Probability Abstracts 47
This document contains abstracts 1097-1152.
They have been mailed on October 29, 1998.
1097. FRACTIONAL BROWNIAN MOTION AND THE MARKOV PROPERTY
Philippe Carmonaand Laure Coutin
Fractional Brownian motion belongs to a class
of long memory Gaussian processes that can be represented
as linear functionals of an infinite dimensional Markov process. This
leads naturaly to:
- An efficient algorithm to approximate the process.
- An infinite dimensional ergodic theorem which applies, in
particular, to functionals of the type
$\int_0^t \phi(V_h(s))\,ds$ where $ V_h(s)=\intot h(t-u)\, dB_u$ with
$B$ a standard Brownian motion.
carmona@cict.fr (hardcopy available)
coutin@cict.fr
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1098. ON RANDOM GRAPH HOMOMORPHISMS INTO Z
Itai Benjamini, Olle H\"aggstr\"om and Elchanan Mossel
Given a bipartite connected finite graph,
G=(V,E) and a vertex v_0 in V,
we consider uniform probability measure on the set of graph homomorphisms
f from V into Z, satisfying f(v_0)=0.
This measure can be viewed as a G-indexed random walk on Z, generalizing
both the usual time-indexed random walk and tree-indexed random walk.
Several general inequalities for G-indexed random walk are derived,
including an upper bound on fluctuations implying that the distance
d(f(u),f(v)) between f(u) and f(v), is stochastically dominated by the
distance to 0 of a simple random walk on Z having run for d(u,v) steps.
Various special cases are
studied. For instance, when G is an n-level regular tree with all
vertices on the last level connected to an additional single vertex, we
show that the expected range of the walk is O(log n). This result can
also be rephrased as a statement about conditional branching random walk.
To prove it, a power-type Pascal triangle is introduced and exploited.
itai@wisdom.weizmann.ac.il
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1099. TOTAL PATH LENGTH FOR RANDOM RECURSIVE TREES
Robert P. Dobrow and James Allen Fill
Total path length for a rooted tree is defined as the
sum of all root-to-node distances. Let $T_n$ be the total
path length for a random recursive tree of order $n$.
Mahmoud showed that $W_n := (T_n - E[T_n])/n$ converges
almost surely and in~$L^2$ to a limiting random
variable $W$. Here we give recurrence
relations for the moments of $W_n$ and of $W$ and show that
$W_n$ converges to $W$ in $L^p$ for each $0 < p < \infty$.
We confirm the conjecture that the distribution of $W$ is
not normal. We also show that the distribution of $W$ is
characterized among all distributions having
zero mean and finite variance by the distributional identity
\[ W \stackrel{d}{=} U(1+W) + (1-U) W^* - {\cal E}(U), \]
where ${\cal E}(x) := - x \ln x - (1-x) \ln (1-x)$ is
the binary entropy function, $U$ is a uniform$(0,1)$
random variable, $W^*$ and $W$ have the same distribution,
and $U,$ $W$, and $W^*$ are mutually independent. Finally,
we derive an approximation for the distribution of $W$
using a Pearson curve density estimator.
Simulations exhibit good accuracy in the approximation.
bdobrow@mathax.truman.edu jimfill@jhu.edu
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1100. MOMENT CONDITIONS FOR A SEQUENCE WITH NEGATIVE DRIFT
TO BE UNIFORMLY BOUNDED IN L^r
Robin Pemantle and Jeffrey S. Rosenthal
Suppose a sequence of random variables $\{ X_n \}$ has negative drift
when above a certain threshold and has increments bounded in $L^p$.
When $p > 2$ this implies that $\E X_n$ is bounded above by a constant
independent of $n$ and the particular sequence $\{ X_n \}$. When $p \leq
2$ there are counterexamples showing this does not hold. In general,
increments bounded in $L^p$ lead to a uniform $L^r$ bound on $X_n^+$
for any $r < p-1$, but not for $r \ge p-1$. These results are motivated
by questions about stability of queueing networks.
pemantle@math.wisc.edu jeff@utstat.toronto.edu
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1101. ON THE CONDITIONED EXIT MEASURES OF SUPER BROWNIAN MOTION
Thomas S. Salisbury and John Verzani
We condition super Brownian motion in a Euclidean domain
using its exit measure. We write the conditioned process both
as an $h$-transform of the unconditioned process (that is, we
give it in terms of a martingale change of measure), and in
terms of a non-homogeneous branchingparticle system with
immigration of mass.
The first such results were due to Evans and Perkins, who
conditioned Euclidean super Brownian motion on
non-extinction, and represented it in terms of an
``immortal particle'' backbone. The analogue of those
results, in a domain, is to condition the exit measure to
have a specified boundary point $z$ in its support. To
parallel the immortal particle representation, this event
must be of probability zero, for which reason we work in
Lipschitz domains in dimension $4$ and higher. In this
case the backbone consists of a Brownian path conditioned
to exit at $z$. More generally, we condition the exit
measure to have each of a fixed finite collection
$\{z_1,\dots,z_n\}$ of boundary points in its support.
This makes the backbone take the form of a tree with $n$
leaves. Our principal result is an explicit construction
of this tree.
In fact, we formulate a general class of $h$-transforms of
super Brownian motion, and derive their backbone
representations using a Palm formula. We show that the
above conditioning gives a member of this class by
analysing the asymptotics of small solutions to an
associated nonlinear PDE.
We conjecture a representation for conditioning the
support of the exit measure to consist exactly of the set
$\{z_1,\dots,z_n\}$. Such conditionings should be minimal
and hence should correspond to elements of the Martin
boundary of super Brownian motion.
salt@nexus.yorku.ca verzani@postbox.csi.cuny.edu
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1102. NON-DEGENERATE CONDITIONINGS OF THE EXIT MEASURES
OF SUPER BROWNIAN MOTION
Thomas S. Salisbury and John Verzani
In an earlier paper, we conditioned the exit measure of
super Brownian motion from a domain to have a fixed
collection $\{z_1,\dots,z_n\}$ of boundary points in its
support. Generalizing results of Evans and Perkins, we
represented the conditioned process both as an
$h$-transform of the unconditioned process (that is, in
terms of a martingale change of measure), and in terms of a
non-homogeneous branching ``backbone'' throwing off
unconditioned mass. We did so in dimensions $4$ and higher,
where the event conditioned on has probability
$0$. In the current paper, we consider similar results in
dimension $2$, where now this conditioning will be
non-singular.
We start by treating the case $n=1$ of a single boundary
target point. We define a general class of $h$-transforms
of super Brownian motion, that includes this case, and use
a Palm formula to represent each of them in terms of a
branching backbone. In contrast to the higher dimensional
case (where with $n=1$ the backbone consisted of a single
path, and with $n>1$ of a non-homogeneous tree with $n$
nodes), the backbone now consists of a homogeneously
branching tree, possibly with infinitely many leaves. We
relate the form of this tree to the classification of
positive solutions to the PDE $\Delta u=4u^2$. We then
treat the general case of $n>1$.
salt@nexus.yorku.ca verzani@postbox.csi.cuny.edu
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1103. THE INVARIANT DISTRIBUTION OF A BAYESIAN TEST STATISTIC
T. Nesan Chelliah
Testing for stationarity in linear regression model has
been studied by several authors. In this paper we have
extended a Bayesian uni variate test statistic and derived its
invariant distribution under spherical symmetry using
group of transformations.
N.Chelliah@mailbox.gu.edu.au
1104. GENERALISED MINIMUM DISPERSION LINEAR ESTIMATOR
T.Nesan Chelliah
In this paper we consider mixed regression estimators
for a random coefficient regression model. We computed and
compared the covariance matrix of restricted and unrestricted
estimators. We also generalised the minimum dispersion
linear estimator and obtained an unbiased estimator
for its covariance matrix.
N.Chelliah@mailbox.gu.edu.au
1105. ASYMPTOTICS FOR VORONOI TESSELLATIONS ON RANDOM SAMPLES
K. McGivney and J. Yukich
Let $V(X_1,...,X_n)$ denote the total edge length of the
Voronoi tessellation on random variables $X_1,...,X_n$. If
$X_1,...,X_n$ are independent and have a common continuous
density $f(x)$ on the unit square which is bounded away
from 0 and infinity, then it is shown in the limit as
n tends to infinity that $V(X_1,...,X_n)/n^{1/2}$ a.s. tends
to $2 \int_{[0,1]^2} (f(x))^{1/2} dx$. In this way we show
that the asymptotics for the Voronoi length functional
resemble those of the classic problems of Euclidean
combinatorial optimization, including the total edge length
of the graphs of the minimal spanning tree, minimal
matching, and traveling salesman problems.
jey0@lehigh.edu
1106. LARGE DEVIATION PRINCIPLES FOR EUCLIDEAN GRAPHS
J. E. Yukich
We show that the total edge length of a large collection of
random Euclidean graphs satisfies a large deviation
principle (LDP). By proving a general LDP for smooth
subadditive Euclidean functionals we deduce large
deviation principles for the total edge length of graphs
representing solutions to problems in Euclidean
combinatorial optimization and computational geometry. We
obtain for example large deviation principles for the
total edge length of the shortest tour through a random
sample, the length of a minimal matching through a random
sample, the length of a minimal spanning tree
through a random sample, and the length of the k-nearest
neighbors graph through a random sample.
jey0@lehigh.edu
1107. APPLICATION OF LOG-SOBOLEV INEQUALITY
TO THE STOCHASTIC DYNAMICS OF UNBOUNDED SPIN SYSTEMS
ON THE LATTICE
Nobuo Yoshida
We consider a ferromagnetic spin system with unbounded
interactions on the $d$-dimensional integer lattice
($d \geq 1$) and investigate the convergence rate for the
associated stochastic dynamics which is sometimes called
the Glauber dynamics. We prove that, if the log-Sobolev
inequality for the finite volume Gibbs states holds
uniformly in both the volume and the boundary condition,
then the finite volume Glauber dynamics relaxes to
equilibrium exponentially fast, uniformly in the volume
whenever it starts from a tempared configuration.
The result strengthens some results obtained earier by
B. Zegarlinski (\cite{Z96}).
The uniformity of log-Sobolev inequality we assume in
this paper is known to be true, for example when $d=1$,
or the coupling constants are sufficiently
small (\cite{BH98, Y97, Z96}).
nobuo@kusm.kyoto-u.ac.jp
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1108. A TRANSACTION COSTS CONVERGENCE RESULT
FOR GENERAL HEDGING STRATEGIES
Laurent Gauthier
This paper generalizes the result of Leland (1985) to hedging
strategies that use not only the underlying but also all kinds
of options. These hedging strategies are a generalization
of static hedging. In addition the result is valid for all
shapes of payoff, including path-dependent. Two cases of
hedging methods are studied. The first one, as in Leland,
assumes rehedging takes place at fixed time intervals. The
second one supposes rehedging takes place when the delta
moves by more than a fixed proportion.
We show that when transaction costs and the time between
transactions go to zero, the price of derivative securities
in this frame converge towards the solution of a non-linear
PDE. The optimal allocation among the derivatives used for
hedging can be found so as to minimize overall transaction
costs.
laurent_gauthier@msn.com
1109. INVESTMENT UNDER UNCERTAINTY: IMPLEMENTATION DELAY
AND EMBEDDED OPTIONS
Laurent Gauthier
One of the major characteristics of the capital budgeting
process is the delay existing between the investment decision and its
implementation. This paper analyses investment decisions under
uncertainty with implementation delay in a unified analytical
framework. We provide closed-form solutions relating
the value of the investment policy and the optimal investment
threshold to the size of the delay. We show that the
implementation lag creates an embedded option for the investor: the option
to abandon the project implementation during this delay.
We derive the value of this embedded option for various exercise
policies corresponding to different levels of freedom with
respect to the abandonment of the project and analyse its effects on
the investment policy of the firm. We follow a
probabilistic approach in our treatment of these computations,
and use the excursion theory of the Brownian Motion.
laurent_gauthier@msn.com
1110. EXTINCTION OF SUPERDIFFUSIONS AND SEMILINEAR PDEs
Eugene Dynkin and Sergei Kuznetsov
A superdiffusion is a measure-valued branching process
associated with a non-linear operator $Lu-\psi(u)$ where
$L$ is a second order elliptic differential operator and
$\psi$ is a function from $\rd\times\R_+$ to $\R_+$.
In the case $L1\le 0$ (the so-called subcritical case),
the expectation of the total mass does not increase and
the mass vanishes at a finite time with probability 1.
Most results on connections between the superdiffusion
and differential equations involving $Lu-\psi(u)$ were
obtained for the subcritical case. In the present paper,
we assume only that $L1$ is a bounded function. In this
more general setting, the probability of extinction can be
smaller than 1, and we show that this happens if and only
if there exists a strictly positive solution of the
equation $Lu-\psi(u)=0$ in $R^d$. We also establish a
relation between the probability of extinction in a domain
$D$ and strictly positive solutions of equation
$Lu-\psi(u)=0$ in $D$ that are equal to 0 on the boundary
of $D$.
ebd1@cornell.edu sek@euclid.colorado.edu
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1111. SEMILINEAR PARABOLIC EQUATIONS, DIFFUSIONS AND
SUPERDIFFUSIONS
Eugene Dynkin
Semilinear equation $\dot u+L^0u=u^\alpha$ where $L^0$ is a
second order elliptic differential operator without zero
order term and $1<\alpha\le 2$ has been studied by the author
(see, e.g., Annals of Probability, Vol. 21, 1993, pp 1185-1262)
by using superdiffusions. In the present paper, we apply
superdiffusions to a more general equation
$\dot u+Lu=\psi(u)$ where $Lu=L^0u+cu$ (with a bounded
coefficient $c$) and $\psi$ belongs to a convex class
which contains $ku^\alpha$ with $1<\alpha\le 2$ and positive
locally bounded coefficient $k$. We also cover a substantially
wider class of functions $\psi$ which do not correspond to
any superdiffusion (for instance, $ku^\alpha$ with
$\alpha>2$). Related problems are treated with the help of
diffusion processes. This approach is useful even in the
linear theory. For instance, the first boundary value
problem for equation $\dot u+Lu=-f$ can be investigated for
a class of domains described in probabilistic terms which
is substantially larger than the class considered in the
literature on PDE's.
ebd1@cornell.edu
1112. ON THE STABILITY OF MEASURE VALUED PROCESSES.
APPLICATIONS TO NON LINEAR FILTERING AND INTERACTING PARTICLE SYSTEMS.
Pierre Del Moral, Alice Guionnet
The asymptotic stability of a class of measure valued
processes arising in nonlinear filtering and genetic
algorithm theory is discussed.
Simple sufficient conditions are given for exponential
decays. These criterion are apply to study the asymptotic
stability of the nonlinear filtering equation and
infinite population models as those arising in Biology.
On the basis of these stability properties we also
propose a uniform convergence theorem for the interacting
particle numerical scheme of the nonlinear filtering
equation introduced in a previous work.
In the last part of this work we propose
a refinement genetic type particle method with periodic
selection dates and we improve the previous uniform
convergence results. We finally discuss the uniform
convergence of particle approximations including branching.
delmoral@cict.fr
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1113. MEASURE VALUED PROCESSES AND INTERACTING PARTICLE SYSTEMS.
APPLICATONS TO NON LINEAR FILTERING PROBLEMS.
Pierre Del Moral
In the paper we study interacting particle approximations
of discrete time and measure valued dynamical systems.
Such systems have arisen in such diverse scientific
disciplines as physics and signal processing.
We give conditions for the so-called particle
density profiles to converge to the desired
distribution when the number of particles is growing.
The strength of our approach is that is applicable to
a large class of measure valued dynamical system arising
in engineering and particularly in nonlinear filtering
problems. Our second objective is to use these results
to solve numerically the nonlinear filtering equation.
Examples arising in fluid mechanics are also given.
delmoral@cict.fr
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1114. CONVERGENCE OF EMPIRICAL PROCESSES FOR INTERACTING PARTICLE
SYSTEMS WITH APPLICATIONS TO NON-LINEAR FILTERING
Pierre Del Moral, Michel Ledoux
In this paper, we investigate the convergence of empirical
processes for a class of interacting particle numerical
schemes arising in biology, genetic algorithms and
advanced signal processing.
The Glivenko-Cantelli and Donsker theorems presented
in this work extend the corresponding statements in
the classical theory and apply to a class of genetic type
particle numerical schemes of the non-linear filtering
equation.
delmoral@cict.fr
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1115. LARGE DEVIATIONS FOR INTERACTING PARTICLE SYSTEMS.
APPLICATIONS TO NON-LINEAR FILTERING
Pierre Del Moral, Alice Guionnet
The non linear filtering problem consists in computing the
conditional distributions of a Markov signal process given
its noisy observations.
The dynamical structure of such distributions can be
modelled by a measure valued dynamical Markov process.
Several random particle approximations were recently
suggested to approximate recursively in time the so-called
non linear filtering equations.
We present an interacting particle system approach
and we develop large deviations principles for the
empirical measures of the particle systems.
We end this paper extending the results to an
interacting particle system approach which includes
branchings.
delmoral@cict.fr guionnet@stats.matups.fr
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1116. CENTRAL LIMIT THEOREM FOR NON-LINEAR FILTERING
AND INTERACTING PARTICLE SYSTEMS
Pierre Del Moral, Alice Guionnet
Several random particle systems approaches were recently
suggested to solve numerically non linear filtering
problems.
The present analysis is concerned with genetic-type
interacting particle systems. Our aim is to study
the fluctuations on path space of such particle
approximating models.
Our basic tools are the Dynkin-Mandelbaum's Theorem
on symmetric statistics and multiple Wiener's integrals.
Several examples of non linear filtering problems that can
be handle in our framework are also given.
delmoral@cict.fr guionnet@stats.matups.fr
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1117. ON THE CONVERGENCE AND THE APPLICATIONS OF THE
GENERALIZED SIMULATED ANNEALING
Pierre Del Moral, Laurent Miclo
The convergence of the generalized simulated annealing
with time-inhomogeneous communication cost functions
is discussed. This study is based on the use of Log-Sobolev
inequalities and semigroup techniques in the spirit of
a previous article by one of the authors.
We also propose a natural test set approach to study the
global minima of the virtual energy. The second part of
the paper is devoted to the application of these results.
First we propose two general Markovian models of genetic
algorithms and we give a simple proof of the convergence
toward the global minima of the fitness function.
Finally we introduce a stochastic algorithm which converges
to the set of the global minima of a given mean cost
optimization problem.
delmoral@cict.fr miclo@cict.fr
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1118. DISCRETE FILTERING USING BRANCHING AND
INTERACTING PARTICLE SYSTEMS
D. Crisan, P. Del Moral and T. Lyons
The stochastic filtering problem deals with the
estimation of the current state of a signal process given
the information supplied by an associate process, usually
called the observation process.
We describe a particle algorithm designed for solving
numerically discrete filtering problems.
The algorithm involves the use of a system of $n$
particles which evolve (mutate) in correlation
with each other (interact) according to law of
the signal process and, at fixed times, give birth
to a number of offsprings depending on
the observation process.
We present several possible branching mechanisms and
prove, in a general context the convergence of the
particle systems (as $n$ tends to $\infty$) to the
conditional distribution of the signal given the
observation.
We then apply the result to the discrete filtering and
give several example when the results can be applied.
d.crisan@ic.ac.uk delmoral@cict.fr t.lyons@ic.ac.uk
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1119. BERRY-ESSEEN INEQUALITIES FOR A RATIO OF SUMS OF RANDOM
VARIABLES
S.Y.Novak
We establish uniform and non-uniform Berry-Esseen-type
inequalities for a ratio of sums of random variables and
for self-normalised sums and Student's statistic.
The topic has divers applications to Nonparametric
Statistics (eg.,density, regression curve, tail index,
extremal index estimation).
Research Report No 98/07. University of Sussex, 1998.
S.Novak@susx.ac.uk
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1120. GAUSSIAN MEASURES OF DILATATIONS OF CONVEX SYMMETRIC SETS
Rafal Latala and Krzysztof Oleszkiewicz
The main result of the paper is the following theorem,
previously also known as an S-conjecture: If $\mu$ is a
centered Gaussian measure on $R^n$ (or more generally on
some separable Banach space $F$), $A$ is a symmetric convex
closed set in $R^n$ and $P$ is a symmetric strip in $R^n$
such that $\mu(A)=\mu(P)$ then $\mu(tA)\geq \mu(tP)$ for
$t\geq 1$ and $\mu(tA)\leq \mu(tP)$ for $0\leq t\leq 1$.
As an application we calculate the best constants in
comparison of moments of Gaussian vectors and derive some
estimates of Gaussian measures of small balls.
rlatala@mimuw.edu.pl koles@mimuw.edu.pl
1121. ON THE ERDOS-RENYI MAXIMUM OF PARTIAL SUMS
S.Y.Novak
We study the distribution of the Erdös-Rényi maximum of
partial sums (MPS). A limit law is found.
In the particular case where MPS is defined over a sequence
of Bernoulli B(1/2) random variables, we estimate the rate
of convergence to that limit law.
S.Novak@susx.ac.uk
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information provided by the author
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1122. ON THE DISTRIBUTION OF THE RATIO OF SUMS
OF RANDOM VARIABLES
S.Y.Novak
We study the distribution of the ratio $\,Z_n\,$ of sums
of random variables.
Berry-Esseen-type estimates of the rate of convergence
in the central limit theorem for $\,Z_n\,$ are given.
In addition, second-order asymptotic expansions for the
first and the second moments of $\,Z^2_n\,$ are presented.
The results are applied to the problems of nonparametric
regression curve estimation, nonparametric tail index
estimation, and nonparametric estimation of a hazard
function.
S.Novak@susx.ac.uk
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1123. SOME COMPARISONS FOR GAUSSIAN PROCESSES
Richard A. Vitale
Among the most important tools for Gaussian processes
are comparison principles. Typically they provide for
the derivation of upper bounds by majorizing a given
process with a second process that is larger in some
sense and that has more tractable properties. Suppose
that $\{X_i, i\in I\}$ and $\{Y_i, i\in I\}$ are two
Gaussian processes indexed on a denumerable $I$ with
$E(X_i -X_j)^2 \leq E(Y_i - Y_j)^2$ for all $i,j\in I$.
We show some variants of the classical comparisons of
Sudakov-Fernique. For any ${\hat{\imath}}\in I$
and constants $m,\{ m_i, i\in I\}$:
$$E\sup_i \{X_i +m_i -X_{\hat{\imath}}, m\}\leq
E\sup_i\{Y_i+m_i-Y_ {\hat{\imath}}, m\},$$
$$E\sup_i\{X_i +m_i\}\leq E\sup_i\{Y_i+m_i\},$$
and for any non--decreasing,
convex $g:R^1\rightarrow R^1$ with $g(-\infty )>-\infty$
or with $\max [ Eg_+(\sup_i\{X_i+m_i\}-X_{\hat{\imath}}),
Eg_+(\sup_i\{Y_i+m_i\}-Y_{\hat{\imath}})] <\infty$:
$$Eg(\sup_i\{X_i+m_i\}-X_{\hat{\imath}})\leq
Eg(\sup_i\{Y_i+m_i\}-Y_{\hat{\imath}}).$$ We relate these
to a set of weakened Slepian--Schlafli assumptions:
if also $EX_i^2\leq EY_i^2$ for all $i\in I$, then
$$\int_t^{\infty} P( \sup_i X_i>s ) ds \leq
\int_t^{\infty} P( \sup_i Y_i >s ) ds$$ for all $t>0$.
rvitale@uconnvm.uconn.edu
1124. WEAK CONVERGENCE OF MULTIVARIATE FRACTIONAL PROCESSES
Domenico Marinucci and Peter M. Robinson
Weak convergence to a form of fractional Brownian motion is
established for a wide class of nonstationary fractionally
integrated multivariate processes. Instrumental for the main
argument is a result of some independent interest on
approximations for partial sums of stationary linear vector
sequences. A functional central limit theorem for smoothed
processes is analyzed under more general assumptions.
marinucc@scec.eco.uniroma1.it
1125. THE MOST VISITED SITES FOR SYMMETRIC STABLE PROCESSES
Richard F. Bass, Nathalie Eisenbaum, and Zhan Shi
Let $X_t$ be a symmetric stable process of index $\alpha\in (1,2]$.
Let $V_t$ be the value of $x$ at which the local time
at time $t$ takes its maximum. We prove that $V_t$
is transient, and give an estimate for the rate of
escape. Our method is to use a type of Ray-Knight theorem
for the local times of stable processes and some
estimates for fractional Brownian motion.
bass@math.uconn.edu nae@ccr.jussieu.fr shi@ccr.jussieu.fr
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1126. TRANSITION DENSITY ESTIMATES FOR DIFFUSION PROCESSES
ON HOMOGENEOUS RANDOM SIERPINSKI CARPETS
Ben Hambly, Takashi Kumagai, Shigeo Kusuoka and Xian Yin Zhou
We consider homogeneous random Sierpinski carpets, a class
of infinitely ramified random fractals which have spatial
symmetry but which do not have exact self-similarity. For
a fixed environment we construct `natural' diffusion
processes on the fractal and obtain upper and lower
estimates of the transition density for the process that
are, up to constants, best possible. By considering the
random case, when the environment is stationary and
ergodic, we deduce estimates of Aronson type.
bmh@maths.ed.ac.uk kumagai@math.ubc.ca
1127. AN $L^p$-VIEW OF THE BAHADUR-KIEFER THEOREM
Miklos Csorgo and Zhan Shi
The classical Bahadur-Kiefer theorem says that, after
suitable normalization, the ratio between the sup-norm of the
empirical difference process and the square root of the
sup-norm of the empirical process converges almost surely to
1. Replacing the sup-norm by suitable $L^p$-norms, we prove
that a similar result holds, but with a different normalizing
function.
Our result also confirms the impossibility of the weak
convergence of the empirical difference process in the space
$D[0,1]$, endowed with the Skorohod $J_1$ topology, with any
normalizing function to any nondegenerate random element of
$D[0,1]$. This fact was first established by Vervaat in 1972.
Postscript file or hardcopy available.
mcsorgo@math.carleton.ca shi@ccr.jussieu.fr
1128. SMOOTHNESS OF HARMONIC FUNCTIONS AND TIME REVERSAL
OF MARKOV PROCESSES
Jean Picard and Catherine Savona
A new probabilistic proof is given for the $C^\infty$
smoothness of solutions of PDEs $Lh=\ell$, when $L$ is a
second order differential operator satisfying the
Hormander condition; this proof is based on a time
reversal result. Then the technique is extended to non
local operators $L$ associated to Markov processes with
jumps, and the smoothness of solutions of $Lh=\ell$ is
again studied.
picard@ucfma.univ-bpclermont.fr
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here.
1129. HOMOGENIZATION OF RANDOM PARABOLIC OPERATOR WITH LARGE POTENTIAL
Fabien Campillo, Marina Kleptsyna and Andrey Piatnitski
We study the averaging problem for a divergence form
random parabolic operators with a large potential and
with coefficients rapidly oscillating both in space and
time variables. We assume that the medium possesses the
periodic microscopic structure while the dynamics of the
system is random and, moreover, diffusive.
campillo@sophia.inria.fr
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1130. MAJORIZATION AND GAUSSIAN CORRELATION
Richard A. Vitale
Using Schur majorization, we show Gaussian correlation
inequalities for two classes of sets: Schur cylinders and
barycentrically ordered sets. As a corollary, we deduce
positive correlation for pairs of centered convex sets
that share the symmetries of a regular simplex.
rvitale@uconnvm.uconn.edu
1131. PERSISTENT RANDOM WALKS IN STATIONARY ENVIRONMENT
Smail Alili
We study the behavior of persistent random walks (r.w.) on
the integers, in a random environment. A complete characterization
of the almost sure limit behavior of these processes, including
the law of large numbers, is obtained. This is done in a general
situation where the environmental sequence of random variables is
stationary and ergodic.
Sz\'asz and T\'oth obtained a central limit theorem when the
ratio ${\mu \over {\lambda}}$, of right- and left-transpassing
probabilities, satisfies ${\mu \over {\lambda}} \leq a<1$ a.s.
(for a given constant $a$). We consider the case where
${\mu \over {\lambda}}$ has wider fluctuations; we shall observe
that an unusual situation arises: the r.w. may converge a.s.
to infinity even with zero drift. Then, we obtain nonclassical
limiting distributions for the r.w. Proofs are based on the introduction
of suitable branching processes in order to count the steps
performed by the r.w.
alili@u-cergy.fr
1132. THE LIOUVILLE PROPERTY AND A CONJECTURE OF DE GIORGI
Martin Barlow, Richard Bass, and C. Gui
We consider bounded entire solutions of the non-linear PDE
$ \Delta u + u - u^3 = 0$ in $R^d$, and prove that under
certain monotonicity conditions these solutions must be
constant on hyperplanes. The proof uses a Liouville theorem
for harmonic functions associated with a non-uniformly
elliptic divergence form operator.
barlow@math.ubc.ca bass@math.uconn.edu gui@math.ubc.ca
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1133. INTRINSIC LOCATION PARAMETER OF A DIFFUSION PROCESS
R. W. R. Darling
For nonlinear functions f of a random vector Y, E[f(Y)] and f(E[Y]) usually
differ. Consequently the mathematical expectation of Y is not intrinsic: when
we change coordinate systems, it is not invariant.This article is about a
fundamental and hitherto neglected property of random vectors of the form Y =
f(X(t)), where X(t) is the value at time t of a diffusion process X: namely
that there exists a measure of location, called the "intrinsic location
parameter" (ILP), which coincides with mathematical expectation only in special
cases, and which is invariant under change of coordinate systems. The
construction uses martingales with respect to the intrinsic geometry of
diffusion processes, and the heat flow of harmonic mappings. We compute
formulas which could be useful to statisticians, engineers, and others who use
diffusion process models; these have immediate application, discussed in a
separate article, to the construction of an intrinsic nonlinear analog to the
Kalman Filter. We present here a numerical simulation of a nonlinear SDE,
showing how well the ILP formula tracks the mean of the SDE for a Euclidean
geometry.
rwrd@afterlife.ncsc.mil
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1134. GEOMETRICALLY INTRINSIC NONLINEAR RECURSIVE FILTERS II: FOUNDATIONS
R. W. R. Darling
This paper contains the technical foundations from stochastic differential
geometry for the construction of geometrically intrinsic nonlinear recursive
filters. A diffusion X on a manifold N is run for a time interval T, with a
random initial condition. There is a single observation consisting of a
nonlinear function of X(T), corrupted by noise, and with values in another
manifold M. The noise covariance of X and the observation covariance themselves
induce geometries on M and N, respectively. Using these geometries we compute
approximate but coordinate-free formulas for the "best estimate" of X(T), given
the observation, and its conditional variance. Calculations are based on use of
Jacobi fields and of "intrinsic location parameters", a notion derived from the
heat flow of harmonic mappings. When any nonlinearity is present, the resulting
formulas are not the same as those for the continuous-discrete Extended Kalman
Filter. A subsidiary result is a formula for computing approximately the
"exponential barycenter" of a random variable S on a manifold, i.e. a point z
such that the inverse image of S under the exponential map at z has mean zero
in the tangent space at z.
rwrd@afterlife.ncsc.mil
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1135. GEOMETRICALLY INTRINSIC NONLINEAR RECURSIVE FILTERS I: ALGORITHMS
R. W. R. Darling
The Geometrically Intrinsic Nonlinear Recursive Filter, or GI Filter, is
designed to estimate an arbitrary continuous-time Markov diffusion process X
subject to nonlinear discrete-time observations. The GI Filter is fundamentally
different from the much-used Extended Kalman Filter (EKF), and its second-order
variants, even in the simplest nonlinear case, in that: (i) It uses a quadratic
function of a vector observation to update the state, instead of the linear
function used by the EKF. (ii) It is based on deeper geometric principles,
which make the GI Filter coordinate-invariant. This implies, for example, that
if a linear system were subjected to a nonlinear transformation f of the
state-space and analyzed using the GI Filter, the resulting state estimates and
conditional variances would be the push-forward under f of the Kalman Filter
estimates for the untransformed system - a property which is not shared by the
EKF or its second-order variants.
The noise covariance of X and the observation covariance themselves induce
geometries on state space and observation space, respectively, and associated
canonical connections. A sequel to this paper develops stochastic differential
geometry results - based on "intrinsic location parameters", a notion derived
from the heat flow of harmonic mappings - from which we derive the
coordinate-free filter update formula. The present article presents the
algorithm with reference to a specific example - the problem of tracking and
intercepting a target, using sensors based on a moving missile. Computational
experiments show that, when the observation function is highly nonlinear, there
exist choices of the noise parameters at which the GI Filter significantly
outperforms the EKF.
rwrd@afterlife.ncsc.mil
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information provided by the author
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1136. SOME MINIMIZATION PROBLEMS FOR THE FREE ANALOGUE OF THE FISHER
INFORMATION
A. Nica, D. Shlyakhtenko, R. Speicher
We consider the free non-commutative analogue Phi^*, introduced by D.
Voiculescu, of the concept of Fisher information for random variables. We
determine the minimal possible value of Phi^*(a,a^*), if a is a non-commutative
random variable subject to the constraint that the distribution of aa^* is
prescribed. More generally, we obtain the minimal possible value of
Phi^*({a_{ij},a_{ij}^*), if {a_{ij}} is a family of non-commutative random
variables such that the distribution of AA^* is prescribed, where A is the
matrix (a_{ij}). The d*d-generalization is obtained from the case d=1 via a
result of independent interest, concerning the minimal value of
Phi^*({a_{ij},a_{ij}^*), when the matrix A=(a_{ij}) and its adjoint have a
given joint distribution. We then show how the minimization results obtained
for Phi^* lead to maximization results concerning the free entropy chi^*, also
defined by Voiculescu.
roland.speicher@urz.uni-heidelberg.de
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1137. MAXIMALITY OF THE MICROSTATES FREE ENTROPY FOR R-DIAGONAL ELEMENTS
A. Nica, D. Shlyakhtenko, R. Speicher
A non-commutative non-selfadjoint random variable z is called R-diagonal, if
its *-distribution is invariant under multiplication by free unitaries: if a
unitary w is *-free from z, then the *-distribution of z is the same as that of
wz. Using Voiculescu's microstates definition of free entropy, we show that the
R-diagonal elements are characterized as having the largest free entropy among
all variables y with a fixed distribution of y^*y. More generally, let Z be a
d*d matrix whose entries are non-commutative random variables X_{ij}. Then the
free entropy of the family {X_{ij}} of the entries of Z is maximal among all Z
with a fixed distribution of Z^*Z, if and only if Z is R-diagonal and is *-free
from the algebra of scalar d*d matrices. The results of this paper are
analogous to the results of our paper "Some minimization problems for the free
analogue of the Fisher information", where we considered the same problems in
the framework of the non-microstates definition of free entropy.
roland.speicher@urz.uni-heidelberg.de
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1138. PROBABILITY AROUND THE QUANTUM GRAVITY. PART 1: PURE PLANAR GRAVITY
V.A. Malyshev
In this paper we study stochastic dynamics which leaves quantum gravity
equilibrium distribution invariant. We start theoretical study of this dynamics
(earlier it was only used for Monte-Carlo simulation). Main new results concern
the existence and properties of local correlation functions in the
thermodynamic limit. The study of dynamics constitutes a third part of the
series of papers where more general class of processes were studied (but it is
self-contained), those processes have some universal significance in
probability and they cover most concrete processes, also they have many
examples in computer science and biology. At the same time the paper can serve
an introduction to quantum gravity for a probabilist: we give a rigorous
exposition of quantum gravity in the planar pure gravity case. Mostly we use
combinatorial techniques, instead of more popular in physics random matrix
models, the central point is the famous $\alpha =-7/2$ exponent.
vadim.malyshev@inria.fr
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information provided by the author
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1139. FINITE TIME EXTINCTION OF SUPER-BROWNIAN MOTIONS WITH CATALYSTS
Donald A. Dawson, Klaus Fleischmann, Carl Mueller
Consider a catalytic super-Brownian motion $X=X^\Gamma$ with finite variance
branching. Here `catalytic' means that branching of the reactant $X$ is only
possible in the presence of some catalyst. Our intrinsic example of a catalyst
is a stable random measure $\Gamma $ on $R$ of index $0< gamma <1$.
Consequently, here the catalyst is located in a countable dense subset of $R$.
Starting with a finite reactant mass $X_0$ supported by a compact set, $X$ is
shown to die in finite time. Our probabilistic argument uses the idea of good
and bad historical paths of reactant `particles' during time periods
$[T_{n},T_{n+1})$. Good paths have a significant collision local time with the
catalyst, and extinction can be shown by individual time change according to
the collision local time and a comparison with Feller's branching diffusion. On
the other hand, the remaining bad paths are shown to have a small expected mass
at time $T_{n+1}$ which can be controlled by the hitting probability of point
catalysts and the collision local time spent on them.
fleischm@wias-berlin.de
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1140. SCALING LIMITS FOR MINIMAL AND RANDOM SPANNING TREES IN TWO DIMENSIONS
Michael Aizenman, Almut Burchard, Charles M. Newman, David B. Wilson
Continuum scaling limits are constructed for a number of stochastic spanning
tree models in two dimensions: the uniformly random spanning tree on $\Z^2$,
the minimal spanning tree on $\Z^2$ (with random edge lengths), and the
Euclidean minimal spanning tree on a set of Poisson points in $\R^2$. The
configurations of the (continuum) spanning tree are described by the collection
of all the finite subtrees, which includes a spanning tree for every finite
collection of points in the plane. The following properties are established for
typical configurations in the scaling limit. The branches of the trees are
given by curves which are regular, in the sense that they admit H\"older
continuous parametrizations, but also rough, as their Hausdorff dimension
exceeds one. Except for endpoints in a non-empty set of Hausdorff dimension
between one and two, the connecting trees are unique and depend continuously on
the endpoints. The spanning tree has a single route to infinity. Typical
configurations exhibit branching at countably many points, where the number of
emanating branches has a non-random finite upper bound. The construction yields
also a scaling limit for the loop erased random walk (LERW) process in two
dimensions. The proofs proceed through the derivation of scale-invariant power
bounds on the probabilities of repeated crossings of annuli.
aizenman@princeton.edu
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1141. MARTIN BOUNDARY AND INTEGRAL REPRESENTATION FOR HARMONIC FUNCTIONS OF
SYMMETRIC STABLE PROCESSES
Zhen-Qing Chen and Renming Song
Martin boundaries and integral representations of positive functions which
are harmonic in a bounded domain $D$ with respect to Brownian motion are well
understood. Unlike the Brownian case, there are two different kinds of
harmonicity with respect to a discontinuous symmetric stable process. One kind
are functions harmonic in $D$ with respect to the whole process $X$, and the
other are functions harmonic in $D$ with respect to the process $X^D$ killed
upon leaving $D$. In this paper we show that for bounded Lipschitz domains, the
Martin boundary with respect to the killed stable process $X^D$ can be
identified with the Euclidean boundary. We further give integral
representations for both kinds of positive harmonic functions. Also given is
the conditional gauge theorem conditioned according to Martin kernels and the
limiting behaviors of the $h$-conditional stable process, where $h$ is a
positive harmonic function of $X^D$. In the case when $D$ is a bounded $C^{1,
1}$ domain, sharp estimate on the Martin kernel of $D$ is obtained.
zchen@math.cornell.edu
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1142. INTRINSIC ULTRACONTRACTIVITY, CONDITIONAL LIFETIMES AND CONDITIONAL
GAUGE FOR SYMMETRIC STABLE PROCESSES ON ROUGH DOMAINS
Zhen-Qing Chen and Renming Song
For a symmetric $\alpha$-stable process $X$ on $\RR^n$ with $0<\alpha <2$,
$n\geq 2$ and a domain $D \subset \RR^n$, let $L^D$ be the infinitesimal
generator of the subprocess of $X$ killed upon leaving $D$. For a Kato class
function $q$, it is shown that $L^D+q$ is intrinsic ultracontractive on a
H\"older domain $D$ of order 0. This is then used to establish the conditional
gauge theorem for $X$ on bounded Lipschitz domains in $\RR^n$. It is also shown
that the conditional lifetimes for symmetric stable process in a H\"older
domain of order 0 are uniformly bounded.
zchen@math.cornell.edu
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information provided by the author
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1143. EQUILIBRIUM PURE STATES AND NONEQUILIBRIUM CHAOS
C.M. Newman, D.L. Stein
We consider nonequilibrium systems such as the Edwards-Anderson Ising spin
glass at a temperature where, in equilibrium, there are presumed to be (two or
many) broken symmetry pure states. Following a deep quench, we argue that as
time goes to infinity, although the system is usually in some pure state
locally, either it never settles permanently on a fixed lengthscale into a
single pure state, or it does but then the pure state depends on both the
initial spin configuration and the realization of the stochastic dynamics. But
this latter case can occur only if there exists an uncountable number of pure
states (for each coupling realization) with almost every pair having zero
overlap. In both cases, almost no initial spin configuration is in the basin of
attraction of a single pure state; that is, the configuration space (resulting
from a deep quench) is all boundary (except for a set of measure zero). We
prove that the former case holds for deeply quenched two-dimensional
ferromagnets. Our results raise the possibility that even if more than one pure
state exists for an infinite system, time averages don't necessarily disagree
with Boltzmann averages.
dls@cosmo.physics.arizona.edu
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1144. FREE PROBABILITY FOR PROBABILISTS
Philippe Biane
This is an introduction to some of the most probabilistic aspects of free
probability theory.
Philippe.Biane@ens.fr
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1145. SOME FUNCTION SPACES RELATED TO THE BROWNIAN MOTION ON SIMPLE NESTED
FRACTALS
Katarzyna Pietruska-Paluba
In this paper we identify the domain of the Dirichlet form associated with
the Brownian motion on simple nested fractals with an integral Lipschitz space.
This result generalizes such an identification on the Sierpi\'nski gasket,
carried on by Jonsson.
kpp@mimuw.edu.pl
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information provided by the author
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1146. POINT PROCESSES AND THE INFINITE SYMMETRIC GROUP. PART IV: MATRIX
WHITTAKER KERNEL
Alexei Borodin
We study a 2-parametric family of probability measures on the space of
countable point configurations on the punctured real line (the points of the
random configuration are concentrated near zero). These measures (or,
equivalently, point processes) have been introduced in Part II (A. Borodin,
math/9804087) in connection with the problem of harmonic analysis on the
infinite symmetric group. The main result of the present paper is a
determinantal formula for the correlation functions.
The formula involves a kernel called the matrix Whittaker kernel. Each of its
two diagonal blocks governs the projection of the process on one of the two
half-lines; the corresponding kernel on the half-line was studied in Part III
(A. Borodin and G. Olshanski, math/9804088).
While the diagonal blocks of the matrix Whitaker kernel are symmetric, the
whole kernel turns out to be $J$-symmetric, i.e., symmetric with respect to a
natural indefinite inner product.
We also discuss a rather surprising connection of our processes with the
recent work by B. Eynard and M. L. Mehta (cond-mat/9710230) on correlations of
eigenvalues of coupled random matrices.
borodine@math.upenn.edu
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1147. POINT PROCESSES AND THE INFINITE SYMMETRIC GROUP. PART V: ANALYSIS OF
THE MATRIX WHITTAKER KERNEL
Grigori Olshanski
The matrix Whittaker kernel has been introduced by A. Borodin in Part IV of
the present series of papers. This kernel describes a point process -- a
probability measure on a space of countable point configurations. The kernel is
expressed in terms of the Whittaker confluent hypergeometric functions. It
depends on two parameters and determines a $J$-symmetric operator $K$ in
$L^2(R_+)\oplus L^2(R_+)$.
It turns out that the operator $K$ can be represented in the form
$L(1+L)^{-1}$, where $L$ is a rather simple integral operator: the kernel of
$L$ is expressed in terms of elementary functions only. This is our main
result; it elucidates the nature of the matrix Whittaker kernel and makes it
possible to directly verify the existence of the associated point process.
Next, we show that the matrix Whittaker kernel can be degenerated to a family
of kernels expressed through the Bessel and Macdonald functions. In this way
one can obtain both the well-known Bessel kernel (which arises in random matrix
theory) and certain interesting new kernels.
borodine@math.upenn.edu
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1148. POINT PROCESSES AND THE INFINITE SYMMETRIC GROUP. PART VI: SUMMARY OF
RESULTS
Alexei Borodin and Grigori Olshanski
We give a summary of the results from Parts I-V (math/9804086, math/9804087,
math/9804088, math/9810013, math/9810014).
Our work originated from harmonic analysis on the infinite symmetric group.
The problem of spectral decomposition for certain representations of this group
leads to a family of probability measures on an infinite-dimensional simplex,
which is a kind of dual object for the infinite symmetric group. To understand
the nature of these measures we interpret them as stochastic point processes on
the punctured real line and compute their correlation functions. The
correlation functions are given by multidimensional integrals which can be
expressed in terms of a multivariate hypergeometric series (the Lauricella
function of type B). It turns out that after a slight modification (`lifting')
of the processes the correlation functions take a common in Random Matrix
Theory (RMT) determinantal form with a certain kernel.
The kernel is expressed through the classical Whittaker functions. It depends
on two parameters and admits a variety of degenerations. They include the
well-known in RMT sine and Bessel kernels as well as some other Bessel-type
kernels which, to our best knowledge, are new.
The explicit knowledge of the correlation functions enables us to derive a
number of conclusions about the initial probability measures.
We also study the structure of our kernel; this finally leads to a
constructive description of the initial measures.
We believe that this work provides a new promising connection between RMT and
Representation Theory.
borodine@math.upenn.edu
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information provided by the author
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1149. THE LINEARIZATION OF THE CENTRAL LIMIT OPERATOR IN FREE PROBABILITY
THEORY
Michael Anshelevich
We interpret the Central Limit Theorem as a fixed point theorem for a certain
operator, and consider the problem of linearizing this operator. In classical
as well as in free probability theory, we consider two methods giving such a
linearization, and interpret the result as a weak form of the CLT. In the
classical case the analysis involves dilation operators; in the free case more
general composition operators appear.
mashel@math.berkeley.edu
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1150. ON THE THERMODYNAMIC LIMIT FOR A ONE-DIMENSIONAL SANDPILE PROCESS
C. Maes, F. Redig, E. Saada, A. Van Moffaert
Considering the standard abelian sandpile model in one dimension, we
construct an infinite volume Markov process corresponding to its thermodynamic
(infinite volume) limit. The main difficulty we overcome is the strong
non-locality of the dynamics. However, using similar ideas as in recent
extensions of the standard Gibbs formalism for lattice spin systems, we can
identify a set of `good' configurations on which the dynamics is effectively
local. We prove that every configuration converges in a finite time to the
unique invariant measure.
annelies.vanmoffaert@fys.kuleuven.ac.be
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1151. THE RESTRICTION OF THE ISING MODEL TO A LAYER
C. Maes, F. Redig, A. Van Moffaert
We discuss the status of recent Gibbsian descriptions of the restriction
(projection) of the Ising phases to a layer. We concentrate on the projection
of the two-dimensional low temperature Ising phases for which we prove a
variational principle.
annelies.vanmoffaert@fys.kuleuven.ac.be
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1152. A GENERAL HSU-ROBBINS-ERDOS TYPE ESTIMATE OF TAIL PROBABILITIES OF SUMS
OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLES
Alexander R. Pruss
Let X_1,X_2,... be a sequence of independent and identically distributed
random variables, and put S_n=X_1+...+X_n. Under some conditions on the
positive sequence tau_n and the positive increasing sequence a_n, we give
necessary and sufficient conditions for the convergence of sum_{n=1}^infty
tau_n P(|S_n|>t a_n) for all t>0, generalizing Baum and Katz's (1965)
generalization of the Hsu-Robbins-Erdos (1947, 1949) law of large numbers, also
allowing us to characterize the convergence of the above series in the case
where tau_n=1/n and a_n=(n log n)^{1/2} for n>1, thereby answering a question
of Spataru. Moreover, some results for non-identically distributed independent
random variables are obtained by a recent comparison inequality. Our basic
method is to use a central limit theorem estimate of Nagaev (1965) combined
with the Hoffman-Jorgensen inequality(1974).
pruss+@pitt.edu
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information provided by the author
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