[Pas] Probabilty Abstrac 87
pas at www.economia.unimi.it
pas at www.economia.unimi.it
Fri Jul 1 13:04:47 CEST 2005
July 1, 2005
Letter 87
Probability Abstract Service
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3205. RANDOM GRAPHS WITH ARBITRARY I.I.D. DEGREES
Remco van der Hofstad and Gerard Hooghiemstra and Dmitri Znamenski
In this paper we study distances and connectivity properties of
random graphs
with an arbitrary i.i.d. degree sequence. When the tail of the degree
distribution is regularly varying with exponent $1-\tau$ there are three
distinct cases: (i) $\tau>3$, where the degrees have finite variance,
(ii)
$\tau\in (2,3)$, where the degrees have infinite variance, but finite
mean, and
(iii) $\tau\in (1,2)$, where the degrees have infinite mean. These
random
graphs can serve as models for complex networks where degree power
laws are
observed. The distances between pairs of nodes in the three cases
mentioned
above have been studied in three previous publications, and we survey
the
results obtained there. Apart from the critical cases $\tau=1$, $
\tau=2$ and
$\tau=3$, this completes the scaling picture. We explain the results
heuristically and describe related work and open problems. We also
compare the
behavior in this model to Internet data, where a degree power law
with exponent
$\tau\approx 2.2$ is observed.
Furthermore, in this paper we derive results concerning the connected
components and the diameter. We give a criterion when there exists a
unique
largest connected component of size proportional to the size of the
graph, and
study sizes of the other connected components. Also, we show that for
$\tau\in
(2,3)$, which is most often observed in real networks, the diameter
in this
model grows much faster than the typical distance between two
arbitrary nodes.
http://front.math.ucdavis.edu/math.PR/0502580
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3206. THE SINGLE SERVER QUEUE AND THE STORAGE MODEL: LARGE DEVIATIONS
AND FIXED POINTS
Moez Draief
We consider the coupling of a single server queue and a storage model
defined
as a Queue/Store model in Draief et al. 2004. We establish that if
the input
variables both arrivals to the queue and to the store satisfy large
deviations
principles and are linked through an {\em exponential tilting} than
the output
variables (departures from each system) satisfy large deviations
principles
with the same rate function. This generalizes to the context of large
deviations the extension of Burke's Theorem derived in Draief et al.
2004.
http://front.math.ucdavis.edu/math.PR/0503016
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3207. SUBEXPONENTIAL ASYMPTOTICS OF HYBRID FLUID AND RUIN MODELS
Bert Zwart and Sem Borst and Krzystof Debicki
We investigate the tail asymptotics of the supremum of X(t)+Y(t)-ct,
where
X={X(t),t\geq 0} and Y={Y(t),t\geq 0} are two independent stochastic
processes.
We assume that the process Y has subexponential characteristics and
that the
process X is more regular in a certain sense than Y. A key issue
examined in
earlier studies is under what conditions the process X contributes to
large
values of the supremum only through its average behavior. The present
paper
studies various scenarios where the latter is not the case, and the
process X
shows some form of ``atypical'' behavior as well. In particular, we
consider a
fluid model fed by a Gaussian process X and an (integrated) On-Off
process Y.
We show that, depending on the model parameters, the Gaussian process
may
contribute to the tail asymptotics by its moderate deviations, large
deviations, or oscillatory behavior.
http://front.math.ucdavis.edu/math.PR/0503482
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3208. DEVIATION INEQUALITIES VIA COUPLING FOR STOCHASTIC PROCESSES
AND RANDOM FIELDS
J.-R. Chazottes and P. Collet and C. Kuelske and F. Redig
We present a new and simple approach to deviation inequalities for
non-product measures, i.e., for dependent random variables. Our
method is based
on coupling. We illustrate our abstract results with chains with
complete
connections and Gibbsian random fields, both at high and low
temperature.
http://front.math.ucdavis.edu/math.PR/0503483
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3209. AN APPROXIMATE SAMPLING FORMULA UNDER GENETIC HITCHHIKING
A. M. Etheridge and P. Pfaffelhuber and A. Wakolbinger
For a genetic locus carrying a strongly beneficial allele which has just
fixed in a large population we study the ancestry at a linked neutral
locus.
During this ''selective sweep'' the linkage between the two loci is
broken up
by recombination, and the ancestry at the neutral locus is modelled by a
structured coalescent in a random background. For large selection
coefficients
$\alpha$ and under an appropriate scaling of the recombination rate,
we derive
a sampling formula with an order of accuracy of $O((\log\alpha)^{-2})
$ in
probability. In particular we see that, with this order of accuracy,
in a
sample of fixed size there are at most two non-singleton families of
individuals which are identi cal by descent at the neutral locus from
the
beginning of the sweep. This refines a formula going back to the work of
Maynard Smith and Haigh, and co mplements recent work of Schweinsberg
and
Durrett on selective sweeps in the Moran model.
http://front.math.ucdavis.edu/math.PR/0503485
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3210. LARGE DEVIATIONS OF A MODIFIED JACKSON NETWORK: STABILITY AND
ROUGH ASYMPTOTICS
Robert D. Foley and David R. McDonald
Consider a modified, stable, two node Jackson network where server 2
helps
server 1 when server 2 is idle. The probability of a large deviation
of the
number of customers at node one can be calculated using the flat
boundary
theory of Schwartz and Weiss [Large Deviations Performance Analysis
(1994),
Chapman and Hall, New York]. Surprisingly, however, these
calculations show
that the proportion of time spent on the boundary, where server 2 is
idle, may
be zero. This is in sharp contrast to the unmodified Jackson network
which
spends a nonzero proportion of time on this boundary.
http://front.math.ucdavis.edu/math.PR/0503487
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3211. BRIDGES AND NETWORKS: EXACT ASYMPTOTICS
Robert D. Foley and David R. McDonald
We extend the Markov additive methodology developed in [Ann. Appl.
Probab. 9
(1999) 110-145, Ann. Appl. Probab. 11 (2001) 596-607] to obtain the
sharp
asymptotics of the steady state probability of a queueing network
when one of
the nodes gets large. We focus on a new phenomenon we call a bridge.
The bridge
cases occur when the Markovian part of the twisted Markov additive
process is
one null recurrent or one transient, while the jitter cases treated
in [Ann.
Appl. Probab. 9 (1999) 110-145, Ann. Appl. Probab. 11 (2001) 596-607]
occur
when the Markovian part is (one) positive recurrent. The asymptotics
of the
steady state is an exponential times a polynomial term in the bridge
case, but
is purely exponential in the jitter case. We apply this theory to a
modified,
stable, two node Jackson network where server two helps server one
when server
two is idle. We derive the sharp asymptotics of the steady state
distribution
of the number of customers queued at each node as the number of
customers
queued at the server one grows large. In so doing we get an intuitive
understanding of the companion paper [Ann. Appl. Probab. 15 (2005)
519-541]
which gives a large deviation analysis of this problem using the flat
boundary
theory in the book by Shwartz and Weiss. Unlike the (unscaled) large
deviation
path of a Jackson network which jitters along the boundary, the
unscaled large
deviation path of the modified network tries to avoid the boundary
where server
two helps server one (and forms a bridge).
http://front.math.ucdavis.edu/math.PR/0503488
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3212. UPPER BOUNDS FOR SPATIAL POINT PROCESS APPROXIMATIONS
Dominic Schuhmacher
We consider the behavior of spatial point processes when subjected to
a class
of linear transformations indexed by a variable T. It was shown in
Ellis [Adv.
in Appl. Probab. 18 (1986) 646-659] that, under mild assumptions, the
transformed processes behave approximately like Poisson processes for
large T.
In this article, under very similar assumptions, explicit upper
bounds are
given for the d_2-distance between the corresponding point process
distributions. A number of related results, and applications to
kernel density
estimation and long range dependence testing are also presented. The
main
results are proved by applying a generalized Stein-Chen method to
discretized
versions of the point processes.
http://front.math.ucdavis.edu/math.PR/0503491
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3213. NOISE STABILITY OF FUNCTIONS WITH LOW INFLUENCES: INVARIANCE
AND OPTIMALITY
Elchanan Mossel and Ryan O'Donnell and Krzysztof Oleszkiewicz
In this paper we study functions with low influences on product
probability
spaces. The analysis of boolean functions with low influences has
become a
central problem in discrete Fourier analysis. It is motivated by
fundamental
questions arising from the construction of probabilistically
checkable proofs
in theoretical computer science and from problems in the theory of
social
choice in economics.
We prove an invariance principle for multilinear polynomials with low
influences and bounded degree; it shows that under mild conditions the
distribution of such polynomials is essentially invariant for all
product
spaces. Ours is one of the very few known non-linear invariance
principles. It
has the advantage that its proof is simple and that the error bounds are
explicit. We also show that the assumption of bounded degree can be
eliminated
if the polynomials are slightly ``smoothed''; this extension is
essential for
our applications to ``noise stability''-type problems.
In particular, as applications of the invariance principle we
prove two
conjectures: the ``Majority Is Stablest'' conjecture from theoretical
computer
science, which was the original motivation for this work, and the
``It Ain't
Over Till It's Over'' conjecture from social choice theory.
http://front.math.ucdavis.edu/math.PR/0503503
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3214. LOGARITHMIC SOBOLEV INEQUALITY FOR LOG-CONCAVE MEASURE FROM
PREKOPA-LEINDLER INEQUALITY
Ivan Gentil
We develop in this paper an amelioration of the method given by S.
Bobkov and
M. Ledoux in GAFA (2000). We prove by Prekopa-Leindler Theorem an
optimal
modified logarithmic Sobolev inequality adapted for all log-concave
measure on
$\dR^n$. This inequality implies results proved by Bobkov and Ledoux,
the
Euclidean Logarithmic Sobolev inequality generalized in the last
years and it
also implies some convex logarithmic Sobolev inequalities for large
entropy.
http://front.math.ucdavis.edu/math.FA/0503476
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3215. EQUILIBRIUM GLAUBER AND KAWASAKI DYNAMICS OF CONTINUOUS
PARTICLE SYSTEMS
Yu. G. Kondratiev and E. Lytvynov and M. R\"ockner
We construct two types of equilibrium dynamics of infinite particle
systems
in a Riemannian manifold $X$. These dynamics are analogs of the Glauber,
respectively Kawasaki dynamics of lattice spin systems. The Glauber
dynamics
now is a process where interacting particles randomly appear and
disappear,
i.e., it is a birth-and-death process in $X$, while in the Kawasaki
dynamics
interacting particles randomly jump over $X$. We establish conditions
on a
priori explicitly given symmetrizing measures and generators of both
dynamics
under which corresponding conservative Markov processes exist.
http://front.math.ucdavis.edu/math.PR/0503042
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3216. THE STEPPING STONE MODEL. II: GENEALOGIES AND THE INFINITE
SITES MODEL
Iljana Zahle and J. Theodore Cox and Richard Durrett
This paper extends earlier work by Cox and Durrett, who studied the
coalescence times for two lineages in the stepping stone model on the
two-dimensional torus. We show that the genealogy of a sample of size
n is
given by a time change of Kingman's coalescent. With DNA sequence
data in mind,
we investigate mutation patterns under the infinite sites model,
which assumes
that each mutation occurs at a new site. Our results suggest that the
spatial
structure of the human population contributes to the haplotype
structure and a
slower than expected decay of genetic correlation with distance
revealed by
recent studies of the human genome.
http://front.math.ucdavis.edu/math.PR/0503512
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3217. RENEWAL THEORY AND COMPUTABLE CONVERGENCE RATES FOR
GEOMETRICALLY ERGODIC MARKOV CHAINS
Peter H. Baxendale
We give computable bounds on the rate of convergence of the transition
probabilities to the stationary distribution for a certain class of
geometrically ergodic Markov chains. Our results are different from
earlier
estimates of Meyn and Tweedie, and from estimates using coupling,
although we
start from essentially the same assumptions of a drift condition
toward a
``small set.'' The estimates show a noticeable improvement on
existing results
if the Markov chain is reversible with respect to its stationary
distribution,
and especially so if the chain is also positive. The method of proof
uses the
first-entrance-last-exit decomposition, together with new
quantitative versions
of a result of Kendall from discrete renewal theory.
http://front.math.ucdavis.edu/math.PR/0503515
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3218. UTILITY MAXIMIZATION WITH A STOCHASTIC CLOCK AND AN UNBOUNDED
RANDOM ENDOWMENT
Gordan Zitkovic
We introduce a linear space of finitely additive measures to treat the
problem of optimal expected utility from consumption under a
stochastic clock
and an unbounded random endowment process. In this way we establish
existence
and uniqueness for a large class of utility-maximization problems
including the
classical ones of terminal wealth or consumption, as well as the
problems that
depend on a random time horizon or multiple consumption instances. As an
example we explicitly treat the problem of maximizing the logarithmic
utility
of a consumption stream, where the local time of an Ornstein-
Uhlenbeck process
acts as a stochastic clock.
http://front.math.ucdavis.edu/math.PR/0503516
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3219. RECONSTRUCTING A TWO-COLOR SCENERY BY OBSERVING IT ALONG A
SIMPLE RANDOM WALK PATH
Heinrich Matzinger
Let {\xi (n)}_{n\in Z} be a two-color random scenery, that is, a random
coloring of Z in two colors, such that the \xi (i)'s are i.i.d.
Bernoulli
variables with parameter \tfrac12. Let {S(n)}_{n\in N} be a symmetric
random
walk starting at 0. Our main result shows that a.s., \xi \circ S (the
composition of \xi and S) determines \xi up to translation and
reflection. In
other words, by observing the scenery \xi along the random walk path
S, we can
a.s. reconstruct \xi up to translation and reflection. This result
gives a
positive answer to the question of H. Kesten of whether one can a.s.
detect a
single defect in almost every two-color random scenery by observing
it only
along a random walk path.
http://front.math.ucdavis.edu/math.PR/0503517
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3220. A DIFFUSION MODEL OF SCHEDULING CONTROL IN QUEUEING SYSTEMS
WITH MANY SERVERS
Rami Atar
This paper studies a diffusion model that arises as the limit of a
queueing
system scheduling problem in the asymptotic heavy traffic regime of
Halfin and
Whitt. The queueing system consists of several customer classes and many
servers working in parallel, grouped in several stations. Servers in
different
stations offer service to customers of each class at possibly
different rates.
The control corresponds to selecting what customer class each server
serves at
each time. The diffusion control problem does not seem to have explicit
solutions and therefore a characterization of optimal solutions via the
Hamilton-Jacobi-Bellman equation is addressed. Our main result is the
existence
and uniqueness of solutions of the equation. Since the model is set
on an
unbounded domain and the cost per unit time is unbounded, the
analysis requires
estimates on the state process that are subexponential in the time
variable. In
establishing these estimates, a key role is played by an integral
formula that
relates queue length and idle time processes, which may be of
independent
interest.
http://front.math.ucdavis.edu/math.PR/0503518
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3221. EXACT AND APPROXIMATE RESULTS FOR DEPOSITION AND ANNIHILATION
PROCESSES ON GRAPHS
Mathew D. Penrose and Aidan Sudbury
We consider random sequential adsorption processes where the
initially empty
sites of a graph are irreversibly occupied, in random order, either
by monomers
which block neighboring sites, or by dimers. We also consider a
process where
initially occupied sites annihilate their neighbors at random times.
We verify
that these processes are well defined on infinite graphs, and derive
forward
equations governing joint vacancy/occupation probabilities. Using
these, we
derive exact formulae for occupation probabilities and pair
correlations in
Bethe lattices. For the blocking and annihilation processes we also
prove
positive correlations between sites an even distance apart, and for
blocking we
derive rigorous lower bounds for the site occupation probability in
lattices,
including a lower bound of 1/3 for Z^2. We also give normal
approximation
results for the number of occupied sites in a large finite graph.
http://front.math.ucdavis.edu/math.PR/0503519
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3222. NEAR-INTEGRATED GARCH SEQUENCES
Istvan Berkes and Lajos Horvath and Piotr Kokoszka
Motivated by regularities observed in time series of returns on
speculative
assets, we develop an asymptotic theory of GARCH(1,1) processes {y_k}
defined
by the equations y_k=\sigma_k\epsilon_k, \sigma_k^2=\omega +\alpha
y_{k-1}^2+\beta \sigma_{k-1}^2 for which the sum \alpha +\beta
approaches unity
as the number of available observations tends to infinity. We call such
sequences near-integrated. We show that the asymptotic behavior of
near-integrated GARCH(1,1) processes critically depends on the sign
of \gamma
:=\alpha +\beta -1. We find assumptions under which the solutions
exhibit
increasing oscillations and show that these oscillations grow
approximately
like a power function if \gamma \leq 0 and exponentially if \gamma
>0. We
establish an additive representation for the near-integrated GARCH(1,1)
processes which is more convenient to use than the traditional
multiplicative
Volterra series expansion.
http://front.math.ucdavis.edu/math.PR/0503520
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3223. ASYMPTOTICS IN RANDOMIZED URN MODELS
Zhi-Dong Bai and Feifang Hu
This paper studies a very general urn model stimulated by designs in
clinical
trials, where the number of balls of different types added to the urn
at trial
n depends on a random outcome directed by the composition at trials
1,2,...,n-1. Patient treatments are allocated according to types of
balls. We
establish the strong consistency and asymptotic normality for both
the urn
composition and the patient allocation under general assumptions on
random
generating matrices which determine how balls are added to the urn.
Also we
obtain explicit forms of the asymptotic variance-covariance matrices
of both
the urn composition and the patient allocation. The conditions on the
nonhomogeneity of generating matrices are mild and widely satisfied in
applications. Several applications are also discussed.
http://front.math.ucdavis.edu/math.PR/0503521
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3224. A BERRY-ESSEEN THEOREM FOR FEYNMAN-KAC AND INTERACTING PARTICLE
MODELS
Pierre Del Moral and Samy Tindel
In this paper we investigate the speed of convergence of the
fluctuations of
a general class of Feynman-Kac particle approximation models. We
design an
original approach based on new Berry-Esseen type estimates for abstract
martingale sequences combined with original exponential concentration
estimates
of interacting processes. These results extend the corresponding
statements in
the classical theory and apply to a class of branching and genealogical
path-particle models arising in nonlinear filtering literature as
well as in
statistical physics and biology.
http://front.math.ucdavis.edu/math.PR/0503522
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3225. PERIODIC COPOLYMERS AT SELECTIVE INTERFACES: A LARGE DEVIATIONS
APPROACH
Erwin Bolthausen and Giambattista Giacomin
We analyze a (1+1)-dimension directed random walk model of a polymer
dipped
in a medium constituted by two immiscible solvents separated by a flat
interface. The polymer chain is heterogeneous in the sense that a single
monomer may energetically favor one or the other solvent. We focus on
the case
in which the polymer types are periodically distributed along the
chain or, in
other words, the polymer is constituted of identical stretches of
fixed length.
The phenomenon that one wants to analyze is the localization at the
interface:
energetically favored configurations place most of the monomers in the
preferred solvent and this can be done only if the polymer sticks
close to the
interface. We investigate, by means of large deviations, the energy-
entropy
competition that may lead, according to the value of the parameters (the
strength of the coupling between monomers and solvents and an asymmetry
parameter), to localization. We express the free energy of the system
in terms
of a variational formula that we can solve. We then use the result to
analyze
the phase diagram.
http://front.math.ucdavis.edu/math.PR/0503523
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3226. HITTING DISTRIBUTIONS OF GEOMETRIC BROWNIAN MOTION
T. Byczkowski and M. Ryznar
Let $\tau$ be the first hitting time of the point 1 by the geometric
Brownian
motion $X(t)= x \exp(B(t)-2\mu t)$ with drift $\mu \geq 0$ starting
from $x>1$.
Here $B(t)$ is the Brownian motion starting from 0 with $E^0 B^2(t) =
2t$. We
provide an integral formula for the density function of the stopped
exponential
functional $A(\tau)=\int_0^\tau X^2(t) dt$ and determine its asymptotic
behaviour at infinity. Although we basically rely on methods
developed in
\cite{BGS}, the present paper also covers the case of arbitrary
drifts $\mu
\geq 0$ and provides a significant unification and extension of
results of the
above-mentioned paper. As a corollary we provide an integral formula
and give
asymptotic behaviour at infinity of the Poisson kernel for half-
spaces for
Brownian motion with drift in real hyperbolic spaces of arbitrary
dimension.
http://front.math.ucdavis.edu/math.PR/0503060
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3227. MASS EXTINCTIONS: AN ALTERNATIVE TO THE ALLEE EFFECT
Rinaldo B. Schinazi
We introduce a spatial stochastic process on the lattice Z^d to model
mass
extinctions. Each site of the lattice may host a flock of up to N
individuals.
Each individual may give birth to a new individual at the same site
at rate
\phi until the maximum of N individuals has been reached at the site.
Once the
flock reaches N individuals, then, and only then, it starts giving
birth on
each of the 2d neighboring sites at rate \lambda(N). Finally,
disaster strikes
at rate 1, that is, the whole flock disappears. Our model shows that,
at least
in theory, there is a critical maximum flock size above which a
species is
certain to disappear and below which it may survive.
http://front.math.ucdavis.edu/math.PR/0503525
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3228. TAIL OF A LINEAR DIFFUSION WITH MARKOV SWITCHING
Benoite de Saporta and Jian-Feng Yao
Let Y be an Ornstein-Uhlenbeck diffusion governed by a stationary and
ergodic
Markov jump process X: dY_t=a(X_t)Y_t dt+\sigma(X_t) dW_t, Y_0=y_0.
Ergodicity
conditions for Y have been obtained. Here we investigate the tail
propriety of
the stationary distribution of this model. A characterization of
either heavy
or light tail case is established. The method is based on a renewal
theorem for
systems of equations with distributions on R.
http://front.math.ucdavis.edu/math.PR/0503527
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3229. THE LONG-RUN BEHAVIOR OF THE STOCHASTIC REPLICATOR DYNAMICS
Lorens A. Imhof
Fudenberg and Harris' stochastic version of the classical replicator
dynamics
is considered. The behavior of this diffusion process in the presence
of an
evolutionarily stable strategy is investigated. Moreover, extinction of
dominated strategies and stochastic stability of strict Nash
equilibria are
studied. The general results are illustrated in connection with a
discrete war
of attrition. A persistence result for the maximum effort strategy is
obtained
and an explicit expression for the evolutionarily stable strategy is
derived.
http://front.math.ucdavis.edu/math.PR/0503529
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3230. OPTIMAL POINTWISE APPROXIMATION OF SDES BASED ON BROWNIAN
MOTION AT DISCRETE POINTS
Thomas Muller-Gronbach
We study pathwise approximation of scalar stochastic differential
equations
at a single point. We provide the exact rate of convergence of the
minimal
errors that can be achieved by arbitrary numerical methods that are
based (in a
measurable way) on a finite number of sequential observations of the
driving
Brownian motion. The resulting lower error bounds hold in particular
for all
methods that are implementable on a computer and use a random number
generator
to simulate the driving Brownian motion at finitely many points. Our
analysis
shows that approximation at a single point is strongly connected to an
integration problem for the driving Brownian motion with a random
weight.
Exploiting general ideas from estimation of weighted integrals of
stochastic
processes, we introduce an adaptive scheme, which is easy to
implement and
performs asymptotically optimally.
http://front.math.ucdavis.edu/math.PR/0503531
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3231. QUANTITATIVE BOUNDS ON CONVERGENCE OF TIME-INHOMOGENEOUS MARKOV
CHAINS
R. Douc and E. Moulines and Jeffrey S. Rosenthal
Convergence rates of Markov chains have been widely studied in recent
years.
In particular, quantitative bounds on convergence rates have been
studied in
various forms by Meyn and Tweedie [Ann. Appl. Probab. 4 (1994)
981-1101],
Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566], Roberts and
Tweedie
[Stochastic Process. Appl. 80 (1999) 211-229], Jones and Hobert
[Statist. Sci.
16 (2001) 312-334] and Fort [Ph.D. thesis (2001) Univ. Paris VI]. In
this
paper, we extend a result of Rosenthal [J. Amer. Statist. Assoc. 90
(1995)
558-566] that concerns quantitative convergence rates for time-
homogeneous
Markov chains. Our extension allows us to consider f-total variation
distance
(instead of total variation) and time-inhomogeneous Markov chains. We
apply our
results to simulated annealing.
http://front.math.ucdavis.edu/math.PR/0503532
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3232. ON STATIONARITY OF LAGRANGIAN OBSERVATIONS OF PASSIVE TRACER
VELOCITY IN A COMPRESSIBLE ENVIRONMENT
Tomasz Komorowski and Grzegorz Krupa
We study the transport of a passive tracer particle in a steady strongly
mixing flow with a nonzero mean velocity. We show that there exists a
probability measure under which the particle Lagrangian velocity
process is
stationary. This measure is absolutely continuous with respect to the
underlying probability measure for the Eulerian flow.
http://front.math.ucdavis.edu/math.PR/0503534
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3233. EXTENDING CHACON-WALSH: MINIMALITY AND GENERALISED STARTING
DISTRIBUTIONS
Alexander Cox
In this paper we consider the Skorokhod embedding problem for general
starting and target measures. In particular, we provide necessary and
sufficient conditions for a stopping time to be minimal in the sense of
Monroe(1972). The resulting conditions have a nice interpretation in the
graphical picture of Chacon and Walsh. Further, we demonstrate how the
construction of Chacon and Walsh can be extended to any (integrable)
starting
and target distributions, allowing the constructions of Azema-Yor,
Vallois and
Jacka to be viewed in this context, and thus extended easily to general
starting and target distributions. In particular, we describe in
detail the
extension of the Azema-Yor embedding in this context, and show that
it retains
its optimality property.
http://front.math.ucdavis.edu/math.PR/0503535
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3234. EXPONENTIAL PENALTY FUNCTION CONTROL OF LOSS NETWORKS
Garud Iyengar and Karl Sigman
We introduce penalty-function-based admission control policies to
approximately maximize the expected reward rate in a loss network. These
control policies are easy to implement and perform well both in the
transient
period as well as in steady state. A major advantage of the penalty
approach is
that it avoids solving the associated dynamic program. However, a
disadvantage
of this approach is that it requires the capacity requested by
individual
requests to be sufficiently small compared to total available
capacity. We
first solve a related deterministic linear program (LP) and then
translate an
optimal solution of the LP into an admission control policy for the loss
network via an exponential penalty function. We show that the penalty
policy is
a target-tracking policy--it performs well because the optimal
solution of the
LP is a good target. We demonstrate that the penalty approach can be
extended
to track arbitrarily defined target sets. Results from preliminary
simulation
studies are included.
http://front.math.ucdavis.edu/math.PR/0503536
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3235. ELEMENTARY BOUNDS ON POINCARE AND LOG-SOBOLEV CONSTANTS FOR
DECOMPOSABLE MARKOV CHAINS
Mark Jerrum and Jung-Bae Son and Prasad Tetali and Eric Vigoda
We consider finite-state Markov chains that can be naturally
decomposed into
smaller ``projection'' and ``restriction'' chains. Possibly this
decomposition
will be inductive, in that the restriction chains will be smaller
copies of the
initial chain. We provide expressions for Poincare (resp. log-Sobolev)
constants of the initial Markov chain in terms of Poincare (resp. log-
Sobolev)
constants of the projection and restriction chains, together with
further a
parameter. In the case of the Poincare constant, our bound is always
at least
as good as existing ones and, depending on the value of the extra
parameter,
may be much better. There appears to be no previously published
decomposition
result for the log-Sobolev constant. Our proofs are elementary and
self-contained.
http://front.math.ucdavis.edu/math.PR/0503537
---------------------------------------------------------------
3236. RUIN PROBABILITIES AND OVERSHOOTS FOR GENERAL LEVY INSURANCE
RISK PROCESSES
Claudia Kluppelberg and Andreas E. Kyprianou and Ross A. Maller
We formulate the insurance risk process in a general Levy process
setting,
and give general theorems for the ruin probability and the asymptotic
distribution of the overshoot of the process above a high level, when
the
process drifts to -\infty a.s. and the positive tail of the Levy
measure, or of
the ladder height measure, is subexponential or, more generally,
convolution
equivalent. Results of Asmussen and Kluppelberg [Stochastic Process.
Appl. 64
(1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28
(1996) 207-226]
for ruin probabilities and the overshoot in random walk and compound
Poisson
models are shown to have analogues in the general setup. The
identities we
derive open the way to further investigation of general renewal-type
properties
of Levy processes.
http://front.math.ucdavis.edu/math.PR/0503539
---------------------------------------------------------------
3237. COMBINATORIAL ASPECTS OF MATRIX MODELS
Alice Guionnet and \'Edouard Maurel-Segala
We show that under reasonably general assumptions, the first order
asymptotics of the free energy of matrix models are generating
functions for
colored planar maps. This is based on the fact that solutions of the
differential Schwinger-Dyson equations are, by nature, generating
functions for
enumerating planar maps, a remark which bypasses the use of Gaussian
calculus.
http://front.math.ucdavis.edu/math.PR/0503064
---------------------------------------------------------------
3238. STABILITY IN DISTRIBUTION OF RANDOMLY PERTURBED QUADRATIC MAPS
AS MARKOV PROCESSES
Rabi Bhattacharya and Mukul Majumdar
Iteration of randomly chosen quadratic maps defines a Markov process:
X_{n+1}=\epsilon_{n+1}X_n(1-X_n), where \epsilon_n are i.i.d. with
values in
the parameter space [0,4] of quadratic maps F_{\theta}(x)=\theta x(1-
x). Its
study is of significance as an important Markov model, with
applications to
problems of optimization under uncertainty arising in economics. In this
article a broad criterion is established for positive Harris
recurrence of X_n.
http://front.math.ucdavis.edu/math.PR/0503540
---------------------------------------------------------------
3239. INTERPLAY BETWEEN DIVIDEND RATE AND BUSINESS CONSTRAINTS FOR A
FINANCIAL CORPORATION
Tahir Choulli and Michael Taksar and Xun Yu Zhou
We study a model of a corporation which has the possibility to choose
various
production/business policies with different expected profits and
risks. In the
model there are restrictions on the dividend distribution rates as
well as
restrictions on the risk the company can undertake. The objective is to
maximize the expected present value of the total dividend
distributions. We
outline the corresponding Hamilton-Jacobi-Bellman equation, compute
explicitly
the optimal return function and determine the optimal policy. As a
consequence
of these results, the way the dividend rate and business constraints
affect the
optimal policy is revealed. In particular, we show that under certain
relationships between the constraints and the exogenous parameters of
the
random processes that govern the returns, some business activities
might be
redundant, that is, under the optimal policy they will never be used
in any
scenario.
http://front.math.ucdavis.edu/math.PR/0503541
---------------------------------------------------------------
3240. LIMIT THEOREMS FOR MIXED MAX-SUM PROCESSES WITH RENEWAL STOPPING
Dmitrii S. Silvestrov and Jozef L. Teugels
This article is devoted to the investigation of limit theorems for mixed
max-sum processes with renewal type stopping indexes. Limit theorems
of weak
convergence type are obtained as well as functional limit theorems.
http://front.math.ucdavis.edu/math.PR/0503543
---------------------------------------------------------------
3241. CONTINUUM PERCOLATION WITH STEPS IN AN ANNULUS
Paul Balister and Bela Bollobas and Mark Walters
Let A be the annulus in R^2 centered at the origin with inner and
outer radii
r(1-\epsilon) and r, respectively. Place points {x_i} in R^2
according to a
Poisson process with intensity 1 and let G_A be the random graph with
vertex
set {x_i} and edges x_ix_j whenever x_i-x_j\in A. We show that if the
area of A
is large, then G_A almost surely has an infinite component. Moreover,
if we fix
\epsilon, increase r and let n_c=n_c(\epsilon) be the area of A when
this
infinite component appears, then n_c\to1 as \epsilon \to 0. This is
in contrast
to the case of a ``square'' annulus where we show that n_c is bounded
away from
1.
http://front.math.ucdavis.edu/math.PR/0503544
---------------------------------------------------------------
3242. A MICROSCOPIC PROBABILISTIC DESCRIPTION OF A LOCALLY REGULATED
POPULATION AND MACROSCOPIC APPROXIMATIONS
Nicolas Fournier and Sylvie Meleard
We consider a discrete model that describes a locally regulated spatial
population with mortality selection. This model was studied in
parallel by
Bolker and Pacala and Dieckmann, Law and Murrell. We first generalize
this
model by adding spatial dependence. Then we give a pathwise
description in
terms of Poisson point measures. We show that different
normalizations may lead
to different macroscopic approximations of this model. The first
approximation
is deterministic and gives a rigorous sense to the number density.
The second
approximation is a superprocess previously studied by Etheridge.
Finally, we
study in specific cases the long time behavior of the system and of its
deterministic approximation.
http://front.math.ucdavis.edu/math.PR/0503546
---------------------------------------------------------------
3243. STABILITY AND THE LYAPOUNOV EXPONENT OF THRESHOLD AR-ARCH MODELS
Daren B. H. Cline and Huay-min H. Pu
The Lyapounov exponent and sharp conditions for geometric ergodicity are
determined of a time series model with both a threshold
autoregression term and
threshold autoregressive conditional heteroscedastic (ARCH) errors.
The conditions require studying or simulating the behavior of a
bounded,
ergodic Markov chain. The method of proof is based on a new approach,
called
the piggyback method, that exploits the relationship between the time
series
and the bounded chain. The piggyback method also provides a means for
evaluating the Lyapounov exponent by simulation and provides a new
perspective
on moments, illuminating recent results for the distribution tails of
GARCH
models.
http://front.math.ucdavis.edu/math.PR/0503547
---------------------------------------------------------------
3244. NORMAL APPROXIMATION FOR HIERARCHICAL STRUCTURES
Larry Goldstein
Given F:[a,b]^k\to [a,b] and a nonconstant X_0 with P(X_0\in [a,b])
=1, define
the hierarchical sequence of random variables {X_n}_{n\ge 0} by
X_{n+1}=F(X_{n,1},...,X_{n,k}), where X_{n,i} are i.i.d. as X_n. Such
sequences
arise from hierarchical structures which have been extensively
studied in the
physics literature to model, for example, the conductivity of a
random medium.
Under an averaging and smoothness condition on nontrivial F, an upper
bound of
the form C\gamma^n for 0<\gamma<1 is obtained on the Wasserstein
distance
between the standardized distribution of X_n and the normal. The
results apply,
for instance, to random resistor networks and, introducing the notion
of strict
averaging, to hierarchical sequences generated by certain
compositions. As an
illustration, upper bounds on the rate of convergence to the normal
are derived
for the hierarchical sequence generated by the weighted diamond
lattice which
is shown to exhibit a full range of convergence rate behavior.
http://front.math.ucdavis.edu/math.PR/0503549
---------------------------------------------------------------
3245. ON THE SUPER REPLICATION PRICE OF UNBOUNDED CLAIMS
Sara Biagini and Marco Frittelli
In an incomplete market the price of a claim f in general cannot be
uniquely
identified by no arbitrage arguments. However, the ``classical'' super
replication price is a sensible indicator of the (maximum selling)
value of the
claim. When f satisfies certain pointwise conditions (e.g., f is
bounded from
below), the super replication price is equal to sup_QE_Q[f], where Q
varies on
the whole set of pricing measures. Unfortunately, this price is often
too high:
a typical situation is here discussed in the examples. We thus define
the less
expensive weak super replication price and we relax the requirements
on f by
asking just for ``enough'' integrability conditions. By building up a
proper
duality theory, we show its economic meaning and its relation with the
investor's preferences. Indeed, it turns out that the weak super
replication
price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi}
is the
class of pricing measures with finite generalized entropy (i.e., E[\Phi
(\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of
the utility
function of the investor.
http://front.math.ucdavis.edu/math.PR/0503550
---------------------------------------------------------------
3246. LIMIT LAWS OF ESTIMATORS FOR CRITICAL MULTI-TYPE GALTON-WATSON
PROCESSES
Zhiyi Chi
We consider the asymptotics of various estimators based on a large
sample of
branching trees from a critical multi-type Galton-Watson process, as
the sample
size increases to infinity. The asymptotics of additive functions of
trees,
such as sizes of trees and frequencies of types within trees, a
higher-order
asymptotic of the ``relative frequency'' estimator of the left
eigenvector of
the mean matrix, a higher-order joint asymptotic of the maximum
likelihood
estimators of the offspring probabilities and the consistency of an
estimator
of the right eigenvector of the mean matrix, are established.
http://front.math.ucdavis.edu/math.PR/0503552
---------------------------------------------------------------
3247. ON SAMPLING OF STATIONARY INCREMENT PROCESSES
J. M. P. Albin
Under a complex technical condition, similar to such used in extreme
value
theory, we find the rate q(\epsilon)^{-1} at which a stochastic
process with
stationary increments \xi should be sampled, for the sampled process
\xi(\lfloor\cdot /q(\epsilon)\rfloor q(\epsilon)) to deviate from \xi
by at
most \epsilon, with a given probability, asymptotically as \epsilon
\downarrow0. The canonical application is to discretization errors in
computer
simulation of stochastic processes.
http://front.math.ucdavis.edu/math.PR/0503554
---------------------------------------------------------------
3248. RECURRENCE OF SIMPLE RANDOM WALK ON $Z^2$ IS DYNAMICALLY SENSITIVE
Christopher Hoffman
Benjamini, Haggstrom, Peres and Steif introduced the concept of a
dynamical
random walk. This is a continuous family of random walks, {S_n(t)}.
Benjamini
et. al. proved that if d=3 or d=4 then there is an exceptional set of
t such
that {S_n(t)} returns to the origin infinitely often. In this paper
we consider
a dynamical random walk on Z^2. We show that with probability one
there exists
t such that {S_n(t)} never returns to the origin. This exceptional
set of times
has dimension one. This proves a conjecture of Benjamini et. al.
http://front.math.ucdavis.edu/math.PR/0503065
---------------------------------------------------------------
3249. SPECTRAL PROPERTIES OF THE TANDEM JACKSON NETWORK, SEEN AS A
QUASI-BIRTH-AND-DEATH PROCESS
D. P. Kroese and W. R. W. Scheinhardt and P. G. Taylor
Quasi-birth-and-death (QBD) processes with infinite ``phase spaces'' can
exhibit unusual and interesting behavior. One of the simplest
examples of such
a process is the two-node tandem Jackson network, with the ``phase''
giving the
state of the first queue and the ``level'' giving the state of the
second
queue. In this paper, we undertake an extensive analysis of the
properties of
this QBD. In particular, we investigate the spectral properties of
Neuts's
R-matrix and show that the decay rate of the stationary distribution
of the
``level'' process is not always equal to the convergence norm of R.
In fact, we
show that we can obtain any decay rate from a certain range by
controlling only
the transition structure at level zero, which is independent of R. We
also
consider the sequence of tandem queues that is constructed by
restricting the
waiting room of the first queue to some finite capacity, and then
allowing this
capacity to increase to infinity. We show that the decay rates for
the finite
truncations converge to a value, which is not necessarily the decay
rate in the
infinite waiting room case. Finally, we show that the probability
that the
process hits level n before level 0 given that it starts in level 1
decays at a
rate which is not necessarily the same as the decay rate for the
stationary
distribution.
http://front.math.ucdavis.edu/math.PR/0503555
---------------------------------------------------------------
3250. NUMBER OF PATHS VERSUS NUMBER OF BASIS FUNCTIONS IN AMERICAN
OPTION PRICING
Paul Glasserman and Bin Yu
An American option grants the holder the right to select the time at
which to
exercise the option, so pricing an American option entails solving an
optimal
stopping problem. Difficulties in applying standard numerical methods to
complex pricing problems have motivated the development of techniques
that
combine Monte Carlo simulation with dynamic programming. One class of
methods
approximates the option value at each time using a linear combination
of basis
functions, and combines Monte Carlo with backward induction to
estimate optimal
coefficients in each approximation. We analyze the convergence of
such a method
as both the number of basis functions and the number of simulated paths
increase. We get explicit results when the basis functions are
polynomials and
the underlying process is either Brownian motion or geometric
Brownian motion.
We show that the number of paths required for worst-case convergence
grows
exponentially in the degree of the approximating polynomials in the
case of
Brownian motion and faster in the case of geometric Brownian motion.
http://front.math.ucdavis.edu/math.PR/0503556
---------------------------------------------------------------
3251. STOCHASTIC CHARACTERIZATION OF HARMONIC MAPS ON RIEMANNIAN
POLYHEDRA
M. A. Aprodu and T. Bouziane
The aim of this paper is to relate the theory of Harmonicity in sense
Korevaar-Schoen and Eells-Fuglede to the notion of a Brownian motion in
riemannian polyhedra achieved by the second author. Firstly, we prove
that
Brownian motions is stochastically continuous Markov processes and
consequently
it has a unique infinitesimal generator on some Banach space.
Secondly, we show
that in some sense, the Brownian motion in Riemannian polyhedra has
as an
infinitesimal generator the "Laplacian". Finally, we show that
harmonic maps,
with target smooth Riemannian manifolds, in the sense of Eells-
Fuglede, are
exactly those which maps Brownian motion in Riemannian polyhedron into a
martingale, while harmonic morphisms are exactly the maps which are
Brownian
preserving paths
http://front.math.ucdavis.edu/math.PR/0503557
---------------------------------------------------------------
3252. CENTRAL LIMIT THEOREMS FOR RANDOM POLYTOPES IN A SMOOTH CONVEX SET
Van Vu
Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n
$ random
points in $K$ independently according to the uniform distribution.
The convex
hull of these points, denoted by $K_n$, is called a {\it random
polytope}. We
prove that several key functionals of $K_n$ satisfy the central limit
theorem
as $n$ tends to infinity.
http://front.math.ucdavis.edu/math.PR/0503559
---------------------------------------------------------------
3253. QUENCHED INVARIANCE PRINCIPLE FOR SIMPLE RANDOM WALK ON TWO-
DIMENSIONAL PERCOLATION CLUSTERS
Noam Berger and Marek Biskup
We consider the simple random walk on a two-dimensional super-critical
infinite percolation cluster and prove that for almost every
configuration it
scales to Brownian motion.
http://front.math.ucdavis.edu/math.PR/0503576
---------------------------------------------------------------
3254. ASYMPTOTIC GENEALOGY OF A CRITICAL BRANCHING PROCESS
Lea Popovic
Consider a continuous-time binary branching process conditioned to have
population size n at some time t, and with a chance p for recording each
extinct individual in the process. Within the family tree of this
process, we
consider the smallest subtree containing the genealogy of the extant
individuals together with the genealogy of the recorded extinct
individuals. We
introduce a novel representation of such subtrees in terms of a point-
process,
and provide asymptotic results on the distribution of this point-
process as the
number of extant individuals increases. We motivate the study within
the scope
of a coherent analysis for an a priori model for macroevolution.
http://front.math.ucdavis.edu/math.PR/0503577
---------------------------------------------------------------
3255. GENERALIZED STOCHASTIC DIFFERENTIAL UTILITY AND PREFERENCE FOR
INFORMATION
Ali Lazrak
This paper develops, in a Brownian information setting, an approach for
analyzing the preference for information, a question that motivates the
stochastic differential utility (SDU) due to Duffie and Epstein
[Econometrica
60 (1992) 353-394]. For a class of backward stochastic differential
equations
(BSDEs) including the generalized SDU [Lazrak and Quenez Math. Oper.
Res. 28
(2003) 154-180], we formulate the information neutrality property as an
invariance principle when the filtration is coarser (or finer) and
characterize
it. We also provide concrete examples of heterogeneity in information
that
illustrate explicitly the nonneutrality property for some GSDUs. Our
results
suggest that, within the GSDUs class of intertemporal utilities, risk
aversion
or ambiguity aversion are inflexibly linked to the preference for
information.
http://front.math.ucdavis.edu/math.PR/0503579
---------------------------------------------------------------
3256. THE RIGHT TIME TO SELL A STOCK WHOSE PRICE IS DRIVEN BY
MARKOVIAN NOISE
Robert C. Dalang and M.-O. Hongler
We consider the problem of finding the optimal time to sell a stock,
subject
to a fixed sales cost and an exponential discounting rate \rho. We
assume that
the price of the stock fluctuates according to the equation dY_t=Y_t(\mu
dt+\sigma\xi(t) dt), where (\xi(t)) is an alternating Markov renewal
process
with values in {\pm1}, with an exponential renewal time. We determine
the
critical value of \rho under which the value function is finite. We
examine the
validity of the ``principle of smooth fit'' and use this to give a
complete and
essentially explicit solution to the problem, which exhibits a
surprisingly
rich structure. The corresponding result when the stock price evolves
according
to the Black and Scholes model is obtained as a limit case.
http://front.math.ucdavis.edu/math.PR/0503580
---------------------------------------------------------------
3257. CONCENTRATION OF NORMALIZED SUMS AND A CENTRAL LIMIT THEOREM
FOR NONCORRELATED RANDOM VARIABLES
Sergey G. Bobkov
For noncorrelated random variables, we study a concentration property
of the
family of distributions of normalized sums formed by sequences of
times of a
given large length.
http://front.math.ucdavis.edu/math.PR/0503583
---------------------------------------------------------------
3258. ANALYSIS OF A CLASS OF LIKELIHOOD BASED CONTINUOUS TIME
STOCHASTIC VOLATILITY MODELS INCLUDING ORNSTEIN-UHLENBECK MODELS IN
FINANCIAL ECONOMICS
Lancelot F. James
In a series of recent papers Barndorff-Nielsen and Shephard introduce an
attractive class of continuous time stochastic volatility models for
financial
assets where the volatility processes are functions of positive
Ornstein-Uhlenbeck(OU) processes. This models are known to be
substantially
more flexible than Gaussian based models. One current problem of this
approach
is the unavailability of a tractable exact analysis of likelihood based
stochastic volatility models for the returns of log prices of stocks.
With this point in mind, the likelihood models of Barndorff-
Nielsen and
Shephard are viewed as members of a much larger class of models. That is
likelihoods based on n conditionally independent Normal random
variables whose
mean and variance are representable as linear functionals of a common
unobserved Poisson random measure. The analysis of these models is
facilitated
by applying the methods in James (2005, 2002), in particular an
Esscher type
transform of Poisson random measures; in conjunction with a special
case of the
Weber-Sonine formula. It is shown that the marginal likelihood may be
expressed
in terms of a multidimensional Fourier-cosine transform. This yields
tractable
forms of the likelihood and also allows a full Bayesian posterior
analysis of
the integrated volatility process. A general formula for the
posterior density
of the log price given the observed data is derived, which could
potentially
have applications to option pricing. We also identify tractable
subclasses,
where inference can be based on a finite number of independent random
variables. It is shown that inference does not necessarily require
simulation
of random measures. Rather, classical numerical integration can be
used in the
most general cases.
http://front.math.ucdavis.edu/math.ST/0503055
---------------------------------------------------------------
3259. MODIFIED LOGARITHMIC SOBOLEV INEQUALITIES IN NULL CURVATURE
Ivan Gentil and Arnaud Guillin and Laurent Miclo
We present a logarithmic Sobolev inequality adapted to a log-concave
measure.
Assume that $\Phi$ is a symmetric convex function on $\dR$ satisfying
$(1+\e)\Phi(x)\leq {x}\Phi'(x)\leq(2-\e)\Phi(x)$ for $x\geq0$ large
enough and
with $\e\in]0,1/2]$. We prove that the probability measure on $\dR$
$\mu_\Phi(dx)=e^{-\Phi(x)}/Z_\Phi dx$ satisfies a modified and adapted
logarithmic Sobolev inequality : there exist three constant $A,B,D>0$
such that
for all smooth $f>0$, \begin{equation*}
\ent{\mu_\Phi}{f^2}\leq A\int H_{\Phi}\PAR{{\frac{f'}{f}}}f^2d\mu_
\Phi,
\text{with} H_{\Phi}(x)= {\begin{array}{rl} \Phi^*\PAR{Bx} &\text{if
}\ABS{x}\geq D, x^2 &\text{if}\ABS{x}\leq D. \end{array} . \end
{equation*}
http://front.math.ucdavis.edu/math.PR/0503585
---------------------------------------------------------------
3260. LENSES IN SKEW BROWNIAN FLOW
Krzysztof Burdzy and Haya Kaspi
We consider a stochastic flow in which individual particles follow skew
Brownian motions, with each one of these processes driven by the same
Brownian
motion. One does not have uniqueness for the solutions of the
corresponding
stochastic differential equation simultaneously for all real initial
conditions. Due to this lack of the simultaneous strong uniqueness
for the
whole system of stochastic differential equations, the flow contains
lenses,
that is, pairs of skew Brownian motions which start at the same point,
bifurcate, and then coalesce in a finite time. The paper contains
qualitative
and quantitative (distributional) results on the geometry of the flow
and
lenses.
http://front.math.ucdavis.edu/math.PR/0503586
---------------------------------------------------------------
3261. WEAK POINCARE INEQUALITIES ON DOMAINS DEFINED BY BROWNIAN ROUGH
PATHS
Shigeki Aida
We prove weak Poincare inequalities on domains which are inverse
images of
open sets in Wiener spaces under continuous functions of Brownian
rough paths.
The result is applicable to Dirichlet forms on loop groups and
connected open
subsets of path spaces over compact Riemannian manifolds.
http://front.math.ucdavis.edu/math.PR/0503587
---------------------------------------------------------------
3262. TIME CHANGES OF SYMMETRIC DIFFUSIONS AND FELLER MEASURES
Masatoshi Fukushima and Ping He and Jiangang Ying
We extend the classical Douglas integral, which expresses the Dirichlet
integral of a harmonic function on the unit disk in terms of its
value on
boundary, to the case of conservative symmetric diffusion in terms of
Feller
measure, by using the approach of time change of Markov processes.
http://front.math.ucdavis.edu/math.PR/0503588
---------------------------------------------------------------
3263. DIFFERENCE PROPHET INEQUALITIES FOR [0,1]-VALUED I.I.D. RANDOM
VARIABLES WITH COST FOR OBSERVATIONS
Holger Kosters
Let X_1,X_2,... be a sequence of [0,1]-valued i.i.d. random
variables, let
c\geq 0 be a sampling cost for each observation and let Y_i=X_i-ic,
i=1,2,....
For n=1,2,..., let M(Y_1,...,Y_n)=E(max_{1\leq i\leq n}Y_i) and
V(Y_1,...,Y_n)=sup_{\tau \in C^n}E(Y_{\tau}), where C^n denotes the
set of all
stopping rules for Y_1,...,Y_n. Sharp upper bounds for the difference
M(Y_1,...,Y_n)-V(Y_1,...,Y_n) are given under various restrictions on
c and n.
http://front.math.ucdavis.edu/math.PR/0503589
---------------------------------------------------------------
3264. UNIQUENESS FOR DIFFUSIONS DEGENERATING AT THE BOUNDARY OF A
SMOOTH BOUNDED SET
Dante DeBlassie
For continuous \gamma, g:[0,1]\to(0,\infty), consider the degenerate
stochastic differential equation dX_t=[1-|X_t|^2]^{1/2}\gamma(|X_t|)
dB_t-g(|X_t|)X_t dt in the closed unit ball of R^n. We introduce a
new idea to
show pathwise uniqueness holds when \gamma and g are Lipschitz and
\frac{g(1)}{\gamma^2(1)}>\sqrt2-1. When specialized to a case studied
by Swart
[Stochastic Process. Appl. 98 (2002) 131-149] with \gamma=\sqrt2 and g
\equiv c,
this gives an improvement of his result. Our method applies to more
general
contexts as well. Let D be a bounded open set with C^3 boundary and
suppose
h:\barD\to R Lipschitz on \barD, as well as C^2 on a neighborhood of
\partial D
with Lipschitz second partials there. Also assume h>0 on D, h=0 on
\partial D
and |\nabla h|>0 on \partial D. An example of such a function is
h(x)=d(x,\partial D). We give conditions which ensure pathwise
uniqueness holds
for dX_t=h(X_t)^{1/2}\sigma(X_t) dB_t+b(X_t) dt in \barD.
http://front.math.ucdavis.edu/math.PR/0503590
---------------------------------------------------------------
3265. MODERATE DEVIATIONS FOR DIFFUSIONS WITH BROWNIAN POTENTIALS
Yueyun Hu and Zhan Shi
We present precise moderate deviation probabilities, in both quenched
and
annealed settings, for a recurrent diffusion process with a Brownian
potential.
Our method relies on fine tools in stochastic calculus, including
Kotani's
lemma and Lamperti's representation for exponential functionals. In
particular,
our result for quenched moderate deviations is in agreement with a
recent
theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003)
571-609]
who studied the corresponding problem for Sinai's random walk in random
environment.
http://front.math.ucdavis.edu/math.PR/0503591
---------------------------------------------------------------
3266. SELF-INTERSECTION LOCAL TIME: CRITICAL EXPONENT, LARGE
DEVIATIONS, AND LAWS OF THE ITERATED LOGARITHM
Richard F. Bass and Xia Chen
If \beta_t is renormalized self-intersection local time for planar
Brownian
motion, we characterize when Ee^{\gamma\beta_1} is finite or infinite
in terms
of the best constant of a Gagliardo-Nirenberg inequality. We prove large
deviation estimates for \beta_1 and -\beta_1. We establish lim sup
and lim inf
laws of the iterated logarithm for \beta_t as t\to\infty.
http://front.math.ucdavis.edu/math.PR/0503592
---------------------------------------------------------------
3267. EXPONENTIAL ASYMPTOTICS AND LAW OF THE ITERATED LOGARITHM FOR
INTERSECTION LOCAL TIMES OF RANDOM WALKS
Xia Chen
Let \alpha ([0,1]^p) denote the intersection local time of p independent
d-dimensional Brownian motions running up to the time 1. Under the
conditions
p(d-2)<d and d\ge 2, we prove lim_{t\to\infty}t^{-1}\log
P\bigl{\alpha([0,1]^p)\ge t^{(d(p-1))/2}\bigr}=-\gamma_{\alpha}(d,p)
with the
right-hand side being identified in terms of the the best constant of
the
Gagliardo-Nirenberg inequality. Within the scale of moderate
deviations, we
also establish the precise tail asymptotics for the intersection
local time
I_n=#{(k_1,...,k_p)\in [1,n]^p;S_1(k_1)=... =S_p(k_p)} run by the
independent,
symmetric, Z^d-valued random walks S_1(n),...,S_p(n). Our results
apply to the
law of the iterated logarithm. Our approach is based on Feynman-Kac
type large
deviation, time exponentiation, moment computation and some
technologies along
the lines of probability in Banach space. As an interesting
coproduct, we
obtain the inequality \bigl(EI_{n_1+... +n_a}^m\bigr)^{1/p}\le \sum_
{k_1+...
+k_a=m\limits_{k_1,...,k_a\ge 0}}\frac{m!}{k_1!...
k_a!}\bigl(EI_{n_1}^{k_1}\bigr)^{1/p}... \bigl(EI_{n_a}^{k_a}\bigr)^
{1/p} in
the case of random walks.
http://front.math.ucdavis.edu/math.PR/0503593
---------------------------------------------------------------
3268. REGULARITY OF SOLUTIONS TO STOCHASTIC VOLTERRA EQUATIONS WITH
INFINITE DELAY
Anna Karczewska and Carlos Lizama
The paper gives necessary and sufficient conditions providing
regularity of
solutions to stochastic Volterra equations with infinite delay on a
$d$-dimensional torus. The harmonic analysis techniques and stochastic
integration in function spaces are used.
http://front.math.ucdavis.edu/math.PR/0503595
---------------------------------------------------------------
3269. A CLASS OF GENERALIZED HYPERBOLIC CONTINUOUS TIME INTEGRATED
STOCHASTIC VOLATILITY LIKELIHOOD MODELS
Lancelot F. James and John W. Lau
This paper discusses and analyzes a class of likelihood models which are
based on two distributional innovations in financial models for stock
returns.
That is, the notion that the marginal distribution of aggregate
returns of
log-stock prices are well approximated by generalized hyperbolic
distributions,
and that volatility clustering can be handled by specifying the
integrated
volatility as a random process such as that proposed in a recent
series of
papers by Barndorff-Nielsen and Shephard (BNS). The BNS models produce
likelihoods for aggregate returns which can be viewed as a subclass
of latent
regression models where one has n conditionally independent Normal
random
variables whose mean and variance are representable as linear
functionals of a
common unobserved Poisson random measure. James (2005b) recently
obtains an
exact analysis for such models yielding expressions of the likelihood
in terms
of quite tractable Fourier-Cosine integrals. Here, our idea is to
analyze a
class of likelihoods, which can be used for similar purposes, but
where the
latent regression models are based on n conditionally independent
models with
distributions belonging to a subclass of the generalized hyperbolic
distributions and whose corresponding parameters are representable as
linear
functionals of a common unobserved Poisson random measure. Our models
are
perhaps most closely related to the Normal inverse Gaussian/GARCH/A-
PARCH
models of Brandorff-Nielsen (1997) and Jensen and Lunde (2001), where
in our
case the GARCH component is replaced by quantities such as INT-OU
processes. It
is seen that, importantly, such likelihood models exhibit quite
different
features structurally. One nice feature of the model is that it
allows for more
flexibility in terms of modelling of external regression parameters.
http://front.math.ucdavis.edu/math.ST/0503056
---------------------------------------------------------------
3270. A LOCAL LIMIT THEOREM FOR DIRECTED POLYMERS IN RANDOM MEDIA:
THE CONTINUOUS AND THE DISCRETE CASE
Vincent Vargas (PMA)
In this article, we consider two models of directed polymers in random
environment: a discrete model and a continuous model. We consider
these models
in dimension greater or equal to 3 and we suppose that the normalized
partition
function is bounded in L^2. Under these assumptions, Sinai proved a
local limit
theorem for the discrete model, using a perturbation expansion. In this
article, we give a new method for proving Sinai's local limit
theorem. This new
method can be transposed to the continuous setting in which we prove
a similar
local limit theorem.
http://front.math.ucdavis.edu/math.PR/0503596
---------------------------------------------------------------
3271. GLOBAL L_2-SOLUTIONS OF STOCHASTIC NAVIER-STOKES EQUATIONS
R. Mikulevicius and B. L. Rozovskii
This paper concerns the Cauchy problem in R^d for the stochastic
Navier-Stokes equation \partial_tu=\Delta u-(u,\nabla)u-\nabla p+f(u)+
[(\sigma,\nabla)u-\nabla \tilde p+g(u)]\circ \dot W, u(0)=u_0,\qquad
divu=0,
driven by white noise \dot W. Under minimal assumptions on regularity
of the
coefficients and random forces, the existence of a global weak
(martingale)
solution of the stochastic Navier-Stokes equation is proved. In the
two-dimensional case, the existence and pathwise uniqueness of a
global strong
solution is shown. A Wiener chaos-based criterion for the existence and
uniqueness of a strong global solution of the Navier-Stokes equations is
established.
http://front.math.ucdavis.edu/math.PR/0503597
---------------------------------------------------------------
3272. CENTRAL LIMIT THEOREMS FOR SEQUENCES OF MULTIPLE STOCHASTIC
INTEGRALS
David Nualart and Giovanni Peccati
We characterize the convergence in distribution to a standard normal
law for
a sequence of multiple stochastic integrals of a fixed order with
variance
converging to 1. Some applications are given, in particular to study the
limiting behavior of quadratic functionals of Gaussian processes.
http://front.math.ucdavis.edu/math.PR/0503598
---------------------------------------------------------------
3273. STOCHASTIC INTEGRAL REPRESENTATION AND REGULARITY OF THE
DENSITY FOR THE EXIT MEASURE OF SUPER-BROWNIAN MOTION
Jean-Francois Le Gall and Leonid Mytnik
This paper studies the regularity properties of the density of the exit
measure for super-Brownian motion with (1+\beta)-stable branching
mechanism. It
establishes the continuity of the density in dimension d=2 and the
unboundedness of the density in all other dimensions where the
density exists.
An alternative description of the exit measure and its density is
also given
via a stochastic integral representation. Results are applied to the
probabilistic representation of nonnegative solutions of the partial
differential equation \Delta u=u^{1+\beta}.
http://front.math.ucdavis.edu/math.PR/0503599
---------------------------------------------------------------
3274. PRECISE ASYMPTOTICS OF SMALL EIGENVALUES OF REVERSIBLE
DIFFUSIONS IN THE METASTABLE REGIME
Michael Eckhoff
We investigate the close connection between metastability of the
reversible
diffusion process X defined by the stochastic differential equation
dX_t=-\nabla F(X_t) dt+\sqrt2\epsilon dW_t,\qquad \epsilon >0, and
the spectrum
near zero of its generator -L_{\epsilon}\equiv \epsilon \Delta -\nabla
F\cdot\nabla, where F:R^d\to R and W denotes Brownian motion on R^d. For
generic F to each local minimum of F there corresponds a metastable
state. We
prove that the distribution of its rescaled relaxation time converges
to the
exponential distribution as \epsilon \downarrow 0 with optimal and
uniform
error estimates. Each metastable state can be viewed as an eigenstate of
L_{\epsilon} with eigenvalue which converges to zero exponentially
fast in
1/\epsilon. Modulo errors of exponentially small order in 1/\epsilon
this
eigenvalue is given as the inverse of the expected metastable
relaxation time.
The eigenstate is highly concentrated in the basin of attraction of the
corresponding trap.
http://front.math.ucdavis.edu/math.PR/0503600
---------------------------------------------------------------
3275. ASYMPTOTIC EXPANSIONS FOR THE LAPLACE APPROXIMATIONS OF SUMS OF
BANACH SPACE-VALUED RANDOM VARIABLES
Sergio Albeverio and Song Liang
Let X_i, i\in N, be i.i.d. B-valued random variables, where B is a real
separable Banach space. Let \Phi be a smooth enough mapping from B
into R. An
asymptotic evaluation of Z_n=E(\exp (n\Phi (\sum_{i=1}^nX_i/n))), up
to a
factor (1+o(1)), has been gotten in Bolthausen [Probab. Theory
Related Fields
72 (1986) 305-318] and Kusuoka and Liang [Probab. Theory Related
Fields 116
(2000) 221-238]. In this paper, a detailed asymptotic expansion of
Z_n as n\to
\infty is given, valid to all orders, and with control on remainders.
The
results are new even in finite dimensions.
http://front.math.ucdavis.edu/math.PR/0503601
---------------------------------------------------------------
3276. MULTIPLICATIVE MONOTONE CONVOLUTIONS
Uwe Franz
Recently, Bercovici has introduced multiplicative convolutions based on
Muraki's monotone independence and shown that these convolution of
probability
measures correspond to the composition of some function of their Cauchy
transforms. We provide a new proof of this fact based on the
combinatorics of
moments. We also give a new characterisation of the probability
measures that
can be embedded into continuous monotone convolution semigroups of
probability
measures on the unit circle and briefly discuss a relation to Galton-
Watson
processes.
http://front.math.ucdavis.edu/math.PR/0503602
---------------------------------------------------------------
3277. EXTREMES ON TREES
Tailen Hsing and Holger Rootzen
This paper considers the asymptotic distribution of the longest edge
of the
minimal spanning tree and nearest neighbor graph on X_1,...,X_{N_n}
where
X_1,X_2,... are i.i.d. in \Re^2 with distribution F and N_n is
independent of
the X_i and satisfies N_n/n\to_p1. A new approach based on spatial
blocking and
a locally orthogonal coordinate system is developed to treat cases
for which F
has unbounded support. The general results are applied to a number of
special
cases, including elliptically contoured distributions, distributions
with
independent Weibull-like margins and distributions with parallel
level curves.
http://front.math.ucdavis.edu/math.PR/0503603
---------------------------------------------------------------
3278. ON THE MONOTONICITY OF THE SPEED OF RANDOM WALKS ON A
PERCOLATION CLUSTER OF TREES
Dayue Chen and Fuxi Zhang
We consider the simple random walk on the infinite cluster of the
Bernoulli
bond percolation of trees, and investigate the relation between the
speed of
the simple random walk and the retaining probability $p$ by studying
three
classes of trees. A sufficient condition is established for Galton-
Watson
trees.
http://front.math.ucdavis.edu/math.PR/0503610
---------------------------------------------------------------
3279. CONTRACTIVE MARKOV SYSTEMS II
Ivan Werner
In this paper, we continue development of the theory of contractive
Markov
systems (CMSs) initiated in \cite{Wer1}. We extend some results from
\cite{Wer1}, \cite{Wer3}, \cite{Wer5} and \cite{Wer6} to the case of
contractive Markov systems with probabilities which have a square
summable
variation by using some ideas of A. Johansson and A. Oeberg \cite
{JO}. In
particular, we show that an irreducible CMS has a unique invariant Borel
probability measure if the vertex sets form an open partition of the
state
space and the restrictions of the probability functions on their
vertex sets
have a square summable variation and are bounded away from zero.
http://front.math.ucdavis.edu/math.PR/0503633
---------------------------------------------------------------
3280. LIMIT THEOREMS FOR ITERATED RANDOM TOPICAL OPERATORS
Glenn Merlet (IRMAR)
Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively
homogeneous) operators. Let $x(n,x\_0)$ be defined by $x(0,x\_0)=x\_0
$ and
$x(n,x\_0)=A(n)x(n-1,x\_0)$. This can modelize a wide range of systems
including, task graphs, train networks, Job-Shop, timed digital
circuits or
parallel processing systems. When A(n) has the memory loss property,
we use the
spectral gap method to prove limit theorems for $x(n,x\_0)$. Roughly
speaking,
we show that $x(n,x\_0)$ behaves like a sum of i.i.d. real variables.
Precisely, we show that with suitable additional conditions, it
satisfies a
central limit theorem with rate, a local limit theorem, a renewal
theorem and a
large deviations principle, and we give an algebraic condition to
ensure the
positivity of the variance in the CLT. When A(n) are defined by
matrices in the
\mp semi-ring, we give more effective statements and show that the
additional
conditions and the positivity of the variance in the CLT are generic.
http://front.math.ucdavis.edu/math.PR/0503634
---------------------------------------------------------------
3281. A PROBABILISTIC APPROACH TO THE GEOMETRY OF THE \ELL_P^N-BALL
Franck Barthe and Olivier Guedon and Shahar Mendelson and Assaf Naor
This article investigates, by probabilistic methods, various geometric
questions on B_p^n, the unit ball of \ell_p^n. We propose
realizations in terms
of independent random variables of several distributions on B_p^n,
including
the normalized volume measure. These representations allow us to
unify and
extend the known results of the sub-independence of coordinate slabs
in B_p^n.
As another application, we compute moments of linear functionals on
B_p^n,
which gives sharp constants in Khinchine's inequalities on B_p^n and
determines
the \psi_2-constant of all directions on B_p^n. We also study the
extremal
values of several Gaussian averages on sections of B_p^n (including
mean width
and \ell-norm), and derive several monotonicity results as p varies.
Applications to balancing vectors in \ell_2 and to covering numbers of
polyhedra complete the exposition.
http://front.math.ucdavis.edu/math.PR/0503650
---------------------------------------------------------------
3282. MOMENT INEQUALITIES FOR FUNCTIONS OF INDEPENDENT RANDOM VARIABLES
Stephane Boucheron and Olivier Bousquet and Gabor Lugosi and Pascal
Massart
A general method for obtaining moment inequalities for functions of
independent random variables is presented. It is a generalization of the
entropy method which has been used to derive concentration
inequalities for
such functions [Boucheron, Lugosi and Massart Ann. Probab. 31 (2003)
1583-1614], and is based on a generalized tensorization inequality
due to
Latala and Oleszkiewicz [Lecture Notes in Math. 1745 (2000) 147-168].
The new
inequalities prove to be a versatile tool in a wide range of
applications. We
illustrate the power of the method by showing how it can be used to
effortlessly re-derive classical inequalities including Rosenthal and
Kahane-Khinchine-type inequalities for sums of independent random
variables,
moment inequalities for suprema of empirical processes and moment
inequalities
for Rademacher chaos and U-statistics. Some of these corollaries are
apparently
new. In particular, we generalize Talagrand's exponential inequality for
Rademacher chaos of order 2 to any order. We also discuss
applications for
other complex functions of independent random variables, such as
suprema of
Boolean polynomials which include, as special cases, subgraph
counting problems
in random graphs.
http://front.math.ucdavis.edu/math.PR/0503651
---------------------------------------------------------------
3283. ON THE STOCHASTIC CALCULUS METHOD FOR SPINS SYSTEMS
Samy Tindel
In this note we show how to generalize the stochastic calculus method
introduced by Comets and Neveu [Comm. Math. Phys. 166 (1995) 549-564]
for two
models of spin glasses, namely, the SK model with external field and the
perceptron model. This method allows to derive quite easily some
fluctuation
results for the free energy in those two cases.
http://front.math.ucdavis.edu/math.PR/0503652
---------------------------------------------------------------
3284. CLOSURES OF EXPONENTIAL FAMILIES
Imre Csiszar and Frantisek Matus
The variation distance closure of an exponential family with a convex
set of
canonical parameters is described, assuming no regularity conditions.
The tools
are the concepts of convex core of a measure and extension of an
exponential
family, introduced previously by the authors, and a new concept of
accessible
faces of a convex set. Two other closures related to the information
divergence
are also characterized.
http://front.math.ucdavis.edu/math.PR/0503653
---------------------------------------------------------------
3285. ONE-DEPENDENT TRIGONOMETRIC DETERMINANTAL PROCESSES ARE TWO-
BLOCK-FACTORS
Erik I. Broman
Given a trigonometric polynomial f:[0,1]\to[0,1] of degree m, one can
define
a corresponding stationary process {X_i}_{i\in Z} via determinants of
the
Toeplitz matrix for f. We show that for m=1 this process, which is
trivially
one-dependent, is a two-block-factor.
http://front.math.ucdavis.edu/math.PR/0503654
---------------------------------------------------------------
3286. ASYMPTOTICS FOR HITTING TIMES
M. Kupsa and Y. Lacroix
In this paper we characterize possible asymptotics for hitting times in
aperiodic ergodic dynamical systems: asymptotics are proved to be the
distribution functions of subprobability measures on the line
belonging to the
functional class {6pt} {-3mm}(A){6mm}F={F:R\to [0,1]:\left\lbrack
\matrixF is
increasing, null on ]-\infty, 0]; \noalignF is continuous and concave;
\noalignF(t)\le t for t\ge 0.\right.}. {6pt} Note that all possible
asymptotics
are absolutely continuous.
http://front.math.ucdavis.edu/math.PR/0503655
---------------------------------------------------------------
3287. KREIN'S SPECTRAL THEORY AND THE PALEY-WIENER EXPANSION FOR
FRACTIONAL BROWNIAN MOTION
Kacha Dzhaparidze and Harry van Zanten
In this paper we develop the spectral theory of the fractional Brownian
motion (fBm) using the ideas of Krein's work on continuous analogous of
orthogonal polynomials on the unit circle. We exhibit the functions
which are
orthogonal with respect to the spectral measure of the fBm and obtain an
explicit reproducing kernel in the frequency domain. We use these
results to
derive an extension of the classical Paley-Wiener expansion of the
ordinary
Brownian motion to the fractional case.
http://front.math.ucdavis.edu/math.PR/0503656
---------------------------------------------------------------
3288. CRITICALITY FOR BRANCHING PROCESSES IN RANDOM ENVIRONMENT
V. I. Afanasyev and J. Geiger and G. Kersting and V. A. Vatutin
We study branching processes in an i.i.d. random environment, where the
associated random walk is of the oscillating type. This class of
processes
generalizes the classical notion of criticality. The main properties
of such
branching processes are developed under a general assumption, known as
Spitzer's condition in fluctuation theory of random walks, and some
additional
moment condition. We determine the exact asymptotic behavior of the
survival
probability and prove conditional functional limit theorems for the
generation
size process and the associated random walk. The results rely on a
stimulating
interplay between branching process theory and fluctuation theory of
random
walks.
http://front.math.ucdavis.edu/math.PR/0503657
---------------------------------------------------------------
3289. EXAMPLES OF MODERATE DEVIATION PRINCIPLE FOR DIFFUSION PROCESSES
A. Guillin} and R. Liptser
Taking into account some likeness of moderate deviations (MD) and
central
limit theorems (CLT), we develop an approach, which made a good
showing in CLT,
for MD analysis of a family $$ S^\kappa_t=\frac{1}{t^\kappa}\int_0^tH
(X_s)ds, \
t\to\infty $$ for an ergodic diffusion process $X_t$ under $0.5<
\kappa<1$ and
appropriate $H$. We mean a decomposition with ``corrector'': $$
\frac{1}{t^\kappa}\int_0^tH(X_s)ds={\rm
corrector}+\frac{1}{t^\kappa}\underbrace{M_t}_{\rm martingale}. $$
and show
that, as in the CLT analysis, the corrector is negligible but in the
MD scale,
and the main contribution in the MD brings the family ``$
\frac{1}{t^\kappa}M_t, t\to\infty. $'' Starting from Bayer and Freidlin,
\cite{BF}, and finishing by Wu's papers \cite{Wu1}-\cite{WuH}, in the
MD study
Laplace's transform dominates. In the paper, we replace the Laplace
technique
by one, admitting to give the conditions, providing the MD, in terms of
``drift-diffusion'' parameters and $H$. However, a verification of these
conditions heavily depends on a specificity of a diffusion model.
That is why
the paper is named ``Examples ...''.
http://front.math.ucdavis.edu/math.PR/0503070
---------------------------------------------------------------
3290. CONFIDENCE INTERVALS FOR NONHOMOGENEOUS BRANCHING PROCESSES
AND POLYMERASE CHAIN REACTIONS
Didier Piau
We extend in two directions our previous results about the sampling
and the
empirical measures of immortal branching Markov processes. Direct
applications
to molecular biology are rigorous estimates of the mutation rates of
polymerase
chain reactions from uniform samples of the population after the
reaction.
First, we consider nonhomogeneous processes, which are more adapted
to real
reactions. Second, recalling that the first moment estimator is
analytically
known only in the infinite population limit, we provide rigorous
confidence
intervals for this estimator that are valid for any finite
population. Our
bounds are explicit, nonasymptotic and valid for a wide class of
nonhomogeneous
branching Markov processes that we describe in detail. In the setting of
polymerase chain reactions, our results imply that enlarging the size
of the
sample becomes useless for surprisingly small sizes. Establishing
confidence
intervals requires precise estimates of the second moment of random
samples.
The proof of these estimates is more involved than the proofs that
allowed us,
in a previous paper, to deal with the first moment. On the other
hand, our
method uses various, seemingly new, monotonicity properties of the
harmonic
moments of sums of exchangeable random variables.
http://front.math.ucdavis.edu/math.PR/0503659
---------------------------------------------------------------
3291. SECTORIAL CONVERGENCE OF U-STATISTICS
Anda Gadidov
In this note we show that almost sure convergence to zero of symmetrized
U-statistics indexed by a linear sector in Z^d_+ is equivalent to
convergence
along the diagonal of Z^d_+, as it is considered in Lata\la and Zinn
[Ann.
Probab. 28 (2000) 1908-1924]. Comparisons with similar results for
sums of
multi-indexed i.i.d. random variables are also made.
http://front.math.ucdavis.edu/math.PR/0503660
---------------------------------------------------------------
3292. A STRONG INVARIANCE PRINCIPLE FOR ASSOCIATED RANDOM FIELDS
Raluca M. Balan
In this paper we generalize Yu's [Ann. Probab. 24 (1996) 2079-2097]
strong
invariance principle for associated sequences to the multi-parameter
case,
under the assumption that the covariance coefficient u(n) decays
exponentially
as n\to \infty. The main tools that we use are the following: the
Berkes and
Morrow [Z. Wahrsch. Verw. Gebiete 57 (1981) 15-37] multi-parameter
blocking
technique, the Csorgo and Revesz [Z. Wahrsch. Verw. Gebiete 31 (1975)
255-260]
quantile transform method and the Bulinski [Theory Probab. Appl. 40
(1995)
136-144] rate of convergence in the CLT.
http://front.math.ucdavis.edu/math.PR/0503661
---------------------------------------------------------------
3293. MODERATE DEVIATION PRINCIPLE FOR ERGODIC MARKOV CHAIN.
LIPSCHITZ SUMMANDS
B. Delyon and A. Juditsky and R. Liptser
For ${1/2}<\alpha<1$, we propose the MDP analysis for family $$
S^\alpha_n=\frac{1}{n^\alpha}\sum_{i=1}^nH(X_{i-1}), n\ge 1, $$ where
$(X_n)_{n\ge 0}$ be a homogeneous ergodic Markov chain, $X_n\in
\mathbb{R}^d$,
when the spectrum of operator $P_x$ is continuous. The vector-valued
function
$H$ is not assumed to be bounded but the Lipschitz continuity of $H$ is
required. The main helpful tools in our approach are Poisson's
equation and
Stochastic Exponential; the first enables to replace the original
family by
$\frac{1}{n^\alpha}M_n$ with a martingale $M_n$ while the second to
avoid the
direct Laplace transform analysis.
http://front.math.ucdavis.edu/math.PR/0503071
---------------------------------------------------------------
3294. DISTANCES IN RANDOM GRAPHS WITH FINITE MEAN AND INFINITE
VARIANCE DEGREES
Remco van der Hofstad and Gerard Hooghiemstra and Dmitri Znamenski
In this paper we study random graphs with independent and identically
distributed degrees of which the tail of the distribution function is
regularly
varying with exponent $\tau\in (2,3)$.
The number of edges between two arbitrary nodes, also called the
graph
distance or hopcount, in a graph with $N$ nodes is investigated when
$N\to
\infty$. When $\tau\in (2,3)$, this graph distance grows like $2\frac
{\log\log
N}{|\log(\tau-2)|}$. In different papers, the cases $\tau>3$ and $\tau
\in
(1,2)$ have been studied. We also study the fluctuations around these
asymptotic means, and describe their distributions. The results
presented here
improve upon results of Reittu and Norros, who prove an upper bound
only.
http://front.math.ucdavis.edu/math.PR/0502581
---------------------------------------------------------------
3295. ON TAIL DISTRIBUTIONS OF SUPREMUM AND QUADRATIC VARIATION OF
LOCAL MARTINGALES
R. Liptser and A. Novikov
We extend some known results relating the distribution tails of a
continuous
local martingale supremum and its quadratic variation to the case of
locally
square integrable martingales with bounded jumps. The predictable and
optional
quadratic variations are involved in the main result.
http://front.math.ucdavis.edu/math.PR/0503072
---------------------------------------------------------------
3296. LIMIT THEOREMS FOR BIPOWER VARIATION IN FINANCIAL ECONOMETRICS
Ole E. Barndorff-Nielsen (DEPT Math Sci) and Svend E. Graversen
(DEPT Math Sci), Jean Jacod (PMA), Neil Shephard (NUFFIELD College)
In this paper we provide an asymptotic analysis of generalised bipower
measures of the variation of price processes in financial economics.
These
measures encompass the usual quadratic variation, power variation and
bipower
variations which have been highlighted in recent years in financial
econometrics. The analysis is carried out under some rather general
Brownian
semimartingale assumptions, which allow for standard leverage effects.
http://front.math.ucdavis.edu/math.PR/0503711
---------------------------------------------------------------
3297. RANDOM WALKS IN A DIRICHLET ENVIRONMENT
Nathana\"el Enriquez and Christophe Sabot
This paper states a law of large numbers for a random walk in a
random iid
environment on ${\mathbb Z}^d$, where the environment follows some
Dirichlet
distribution. Moreover, we give explicit bounds for the asymptotic
velocity of
the process and also an asymptotic expansion of this velocity at low
disorder.
http://front.math.ucdavis.edu/math.PR/0503713
---------------------------------------------------------------
3298. RANDOM WALKS IN A RANDOM ENVIRONMENT
S R S Varadhan
Random walks as well as diffusions in random media are considered.
Methods
are developed that allow one to establish large deviation results for
both the
`quenched' and the `averaged' case.
http://front.math.ucdavis.edu/math.PR/0503089
---------------------------------------------------------------
3299. RANDOM TREES AND GENERAL BRANCHING PROCESSES
Anna Rudas and Balint Toth and Benedek Valko
We consider a model of random tree growth, where at each time unit a new
vertex is added and attached to an already existing vertex chosen at
random.
The probability with which a vertex with degree $k$ is chosen is
proportional
to $w(k)$, where the weight function $w$ is the parameter of the model.
In the papers of B. Bollobas, O. Riordan, J. Spencer, G. Tusnady,
and,
independently, Mori, the asymptotic degree distribution is obtained
for a model
that is equivalent to the special case of ours, when the weight
function is
linear. The proof therein strongly relies on the linear choice of $w$.
We give the asymptotical degree distribution for a wide range of
weight
functions. Moreover, we provide the asymptotic distribution of the
tree itself
as seen from a randomly selected vertex. The latter approach is new
and gives
full insight to the limiting structure of the tree.
Our proof relies on the fact that considering the evolution of the
random
tree in continuous time, the process may be viewed as a general
branching
process, this way classical results can be applied.
http://front.math.ucdavis.edu/math.PR/0503728
---------------------------------------------------------------
3300. MIXED POISSON APPROXIMATION OF NODE DEPTH DISTRIBUTIONS IN
RANDOM BINARY SEARCH TREES
Rudolf Grubel and Nikolce Stefanoski
We investigate the distribution of the depth of a node containing a
specific
key or, equivalently, the number of steps needed to retrieve an item
stored in
a randomly grown binary search tree. Using a representation in terms
of mixed
and compounded standard distributions, we derive approximations by
Poisson and
mixed Poisson distributions; these lead to asymptotic normality
results. We are
particularly interested in the influence of the key value on the
distribution
of the node depth. Methodologically our message is that the explicit
representation may provide additional insight if compared to the
standard
approach that is based on the recursive structure of the trees.
Further, in
order to exhibit the influence of the key on the distributional
asymptotics, a
suitable choice of distance of probability distributions is
important. Our
results are also applicable in connection with the number of
recursions needed
in Hoare's [Comm. ACM 4 (1961) 321-322] selection algorithm Find.
http://front.math.ucdavis.edu/math.PR/0503738
---------------------------------------------------------------
3301. ON FRACTIONAL TEMPERED STABLE MOTION
C. Houdr\'e and R. Kawai
Fractional tempered stable motion (fTSm)} is defined and studied.
FTSm has
the same covariance structure as fractional Brownian motion, while
having tails
heavier than Gaussian but lighter than stable. Moreover, in short
time it is
close to fractional stable L\'evy motion, while it is approximately
fractional
Brownian motion in long time. A series representation of fTSm is
derived and
used for simulation and to study some of its sample path properties.
http://front.math.ucdavis.edu/math.PR/0503741
---------------------------------------------------------------
3302. ON LAYERED STABLE PROCESSES
C. Houdr\'e and R. Kawai
Layered stable (multivariate) distributions and processes are defined
and
studied. A layered stable process combines stable trends of two
different
indices, one of them possibly Gaussian. More precisely, in short
time, it is
close to a stable process while, in long time, it approximates
another stable
(possibly Gaussian) process. We also investigate the absolute
continuity of a
layered stable process with respect to its short time limiting stable
process.
A series representation of layered stable processes is derived,
giving insights
into both the structure of the sample paths and of the short and long
time
behaviors. This series is further used for sample paths simulation.
http://front.math.ucdavis.edu/math.PR/0503742
---------------------------------------------------------------
3303. MEASURE FREE MARTINGALES
Rajeeva L Karandikar and M G Nadkarni
We give a necessary and sufficient condition on a sequence of
functions on a
set $\Omega$ under which there is a measure on $\Omega$ which renders
the given
sequence of functions a martingale. Further such a measure is unique
if we
impose a natural maximum entropy condition on the conditional
probabilities.
http://front.math.ucdavis.edu/math.PR/0503099
---------------------------------------------------------------
3304. METRIC STABILITY FOR RANDOM WALKS (WITH APPLICATIONS IN
RENORMALIZATION THEORY)
Carlos G. Moreira (IMPA-Brazil) Daniel Smania (ICMC-USP-Brazil)
Consider deterministic random walks F: I x Z -> I x Z, defined by
F(x,n)=(f(x), K(x)+n), where f is an expanding Markov map on the
interval I and
K: I->Z. We study the universality (stability) of ergodic (for instance,
recurrence and transience), geometric and multifractal properties in
the class
of perturbations of the type G(x,n)=(f_n(x), L(x,n)+n) which are
topologically
conjugate with F and f_n are expanding maps exponentially close to f
when |n|
goes to infinity. We give applications of these results in the study
of the
regularity of conjugacies between (generalized) infinitely
renormalizable maps
of the interval and the existence of wild attractors for one-
dimensional maps.
http://front.math.ucdavis.edu/math.DS/0503736
---------------------------------------------------------------
3305. THE JAMMED PHASE OF THE BIHAM-MIDDLETON-LEVINE TRAFFIC MODEL
Omer Angel and Alexander E Holroyd and James B Martin
Initially a car is placed with probability p at each site of the
two-dimensional integer lattice. Each car is equally likely to be
East-facing
or North-facing, and different sites receive independent assignments.
At odd
time steps, each North-facing car moves one unit North if there is a
vacant
site for it to move into. At even time steps, East-facing cars move
East in the
same way. We prove that when p is sufficiently close to 1 traffic is
jammed, in
the sense that no car moves infinitely many times. The result extends to
several variant settings, including a model with cars moving at
random times,
and higher dimensions.
http://front.math.ucdavis.edu/math.PR/0504001
---------------------------------------------------------------
3306. BSDE WITH QUADRATIC GROWTH AND UNBOUNDED TERMINAL VALUE
Philippe Briand (IRMAR) and Ying Hu (IRMAR)
In this paper, we study the existence of solution to BSDE with quadratic
growth and unbounded terminal value. We apply a localization
procedure together
with a priori bounds. As a byproduct, we apply the same method to
extend a
result on BSDEs with integrable terminal condition.
http://front.math.ucdavis.edu/math.PR/0504002
---------------------------------------------------------------
3307. THE HEAT EQUATION WITH MULTIPLICATIVE STABLE L\'EVY NOISE
Carl Mueller and Leonid Mytnik and Aurel Stan
We study the heat equation with a random potential term. The
potential is a
one-sided stable noise, with positive jumps, which does not depend on
time. To
avoid singularities, we define the equation in terms of a
construction similar
to the Skorokhod integral or Wick product. We give a criterion for
existence
based on the dimension of the space variable, and the parameter p of
the stable
noise. Our arguments are different for p<1 and p>1.
http://front.math.ucdavis.edu/math.PR/0504027
---------------------------------------------------------------
3308. THE FULL SCALING LIMIT OF TWO-DIMENSIONAL CRITICAL PERCOLATION
Federico Camia and Charles M. Newman
We use SLE(6) paths to construct a process of continuum nonsimple
loops in
the plane and prove that this process coincides with the full
continuum scaling
limit of 2D critical site percolation on the triangular lattice --
that is, the
scaling limit of the set of all interfaces between different
clusters. Some
properties of the loop process, including conformal invariance, are also
proved. In the main body of the paper these results are proved while
assuming,
as argued by Schramm and Smirnov, that the percolation exploration path
converges in distribution to the trace of chordal SLE(6). Then, in a
lengthy
appendix, a detailed proof is provided for this convergence to SLE
(6), which
itself relies on Smirnov's result that crossing probabilities
converge to
Cardy's formula.
http://front.math.ucdavis.edu/math.PR/0504036
---------------------------------------------------------------
3309. MINIMAX AND ADAPTIVE ESTIMATION OF THE WIGNER FUNCTION IN
QUANTUM HOMODYNE TOMOGRAPHY WITH NOISY DATA
Cristina Butucea (PMA and MODALX) and Madalin Guta and Luis Artiles
We estimate the quantum state of a light beam from results of quantum
homodyne measurements performed on identically prepared quantum
systems. The
state is represented through the Wigner function, a density on R2
which may
take negative values but must respect intrinsic positivity
constraints imposed
by quantum physics. The effect of the losses due to detection
inefficiencies
which are always present in a real experiment is the addition to the
tomographic data of independent Gaussian noise. We construct a kernel
estimator
for the Wigner function and prove that it is minimax efficient for the
pointwise risk over a class of infinitely differentiable functions.
For the L2
risk, we compute the upper bounds of a truncated kernel estimator
over the same
classes, restricted to functions with sub-Gaussian asymptotic
behaviour. We
construct adaptive estimators, i.e. which do not depend on the
smoothness
parameters, and prove that in some set-ups they attain the minimax
rates for
the corresponding smoothness class.
http://front.math.ucdavis.edu/math.PR/0504058
---------------------------------------------------------------
3310. POINT PROCESS MODEL OF 1/F NOISE VERSUS A SUM OF LORENTZIANS
B. Kaulakys and V. Gontis and and M. Alaburda
We present a simple point process model of $1/f^{\beta}$ noise, covering
different values of the exponent $\beta$. The signal of the model
consists of
pulses or events. The interpulse, interevent, interarrival,
recurrence or
waiting times of the signal are described by the general Langevin
equation with
the multiplicative noise and stochastically diffuse in some interval
resulting
in the power-law distribution. Our model is free from the requirement
of a wide
distribution of relaxation times and from the power-law forms of the
pulses. It
contains only one relaxation rate and yields $1/f^ {\beta}$ spectra
in a wide
range of frequency. We obtain explicit expressions for the power
spectra and
present numerical illustrations of the model. Further we analyze the
relation
of the point process model of $1/f$ noise with the Bernamont-Surdin-
McWhorter
model, representing the signals as a sum of the uncorrelated
components. We
show that the point process model is complementary to the model based
on the
sum of signals with a wide-range distribution of the relaxation
times. In
contrast to the Gaussian distribution of the signal intensity of the
sum of the
uncorrelated components, the point process exhibits asymptotically a
power-law
distribution of the signal intensity. The developed multiplicative point
process model of $1/f^{\beta}$ noise may be used for modeling and
analysis of
stochastic processes in different systems with the power-law
distribution of
the intensity of pulsing signals.
http://front.math.ucdavis.edu/cond-mat/0504025
---------------------------------------------------------------
3311. A RANDOM WALK PROOF OF THE ERDOS-TAYLOR CONJECTURE
Jay Rosen
For the simple random walk in Z^2 we study those points which are
visited an
unusually large number of times, and provide a new proof of the Erdos-
Taylor
conjecture describing the number of visits to the most visited point.
http://front.math.ucdavis.edu/math.PR/0503108
---------------------------------------------------------------
3312. WHAT IS ALWAYS STABLE IN NONLINEAR FILTERING?
P. Chigansky and R. Liptser
This note addresses certain stability properties of the nonlinear
filtering
equation in discrete time. The available positive and negative
results indicate
that much depends on the structure of the signal state space, its
ergodic
properties and observations regularity. We show that certain predicting
estimates are stable under surprisingly general assumptions.
http://front.math.ucdavis.edu/math.PR/0504094
---------------------------------------------------------------
3313. HOW LIKELY IS AN I.I.D. DEGREE SEQUENCE TO BE GRAPHICAL?
Richard Arratia and Thomas M. Liggett
Given i.i.d. positive integer valued random variables D_1,...,D_n,
one can
ask whether there is a simple graph on n vertices so that the degrees
of the
vertices are D_1,...,D_n. We give sufficient conditions on the
distribution of
D_i for the probability that this be the case to be asymptotically 0,
{1/2} or
strictly between 0 and {1/2}. These conditions roughly correspond to
whether
the limit of nP(D_i\geq n) is infinite, zero or strictly positive and
finite.
This paper is motivated by the problem of modeling large communications
networks by random graphs.
http://front.math.ucdavis.edu/math.PR/0504096
---------------------------------------------------------------
3314. THE UNIVERSALITY CLASSES IN THE PARABOLIC ANDERSON MODEL
Remco van der Hofstad and Wolfgang Koenig and Peter Moerters
We discuss the long time behaviour of the parabolic Anderson model, the
Cauchy problem for the heat equation with random potential on $\Z^d$. We
consider general i.i.d. potentials and show that exactly \emph{four}
qualitatively different types of intermittent behaviour can occur.
These four
universality classes depend on the upper tail of the potential
distribution:
(1) tails at $\infty$ that are thicker than the double-exponential
tails, (2)
double-exponential tails at $\infty$ studied by G\"artner and
Molchanov, (3) a
new class called \emph{almost bounded potentials}, and (4) potentials
bounded
from above studied by Biskup and K\"onig. The new class (3), which
contains
both unbounded and bounded potentials, is studied in both the
annealed and the
quenched setting. We show that intermittency occurs on unboundedly
increasing
islands whose diameter is slowly varying in time. The characteristic
variational formulas describing the optimal profiles of the potential
and of
the solution are solved explicitly by parabolas, respectively, Gaussian
densities.
http://front.math.ucdavis.edu/math.PR/0504102
---------------------------------------------------------------
3315. INVARIANCE PRINCIPLES FOR LABELED MOBILES AND BIPARTITE PLANAR
MAPS
Jean-Fran\c{c}ois Marckert (LM-Versailles) and Gr\'{e}gory Miermont
(LM-Orsay)
A class of labeled trees, called mobiles, was introduced by Bouttier-di
Francesco and Guitter in order to generalize the bijective studies of
planar
maps initiated by Cori-Vauquelin and Schaeffer. We prove an invariance
principle for rescaled random mobiles associated with bipartite
random planar
maps under a Boltzmann distribution. We infer that the latter
converge in a
certain sense to the Brownian map introduced by Marckert and
Mokkadem, which
encompasses results of Chassaing and Schaeffer on quadrangulations
(although in
a slightly different context). These results are derived from a new
invariance
principle for a class of two-type Galton-Watson trees coupled with a
spatial
motion, which are shown to converge to the Brownian snake.
http://front.math.ucdavis.edu/math.PR/0504110
---------------------------------------------------------------
3316. TRACY-WIDOM LIMIT FOR THE LARGEST EIGENVALUE OF A LARGE CLASS
OF COMPLEX WISHART MATRICES
Noureddine El Karoui
We study the limiting behavior of the largest eigenvalue of a large
class of
complex Wishart matrices. In other words, let X be an n*p matrix, and
let its
rows be i.i.d complex normal N_{C}(0,Sigma_p). We denote by H_p the
spectral
distribution of Sigma_p, and call lambda_i's its ordered eigenvalues.
Let us
call l_i's the ordered eigenvalues of X^*X and c the unique root in
[0,1/lambda_1(Sigma_p)) of the equation
\int ((lambda c)/(1-\lambda c))^2 dH_p(lambda) = n/p. The main
result of this
paper is that, under technical conditions on (Sigma_p,n,p), we have,
when
n->\infty,
(l_1(X^*X)-n mu)/(n^{1/3} sigma) -> TW_2 .
We give explicit formulas for mu and sigma, that depend non
trivially on c.
Here TW_2 denotes the Tracy-Widom law appearing in the study of the
Gaussian
Unitary Ensemble.
This theorem applies to a number of covariance models found in
applications,
including well-behaved Toeplitz matrices and covariance matrices
whose spectral
distribution is a sum of atoms (under some conditions on the mass of the
atoms). Generalizations of the theorem to certain spiked versions of
models in
G and a.s statements about l_1/n are given. Most known examples of
convergence
of the largest eigenvalue of a complex sample covariance matrix to this
Tracy-Widom law are subcases of this result.
http://front.math.ucdavis.edu/math.PR/0503109
---------------------------------------------------------------
3317. DETERMINANTAL PROCESSES AND INDEPENDENCE
J. Ben Hough and Manjunath Krishnapur and Yuval Peres and Balint Virag
We give a probabilistic introduction to determinantal and permanental
point
processes. Determinantal processes arise in physics (fermions,
eigenvalues of
random matrices) and in combinatorics (nonintersecting paths, random
spanning
trees). They have the striking property that the number of points in
a region
$D$ is a sum of independent Bernoulli random variables, with
parameters which
are eigenvalues of the relevant operator on $L^2(D)$. Moreover, any
determinantal process can be represented as a mixture of determinantal
projection processes. We give a simple explanation for these known
facts, and
establish analogous representations for permanental processes, with
geometric
variables replacing the Bernoulli variables. These representations
lead to
simple proofs of existence criteria and central limit theorems, and
unify known
results on the distribution of absolute values in certain processes with
radially symmetric distributions.
http://front.math.ucdavis.edu/math.PR/0503110
---------------------------------------------------------------
3318. RANDOM WALK ON THE INCIPIENT INFINITE CLUSTER ON TREES
Martin T. Barlow and Takashi Kumagai
Let ${\cal G}$ be the incipient infinite cluster (IIC) for
percolation on a
homogeneous tree of degree $n_0+1$. We obtain estimates for the
transition
density of the continuous time simple random walk $Y$ on ${\cal G}$; the
process satisfies anomalous diffusion and has spectral dimension 4/3.
http://front.math.ucdavis.edu/math.PR/0503118
---------------------------------------------------------------
3319. QUANTITATIVE CONCENTRATION INEQUALITIES FOR EMPIRICAL MEASURES
ON NON-COMPACT SPACES
Francois Bolley and Arnaud Guillin and Cedric Villani
We establish some quantitative concentration estimates for the empirical
measure of many independent variables, in transportation distances.
As an
application, we provide some error bounds for particle simulations in
a model
mean field problem. The tools include coupling arguments, as well as
regularity
and moments estimates for solutions of certain diffusive partial
differential
equations.
http://front.math.ucdavis.edu/math.PR/0503123
---------------------------------------------------------------
3320. ON THE BIAS OF TRACEROUTE SAMPLING; OR, POWER-LAW DEGREE
DISTRIBUTIONS IN REGULAR GRAPHS
Dimitris Achlioptas and Aaron Clauset and David Kempe and and
Cristopher Moore
Understanding the structure of the Internet graph is a crucial step for
building accurate network models and designing efficient algorithms for
Internet applications. Yet, obtaining its graph structure is a
surprisingly
difficult task, as edges cannot be explicitly queried. Instead,
empirical
studies rely on traceroutes to build what are essentially single-source,
all-destinations, shortest-path trees. These trees only sample a
fraction of
the network's edges, and a recent paper by Lakhina et al. found
empirically
that the resuting sample is intrinsically biased. For instance, the
observed
degree distribution under traceroute sampling exhibits a power law
even when
the underlying degree distribution is Poisson.
In this paper, we study the bias of traceroute sampling
systematically, and,
for a very general class of underlying degree distributions,
calculate the
likely observed distributions explicitly. To do this, we use a
continuous-time
realization of the process of exposing the BFS tree of a random graph
with a
given degree distribution, calculate the expected degree distribution
of the
tree, and show that it is sharply concentrated. As example
applications of our
machinery, we show how traceroute sampling finds power-law degree
distributions
in both delta-regular and Poisson-distributed random graphs. Thus,
our work
puts the observations of Lakhina et al. on a rigorous footing, and
extends them
to nearly arbitrary degree distributions.
http://front.math.ucdavis.edu/cond-mat/0503087
---------------------------------------------------------------
3321. THE CRITICAL ISING MODEL ON TREES, CONCAVE RECURSIONS AND
NONLINEAR CAPACITY
Robin Pemantle and Yuval Peres
We consider the Ising model on a general tree under various boundary
conditions: all plus, free and spin-glass. In each case, we determine
when the
root is influenced by the boundary values in the limit as the
boundary recedes
to infinity. We obtain exact capacity criteria that govern behavior
at critical
temperatures. For plus boundary conditions, an $L^3$ capacity arises. In
particular, on a spherically symmetric tree that has $n^c b^n$
vertices at
level $n$ (up to bounded factors), we prove that there is a unique Gibbs
measure for the ferromagnetic Ising model if and only if $c$ is at
most 1/2.
Our proofs are based on a new link between nonlinear recursions on
trees and
$L^p$ capacities.
http://front.math.ucdavis.edu/math.PR/0503137
---------------------------------------------------------------
3322. HOW LARGE A DISC IS COVERED BY A RANDOM WALK IN $N$ STEPS?
Amir Dembo and Yuval Peres and Jay Rosen
We show that the largest disc covered by a simple random walk on the
planar
square lattice after $n$ steps has radius $n^{1/4+o(1)}$, thus
resolving an
open problem of P. R\'ev\'esz (1990). We also show that almost
surely, for
infinitely many values of $n$ it takes about $n^{1/2+o(1)}$ steps
after step
$n$ for the random walk to reach the first previously unvisited site
(and the
exponent 1/2 is sharp). This resolves a problem raised by P. R\'ev
\'esz (1993).
Additional results on multiple covering are obtained as well.
http://front.math.ucdavis.edu/math.PR/0503139
---------------------------------------------------------------
3323. INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS OF
ORNSTEIN-UHLENBECK TYPE
Siva R. Athreya and Richard F. Bass and Maria Gordina and Edwin A.
Perkins
We consider the operator $$\sL f(x)=\tfrac12 \sum_{i,j=1}^\infty
a_{ij}(x)\frac{\del^2 f}{\del x_i \del x_j}(x)-\sum_{i=1}^\infty
\lam_i x_i
b_i(x) \frac{\del f}{\del x_i}(x).$$ We prove existence and
uniqueness of
solutions to the martingale problem for this operator under appropriate
conditions on the $a_{ij}, b_i$, and $\lam_i$. The process
corresponding to
$\sL$ solves an infinite dimensional stochastic differential equation
similar
to that for the infinite dimensional Ornstein-Uhlenbeck process.
http://front.math.ucdavis.edu/math.PR/0503165
---------------------------------------------------------------
3324. ON CHORDAL AND BILATERAL SLE IN MULTIPLY CONNECTED DOMAINS
Robert O. Bauer and Roland M. Friedrich
We discuss the possible candidates for conformally invariant random
non-self-crossing curves which begin and end on the boundary of a
multiply
connected planar domain, and which satisfy a Markovian-type property. We
consider both, the case when the curve connects a boundary component
to itself
(chordal), and the case when the curve connects two different boundary
components (bilateral). We establish appropriate extensions of Loewner's
equation to multiply connected domains for the two cases. We show
that a curve
in the domain induces a motion on the boundary and that this motion
is enough
to first recover the motion of the moduli of the domain and then,
second, the
curve in the interior. For random curves in the interior we show that
the
induced random motion on the boundary is not Markov if the domain is
multiply
connected, but that the random motion on the boundary together with
the random
motion of the moduli forms a Markov process. In the chordal case, we
show that
this Markov process satisfies Brownian scaling and discuss how this
limits the
possible conformally invariant random non-self-crossing curves. We
show that
the possible candidates are labeled by a real constant and a function
homogeneous of degree minus one which describes the interaction of
the random
curve with the boundary. We show that the random curve has the locality
property if the interaction term vanishes and the real parameter
equals six.
http://front.math.ucdavis.edu/math.PR/0503178
---------------------------------------------------------------
3325. FROM N-PARAMETER FRACTIONAL BROWNIAN MOTIONS TO N-PARAMETER
MULTIFRACTIONAL BROWNIAN MOTIONS
E. Herbin
Multifractional Brownian motion is an extension of the well-known
fractional
Brownian motion where the Holder regularity is allowed to vary along
the paths.
In this paper, two kind of multi-parameter extensions of mBm are
studied: one
is isotropic while the other is not. For each of these processes, a
moving
average representation, a harmonizable representation, and the
covariance
structure are given. The Holder regularity is then studied. In
particular, the
case of an irregular exponent function H is investigated. In this
situation,
the almost sure pointwise and local Holder exponents of the multi-
parameter mBm
are proved to be equal to the correspondent exponents of H.
Eventually, a local
asymptotic self-similarity property is proved. The limit process can
be another
process than fBm.
http://front.math.ucdavis.edu/math.PR/0503182
---------------------------------------------------------------
3326. EXAMPLES OF GROUPS THAT ARE MEASURE EQUIVALENT TO THE FREE GROUP
Damien Gaboriau (UMPA-ENSL)
Measure Equivalence (ME) is the measure theoretic counterpart of
quasi-isometry. This field grew considerably during the last years,
developing
tools to distinguish between different ME classes of countable
groups. On the
other hand, contructions of ME equivalent groups are very rare. We
present a
new method, based on a notion of measurable free-factor, and we apply
it to
exhibit a new family of groups that are measure equivalent to the
free group.
We also present a quite extensive survey on results about Measure
Equivalence
for countable groups.
http://front.math.ucdavis.edu/math.DS/0503181
---------------------------------------------------------------
3327. ORTHOGONAL POLYNOMIALS AND FLUCTUATIONS OF RANDOM MATRICES
Timothy Kusalik and James A. Mingo and and Roland Speicher
In this paper we establish a connection between the fluctuations of
Wishart
random matrices, shifted Chebyshev polynomials, and planar diagrams
whose
linear span form a basis for the irreducible representations of the
annular
Temperly-Lieb algebra.
http://front.math.ucdavis.edu/math.OA/0503169
---------------------------------------------------------------
3328. COUNTING CONNECTED GRAPHS ASYMPTOTICALLY
Remco van der Hofstad and Joel Spencer
We find the asymptotic number of connected graphs with $k$ vertices and
$k-1+l$ edges when $k,l$ approach infinity, reproving a result of
Bender,
Canfield and McKay. We use the {\em probabilistic method}, analyzing
breadth-first search on the random graph $G(k,p)$ for an appropriate
edge
probability $p$. Central is analysis of a random walk with fixed
beginning and
end which is tilted to the left.
http://front.math.ucdavis.edu/math.CO/0502579
---------------------------------------------------------------
3329. ON Q-FUNCTIONAL EQUATIONS AND EXCURSION MOMENTS
Christoph Richard
We analyse q-functional equations arising from tree-like combinatorial
structures, which are counted by size, internal path length and certain
generalisations thereof. The corresponding counting parameters are
labelled by
an integer k>1. We show the existence of a joint limit distribution
for these
parameters in the limit of infinite size, if the size generating
function has a
square root as dominant singularity. The limit distribution coincides
with that
of integrals of (k-1)th powers of the standard Brownian excursion.
Our method
yields a recursion for the moments of the joint distribution and
admits an
extension to other types of singularities.
http://front.math.ucdavis.edu/math.CO/0503198
---------------------------------------------------------------
3330. A SET-INDEXED FRACTIONAL BROWNIAN MOTION
E. Herbin and E. Merzbach
We define and prove the existence of a fractional Brownian motion
indexed by
a collection of closed subsets of a measure space. This process is a
generalization of the set-indexed Brownian motion, when the condition of
independance is relaxed. Relations with the Levy fractional Brownian
motion and
with the fractional Brownian sheet are studied. We prove stationarity
of the
increments and a property of self-similarity with respect to the
action of
solid motions. Regularity conditions are exhibited. Finally, behavior
of the
set-indexed fractional Brownian motion along increasing paths is
analysed.
http://front.math.ucdavis.edu/math.PR/0503211
---------------------------------------------------------------
3331. ENTROPY-DRIVEN PHASE TRANSITION IN A POLYDISPERSE HARD-RODS
LATTICE SYSTEM
Dmitry Ioffe and Yvan Velenik (LMRS) and Milos Zahradnik
We study a system of rods on the 2d square lattice, with hard-core
exclusion.
Each rod has a length between 2 and N. We show that, when N is
sufficiently
large, and for suitable fugacity, there are several distinct Gibbs
states, with
orientational long-range order. This is in sharp contrast with the
case N=2
(the monomer-dimer model), for which Heilmann and Lieb proved absence
of phase
transition at any fugacity. This is the first example of a pure hard-
core
system with phases displaying orientational order, but not
translational order;
this is a fundamental characteristic feature of liquid crystals.
http://front.math.ucdavis.edu/math.PR/0503222
---------------------------------------------------------------
3332. AN INDUCTIVE PROOF OF THE BERRY-ESSEEN THEOREM FOR CHARACTER
RATIOS
Jason Fulman
Bolthausen used a variation of Stein's method to give an inductive
proof of
the Berry-Esseen theorem for sums of independent, identically
distributed
random variables. We modify this technique to prove a Berry-Esseen
theorem for
character ratios of a random representation of the symmetric group on
transpositions. An analogous result is proved for Jack measure on
partitions.
http://front.math.ucdavis.edu/math.CO/0503227
---------------------------------------------------------------
3333. MAX-SEMI-SELFDECOMPOSABLE LAWS AND RELATED PROCESSES
S Satheesh and E Sandhya
Methods of construction of Max-semi-selfdecompsable laws are given.
Implications of this method in random time changed extremal processes
are
discussed. Max-autoregressive model is introduced and characterized
using the
max-semi-selfdecompsable laws and exponential max-semi-stable laws. Some
comments regarding the infinite divisibility of semi-stable and max-
semi-stable
laws are given.
http://front.math.ucdavis.edu/math.PR/0503232
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3334. DISCRETE INTERPOLATION BETWEEN MONOTONE PROBABILITY AND FREE
PROBABILITY
Romuald Lenczewski and Rafal Salapata
We construct a sequence of states called m-monotone product states
which give
a discrete interpolation between the monotone product of states of
Muraki and
the free product of states of Avitzour and Voiculescu in free
probability. We
derive the associated basic limit theorems and develop the
combinatorics based
on non-crossing ordered partitions with monotone order starting from
depth m.
The Hilbert space representations of the limit mixed moments in the
invariance
principle lead to m-monotone Gaussian operators living in m-monotone
Fock
spaces, which are truncations of the free Fock space over the square-
integrable
functions on the non-negative real line (m=1 gives the monotone Fock
space). A
new type of combinatorics of inner blocks leads to explicit formulas
for the
mixed moments of m-monotone Gaussian operators, which are new even in
the case
of monotone independent Gaussian operators with arcsine distributions.
http://front.math.ucdavis.edu/math.QA/0502570
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3335. RIFFLE SHUFFLES OF DECKS WITH REPEATED CARDS
Mark Conger and D. Viswanath
By a well-known result of Bayer and Diaconis, the maximum entropy
model of
the common riffle shuffle implies that the number of riffle shuffles
necessary
to mix a standard deck of 52 cards is either 7 or 11 -- with the
former number
applying when the metric used to define mixing is the total variation
distance
and the later when it is the separation distance. This and other related
results assume all 52 cards in the deck to be distinct and require
all $52!$
permutations of the deck to be almost equally likely for the deck to be
considered well mixed. In many instances, not all cards in the deck are
distinct and only the sets of cards dealt out to players, and not the
order in
which they are dealt out to each player, needs to be random. We derive
transition probabilities under riffle shuffles between decks with
repeated
cards to cover some instances of the type just described. We focus on
decks
with cards all of which are labeled either 1 or 2 and describe the
consequences
of having a symmetric starting deck of the form $1,...,1,2...,2$ or
$1,2,...,
1,2$. Finally, we consider mixing times for common card games.
http://front.math.ucdavis.edu/math.PR/0503233
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3336. BERMUDAN OPTION PRICING BASED ON PIECEWISE HARMONIC
INTERPOLATION AND THE R\'EDUITE
Frederik S. Herzberg
We consider an iterative Bermudan option pricing algorithm based on
piecewise
harmonic interpolation and give an explicit constructive
characterisation of
the smallest fixed point of the iteration step as the approximate
price of the
perpetual Bermudan option. The same arguments work for a related
iterative
algorithm based on the approximation of subharmonic functions via the
r\'eduite
associated with a given closed $F_{\sigma}$ subset of $\RR^d$.
http://front.math.ucdavis.edu/math.PR/0503234
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3337. A BRIEF NOTE ON THE SOUNDNESS OF BERMUDAN OPTION PRICING VIA
CUBATURE
Frederik S. Herzberg
The subject of this study is an iterative Bermudan option pricing
algorithm
based on (high-dimensional) cubature. We show that the sequence of
Bermudan
prices (as functions of the underlying assets' logarithmic start prices)
resulting from the iteration is bounded and increases monotonely to the
approximate perpetual Bermudan option price; the convergence is
linear in the
supremum norm with the discount factor being the convergence factor.
Furthermore, we prove a characterisation of this approximated perpetual
Bermudan price as the smallest fixed point of the iteration procedure.
http://front.math.ucdavis.edu/math.PR/0503235
---------------------------------------------------------------
3338. SPHERICAL ASYMPTOTICS FOR THE ROTOR-ROUTER MODEL IN Z^D
Lionel Levine and Yuval Peres
The rotor-router model is a deterministic analogue of random walk
invented by
Jim Propp. It can be used to define a deterministic aggregation model
analogous
to internal diffusion limited aggregation. We prove an isoperimetric
inequality
for the exit time of simple random walk from a finite region in Z^d,
and use
this to prove that the shape of the rotor-router aggregation model in
Z^d,
suitably rescaled, converges to a Euclidean ball in R^d.
http://front.math.ucdavis.edu/math.PR/0503251
---------------------------------------------------------------
3339. SOME EXPLICIT KREIN REPRESENTATIONS OF CERTAIN SUBORDINATORS,
INCLUDING THE GAMMA PROCESS
Catherine Donati-Martin (PMA) and Marc Yor (PMA)
We give a representation of the Gamma subordinator as a Krein
functional of
Brownian motion, using the known representations for stable
subordinators and
Esscher transforms. In particular, we have obtained Krein
representations of
the subordinators which govern the two parameter Poisson-Dirichlet
family of
distributions.
http://front.math.ucdavis.edu/math.PR/0503254
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3340. AN INVARIANCE PRINCIPLE FOR CONDITIONED TREES
Jean-Francois Le Gall (DMA-ENS Paris)
We consider Galton-Watson trees associated with a critical offspring
distribution and conditioned to have exactly $n$ vertices. These
trees are
embedded in the real line by affecting spatial positions to the
vertices, in
such a way that the increments of the spatial positions along edges
of the tree
are independent variables distributed according to a symmetric
probability
distribution on the real line. We then condition on the event that
all spatial
positions are nonnegative. Under suitable assumptions on the offspring
distribution and the spatial displacements, we prove that these
conditioned
spatial trees converge as $n\to\infty$, modulo an appropriate rescaling,
towards the conditioned Brownian tree that was studied in previous work.
Applications are given to asymptotics for random quadrangulations.
http://front.math.ucdavis.edu/math.PR/0503263
---------------------------------------------------------------
3341. ON GENERALIZED COMPUTABLE UNIVERSAL PRIORS AND THEIR CONVERGENCE
Marcus Hutter
Solomonoff unified Occam's razor and Epicurus' principle of multiple
explanations to one elegant, formal, universal theory of inductive
inference,
which initiated the field of algorithmic information theory. His
central result
is that the posterior of the universal semimeasure M converges
rapidly to the
true sequence generating posterior mu, if the latter is computable.
Hence, M is
eligible as a universal predictor in case of unknown mu. The first
part of the
paper investigates the existence and convergence of computable universal
(semi)measures for a hierarchy of computability classes: recursive,
estimable,
enumerable, and approximable. For instance, M is known to be
enumerable, but
not estimable, and to dominate all enumerable semimeasures. We
present proofs
for discrete and continuous semimeasures. The second part
investigates more
closely the types of convergence, possibly implied by universality: in
difference and in ratio, with probability 1, in mean sum, and for
Martin-Loef
random sequences. We introduce a generalized concept of randomness for
individual sequences and use it to exhibit difficulties regarding
these issues.
In particular, we show that convergence fails (holds) on generalized-
random
sequences in gappy (dense) Bernoulli classes.
http://front.math.ucdavis.edu/cs.LG/0503026
---------------------------------------------------------------
3342. THE STABLE MANIFOLD THEOREM FOR SEMILINEAR STOCHASTIC
EVOLUTION EQUATIONS AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS I:
THE STOCHASTIC
SEMIFLOW
Salah-Eldin A Mohammed and Tusheng Zhang and Huaizhong Zhao
The main objective of this work is to characterize the pathwise local
structure of solutions of semilinear stochastic evolution equations
(see's) and
stochastic partial differential equations (spde's) near stationary
solutions.
Such characterization is realized through the long-term behavior of the
solution field near stationary points. The analysis falls in two
parts I, II.
In Part I (this paper), we prove a general existence and compactness
theorem
for $C^k$-cocycles of semilinear see's and spde's. Our results cover
a large
class of semilinear see's as well as certain semilinear spde's with
non-Lipschitz terms such as stochastic reaction diffusion equations
and the
stochastic Burgers equation with additive infinite-dimensional noise.
In Part
II of this work ([M-Z-Z]), we establish a local stable manifold
theorem for
non-linear see's and spde's.
http://front.math.ucdavis.edu/math.PR/0503320
---------------------------------------------------------------
3343. THE STABLE MANIFOLD THEOREM FOR SEMILINEAR STOCHASTIC
EVOLUTION EQUATIONS AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
II: EXISTENCE OF
STABLE AND UNSTABLE MANIFOLDS
Salah-Eldin A. Mohammed and Tusheng Zhang and Huaizhong Zhao
This article is a sequel to [M.Z.Z.1] aimed at completing the
characterization of the pathwise local structure of solutions of
semilinear
stochastic evolution equations (see's) and stochastic partial
differential
equations (spde's) near stationary solutions. Stationary solution are
viewed as
random points in the infinite-dimensional state space, and the
characterization
is expressed in terms of the almost sure long-time behavior of
trajectories of
the equation in relation to the stationary solution. More
specifically, we
establish local stable manifold theorems for semilinear see's and spde's
(Theorems 4.1-4.4). These results give smooth stable and unstable
manifolds in
the neighborhood of a hyperbolic stationary solution of the underlying
stochastic equation. The stable and unstable manifolds are
stationary, live in
a stationary tubular neighborhood of the stationary solution and are
asymptotically invariant under the stochastic semiflow of the see/
spde. The
proof uses infinite-dimensional multiplicative ergodic theory
techniques and
interpolation arguments (Theorem 2.1).
http://front.math.ucdavis.edu/math.PR/0503321
---------------------------------------------------------------
3344. BOUNDARY HARNACK PRINCIPLE FOR FRACTIONAL POWERS OF LAPLACIAN
ON THE SIERPINSKI CARPET
Andrzej Stos (LMP-Clermont)
We prove the Boundary Harnack Principle related to fractional powers of
Laplacian for some natural regions in the two-dimensional Sierpinski
carpet.
This is a natual application of a probabilistic method based on the
Ikeda-Watanabe formula
http://front.math.ucdavis.edu/math.PR/0503333
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3345. A NOTE ON EXACT LIKELIHOODS OF THE CARR-WU MODELS FOR LEVERAGE
EFFECTS AND VOLATILITY IN FINANCIAL ECONOMICS
Lancelot F. James
Recently Carr and Wu (2004, 2005) and also Huang and Wu (2004) show
that most
stochastic processes used in traditional option pricing models can be
cast as
special cases of time-changed L\'evy processes. In particular these
are models
which can be tailored to exhibit correlated jumps in both the log
price of
assets and the instantaneous volatility. Naturally similar to a
recent work of
Barndorff-Nielsen and Shephard (2001a, b), such models may be used in a
likelihood based framework. These likelihoods are based on the
unobserved
integrated volatility, rather than the instantaneous volatility.
James (2005)
establishes general results for the likelihood and estimation of a
large class
of such models which include possible leverage effects. In this note
we show
that exact expressions for likelihood models based on generalizations
of Carr
and Wu (2005) and Huang and Wu (2005), follow essentially from the
arguments in
Theorem 5.1 in James (2005) with some slight modification. This
serves to
formally verify a claim made by James (2005).
http://front.math.ucdavis.edu/math.ST/0503314
---------------------------------------------------------------
3346. POISSON KERNELS OF HALF-SPACES IN REAL HYPERBOLIC SPACES
T. Byczkowski and P. Graczyk and A. Stos
We provide an integral formula for the Poisson kernel of half-spaces for
Brownian motion in real hyperbolic space $\H^n$. This enables us to find
asymptotic properties of the kernel. Our starting point is the
formula for its
Fourier transform. When $n=3$, 4 or 6 we give an explicit formula for
the
Poisson kernel itself. In the general case we give various
asymptotics and show
convergence to the Poisson kernel of $\H^n$.
http://front.math.ucdavis.edu/math.PR/0503372
---------------------------------------------------------------
3347. DOOB'S MAXIMAL IDENTITY, MULTIPLICATIVE DECOMPOSITIONS AND
ENLARGEMENTS OF FILTRATIONS
A. Nikeghbali and M. Yor
In the theory of progressive enlargements of filtrations, the
supermartingale
$Z_{t}=\mathbf{P}(g>t\mid \mathcal{F}_{t}) $ associated with an
honest time
$g$, and its additive (Doob-Meyer) decomposition, play an essential
role. In
this paper, we propose an alternative approach, using a multiplicative
representation for the supermartingale $Z_{t}$, based on Doob's maximal
identity. We thus give new examples of progressive enlargements.
Moreover, we
give, in our setting, a proof of the decomposition formula for
martingales,
using initial enlargement techniques, and use it to obtain some path
decompositions given the maximum or minimum of some processes.
http://front.math.ucdavis.edu/math.PR/0503386
---------------------------------------------------------------
3348. AN ANNIHILATING-BRANCHING PARTICLE MODEL FOR THE HEAT EQUATION
WITH AVERAGE TEMPERATURE ZERO
Krzysztof Burdzy and Jeremy Quastel
We consider two species of particles performing random walks in a
domain in
Euclidean space with reflecting boundary conditions, which annihilate on
contact. In addition there is a conservation law so that the total
number of
particles of each type is preserved: When the two particles of different
species annihilate each other, particles of each species, chosen at
random,
give birth. We assume initially equal numbers of each species and
show that the
system has a diffusive scaling limit in which the densities of the
two species
are well approximated by the positive and negative parts of the
solution of the
heat equation normalized to have constant $L^1$ norm. In particular,
the higher
Neumann eigenfunctions appear as asymptotically stable states at the
diffusive
time scale.
http://front.math.ucdavis.edu/math.PR/0503395
---------------------------------------------------------------
3349. THE REVERSIBLE NEAREST PARTICLE SYSTGEMS ON A FINITE INTERVAL
Dayue Chen and Juxin Liu and Fuxi Zhang
In this paper we study a one-parameter family of attractive reversible
nearest particle system on a finite interval. As the length of the
interval
increases, the time that the nearest particle system first hits the
empty set
increases in different order, from logarithmic to exponential,
according to the
intensity of interaction. In particular, at the critical case, the first
hitting time increases in a polynomial order.
http://front.math.ucdavis.edu/math.PR/0503409
---------------------------------------------------------------
3350. INSIDE SINGULARITY SETS OF RANDOM GIBBS MEASURES
Julien Barral and Stephane Seuret
We evaluate the scale at which the multifractal structure of some random
Gibbs measures becomes discernible. The value of this scale is
obtained through
what we call the growth speed in H\"older singularity sets of a Borel
measure.
This growth speed yields new information on the multifractal behavior
of the
rescaled copies involved in the structure of statistically self-
similar Gibbs
measures. Our results are useful to understand the multifractal
nature of
various heterogeneous jump processes.
http://front.math.ucdavis.edu/math.PR/0503420
---------------------------------------------------------------
3351. RENEWAL OF SINGULARITY SETS OF STATISTICALLY SELF-SIMILAR MEASURES
Julien Barral and Stephane Seuret
This paper investigates new properties concerning the multifractal
structure
of a class of statistically self-similar measures. These measures
include the
well-known Mandelbrot multiplicative cascades, sometimes called
independent
random cascades. We evaluate the scale at which the multifractal
structure of
these measures becomes discernible. The value of this scale is
obtained through
what we call the growth speed in H\"older singularity sets of a Borel
measure.
This growth speed yields new information on the multifractal behavior
of the
rescaled copies involved in the structure of statistically self-similar
measures. Our results are useful to understand the multifractal
nature of
various heterogeneous jump processes.
http://front.math.ucdavis.edu/math.PR/0503421
---------------------------------------------------------------
3352. A POLYHEDRAL MARKOV FIELD - PUSHING THE ARAK-SURGAILIS
CONSTRUCTION INTO THREE DIMENSIONS
Tomasz Schreiber
The purpose of the paper is to construct a polyhedral Markov field in
${\mathbb R}^3$ in analogy with the planar construction of the
original Arak
(1982) polygonal Markov field. We provide a dynamic construction of
the process
in terms of evolution of two-dimensional multi-edge systems tracing
polyhedral
boundaries of the field in three-dimensional time-space. We also give
a general
algorithm for simulating Gibbsian modifications of the constructed
polyhedral
field.
http://front.math.ucdavis.edu/math.PR/0503429
---------------------------------------------------------------
3353. BAYSIAN INFERENCE VIA CLASSES OF NORMALIZED RANDOM MEASURES
Lancelot F. James and Antonio Lijoi and Igor Pruenster
One of the main research areas in Bayesian Nonparametrics is the
proposal and
study of priors which generalize the Dirichlet process. Here we exploit
theoretical properties of Poisson random measures in order to provide a
comprehensive Bayesian analysis of random probabilities which are
obtained by
an appropriate normalization. Specifically we achieve explicit and
tractable
forms of the posterior and the marginal distributions, including an
explicit
and easily used description of generalizations of the important
Blackwell-MacQueen P\'olya urn distribution. Such simplifications are
achieved
by the use of a latent variable which admits quite interesting
interpretations
which allow to gain a better understanding of the behaviour of these
random
probability measures. It is noteworthy that these models are
generalizations of
models considered by Kingman (1975) in a non-Bayesian context. Such
models are
known to play a significant role in a variety of applications including
genetics, physics, and work involving random mappings and assemblies.
Hence our
analysis is of utility in those contexts as well. We also show how
our results
may be applied to Bayesian mixture models and describe computational
schemes
which are generalizations of known efficient methods for the case of the
Dirichlet process. We illustrate new examples of processes which can
play the
role of priors for Bayesian nonparametric inference and finally point
out some
interesting connections with the theory of generalized gamma
convolutions
initiated by Thorin and further developed by Bondesson.
http://front.math.ucdavis.edu/math.ST/0503394
---------------------------------------------------------------
3354. A STOCHASTIC APPROXIMATION ALGORITHM WITH MULTIPLICATIVE STEP
SIZE ADAPTATION
Alexander Plakhov and Pedro Cruz
An algorithm of searching a zero of an unknown undimensional function is
considered, measured at a point x with some error. The step sizes are
random
positive values and are calculated according to the rule: if two
consecutive
iterations are in same direction step is multiplied by u>1,
otherwise, it is
multiplied by 0<d<1. The function may have one or more zeros; the
random values
are independent and identically distributed, with zero mean and finite
variance. Under some additional assumptions on the conditions on the two
parameters u and d almost sure convergence of the sequence as well as
under
some conditions is guaranteed almost sure divergence. In particular,
if the
error distribuition as median 0 and zero probability for particular
poinst then
it is established that for ud<1, convergence takes place, and for ud>1,
divergence. Due to the multiplicative rule of updating of the step,
it is
natural to expect that the sequence converges rapidly: like a geometric
progression (if convergence takes place), but the limit value may not
coincide
with, but instead, approximates one of zeros of the function. By
adjusting the
parameters u and d, one can reach necessary precision of
approximation; higher
precision is obtained at the expense of lower convergence rate.
http://front.math.ucdavis.edu/math.ST/0503434
---------------------------------------------------------------
3355. ON APPROXIMATE PATTERN MATCHING FOR A CLASS OF GIBBS RANDOM FIELDS
J.R. Chazottes and F. Redig and E. Verbitskiy
We prove an exponential approximation for the law of approximate
occurrence
of typical patterns for a class of Gibbsian sources on the lattice $
\mathbb
Z^d$, $d\ge 2$. From this result, we deduce a law of large numbers
and a large
deviation result for the the waiting time of distorted patterns.
http://front.math.ucdavis.edu/math.PR/0503008
---------------------------------------------------------------
3356. THE BASIC REPRESENTATION OF THE CURRENT GROUP O(N,1)^X IN THE
L^2 SPACE OVER THE GENERALIZED LEBESGUE MEASURE
A.M.Vershik and M.I.Graev
We give the realization of the representation of the current group O
(n,1)^X
where X is a manifold, in the Hilbert space of L^2(F,\nu) of
functionals on the
the space F of the generalized functions on the manifold X which are
square
integrable over measure \nu which is related to a distinguish Levy
process with
values in R^{n-1} which generalized one dimensional gamma process.
Unipotent
subgroup of the group O(n,1)^X acts as the group of multiplicators.
Measure \nu
is sigma-finite and invariant under the action current group O(n-1)
^X. Ther
case of n=2 (SL(2,R^X)) was considered before in the series of papers
starting
from the article Vershik-Gel'fand-Graev (1973).
http://front.math.ucdavis.edu/math.RT/0503404
---------------------------------------------------------------
3357. DYNAMIC IMPORTANCE SAMPLING FOR UNIFORMLY RECURRENT MARKOV CHAINS
Paul Dupuis and Hui Wang
Importance sampling is a variance reduction technique for efficient
estimation of rare-event probabilities by Monte Carlo. In standard
importance
sampling schemes, the system is simulated using an a priori fixed
change of
measure suggested by a large deviation lower bound analysis. Recent
work,
however, has suggested that such schemes do not work well in many
situations.
In this paper we consider dynamic importance sampling in the setting of
uniformly recurrent Markov chains. By ``dynamic'' we mean that in the
course of
a single simulation, the change of measure can depend on the outcome
of the
simulation up till that time. Based on a control-theoretic approach
to large
deviations, the existence of asymptotically optimal dynamic schemes is
demonstrated in great generality. The implementation of the dynamic
schemes is
carried out with the help of a limiting Bellman equation. Numerical
examples
are presented to contrast the dynamic and standard schemes.
http://front.math.ucdavis.edu/math.PR/0503454
---------------------------------------------------------------
3358. THE EXIT PROBLEM FOR DIFFUSIONS WITH TIME-PERIODIC DRIFT AND
STOCHASTIC RESONANCE
Samuel Herrmann and Peter Imkeller
Physical notions of stochastic resonance for potential diffusions in
periodically changing double-well potentials such as the spectral power
amplification have proved to be defective. They are not robust for
the passage
to their effective dynamics: continuous-time finite-state Markov chains
describing the rough features of transitions between different
domains of
attraction of metastable points. In the framework of one-dimensional
diffusions
moving in periodically changing double-well potentials we design a
new notion
of stochastic resonance which refines Freidlin's concept of quasi-
periodic
motion. It is based on exact exponential rates for the transition
probabilities
between the domains of attraction which are robust with respect to
the reduced
Markov chains. The quality of periodic tuning is measured by the
probability
for transition during fixed time windows depending on a time scale
parameter.
Maximizing it in this parameter produces the stochastic resonance
points.
http://front.math.ucdavis.edu/math.PR/0503455
---------------------------------------------------------------
3359. LEARNING MIXTURES OF SEPARATED NONSPHERICAL GAUSSIANS
Sanjeev Arora and Ravi Kannan
Mixtures of Gaussian (or normal) distributions arise in a variety of
application areas. Many heuristics have been proposed for the task of
finding
the component Gaussians given samples from the mixture, such as the EM
algorithm, a local-search heuristic from Dempster, Laird and Rubin
[J. Roy.
Statist. Soc. Ser. B 39 (1977) 1-38]. These do not provably run in
polynomial
time. We present the first algorithm that provably learns the component
Gaussians in time that is polynomial in the dimension. The Gaussians
may have
arbitrary shape, but they must satisfy a ``separation condition''
which places
a lower bound on the distance between the centers of any two component
Gaussians. The mathematical results at the heart of our proof are
``distance
concentration'' results--proved using isoperimetric inequalities--which
establish bounds on the probability distribution of the distance
between a pair
of points generated according to the mixture. We also formalize the more
general problem of max-likelihood fit of a Gaussian mixture to
unstructured
data.
http://front.math.ucdavis.edu/math.PR/0503457
---------------------------------------------------------------
3360. FAST SIMULATION OF NEW COINS FROM OLD
Serban Nacu and Yuval Peres
Let S\subset (0,1). Given a known function f:S\to (0,1), we consider the
problem of using independent tosses of a coin with probability of
heads p
(where p\in S is unknown) to simulate a coin with probability of
heads f(p). We
prove that if S is a closed interval and f is real analytic on S,
then f has a
fast simulation on S (the number of p-coin tosses needed has exponential
tails). Conversely, if a function f has a fast simulation on an open
set, then
it is real analytic on that set.
http://front.math.ucdavis.edu/math.PR/0503458
---------------------------------------------------------------
3361. STRUCTURE OF LARGE RANDOM HYPERGRAPHS
R. W. R. Darling and J. R. Norris
The theme of this paper is the derivation of analytic formulae for
certain
large combinatorial structures. The formulae are obtained via fluid
limits of
pure jump-type Markov processes, established under simple conditions
on the
Laplace transforms of their Levy kernels. Furthermore, a related
Gaussian
approximation allows us to describe the randomness which may persist
in the
limit when certain 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 identifiable if every one of its
vertices is
identifiable. Our analytic formulae describe the asymptotics of the
number of
identifiable vertices and the number of identifiable hyperedges for a
Poisson(\beta) random hypergraph \Lambda on a set V of N vertices, in
the limit
as N\to \infty. Here \beta is a formal power series with nonnegative
coefficients \beta_0,\beta_1,..., and (\Lambda(A))_{A\subseteq V} are
independent Poisson random variables such that \Lambda(A), the number of
hyperedges on A, has mean N\beta_j/\pmatrixN j whenever |A|=j.
http://front.math.ucdavis.edu/math.PR/0503460
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3362. LARGE DEVIATIONS FOR TEMPLATE MATCHING BETWEEN POINT PROCESSES
Zhiyi Chi
We study the asymptotics related to the following matching criteria
for two
independent realizations of point processes X\sim X and Y\sim Y.
Given l>0,
X\cap [0,l) serves as a template. For each t>0, the matching score
between the
template and Y\cap [t,t+l) is a weighted sum of the Euclidean
distances from
y-t to the template over all y\in Y\cap [t,t+l). The template
matching criteria
are used in neuroscience to detect neural activity with certain
patterns. We
first consider W_l(\theta), the waiting time until the matching score
is above
a given threshold \theta. We show that whether the score is scalar- or
vector-valued, (1/l)\log W_l(\theta) converges almost surely to a
constant
whose explicit form is available, when X is a stationary ergodic
process and Y
is a homogeneous Poisson point process. Second, as l\to\infty, a strong
approximation for -\log [\Pr{W_l(\theta)=0}] by its rate function is
established, and in the case where X is sufficiently mixing, the
rates, after
being centered and normalized by \sqrtl, satisfy a central limit
theorem and
almost sure invariance principle. The explicit form of the variance
of the
normal distribution is given for the case where X is a homogeneous
Poisson
process as well.
http://front.math.ucdavis.edu/math.PR/0503463
---------------------------------------------------------------
3363. RANDOM K-SAT: TWO MOMENTS SUFFICE TO CROSS A SHARP THRESHOLD
Dimitris Achlioptas and Cristopher Moore
Many NP-complete constraint satisfaction problems appear to undergo a
"phase
transition'' from solubility to insolubility when the constraint
density passes
through a critical threshold. In all such cases it is easy to derive
upper
bounds on the location of the threshold by showing that above a
certain density
the first moment (expectation) of the number of solutions tends to
zero. We
show that in the case of certain symmetric constraints, considering
the second
moment of the number of solutions yields nearly matching lower bounds
for the
location of the threshold. Specifically, we prove that the threshold
for both
random hypergraph 2-colorability (Property B) and random Not-All-
Equal k-SAT is
2^{k-1} ln 2 -O(1). As a corollary, we establish that the threshold
for random
k-SAT is of order Theta(2^k), resolving a long-standing open problem.
http://front.math.ucdavis.edu/cond-mat/0310227
---------------------------------------------------------------
3364. DISTRIBUTION OF THE SIZE OF A LARGEST PLANAR MATCHING AND
LARGEST PLANAR SUBGRAPH IN RANDOM BIPARTITE GRAPHS
Marcos Kiwi and Martin Loebl
We address the following question: When a randomly chosen regular
bipartite
multi--graph is drawn in the plane in the ``standard way'', what is the
distribution of its maximum size planar matching (set of non--
crossing disjoint
edges) and maximum size planar subgraph (set of non--crossing edges
which may
share endpoints)? The problem is a generalization of the Longest
Increasing
Sequence (LIS) problem (also called Ulam's problem). We present
combinatorial
identities which relate the number of $r$-regular bipartite multi--
graphs with
maximum planar matching (maximum planar subgraph)of at most $d$ edges
to a
signed sum of restricted lattice walks in $\ZZ^d$, and to the number
of pairs
of standard Young tableaux of the same shape and with a ``descend--
type''
property. Our results are obtained via generalizations of two
combinatorial
proofs through which Gessel's identity can be obtained (an identity
that is
crucial in the derivation of a bivariate generating function
associated to the
distribution of LISs, and key to the analytic attack on Ulam's problem).
http://front.math.ucdavis.edu/math.CO/0503465
---------------------------------------------------------------
3365. THE SHANNON INFORMATION OF FILTRATIONS AND THE ADDITIONAL
LOGARITHMIC UTILITY OF INSIDERS
Stefan Ankirchner and Steffen Dereich and Peter Imkeller
The background for the general mathematical link between utility and
information theory investigated in this paper is a simple financial
market
model with two kinds of small traders: less informed traders and
insiders,
whose extra information is represented by an enlargement of the other
agents'
filtration. The expected logarithmic utility increment, i.e. the
difference of
the insider's and the less informed trader's expected logarithmic
utility is
described in terms of the information drift, i.e. the drift one has to
eliminate in order to perceive the price dynamics as a martingale
from the
insider's perspective. On the one hand, we describe the information
drift in a
very general setting by natural quantities expressing the
probabilistic better
informed view of the world. This on the other hand allows us to
identify the
additional utility by entropy related quantities known from
information theory.
In particular, in a complete market in which the insider has some fixed
additional information during the entire trading interval, its utility
increment can be represented by the Shannon information of his extra
knowledge.
For general markets, and in some particular examples, we provide
estimates of
maximal utility by information inequalities.
http://front.math.ucdavis.edu/math.PR/0503013
---------------------------------------------------------------
3366. DIFFUSION MAPS, SPECTRAL CLUSTERING AND REACTION COORDINATES
OF DYNAMICAL SYSTEMS
Boaz Nadler and Stephane Lafon and Ronald R. Coifman and Ioannis
G. Kevrekidis
A central problem in data analysis is the low dimensional
representation of
high dimensional data, and the concise description of its underlying
geometry
and density. In the analysis of large scale simulations of complex
dynamical
systems, where the notion of time evolution comes into play,
important problems
are the identification of slow variables and dynamically meaningful
reaction
coordinates that capture the long time evolution of the system. In
this paper
we provide a unifying view of these apparently different tasks, by
considering
a family of {\em diffusion maps}, defined as the embedding of complex
(high
dimensional) data onto a low dimensional Euclidian space, via the
eigenvectors
of suitably defined random walks defined on the given datasets.
Assuming that
the data is randomly sampled from an underlying general probability
distribution $p(\x)=e^{-U(\x)}$, we show that as the number of
samples goes to
infinity, the eigenvectors of each diffusion map converge to the
eigenfunctions
of a corresponding differential operator defined on the support of the
probability distribution. Different normalizations of the Markov
chain on the
graph lead to different limiting differential operators. One
normalization
gives the Fokker-Planck operators with the same potential U(x), best
suited for
the study of stochastic differential equations as well as for
clustering.
Another normalization gives the Laplace-Beltrami (heat) operator on the
manifold in which the data resides, best suited for the analysis of the
geometry of the dataset, regardless of its possibly non-uniform density.
http://front.math.ucdavis.edu/math.NA/0503445
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3367. TRADING STRATEGY ADIPTED OPTIMIZATION OF EUROPEAN CALL OPTION
Toshio Fukumi
Optimal pricing of European call option is described by linear
stochastic
differential equation. Trading strategy given by a twin of stochastic
variables
was integrated w.r.t. Black-Scholes formula to adopt optimal pricing to
tarading strategy.
http://front.math.ucdavis.edu/math.OC/0503444
---------------------------------------------------------------
3368. CHARACTERIZATION OF ARBITRAGE-FREE MARKETS
Eva Strasser
The present paper deals with the characterization of no-arbitrage
properties
of a continuous semimartingale. The first main result, Theorem
\refMainTheoremCharNA, extends the no-arbitrage criterion by Levental
and
Skorohod [Ann. Appl.
Probab. 5 (1995) 906-925] from diffusion processes to arbitrary
continuous
semimartingales. The second main result, Theorem 2.4, is a
characterization of
a weaker notion of no-arbitrage in terms of the existence of
supermartingale
densities. The pertaining weaker notion of no-arbitrage is equivalent
to the
absence of immediate arbitrage opportunities, a concept introduced by
Delbaen
and Schachermayer [Ann. Appl. Probab. 5 (1995) 926-945]. Both results
are
stated in terms of conditions for any semimartingales starting at
arbitrary
stopping times \sigma. The necessity parts of both results are known
for the
stopping time \sigma=0 from Delbaen and Schachermayer [Ann. Appl.
Probab. 5
(1995) 926-945]. The contribution of the present paper is the proofs
of the
corresponding sufficiency parts.
http://front.math.ucdavis.edu/math.PR/0503473
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3369. GAUSSIAN LIMITS FOR RANDOM MEASURES IN GEOMETRIC PROBABILITY
Yu. Baryshnikov and J. E. Yukich
We establish Gaussian limits for general measures induced by binomial
and
Poisson point processes in d-dimensional space. The limiting Gaussian
field has
a covariance functional which depends on the density of the point
process. The
general results are used to deduce central limit theorems for
measures induced
by random graphs (nearest neighbor, Voronoi and sphere of influence
graph),
random sequential packing models (ballistic deposition and spatial
birth-growth
models) and statistics of germ-grain models.
http://front.math.ucdavis.edu/math.PR/0503474
---------------------------------------------------------------
3370. ON THE DISTRIBUTION OF THE MAXIMUM OF A GAUSSIAN FIELD WITH D
PARAMETERS
Jean-Marc Azais and Mario Wschebor
Let I be a compact d-dimensional manifold, let X:I\to R be a Gaussian
process
with regular paths and let F_I(u), u\in R, be the probability
distribution
function of sup_{t\in I}X(t). We prove that under certain regularity and
nondegeneracy conditions, F_I is a C^1-function and satisfies a certain
implicit equation that permits to give bounds for its values and to
compute its
asymptotic behavior as u\to +\infty. This is a partial extension of
previous
results by the authors in the case d=1. Our methods use strongly the
so-called
Rice formulae for the moments of the number of roots of an equation
of the form
Z(t)=x, where Z:I\to R^d is a random field and x is a fixed point in
R^d. We
also give proofs for this kind of formulae, which have their own
interest
beyond the present application.
http://front.math.ucdavis.edu/math.PR/0503475
---------------------------------------------------------------
3371. HEAVY TRAFFIC ANALYSIS OF OPEN PROCESSING NETWORKS WITH
COMPLETE RESOURCE POOLING: ASYMPTOTIC OPTIMALITY OF DISCRETE REVIEW
POLICIES
Baris Ata and Sunil Kumar
We consider a class of open stochastic processing networks, with
feedback
routing and overlapping server capabilities, in heavy traffic. The
networks we
consider satisfy the so-called complete resource pooling condition and
therefore have one-dimensional approximating Brownian control problems.
We propose a simple discrete review policy for controlling such
networks.
Assuming 2+\epsilon moments on the interarrival times and
processing times,
we provide a conceptually simple proof of asymptotic optimality of
the proposed
policy.
http://front.math.ucdavis.edu/math.PR/0503477
---------------------------------------------------------------
3372. A CHARACTERIZATION OF THE OPTIMAL RISK-SENSITIVE AVERAGE COST
IN FINITE CONTROLLED MARKOV CHAINS
Rolando Cavazos-Cadena and Daniel Hernandez-Hernandez
This work concerns controlled Markov chains with finite state and action
spaces. The transition law satisfies the simultaneous Doeblin
condition, and
the performance of a control policy is measured by the (long-run)
risk-sensitive average cost criterion associated to a positive, but
otherwise
arbitrary, risk sensitivity coefficient. Within this context, the
optimal
risk-sensitive average cost is characterized via a minimization
problem in a
finite-dimensional Euclidean space.
http://front.math.ucdavis.edu/math.PR/0503478
---------------------------------------------------------------
3373. LARGE DEVIATIONS OF THE EMPIRICAL VOLUME FRACTION FOR
STATIONARY POISSON GRAIN MODELS
Lothar Heinrich
We study the existence of the (thermodynamic) limit of the scaled
cumulant-generating function L_n(z)=|W_n|^{-1}\logE\exp{z|\Xi\cap
W_n|} of the
empirical volume fraction |\Xi\cap W_n|/|W_n|, where |\cdot| denotes the
d-dimensional Lebesgue measure. Here \Xi=\bigcup_{i\ge1}(\Xi_i+X_i)
denotes a
d-dimensional Poisson grain model (also known as a Boolean model)
defined by a
stationary Poisson process \Pi_{\lambda}=\sum_{i\ge1}\delta_{X_i} with
intensity \lambda >0 and a sequence of independent copies \Xi_1,
\Xi_2,... of a
random compact set \Xi_0. For an increasing family of compact convex
sets {W_n,
n\ge1} which expand unboundedly in all directions, we prove the
existence and
analyticity of the limit lim_{n\to\infty}L_n(z) on some disk in the
complex
plane whenever E\exp{a|\Xi_0|}<\infty for some a>0. Moreover, closely
connected
with this result, we obtain exponential inequalities and the exact
asymptotics
for the large deviation probabilities of the empirical volume
fraction in the
sense of Cram\'er and Chernoff.
http://front.math.ucdavis.edu/math.PR/0503479
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