$ converges $P-a.s.$ and in $L^p$ to $\frac{14}{15}
T(-1)$ as $n \to \infty$ over sets of the form
$\{\max\limits_t B < K \}$.
mcdonald@math.uiuc.edu
