Probability Abstracts 29
This document contains abstracts 571-592.
They have been mailed on September 29, 1995.
Click here to see the
list of all abstract titles.
571. APPROXIMATIONS AND UPPER BOUNDS ON PROBABILITIES OF LARGE
DEVIATIONS IN THE PROBLEM OF RUIN WITHIN FINITE TIME
Vsevolod K. Malinovskii
In the framework of Andersen's risk model, a new asymptotic expression
and upper bounds on probabilities of ruin after time $t(u)\gg {\mu_T}
{\mu_X}^{-1} u$ and before time $0<t(u)\ll {\mu_T}{\mu_X}^{-1} u$, as
the initial risk reserve $u$ increases to infinity, are suggested. This
result complements the classical normal-type approximation for the
probability of ruin within finite time and is designed as its large
deviations counterpart. The main technical device of the paper, which
is of independent interest, are the upper bounds and the asymptotic
expressions for the probabilities of large deviations of the stopped
random walks, developed under low moment conditions.
malinov@vsevo.mian.su
572. THE CONSENSUS TIMES OF THE MAJORITY VOTE PROCESS ON A TORUS
Dayue Chen
We study the majority vote process on a 2-dimensional torus in which
every voter adopts the minority of opinion with small probability $\delta$.
By analyzing the hierarchic structure of its recurrent classes in detail,
we identify the exponent that the mean of consensus time is asymptotically
(1/$\delta$) to that exponent as $\delta$ goes to 0.
The proof is by a formula of mean exit time and by the metastable theory of
Markov chains developed in the study of the stochastic Ising model.
dayue@stat.pku.edu.cn (AMS TeX available)
573. OPERATOR-STABLE PROCESSES AND OPERATOR FRACTIONAL STABLE MOTIONS
Makoto Maejima
In all literature up tp now, "operator-stable process" {X(t)} means L\'evy
process (that is a process with independent and stationary increments) such
that the distribution of X(1) operator-stable. On the other hand, in the
recent theory of stable processes, following the definitions of Gaussian
processes and Brownian motion that if any finite dimensional distribution of
a process is Gaussian, it is defined as a Gaussian process, and further if
it is L\'evy process, it is called Brownian motion, we use the term "stable
process" in the following way. If any finite dimensional distribution of a
process is stable, it is defined as a stable process, and further if it is
L\'evy process, it is called stable motion (or L\'evy stable motion).
Then a natural question arises: How can we define d-dimensional processes
(which are not L\'evy process but) whose finite-dimensional distributions
are operator-stable? One might define "operator-stable process" as processes
whose finite-dimensional distributions are operator-stable. However, this
seems incomplete as a definition. For, the marginal distributions of
operator-stable processes are not necessarily operator-stable or stable.
In the present paper, the author proposes a new notion of operator-stable
processes and show some examples of operator-stable processes in this
sense.
maejima@math.keio.ac.jp (AMSTex file or hardcopy available)
574. RATES OF CONVERGENCE IN THE OPERATOR-STABLE LIMIT THEOREM
Makoto Maejima and S.T. Rachev
Suppose that $\bold R^d$-valued random vector $\theta$ is strictly
operator-stable in the sense that $\hat\mu$, the characteristic function
of $\theta$, satisfies $\hat\mu (z)^t=\hat\mu (t^{B^*}z)$ for some
invertible linear operator $B$ on $\bold R^d$ and suppose that for
$\{X_i\}$, i.i.d. random vectors in $\bold R^d$, $n^{-B}\sum^n_{i=1} X_i
\overset w \to \to \theta $.
In the present paper, the authors investigate the rates of convergence of it
in terms of several probability metrics.
maejima@math.keio.ac.jp (AMSTex file or hardcopy available)
575. LIMIT THEOREMS RELATED TO A CLASS OF OPERATOR-SELF-SIMILAR PROCESSES
Makoto Maejima
An $\bold R^d$-valed stochastic process $\{ X(t)\}_{t\ge 0}$ is said
to be operator-self-similar if there exists a linear operator $D$ on
$\bold R^d$ such taht for each $t>0$, $\{ X(ct)\} \overset {f.d.} \to =
\{ c^DX(t)\}$. Let $Z_B$ be a purely non-Gaussian operator-stable
random vector with exponent $B$ and let $\{\xi (k)\}_{k\in\bold Z}$
be i.i.d. $\bold R ^d$-valued random vectors such that $n^{-B}\sum_{k=1}
^n\xi (k)\overset w \to\to Z_B$. Let $\{ S_n\}_{n=0}^{\infty}$ be an
integer-valued random walk independent of $\{\xi (k)\}$ such that
$n^{-1/\alpha}S_n\overset w \to\to Z_{\alpha}$, where $Z_{\alpha}$ is
one-dimensional $\alpha$-stable with $1<\le\alpha\le 2$.
In the present paper, the author proves the weak convergence of $\sum_{k=1}
^{[nk]}\xi (S_k)$ under suitable normalization to construct a new class of
operator-self-similar processes. Some new estimates for $P(\| n^{-B}\xi (1)\|
\in A)$ are essentially needed for the proof.
maejima@math.keio.ac.jp (AMSTex file or hardcopy available)
576. SYMMETRIC GIBBS MEASURES
Karl Petersen and Klaus Schmidt
We prove that certain Gibbs measures on subshifts of finite type are
nonsingular and ergodic for certain countable equivalence relations,
including the orbit relation of the adic transformation (the same as
equality after a permutation of finitely many coordinates). The relations
we consider are defined by cocycles taking values in groups, including
some nonabelian ones. This generalizes (half of) the identification of the
invariant ergodic probability measures for the Pascal adic transformation
as exactly the Bernoulli measures---a version of de Finetti's Theorem.
Generalizing the other half, we characterize the measures on subshifts of
finite type that are invariant under both the adic and the shift as the
Gibbs measures whose potential functions depend on only a single
coordinate. There are connections with and implications for
exchangeability, ratio limit theorems for transient Markov chains,
interval splitting procedures, `canonical' Gibbs states, and the
triviality of remote sigma-fields finer than the usual tail field.
petersen@math.unc.edu
klaus.schmidt@univie.ac.at
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577. DETECTING A SINGLE DEFECT IN A SCENERY BY OBSERVING THE
SCENERY ALONG A RANDOM WALK PATH
Harry Kesten
A {\it scenery} on $\Bbb Z$ is a map $\xi:{\Bbb Z} \to
\{0,\dots,k-1\}$; we think of $\xi$ as a coloring of $\Bbb Z$,
which assigns to each point of $\Bbb Z$ one of $k$ colors. For a
given scenery $\xi$, denote by $\hat \xi$ a scenery obtained from
$\xi$ by changing $\xi(0)$ only. Let $\{S_n\}_{n \ge 0}$ be a
simple symmetric random walk on $\Bbb Z$, starting at the
origin. Assume that we observe one of the two sequences
$\{\xi(S_n)\}_{n \ge 0}$ or $\hat \xi(S_n)\}_{n \ge 0}$,
without being told which of the two sequences is observed. If
$\xi$ is known, can we decide (with zero probability of error)
on the basis of these observations which of the two sequences
was observed ? We prove that this can be done for `almost all'
$\xi$, when $ k \ge 5$.
kesten@math.cornell.edu
578. NONSYMMETRIC ORNSTEIN-UHLENBECK SEMIGROUP AS SECOND QUANTIZED OPERATOR
A.Chojnowska-Michalik and B.Goldys
Necessary and sufficient conditions are given for hypercontractivity,
compactness, Hilbert-Schmidt and smoothing properties of nonsymmetric
Ornstein-Uhlenbeck semigroup on a Hilbert space. We provide also necessary and
suficient conditions for the time differentiability of the semigroup.
To prove this we show that the Ornstein-Uhlenbeck semigroup can be obtained by
the second quantization procedure.
beng@solution.maths.unsw.EDU.AU
579. ON AN INTERACTING PARTICLE SYSTEM MODELING AN EPIDEMIC
Rinaldo Schinazi
We consider an interacting particle system on $Z^d$ to
model an epidemic. Each site of $Z^d$ can be in either one of three states:
empty, healthy or infected. Healthy and infected individuals give birth
at different rates to healthy individuals on empty sites.
Healthy individuals get infected by infected individuals.
Infected and healthy individuals die at different rates.
We prove that in dimension 1 and with nearest-neighbor interactions
the epidemic may persist forever if and only if the infection rate
and the rate at which infected individuals give birth
to healthy individuals are high enough.
schinazi@vision.uccs.edu
580. A CONTACT PROCESS WITH A SINGLE INHOMOGENEOUS SITE
Rinaldo Schinazi
The one-dimensional basic contact process is a Markov process for which
particles give birth on vacant nearest neighbor sites at
rate $\lambda>0$ and particles die at rate one.
We introduce a one-dimensional contact process with a single inhomogeneous site:
the evolution is as above except that a particle located at the origin does not
die.
Let $\lambda_c$ be the critical value of the basic contact process.
We show that for $\lambda\not =\lambda_c$ the upper invariant measures of
the inhomogeneous contact process and the basic contact process coincide
except at a finite number of sites.
The behavior at $\lambda=\lambda_c$ is much more interesting:
the upper invariant measure of the inhomogeneous
contact process concentrates on configurations with infinitely many
particles while it is known that the critical basic contact process dies out.
So a single inhomogeneity may provoke a perturbation unbounded in space.
As a byproduct of our analysis
we prove that the connectivity probabilities of the critical basic
contact process are not summable.
We also give a biological interpretation of this model.
schinazi@vision.uccs.edu
581. THE METRIC OF LARGE DEVIATION CONVERGENCE
Tiefeng Jiang and Amir Dembo
We construct a metric space of set functions (Q,d)
such that a sequence {P_n} of Borel probability measures
on a metric space (X,d^*) satisfies the Large Deviation Principle (LDP)
with speed {a_n} and a good rate function if and only if the
sequence {P_n^{a_n} } converges in (Q,d).
Weak convergence of probability measures is another special
case of convergence in (Q,d).
Properties related to the LDP and to weak convergence
are then characterized in terms of (Q,d).
amir@playfair.stanford.edu (LaTeX or Postscript file available)
582. THE TWO-PARAMETER POISSON-DIRICHLET DISTRIBUTION
DERIVED FROM A STABLE SUBORDINATOR
Jim Pitman and Marc Yor
The two-parameter Poisson-Dirichlet distribution, denoted \PD$(\alpha,\theta)$,
is a distribution on the set of decreasing positive sequences with sum 1.
The usual Poisson-Dirichlet distribution with a single parameter $\theta$,
introduced by Kingman, is \PD$(0,\theta)$. Known properties of \PD$(0,\theta)$,
including the Markov chain description due to Vershik-Shmidt-Ignatov,
are generalized to the two-parameter case. The size-biased random permutation
of \PD$(\alpha,\theta)$ is a simple residual allocation model proposed by Engen
in the context of species diversity, and rediscovered by Perman and the authors
in the study of excursions of Brownian motion and Bessel processes.
For $0 < \alpha < 1$, \PD$(\alpha,0)$ is the asymptotic distribution of ranked
lengths of excursions of a Markov chain away from a state whose recurrence time
distribution is in the domain of attraction of a stable law of index $\alpha$.
Formulae in this case trace back to work of Darling, Lamperti and Wendel
in the 1950's and 60's. The distribution of ranked lengths of excursions of a
one-dimensional Brownian motion is \PD$(1/2,0)$, and the corresponding
distribution for Brownian bridge is \PD$(1/2,1/2)$. The \PD$(\alpha,0)$ and
\PD$(\alpha,\alpha)$ distributions admit a similar interpretation in terms of
the ranked lengths of excursions of a semi-stable Markov process whose zero set
is the range of a stable subordinator of index $\alpha$.
Technical Report No. 427, Department of Statistics, U.C. Berkeley
pitman@stat.Berkeley.EDU
583. RANDOM DISCRETE DISTRIBUTIONS DERIVED FROM SELF-SIMILAR RANDOM SETS
Jim Pitman and Marc Yor
A model is proposed for a decreasing sequence of random variables
$(V_1, V_2, \cdots)$ with $\sum_n V_n = 1$, which generalizes the
Poisson-Dirichlet distribution and the distribution of ranked lengths of
excursions of a Brownian motion or recurrent Bessel process.
Let $V_n$ be the length of the $n$th longest component interval of
$[0,1]\backslash Z$, where $Z$ is an a.s. non-empty random closed of
$(0,\infty)$ of Lebesgue measure $0$, and $Z$ is self-similar, i.e.
$cZ$ has the same distribution as $Z$ for every $c > 0$.
Then for $0 \le a < b \le 1$ the expected number of $n$'s such that
$V_n \in (a,b)$ equals $\int_a^b v^{-1} F(dv)$ where the structural distribution
$F$ is identical to the distribution of $1 - \sup ( Z \cap [0,1] )$. Then
$F(dv) = f(v)dv$ where $(1-v) f(v)$ is a decreasing function of $v$, and every
such probability distribution $F$ on $[0,1]$ can arise from this construction.
Technical Report No. 436, Department of Statistics, U.C. Berkeley
pitman@stat.Berkeley.EDU
584. OPTION REPLICATION WITH TRANSACTION COSTS: GENERAL DIFFUSION LIMITS
Hyungsok Ahn, Mohit Dayal, Eric Grannan and Glen Swindle
Transaction costs preclude the construction of hedging strategies
for general contingent claims. Leland (1985) introduced the concept
of diffusion limits of hedging strategies in a small transaction cost
limit. This paper establishes such diffusion limits for very general
hedging strategies and also establishes leading order asymptotic
expressions for the replication error. In addition to subsuming
previously considered temporal strategies, the results in this paper
yield new results, namely expressions for replication errors of stock
price strategies and a variety of ``renewal'' strategies. Most importantly,
this paper provides a unified methodology for calculating hedging strategies
and replication errors in the small transaction cost limit. This is an
essential component of optimization methods, when, for example one is
trying to minimize replication error for a given initial portfolio value.
swindle@pstat.ucsb.edu (hardcopy or Postscript available)
585. A MICROSCOPIC MODEL FOR THE BURGERS EQUATION AND LONGEST
INCREASING SUBSEQUENCES
Timo Seppalainen
We introduce an interacting random process
related to Ulam's problem, or finding
the limit of the normalized longest
increasing subsequence of a random permutation.
The process describes the evolution of a
configuration of sticks on the sites of the
one-dimensional integer lattice.
Our main result is a hydrodynamic scaling limit:
The empirical stick profile converges
to a weak solution of the inviscid Burgers equation under
a scaling of lattice space and time.
The stick process is an alternative view of
Hammersley's particle system recently used by Aldous
and Diaconis to give a new solution to Ulam's problem.
Along the way to the scaling limit we also
produce a solution to this question. The heart
of the proof is that
individual paths of the stochastic process
evolve under a semigroup action which under the scaling
turns into the corresponding action for
the Burgers equation. This semigroup action
for nonlinear scalar conservation laws in one space
variable is developed
in a separate appendix, where we give an existence result
and a uniqueness criterion for solutions of
such equations with initial
data given by a Radon measure.
seppalai@iastate.edu (hardcopy and AMS-TeX file available)
586. AN OPTIMAL TERMINATION TESTING PROCEDURE FOR DISCRETE
EVENT SIMULATIONS
Phillip Feldman, Adviti Muni and Glen Swindle
In this paper we consider discrete-event simulations
which yield results until a termination condition
is satisfied. The simulation can proceed beyond this
time, but no useful information is generated. The time
at which the termination condition will be satisied is
not known initially, and is taken to be randomly
distributed with some prescribed density. It is
necessary, therefore, to periodically check the
termination condition, and this consumes CPU time.
The question that we address is how to distribute checking
times to minimize expected CPU expenditure. We do this
by taking a limit in which the cost of checking is small,
and then minimizing the limiting expected CPU expenditure.
In general uniformly distributed checking times are not optimal.
The layouts of checking times which are generated by our
minimization procedure can significantly outperform constant
checking intervals.
muni@pstat.ucsb.edu (hardcopy or PostScript available)
587. SECOND ORDER REGULAR VARIATION, CONVOLUTION AND THE CENTRAL
LIMIT THEOREM
J. Geluk, L. de Haan, S. Resnick and C. St\u{a}ric\u{a}
Second order regular variation is a refinement of the concept of regular
variation which is useful for studying rates of convergence in extreme value
theory and asymptotic normality of tail estimators. For a distribution tail
$1-F$ which possesses second order regular variation, we discuss how this
property is inherited by $1-F^2$ and $1-F^{*2}$. We also discuss the
relationship of central limit behavior of tail empirical processes,
asymptotic normality of Hill's estimator and second order regular variation.
sid\@orie.cornell.edu starica\@orie.cornell.edu
dehaan\@cs.few.eur.nl geluk\@wstat.few.eur.nl
588. SAMPLE PATH LARGE DEVIATIONS FOR SUPER-BROWNIAN MOTION
Alexander Schied
Two large deviation principles of Freidlin-Wentzell type for rescaled
super-Brownian motion are derived. For one of the appearing rate functions
an integral representation is given and interpreted as "Kakutani-Hellinger
energy". As a tool we develop estimates for the Laplace functionals of
(historical) super-Brownian motion and certain maximal inequalities. Also it
is shown that the H\"older norm of index $\alpha<1/2$ of the process
$t\mapsto <f,X_t>$ possesses some finite exponential moments provided
the function $f$ is smooth.
schied@mathematik.hu-berlin.de (Postscript or dvi file available)
589. HEAVY TAIL MODELLING AND TELETRAFFIC DATA
Sidney I. Resnick
Huge data sets from the teletraffic industry exhibit many
non-standard characteristics such as heavy tails and long range dependence.
Various estimation methods for heavy tailed time series with positive
innovations are reviewed. These include parameter estimation and model
identification methods for autoregressions and moving averages. Parameter
estimation methods include those of Yule-Walker and the linear programming
estimators of Feigin and Resnick as well estimators for tail heaviness such
as the Hill estimator and the qq-estimator. Examples are given using call
holding data and interarrivals between packet transmissions on a computer
network. The limit theory makes heavy use of point process techniques and
random set theory.
sid@orie.cornell.edu (Technical report TR1134 available at "url" below)
- To see some information provided by the author
click here.
590. DOMINATION BY PRODUCT MEASURES
T. M. Liggett, R. H. Schonmann and A. M. Stacey
We consider families of $\{0,1\}$-valued random variables indexed
by the vertices of countable graphs with bounded degree.
First we show that if these random variables satisfy the
property that conditioned on what happens outside of the
neighbourhood of each given site, the probability of seeing a 1
at this site is at least a value $p$ which is large enough, then this
random field dominates a product measure with positive density.
Moreover the density of this dominated product mesure can be made
arbitrarily close to 1, provided that $p$ is close enough to 1.
Next we address the issue of obtaining the critical value of $p$,
defined as the threshold above which the domination by positive-density
product measures is assured. For the graphs which have as vertices the
integers and edges connecting vertices which are separated by no more
than $k$ units, this critical value is shown to be $1-k^k/(k+1)^{k+1}$,
and a discontinuous transition is shown to occur. Similar
critical values of $p$ are found for other classes of probability
measures on $\{0,1\}^Z$. For the class of $k$-dependent measures
the critical value is again $1-k^k/(k+1)^{k+1}$, with a discontinuous
transition. For the class of two-block factors the critical value
is shown to be 1/2 and a continuous transition is shown to take place
in this case. Thus both the critical value and the nature of the
transition are different in the two-block factor and $1$-dependent cases.
tml@math.ucla.edu rhs@math.ucla.edu stacey@math.ucla.edu
591. DYNAMICAL PERCOLATION
Olle H\"aggstr\"om, Yuval Peres and Jeffrey E. Steif
We study bond percolation evolving in time in such a way that the edges
turn on and off independently according to a continuous time stationary
2-state Markov chain. Asking whether an infinite open cluster exists
for a.e. $t$ reduces (by Fubini's Theorem) to ordinary bond percolation.
We ask whether ``a.e. $t$'' can be replaced by ``every $t$'' and show that
for sub- and supercritical percolation the answer is yes (for any graph),
while at criticality the answer is no for certain graphs. For instance,
there exist graphs which do not percolate at criticality for a.e. $t$,
but do percolate for some exceptional $t$. We show that for $\Z^d$,
$d\geq 19$, there is a.s. no infinite open cluster for all $t$ at
criticality. We give a sharp criterion for a general tree to have an
infinite open cluster for some $t$, in terms of the effective conductance
of the tree (analogous to a criterion of R. Lyons for ordinary percolation
on trees). Finally, we compute the Hausdorff dimension of the set of times
for which an infinite open cluster exists on a spherically symmetric tree.
olleh@math.chalmers.se, peres@stat.berkeley.edu, steif@math.chalmers.se
592. EIGENVALUE EXPANSIONS FOR BROWNIAN MOTION WITH AN APPLICATION
TO OCCUPATION TIMES
Richard Bass and Krzysztof Burdzy
Let $B$ be a Borel subset of $R^d$ with
finite volume. We give an eigenvalue expansion for
the transition densities of Brownian motion killed
on exiting $B$. Let $A_1$ be the time spent
by Brownian motion in a closed cone with vertex $0$
until time one. We show that
$\lim_{u\to 0} \log P^0(A_1 <u) /\log u = 1/\xi$
where $\xi$ is defined in terms of the first eigenvalue of the Laplacian
in a compact domain. Eigenvalues of the Laplacian in open and closed
sets are compared.
bass@math.washington.edu or burdzy@math.washington.edu
(plain TeX file available)
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click here.
