Probability Abstracts 46
This document contains abstracts 1034-1096.
They have been mailed on August 26, 1998.
1034. CLASSICAL MARKOV PROCESSES FROM QUANTUM LEVY PROCESSES
Uwe Franz
We show how classical Markov processes can be obtained from quantum
L\'evy processes. It is shown that quantum L\'evy processes are
quantum Markov processes, and sufficient conditions for
restrictions to subalgebras to remain quantum Markov processes are
given. A classical Markov process (which has the same time-ordered moments
as the quantum process in the vacuum state) exists whenever we can restrict
to a commutative subalgebra without loosing the quantum Markov property.
Several examples, including the Az\'ema martingale, are presented with
explicit calculations. In particular, the action of the generator of the
classical Markov processes on polynomials or their moments are
calculated using Hopf algebra duality.
franz@math.u-strasbg.fr
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information provided by the author
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1035. FUNCTIONAL CALCULUS FOR DIRICHLET FORMS
Kazuhiro Kuwae
Our first aim of this paper is to clarify the analytic
structure of the local space (i.e. localized space in the broad sense)
in the framework of semi-Dirichlet forms on a
(not necessarily locally compact) separable metric space.
The second purpose is to give
the chain rule for the functions in local space
which is described in terms of energy measure of continuous
part in the framework of symmetric quasi-regular Dirichlet forms.
We also propose the stochastic integrals for martingale additive functionals
locally of finite energy and the chain rule of
stochastic integrals for these functionals.
LaTeX file or hard copy is available.
kuwae@is.saga-u.ac.jp
1036. SOBOLEV SPACES, LAPLACIAN, AND HEAT KERNEL ON ALEXANDROV SPACES
Kazuhiro Kuwae, Yoshiroh Machigashira, Takashi Shioya
Consider a Gromov-Hausdorff limit point for the class of
n-dimensional closed Riemannaa manifolds with a uniform lower
bound of sectional curvature and a uniform upper bound of diameter.
The limits consist of Alexandrov space introduced by
A.D. Alexandrov (1951). This space is no lomger than topological
manifold. Otsu-Shioya (1994) discoverd the $C^1$-structure
for Alexandrov space by deleting the singular set.
The singular set may be dense in the Alexandrov sapce, so the
treatment is different from the case of Lipschitz Riemannan manifold.
This paper is the first study on the Sobolev space, the Laplacian,
and the heat equation on Alexandrov spaces. We propose a notion of
DC-structure (DC means "difference of concave function)
on Alexandrov space.
The underlying measure is the Haussdorf measure.
The Dirichlet form is strongly local, regular and
has a special standard core which consists of
DC-functions with compact support. Further we show that the singular set
out of the boundary is 1-capacity 0.
We prove the compactness of the imbedding of the Sobolev space
$W_0^{1,2}(\Omega)$ into $L^2(\Omega)$ for any relatively compact
open subset $\Omega$ of an Alexandrov space. As a corollary, the
generator of the canonial Dirichlet form on $W_0^{1,2}(\Omega)$ has
discrete spectrum. The generator can be approximated by the Laplacian
induced from the DC-structure. We also prove the existence of the locally
H\"older continuous Dirichlet heat kernel.
AMS-LaTeX file or hard copy is available.
kuwae@is.saga-u.ac.jp
shioya@math.kyushu-u.ac.jp
machi@cc.osaka-kyoiku.ac.jp
1037. REFLECTED DIRICHLET FORMS AND UNIQUENESS OF SILVERSTEIN'S
EXTENSIONS
Kazuhiro Kuwae
Reflected Dirichlet space for quasi-regular
Dirichlet forms is presented in this paper. We show the closedness
of the (active) reflected Dirichlet forms without using
the first definition of reflected Dirichlet space by
Silverstein(1974)
and the characterization by Chen (1992). We also
show the maximality of (active) reflected Dirichlet space
in the class of Silverstein's extensions
and consider the uniqueness problem which extend the
recent results of Kawabata-Takeda (1996).
Only the techniques of the transfer method and the change of underlying
measures are used.
LaTeX file or hard copy is available.
kuwae@is.saga-u.ac.jp
1038. LARGE DEVIATIONS FOR SMALL NOISE DIFFUSIONS WITH
DISCONTINUOUS STATISTICS
Michelle Boue, Paul Dupuis, and Richard S. Ellis
This paper proves the large deviation principle for a class of
non-degenerate small noise diffusions with discontinuous drift
and with state-dependent diffusion matrix. The proof is based
on a variational representation for functionals of strong
solutions of stochastic differential equations and on weak
convergence methods.
boue@math.umass.edu dupuis@cfm.brown.edu rsellis@math.umass.edu
1039. USING TWO-PARAMETER APPROXIMATIONS TO PROVE
LARGE DEVIATION PRINCIPLES
Christopher Boucher and Richard S. Ellis
A new method is introduced for proving process-level
large deviation results. Based on two-parameter
approximations, this method is motivated by applications
to the equilibrium statistical mechanics of two-dimensional
fluid turbulence. The method is explained in the context
of a process-level generalization of Cramer's Theorem
and is then applied to a more complicated process
involving weighted averages of i.i.d. random variables.
boucher@math.umass.edu rsellis@math.umass.edu
1040. DERIVATION OF MAXIMUM ENTROPY PRINCIPLES
IN TWO-DIMENSIONAL TURBULENCE VIA LARGE DEVIATIONS
Christopher Boucher, Richard S. Ellis, and Bruce Turkington
The continuum limit of lattice models arising in two-dimensional
turbulence is analyzed by means of the theory of large deviations.
In particular, the Miller-Robert continuum model of equilibrium
states in an ideal fluid and a modification of that model due to
Turkington are examined in a unified framework, and the maximum
entropy principles that govern these models are rigorously derived
by a new method. In this method, a doubly indexed, measure-valued
random process is introduced to represent the coarse-grained vorticity
field. The natural large deviation principle for this process is
established and is then used to derive the equilibrium conditions
satisfied by the most probable macrostates in the continuum models.
The physical implications of these results are discussed, and some
modeling issues of importance to the theory of long-lived, large-scale
coherent vortices in turbulent flows are clarified.
boucher@math.umass.edu rsellis@math.umass.edu turk@math.umass.edu
1041. CENTRAL LIMIT THEOREM FOR THE HIERARCHY OF FREENESS
Uwe Franz and Romuald Lenczewski
The central limit theorem and the invariance principle
for the hierarchy of freeness are studied. We show that for given
$m\in N$ the limit law can be expressed in terms of
non-crossing partitions of depth smaller or equal to $m$.
For ${\cal A}=C[x]$, we solve the associated
moment problems and find explicitly
the discrete limit measures $\mu^{(m)}$
which approximate weakly the Wigner measure.
franz@math.u-strasbg.fr, lenczew@im.pwr.wroc.pl
1042. THE GNS CONSTRUCTION FOR THE HIERARCHY OF FREENESS
Uwe Franz, Romuald Lenczewski, and Michael Sch\"urmann
The GNS representation of the hierarchy of freeness is constructed.
It is shown that the $m$-free product corresponds to the
truncation of order $m$ of the free product representation.
This is done by means of introducing suitable intertwining
operators between the underlying pre-Hilbert spaces.
This approach leads to the representation of free random
variables as strongly convergent series.
franz@math.u-strasbg.fr, schurman@math.u-strasbg.fr, lenczew@im.pwr.wroc.pl
1043. OPERATOR SEMI-SELFDECOMPOSABILITY, (C,Q)-DECOMPOSABILITY
AND RELATED NESTED CLASSES
Makoto Maejima, Ken-iti Sato and Toshiro Watanabe
There are two types of generalizations of selfdecomposability of
probability measures on $\bold R^d$, $d\ge1$: the $c$-decomposability and
the $C$-decomposability of Lo\`eve and Bunge on the one hand, and the
semi-selfdecomposability of Maejima-Naito on the other. The latter
implies infinite divisibility but the former does not in general.
For $d\ge2$ introduction of operator (matrix) normalizations yields
four kinds of classes of distributions on $\bold R^d$: $L_0(b,Q)$,
$\widetilde L_0(b,Q)$, $L_0(C,Q)$, and $\widetilde L_0(C,Q)$,
where $0<b<1$, $Q$ is a $d\times d$ matrix with eigenvalues having
positive real parts, and $C$ is a closed multiplicative subsemigroup
of $[0,1]$ containing 0 and 1. Further, each of these classes
generates the Urbanik-Sato type decreasing sequence of its
subclasses. Characterizations and relations of these classes and
subclasses are established. They complement and generalize results of
Bunge, Jurek, Maejima-Naito, and Sato-Yamazato.
maejima@math.keio.ac.jp
1044. COMPLETELY OPERATOR SEMI-SELFDECOMPOSABLE DISTRIBUTIONS
Makoto Maejima, Ken-iti Sato and Toshiro Watanabe
The class $L_{\infty}(b,Q)$ of completely operator semi-selfdecomposable
distributions on $\bold R^d$ for $b$ and $Q$ is studied.
Here $0<b<1$ and $Q$ is a $d\times d$ matrix whose eigenvalues have
positive real parts.
This is the limiting class of the decreasing sequence of classes
$L_m(b,Q)$, $m=-1,0,1,\dots$, where $L_{-1}(b,Q)$ is the class of all
infinitely divisible distributions on $\bold R^d$ and $L_m(b,Q)$ is
defined inductively as the class of distributions $\mu$ with
characteristic function $\widehat\mu(z)$ satisfying
$\widehat\mu(z)=\widehat\mu(b^{Q'}z)\widehat\rho(z)$
for some $\rho\in L_{m-1}(b,Q)$. $Q'$ is the transpose of $Q$.
Distributions in $L_{\infty}(b,Q)$ are characterized in terms of
Gaussian covariance matrices and L\'evy measures.
The connection with the class $OSS(b,Q)$ of operator semi-stable
distributions on $\bold R^d$ for $b$ and $Q$ is established.
maejima@math.keio.ac.jp
1045. A PHASE TRANSITION IN RANDOM COIN TOSSING
David Levin, Robin Pemantle and Yuval Peres
Suppose that a coin with bias $\theta$ is tossed at renewal times of a
renewal process, and a fair coin is tossed at all other times. Let
$\mu_\theta$ be the distribution of the observed sequence of coin tosses,
and let $u_n$ denote the chance of a renewal at time $n$. Harris and
Keane~(1997) showed that if ${u_n}$ are not square-summable, then
$\mu_\theta$ and $\mu_0$ are singular, while if ${u_n}$ are square-summable
and the bias $\theta$ is small enough, then $\mu_\theta$ is absolutely
continuous with respect to $\mu_0$. They conjectured that absolute
continuity should not depend on $\theta$, but only on the square-summability
of ${u_n}$. We show that in fact, the power law governing the decay of
${u_n}$ is crucial, and for some renewal sequences, there is a phase
transition at a critical parameter $\theta_c \in (0,1)$:
for $ |\theta|<\theta_c$ the measures $\mu_\theta$ and $\mu_0$ are mutually
absolutely continuous, but for $|\theta|>\theta_c$, they are singular.
Moreover, for these sequences, the unknown bias can be recovered from
the observations if it is large enough. When ${u_n}$ are not
square-summable, the unknown bias can always be recovered.
levin@stat.berkeley.edu pemantle@math.wisc.edu peres@stat.berkeley.edu
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information provided by the author
click here.
- Or
here.
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here.
1046. MARKOV CHAINS IN A FIELD OF TRAPS
Robin Pemantle and Stas Volkov
We consider a Markov chain on a countable state space, on which is
placed a random field of traps, and ask whether the chain gets trapped
almost surely. We show that the quenched problem (when the traps are
fixed) is equivalent to the annealed problem (when the traps are updated
each unit of time) and give a criterion for almost sure trapping versus
positive probability of non-trapping. The hypotheses on the Markov chain
are minimal, and in particular, our results encompass the results of
den Hollander, Menshikov and Volkov (1995).
pemantle@math.wisc.edu , svolkov@ssc.wisc.edu
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information provided by the author
click here.
1047. VERTEX-REINFORCED RANDOM WALK ON Z HAS FINITE RANGE
Robin Pemantle and Stas Volkov
A stochastic process called Vertex-Reinforced Random Walk (VRRW) is
defined in Pemantle (1988a). We consider this process in the case
where the underlying graph is an infinite chain (i.e., the one-dimensional
integer lattice). We show that the range is almost surely finite,
that at least 5 points are visited infinitely often almost surely,
and that with positive probability the range contains exactly 5 points.
There are always points visited infinitely often but at a set of times
of zero density, and we show that the number of visits to such a point
to time $n$ may be asymptotically $n^\alpha$ for a dense set of
values $\alpha \in (0,1)$. The power law analysis relies on analysis
of a related urn model.
pemantle@math.wisc.edu , svolkov@ssc.wisc.edu
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information provided by the author
click here.
1048. ABSENCE OF MUTUAL UNBOUNDED GROWTH FOR ALMOST ALL PARAMETER VALUES
IN THE TWO-TYPE RICHARDSON MODEL
Olle H\"aggstr\"om and Robin Pemantle
We study the two-type Richardson model on ${\bf Z}^d$, $d\geq 2$,
in the asymmetric case where
the two particle types have different infection rates. Starting with a
single particle of each type, and fixing the infection rate for one
of the types, we show that mutual unbounded growth has probability $0$ for
all but at most countably many values of the other type's infection rate.
olleh@math.chalmers.se , pemantle@math.wisc.edu
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information provided by the author
click here.
1049. MANIFOLD-VALUED MARTINGALES, CHANGES OF PROBABILITIES,
AND SMOOTHNESS OF FINELY HARMONIC MAPS
Marc Arnaudon, Xue-Mei Li and Anton Thalmaier
This paper is concerned with regularity results for starting points
of continuous manifold-valued martingales with fixed terminal value
under a possibly singular change of probability.
In particular, if the martingales live in a small neighbourhood
of a point and if the stochastic logarithm $M$ of the change
of probability varies in some Hardy space $H_r$ for sufficiently
large $r<2$, then the starting point is differentiable at $M=0$.
As an application, our results imply that continuous finely harmonic
maps between manifolds are smooth, and the differentials have
stochastic representations not involving derivatives.
This gives a probabilistic alternative to the coupling technique
used by Kendall (1994) to prove smoothness of finely harmonic maps.
arnaudon@math.u-strasbg.fr, xmli@math.uconn.edu,
anton.thalmaier@mathematik.uni-regensburg.de (postscript file available)
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information provided by the author
click here.
- Or
here.
1050. ON A CONJECTURE OF D.G.KENDALL CONCERNING THE PLANAR CROFTON CELL
AND ON ITS BROWNIAN COUNTERPART
Andre Goldman
Consider a homogeneous Poisson line process in the plane and the associated
random convex tesselation. We obtain a precise estimation of the distribution
of the "large aera" of the Crofton cell (the random polygon containing the
origin). This result implies that, expressed in term of eigenvalues, the
large Crofton cell are nearly circular (which sustain a conjecture stated
by D.G.Kendall in the forties). We deduce also from this the precice
asymptotic behavior of the Laplace transform of the perimeter of the convex
hull of Brownian motion run till time 1.This last result implies that the
small convex hulls of Brownian motion are nearly circular.
goldman@jonas.univ-lyon1.fr
1051. NON-SYMMETRIC APPROXIMATIONS FOR MANIFOLD-VALUED SEMIMARTINGALES
Serge Cohen and Anne Estrade
We study general approximations of continuous semimartingales in a
manifold. Classically the limits of integrals with respect to the
approximated semimartingales yield Stratonovich integrals. Nevertheless
several authors have remarked that a skew-symmetric extra-term may
appear for specific approximations when the manifold is a vector space.
We give the geometric meaning of the
skew-symmetric term and an interpretation in term of a
``second order non-symmetric intrinsic calculus''. This stochastic
non-symmetric calculus is further extended to stochastic differential equations
between manifolds. A particular emphasis
is pointed on the role of interpolators in approximations.
cohen@math.uvsq.fr estrade@labomath.univ-orleans.fr
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information provided by the author
click here.
1052. THE THEORY OF LARGE DEVIATIONS: FROM BOLTZMANN'S 1877
CALCULATION TO EQUILIBRIUM MACROSTATES IN 2D TURBULENCE
Richard S. Ellis
After presenting some basic ideas in the theory of large
deviations, this paper applies the theory to a number of problems
in statistical mechanics. These include deriving the form
of the Gibbs state for a random ideal gas; describing
probabilistically the phase transition in the Curie-Weiss
model of a ferromagnet; and deriving variational formulas
that describe the equilibrium macrostates in models
of two-dimensional turbulence. A general approach
to the large deviation analysis of models in statistical
mechanics is also formulated.
rsellis@math.umass.edu
1053. CONSTRUCTION OF DIFFUSIONS ON CONFIGURATION SPACES
Zhi--Ming Ma and Michael R\"ockner
We show that any square field operator on a measurable state space $E$ can
be lifted by a natural procedure to a square field operator on the
corresponding (multiple) configuration space $\bar\Gamma_E$. We then
show the closability of the associated lifted
(pre--)Dirichlet forms $\Emug$ on $L^2(\bar\Gamma_E;\mu)$ for a large class of
measures $\mu$ on $\barge$ (without assuming an integration by parts formula)
generalizing all corresponding results known so far. Subsequently,
we prove that under mild conditions the Dirichlet forms
$\Emug$ are quasi--regular, and that hence there
exist associated diffusions on $\bar\Gamma_E$, provided $E$ is a complete
separable metric space and $\bar\Gamma_E$ is equipped with a suitable topology,
which is the vague topology if $E$ is locally compact. We discuss
applications to the case where $E$ is a finite dimensional manifold yielding
an existence result on diffusions on $\bar\Gamma_E$ which was already
announced in \cite{AKR96a, AKR96b},
resp.~used in \cite{AKR98, AKR97b}. Furthermore, we present applications in
particular detail to the case where $E$ is the free loop space.
roeckner@mathematik.uni-bielefeld.de
1054. ROBUST PHASE TRANSITIONS FOR SPHERICAL AND OTHER MODELS ON
GENERAL TREES
Robin Pemantle and Jeffrey Steif
We study several statistical mechanical models on a general tree.
Particular attention is devoted to the spherical models, where the
state space is the $d$--dimensional unit sphere and the interactions
are proportional to the cosines of the angles between neighboring
spins. The phenomenon of interest here is the classification of
phase transition (non-uniqueness of the Gibbs state) according to
whether it is {\it robust}. In many cases, including all of the
spherical and Potts models, occurrence of robust phase transition
is determined by the geometry (branching number) of the tree in a
way that parallels the situation with independent percolation and
usual phase transition for the Ising model. The critical values for
robust phase transition for the spherical and Potts models are also
calculated exactly. In some cases, such as the $q\ge 3$ Potts model,
robust phase transition and usual phase transition do not coincide,
while in other cases, such as the spherical models, we conjecture that
robust phase transition and usual phase transition are equivalent.
In addition, we show that symmetry breaking is equivalent to the
existence of a phase transition, a fact believed but not known for
the rotor model on $Z^2$.
pemantle@math.wisc.edu steif@math.chalmers.se
1055. SCATTERING PROBLEM FOR LOCAL PERTURBATIONS OF THE FREE QUANTUM GAS
Yu.~G.~Kondratiev, A.~Yu.~Konstantinov, M.~R\"ockner and G.~V.~Shchepan'uk
Scattering theory for perturbations of the intrinsic Dirichlet
(Laplace-Beltrami) operator $H_0=-\div^\Ga\na^\Ga$ on $L_2(\Ga,\pi_z)$,
{\it ie} the space of $\pi_z$-square integrable functions on the
configuration space $\Ga$ over $\BbR^d$, is studied. Here $\pi_z$
denotes Poisson measure with intensity $z$. We show that for an
arbitrary regular non-zero potential $V$ the standard wave operators
$W^{\pm}(H_0,H_0+V)$ do not exist, and propose to consider Dirichlet
operators of perturbed Poisson measures instead of potential
perturbations of the Hamiltonian $H_0$. As case studies, cylindric
smooth densities and finite volume Gibbs perturbations of the Poisson
measure are considered. In these cases the existence of the corresponding
wave operators is proved.
kondratiev@wiener.iam.uni-bonn.de
1056. UNIQUENESS FOR GIBBS STATES OF QUANTUM LATTICES IN SMALL MASS REGIME
Sergio Albeverio, Yuri Kondratiev, Yuri Kozitsky and Michael R\"ockner
The model of interacting quantum particles of mass $m$
performing anharmonic one--dimensional oscillations
around their unstable equilibrium positions, which form the $d$--
dimensional simple cubic lattice $\Z^d$, is considered. For this
model, it is proved that for every fixed value of the temperature
$\b^{-1}$, there exists a positive $m_{*}(\b )$ such that if
$m \in (0, m_{*}(\b))$, then the set of tempered Gibbs
states consists of exactly one element.
kondratiev@wiener.iam.uni-bonn.de
1057. THICK POINTS FOR SPATIAL BROWNIAN MOTION: MULTIFRACTAL ANALYSIS OF
OCCUPATION MEASURE
Amir Dembo, Yuval Peres, Jay Rosen and Ofer Zeitouni
Let $T(x,r)$ denote the total occupation measure of the ball of radius
$r$ centered at $x$ for Brownian motion in $R^3$. We prove that the
supremum over the unit ball in $R^3$ of $T(x,r)/(r^2|\log r|)$, tends
to $16/\pi^2$ a.s. as $r$ tends to $0$, thus solving a problem posed by
Taylor in 1974. Furthermore, for any $a$ in $(0,16/\pi^2)$, the Hausdorff
dimension of the set of ``thick points'' $x$ for which
$\limsup_{r \rightarrow 0} T(x,r)/(r^2|\log r|)=a$, is almost surely
$2-a\pi^2/8$.
This is the correct scaling to obtain a nondegenerate ``multifractal spectrum''
for Brownian occupation measure. Analogous results hold for Brownian motion
in any dimension $d>3$. These results are related to the LIL of Ciesielski and
Taylor (1962) for the Brownian occupation measure of small balls, in the same
way that L\'{e}vy's uniform modulus of continuity, and the formula of Orey and
Taylor (1974) for the dimension of ``fast points'', are related to the usual
LIL. We also show that the $\liminf$ scaling of $T(x,r)$ is quite different:
we exhibit non-random $c_1,c_2>0$, such that
$c_1 < \sup_x \liminf_{r \rightarrow 0} T(x,r)/r^2 < c_2 $, a.s.
In the course of our work we provide a general framework for obtaining lower
bounds on the Hausdorff dimension of random fractals of `limsup type'.
jrosen3@idt.net peres@stat.berkeley.edu amir@playfair.stanford.edu
zeitouni@ee.technion.ac.il (ps file available)
1058. CROSSING ESTIMATES AND CONVERGENCE OF DIRICHLET FUNCTIONS
ALONG RANDOM WALK AND DIFFUSION PATHS
Alano Ancona, Russell Lyons and Yuval Peres
Let $\{X_n\}$ be a transient reversible Markov chain and let $f$ be a
function on the state space with finite Dirichlet energy. We prove crossing
inequalities for the process $\{f(X_n)\}_{n \ge 1}$ and show that it
converges almost surely and in $L^2$. Analogous results are also
established for reversible diffusions on Riemannian manifolds.
ancona@matups.math.u-psud.fr rdlyons@indiana.edu peres@stat.berkeley.edu
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information provided by the author
click here.
- Or
here.
1059. HOW TO MAKE A HILL PLOT
Holger Drees, Laurens de Haan and Sidney Resnick
An abundance of high quality data sets requiring heavy tailed models
necessitates reliable methods of estimating the shape parameter
governing the degree of tail heaviness. The Hill estimator is a
popular method for doing this but its practical use is encumbered by
several difficulties. We show that an alternative method of plotting
Hill estimator values is more revealing than the standard method
unless the underlying data comes from a Pareto distribution.
hdrees@mi.uni-koeln.de ldhaan@few.eur.nl sid@orie.cornell.edu
1060. ON EXCURSION SETS, TUBE FORMULAE, AND MAXIMA OF RANDOM FIELDS
Robert J. Adler
This is rather rambling review of what, with a few notable exceptions,
has been a rather dormant area for over a decade. It concentrates on the
septuagenarian problem of finding good approximations for the excursion
probability
$P\{\sup_{t\in T} X_t \geq \lambda\}$
where $\lambda$ is large, $X$ is a Gaussian, or ``Gaussian-like'', process
over a region $T\subset \Re^N$, and, generally, $N>1$.
A quarter of a century ago there was a flurry of papers out of various schools
linking this problem to the geometrical properties of random field sample
paths. My own papers made the link via Euler characteristics of the excursion
sets $\{t\in T:X_t\geq\lambda\}$. A decade ago, Aldous popularised the Poisson
clumping heuristic for computing excursion probabilties in a wide variety of
scenarios, including the Gaussian. Over the past few years, Keith Worsley has
been the driving force behind the computation of many new Euler characteristic
functionals, primarily driven by applications in medical imaging. There has
also been a parallel development of techniques in the astrophysical
literature. Meanwhile, somewhat closer to home, Hotelling's 1939 ``tube
formulae" have seen a renaissance as sophisticated statistical hypothesis
testing problems led to their reapplication towards computing excursion
probabilities.
The aim of the present paper is to look again at many of these results, and
tie them together in new ways to obtain a few new results and, hopefully,
considerable new insight. The ``Punchline of this paper", which relies
heavily on a recent result of Piterbarg,is given in Section 6.6:
"In computing excursion probabilities for smooth enough
Gaussian random fields over reasonable enough regions, the expected Euler
characteristic of the corresponding excursion sets gives an approximation,
for large levels, that is accurate to as many terms as there are in its
expansion."
robert@ieadler.technion.ac.il, robert@adler.stat.unc.edu,
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information provided by the author
click here.
1061. ON CHANGES OF MEASURE FOR SUPER BROWNIAN MOTION
Robert J. Adler and Srikanth K. Iyer
Let X be a superprocess under the measure P. We show the existence of
probability measures which are absolutely continuous with respect to P,
and whose Radon-Nikodym derivatives are suitably normalized functions of
the self intersection local time of X. These measures correspond to
measure valued processes exhibiting a certain amount of (reinforcing) self
interaction in terms of both the particle motion and the branching
mechanism.
skiyer@iitk.ernet.in,robert@ieadler.technion.ac.il, robert@adler.stat.unc.edu
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information provided by the author
click here.
1062. ANALYSING STABLE TIME SERIES
Robert J. Adler, Raisa E. Feldman, and Colin Gallagher
We describe how to take a stable, ARMA, time series through the various
stages of model identification, parameter estimation, and diagnostic
checking, and accompany the discussion with a goodly number of large scale
simulations that show which methods do and do not work, and where
some of the pitfalls and problems associated with stable time series
modelling lie.
epstein@pstat.ucsb.edu, robert@ieadler.technion.ac.il,
robert@adler.stat.unc.edu
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information provided by the author
click here.
1063. IBM, SIBM and IBS
John Verzani and Robert J. Adler
We construct a super iterated Brownian motion (SIBM) from a historical
version of an iterated Brownian motion (IBM) using an iterated Brownian
snake (IBS).
It is shown that the range of super iterated Brownian motion is qualitatively
quite different from that of super Brownian motion in that there are points
with explosions in the branching. However, at a fixed time the support of
super iterated Brownian motion has an exact Hausdorff measure function
that is the same (up to a constant) as that of super Brownian motion at a
fixed time.
verzani@postbox.csi.cuny.edu, robert@ieadler.technion.ac.il,
robert@adler.stat.unc.edu,
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information provided by the author
click here.
1064. NONLINEAR MODELS FOR TIME SERIES USING MIXTURES OF AUTOREGRESSIVE MODELS
Assaf J. Zeevi, Ronny Meir, and Robert J. Adler
We consider a novel non-linear model for time series analysis. The study
of this model emphasizes both theoretical aspects as well as practical
applicability. The architecture of the model is demonstrated to be
sufficiently rich, in the sense of approximating unknown functional
forms, yet it retains some of the simple and intuitive characteristics
of linear models. A comparison to some more established non-linear models
will be emphasized, and theoretical issues are backed by prediction results
for benchmark time series, as well as computer generated data sets.
Efficient estimation algorithms are seen to be applicable, made possible
by the mixture based structure of the model. Large sample properties of
We consider a novel class of non-linear models for time series analysis.
The proposed model uses mixtures of local autoregressive models;
we term this class the MixAR. The MixAR model is constructed so that locally
(in the state space), the process follows a linear autoregressive structure.
In addition, there is a state dependent probability distribution defined over
the different AR models.
This structure is realized via Markov process with transition kernel that
is a mixture of Gaussians with non-constant, state dependent, class
probabilities. This in turn endows the model with many attractive features,
that will be illustrated in this study.
Several theoretical aspects are investigated; sufficient conditions for
stochastic stability, parameter estimation algorithms, as well as
the universality of the model in approximating underlying prediction
functions. A comparison to some more established non-linear models
(including neural networks), and numerical prediction results for
benchmark time series, complement the theoretical results.
azeevi@isl.stanford.edu, rmeir@dumbo.technion.ac.il
robert@ieadler.technion.ac.il, robert@adler.stat.unc.edu,
1065. CONSTRUCTING AND COUNTING PHYLOGENETIC INVARIANTS
Steven N. Evans and Xiaowen Zhou
The method of invariants is an approach to the problem of reconstructing
the phylogenetic tree of a collection of $m$
taxa using nucleotide sequence
data. Models for the respective probabilities of the
$4^m$ possible vectors of bases at a given site will have
unknown parameters that describe the random walk by which substitution
occurs along the branches of a putative phylogenetic tree.
An invariant is a polynomial in these probabilities that, for a given
phylogeny, is zero for all choices of the substitution mechanism
parameters. If the invariant is typically non--zero for another
phylogenetic tree, then estimates of the invariant can be used
as evidence to support one phylogeny over another.
Previous work of
Evans and Speed showed that, for certain commonly used substitution models,
the problem of finding a minimal generating
set for the ideal of invariants can be reduced to the linear algebra
problem of finding a basis for a certain lattice
(that is, a free $\bZ$-module).
They also conjectured that the cardinality of such a generating set
can be computed using a simple ``degrees of freedom'' formula.
We verify this conjecture. Along the way, we explain in detail how
the observations of Evans and Speed lead to a simple, computationally
feasible algorithm for constructing a minimal generating set.
evans@stat.berkeley.edu
- To see a preprint or other
information provided by the author
click here.
1066. KINGMAN'S COALESCENT AS A RANDOM METRIC SPACE
Steven N. Evans
Kingman's coalescent is a Markov process that originally
arose in population genetics. It has state--space
the collection of partitions of the positive integers. Its initial state
is the trivial partition of singletons and it evolves
by successive pairwise mergers of blocks. The coalescent
induces a metric on the positive integers: the distance
between two integers is the time until they both belong
to the same block. We investigate the completion of this
(random) metric space. We show that almost surely it is
a compact metric space with Hausdorff and packing dimension
both $1$, and it has positive capacities in precisely the same
gauges as the unit interval.
evans@stat.berkeley.edu
- To see a preprint or other
information provided by the author
click here.
1067. A STRONG LAW FOR THE LONGEST EDGE OF THE MINIMAL SPANNING TREE
Mathew D. Penrose
Suppose $X1, X2, X3, ...$ are independent random
points in 2 or more dimensions, with common density $f$,
having connected compact support $A$ with smooth boundary
$dU$. Assume the restriction of $f$ to $A$ is continuous.
Let $M_n$ denote the smallest $r$ such that the union
of balls of diameter $r$ centred at the
the first $n$ points is connected.
We derive an almost sure limit for $(n/ \log n) M_n^d$.
Mathew.Penrose@durham.ac.uk
1068. CENTRAL LIMIT THEOREMS FOR K-NEAREST NEIGHBOUR DISTANCES
Mathew D. Penrose
Let $X1, X2, X3, ... $ be indepedent $d$-dimensional variables
with common density function $f$. Let $R_{i,k,n}$ be the distance
from $Xi$ to its $k$-th nearest neighbour in $\{X1, ... ,Xn\}$.
Suppose $(k_n)$ is a sequence with $1 << k_n << n^{2/(2+d)}$ as
$n$ tends to infinity (or $ 1 << k_n << n^{2/3}$ for a uniform
distribution).
Subject to conditions on $f$, we find a central limit theorem
(in the large-$n$ limit) for a time-change of the counting process with jumps
at the points $f(Xi) R_{i,k_n,n}^d$, $1 \leq i \leq n$.
Mathew.Penrose@durham.ac.uk
1069. POISSON LIMITS FOR PAIRWISE AND AREA INTERACTION POINT PROCESSES
S. Rao Jammalamadaka and Mathew D. Penrose
Suppose $n$ particles $x_i$ in a region of the plane
(possibly representing biological individuals - trees, organisms)
have a joint density proportional to
$\exp \{ -\sum_{i<j} \phi(n(x_i - x_j))\}$,
with $\phi$ a specified function of compact support.
We obtain a Poisson process limit for the collection of
rescaled interparticle distances as $n$ becomes large.
We give corresponding results for the case of several types
of particles, representing different species, and also for
the area-interaction (Widom-Rowlinson) point process of interpenetrating
spheres.
Mathew.Penrose@durham.ac.uk
1070. CONCRETE REPRESENTATION OF MARTINGALES
Stephen J. Montgomery-Smith
Let (f_n) be a mean zero vector valued martingale sequence. Then there exist
vector valued functions (d_n) from [0,1]^n such that int_0^1 d_n(x_1,...,x_n)
dx_n = 0 for almost all x_1,...,x_{n-1}, and such that the law of (f_n) is the
same as the law of (sum_{k=1}^n d_k(x_1,...,x_k)) . Similar results for tangent
sequences and sequences satisfying condition (C.I.) are presented. We also
present a weaker version of a result of McConnell that provides a Skorohod like
representation for vector valued martingales. This paper may be found at
http://math.missouri.edu/~stephen/preprints
stephen@cauchy.math.missouri.edu
- To see a preprint or other
information provided by the author
click here.
1071. INSURANCE POLICY VALUE AND PARETO-OPTIMAL RETENTION IN THE HYPOTHESIS OF
RARE LOSS EVENTS
Renato Ghisellini
In the hypothesis of rare loss events, the general expression of the policy
value has been determined as a functional of the "expected frequency / loss
severity" function and of the retention function. Exponential disutility has
been chosen after mathematical characterization of some of its economical
aspects, where functional properties of quasiarithmetic averages have been
used. By means of variational techniques, in the case of a risk neutral Insurer
the Pareto-optimal retention function has been finally determined.
sistemi.srl@galactica.it
- To see a preprint or other
information provided by the author
click here.
1072. ON THE NUMBER OF INCIPIENT SPANNING CLUSTERS
Michael Aizenman
In critical percolation models, in a large cube there will typically be more
than one cluster of comparable diameter. In 2D, the probability of $k>>1$
spanning clusters is of the order $e^{-\alpha k^{2}}$. In dimensions d>6,
when $\eta = 0$ the spanning clusters proliferate: for $L\to \infty$ the
spanning probability tends to one, and there typically are $ \approx L^{d-6}$
spanning clusters of size comparable to $|\C_{max}| \approx L^4$. The rigorous
results confirm a generally accepted picture for d>6, but also correct some
misconceptions concerning the uniqueness of the dominant cluster. We
distinguish between two related concepts: the Incipient Infinite Cluster, which
is unique partly due to its construction, and the Incipient Spanning Clusters,
which are not. The scaling limits of the ISC show interesting differences
between low (d=2) and high dimensions. In the latter case (d>6 ?) we find
indication that the double limit: infinite volume and zero lattice spacing,
when properly defined would exhibit both percolation at the critical state and
infinitely many infinite clusters.
aizenman@Princeton.EDU (Michael Aizenman)
- To see a preprint or other
information provided by the author
click here.
1073. THE CONTINUOUS SPIN RANDOM FIELD MODEL: FERROMAGNETIC ORDERING IN D>=3
Christof Kuelske
We investigate the Gibbs-measures of ferromagnetically coupled continuous
spins in double-well potentials subjected to a random field (our specific
example being the $\phi^4$ theory), showing ferromagnetic ordering in $d\geq 3$
dimensions for weak disorder and large energy barriers. We map the random
continuous spin distributions to distributions for an Ising-spin system by
means of a single-site coarse-graining method described by local transition
kernels. We derive a contour- representation for them with notably positive
contour activities and prove their Gibbsianness. This representation is shown
to allow for application of the discrete-spin renormalization group developed
by Bricmont/Kupiainen implying the result in $d\geq 3$.
kuelske@wias-berlin.de
- To see a preprint or other
information provided by the author
click here.
1074. FACTOR MAPS BETWEEN TILING DYNAMICAL SYSTEMS
Karl Petersen
We show that there is no Curtis-Hedlund-Lyndon Theorem for factor maps
between tiling dynamical systems: there are codes between such systems which
cannot be achieved by working within a finite window. By considering
1-dimensional tiling systems, which are the same as flows under functions on
subshifts with finite alphabets of symbols, we construct a `simple' code which
is not `local', a local code which is not simple, and a continuous code which
is neither local nor simple.
petersen@math.unc.edu
- To see a preprint or other
information provided by the author
click here.
1075. UNITARY BROWNIAN MOTIONS ARE LINEARIZABLE
Boris Tsirelson
Brownian motions in the infinite-dimensional group of all unitary operators
are studied under strong continuity assumption rather than norm continuity.
Every such motion can be described in terms of a countable collection of
independent one-dimensional Brownian motions. The proof involves continuous
tensor products and continuous quantum measurements. A by-product: a Brownian
motion in a separable F-space (not locally convex) is a Gaussian process.
tsirel@math.tau.ac.il
- To see a preprint or other
information provided by the author
click here.
1076. SOME PROPERTIES OF THE RANGE OF SUPER-BROWNIAN MOTION
Jean-Fran\c{c}ois Delmas
We consider a super-Brownian motion $X$. Its canonical measures can be
studied through the path-valued process called the Brownian snake. We obtain
the limiting behavior of the volume of the $\epsilon$-neighborhood for the
range of the Brownian snake, and as a consequence we derive the analogous
result for the range of super-Brownian motion and for the support of the
integrated super-Brownian excursion. Then we prove the support of $X_t$ is
capacity-equivalent to $[0,1]^2$ in $\R^d$, $d\geq 3$, and the range of $X$, as
well as the support of the integrated super-Brownian excursion are
capacity-equivalent to $[0,1]^4$ in $\R^d$, $d\geq 5$.
delmas@msri.org
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information provided by the author
click here.
1077. CHARACTERIZATION OF G-REGULARITY FOR SUPER-BROWNIAN MOTION AND
CONSEQUENCES FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS
Jean-Fran\c{c}ois Delmas and Jean-St\'ephane Dhersin
We give a characterization of G-regularity for super-Brownian motion and the
Brownian snake. More precisely, we define a capacity on $E=(0,\infty)\times
\R^d$, which is not invariant by translation. We then prove that the hitting
probability of a Borel set $A\subset E$ for the graph of the Brownian snake
starting at $(0,0)$ is comparable, up to multiplicative constants, to its
capacity. This implies that super-Brownian motion started at time 0 at the
Dirac mass $\delta_0$ hits immediately $A$ (that is $(0,0)$ is G-regular for
$A^c$) if and only if its capacity is infinite. As a direct consequence, if
$Q\subset E$ is a domain such that $(0,0)\in \partial Q$, we give a necessary
and sufficient condition for the existence on $Q$ of a positive solution of
$\partial_t u+{1/2}\Delta u =2u^2$ which blows up at $(0,0)$. We also give an
estimation of the hitting probabilities for the support of super-Brownian
motion at fixed time. We prove that if $d\geq 2$, the support of super-Brownian
motion is intersection-equivalent to the range of Brownian motion.
delmas@msri.org
- To see a preprint or other
information provided by the author
click here.
1078. EQUILIBRIUM CONDITION IN INSURANCE PRICING: A PARTICULAR CASE
Renato Ghisellini
Equilibrium pricing has been proven to underlie the rational Insured
expectancy of premia additivity for composition of policies fully covering
independent risks.
sistemi.srl@galactica.it
- To see a preprint or other
information provided by the author
click here.
1079. LOSS NETWORK REPRESENTATION OF PEIERLS CONTOURS
Roberto Fern\'andez, Pablo A. Ferrari, Nancy L. Garcia
We present a probabilistic approach for the study of systems with exclusions,
in the regime traditionally studied via cluster-expansion methods. In this
paper we focus on its application for the gases of Peierls contours found in
the study of the Ising model at low temperatures, but most of the results are
general. We realize the equilibrium measure as the invariant measure of a
loss-network process whose existence is ensured by a subcriticality condition
of a dominant branching process. In this regime, the approach yields, besides
existence and uniqueness of the measure, properties such as exponential space
convergence and mixing, and a central limit theorem. The loss network converges
exponentially fast to the equilibrium measure, without metastable traps. This
convergence is faster at low temperatures, where it leads to the proof of an
asymptotic Poisson distribution of contours. Our results on the mixing
properties of the measure are comparable to those obtained with
``duplicated-variables expansion'', used to treat systems with disorder and
coupled map lattices. It works in a larger region of validity than usual
cluster-expansion formalisms, and it is not tied to the analyticity of the
pressure. In fact, it does not lead to any kind of expansion for the latter,
and the properties of the equilibrium measure are obtained without resorting to
combinatorial or complex analysis techniques.
pablo@ime.usp.br
- To see a preprint or other
information provided by the author
click here.
1080. DECAY OF CORRELATIONS FOR NON H\"{O}LDERIAN DYNAMICS. A COUPLING
APPROACH
Xavier Bressaud, Roberto Fern\'andez and Antonio Galves
We present an upper bound on the mixing rate of the equilibrium state of a
dynamical systems defined by the one-sided shift and a non H\"{o}lder potential
of summable variations. The bound follows from an estimation of the relaxation
speed of chains with complete connections with summable decay, which is
obtained via a explicit coupling between pairs of chains with different
histories.
rf@ime.usp.br
- To see a preprint or other
information provided by the author
click here.
1081. COMPLETE RESOLUTION OF THE EPR-BELL PARADOX
Laszlo E. Szabo
Reichenbach's Common Cause Principle claims that if there is correlation
between two events and none of them is directly causally influenced by the
other, then there must exist a third event that can, as a common cause, account
for the correlation. The EPR-Bell paradox consists in the problem that we
observe correlations between spatially separated events in the EPR-experiments,
which do not admit common-cause-type explanation; and it must therefore be
inevitably concluded, that, contrary to relativity theory, in the realm of
quantum physics there exists action at a distance, or at least superluminal
causal propagation is possible; that is, either relativity theory or
Reichenbachs common cause principle fails. This paradox will be completely
resolved in this paper. By means of closer analyses of the concept of common
cause and a more precise reformulation of the EPR experimental scenario, I will
sharpen the conclusion we can draw from the violation of Bell's inequalities.
It will be explicitly shown that the correlations we encounter in the EPR
experiment could have common causes; that is, Reichenbach's Common Cause
Principle does not fail in quantum mechanics. Moreover, these common causes are
entirely compatible with locality principle of relativity.
szabol@caesar.elte.hu
- To see a preprint or other
information provided by the author
click here.
1082. ON REICHENBACH'S COMMON CAUSE PRINCIPLE AND REICHENBACH'S NOTION OF
COMMON CAUSE
G. Hofer-Szabo, M. Redei, L. E. Szabo
It is shown that, given any finite set of pairs of random events in a Boolean
algebra which are correlated with respect to a fixed probability measure on the
algebra, the algebra can be extended in such a way that the extension contains
events that can be regarded as common causes of the correlations in the sense
of Reichenbach's definition of common cause. It is shown, further, that, given
any quantum probability space and any set of commuting events in it which are
correlated with respect to a fixed quantum state, the quantum probability space
can be extended in such a way that the extension contains common causes of all
the selected correlations, where common cause is again taken in the sense of
Reichenbach's definition. It is argued that these results very strongly
restrict the possible ways of disproving Reichenbach's Common Cause Principle.
szabol@caesar.elte.hu
- To see a preprint or other
information provided by the author
click here.
1083. ANALYSIS AND PROBABILITY OVER INFINITE EXTENSIONS OF A LOCAL FIELD
Anatoly N. Kochubei
We consider an infinite extension $K$ of a local field of zero characteristic
which is a union of an increasing sequence of finite extensions. $K$ is
equipped with an inductive limit topology; its conjugate $\bar{K}$ is a
completion of $K$ with respect to a topology given by certain explicitly
written seminorms. We construct and study a Gaussian measure, a Fourier
transform, a fractional differentiation operator and a cadlag Markov process on
$\bar{K}$. If we deal with Galois extensions then all these objects are
Galois-invariant.
ank@ank.kiev.ua
- To see a preprint or other
information provided by the author
click here.
1084. FRACTIONAL DIFFERENTIATION OPERATOR OVER AN INFINITE EXTENSION OF A
LOCAL FIELD
Anatoly N. Kochubei
We study a fractional differentiation operator for functions on the conjugate
space to an infinite extension of a local field of zero characteristic which is
a union of an increasing sequence of finite extensions. In particular, a
representation in the form of a hypersingular integral operator is obtained. It
is also shown that the corresponding diffusion measure is not absolutely
continuous with respect to the Gaussian measure.
ank@ank.kiev.ua
- To see a preprint or other
information provided by the author
click here.
1085. FINITE DIMENSIONAL APPROXIMATIONS TO WIENER MEASURE AND PATH INTEGRAL
FORMULAS ON MANIFOLDS
Lars Andersson, Bruce K. Driver
Certain natural geometric approximation schemes are developed for Wiener
measure on a compact Riemannian manifold. These approximations closely mimic
the informal path integral formulas used in the physics literature for
representing the heat semi-group on Riemannian manifolds. The path space is
approximated by finite dimensional manifolds consisting of piecewise geodesic
paths adapted to partitions $P$ of $[0,1]$. The finite dimensional manifolds of
piecewise geodesics carry both an $H^{1}$ and a $L^{2}$ type Riemannian
structures $G^i_P$. It is proved that as the mesh of the partition tends to
$0$,
$$ 1/Z_P^i e^{- 1/2 E(\sigma)} Vol_{G^i_P}(\sigma) \to
\rho_i(\sigma)\nu(\sigma) $$
where $E(\sigma )$ is the energy of the piecewise geodesic path $\sigma$, and
for $i=0$ and $1$, $Z_P^i$ is a ``normalization'' constant, $Vol_{G^i_P}$ is
the Riemannian volume form relative $G^i_P$, and $\nu$ is Wiener measure on
paths on $M$. Here $\rho_1 = 1$ and
$$ \rho_0 (\sigma) = \exp( -1/6 \int_0^1 Scal(\sigma(s))ds ) $$
where $Scal$ is the scalar curvature of $M$. These results are also shown to
imply the well know integration by parts formula for the Wiener measure.
larsa@math.kth.se
- To see a preprint or other
information provided by the author
click here.
1086. NON-GAUSSIAN SURFACE PINNED BY A WEAK POTENTIAL
J.-D. Deuschel and Y. Velenik
We consider a model of a two-dimensional interface of the SOS type, with
finite-range, even, strictly convex, twice continuously differentiable
interactions. We prove that, under an arbitrarily weak potential favouring
zero-height, the surface has finite mean square heights. We consider the cases
of both square well and $\delta$ potentials. These results extend previous
results for the case of nearest-neighbours Gaussian interactions in \cite{DMRR}
and \cite{BB}. We also obtain estimates on the tail of the height distribution
implying, for example, existence of exponential moments. In the case of the
$\delta$ potential, we prove a spectral gap estimate for linear functionals. We
finally prove exponential decay of the two-point function (1) for strong
$\delta$-pinning and the above interactions, and (2) for arbitrarily weak
$\delta$-pinning, but with finite-range Gaussian interactions.
velenik@sidpsg.epfl.ch
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information provided by the author
click here.
1087. A NOTE ON THE DECAY OF CORRELATIONS UNDER $\DELTA$-PINNING
D.Ioffe and Y.Velenik
We prove that for a class of massless $\nabla\phi$ interface models on
$\Ztwo$ an introduction of an arbitrary small pinning self-potential leads to
exponential decay of correlation, or, in other words, to creation of mass.
velenik@sidpsg.epfl.ch
- To see a preprint or other
information provided by the author
click here.
1088. THE REPEATED SOLICITATION MODEL
R. W. R. Darling
This paper presents a probabilistic analysis of what we call the "repeated
solicitation model". To give a specific context, suppose B is a direct
marketing company with a list of S sales prospects. At epoch 1, B sends a
solicitation to every prospect on the list, and elicits X(1) replies. The
company deletes the respondents from the list, and at epoch 2 sends a
solicitation to the other prospects, of whom X(2) respond, and so on. This
continues until an epoch n such that X(n) = 0, which we call epoch T, and then
B makes no further solicitations. We seek - The probability distribution of T;
- The distribution of the total number of respondents; - The expected total
number of solicitations. All three quantities are explicitly computed, assuming
that (a) prospects' response times are independent, and (b) S is Poisson
distributed.
rwrd@afterlife.ncsc.mil
- To see a preprint or other
information provided by the author
click here.
1089. UNREAL PROBABILITIES: PARTIAL TRUTH WITH CLIFFORD NUMBERS
Carlos C. Rodriguez
This paper introduces and studies the basic properties of Clifford algebra
valued conditional measures.
cr569@csc.albany.edu
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information provided by the author
click here.
1090. CONTINUITY PROPERTIES OF SCHR\"ODINGER SEMIGROUPS WITH MAGNETIC FIELDS
Kurt Broderix, Dirk Hundertmark, Hajo Leschke
The objects of the present study are one-parameter semigroups generated by
Schr\"odinger operators with fairly general electromagnetic potentials. More
precisely, we allow scalar potentials from the Kato class and impose on the
vector potentials only local Kato-like conditions. The configuration space is
supposed to be an arbitrary open subset of multi-dimensional Euclidean space;
in case that it is a proper subset, the Schr\"odinger operator is rendered
symmetric by imposing Dirichlet boundary conditions. We discuss the continuity
of the image functions of the semigroup and show local-norm-continuity of the
semigroup in the potentials. Finally, we prove that the semigroup has a
continuous integral kernel given by a Brownian-bridge expectation. Altogether,
the article is meant to extend some of the results in B. Simon's landmark paper
(Bull. Amer. Math. Soc. (N.S.) {\bf 7}, 447-526 (1982)) to non-zero vector
potentials and more general configuration spaces.
broderix@theorie.physik.uni-goettingen.de
- To see a preprint or other
information provided by the author
click here.
1091. PROBABILITY AND ENTROPY IN QUANTUM THEORY
Ariel Caticha
Entropic arguments are shown to play a central role in the foundations of
quantum theory. We prove that probabilities are given by the modulus squared of
wave functions, and that the time evolution of states is linear and also
unitary.
ariel@cnsvax.albany.edu
- To see a preprint or other
information provided by the author
click here.
1092. PARKING CARS WITH SPIN BUT NO LENGTH
Yoshiaki Itoh and Larry Shepp
The car parking problem is a one-dimensional model of random packing.
Cars arrive to park on a block of length $x$, sequentially. Each car has,
independently, spin up or spin down, w.p. $0 < p \le 1$, for spin up, and
$q = 1-p$, for spin down, respectively. Each car tries to park at a uniformly
distributed random point $t \in [0,x]$. If $t$ is within distance 1 of the
location of a previously parked car of the {\it same} spin, or within distance
$a$ of the location of a previously parked car of the opposite spin then the
new car leaves without parking and the next car arrives, until saturation,
when no more cars can park on $[0,x]$. We study the problem analytically
as well as numerically. The expected number of up spins $c(p,a)$ per unit
length for sufficiently large $x$ is neither monotonic in $p$ for fixed $a$,
nor is it monotone in $a$ for fixed $p$, in general. An intuitive explanation
is given for this non-monotonicity.
itoh@ism.ac.jp shepp@stat.rutgers.edu
1093. PATHWISE UNIQUENESS FOR REFLECTING BROWNIAN MOTION IN EUCLIDEAN DOMAINS
Richard F. Bass and Elton P. Hsu
For bounded $C^{1,\alpha}$ domains in $R^d$ we show pathwise
uniqueness for the Skorokhod equation
$$dX_t=dW_t+(1/2)\nu(X_t) dL_t,$$
where $\nu$ is the inward pointing normal vector,
$L_t$ is a continuous, nondecreasing process that increases
only when $X_t$ is on the boundary, and $W_t$ is a standard
$d$-dimensional Brownian motion. Our method is to prove
strong existence and weak uniqueness and then to show that
those together imply pathwise uniqueness.
bass@math.uconn.edu
- To see a preprint or other
information provided by the author
click here.
1094. NUMERICAL CALCULATION OF STABLE DENSITIES AND DISTRIBUTION FUNCTIONS
John Nolan
Computational formulas are given for stable densities and
distribution functions in Zolotarev's (M) parameterization.
These formulas are used in a software package called
STABLE the calculate general stable densities, distribution
functions, and quantiles.
The original paper appeared in Comm. in Stat, Stochastic
Models, vol. 13, 759-774 (1997).
The new version of STABLE includes a fast, precomputed
cubic spline approximation to stable densities and a
routine for computing maximum likelihood estimators
of all four stable parameters. A Windows 95 version of
the program can be downloaded from the website below.
jpnolan@american.edu
- To see a preprint or other
information provided by the author
click here.
1095. MODELLING ENVIROMENTAL AND GENETIC EFFECTS: A RANDOM COEFFICIENT
REGRESSION APPROACH
Nesan Chelliah
In this paper a random coefficient regression model has been used
to model quantitative traits in pedigree analysis.
The genetic and the environmental effects has been taken into account.
We review a new estimator for estimating the variance components
and derived the distribution of the pedigree statistics used in
testing for outlying observations.
N.Chelliah@mailbox.gu.edu.au
1096. FRACTIONAL DERIVATIVES, NON-SYMMETRIC AND TIME-DEPENDENT
DIRICHLET FORMS, AND THE DRIFT FORM
Niels Jacob and Rene L. Schilling
Using fractional derivatives we show that the drift form
$\int_{-\infty}^\infty u(x) \, {dv(x) \over dx}\,dx$ can be
approximated by non-symmetric Dirichlet forms. A similar
result holds for the drift form in higher dimensions and
with variable coefficients, if the coefficient functions
satisfy certain regularity and commutator conditions. Since
time-dependent Dirichlet forms (in the sense of Y. Oshima)
can be interpreted as sums of a drift form (in $\tau$-direction)
and a mixture of $\tau$-parametrized Dirichlet forms over
$\mathbb{R}^n$, our results show that time-dependent Dirichlet
forms arise as limits of ordinary non-symmetric Dirichlet
forms in $\mathbb{R}\times\mathbb{R}^n$-space. An abstract
result on fractional powers of Markov generators allows to
extend this observation to generalized Dirichlet forms.
Another consequence is that the bilinear form induced by
an arbitrary Levy process is the limit of non-symmetric
Dirichlet forms.
jacob@informatik.unibw-muenchen.de rls@maths.ntu.ac.uk
