Probability Abstracts 70
This document contains abstracts 2008-2039.
They have been mailed on August 31, 2002.
2008. SOME NON-LINEAR S.P.D.E.'S THAT ARE SECOND ORDER IN TIME
Robert C. Dalang and Carl Mueller
We extend Walsh's theory of martingale measures in order to deal with
hyperbolic stochastic partial differential equations that are second order in
time, such as the wave equation and the beam equation, and driven by spatially
homogeneous Gaussian noise. For such equations, the fundamental solution can be
a distribution in the sense of Schwartz, which appears as an integrand in the
reformulation of the s.p.d.e. as a stochastic integral equation. Our approach
provides an alternative to the Hilbert space integrals of Hilbert-Schmidt
operators. We give several examples, including the beam equation and the wave
equation, with nonlinear multiplicative noise terms.
cmlr@math.rochester.edu
2009. A SIMPLE PROOF OF A RESULT OF A. NOVIKOV
Nicolai Krylov
We give simple proofs that for a continuous local martingale M_t:
1) \liminf_{\epsilon->0} \epsilon \log Ee^{(1-\epsilon) <M>_\infty /2} <
\infty ==> E\exp(M_\infty - <M>_\infty /2) = 1, 2) \liminf_{\epsilon->0}
\epsilon \log\sup_{t>=0} Ee^{(1-\epsilon)M_t/2} < \infty ==> E\exp(M_\infty -
<M>_\infty /2) = 1 .
krylov@math.umn.edu
2010. STOCHASTIC APPORTIONMENT
Geoffrey Grimmett
The problem of how to allocate to states the seats in the US House of
Representatives is the most studied instance of what is termed the
`apportionment problem'. We propose a new method of apportionment which is
stochastic, which meets the quota condition, and which is fair in the sense of
expectations. Two sources of systematic unfairness are identified, firstly the
lower bound condition (every state shall receive at least one seat), and
secondly the lower quota condition (every state shall receive at least the
integer part of its quota).
g.r.grimmett@statslab.cam.ac.uk
2011. ON-LINE TRACKING OF A SMOOTH REGRESSION FUNCTION
L. Goldentayer, R. Liptser
We construct an on-line estimator with equidistant design for tracking a
smooth function from Stone-Ibragimov-Khasminskii class. This estimator has the
optimal convergence rate of risk to zero in sample size. The procedure for
setting coefficients of the estimator is controlled by a single parameter and
has a simple numerical solution. The off-line version of this estimator allows
to eliminate a boundary layer. Simulation results are given.
liptser@eng.tau.ac.il
2012. MINIMUM ABERRATION DESIGNS OF RESOLUTION III
Jesus Juan J. Gabriel Palomo
In this article we prove several important properties of 2^{k-p} minimum
aberration (MA) designs with k>2, where n=2^{k-p} is the number of runs. We
develop a simple method to build MA designs of resolution III. Furthermore, we
introduce a simple relationship, based on product of polynomials, for computing
their word-length patterns.
jjuan@ingor.etsii.upm.es
2013. PERCOLATION ON FINITE GRAPHS AND ISOPERIMETRIC INEQUALITIES
Noga Alon, Itai Benjamini and Alan Stacey
Consider a uniform expanders family $G_n$ with a uniform bound on the
degrees. It is shown that for any $p$ and $c >0$, a random subgraph of $G_n$
obtained by retaining each edge, randomly and independently, with probability
$p$, will have at most one cluster of size at least $c|G_n|$, with probability
going to one, uniformly in $p$ and the size of the expander. The method from
Ajtai, Koml\'os and Szemer\'edi [AKS] is applied to obtain some new results
about the critical probability for the emergence of a giant component in random
subgraphs of finite regular expanding graphs of high girth, as well as a simple
proof of a result of Kesten [Ke] about the critical probability for bond
percolation in high dimensions. Several problems and conjectures regarding
percolation on finite transitive graphs are presented.
itai@wisdom.weizmann.ac.il
2014. UNIFORM INFINITE PLANAR TRIANGULATIONS
Omer Angel, Oded Schramm
The existence of the weak limit as $n\to\infty$ of the uniform measure on
rooted triangulations of the sphere with n vertices is proved. Some properties
of the limit are studied. In particular, the limit is a probability measure on
random triangulations of the plane.
omer@wisdom.weizmann.ac.il
2015. DEFAULT LOGIC IN A COHERENT SETTING
Giulianella Coletti, Romano Scozzafava, Barbara Vantaggi
In this talk - based on the results of a forthcoming paper (Coletti,
Scozzafava and Vantaggi 2002), presented also by one of us at the Conference on
"Non Classical Logic, Approximate Reasoning and Soft-Computing" (Anacapri,
Italy, 2001) - we discuss the problem of representing default rules by means of
a suitable coherent conditional probability, defined on a family of conditional
events. An event is singled-out (in our approach) by a proposition, that is a
statement that can be either true or false; a conditional event is consequently
defined by means of two propositions and is a 3-valued entity, the third value
being (in this context) a conditional probability.
romscozz@dmmm.uniroma1.it
2016. MIXING TIMES OF THE BIASED CARD SHUFFLING AND THE ASYMMETRIC EXCLUSION
PROCESS
Itai Benjamini, Noam Berger, Christopher Hoffman and Elchanan Mossel
Consider the following method of card shuffling. Start with a deck of $N$
cards numbered 1 through N. Fix a parameter $p$ between 0 and 1. In this model
a ``shuffle'' consists of uniformly selecting a pair of adjacent cards and then
flipping a coin that is heads with probability p. If the coin comes up heads
then we arrange the two cards so that the lower numbered card comes before the
higher numbered card. If the coin comes up tails then we arrange the cards with
the higher numbered card first. In this paper we prove that for all p not equal
to 1/2, the mixing time of this card shuffling is O(N^2), as conjectured by
Diaconis and Ram [DR]. A novel feature of our proof is that the analysis of an
infinite (asymmetric exclusion) process plays an essential role in bounding the
mixing time of a finite process.
itai@wisdom.weizmann.ac.il
2017. THE SIMPLEST NEAREST-NEIGHBOR SPIN SYSTEM ON REGULAR GRAPHS:TIME
DYNAMICS OF THE MEAN COVERAGE FUNCTION
Boris L. Granovsky
We establish a characterization of the class of the simplest nearest neighbor
spin systems possesing the mean coverage function (mcf) that obeys a second
order differential equation, and derive explicit expressions for the mcf's of
the above models. Based on these expressions, the problem of ergodicity of the
models is studied and bounds for their spectral gaps are obtained.
mar18aa@techunix.technion.ac.il
2018. ASYMPTOTIC FORMULA FOR A PARTITION FUNCTION OF REVERSIBLE COAGULATION
-FRAGMENTATION PROCESSES
Gregory Freiman, Boris Granovsky
We construct a probability model seemingly unrelated to the considered
stochastic process of coagulation and fragmentation. By proving for this model
the local limit theorem, we establish the asymptotic formula for the partition
function of the equilibrium measure for a wide class of parameter functions of
the process. This formula proves the conjecture stated in [dgg] for the above
class of processes. The method used goes back to A.Khintchine.
mar18aa@techunix.technion.ac.il
2019. CONFORMAL INVARIANCE AND STOCHASTIC LOEWNER EVOLUTION PREDICTIONS FOR
THE 2D SELF-AVOIDING WALK - MONTE CARLO TESTS
Tom Kennedy
Simulations of the self-avoiding walk (SAW) are performed in a half-plane and
a cut-plane (the complex plane with the positive real axis removed) using the
pivot algorithm. We test the conjecture of Lawler, Schramm and Werner that the
scaling limit of the two-dimensional SAW is given by Schramm's Stochastic
Loewner Evolution (SLE). The agreement is found to be excellent. The
simulations also test the conformal invariance of the SAW since conformal
invariance would imply that if we map the walks in the cut-plane into the half
plane using the conformal map z -> sqrt(z), then the resulting walks will have
the same distribution as the SAW in the half plane. The simulations show
excellent agreement between the distributions.
tgk@math.arizona.edu
2020. RIGOROUS ANALYSIS OF DISCONTINUOUS PHASE TRANSITIONS VIA MEAN-FIELD
BOUNDS
Marek Biskup and Lincoln Chayes
We consider a variety of nearest-neighbor spin models defined on the
d-dimensional hypercubic lattice Z^d. Our essential assumption is that these
models satisfy the condition of reflection positivity. We prove that whenever
the associated mean-field theory predicts a discontinuous transition, the
actual model also undergoes a discontinuous transition (which occurs near the
mean-field transition temperature), provided the dimension is sufficiently
large or the first-order transition in the mean-field model is sufficiently
strong. As an application of our general theory, we show that for d
sufficiently large, the 3-state Potts ferromagnet on Z^d undergoes a
first-order phase transition as the temperature varies. Similar results are
established for all q-state Potts models with q>=3, the r-component cubic
models with r>=4 and the O(N)-nematic liquid-crystal models with N>=3.
biskup@math.ucla.edu
2021. MAXIMIN SETTING FOR INVESTMENT PROBLEMS AND FIXED INCOME MANAGEMENT WITH
OBSERVABLE BUT NON-PREDICTABLE PARAMETERS
Nikolai Dokuchaev
We study optimal investment problem for a diffusion market consisting of a
finite number of risky assets (for example, bonds, stocks and options). Risky
assets evolution is described by Ito's equation, and the number of risky assets
can be larger than the number of driving Brownian motions. We assume that the
risk-free rate, the appreciation rates and the volatility of the stocks are all
random; they are not necessary adapted to the driving Brownian motion, and
their distributions are unknown, but they are supposed to be currently
observable. Admissible strategies are based on current observations of the
stock prices and the aforementioned parameters. The optimal investment problem
is stated as a problem with a maximin performance criterion. This criterion is
to ensure that a strategy is found such that the minimum of utility over all
distributions of parameters is maximal. Then the maximin problem is solved for
a very general case via solution of a linear parabolic equation.
ndokuch@uwimona.edu.jm
2022. OPTIMAL PORTFOLIO SELECTION AND COMPRESSION IN AN INCOMPLETE MARKET
Nikolai Dokuchaev and Ulrich Haussmann
We investigate an optimal investment problem with a general performance
criterion which, in particular, includes discontinuous functions. Prices are
modeled as diffusions and the market is incomplete. We find an explicit
solution for the case of limited diversification of the portfolio, i.e. for the
portfolio compression problem. By this we mean that an admissible strategies
may include no more than m different stocks concurrently, where m may be less
than the total number n of available stocks.
ndokuch@uwimona.edu.jm
2023. CLUSTERING IN COAGULATION -FRAGMENTATION PROCESSES, RANDOM COMBINATORIAL
STRUCTURES AND ADDITIVE NUMBER SYSTEMS: ASYMPTOTIC FORMULAE AND ZERO -ONE LAW
Gregory Freiman, Boris Granovsky
We develop a unified approach to the problem of clustering in the three
different fields of applications, as indicated in the title the paper. The
approach is based on Khintchine's probabilistic method that grew out of the
Darwin-Fawler method in statistical physics. Our main result is the derivation
of asymptotic formulae for probabilities of clusters (= groups) of certain
sizes as the number of particles goes to infinity. Based on these formulae we
prove the zero-one law for the distribution of the largest cluster and
establish the threshold function in the phase transition from 0 to 1 in the
above law.
mar18aa@techunix.technion.ac.il
2024. COMPARISON OF DIFFERENT GOODNESS-OF-FIT TESTS
B. Aslan, G. Zech
Various distribution free goodness-of-fit test procedures have been extracted
from literature. We present two new binning free tests, the univariate
three-region-test and the multivariate energy test. The power of the selected
tests with respect to different slowly varying distortions of experimental
distributions are investigated. None of the tests is optimum for all
distortions. The energy test has high power in many applications and is
superior to the chi^2 test.
aslan@elfi1.physik.uni-siegen.de
2025. ON SMALLEST TRIANGLES
Geoffrey Grimmett and Svante Janson
Pick n points independently at random in R^2, according to a prescribed
probability measure mu, and let D^n_1 <= D^n_2 <= ... be the areas of the
binomial n choose 3 triangles thus formed, in non-decreasing order. If mu is
absolutely continuous with respect to Lebesgue measure, then, under weak
conditions, the set {n^3 D^n_i : i >= 1} converges as n --> infinity to a
Poisson process with a constant intensity c(mu). This result, and related
conclusions, are proved using standard arguments of Poisson approximation, and
may be extended to functionals more general than the area of a triangle. It is
proved in addition that, if mu is the uniform probability measure on the region
S, then c(mu) <= 2/|S|, where |S| denotes the area of S. Equality holds in that
c(mu) = 2/|S| if S is convex, and essentially only then. This work generalizes
and extends considerably the conclusions of a recent paper of Jiang, Li, and
Vitanyi.
g.r.grimmett@statslab.cam.ac.uk
2026. DIFFRACTION AND PALM MEASURE OF POINT PROCESSES
Jean-Baptiste Gou\'er\'e
Using the Palm measure notion, we prove the existence of the diffraction
measure of all stationary and ergodic point processes. We get precise
expressions of those measures in the case of specific processes : stochastic
subsets of Z^d, sets obtained by the ``cut-and-project'' method.
jean-baptiste.gouere@univ-lyon1.fr
2027. GROWTH AND PERCOLATION ON THE UNIFORM INFINITE PLANAR TRIANGULATION
Omer Angel
A construction as a growth process for sampling of the uniform infinite
planar triangulation (UIPT), defined in a previous paper, is given. The
construction is algorithmic in nature, and is an efficient method of sampling a
portion of the UIPT.
By analyzing the progress rate of the growth process we show that a.s. the
UIPT has growth rate r^4 up to polylogarithmic factors, confirming heuristic
results from the physics literature. Additionally, the boundary component of
the ball of radius r separating it from infinity a.s. has growth rate r^2 up to
polylogarithmic factors. It is also shown that the properly scaled size of a
variant of the free triangulation of an m-gon converges in distribution to an
asymmetric stable random variable of type 1/2.
By combining Bernoulli site percolation with the growth process for the UIPT,
it is shown that a.s. the critical probability p_c=1/2 and that at p_c
percolation does not occur.
omer@wisdom.weizmann.ac.il
2028. GOOD LOCAL BOUNDS FOR SIMPLE RANDOM WALKS
Christine Ritzmann
We give a local central limit theorem for simple random walks on Z^d,
including Gaussian error estimates. The detailed proof combines standard large
deviation techniques with Cramer-Edgeworth expansions for lattice
distributions.
chritz@amath.unizh.ch
2029. LOWNER'S EQUATION IN NONCOMMUTATIVE PROBABILITY
Robert O. Bauer
Using concepts of noncommutative probability we show that the Lowner's
evolution equation can be viewed as providing a map from paths of measures to
paths of probability measures. We show that the fixed point of the Lowner map
is the convolution semigroup of the semicircle law in the chordal case, and its
multiplicative analogue in the radial case. We further show that the Lowner
evolution ``spreads out'' the distribution and that it gives rise to a Markov
process.
rbauer@math.uiuc.edu
2030. CHARACTERIZATION OF MARKOV SEMIGROUPS ON R ASSOCIATED
TO SOME FAMILIES OF ORTHOGONAL POLYNOMIALS
Dominique Bakry and Olivier Mazet
We give a characterization of the eigenvalues of Markov
operators which admits an orthogonal polynomial basis as
eigenfunctions, in the Hermite and the Laguerre cases, as
well as for the sequence of orthogonal polynomials
associated to some probability measures on $\N$. In the
Hermite case, we then give a description of the path of
the associated Markov processes, as well as a geometrical
interpretation.
Olivier.Mazet@insa-lyon.fr
- To see a preprint or other
information provided by the author
click here.
2031. RANDOM FIELDS AND PROBABILITY DISTRIBUTIONS WITH GIVEN MARGINALS
ON RANDOMLY CORRELATED SYSTEMS: A GENERAL METHOD AND A PROBLEM
FROM THEORETICAL NEUROSCIENCE
Carlo Fulvi Mari
A class of families of marginal probabilities on sets of discrete
random variables is studied and a necessary and sufficient condition
for the consistency of the given marginals is provided. This result
allows one to verify the consistency of the marginals through a
Boltzmann statistical analysis.
The procedure is then applied in order to verify the hypotheses
assumed in a recent model of neocortical associative areas, according
to which connected modules of neurons are simultaneously active with
probability higher than chance, and inter-modular connections are very
diluted. The verification becomes a typical problem of extremely
diluted spin systems in Boltzmann-Gibbs ensemble. The results
presented here justify the assumptions made in the neuroscientific
theory, and an upper bound to the inter-modular activity correlation
is found.
cfm5@le.ac.uk
- To see a preprint or other
information provided by the author
click here.
2032. CONFIDENCE REGIONS OF THE PARAMETERS IN THE CASE
OF A LINEAR REGRESSION MODEL WITH DEPENDENT ERRORS
R. Kassa and A. Dahmani
In this work, we establish exponential inequalities that allow us
to construct a confidence regions, characterized by ellipses,
for the least sqares estimates of the parameters in the case
of linear regression with psi and alpha-mixing.
rabah_kassa2002@yahoo.fr
- To see a preprint or other
information provided by the author
click here.
2033. THE GENERAL OPTIMAL $L^p$-EUCLIDEAN LOGARITHMIC SOBOLEV
INEQUALITY BY HAMILTON-JACOBI EQUATIONS
Ivan Gentil
We prove a general optimal $L^p$-Euclidean logarithmic Sobolev
inequality by using Prekopa-Leindler inequality and
Hamilton-Jacobi equation. In particular we generalize the
inequality proved by Del-Pino and Dolbeault.
gentil@cict.fr
- To see a preprint or other
information provided by the author
click here.
2034. REAL HARMONIZABLE MULTIFRACTIONAL LÉVY MOTIONS
Céline Lacaux
In this paper, the class of Real Harmonizable Multifractional
Lévy Motions (in short RHMLMs) is introduced. This class is
a generalization of the Multifractional Brownian Motion (in
short MBM) and of the class of Real Harmonizable Fractional
Lévy Motions. One of its main interest is that it contains
some non-Gaussian fields which share many properties with the
MBM.RHMLMs have locally Holder sample paths and their
Holder exponent is allowed to vary along the trajectory.
Moreover these fields are locally asymptotically self-similar.
The multifractional function can be estimated with the
localized generalized quadratic variations.
Celine.Lacaux@math.ups-tlse.fr
- To see a preprint or other
information provided by the author
click here.
2035. LARGE NOISE ASYMPTOTICS FOR ONE-DIMENSIONAL DIFFUSIONS
Szymon Peszat and Francesco Russo
We establish a law of large numbers and a central limit
theorem for a class of additive functionals related to the
solution of a one-dimensional stochastic differential
equation perturbed by a large noise.
russo@math.univ-paris13.fr napeszat@cyf-kr.edu.pl
- To see a preprint or other
information provided by the author
click here.
2036. PRICING AND HEDGING IN INCOMPLETE MARKETS:
FUNDAMENTAL THEOREMS AND ROBUST UTILITY MAXIMIZATION
Jeremy Staum
We prove fundamental theorems of asset pricing for good-deal bounds in incomplete markets,
relating arbitrage-freedom and uniqueness of prices to existence and uniqueness of a
pricing kernel with appropriate properties. The technology employed is duality of convex
optimization in locally convex linear topological spaces. The concepts investigated are
closely related to convex and coherent risk measures, exact functionals, and coherent
lower previsions in the theory of imprecise probabilities. We apply the results to analyze
a specific method for constructing good-deal bounds, based on robust expected utility
involving a unanimity rather than a maxmin criterion.
staum@orie.cornell.edu
2037. HOMOGENEOUS RANDOM MEASURES AND
STRONGLY SUPERMEDIAN KERNELS OF A MARKOV PROCESS
P. J. Fitzsimmons and R. K. Getoor
The potential kernel of a positive left additive functional
(of a strong Markov process $X$) maps positive functions to
strongly supermedian functions and satisfies a variant
of the classical domination principle of potential theory.
Such a kernel $V$ is called a regular strongly supermedian
kernel in recent work of L. Beznea and N. Boboc.
We establish the converse: Every regular strongly supermedian
kernel $V$ is the potential kernel of a random measure
homogeneous on $[0,\infty[$. Under additional finiteness
conditions such random measures give rise to left additive
functionals. We investigate such random measures, their
potential kernels, and their associated characteristic
measures. Given a left additive functional $A$ (not
necessarily continuous), we give an explicit construction of
a simple Markov process $Z$ whose resolvent has initial
kernel equal to the potential kernel $U_A$. The theory we
develop is the probabilistic counterpart of the work of
Beznea and Boboc. Our main tool is the Kuznetsov process
associated with $X$ and a given excessive measure $m$.
pfitzsim@ucsd.edu rgetoor@ucsd.edu
- To see a preprint or other
information provided by the author
click here.
2038. PROBABILISTIC INTERPRETATION AND PARTICLE METHOD FOR VORTEX EQUATIONS WITH
NEUMANN'S BOUNDARY CONDITIONS
Benjamin Jourdain and Sylvie M\'el\'eard
We are interested in proving the convergence of Monte-Carlo approximations
for vortex equations in bounded domains of $R^2$ with Neumann's condition on the
boundary. This work is the first step to justify theorically some
numerical algorithms for Navier-Stokes equations in bounded domains
with a no-slip condition.
We prove that the vortex equation has a unique solution in an
appropriate space and can be interpreted in a
probabilistic point of view through a nonlinear reflected process with
space-time random births on the boundary of the domain.
Next, we approximate the solution $w$ of this vortex equation
by the weighted empirical measure of interacting diffusive particles with
normal reflecting boundary conditions and
space-time random births on the boundary. The weights are related to
the initial data and to the Neumann condition. We can deduce from
this result a simple stochastic particle
algorithm to approximate $w$.
jourdain@cermics.enpc.fr sylm@ccr.jussieu.fr
2039. A REMARK ON HYPERCONTRACTIVITY AND TAIL INEQUALITIES
FOR THE LARGEST EIGENVALUES OF RANDOM MATRICES
Michel Ledoux
We point out a simple argument relying on hypercontractive
bounds to describe tail inequalities on the distribution of
the largest eigenvalues of random matrices at the rate given
by the Tracy-Widom distribution. The result is illustrated
on the known examples of the Gaussian and Laguerre unitary
ensembles. The argument may be applied to describe the
generic tail behavior of eigenfunction measures of
hypercontractive operators.
ledoux@math.ups-tlse.fr
- To see a preprint or other
information provided by the author
click here.
