Probability Abstracts 60
This document contains abstracts 1626-1665.
They have been mailed on December 31, 2000.
1626. CRITICAL BEHAVIOR OF THE MASSLESS FREE FIELD AT THE DEPINNING TRANSITION
Erwin Bolthausen and Yvan Velenik
We consider the d-dimensional massless free field localized by a
delta-pinning of strength e. We study the asymptotics of the variance of the
field, and of the decay-rate of its 2-point function, as e goes to zero, for
general Gaussian interactions. Physically speaking, we thus rigorously obtain
the critical behavior of the transverse and longitudinal correlation lengths of
the corresponding d+1-dimensional effective interface model in a non-mean-field
regime. We also describe the set of pinned sites at small e, for a broad class
of d-dimensional massless models.
velenik@cmi.univ-mrs.fr
- This article is available
from the xxx mathematics archive as
math.PR/0010291.
1627. UNBIASED FINITE PLANAR GRAPHS ARE ASYMPTOTICALLY RECURRENT
Itai Benjamini and Oded Schramm
Suppose that $G_j$ is a sequence of finite connected planar graphs, and in
each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We
introduce the notion of a distributional limit $G$ of such graphs. Assume that
the vertex degrees of the vertices in $G_j$ are bounded, and the bound does not
depend on $j$. Then after passing to a subsequence, the limit exists, and is a
random rooted graph $G$. We prove that with probability one $G$ is recurrent.
The proof involves the Circle Packing Theorem. The motivation for this work
comes from the theory of random spherical triangulations.
schramm@microsoft.com
- This article is available
from the xxx mathematics archive as
math.PR/0011019.
1628. STATISTICAL ANALYSIS OF THE INHOMOGENEOUS TELEGRAPHER'S PROCESS
Stefano M. Iacus
We consider a problem of estimation for the telegrapher's process on the
line, say X(t), driven by a Poisson process with non constant rate. It turns
out that the finite-dimensional law of the process X(t) is a solution to the
telegraph equation with non constant coefficients. We give the explicit law
P(theta) of the process X(t) for a parametric class of intensity functions for
the Poisson process. We propose an estimator for the parameter theta of
P(theta) and we discuss its properties as a first attempt to apply statistics
to these models.
stefano.iacus@unimi.it
- This article is available
from the xxx mathematics archive as
math.PR/0011059.
1629. A NEW PROPERTY OF ABSORBED DIFFUSIONS
Nikolai Dokuchaev
We consider stochastic diffusion processes absorbed at the boundary of a
domain. It is shown that there exist initial distributions which ensure a given
decreasing of density of the absorbed process.
dokuchaev@pobox.spbu.ru
- This article is available
from the xxx mathematics archive as
math.PR/0011089.
1630. ON THE MIXING TIME OF SIMPLE RANDOM WALK ON THE SUPER CRITICAL
PERCOLATION CLUSTER
Itai Benjamini and Elchanan Mossel
It is shown that the mixing time of simple random walk on the super critical
cluster inside the $n\times...\times n$ box in $\Z^d$ is $\theta(n^2)$, and
that the Cheeger constant of the cluster inside the box is $\theta(n^{-1})$.
mossel@microsoft.com
- This article is available
from the xxx mathematics archive as
math.PR/0011092.
1631. SOME COMPARISONS FOR GAUSSIAN PROCESSES
Richard A. Vitale
Extensions and variants are given for the well-known comparison principle for
Gaussian processes based on ordering by pairwise distance.
rvitale@uconnvm.uconn.edu
- This article is available
from the xxx mathematics archive as
math.PR/0011093.
1632. A GROWTH MODEL IN A RANDOM ENVIRONMENT
Janko Gravner, Craig A. Tracy, Harold Widom
We consider a model of interface growth in two dimensions, given by a height
function on the sites of the one--dimensional integer lattice. According to the
discrete time update rule, the height above the site $x$ increases to the
height above $x-1$, if the latter height is larger; otherwise the height above
$x$ increases by 1 with probability $p_x$. We assume that $p_x$ are chosen
independently at random with a common distribution $F$, and that the initial
state is such that the origin is far above the other sites. We explicitly
identify the asymptotic shape and prove that, in the pure regime, the
fluctuations about that shape, normalized by the square root of time, are
asymptotically normal. This contrasts with the quenched version: conditioned on
the environment, and normalized by the cube root of time, the fluctuations
almost surely approach a distribution known from random matrix theory.
gravner@math.ucdavis.edu
- This article is available
from the xxx mathematics archive as
math.PR/0011150.
1633. NON-INTERSECTING PATHS, RANDOM TILINGS AND RANDOM MATRICES
Kurt Johansson
We investigate certain measures induced by families of non-intersecting paths
in domino tilings of the Aztec diamond, rhombus tilings of an abc-hexagon, a
dimer model on a cylindrical brick lattice and a growth model. The measures
obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as
the eigenvalue measures in random matrix theory like GUE, which can in fact be
obtained from non-intersecting Brownian motions. The derivations of the
measures are based on the Karlin-McGregor or Lindstr\"om-Gessel-Viennot method.
We use the measure to show some asymptotic results for the models.
kurtj@math.kth.se
- This article is available
from the xxx mathematics archive as
math.PR/0011250.
1634. A GENERALISED INDUCTIVE APPROACH TO THE LACE EXPANSION
Remco van der Hofstad, Gordon Slade
The lace expansion is a powerful tool for analysing the critical behaviour of
self-avoiding walks and percolation. It gives rise to a recursion relation
which we abstract and study using an adaptation of the inductive method
introduced by den Hollander and the authors. We give conditions under which the
solution to the recursion relation behaves as a Gaussian, both in Fourier space
and in terms of a local central limit theorem. These conditions are shown
elsewhere to hold for sufficiently spread-out models of networks of
self-avoiding walks in dimensions $d>4$, and for critical oriented percolation
in dimensions $d+1>5$, providing a unified approach and an essential ingredient
for a detailed analysis of the branching behaviour of these models.
R.W.vanderHofstad@its.tudelft.nl
- This article is available
from the xxx mathematics archive as
math.PR/0012026.
1635. THE DIAMETER OF LONG-RANGE PERCOLATION CLUSTERS ON FINITE CYCLES
Itai Benjamini and Noam Berger
Bounds for the diameter and for the expansion of long-range percolation
clusters on the cycle $\Z / N\Z$ are given.
noam@stat.berkeley.edu
- This article is available
from the xxx mathematics archive as
math.PR/0012070.
1636. LARGE DEVIATION PRINCIPLES AND COMPLETE EQUIVALENCE AND NONEQUIVALENCE
RESULTS FOR PURE AND MIXED ENSEMBLES
R. S. Ellis, K. Haven, and B. Turkington
We consider a general class of statistical mechanical models of coherent
structures in turbulence, which includes models of two-dimensional fluid
motion, quasi-geostrophic flows, and dispersive waves. First, large deviation
principles are proved for the canonical ensemble and the microcanonical
ensemble. For each ensemble the set of equilibrium macrostates is defined as
the set on which the corresponding rate function attains its minimum of 0. We
then present complete equivalence and nonequivalence results at the level of
equilibrium macrostates for the two ensembles.
rsellis@math.umass.edu
- This article is available
from the xxx mathematics archive as
math.PR/0012081.
1637. ESTIMATING THE P-VARIATION INDEX OF A SAMPLE FUNCTION: AN APPLICATION TO
FINANCIAL DATA SET
R. Norvaisa, D. M. Salopek
This paper modifies a box-counting method of estimating a fractal dimension
of a graph, and applies it to estimate the roughness of a sample function of a
stochastic process such as a Levy process or a Gaussian process with stationary
increments
rnorvais@fields.utoronto.ca
- This article is available
from the xxx mathematics archive as
math.PR/0012098.
1638. STOCHASTIC DIFFERENTIAL EQUATIONS FOR TRACE-CLASS OPERATORS AND QUANTUM
CONTINUAL MEASUREMENTS
Alberto Barchielli, Anna Maria Paganoni
The theory of measurements continuous in time in quantum mechanics (quantum
continual measurements) has been formulated by using the notions of instrument
and positive operator valued measure, functional integrals, quantum stochastic
differential equations and classical stochastic differential equations (SDE's).
Various types of SDE's are involved, and precisely linear and non linear
equations for vectors in Hilbert spaces and for trace-class operators. All such
equations contain either a diffusive part, or a jump one, or both.
In this paper we introduce a class of linear SDE's for trace-class operators,
relevant to the theory of continual measurements, and we recall how such SDE's
are related to instruments and master equations and, so, to the general
formulation of quantum mechanics. We do not present the Hilbert space
formulation of such SDE's and we make some mathematical simplifications: no
time dependence is introduced into the coefficients and only bounded operators
on the Hilbert space of the quantum system are considered. Then we introduce
the notion of a posteriori state and the non linear SDE satisfied by such
states and we give conditions from which such equation is assured to preserve
pure states and to send any mixed state into a pure one for large times.
Finally we review the known results about the existence and uniqueness of
invariant measures in the purely diffusive case and we give some concrete
examples of physical systems.
barchielli@mate.polimi.it
- This article is available
from the xxx mathematics archive as
math.PR/0012226.
1639. HYDRODYNAMIC EQUATION FOR A DEPOSITION MODEL
Balint Toth and Wendelin Werner
We show that the two-component system of hyperbolic conservation laws
$\partial_t \rho + \partial_x (\rho u) =0 = \partial_t u + \partial_x \rho$
appears naturally in the formally computed hydrodynamic limit of some randomly
growing interface models, and we study some properties of this system. We show
that the two-component system of hyperbolic conservation laws $\partial_t \rho
+ \partial_x (\rho u) =0 = \partial_t u + \partial_x \rho$ appears naturally in
the formally computed hydrodynamic limit of some randomly growing interface
models, and we study some properties of this system.
wendelin.werner@math.u-psud.fr
- This article is available
from the xxx mathematics archive as
math.PR/0012232.
1640. A SIGNAL-RECOVERY SYSTEM: ASYMPTOTIC PROPERTIES AND CONSTRUCTION OF AN
INFINITE VOLUME LIMIT
Jacob van den Berg, Balint Toth
We consider a linear sequence of `nodes', each of which can be in state 0
(`off') or 1 (`on'). Signals from outside are sent to the rightmost node and
travel instantaneously as far as possible to the left along nodes which are
`on'. These nodes are immediately switched off, and become on again after a
recovery time. The recovery times are independent exponentially distributed
random variables.
We present properties for finite systems and use some of these properties to
construct an infinite-volume extension, with signals `coming from infinity'.
This construction is related to a question by D. Aldous and we expect that it
sheds some light on, and stimulates further investigation of, that question.
balint@renyi.hu
- This article is available
from the xxx mathematics archive as
math.PR/0012237.
1641. PERPETUAL OPTIONS FOR LEVY PROCESSES IN THE BACHELIER MODEL
Ernesto Mordecki
Solution to the optimal stopping problem
V(x)=sup{E[exp(-dT)g(x+X(T))]; T is a stopping time}
is given, where X is a Lévy process, d, a constant greather
or equal to 0 is a discount rate, and the reward function
g takes the form gc(x)=max(x-K,0) or gp(x)=max(K-x,0).
Results, interpreted as option prices of perpetual options
in Bachelier's model are expressed in terms of the
distribution of the overall supremum in case g=gc and
overall infimum in case g=gp of the process X killed at
rate d. Closed form solutions are obtained under mixed
exponentially distributed positive jumps with arbitrary
negative jumps for gc, and under arbitrary positive jumps
and mixed exponentially distributed negative jumps for gp.
In case g=gc a prophet inequality comparing prices of
perpetual look-back call options and perpetual call
options is obtained.
mordecki@cmat.edu.uy
- To see a preprint or other
information provided by the author
click here.
1642. ON CONSISTENCY OF KERNEL DENSITY ESTIMATORS FOR RANDOMLY
CENSORED DATA: RATES HOLDING UNIFORMLY OVER ADAPTIVE INTERVALS
Evarist Gin\'e and Armelle Guillou
In the usual right-censored data situation, let
$f_n$, $n\in{\bf N}$, denote the convolution
of the Kaplan-Meier product limit estimator
with the kernels $a_n^{-1}K(\cdot/a_n)$, where
$K$ is a smooth probability density with bounded
support and $a_n\to0$. That is, $f_n$ is the usual
kernel density estimator based on Kaplan-Meier.
Let $\bar{f}_n$ denote the convolution of the
distribution of the uncensored data, which is assumed
to have a bounded density, with the same kernels.
For each $n$, let $J_n$ denote the half line
with right end point $Z_{n,n(1-\varepsilon_n)}-a_n$,
where $\varepsilon_n\to0$ and, for each $m$,
$Z_{n,m}$ is the $m$-th order statistic of the
censored data. It is shown that, under some mild
conditions on $a_n$ and
$\varepsilon_n$, $\sup_{J_n}|f_n(t)-\bar{f}_n(t)|$
converges a.s. to zero as $n\to\infty$ at
least as fast as
$\sqrt{|\log(a_n\wedge\varepsilon_n)|/(na_n\varepsilon_n)}$.
For $\varepsilon_n=$constant, this rate compares,
up to constants, with the exact rate for fixed intervals.
gine@uconnvm.uconn.edu guillou@ccr.jussieu.fr
- To see a preprint or other
information provided by the author
click here.
1643. RATES OF STRONG UNIFORM CONSISTENCY FOR
MULTIVARIATE KERNEL DENSITY ESTIMATORS
Evarist Gin\'e and Armelle Guillou
Let $f_n$ denote the usual kernel density estimator
in several dimensions. It is shown that if $\{a_n\}$
is a regular band sequence,
$K$ is a bounded square integrable kernel of several
variables, of bounded variation, and if the data
consist of an i.i.d. sample from a distribution
possessing a bounded density $f$ with respect
to Lebesgue measure on
${\bf R}^d$, then
$\limsup_{n\to\infty}{\sqrt{na_n^d\over\log a_n^{-1}}
\sup_{t\in {\bf R}^d}}|f_n(t)-Ef_n(t)|\le
C\sqrt{\|f\|_\infty\int K^2(x)dx}$ a.s.
for some absolute constant $C$ that depends only on
$d$. With some additional but still weak conditions,
it is proved that the above sequence of normalized
suprema converges a.s. to
$\sqrt{2d\|f\|_\infty\int K^2(x)dx}$. Neither of the
two results require $f$ to be strictly positive.
These results improve upon, and extend to several
dimensions, results by Silverman (1978) for
univariate densities.
gine@uconnvm.uconn.edu guillou@ccr.jussieu.fr
- To see a preprint or other
information provided by the author
click here.
1644. HOW TO FIND AN EXTRA HEAD: OPTIMAL RANDOM SHIFTS OF
BERNOULLI AND POISSON RANDOM VARIABLES
Alexander E. Holroyd and Thomas M. Liggett
We consider the following problem: given an i.i.d. family of
Bernoulli random variables indexed by Z^d, find a random
occupied site X in Z^d such that relative to X, the other
random variables are still i.i.d. Bernoulli. Results of
Thorisson (1996) imply that such an X exists for all d.
Liggett (2000) proved that for d=1, there exists an X with
tails P(|X|\geq t) of order c t^{-1/2}, but none with finite
1/2-th moment. We prove that for general d there exists a
solution with tails of order c t^{-d/2}, while for d=2 there
is none with finite first moment. We also prove analogous
results for a continuum version of the same problem.
Finally we prove a result which strongly suggests that the
tail behavior mentioned above is the best possible for all d.
holroyd@math.ucla.edu tml@math.ucla.edu
- To see a preprint or other
information provided by the author
click here.
1645. MULTIPLE-INPUT HEAVY-TRAFFIC REAL-TIME QUEUES
Lukasz Kruk, John Lehoczky, Steven Shreve and Shu-Ngai Yeung
A single queueing station which serves $K$ input streams
is considered. Each stream is an independent renewal
process, with customers having random lead-times. Customers
are served by processor sharing across streams. Within each
stream, two disciplines are considered - earliest-deadline
-first and first-in-first-out. The set of current lead times
of the $K$ streams is modeled as a $K$-dimensional vector
of random counting measures on the real line, and the limit
of this vector of measure-valued processes is obtained
under heavy traffic conditions.
kruk+@andrew.cmu.edu jpl@stat.cmu.edu shreve@cmu.edu syeung@stat.cmu.edu
- To see a preprint or other
information provided by the author
click here.
1646. ON RECURRENT AND TRANSIENT SETS OF
INHOMOGENEOUS SYMMETRIC RANDOM WALKS
Giambattista Giacomin and Gustavo Posta
We consider a continuous time random walk on the
$d$--dimensional lattice $Z^d$: the jump rates are time
dependent, but symmetric and strongly elliptic with
ellipticity constants independent of time. We investigate
the implications of heat kernels estimates on
recurrence--transience properties of the walk and we give
conditions for recurrence as well as for transience: we
give applications of these conditions and discuss them in
relation with the (optimal) Wiener test available in the
time independent context. Our approach relies on estimates
on the time spent by the walk in a set and on a $0$--$1$
law. We show also that, still via heat kernel estimates,
one can avoid using a $0$--$1$ law, achieving this way
quantitative estimates on more general hitting
probabilities.
giacomin@math.jussieu.fr guspos@mate.polimi.it
- To see a preprint or other
information provided by the author
click here.
1647. THE LARGE DEVIATION PRINCIPLE FOR COARSE-GRAINED PROCESSES
Richard S. Ellis, Kyle Haven and Bruce Turkington
The large deviation principle is proved for a class of
$L^2$-valued processes that arise from the coarse-graining
of a random field. Coarse-grained processes of this kind
form the basis of the analysis of local mean-field models
in statistical mechanics by exploiting the long-range
nature of the interaction function defining such models.
In particular, the large deviation principle is used in a
companion paper to derive the variational principles that
characterize equilibrium macrostates in statistical models
of two-dimensional and quasi-geostrophic turbulence. Such
macrostates correspond to large-scale, long-lived flow
structures, the description of which is the goal of the
statistical equilibrium theory of turbulence. The large
deviation bounds for the coarse-grained process under
consideration are shown to hold with respect to the strong
$L^2$ topology, while the associated rate function is
proved to have compact level sets with respect to the weak
topology. This compactness property is nevertheless
sufficient to establish the existence of equilibrium
macrostates for both the microcanonical and canonical
ensembles.
rsellis@math.umass.edu
- To see a preprint or other
information provided by the author
click here.
1648. LARGE DEVIATION PRINCIPLES AND COMPLETE EQUIVALENCE AND
NONEQUIVALENCE RESULTS FOR PURE AND MIXED ENSEMBLES
Richard S. Ellis, Kyle Haven and Bruce Turkington
We consider a general class of statistical mechanical
models of coherent structures in turbulence, which
includes models of two-dimensional fluid motion,
quasi-geostrophic flows, and dispersive waves. First,
large deviation principles are proved for the canonical
ensemble and the microcanonical ensemble. For each
ensemble the set of equilibrium macrostates is defined
as the set on which the corresponding rate function
attains its minimum of 0. We then present complete
equivalence and nonequivalence results at the level of
equilibrium macrostates for the two ensembles.
Microcanonical equilibrium macrostates are characterized
as the solutions of a certain constrained minimization
problem, while canonical equilibrium macrostates are
characterized as the solutions of an unconstrained
minimization problem in which the constraint in the first
problem is replaced by a Lagrange multiplier. The
analysis of equivalence and nonequivalence of ensembles
reduces to the following question in global optimization.
What are the relationships between the set of solutions
of the constrained minimization problem that characterizes
microcanonical equilibrium macrostates and the set of
solutions of the unconstrained minimization problem that
characterizes canonical equilibrium macrostates?
In general terms, our main result is that a necessary and
sufficient condition for equivalence of ensembles to hold
at the level of equilibrium macrostates is that it holds
at the level of thermodynamic functions, which is the case
if and only if the microcanonical entropy is concave. The
necessity of this condition is new and has the following
striking formulation. If the microcanonical entropy is
not concave at some value of its argument, then the
ensembles are nonequivalent in the sense that the
corresponding set of microcanonical equilibrium macrostates
is disjoint from any set of canonical equilibrium
macrostates. We point out a number of models of physical
interest in which nonconcave microcanonical entropies arise.
We also introduce a new class of ensembles called mixed
ensembles, obtained by treating a subset of the dynamical
invariants canonically and the complementary set
microcanonically. Such ensembles arise naturally in
applications where there are several independent dynamical
invariants, including models of dispersive waves for the
nonlinear Schroedinger equation. Complete equivalence
and nonequivalence results are presented at the level of
equilibrium macrostates for the pure canonical, the pure
microcanonical, and the mixed ensembles.
rsellis@math.umass.edu
- To see a preprint or other
information provided by the author
click here.
1649. BRWONIAN MOTION WITH SINGULAR DRIFT
Richard F. Bass and Zhen-Qing Chen
We consider the stochastic differential equation
$$dX_t=dW_t+dA_t,$$
where $W_t$ is $d$-dimensional Brownian motion and
the $i$th component of $A_t$ is a process of bounded
variation that stands in the same relationship to a
measure $\pi_i$ as $\int_0^t f(X_s) ds$ does to the
measure $f(x) dx$. We prove weak existence and uniqueness
for the above stochastic differential equation when
the measures $\pi_i$ are members of a certain large
class.
bass@math.uconn.edu zchen@math.washington.edu
- To see a preprint or other
information provided by the author
click here.
1650. NONEQUIVALENT STATISTICAL EQUILIBRIUM ENSEMBLES AND
REFINED STABILITY THEOREMS FOR MOST PROBABLE FLOWS
Richard S. Ellis, Kyle Haven and Bruce Turkington
Statistical equilibrium models of coherent structures in
two-dimensional and barotropic quasi-geostrophic turbulence
are formulated using canonical and microcanonical ensembles,
and the equivalence or nonequivalence of ensembles is
investigated for these models. The main results show that
models in which the global invariants are treated
microcanonically give richer families of equilibria than
models in which they are treated canonically. Such global
invariants are those conserved quantities for ideal dynamics
which depend on the large scales of the motion; they include
the total energy and circulation. For each model a variational
principle that characterizes its equilibrium states is derived
by invoking large deviations techniques to evaluate the
continuum limit of the probabilistic lattice model. An
analysis of the two different variational principles resulting
from the canonical and microcanonical ensembles reveals that
their equilibrium states coincide only when the microcanonical
entropy function is concave. These variational principles also
furnish Lyapunov functionals from which the nonlinear stability
of the mean flows can be deduced. While in the canonical model
the well-known Arnold stability theorems are reproduced, in the
microcanonical model more refined theorems are obtained which
extend known stability criteria when the microcanonical and
canonical ensembles are not equivalent. A numerical example
pertaining to geostrophic turbulence over topography in a zonal
channel is included to illustrate the general results.
rsellis@math.umass.edu
- To see a preprint or other
information provided by the author
click here.
1651. ONE-SIDED RANDOM WALK WITH WEAK PINNING:
PATHWISE DESCRIPTIONS OF THE PHASE TRANSITION
Yasuki Isozaki and Nobuo Yoshida
We consider a one-dimensional random walk which is conditioned
to stay non-negative and is ``weakly pinned'' to zero.
This model is known to exhibit a phase transition as the strength
of the weak pinning varies.
We prove path space limit theorems which describe the macroscopic shape
of the path for all values of the pinning strength.
If the pinning is less than (resp. equal to) the critical strength,
then the limit process is the Brownian meander
(resp. reflecting Brownian motion). If the pinning strength is supercritical,
then the limit process is a positively recurrent
Markov chain with a strong mixing property.
yasuki@math.sci.osaka-u.ac.jp nobuo@kusm.kyoto-u.ac.jp
- To see a preprint or other
information provided by the author
click here.
1652. THE EMPIRICAL PROCESS FOR BIVARIATE SEQUENCES WITH LONG MEMORY
Domenico Marinucci
We establish functional central limit theorems for
the empirical process of bivariate stationary long range
dependent sequences under Gaussian subordination and
linearity assumptions. We extend these results to allow for
the presence of unknown marginal distributions, estimated
parameters and regression residuals; applications to
nonparametric statistical inference are also illustrated.
marinucc@scec.eco.uniroma1.it
1653. GAUSSIAN SEMIPARAMETRIC ESTIMATION FOR RANDOM FIELDS WITH SINGULAR SPECTRUM
Domenico Marinucci
We analyze the asymptotic behaviour of the tapered
discrete Fourier transforms for random fields with
singular spectrum. The results are used to establish
consistency and asymptotic normality for semiparametric
estimates of the singularity parameter under broad
conditions.
marinucc@scec.eco.uniroma1.it
1654. A NETWORK TRAFFIC MODEL WITH RANDOM TRANSMISSION RATE
Krishanu Maulik, Sidney Resnick and Holger Rootz\'en
The infinite source Poisson network model assumes sources begin
data transmissions at Poisson time points and continue for random
lengths of time. The random transmission times have such heavy
tails that the variance
is infinite. Transmission rates have been assumed non-random and,
usually constant. However, analysis of network data suggests
that the transmission rate is also a random variable with a heavy
tail. So we consider an infinite source Poisson model with sources
transmitting for a random length of time at a random rate. Both
the rates and lengths have infinite variance but finite mean and
are assumed asymptotically independent, a concept made
precise. We carefully discuss equivalent formulations of
asymptotic independence and prove a limit theorem for the input
process showing that the centered process under a suitable scaling
converges to a totally skewed stable Levy motion in the sense of
finite dimensional distributions.
km75@cornell.edu sid@orie.cornell.edu rootzen@math.chalmers.se
- To see a preprint or other
information provided by the author
click here.
1655. A SINGLE CHANNEL ON/OFF MODEL WITH TCP-LIKE CONTROL
Milan Borkovec, Amites Dasgupta, Sidney Resnick, and Gennady Samorodnitsky
We model behavior of a TCP-like source transmitting
over a single channel to a server that processes work at constant
rate r. Transmission by the source follows an on/off mechanism.
When the overall load in the system is
below a critical constant $\gamma$, transmission rates increase
linearly but when the load exceeds $\gamma$, then transmission rates
decrease geometrically
fast. We study the system by means of an embedded
Markov chain which gives the buffer content at the start of
transmissions. Attention is paid to the time necessary to transmit
a file of size L and both the
tail behavior and expectation of the distribution of
file transmission time are considered.
amites_v@www.isical.ac.in {borkovec, sid, gennady}@orie.cornell.edu}
- To see a preprint or other
information provided by the author
click here.
1656. A GENERALISED INDUCTIVE APPROACH TO THE LACE EXPANSION
Remco van der Hofstad, Gordon Slade
The lace expansion is a powerful tool for analysing
the critical behaviour of self-avoiding walks and
percolation. It gives rise to a recursion relation
which we abstract and study using an adaptation of
the inductive method introduced by den Hollander and
the authors. We give conditions under which the
solution to the recursion relation behaves as a
Gaussian, both in Fourier space and in terms of a
local central limit theorem. These conditions are
shown elsewhere to hold for sufficiently spread-out
models of networks of self-avoiding walks in
dimensions $d>4$, and for critical oriented percolation
in dimensions $d+1>5$, providing a unified approach and
an essential ingredient for a detailed analysis of the
branching behaviour of these models.
R.W.vanderHofstad@its.tudelft.nl slade@math.ubc.ca
- To see a preprint or other
information provided by the author
click here.
- Or
here.
1657. CRITICAL TWO-POINT FUNCTIONS AND THE LACE EXPANSION
FOR SPREAD-OUT HIGH DIMENSIONAL PERCOLATION AND RELATED MODELS
Takashi Hara, Remco van der Hofstad and Gordon Slade
We consider spread-out models of self-avoiding walk,
bond percolation, lattice trees and bond lattice
animals on $\Zd$, having long finite-range connections,
above their upper critical dimensions $d=4$ (self-avoiding
walk), $d=6$ (percolation) and $d=8$ (trees and animals).
The two-point functions for these models are respectively
the generating function for self-avoiding walks from the
origin to $x \in \Zd$, the probability of a connection
from $0$ to $x$, and the generating function for lattice
trees or lattice animals containing $0$ and $x$. We use
the lace expansion to prove that for sufficiently
spread-out models above the upper critical dimension,
the two-point function of each model decays, at the
critical point, as a multiple of $|x|^{2-d}$ as
$x \to \infty$. We use a new unified method to prove
convergence of the lace expansion. The method is based
on $x$-space methods rather than the Fourier transform.
Our results also yield unified and simplified proofs of
the bubble condition for self-avoiding walk, the triangle
condition for percolation, and the square condition for
lattice trees and lattice animals, for sufficiently
spread-out models above the upper critical dimension.
hara@math.nagoya-u.ac.jp R.W.vanderHofstad@its.tudelft.nl
slade@math.ubc.ca
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
1658. LAWS OF THE ITERATED LOGARITHM FOR THE RANGE OF
RANDOM WALKS IN TWO AND THREE DIMENSIONS
Richard F. Bass and Takashi Kumagai
Let $S_n$ be a random walk in ${\bf Z}^d$ and let $R_n$ be the range of $S_n$.
We prove an almost sure invariance principle for $R_n$ when $d=3$ and a law
of the iterated logarithm for $R_n$ when $d=2$.
bass@math.uconn.edu kumagai@i.kyoto-u.ac.jp
- To see a preprint or other
information provided by the author
click here.
- Or
here.
1659. SKEW CONVOLUTION SEMIGROUPS AND RELATED
IMMIGRATION PROCESSES
Zeng-Hu Li
Let $M(E)$ be the space of finite Borel measures on a topological
Lusin space $E$. Suppose that $(Q_t)_{t\ge0}$ is the semigroup of a
measure-valued branching process $X$ with state space $M(E)$. A
class of probability measures $(N_t)_{t\ge0}$ on $M(E)$ is called a
{\it skew convolution semigroup} associated with $(Q_t)_{t\ge0}$ if
it satisfies $N_{r+t} = (N_rQ_t)*N_t$ for $r,t\ge0$. Given a skew
convolution semigroup $(N_t)_{t\ge0}$ we may define a transition
semigroup $(Q_t^N)_{t\ge0}$ on $M(E)$ by $Q^N_t(\mu,\cdot) :=
Q_t(\mu,\cdot)*N_t$. A Markov process is naturally called an {\it
immigration process} associated with $X$ if it has transition
semigroup $(Q^N_t)_{t\ge0}$. The inhomogeneous versions of the
definitions can be given obviously. The above formulation of the
immigration process includes essentially all the models of Dawson
(1993), Dawson and Ivanoff (1978), Dynkin (1991), Gorostiza and
Lopez-Mimbela (1990), Konno and Shiga (1988) and Li (1992). It also
contains some new kinds of immigration processes which had never
been studied before. In this work, we first prove that a general
inhomogeneous skew convolution semigroup may be decomposed into
three components, which involve respectively a countable family of
entrance laws, a countable family of closed entrance laws, and a
continuum family of infinitely divisible probability entrance laws
together with a diffuse measure on the index space. Then we give a
construction for the corresponding immigration process by picking up
measure-valued paths with random times of birth and death. Our
construction is based on the observation that any skew convolution
semigroup is determined by a continuous increasing measure-valued
path $(\gamma_t)_{t\ge0}$ and an entrance rule $(G_t)_{t\ge 0}$.
This fact yields a natural decomposition of the immigration into two
parts; the deterministic part represented by $(\gamma_t) _{t\ge0}$
and the random part determined by $(G_t)_{t\ge0}$. The latter is
usually an inhomogeneous immigration process and can be constructed
by summing up paths $\{w_t: \alpha<t<\beta\}$ in the associated
Kuznetsov process. By analyzing the asymptotic behavior of the paths
$\{w_t: \alpha<t<\beta\}$ near the birth time $\alpha = \alpha(w)$,
we show that almost all these paths start propagation in an
extension $E^T_D$ of the underlying space, including those growing
up at points in this space from the null measure. Those combined
with our construction of the immigration process give a heuristic
description of the immigration phenomenon. As additional
applications of the construction, we give formulations of some well-
known results on excessive measures in terms of stationary
immigration processes. (Tex files available upon request.)
liz@math.carleton.ca
1660. CONVERGENCE OF THE POINCARE CONSTANT
Oliver Johnson
The Poincare constant $R(Y)$ of a random variable $Y$
relates the norm of a function $g$ and its derivative
$g'$ in $L^2(Y)$. Since $R(Y) - Var(Y)$ is positive, with
equality if and only if $Y$ is normal, $R(Y)$ acts as a
distance from the normal distribution. In this paper we
provide conditions for convergence of $R(Y) - Var(Y)$ in
the Central Limit Theorem. Furthermore, we show that $R(Y)$
is finite for discrete mixtures of normals, allowing us
to add rates to the proof of Central Limit Theorem in the
sense of relative entropy.
otj1000@cam.ac.uk
- To see a preprint or other
information provided by the author
click here.
1661. A LAW OF LARGE NUMBERS AND A CENTRAL LIMIT THEOREM FOR
BIASED RANDOM MOTIONS IN RANDOM ENVIRONMENT
Lian Shen
We discuss a class of biased random motions in a random
environment on $\entier{d}$, $d\geq 1$, which have
reversible transition kernels when the environment is
fixed. The main aim is to derive a strong law of large
numbers and a functional central limit theorem for this
class of models. The technique of the environment viewed
from the particle does not seem to apply well in this
setting. Our approach is based on the technique of
introducing certain times similar to the regeneration times
in the work concerning random walks in random environment
by Sznitman-Zerner. With the help of these times we are
able to construct an ergodic Markov structure.
lian@math.ethz.ch
- To see a preprint or other
information provided by the author
click here.
1662. SKEW CONVOLUTION SEMIGROUPS AND RELATED IMMIGRATION
PROCESSES
Zeng-Hu Li
Let $M(E)$ be the space of finite Borel measures on a topological
Lusin space $E$. Suppose that $(Q_t)_{t\ge0}$ is the semigroup of a
measure-valued branching process $X$ with state space $M(E)$. A
class of probability measures $(N_t)_{t\ge0}$ on $M(E)$ is called a
{\it skew convolution semigroup} associated with $(Q_t)_{t\ge0}$ if
it satisfies $N_{r+t} = (N_rQ_t)*N_t$ for $r,t\ge0$. Given a skew
convolution semigroup $(N_t)_{t\ge0}$ we may define a transition
semigroup $(Q_t^N)_{t\ge0}$ on $M(E)$ by $Q^N_t(\mu,\cdot) :=
Q_t(\mu,\cdot)*N_t$. A Markov process is naturally called an {\it
immigration process} associated with $X$ if it has transition
semigroup $(Q^N_t)_{t\ge0}$. The inhomogeneous versions of the
definitions can be given obviously. The above formulation of the
immigration process includes essentially all the models of Dawson
(1993), Dawson and Ivanoff (1978), Dynkin (1991), Gorostiza and
Lopez-Mimbela (1990), Konno and Shiga (1988) and Li (1992). It also
contains some new kinds of immigration processes which had never
been studied before. In this work, we first prove that a general
inhomogeneous skew convolution semigroup may be decomposed into
three components, which involve respectively a countable family of
entrance laws, a countable family of closed entrance laws, and a
continuum family of infinitely divisible probability entrance laws
together with a diffuse measure on the index space. Then we give a
construction for the corresponding immigration process by picking up
measure-valued paths with random times of birth and death. Our
construction is based on the observation that any skew convolution
semigroup is determined by a continuous increasing measure-valued
path $(\gamma_t)_{t\ge0}$ and an entrance rule $(G_t)_{t\ge 0}$.
This fact yields a natural decomposition of the immigration into two
parts; the deterministic part represented by $(\gamma_t) _{t\ge0}$
and the random part determined by $(G_t)_{t\ge0}$. The latter is
usually an inhomogeneous immigration process and can be constructed
by summing up paths $\{w_t: \alpha<t<\beta\}$ in the associated
Kuznetsov process. By analyzing the asymptotic behavior of the paths
$\{w_t: \alpha<t<\beta\}$ near the birth time $\alpha = \alpha(w)$,
we show that almost all these paths start propagation in an
extension $E^T_D$ of the underlying space, including those growing
up at points in this space from the null measure. Those combined
with our construction of the immigration process give a heuristic
description of the immigration phenomenon. As additional
applications of the construction, we give formulations of some well-
known results on excessive measures in terms of stationary
immigration processes. (Tex files available upon request.)
liz@math.carleton.ca
1663. ON THE ALMOST SURE CENTRAL LIMIT THEOREM FOR A CLASS OF $Z^d$-ACTIONS
Mikhail Gordin and Michel Weber
As an extension of earlier papers on stationary sequences, a concept of
weak dependence for strictly stationary random fields is introduced in
terms of
so-called homoclinic transformations. Under assumptions made within the
framework of this concept a form of the almost sure central limit theorem
(ASCLT) is established for random fields arising from a class of algebraic
$\Z^d$-actions on compact abelian groups. As an auxillary result, the
central limit theorem is proved via Ch. Stein's method. The next stage of
the proof includes some estimates which are specific for ASCLT. Both steps
are based on the use of homoclinic transformations.
gordin@pdmi.ras.ru weber@math.u-strasbg.fr
1664. REGULARISATION SPECTRALE ET PROPRIETES METRIQUES DES MOYENNES MOBILES
Mikhail Lifschits and Michel Weber
We develop a spectral regularization technique for moving averages
$B_n^{U,\phi}= {1\over n} \sum_{j=\phi(n)}^{\phi(n)+n-1} U^j$, where $\phi$
is a nondecreasing map and $U:H\ra H$ is a contraction of a Hilbert space
$(H,\|\cdot\|)$. We obtain a spectral regularization inequality which
allows to evaluate efficiently the increments $\|B_m^{U,\phi}(f) -
B_n^{U,\phi}(f)\|$,\ $f\in H$, by means of $\mh[{1\over m},{1\over n}]$
where $\mh$ is a properly regularized version of the spectral measure of
$f$ with respect to $U$. We apply this inequality to investigation of
metric properties of the sets
of moving averages $\{ B_n^{U,\phi}(f), n\in \N \}$ with fixed $f\in H$ and
$\N\subset\NN$. In particular, we obtain estimates of the associated
covering numbers as well as of the related Littlewood-Paley-type square
functions. This work extends our previous results concerning the case of
classical
averages ($\phi(n)=0$). Since it is well known that the structure of
general moving averages is more complicated, there is no surprise that the
general results we obtain are sometimes less complete than their classical
counterparts, and need suitable moment assumptions on the spectral measure
(depending on the growth of the shift function $\phi$). Nevertheless, when
applied to classical situation, our estimates still yield optimal bounds.
lifts@mail.rcom.ru , weber@math.u-strasbg.fr
1665. A SURVEY ON RIEMANN SUMS
Jean-Jacques Ruch and Michel Weber
The study of almost sure convergence of Riemann sums is a fascinating
question which has connections with various problems from Number Theory,
among them the Riemann hypothesis through its link with Farey sequences.
Moreover, it has been known since the fundamental paper of Rudin, that the
convergence almost everywhere of Riemann sums, along a given subsequence of
positive integers, definitively relies on the arithmetical properties of
the subsequence. The arithmetical characterization of that property is an
open and certainly hard question. The study of Riemann sums has for years
been an object of constant interest from analysts, ergodicians, and number
theorists. It even seems, that its power of attraction has grown much
during this last decade. This is the reason of the present survey. Our
motivation inwriting it, was to propose a text to the interested reader,
giving a direct access to the main results of that theory, as well as an
easy understanding, as far as possible each time in each case, of the
various methods elaborated by the authors of these results.
ruch@math.u-bordeaux.fr , weber@math.u-strasbg.fr
