Probability Abstracts 65

1799. ASYMPTOTIC ACCURACY OF THE JACKKNIFE VARIANCE ESTIMATOR FOR CERTAIN SMOOTH STATISTICS

Alex D Gottlieb

We show that that the jackknife variance estimator $v_{jack}$ and the the
infinitesimal jackknife variance estimator are asymptotically equivalent if the
functional of interest is a smooth function of the mean or a smooth trimmed
L-statistic. We calculate the asymptotic variance of $v_{jack}$ for these
functionals.

gottlieb@math.berkeley.edu

• This article is available from the xxx mathematics archive as math.PR/0109002.

1800. VERTEX IDENTIFIABILITY IN LARGE RANDOM HYPERGRAPHS

R.W.R. Darling and J.R. Norris

The theme of this paper is the derivation of analytic formulae for the size
of combinatorial structures in the limit as the number of vertices tends to
infinity. Such formulae are obtained via fluid limits of pure jump type Markov
processes, when the Laplace transforms of their Levy kernels converge.
Furthermore we describe the remaining stochasticity when parameters take
critical values. Our method is quite general, but is applied here to vertex
identifiability in random hypergraphs. A vertex v is identifiable in n steps if
there is a hyperedge containing v all of whose other vertices are identifiable
in fewer steps. We say that a hyperedge is solvable if every one of its
vertices is identifiable. Our analytic formulae describe the asymptotics of the
number of identifiable vertices and the number of solvable hyperedges for a
large random hypergraph.

j.r.norris@statslab.cam.ac.uk

• This article is available from the xxx mathematics archive as math.PR/0109020.

1801. ANISOTROPIC CONTACT PROCESS ON HOMOGENEOUS TREES

Irene Hueter

The existence of a weak survival region is established for the anisotropic
symmetric contact process on a homogeneous tree T_{2d} of degree 2d > 2: For
parameter values in a certain connected region of positive Lebesgue measure,
the population survives forever with positive probability but ultimately
vacates every finite subset of the tree with probability one. In this phase,
infection trails must converge to the geometric boundary \Omega of the tree.
The random subset \Lambda of the boundary consisting of all ends of the tree in
which the infection survives, called the limit set of the process, is shown to
have Hausdorff dimension no larger than one half the Hausdorff dimension of the
entire geometric boundary. In addition, there is strict inequality at the
transition between weak and strong survival except when the contact process is
isotropic. It is further shown that in all cases there is a distinguished
probability measure \mu, supported by \Omega, such that the Hausdorff dimension
of \Lambda \cap \Omega_{\mu}, where \Omega_{\mu} is the set of \mu-generic
points of \Omega, converges to one half the Hausdorff dimension of \Omega_{\mu}
at the phase separation points. Exact formulae for the Hausdorff dimensions of
\Lambda and \Lambda \cap \Omega_{\mu} are obtained. We also prove that the
contact process at the transition between extinction and weak survival does not
survive. The method developed shows that the contact process at the phase
transition to strong survival survives weakly for d > 1.

hueter@math.ufl.edu

• This article is available from the xxx mathematics archive as math.PR/0109047.

1802. RIGIDITY OF THE INTERFACE FOR PERCOLATION AND RANDOM-CLUSTER MODELS

Guy Gielis and Geoffrey Grimmett

We study conditioned random-cluster measures with edge-parameter p and
cluster-weighting factor q satisfying q \ge 1. The conditioning corresponds to
mixed boundary conditions for a spin model. Interfaces may be defined in the
sense of Dobrushin, and these are proved to be `rigid' in the thermodynamic
limit, in three dimensions and for sufficiently large values of p. This implies
the existence of non-translation-invariant (conditioned) random-cluster
measures in three dimensions. The results are valid in the special case q=1,
thus indicating a property of three-dimensional percolation not previously
noted.

g.r.grimmett@statslab.cam.ac.uk

• This article is available from the xxx mathematics archive as math.PR/0109103.

1803. EXPONENTIAL MIXING PROPERTIES OF STOCHASTIC PDES THROUGH ASYMPTOTIC COUPLING

Martin Hairer

We consider parabolic stochastic partial differential equations driven by
white noise in time. We prove exponential convergence of the transition
probabilities towards a unique invariant measure under suitable conditions.
These conditions amount essentially to the fact that the equation transmits the
noise to all its determining modes. Several examples are investigated,
including some where the noise does not act on every determining mode directly.

martin.hairer@math.unige.ch

• This article is available from the xxx mathematics archive as math.PR/0109115.

1804. CRITICAL EXPONENTS FOR TWO-DIMENSIONAL PERCOLATION

Stanislav Smirnov, Wendelin Werner

We show how to combine Kesten's scaling relations, the determination of
critical exponents associated to the stochastic Loewner evolution process by
Lawler, Schramm, and Werner, and Smirnov's proof of Cardy's formula, in order
to determine the existence and value of critical exponents associated to
percolation on the triangular lattice.

wendelin.werner@math.u-psud.fr

• This article is available from the xxx mathematics archive as math.PR/0109120.

1805. CLAIRVOYANT SCHEDULING OF RANDOM WALKS

Peter Gacs

Two infinite walks on the same finite graph are called compatible if it is
possible to introduce delays into them in such a way that they never collide.
About 10 years ago, Peter Winkler asked the question: for which graphs are two
independent walks compatible with positive probability. Up to now, no such
graphs were found. We show in this paper that large complete graphs have this
property. The question is equivalent to a certain dependent percolation with a
power-law behavior: the probability that the origin is blocked at distance n
but not closer decreases only polynomially fast and not, as usual,
exponentially.

gacs@cs.bu.edu

• This article is available from the xxx mathematics archive as math.PR/0109152.

1806. AN INFORMATION-THEORETIC CENTRAL LIMIT THEOREM FOR FINITELY SUSCEPTIBLE FKG SYSTEMS

Oliver Johnson

We adapt arguments concerning entropy-theoretic convergence from the
independent case to the case of FKG random variables. FKG systems are chosen
since their dependence structure is controlled through covariance alone, though
in the sequel we use many of the same arguments for weakly dependent random
variables. As in previous work of Barron and Johnson, we consider random
variables perturbed by small normals, since the FKG property gives us control
of the resulting densities. We need to impose a finite susceptibility condition
-- that is, the covariance between one random variable and the sum of all the
random variables should remain finite.

o.johnson@statslab.cam.ac.uk

• This article is available from the xxx mathematics archive as math.PR/0109156.

1807. REGULARITY OF QUASI-STATIONARY MEASURES FOR SIMPLE EXCLUSION IN DIMENSION D >= 5

Amine Asselah, Pablo A. Ferrari

We consider the symmetric simple exclusion process on Z^d, for d>= 5, and
study the regularity of the quasi-stationary measures of the dynamics
conditionned on not occupying the origin. For each \rho\in ]0,1[, we establish
uniqueness of the density of quasi-stationary measures in L^2(d\nur), where
\nur is the stationary measure of density \rho. This, in turn, permits us to
obtain sharp estimates for P_{\nur}(\tau>t), where \tau is the first time the
origin is occupied.

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/0109189.

1808. SEGAL-BARGMANN TRANSFORMS OF ONE-MODE INTERACTING FOCK SPACES ASSOCIATED WITH GAUSSIAN AND POISSON MEASURES

Nobuhiro Asai, Izumi Kubo and Hui-Hsiung Kuo

Let $\mu_{g}$ and $\mu_{p}$ denote the Gaussian and Poisson measures on
${\Bbb R}$, respectively. We show that there exists a unique measure
$\widetilde{\mu}_{g}$ on ${\Bbb C}$ such that under the Segal-Bargmann
transform $S_{\mu_g}$ the space $L^2({\Bbb R},\mu_g)$ is isomorphic to the
space ${\cal H}L^2({\Bbb C}, \widetilde{\mu}_{g})$ of analytic $L^2$-functions
on ${\Bbb C}$ with respect to $\widetilde{\mu}_{g}$. We also introduce the
Segal-Bargmann transform $S_{\mu_p}$ for the Poisson measure $\mu_{p}$ and
prove the corresponding result. As a consequence, when $\mu_{g}$ and $\mu_{p}$
have the same variance, $L^2({\Bbb R},\mu_g)$ and $L^2({\Bbb R},\mu_p)$ are
isomorphic to the same space ${\cal H}L^2({\Bbb C}, \widetilde{\mu}_{g})$ under
the $S_{\mu_g}$ and $S_{\mu_p}$-transforms, respectively. However, we show that
the multiplication operators by $x$ on $L^2({\Bbb R}, \mu_g)$ and on $L^2({\Bbb
R}, \mu_p)$ act quite differently on ${\cal H}L^2({\Bbb C},
\widetilde{\mu}_{g})$.

asai@iias.or.jp

• This article is available from the xxx mathematics archive as math.PR/0110011.

1809. THE ASYMMETRIC ONE-DIMENSIONAL CONSTRAINED ISING MODEL

David Aldous and Persi Diaconis

We study a reversible one-dimensional spin system with Bernoulli(p)
stationary distribution, in which a site can flip only if the site to its left
is in state +1. Such models have been used as simple exemplars of systems
exhibiting slow relaxation. We give fairly sharp estimates of the spectral gap
as p decreases to zero. The method uses Poincare comparison with a long-range
process which is analyzed by probabilistic methods (coupling,
supermartingales).

aldous@stat.berkeley.edu

• This article is available from the xxx mathematics archive as math.PR/0110023.

1810. RANDOMNESS

Paul M.B. Vitanyi

Here we present in a single essay a combination and completion of the several
aspects of the problem of randomness of individual objects which of necessity
occur scattered in our texbook "An Introduction to Kolmogorov Complexity and
Its Applications" (M. Li and P. Vitanyi), 2nd Ed., Springer-Verlag, 1997.

paul.vitanyi@cwi.nl

• This article is available from the xxx mathematics archive as math.PR/0110086.

1811. THE GENERALIZED SPIKE PROCESS, SPARSITY, AND STATISTICAL INDEPENDENCE

Naoki Saito

A basis under which a given set of realizations of a stochastic process can
be represented most sparsely (the so-called best sparsifying basis (BSB)) and
the one under which such a set becomes as less statistically dependent as
possible (the so-called least statistically-dependent basis (LSDB)) are
important for data compression and have generated interests among computational
neuroscientists as well as applied mathematicians. Here we consider these bases
for a particularly simple stochastic process called ``generalized spike
process'', which puts a single spike--whose amplitude is sampled from the
standard normal distribution--at a random location in the zero vector of length
$\ndim$ for each realization.
  Unlike the ``simple spike process'' which we dealt with in our previous paper
and whose amplitude is constant, we need to consider the kurtosis-maximizing
basis (KMB) instead of the LSDB due to the difficulty of evaluating
differential entropy and mutual information of the generalized spike process.
By computing the marginal densities and moments, we prove that: 1) the BSB and
the KMB selects the standard basis if we restrict our basis search within all
possible orthonormal bases in ${\mathbb R}^n$; 2) if we extend our basis search
to all possible volume-preserving invertible linear transformations, then the
BSB exists and is again the standard basis whereas the KMB does not exist.
Thus, the KMB is rather sensitive to the orthonormality of the transformations
under consideration whereas the BSB is insensitive to that. Our results once
again support the preference of the BSB over the LSDB/KMB for data compression
applications as our previous work did.

saito@math.ucdavis.edu

• This article is available from the xxx mathematics archive as math.PR/0110103.

1812. 2D MODELS OF STATISTICAL PHYSICS WITH CONTINUOUS SYMMETRY: THE CASE OF SINGULAR INTERACTIONS

Dima Ioffe, Senya Shlosman and Yvan Velenik

We show the absence of continuous symmetry breaking in 2D lattice systems
without any smoothness assumptions on the interaction. We treat certain cases
of interactions with integrable singularities. We also present cases of
singular interactions with continuous symmetry, when the symmetry is broken in
the thermodynamic limit.

velenik@cmi.univ-mrs.fr

• This article is available from the xxx mathematics archive as math.PR/0110127.

1813. THE KRUSKAL COUNT

Jeffrey C. Lagarias, Eric Rains and Robert J. Vanderbei

The Kruskal Count is a card trick invented by Martin J. Kruskal in which a
magician "guesses" a card selected by a subject according to a certain counting
procedure. With high probability the magician can correctly "guess" the card.
The success of the trick is based on a mathematical principle related to
coupling methods for Markov chains. This paper analyzes in detail two
simplified variants of the trick and estimates the probability of success. The
model predictions are compared with simulation data for several variants of the
actual trick.

jcl@research.att.com

• This article is available from the xxx mathematics archive as math.PR/0110143.

1814. STRONG AND WEAK MEAN VALUE PROPERTIES ON TREES

Fabio Zucca

We consider the mean value properties for finite variation measures with
respect to a Markov operator in a discrete environnement. We prove equivalent
conditions for the weak mean value property in the case of general Markov
operators and for the strong mean value property in the case of transient
Markov operators adapted to a tree structure. In this last case, conditions for
the equivalence between weak and strong mean value properties are given.

fabio.zucca@math.u-cergy.fr

• This article is available from the xxx mathematics archive as math.PR/0110145.

1815. TANGENT PROCESSES ON WIENER SPACE

Y. Hu, A.S. Ustunel, M. Zakai

This paper deals with the study of the Malliavin calculus of Euclidean
motions on Wiener space, (i.e. transformations induced by general measure
preserving transformations, called `rotations', and H-valued shifts) and the
associated flows on abstract Wiener spaces.

zakai@ee.technion.ac.il

• This article is available from the xxx mathematics archive as math.PR/0110161.

1816. SOME ERGODIC THEOREMS FOR RANDOM ROTATIONS ON WIENER SPACE

A.S. Ustunel, M. Zakai

In this paper we study ergodicity and mixing property of some measure
preserving transformations on the Wiener space (W,H,\mu) which are generated by
some random unitary operators defined on the Cameron-Martin space H.

zakai@ee.technion.ac.il

• This article is available from the xxx mathematics archive as math.PR/0110162.

1817. FINITE VOLUME APPROXIMATION OF THE EFFECTIVE DIFFUSION MATRIX: THE CASE OF INDEPENDENT BOND DISORDER

Pietro Caputo and Dima Ioffe

Consider uniformly elliptic random walk on $\bbZ^d$ with independent jump
rates across nearest neighbour bonds of the lattice. We show that the infinite
volume effective diffusion matrix can be almost surely recovered as the limit
of finite volume periodized effective diffusion matrices.

caputo@mat.uniroma3.it

• This article is available from the xxx mathematics archive as math.PR/0110215.

1818. THE SHAPE THEOREM FOR THE FROG MODEL WITH RANDOM INITIAL CONFIGURATION

O.S.M.Alves, F.P.Machado, S.Yu.Popov and K.Ravishankar

We prove a shape theorem for a growing set of simple random walks on Z^d,
known as frog model. The dynamics of this process is described as follows:
There are active particles, which perform independent discrete time SRWs, and
sleeping particles, which do not move. When a sleeping particle is hit by an
active particle, the former becomes active as well. Initially, a random number
of particles is placed into each site. At time 0 all particles are sleeping,
except for those placed at the origin. We prove that the set of all sites
visited by active particles, rescaled by the elapsed time, converges to a
compact convex set.

fmachado@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/0110280.

1819. TRANSIENCE, RECURRENCE AND CRITICAL BEHAVIOR FOR LONG-RANGE PERCOLATION

Noam Berger

We study the behavior of the random walk on the infinite cluster of
independent long range percolation in dimensions $d=1,2$, where $x$ and $y$ a
re connected with probability $\sim\beta/\|x-y\|^{-s}$. We show that when
$d<s<2d$ the walk is transient, and when $s\geq 2d$, the walk is recurrent. The
proof of transience is based on a renormalization argument. As a corollary of
this renormalization argument, we get that for every dimension $d$, if
$d<s<2d$, then critical percolation has no infinite clusters. This result is
extended to the free random cluster model. A second corollary is that when
$d\geq 2$ and $d<s<2d$ we can erase all long enough bonds and still have an
infinite cluster. The proof of recurrence in two dimensions is based on general
stability results for recurrence in random electrical networks. In particular,
we show that i.i.d. conductances on a recurrent graph of bounded degree yield a
recurrent electrical network.

noam@stat.berkeley.edu

• This article is available from the xxx mathematics archive as math.PR/0110296.

1820. THE EFFECT OF ADDITIVE NOISE ON DYNAMICAL HYSTERESIS

Nils Berglund and Barbara Gentz

We investigate the properties of hysteresis cycles produced
by a one-dimensional, periodically forced Langevin equation.
We show that depending on amplitude and frequency of the 
forcing and on noise intensity, there are three 
qualitatively different types of hysteresis cycles. Below a
critical noise intensity, the random area enclosed by 
hysteresis cycles is concentrated near the deterministic 
area, which is different for small and large driving 
amplitude. Above this threshold, the area of typical
hysteresis cycles depends, to leading order, only on the 
noise intensity. In all three regimes, we derive 
mathematically rigorous estimates for expectation, variance, 
and the probability of deviations of the hysteresis area 
from its typical value.  

berglund@math.ethz.ch gentz@wias-berlin.de 

• To see a preprint or other information provided by the author click here.
• Or here.
• Or here.

1821. UNIVERSAL BOUNDS ON THE SELFAVERAGING OF RANDOM DIFFRACTION MEASURES

Christof Kuelske

We consider diffraction at random point scatterers on 
general discrete point sets in $\R^\nu$, restricted to a 
finite volume. We allow for random amplitudes and random 
dislocations of the scatterers. 
We investigate the speed of convergence of the random 
scattering measures applied to an observable towards its 
mean, when the finite volume tends to infinity. 
We give an explicit universal large deviation upper bound 
that is exponential in the number of scatterers. The rate 
is given in terms of a universal function that depends 
on the point set only through the minimal distance between 
points, and on the observable only through a suitable 
Sobolev-norm. 
Our proof uses a cluster expansion and also provides a 
central limit theorem.

kuelske@wias-berlin.de

• To see a preprint or other information provided by the author click here.

1822. ASYMPTOTICS OF REGRESSIONS WITH STATIONARY AND NONSTATIONARY RESIDUALS

 R. A. Maller

A comprehensive description of the possible limiting behaviour of 
normalised pseudo-MLEs of the 
coefficients in a discrete-time autoregressive process 
with non-stochastic regressors
 for all cases: stationary, unit root and explosive situations,
is given.
Only a finite second moment is required, and we allow a wide class of 
nonstochastic regressors which need only be 
uniformly asymptotically negligible and not too `regular', in a certain sense.
A new version of the  functional central limit theorem is 
developed to handle the unit root case in the generality we need, and 
for the explosive case we apply results 
of Kwapie\'n and Woyczy\'nski on the convergence of random series.
Under our assumptions, the limiting distribution of the normalised estimator of
the  regression coefficient is shown to be normal,
regardless of the value of  the autocorrelation coefficient,
and the normalised estimator of the autocorrelation coefficient 
always has a limiting 
distribution 
and is asymptotically independent of the  regression coefficient estimator.
Another major outcome is that the normalisation for the estimators 
can always be based on the sample information matrix.

rmaller@ecel.uwa.edu.au

• To see a preprint or other information provided by the author click here.

1823. ASYMPTOTICS FOR THE SPECTRAL AND WALK DIMENSION AS FRACTALS APPROACH EUCLIDEAN SPACE

Ben M. Hambly and Takashi Kumagai

We discuss the behaviour of the dynamic dimension exponents for
families of fractals based on the Sierpinski gasket and carpet. As the
length scale factor for the family tends to infinity the lattice
approximations to the fractals look more like the tetrahedral or cubic
lattice in Euclidean space and the fractal dimension converges to that
of the embedding space. However, in the Sierpinski gasket case, the
spectral dimension converges to two for all dimensions. In two
dimensions we prove a conjecture made in the physics literature
concerning the rate of convergence.  On the other hand, for natural
families of Sierpinski carpets, the spectral dimension converges to
the dimension of the embedding Euclidean space. In general we
demonstrate that for both cases of finitely and infinitely ramified
fractals, a variety of asymptotic values for the spectral dimension
can be achieved.

hambly@maths.ox.ac.uk  kumagai@kurims.kyoto-u.ac.jp

• To see a preprint or other information provided by the author click here.

1824. BROWNIAN-TIME PROCESSES: THE PDE CONNECTION II AND THE CORRESPONDING FEYNMAN-KAC FORMULA

Hassan Allouba

We delve deeper into our study of the connection of 
Brownian-time processes (BTPs) to fourth order parabolic PDEs, which we 
introduced in a recent joint article with W.~Zheng.   
Probabilistically, BTPs and their cousins BTPs with excursions form a 
unifying class of interesting stochastic processes that includes 
the celebrated IBM of Burdzy and other new intriguing processes, 
and is also connected to the Markov snake of Le Gall.   
BTPs also offer a new connection of probability to PDEs that is 
fundamentally different from the  Markovian one. They solve 
fourth order PDEs in which the initial function plays an important 
role in the PDE itself, not only as initial data. We connect  
two such types of interesting  and new PDEs to BTPs.  
The first is obtained by running the BTP and then integrating 
along its path, and the second type of PDEs is related 
to what we call the Feynman-Kac formula for BTPs.  A special 
case of the second type is a step towards a probabilistic 
solution to initially perturbed linearized Cahn-Hilliard and 
Kuramoto-Sivashinsky type PDEs, which we tackle in an upcoming paper.

allouba@indiana.edu

• To see a preprint or other information provided by the author click here.

1825. MARKET PRICE OF RISK AND RANDOM FIELD DRIVEN MODELS OF TERM STRUCTURE: A SPACE-TIME CHANGE OF MEASURE LOOK

Hassan Allouba and Victor Goodman

No-arbitrage models of term structure have the feature that 
the return on zero-coupon bonds is the sum of the short rate 
and the product of volatility and market price of risk.  
Well known models restrict the behavior of the market price of risk 
so that it is not dependent on the type of asset being modeled.  
We show that the models recently proposed by Goldstein and 
Santa-Clara and Sornette, among others, allow the market price 
of risk to depend on characteristics of each asset, and we
quantify this dependence. A key tool in our analysis is a very 
general space-time change of measure theorem, proved by the 
first author in earlier work, and covers continuous orthogonal 
local martingale measures including space-time white noise.

allouba@indiana.edu    goodmanv@indiana.edu

• To see a preprint or other information provided by the author click here.

1826. GALUBER DYNAMICS OF THE RANDOM ENERGY MODEL 1. METASTABLE MOTION ON THE EXTREME STATES

Gerard ben Arous, Anton Bovier, and Veronique Gayrard

We investigate the long-time behavior of the Glauber dynamics for the
random energy model below the critical temperature. 
We give very precise estimates on the motion of the process  to and between 
the states of extremal energies.  We show that when disregarding time, the
consecutive steps of the process on these states are governed by a Markov 
chain 
that jumps uniformly on all possible states. The mean times of these jumps 
are also computed very precisely and are seen to be asymptotically independent
of the terminal point. A first indicator of aging is the observation that
the mean time of arrival in the set of states that have waiting times of order
$T $ is itself of order $T$. The estimates proven in this paper will furnish
crucial input for a follow-up paper where aging is analysed in full detail.

Gerard.Benarous@epfl.ch, bovier@wias-berlin.de, Veronique.Gayrard@epfl.ch

• To see a preprint or other information provided by the author click here.

1827. GALUBER DYNAMICS OF THE RANDOM ENERGY MODEL 2. AGING BELOW THE CRITICAL TEMPERATURE

Gerard Ben Arous, Anton Bovier, Veronique Gayrard

We investigate the long-time behavior of the Glauber dynamics for the
random energy model below the critical temperature. We establish that
for a suitably chosen timescale that diverges with the size of the system, 
one can prove that a natural  autocorrelation function
exhibits aging. Moreover, we show that the
long-time  asymptotics of this function coincide with those of the so-called 
``REM-like trap model'' proposed by Bouchaud and Dean. Our results rely on 
very precise estimates on the distribution of transition times of the process
between different states of extremely low energy.

Gerard.Benarous@epfl.ch, bovier@wias-berlin.de, Veronique.Gayrard@epfl.ch

• To see a preprint or other information provided by the author click here.

1828. SOME REMARKS FOR STABLE-LIKE JUMP PROCESSES ON FRACTALS

Takashi Kumagai

We summarize recent work on non-local Dirichlet forms on fractals 
whose corresponding processes are stable-like jump processes. 
Especially, we introduce three natural non-local Dirichlet forms 
on d-sets and prove that these forms are equivalent. 

kumagai@kurims.kyoto-u.ac.jp

• To see a preprint or other information provided by the author click here.

1829. CONVERGENCE TIME TO EQUILIBRIUM FOR NONHOMOGENEOUS STOCHASTICALLY MONOTONE MARKOV CHAINS

Francois Simonot and Anatoli Manita

Let $P$ be the transition probability matrix of some
stochastically monotone Markov chain defined on $Z _{+}$,
we consider the minimal convergence time to equilibrium
corresponding to increasing truncations $P_{N}$,
up to state $N$, of $P$. For a space inhomogeneous random walk
with state dependent jumps $A^{i}$ having nonpositive means 
we prove that the minimal convergence time to equilibrium 
can take any value from $N$ to $N^{2}$ according to the behaviour 
of the sequence $(E A^{i})_{i\geq 0}$. As particular cases, 
we derive results for two emblematic classes of Markov chains:
for a homogeneous random walk and for a nonhomogeneous (in space)
discrete time birth-death process with asymptotically vanishing 
nonpositive drift.

francois.simonot@esstin.uhp-nancy.fr   manita@mech.math.msu.su

• To see a preprint or other information provided by the author click here.
• Or here.

1830. A NOTE ON THE TWO-SIDED REGULATED RANDOM WALK

Anatoli Manita and Francois Simonot

In this paper we deal with the two-sided regulated random walk 
defined by the relation $X_N(t+1)=\min (N,\max(0,X_N(t)+A(t+1)))$
when $A\geq -1$, $E(A)=0$ and $E(r^A)<+\infty$ for an $r>1$.
Let $\pi_N$ be its stationary distribution,$F_N(x)=\pi_N([0,Nx])$
and $G(x)$ the d.f. of a uniform r.v. on $[0,1]$. We address the
asymptotic behaviour of $\pi_N$ as $N$ tends to infinity and 
we show that $1/N$ is the exact convergence rate of $F_N$ to $G$.
This result improves (in the particular case considered) earlier
results claiming that $\lim_N \| F_N-G \|_\infty =0$. 
We conclude by developping similar results for a class of Markov
chains which are, roughly speaking, finite perturbation of
$(X_N(t);t\geq 0)$.

manita@mech.math.msu.su  francois.simonot@esstin.uhp-nancy.fr  

• To see a preprint or other information provided by the author click here.
• Or here.

1831. THE INTERSECTIVE ASCLT

Rita Giuliano Antonini and Michel Weber

In a recent work of the second named author on the Almost Sure Central
Limit Theorem (ASCLT),
we showed the usefulness of the concept of quasi-orthogonal system of
random variables introduced
by Bellman and later developped by Kac, Salem and Zygmund. In this paper,
we propose an optimal
formulation of the ASCLT by using this idea and some correlation
inequalities for sums of independent
random variables. We also introduce and develop the notion of "intersective
ASCLT" by proving some new results
generalizing and improving substancially  the classical formulation of the
ASCLT.

giuliano@dm.unipi.it , weber@math.u-strasbg.fr

1832. THE CONCENTRATION OF MEASURE PHENOMENON

Michel Ledoux

This book presents the basic aspects of the
concentration of measure phenomenon that was put forward in the early
seventies, and emphasized since then, by V. Milman in asymptotic geometric
analysis. It has now become of powerful interest in applications,
in various areas such as geometry, functional analysis and infinite
dimensional integration, discrete mathematics and complexity theory, and
probability theory. This book is concerned with the basic techniques and
examples of the concentration of measure phenomenon. A particular
emphasis has been put on geometric, functional and probabilistic
tools to reach and describe measure concentration in a number of settings,
as well as on M. Talagrand's investigation of concentration
in product spaces and its application in discrete mathematics
and probability theory.

Mathematical Surveys and Monographs 89 (181 p.) AMS 2001

ledoux@cict.fr

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