[Pas] Probability Abstracts 93
pas at www2.economia.unimi.it
pas at www2.economia.unimi.it
Tue Aug 1 10:02:33 CEST 2006
Aug 1st, 2006
Letter 93
Probability Abstract Service
Abstracts from May-1-2006 to Jul-31-2006
html version here: http://www2.economia.unimi.it/PAS/Letters/
letter_93.shtml
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Note: this PAS letter cover three months instead of only two.
This was due to a PAS server update during July 2006 which
cause delay. Next PAS letter will have the same bimonthly
posting.
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4255. ESTIMATION IN SPIN GLASSES: A FIRST STEP
Sourav Chatterjee
The Sherrington-Kirkpatrick model of spin glasses, the Hopfield model of
neural networks, and the Ising spin glass are all models of binary data
belonging to the one-parameter exponential family with quadratic
sufficient
statistic. Under bare minimal conditions, we establish the
consistency of the
maximum pseudolikelihood estimate of the natural parameter in this
family, even
at critical temperatures. Since very little is known about the low
and critical
temperature regimes of these extremely difficult models, the proof
requires
several new ideas. The author's version of Stein's method is a
particularly
useful tool. One goal of this paper is to introduce these techniques
into the
realm of mathematical statistics through an example.
http://front.math.ucdavis.edu/math.PR/0604634
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4256. A DELAYED BLACK AND SCHOLES FORMULA I
Mercedes Arriojas and Yaozhong Hu and Salah-Eldin Mohammed and
Gyula Pap
In this article we develop an explicit formula for pricing European
options
when the underlying stock price follows a non-linear stochastic
differential
delay equation (sdde). We believe that the proposed model is
sufficiently
flexible to fit real market data, and is yet simple enough to allow
for a
closed-form representation of the option price. Furthermore, the model
maintains the no-arbitrage property and the completeness of the
market. The
derivation of the option-pricing formula is based on an equivalent
martingale
measure.
http://front.math.ucdavis.edu/math.PR/0604640
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4257. A DELAYED BLACK AND SCHOLES FORMULA II
Mercedes Arriojas and Yaozhong Hu and Salah-Eldin Mohammed and
Gyula Pap
This article is a sequel to [A.H.M.P]. In [A.H.M.P], we develop an
explicit
formula for pricing European options when the underlying stock price
follows a
non-linear stochastic delay equation with fixed delays in the drift and
diffusion terms. In this article, we look at models of the stock price
described by stochastic functional differential equations with
variable delays.
We present a class of examples of stock dynamics with variable delays
that
permit an explicit form for the option pricing formula. As in
[A.H.M.P], the
market is complete with no arbitrage. This is achieved through the
existence of
an equivalent martingale measure. In subsequent work, the authors
intend to
test the models in [A.H.M.P] and the present article against real
market data.
http://front.math.ucdavis.edu/math.PR/0604641
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4258. THE HECKMAN-OPDAM MARKOV PROCESSES
Bruno Schapira (MAPMO and PMA)
We introduce and study the natural counterpart of the Dunkl Markov
processes
in a negatively curved setting. We give a semimartingale
decomposition of the
radial part, and some properties of the jumps. We prove also a law of
large
numbers, a central limit theorem, and the convergence of the
normalized process
to the Dunkl process. Eventually we describe the asymptotic behavior
of the
infinite loop as it was done by Anker, Bougerol and Jeulin in the
symmetric
spaces setting in \cite{ABJ}.
http://front.math.ucdavis.edu/math.PR/0605020
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4259. TWO-DIMENSIONAL CRITICAL PERCOLATION: THE FULL SCALING LIMIT
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.
http://front.math.ucdavis.edu/math.PR/0605035
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4260. GENERALIZATION OF THE BOREL-CANTELLI LEMMA
Alexei Stepanov
In the present note a generalization of Borel-Cantelli Lemma is
proposed.
http://front.math.ucdavis.edu/math.ST/0605007
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4261. TUG-OF-WAR AND THE INFINITY LAPLACIAN
Yuval Peres and Oded Schramm and Scott Sheffield and David Wilson
We prove that every bounded Lipschitz function F on a subset Y of a
length
space X admits a tautest extension to X, i.e., a unique Lipschitz
extension u
for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X
that do
not intersect Y.
This was previously known only for bounded domains R^n, in which
case u is
infinity harmonic, that is, a viscosity solution to Delta_infty u =
0. We also
prove the first general uniqueness results for Delta_infty u = g on
bounded
subsets of R^n (when g is uniformly continuous and bounded away from
zero), and
analogous results for bounded length spaces.
The proofs rely on a new game-theoretic description of u. Let
u^epsilon(x) be
the value of the following two-player zero-sum game, called tug-of-
war: fix
x_0=x \in X minus Y. At the kth turn, the players toss a coin and the
winner
chooses an x_k with d(x_k, x_{k-1})< \epsilon. The game ends when x_k
is in Y,
and player one's payoff is
F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i)
We show that the u^\epsilon converge uniformly to u as epsilon
tends to zero.
Even for bounded domains in R^n, the game theoretic description of
infinity-harmonic functions yields new intuition and estimates; for
instance,
we prove power law bounds for infinity-harmonic functions in the unit
disk with
boundary values supported in a delta-neighborhood of a Cantor set on
the unit
circle.
http://front.math.ucdavis.edu/math.AP/0605002
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4262. OPERATORS ASSOCIATED WITH THE SOFT AND HARD SPECTRAL EDGES OF
UNITARY ENSEMBLES
Gordon Blower
Using Hankel operators and shift-invariant subspaces on Hilbert
space, this
paper develops the theory of the operators associated with soft and
hard edges
of eigenvalue distributions of random matrices. Tracy and Widom
introduced a
projection operator $W$ to describe the soft edge of the spectrum of the
Gaussian unitary ensemble. The subspace $WL^2$ is simply invariant
under the
translation semigroup $e^{itD}$ $(t\geq 0)$ and invariant under the
Schr\"odinger semigroup $e^{it(D^2+x)}$ $(t\geq 0)$; these properties
characterize $WL^2$ via Beurling's theorem. The Jacobi ensemble of
random
matrices has positive eigenvalues which tend to accumulate near to
the hard
edge at zero. This paper identifies a pair of unitary groups that
satisfy the
von Neumann--Weyl anti-commutation relations and leave invariant certain
subspaces of $L^2(0,\infty)$ which are invariant for operators with
Jacobi
kernels. Such Tracy--Widom operators are reproducing kernels for
weighted Hardy
spaces, known as Sonine spaces. Periodic solutions of Hill's equation
give a
new family of Tracy--Widom type operators.
http://front.math.ucdavis.edu/math.FA/0605010
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4263. A CENTRAL LIMIT THEOREM FOR CONVEX SETS
B. Klartag
We show that there exists a sequence $\eps_n \searrow 0$ for which the
following holds: Let $K \subset \RR^n$ be a compact, convex set with a
non-empty interior. Let $X$ be a random vector that is distributed
uniformly in
$K$. Then there exists a unit vector $\theta$ in $\RR^n$, $t_0 \in \RR
$ and
$\sigma > 0$ such that \begin{equation}
\sup_{A \subset \RR} | Prob \{< X, \theta > \in A \} - \frac{1}
{\sqrt{2 \pi
\sigma}} \int_A e^{-\frac{(t - t_0)^2}{2 \sigma^2}} dt | \leq \eps_n,
\end{equation} where the supremum runs over all measurable sets $A
\subset
\RR$, and where $<\cdot, \cdot >$ denotes the usual scalar product in
$\RR^n$.
Moreover, under the additional assumptions that the expectation of $X
$ is zero
and that the covariance matrix of $X$ is the identity matrix, we
argue that
most unit vectors $\theta$ satisfy ($\dagger$), with $t_0 = 0$ and $
\sigma =
1$. Thus, typical one-dimensional marginal distributions of high-
dimensional,
isotropic, convex sets are approximately gaussian. This proves a basic
conjecture in asymptotic convex geometry, that was put forward by
Anttila, Ball
and Perissinaki and by Brehm and Voigt. We also discuss normal
approximation
for multi-dimensional marginal distributions of uniform measures on
convex
sets.
http://front.math.ucdavis.edu/math.MG/0605014
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4264. PRICING WITH COHERENT RISK
Alexander S. Cherny
This paper deals with applications of coherent risk measures to
pricing in
incomplete markets. Namely, we study the No Good Deals pricing
technique based
on coherent risk. Two forms of this technique are presented: one
defines a good
deal as a trade with negative risk; the other one defines a good deal
as a
trade with unusually high RAROC. For each technique, the fundamental
theorem of
asset pricing and the form of the fair price interval are presented.
The model
considered includes static as well as dynamic models, models with an
infinite
number of assets, models with transaction costs, and models with
portfolio
constraints. In particular, we prove that in a model with proportional
transaction costs the fair price interval converges to the fair price
interval
in a frictionless model as the coefficient of transaction costs tends
to zero.
Moreover, we study some problems in the ``pure'' theory of risk
measures: we
present a simple geometric solution of the capital allocation problem
and apply
it to define the coherent risk contribution. The mathematical tools
employed
are probability theory, functional analysis, and finite-dimensional
convex
analysis.
http://front.math.ucdavis.edu/math.PR/0605049
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4265. ON THE RANGE OF THE SIMPLE RANDOM WALK BRIDGE ON GROUPS
Itai Benjamini and Roey Izkovsky and Harry Kesten
Let G be a vertex transitive graph. A study of the range of simple
random
walk on G and of its bridge is proposed. While it is expected that on
a graph
of polynomial growth the sizes of the range of the unrestricted
random walk and
of its bridge are the same in first order, this is not the case on
some larger
graphs such as regular trees. Of particular interest is the case when
G is the
Cayley graph of a group. In this case we even study the range of a
general
symmetric (not necessarily simple) random walk on G. We hope that the
few
examples for which we calculate the first order behavior of the range
here will
help to discover some relation between the group structure and the
behavior of
the range. Further problems regarding bridges are presented.
http://front.math.ucdavis.edu/math.PR/0605050
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4266. EQUILIBRIUM WITH COHERENT RISK
Alexander S. Cherny
This paper is the continuation of "Pricing with coherent risk" and
deals with
further applications of coherent risk measures to problems of
finance. First,
we study the optimization problem. Three forms of this problem are
considered.
Furthermore, the results obtained are applied to the optimality
pricing. Again
three forms of this technique are considered. Finally, we study the
equilibrium
problem both in the unconstrained and in the constrained forms. We
establish
the equivalence between the global and the competitive optima and
give a dual
description of the equilibrium. Moreover, we provide an explicit
geometric
solution of the constrained equilibrium problem. Most of the results are
presented on two levels: on a general level the results have a
probabilistic
form; for a static model with a finite number of assets, the results
have a
geometric form.
http://front.math.ucdavis.edu/math.PR/0605051
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4267. LARGE DEVIATIONS AND A KRAMERS' TYPE LAW FOR SELF-STABILIZING
DIFFUSIONS
Samuel Herrmann and Peter Imkeller and Dierk Peithmann
We investigate exit times from domains of attraction for the motion of a
self-stabilized particle travelling in a geometric (potential type)
landscape
and perturbed by Brownian noise of small amplitude. Self-
stabilization is
mediated by an ensemble-average attraction adding on to the individual
potential drift, where the particle is supposed to be suspended in a
large
population of identical ones. A Kramers' type law for the particle's
exit from
the potential's domains of attraction and a large deviations
principle for the
self-stabilizing diffusion are proved. It turns out that the exit law
for the
self-stabilizing diffusion coincides with the exit law of a potential
diffusion
without self-stabilization with a drift component perturbed by average
attraction. We show that self-stabilization may substantially delay
the exit
from domains of attraction, and that the exit location may be completely
different.
http://front.math.ucdavis.edu/math.PR/0605053
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4268. OPTIMAL STOPPING OF HUNT AND L\'EVY PROCESSES
Ernesto Mordecki and Paavo Salminen
The optimal stopping problem for a Hunt processes on $\R$ is
considered via
the representation theory of excessive functions. In particular, we
focus on
infinite horizon (or perpetual) problems with one-sided structure,
that is,
there exists a point $x^*$ such that the stopping region is of the form
$[x^*,+\infty)$. Corresponding results for two-sided problems are also
indicated. The main result is a spectral representation of the value
function
in terms of the Green kernel of the process. Specializing in L\'evy
processes,
we obtain, by applying the Wiener-Hopf factorization, a general
representation
of the value function in terms of the maximum of the L\'evy process. To
illustrate the results, an explicit expression for the Green kernel
of Brownian
motion with exponential jumps is computed and some optimal stopping
problems
for Poisson process with positive exponential jumps and negative
drift are
solved.
http://front.math.ucdavis.edu/math.PR/0605054
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4269. SUR LE NOMBRE DE POINTS VISIT\'{E}S PAR UNE MARCHE AL\'{E}
ATOIRE SUR UN AMAS INFINI DE PERCOLATION
Clement Rau (LATP)
In this article, we consider random walk on the infinite cluster of bond
percolation on $\Z^d (d \geq 2)$. We show that the Laplace
transformation of
the number of visited points $N\_n$, has a behaviour as the random
walk was on
$\Z^d$. More precisely, for all $0<\alpha<1$, we proved that there exist
constants $C\_i$ and $C\_s$ such that for all infinite cluster that
contains
the origin, we have: $$ e^{-C\_i n^{\frac{d}{d+2}}} \leq \E\_0^{\omega}
(\alpha^{N\_n}) \leq e^{-C\_sn^{\frac{d}{d+2}}}.$$ Our approach is
based on
finding an isoperimetric inequalities on the infinite cluster, lifted
on a
wreath product which give good behaviour. The problem of the
isoperimetry on
wreath product was already raised by A.Ershler.
http://front.math.ucdavis.edu/math.PR/0605056
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4270. COHERENT MEASUREMENT OF FACTOR RISKS
Alexander S. Cherny and Dilip B. Madan
We propose a new procedure for the risk measurement of large
portfolios. It
employs the following objects as the building blocks: - coherent risk
measures
introduced by Artzner, Delbaen, Eber, and Heath; - factor risk measures
introduced in this paper, which assess the risks driven by particular
factors
like the price of oil, S&P500 index, or the credit spread; - risk
contributions
and factor risk contributions, which provide a coherent alternative
to the
sensitivity coefficients.
We also propose two particular classes of coherent risk measures
called Alpha
V at R and Beta V at R, for which all the objects described above admit an
extremely
simple empirical estimation procedure. This procedure uses no model
assumptions
on the structure of the price evolution.
Moreover, we consider the problem of the risk management on a
firm's level.
It is shown that if the risk limits are imposed on the risk
contributions of
the desks to the overall risk of the firm (rather than on their
outstanding
risks) and the desks are allowed to trade these limits within a firm,
then the
desks automatically find the globally optimal portfolio.
http://front.math.ucdavis.edu/math.PR/0605062
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4271. PRICING AND HEDGING IN INCOMPLETE MARKETS WITH COHERENT RISK
Alexander S. Cherny and Dilip B. Madan
We propose a pricing technique based on coherent risk measures, which
enables
one to get finer price intervals than in the No Good Deals pricing.
The main
idea consists in splitting a liability into several parts and selling
these
parts to different agents. The technique is closely connected with the
convolution of coherent risk measures and equilibrium considerations.
Furthermore, we propose a way to apply the above technique to the
coherent
estimation of the Greeks.
http://front.math.ucdavis.edu/math.PR/0605064
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4272. CAPM, REWARDS, AND EMPIRICAL ASSET PRICING WITH COHERENT RISK
Alexander S. Cherny and Dilip B. Madan
The paper has 2 main goals: 1. We propose a variant of the CAPM based on
coherent risk. 2. In addition to the real-world measure and the risk-
neutral
measure, we propose the third one: the extreme measure. The
introduction of
this measure provides a powerful tool for investigating the relation
between
the first two measures. In particular, this gives us - a new way of
measuring
reward; - a new approach to the empirical asset pricing.
http://front.math.ucdavis.edu/math.PR/0605065
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4273. ITO MAPS AND ANALYSIS ON PATH SPACES
K. D. Elworthy and Xue-Mei Li
We consider versions of Malliavin calculus on path spaces of compact
manifolds with diffusion measures, defining Gross-Sobolev spaces of
differentiable functions and proving their intertwining with solution
maps, I,
of certain stochastic differential equations. This is shown to shed
light on
fundamental uniqueness questions for this calculus including
uniqueness of the
closed derivative operator $d$ and Markov uniqueness of the associated
Dirichlet form. A continuity result for the divergence operator by
Kree and
Kree is extended to this situation. The regularity of conditional
expectations
of smooth functionals of classical Wiener space, given I, is
considered and
shown to have strong implications for these questions. A major role
is played
by the (possibly sub-Riemannian) connections induced by stochastic
differential
equations: Damped Markovian connections are used for the covariant
derivatives.
http://front.math.ucdavis.edu/math.PR/0605089
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4274. COMPRESSING REDUNDANT INFORMATION IN MARKOV CHAINS
Giacomo Aletti
Given a strongly stationary Markov chain and a finite set of stopping
rules,
we prove the existence of a polynomial algorithm which projects the
Markov
chain onto a minimal Markov chain without redundant information. Markov
complexity is hence defined and tested on some classical problems.
http://front.math.ucdavis.edu/math.PR/0605099
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4275. EXPECTED NUMBER OF LOCAL MAXIMA OF SOME GAUSSIAN RANDOM
POLYNOMIALS
S. Shemehsavar and S. Rezakhah
Let $Q_n(x)=\sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraic
polynomial where
the coefficients $A_0,A_1,... $ form a sequence of centered Gaussian
random
variables. Moreover, assume that the increments $\Delta_j=A_j-A_{j-1}$,
$j=0,1,2,...$ are independent, $A_{-1}=0$. The coefficients can be
considered
as $n$ consecutive observations of a Brownian motion. We study the
asymptotic
behaviour of the expected number of local maxima of $Q_n(x)$ below level
$u=O(n^k)$, for some $k>0$.
http://front.math.ucdavis.edu/math.PR/0605116
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4276. ANCHORED CRITICAL PERCOLATION CLUSTERS AND 2-D ELECTROSTATICS
P. Kleban and J. J. H. Simmons and and R. M. Ziff
We consider the densities of clusters, at the percolation point of a
two-dimensional system, which are anchored in various ways to an
edge. These
quantities are calculated by use of conformal field theory and computer
simulations. We find that they are given by simple functions of the
potentials
of 2-D electrostatic dipoles, and that a kind of superposition {\it cum}
factorization applies. Our results broaden this connection, already
known from
previous studies, and we present evidence that it is more generally
valid. An
exact result similar to the Kirkwood superposition approximation
emerges.
http://front.math.ucdavis.edu/cond-mat/0605120
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4277. THE CONFIGURATIONAL MEASURE ON MUTUALLY AVOIDING SLE PATHS
Michael J. Kozdron (University of Regina) and Gregory F. Lawler
(Cornell University)
We define multiple chordal SLEs in a simply connected domain by
considering a
natural configurational measure on paths. We show how to construct these
measures so that they are conformally covariant and satisfy certain
boundary
perturbation and Markov properties, as well as a cascade relation. As an
example of our construction, we derive the scaling limit of Fomin's
identity in
the case of two paths directly; that is, we prove that the
probability that an
SLE(2) and a Brownian excursion do not intersect can be given in
terms of the
determinant of the excursion hitting matrix. Finally, we define the
lambda-SAW,
a one-parameter family of measures on self-avoiding walks on Z^2.
http://front.math.ucdavis.edu/math.PR/0605159
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4278. LOOP-FREE MARKOV CHAINS AS DETERMINANTAL POINT PROCESSES
Alexei Borodin
We show that any loop-free Markov chain on a discrete space can be
viewed as
a determinantal point process. As an application we prove central limit
theorems for the number of particles in a window for renewal
processes and
Markov renewal processes with Bernoulli noise.
http://front.math.ucdavis.edu/math.PR/0605168
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4279. BEHAVIOR OF A SECOND CLASS PARTICLE IN HAMMERSLEY'S PROCESS
Eric Cator and Sergei Dobrynin
In the case of a rarefaction fan in a non-stationary Hammersley
process, we
explicitly calculate the asymptotic behavior of the process as we
move out
along a ray, and the asymptotic distribution of the angle within the
rarefaction fan of a second class particle and a dual second class
particle.
Furthermore, we consider a stationary Hammersley process and use the
previous
results to show that trajectories of a second class particle and a
dual second
class particles touch with probability one, and we give some
information on the
area enclosed by the two trajectories, up until the first
intersection point.
This is linked to the area of influence of an added Poisson point in
the plane.
http://front.math.ucdavis.edu/math.PR/0605199
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4280. RANDOM MATRIX CENTRAL LIMIT THEOREMS FOR NON-INTERSECTING
RANDOM WALKS
Jinho Baik and Toufic Suidan
We consider non-intersecting random walks satisfying the condition
that the
increments have a finite moment generating function. We prove that in
a certain
limiting regime where the number of walks and the number of time
steps grow to
infinity, several limiting distributions of the walks at the mid-time
behave as
the eigenvalues of random Hermitian matrices as the dimension of the
matrices
grows to infinity.
http://front.math.ucdavis.edu/math.PR/0605212
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4281. ON THE BEHAVIOR OF RANDOM WALK AROUND HEAVY POINTS
Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
Consider a symmetric aperiodic random walk in $Z^d$, $d\geq 3$. There
are
points (called heavy points) where the number of visits by the random
walk is
close to its maximum. We investigate the local times around these
heavy points
and show that they converge to a deterministic limit as the number of
steps
tends to infinity.
http://front.math.ucdavis.edu/math.PR/0605221
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4282. $T^{1/3}$ SUPERDIFFUSIVITY OF FINITE-RANGE ASYMMETRIC
EXCLUSION PROCESSES ON $\MATHBB Z$
Jeremy Quastel and Benedek Valko
We consider finite-range asymmetric exclusion processes on $\mathbb Z
$ with
non-zero drift. The diffusivity $D(t)$ is expected to be of $O(t^
{1/3})$. We
prove that $D(t)\ge Ct^{1/3}$ in the weak (Tauberian) sense that $
\int_0^\infty
e^{-\lambda t}tD(t)dt \ge C\lambda^{-7/3}$ as $\lambda\to 0$. The
proof employs
the resolvent method to make a direct comparison with the totally
asymmetric
simple exclusion process, for which the result is a consequence of
the scaling
limit for the two-point function recently obtained by Ferrari and
Spohn. When
$p(z)\ge p(-z)$ for each $z>0$, we show further that $tD(t)$ is
monotone, and
hence we can conclude that $D(t)\ge Ct^{1/3}(\log t)^{-7/3}$ in the
usual
sense.
http://front.math.ucdavis.edu/math.PR/0605266
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4283. THE MULTIPARAMETER FRACTIONAL BROWNIAN MOTION
Erick Herbin and Ely Merzbach
We define and study the multiparameter fractional Brownian motion. This
process is a generalization of both the classical fractional Brownian
motion
and the multiparameter Brownian motion, when the condition of
independence is
relaxed. Relations with the L\'evy fractional Brownian motion and
with the
fractional Brownian sheet are discussed. Different notions of
stationarity of
the increments for a multiparameter process are studied and applied
to the
fractional property. Using self-similarity we present a
characterization for
such processes. Finally, behavior of the multiparameter fractional
Brownian
motion along increasing paths is analysed.
http://front.math.ucdavis.edu/math.PR/0605279
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4284. MULTISERVER QUEUEING SYSTEMS WITH RETRIALS AND ABANDONMENTS AND
THEIR APPLICATION TO CALL CENTERS
Vyacheslav M. Abramov
The paper studies multiserver retrial queueing systems with $m$ servers.
Arrival process is a quite general point process. An arriving
customer occupies
one of free servers. If upon arrival all servers are busy, then the
customer
waits for his service in orbit, and after random time retries more
and more to
occupy a server. The orbit has one waiting space only, and arriving
customer,
who finds all servers busy and the waiting space occupied, abandons
the system.
Time intervals between possible retrials are assumed to have arbitrary
distribution (the retrial scheme is exactly explained in the paper).
The paper
provides analysis of this system. Specifically the paper studies
optimal number
of servers to decrease the loss proportion to a given value. The
representation
obtained for loss proportion enables us to solve the problem
numerically. The
algorithm for numerical solution includes effective simulation, which
meets the
challenge of rare events problem in simulation. Application of the
results to
call centers is discussed as well.
http://front.math.ucdavis.edu/math.PR/0605285
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4285. A LIMIT THEOREM FOR THE MAXIMAL INTERPOINT DISTANCE OF A RANDOM
SAMPLE IN THE UNIT BALL
Michael Mayer and Ilya Molchanov
We prove a limit theorem for the the maximal interpoint distance
(also called
the diameter) for a sample of n i.i.d. points in the unit ball of
dimension 2
or more. The exact form of the limit distribution and the required
normalisation are derived using assumptions on the tail of the
interpoint
distance for two i.i.d. points. The results are specialised for the
cases when
the points have spherical symmetric distributions, in particular, are
uniformly
distributed in the whole ball and on its boundary.
http://front.math.ucdavis.edu/math.PR/0605289
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4286. CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
Oded Schramm and Scott Sheffield
We prove that the chordal contour lines of the discrete Gaussian free
field
converge to forms of SLE(4). Specifically, there is a constant lambda
> 0 such
that when h is an interpolation of the discrete Gaussian free field
on a Jordan
domain -- with boundary values -lambda on one boundary arc and lambda
on the
complementary arc -- the zero level line of h joining the endpoints
of these
arcs converges to SLE(4) as the domain grows larger. If instead the
boundary
values are -a < 0 on the first arc and b > 0 on the complementary
arc, then the
convergence is to SLE(4;a/lambda-1,b/lambda-1), a variant of SLE(4).
http://front.math.ucdavis.edu/math.PR/0605337
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4287. TOWARD THE BEST CONSTANT FACTOR FOR THE RADEMACHER-GAUSSIAN
TAIL COMPARISON
Iosif Pinelis
Let S_n:=a_1\vp_1+...+a_n\vp_n, where \vp_1,...,\vp_n are independent
Rademacher random variables (r.v.'s) and a_1,...,a_n are any real
numbers such
that a_1^2+...+a_n^2=1. Let Z be a standard normal r.v. It is proved
that the
best constant factor c in inequality
\P(S_n>x) \leq c\P(Z>x) for all x in \R is between two explicitly
defined
absolute constants c_1 and c_2 such that c_1<c_2 \approx 1.01c_1.
http://front.math.ucdavis.edu/math.PR/0605340
---------------------------------------------------------------
4288. GENERALIZED ENTROPY POWER INEQUALITIES AND MONOTONICITY
PROPERTIES OF INFORMATION
Mokshay Madiman and Andrew Barron
New families of Fisher information and entropy power inequalities for
sums of
independent random variables are presented. These inequalities relate
the
information in the sum of n independent random variables to the
information
contained in sums over subsets of the random variables, for an arbitrary
collection of subsets. As a consequence, a simple proof of the
monotonicity of
information in central limit theorems is obtained, both in the
setting of
i.i.d. summands as well as in the more general setting of independent
summands
with variance-standardized sums.
http://front.math.ucdavis.edu/cs.IT/0605047
---------------------------------------------------------------
4289. WEAK APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS AND
APPLICATION TO DERIVATIVE PRICING
Syoiti Ninomiya and Nicolas Victoir
The authors present a new simple algorithm to approximate weakly
stochastic
differential equations in the spirit of [1] and [2]. They apply it to
the
problem of pricing Asian options under the Heston stochastic
volatility model,
and compare it with other known methods. It is shown that the
combination of
the suggested algorithm and quasi-Monte Carlo methods makes computations
extremely fast.
[1] Shigeo Kusuoka, ``Approximation of Expectation of Diffusion
Process and
Mathematical Finance,'' Advanced Studies in Pure Mathematics,
Proceedings of
Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp.
147--165. [2] Terry Lyons and Nicolas Victoir, ``Cubature on Wiener
Space,''
Proceedings of the Royal Society of London. Series A. Mathematical
and Physical
Sciences 460 (2004), pp. 169--198.
http://front.math.ucdavis.edu/math.PR/0605361
---------------------------------------------------------------
4290. THE FREIDLIN-WENTZELL LDP WITH RAPIDLY GROWING COEFFICIENTS
P. Chigansky and R. Liptser
The Large Deviations Principle (LDP) is verified for a homogeneous
diffusion
process with respect to a Brownian motion $B_t$, $$
X^\eps_t=x_0+\int_0^tb(X^\eps_s)ds+ \eps\int_0^t\sigma(X^\eps_s)dB_s,
$$ where
$b(x)$ and $\sigma(x)$ are are locally Lipschitz functions with super
linear
growth. We assume that the drift is directed towards the origin and
the growth
rates of the drift and diffusion terms are properly balanced.
Nonsingularity of
$a=\sigma\sigma^*(x)$ is not required.
http://front.math.ucdavis.edu/math.PR/0605365
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4291. ESTIMATES OF GREEN FUNCTION FOR SOME PERTURBATIONS OF
FRACTIONAL LAPLACIAN
Tomasz Grzywny and Micha{\l} Ryznar
Suppose that Y(t) is a d-dimensional Levy symmetric process for which
its
Levy measure differs from the Levy measure of the isotropic alpha-stable
process (0<alpha<2) by a finite signed measure. For a bounded
Lipschitz set D
we compare the Green functions of the process Y and its stable
counterpart. We
prove a few comparability results either one sided or two sided.
Assuming an
additional condition about the difference of the densities of the Levy
measures, namely that it is of order of |x|^{-d+varrho} as x near 0,
where
varrho>0, we prove that the Green functions are comparable, provided
D is
connected.
These results apply for example to alpha-stable relativistic
process. This
process was studied in recent years. In the paper we also considered one
dimensional case for alpha<= 1 and proved that the Green functions
for an open
and bounded interval are comparable.
http://front.math.ucdavis.edu/math.PR/0605370
---------------------------------------------------------------
4292. POISSON APPROXIMATIONS FOR THE ISING MODEL
David Coupier
A $d$-dimensional Ising model on a lattice torus is considered. As
the size
$n$ of the lattice tends to infinity, a Poisson approximation is
given for the
distribution of the number of copies in the lattice of any given local
configuration, provided the magnetic field $a=a(n)$ tends to $-\infty
$ and the
pair potential $b$ remains fixed. Using the Stein-Chen method, a
bound is given
for the total variation error in the ferromagnetic case.
http://front.math.ucdavis.edu/math.PR/0605395
---------------------------------------------------------------
4293. AN EXPLICIT BOUND ON THE LOGARITHMIC SOBOLEV CONSTANT OF
WEAKLY DEPENDENT RANDOM VARIABLES
Katalin Marton
We prove logarithmic Sobolev inequality for measures $$
q^n(x^n)=\text{dist}(X^n)=\exp\bigl(-V(x^n)\bigr), \quad x^n\in \Bbb
R^n, $$
under the assumptions that: (i) the conditional distributions $$ Q_i
(\cdot|
x_j, j\neq i)=\text{dist}(X_i| X_j= x_j, j\neq i) $$ satisfy a
logarithmic
Sobolev inequality with a common constant $\rho$, and (ii) they also
satisfy
some condition expressing that the mixed partial derivatives of the
Hamiltonian
$V$ are not too large relative to $\rho$. \bigskip Condition (ii) has
the form
that the norms of some matrices defined in terms of the mixed partial
derivatives of $V$ do not exceed $1/2\cdot\rho\cdot(1-\de)$. The
logarithmic
Sobolev constant of $q^n$ can then be estimated from below by
$1/2\cdot\rho\cdot\delta$. This improves on earlier results by Th.
Bodineau and
B. Helffer, by giving an explicit bound, for the logarithmic Sobolev
constant
for $q^n$.
http://front.math.ucdavis.edu/math.PR/0605397
---------------------------------------------------------------
4294. POISSON LIMITS FOR EMPIRICAL POINT PROCESSES
Andr\'{e} Dabrowski and Gail Ivanoof and Rafal Kulik
Define the scaled empirical point process on an independent and
identically
distributed sequence $\{Y_i: i\le n\}$ as the random point measure
with masses
at $a_n^{-1} Y_i$. For suitable $a_n$ we obtain the weak limit of
these point
processes through a novel use of a dimension-free method based on the
convergence of compensators of multiparameter martingales. The method
extends
previous results in several directions. We obtain limits at points
where the
density of $Y_i$ may be zero, but has regular variation. The joint
limit of the
empirical process evaluated at distinct points is given by
independent Poisson
processes. These results also hold for multivariate $Y_i$ with little
additional effort. Applications are provided both to nearest-
neighbour density
estimation in high dimensions, and to the asymptotic behaviour of
multivariate
extremes such as those arising from bivariate normal copulas.
http://front.math.ucdavis.edu/math.PR/0605400
---------------------------------------------------------------
4295. DECAY PROPERTIES OF THE CONNECTIVITY FOR MIXED LONG RANGE
PERCOLATION MODELS ON $\Z^D$
Gastao A. Braga and Leandro M. Cioletti and Remy Sanchis
In this paper we consider mixed short-long range independent bond
percolation
models on $\Z^d$. Let $p_{uv}$ be the probability that the edge $(u,v)
$ will be
open. Successive applications of the Simon-Lieb inequality at a fixed
length
scale generates convolutions of $p_{uv}$ with itself which yields, in
the
perturbative regime, that the long distance behavior of the connectivity
$\tau_{xy}$ is governed by the probability $p_{xy}$. Allowing a $x,y$-
dependent
length scale and using a multi-scale analysis due to Aizenman and
Newman, decay
properties of $\tau_{xy}$ are obtained up to the critical point.
http://front.math.ucdavis.edu/math-ph/0605047
---------------------------------------------------------------
4296. UNIVERSALITY FOR THE DISTANCE IN FINITE VARIANCE RANDOM GRAPHS:
EXTENDED VERSION
Henri van den Esker and Remco van der Hofstad and Gerard Hooghiemstra
The asymptotic behavior of the graph distance between two uniformly
chosen
nodes in the configuration model is generalized to a wide class of
random
graphs, where the degrees have finite variance. Among others, this class
contains the Poissonian random graph and the generalized random graph
(including the classical Erd\H{o}s-R\'enyi graph).
We prove that the graph distance grows like $\log_\nu N$, when the
base of
the logarithm equals $\nu = E[\Lambda^2]/E[\Lambda]$, where $\Lambda$
is a
positive random variable with $P(\Lambda> x)\leq cx^{1-\tau}$, for some
constant $c$ and some power-law exponent $\tau>3$. In addition, the
random
fluctuations around this asymptotic mean $\log_\nu N$ are
characterized and
shown to be uniformly bounded.
The proof of this result uses that the graph distance of all
members of the
class can be coupled successfully to the graph distance in the
Poissonian
random graph.
http://front.math.ucdavis.edu/math.PR/0605414
---------------------------------------------------------------
4297. SMALL DEVIATIONS OF GAUSSIAN RANDOM FIELDS IN $L_Q$--SPACES
Mikhail Lifshits and Werner Linde and Zhan Shi
We investigate small deviation properties of Gaussian random fields
in the
space $L_q(\R^N,\mu)$ where $\mu$ is an arbitrary finite compactly
supported
Borel measure. Of special interest are hereby "thin" measures $\mu$,
i.e.,
those which are singular with respect to the $N$--dimensional
Lebesgue measure;
the so--called self--similar measures providing a class of typical
examples.
For a large class of random fields (including, among others,
fractional
Brownian motions), we describe the behavior of small deviation
probabilities
via numerical characteristics of $\mu$, called mixed entropy,
characterizing
size and regularity of $\mu$.
For the particularly interesting case of self--similar measures $
\mu$, the
asymptotic behavior of the mixed entropy is evaluated explicitly. As a
consequence, we get the asymptotic of the small deviation for $N$--
parameter
fractional Brownian motions with respect to $L_q(\R^N,\mu)$--norms.
While the upper estimates for the small deviation probabilities
are proved by
purely probabilistic methods, the lower bounds are established by
analytic
tools concerning Kolmogorov and entropy numbers of H\"older operators.
http://front.math.ucdavis.edu/math.PR/0605417
---------------------------------------------------------------
4298. IMBALANCE ATTRACTORS FOR A STRATEGIC MODEL OF MARKET
MICROSTRUCTURE
Ted Theodosopoulos and Ming Yuen
In this paper we extend the series of our studies on the properties
of an
interacting particle model for market microstructure. In our earlier
work we
defined a Markov process on the majority opinion of the agents,
obtained the
transition probabilities and analyzed the martingale properties of
the ensuing
wealth process. Here we relax the assumption on the choices of
individual
agents by allowing mixed strategies, offering opportunities for the
agents to
gain intermediate submartingale exposure for their individual wealth
processes.
We develop a novel two-dimensional spin system to model the critical
regions of
the wealth process as a reflection of the agents' behaviors. We exhibit
strategic conflicts between individual market participants and the
market as a
whole, and identify a new source of uncertainty arising from `reinforced
expectations'.
http://front.math.ucdavis.edu/math.PR/0605421
---------------------------------------------------------------
4299. GENERALIZED 3G THEOREM AND APPLICATION TO RELATIVISTIC STABLE
PROCESS ON NON-SMOOTH OPEN SETS
Panki Kim and Young-Ran Lee
Let G(x,y) and G_D(x,y) be the Green functions of rotationally invariant
symmetric \alpha-stable process in R^d and in an open set D
respectively, where
0<\alpha < 2. The inequality G_D(x,y)G_D(y,z)/G_D(x,z) \le c(G(x,y)+G
(y,z)) is
a very useful tool in studying (local) Schrodinger operators. When
the above
inequality is true with a constant c=c(D)>0, then we say that the 3G
theorem
holds in D.
In this paper, we establish a generalized version of 3G theorem
when D is a
bounded \kappa-fat open set, which includes a bounded John domain.
The 3G we
consider is of the form G_D(x,y)G_D(z,w)/G_D(x,w), where y may be
different
from z. When y=z, we recover the usual 3G.
The 3G form G_D(x,y)G_D(z,w)/G_D(x,w) appears in non-local
Schrodinger
operator theory. Using our generalized 3G theorem, we give a concrete
class of
functions belonging to the non-local Kato class, introduced by Chen
and Song,
on \kappa-fat open sets.
As an application, we discuss relativistic \alpha-stable processes
(relativistic Hamiltonian when \alpha=1) in \kappa-fat open sets. We
identify
the Martin boundary and the minimal Martin boundary with the
Euclidean boundary
for relativistic \alpha-stable processes in \kappa-fat open sets.
Furthermore,
we show that relative Fatou type theorem is true for relativistic stable
processes in \kappa-fat open sets.
The main results of this paper hold for a large class of symmetric
Markov
processes, as are illustrated in the last section of this paper. We also
discuss the generalized 3G theorem for a large class of symmetric
stable Levy
processes.
http://front.math.ucdavis.edu/math.PR/0605422
---------------------------------------------------------------
4300. SUFFICIENT CONDITIONS FOR THE INVERTIBILITY OF ADAPTED
PERTURBATIONS OF IDENTITY ON THE WIENER SPACE
Ali Suleyman Ustunel and Moshe Zakai
Let $(W,H,\mu)$ be the classical Wiener space. Assume that $U=I_W+u$
is an
adapted perturbation of identity, i.e., $u:W\to H$ is adapted to the
canonical
filtration of $W$. We give some sufficient analytic conditions on $u$
which
imply the invertibility of the map $U$. In particular it is shown
that if $u\in
\DD_{p,1}(H)$ is adapted and if $\exp({1/2}\|\nabla u\|_2^2-\delta u)\in
L^q(\mu)$, where $p^{-1}+q^{-1}=1$, then $I_W+u$ is almost surely
invertible.
As a consequence, if, there exists an integer $k\geq 1$ such that $\|
\nabla^k
u\|_{H^{\otimes(k+1)}}\in L^\infty(\mu)$, then $I_W+u$ is again
almost surely
invertible.
http://front.math.ucdavis.edu/math.PR/0605433
---------------------------------------------------------------
4301. RESAMPLING FROM THE PAST TO IMPROVE ON MCMC ALGORITHMS
Yves F. Atchade
We introduce the idea that resampling from past observations in a Markov
Chain Monte Carlo sampler can fasten convergence. We prove that proper
resampling from the past does not disturb the limit distribution of the
algorithm. We illustrate the method with two examples. The first on a
Bayesian
analysis of stochastic volatility models and the other on Bayesian
phylogeny
reconstruction.
http://front.math.ucdavis.edu/math.ST/0605452
---------------------------------------------------------------
4302. INFINITELY DIVISIBILITY OF SOLUTIONS OF SOME SEMI-STABLE
INTEGRO-DIFFERENTIAL EQUATIONS AND EXPONENTIAL FUNCTIONALS OF LEVY
PROCESSES
Pierre Patie
We provide the increasing $q$-harmonic functions associated to
spectrally
negative semi-stable Feller semigroups, which have been introduced by
Lamperti.
The functions are expressed in terms of a new family of power series
which
includes, for instance, the modified Bessel functions of the first
kind and
some new generalization of the Mittag-Leffler function. Then, we show
that some
specific combinations of these functions are Laplace transforms of
selfdecomposable or infinitely divisible distributions concentrated
on the
positive line. In particular, this generalizes the result of Hartman
in the
case of the Bessel semigroup. Finally, when the Levy process has a
negative
mean, we compute the associated decreasing $q$-harmonic functions and
derive
the Laplace transform of the exponential functionals.
http://front.math.ucdavis.edu/math.PR/0605453
---------------------------------------------------------------
4303. HYBRID DYNAMICS FOR CURRENCY MODELING
Ted Theodosopoulos and Alex Trifunovic
We present a simple hybrid dynamical model as a tool to investigate
behavioral strategies based on trend following. The multiplicative
symbolic
dynamics are generated using a lognormal diffusion model for the at-
the-money
implied volatility term structure. Thus, are model exploits
information from
derivative markets to obtain qualititative properties of the return
distribution for the underlier. We apply our model to the JPY-USD
exchange rate
and the corresponding 1mo., 3mo., 6mo. and 1yr. implied volatilities.
Our
results indicate that the modulation of autoregressive trend
following using
derivative-based signals significantly improves the fit to the
distribution of
times between successive sign flips in the underlier time series.
http://front.math.ucdavis.edu/math.PR/0605457
---------------------------------------------------------------
4304. ON STABLE PARETO LAWS IN A HIERARCHICAL MODEL OF ECONOMY
Alexander M. Chebotarev
This study considers a model of the income distribution of agents whose
pairwise interaction is asymmetric and price-invariant. Asymmetric
transactions
are typical for chain-trading groups who arrange their business such
that
commodities move from senior to junior partners and money moves in
the opposite
direction. The price-invariance of transactions means that the
probability of a
pairwise interaction is a function of the ratio of incomes, which is
independent of the price scale or absolute income level. These two
features
characterize the hierarchical model. The income distribution in this
class of
models is a well-defined double-Pareto function, which possesses
Pareto tails
for the upper and lower incomes. For gross and net upper incomes, the
model
predicts definite values of the Pareto exponents, $a_{\rm gross}$ and
$a_{\rm
net}$, which are stable with respect to quantitative variation of the
pair-interaction. The Pareto exponents are also stable with respect
to the
choice of a demand function within two classes of status-dependent
behavior of
agents: linear demand ($a_{\rm gross}=1$, $a_{\rm net}=2$) and
unlimited slowly
varying demand ($a_{\rm gross}=a_{\rm net}=1$). For the sigmoidal
demand that
describes limited returns, $a_{\rm gross}=a_{\rm net}=1+\alpha$, with
some
$\alpha>0$ satisfying a transcendental equation. The low-income
distribution
may be singular or vanishing in the neighborhood of the minimal
income; in any
case, it is $L_1$-integrable and its Pareto exponent is given
explicitly.
The theory used in the present study is based on a simple balance
equation
and new results from multiplicative Markov chains and exponential
moments of
random geometric progressions.
http://front.math.ucdavis.edu/math.PR/0605461
---------------------------------------------------------------
4305. STABILITY OF PROCESSOR SHARING NETWORKS WITH SIMULTANEOUS
RESOURCE REQUIREMENTS
Jennie Hansen and Cian Reynolds and Stan Zachary
We study the phenomenon of entrainment in processor sharing networks,
whereby, while individual network resources have sufficient capacity
to met
demand, the requirement for simultaneous availability of resources
means that a
network may nevertheless be unstable. We show that instability occurs
through
poor control, and that, for a variety of network topologies, only small
modifications to controls are required in order to ensure stability. For
controls which possess a natural monotonicity property, we give some new
results for the classification of the corresponding Markov processes,
which
lead to conditions both for stability and for instability.
http://front.math.ucdavis.edu/math.PR/0605477
---------------------------------------------------------------
4306. ON THE OCCUPATION MEASURE OF SUPER-BROWNIAN MOTION
J.F. Le Gall and M. Merle
We derive the asymptotic behavior of the occupation measure of the
unit ball,
for super-Brownian motion started from the Dirac measure at a distant
point x
and conditioned to hit the unit ball. In the critical dimension d=4,
we obtain
a limiting exponential distribution for the ratio of the occupation
measure
over log(|x|).
http://front.math.ucdavis.edu/math.PR/0605482
---------------------------------------------------------------
4307. RANDOM REAL TREES
J.F. Le Gall
We survey recent developments about random real trees, whose
prototype is the
Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly
explain
the formalism of real trees, which yields a neat presentation of the
theory and
in particular of the relations between discrete Galton-Watson trees and
continuous random trees. We then discuss the particular class of self-
similar
random real trees called stable trees, which generalize the CRT. We
review
several important results concerning stable trees, including their
branching
property, which is analogous to the well-known property of Galton-
Watson trees,
and the calculation of their fractal dimension. We then consider
spatial trees,
which combine the genealogical structure of a real tree with spatial
displacements, and we explain their connections with superprocesses.
In the
last section, we deal with a particular conditioning problem for
spatial trees,
which is closely related to asymptotics for random planar
quadrangulations.
http://front.math.ucdavis.edu/math.PR/0605484
---------------------------------------------------------------
4308. AN ALGEBRAIC APPROACH OF POLYA PROCESSES
Nicolas Pouyanne (LM-Versailles)
P\'olya processes are natural generalization of P\'olya-Eggenberger urn
models. This article presents a new approach of their asymptotic
behaviour {\it
via} moments, based on the spectral decomposition of a suitable finite
difference operator on polynomial functions. Especially, it provides new
results for {\it large} processes (a P\'olya process is called {\it
small} when
1 is simple eigenvalue of its replacement matrix and when any other
eigenvalue
has a real part $\leq 1/2$; otherwise, it is called large).
http://front.math.ucdavis.edu/math.CO/0605472
---------------------------------------------------------------
4309. ON THE LIKELIHOOD OF COMPARABILITY IN BRUHAT ORDER
Adam Hammett and Boris Pittel
The poset of permutations of [n] under Bruhat ordering is studied. We
give
nontrivial upper and lower bounds for the number of comparable pairs of
permutations in both the weak and strong versions of this order. In
light of
numerical experiments, we conjecture that in either case the upper
bound is
qualitatively close to the actual number of comparable pairs.
http://front.math.ucdavis.edu/math.PR/0605490
---------------------------------------------------------------
4310. LARGE DEVIATIONS FOR WEIGHTED EMPIRICAL MEAN WITH OUTLIERS
Myl\`ene Ma\"{\i}da and Jamal Najim and Sandrine P\'ech\'e
We study in this article large deviations for the empirical mean of iid
random vectors with some deterministic weights, whose empirical
measure weakly
converges to some compactly support probability distribution. The
scope of this
paper is to study the effect on the LDP of outliers, that is
sequences of
weights that remain far from the support of the limiting measure.
http://front.math.ucdavis.edu/math.PR/0605491
---------------------------------------------------------------
4311. ZERO-ONE LAWS FOR BINARY RANDOM FIELDS
David Coupier and Paul Doukhan and Bernard Ycart
A set of binary random variables indexed by a lattice torus is
considered.
Under a mixing hypothesis, the probability of any proposition
belonging to the
first order logic of colored graphs tends to 0 or 1, as the size of
the lattice
tends to infinity. For the particular case of the Ising model with
bounded pair
potential and surface potential tending to $-\infty$, the threshold
functions
of local propositions are computed, and sufficient conditions for the
zero-one
law are given.
http://front.math.ucdavis.edu/math.PR/0605502
---------------------------------------------------------------
4312. ON CLASSES OF NON-GAUSSIAN ASYMPTOTIC MINIMIZERS IN ENTROPIC
UNCERTAINTY PRINCIPLES
S. Zozor and C. Vignat
In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty
principle and its cases of equality. This Shannon entropic version of
the
well-known Heisenberg uncertainty principle can be used when dealing
with
variables that admit no variance. In this paper, we extend this
uncertainty
principle to Renyi entropies. We recall that in both Shannon and
Renyi cases,
and for a given dimension n, the only case of equality occurs for
Gaussian
random vectors. We show that as n grows, however, the bound is also
asymptotically attained in the cases of n-dimensional Student-t and
Student-r
distributions. A complete analytical study is performed in a special
case of a
Student-t distribution. We also show numerically that this effect
exists for
the particular case of a n-dimensional Cauchy variable, whatever the
Renyi
entropy considered, extending the results of Abe and illustrating the
analytical asymptotic study of the student-t case. In the Student-r
case, we
show numerically that the same behavior occurs for uniformly distributed
vectors. These particular cases and other ones investigated in this
paper are
interesting since they show that this asymptotic behavior cannot be
considered
as a "Gaussianization" of the vector when the dimension increases.
http://front.math.ucdavis.edu/math.PR/0605510
---------------------------------------------------------------
4313. PHASE TRANSITIONS IN A PIECEWISE EXPANDING COUPLED MAP LATTICE
WITH LINEAR NEAREST NEIGHBOUR COUPLING
Jean-Baptiste Bardet (IRMAR) and Gerhard Keller
We construct a mixing continuous piecewise linear map on [-1,1] with the
property that a two-dimensional lattice made of these maps with a
linear north
and east nearest neighbour coupling admits a phase transition. We
also provide
a modification of this construction where the local map is an expanding
analytic circle map. The basic strategy is borroughed from [Gielis-
MacKay
(2000)], namely we compare the dynamics of the CML to those of a
probabilistic
cellular automaton of Toom's type.
http://front.math.ucdavis.edu/math.DS/0605501
---------------------------------------------------------------
4314. POTENTIAL THEORY OF TRUNCATED STABLE PROCESSES
Panki Kim and Renming Song
For any 0 < alpha <2, a truncated symmetric alpha-stable process is a
symmetric Levy process in R^d with a Levy density given by c|x|^{-d-
alpha}
1_{|x|< 1} for some constant c. In this paper we study the potential
theory of
truncated symmetric stable processes in detail. We prove a Harnack
inequality
for nonnegative harmonic nonnegative functions these processes. We also
establish a boundary Harnack principle for nonnegative functions
which are
harmonic with respect to these processes in bounded convex domains.
We give an
example of a non-convex domain for which the boundary Harnack
principle fails.
http://front.math.ucdavis.edu/math.PR/0605533
---------------------------------------------------------------
4315. EXPONENTIAL APPROXIMATION BY EXCHANGEABLE PAIRS AND SPECTRAL
GRAPH THEORY
Sourav Chatterjee and Jason Fulman
A general Berry-Esseen bound is obtained for the exponential
distribution
using Stein's method of exchangeable pairs. As an application, an
error term is
derived for Hora's result that the spectrum of the Bernoulli-Laplace
Markov
chain has an exponential limit. This is the first use of Stein's
method to
study the spectrum of a graph with a non-normal limit.
http://front.math.ucdavis.edu/math.PR/0605552
---------------------------------------------------------------
4316. ON DUAL PROCESSES OF NON-SYMMETRIC DIFFUSIONS WITH MEASURE-
VALUED DRIFTS
Panki Kim and Renming Song
In this paper, we study properties of the dual process and
Schrodinger-type
operators of a non-symmetric diffusion with measure-valued drift. Let
mu=(mu^1,..., mu^d) be such that each mu^i is a signed measure on R^d
belonging
to the Kato class K_{d, 1}. We show that a killed diffusion process with
measure-valued drift in any bounded domain has a dual process with
respect to a
certain reference measure. For an arbitrary bounded domain, we show
that a
scale invariant Harnack inequality is true for the dual process. We
also show
that, if the domain is bounded C^{1,1}, the boundary Harnack
principle for the
dual process is true and the (minimal) Martin boundary for the dual
process can
be identified with the Euclidean boundary. It is also shown that the
harmonic
measure for the dual process is locally comparable to that of the h-
conditioned
Brownian motion with h being the ground state. Under the gaugeability
assumption, if the domain is bounded Lipschitz, the (minimal) Martin
boundary
for the Schrodinger operator obtained from the diffusion with measure-
value
drift can be identified with the Euclidean boundary.
http://front.math.ucdavis.edu/math.PR/0605556
---------------------------------------------------------------
4317. ESTIMATES ON GREEN FUNCTIONS AND SCHRODINGER-TYPE EQUATIONS
FOR NON-SYMMETRIC DIFFUSIONS WITH MEASURE-VALUED DRIFTS
Panki Kim and Renming Song
In this paper, we establish sharp two-sided estimates for the Green
functions
of non-symmetric diffusions with measure-valued drifts in bounded
Lipschitz
domains. As consequences of these estimates, we get a 3G type theorem
and a
conditional gauge theorem for these diffusions in bounded Lipschitz
domains. We
also establish two-sided estimates for the heat kernels of
Schrodinger-type
operators with measure-valued potential in bounded C^{1,1}-domains
and a scale
invariant boundary Harnack principle for the positive harmonic
functions with
respect to Schrodinger-type operators in bounded Lipschitz domains.
http://front.math.ucdavis.edu/math.PR/0605557
---------------------------------------------------------------
4318. ON TAYLOR DISPERSION IN OSCILLATORY CHANNEL FLOWS
Kalvis M. Jansons
We revisit Taylor dispersion in oscillatory flows at zero Reynolds
number,
giving an alternative method of calculating the Taylor dispersivity
that is
easier to use with computer algebra packages to obtain exact
expressions. We
consider the effect of out-of-phase oscillatory shear and Poiseuille
flow, and
show that the resulting Taylor dispersivity is independent of the phase
difference. We also determine exact expressions for several examples of
oscillatory power-law fluid flows.
http://front.math.ucdavis.edu/math.PR/0605561
---------------------------------------------------------------
4319. PARTITION FUNCTION OF PERIODIC ISORADIAL DIMER MODELS
B\'eatrice de Tili\`ere
Isoradial dimer models were introduced in \cite{Kenyon3} - they
consist of
dimer models whose underlying graph satisfies a simple geometric
condition, and
whose weight function is chosen accordingly. In this paper, we prove a
conjecture of \cite{Kenyon3}, namely that for periodic isoradial
dimer models,
the growth rate of the toroidal partition function has a simple explicit
formula involving the local geometry of the graph only. This is a
surprising
feature of periodic isoradial dimer models, which does not hold in
the general
periodic dimer case \cite{KOS}.
http://front.math.ucdavis.edu/math.PR/0605583
---------------------------------------------------------------
4320. MODELLING DERIVATIVES PRICING MECHANISMS WITH THEIR GENERATING
FUNCTIONS
Shige Peng
In this paper we study dynamic pricing mechanisms of financial
derivatives. A
typical model of such pricing mechanism is the so-called g--
expectation defined
by solutions of a backward stochastic differential equation with g as
its
generating function. Black-Scholes pricing model is a special linear
case of
this pricing mechanism. We are mainly concerned with two types of
pricing
mechanisms in an option market: the market pricing mechanism through
which the
market prices of options are produced, and the ask-bid pricing mechanism
operated through the system of market makers. The later one is a typical
nonlinear pricing mechanism. Data of prices produced by these two
pricing
mechanisms are usually quoted in an option market.
We introduce a criteria, i.e., the domination condition (A5) in
(2.5) to
test if a dynamic pricing mechanism under investigation is a g--pricing
mechanism. This domination condition was statistically tested using
CME data
documents. The result of test is significantly positive. We also
provide some
useful characterizations of a pricing mechanism by its generating
function.
http://front.math.ucdavis.edu/math.PR/0605599
---------------------------------------------------------------
4321. LARGE DEVIATIONS FOR SUMS DEFINED ON A GALTON-WATSON PROCESS
Klaus Fleischmann and Vitali Wachtel
In this paper we study the large deviation behavior of sums of i.i.d.
random
variables X_i defined on a supercritical Galton-Watson process Z. We
assume the
finiteness of the moments EX_1^2 and EZ_1log Z_1. The underlying
interplay of
the partial sums of the X_i and the lower deviation probabilities of
Z is
clarified. Here we heavily use lower deviation probability results on
Z we
recently published in [FW06].
http://front.math.ucdavis.edu/math.PR/0605617
---------------------------------------------------------------
4322. SPATIAL BIRTH AND DEATH PROCESSES AS SOLUTIONS OF STOCHASTIC
EQUATIONS
Nancy L. Garcia and Thomas G. Kurtz
Spatial birth and death processes are obtained as solutions of a
system of
stochastic equations. The processes are required to be locally
finite, but may
involve an infinite population over the full (noncompact) type space.
Conditions are given for existence and uniqueness of such solutions,
and for
temporal and spatial ergodicity. For birth and death processes with
constant
death rate, a sub-criticality condition on the birth rate implies
that the
process is ergodic and converges exponentially fast to the stationary
distribution.
http://front.math.ucdavis.edu/math.PR/0605620
---------------------------------------------------------------
4323. THE LARGEST EIGENVALUE OF RANK ONE DEFORMATION OF LARGE WIGNER
MATRICES
Delphine F\'eral and Sandrine P\'ech\'e
The purpose of this paper is to establish universality of the
fluctuations of
the largest eigenvalue of some non necessarily Gaussian complex
Deformed Wigner
Ensembles. The real model is also considered. Our approach is close
to the one
used by A. Soshnikov in the investigations of classical real or
complex Wigner
Ensembles. It is based on the computation of moments of traces of
high powers
of the random matrices under consideration.
http://front.math.ucdavis.edu/math.PR/0605624
---------------------------------------------------------------
4324. ON THE MAXIMUM QUEUE LENGTH IN THE SUPERMARKET MODEL
Malwina J. Luczak and Colin McDiarmid
There are $n$ queues, each with a single server. Customers arrive in a
Poisson process at rate $\lambda n$, where $0<\lambda<1$. Upon
arrival each
customer selects $d\geq2$ servers uniformly at random, and joins the
queue at a
least-loaded server among those chosen. Service times are independent
exponentially distributed random variables with mean 1. We show that
the system
is rapidly mixing, and then investigate the maximum length of a queue
in the
equilibrium distribution. We prove that with probability tending to 1 as
$n\to\infty$ the maximum queue length takes at most two values, which
are
$\ln\ln n/\ln d+O(1)$.
http://front.math.ucdavis.edu/math.PR/0605639
---------------------------------------------------------------
4325. THE SIZE OF COMPONENTS IN CONTINUUM NEAREST-NEIGHBOR GRAPHS
Iva Kozakova and Ronald Meester and Seema Nanda
We study the size of connected components of random nearest-neighbor
graphs
with vertex set the points of a homogeneous Poisson point process in
${\mathbb{R}}^d$. The connectivity function is shown to decay
superexponentially, and we identify the exact exponent. From this we
also
obtain the decay rate of the maximal number of points of a path
through the
origin. We define the generation number of a point in a component and
establish
its asymptotic distribution as the dimension $d$ tends to infinity.
http://front.math.ucdavis.edu/math.PR/0605640
---------------------------------------------------------------
4326. DYNAMICAL STABILITY OF PERCOLATION FOR SOME INTERACTING
PARTICLE SYSTEMS AND $\EPSILON$-MOVABILITY
Erik I. Broman and Jeffrey E. Steif
In this paper we will investigate dynamic stability of percolation
for the
stochastic Ising model and the contact process. We also introduce the
notion of
downward and upward $\epsilon$-movability which will be a key tool
for our
analysis.
http://front.math.ucdavis.edu/math.PR/0605641
---------------------------------------------------------------
4327. MONOTONICITY, ASYMPTOTIC NORMALITY AND VERTEX DEGREES IN RANDOM
GRAPHS
Svante Janson
We exploit a result by Nerman which shows that conditional limit
theorems
hold when a certain monotonicity condition is satisfied. Our main
result is an
application to vertex degrees in random graphs where we obtain
asymptotic
normality for the number of vertices with a given degree in the
random graph
G(n,m) with a fixed number of edges from the corresponding result for
the
random graph G(n,m) with independent edges. We give also some simple
applications to random allocations and to spacings.
Finally, inspired by these results but logically independent from
them, we
investigate whether a one-sided version of the Cramer-Wold theorem
holds. We
show that such a version holds under a weak supplementary condition,
but not
without it.
http://front.math.ucdavis.edu/math.PR/0605642
---------------------------------------------------------------
4328. COMPARISON OF WEIGHTED AND UNWEIGHTED HISTOGRAMS
N.D. Gagunashvili
Two modifications of the chi square test for comparing usual
(unweighted) and
weighted histograms and two weighted histograms are proposed. Numerical
examples illustrate an application of the tests for the histograms with
different statistics of events. Proposed tests can be used for the
comparison
of experimental data histograms against simulated data histograms and
two
simulated data histograms.
http://front.math.ucdavis.edu/physics/0605123
---------------------------------------------------------------
4329. INTERMITTENCY ON CATALYSTS: SYMMETRIC EXCLUSION
J. Gaertner and F. den Hollander and G. Maillard
We continue our study of intermittency for the parabolic Anderson
equation
$\partial u/\partial t = \kappa\Delta u + \xi u$, where $u\colon \Z^d
\times
[0,\infty)\to\R$, $\kappa$ is the diffusion constant, $\Delta$ is the
discrete
Laplacian, and $\xi\colon \Z^d\times [0,\infty)\to\R$ is a space-time
random
medium. The solution of the equation describes the evolution of a
``reactant''
$u$ under the influence of a ``catalyst'' $\xi$.
In this paper we focus on the case where $\xi$ is exclusion with a
symmetric
random walk transition kernel, starting from equilibrium with density
$\rho\in
(0,1)$. We consider the annealed Lyapunov exponents, i.e., the
exponential
growth rates of the successive moments of $u$. We show that these
exponents are
trivial when the random walk is recurrent, but display an interesting
dependence on the diffusion constant $\kappa$ when the random walk is
transient, with qualitatively different behavior in different
dimensions.
Special attention is given to the asymptotics of the exponents for
$\kappa\to\infty$, which is controlled by moderate deviations of $\xi$
requiring a delicate expansion argument.
In G\"artner and den Hollander \cite{garhol04} the case where $\xi
$ is a
Poisson field of independent (simple) random walks was studied. The
two cases
show interesting differences and similarities. Throughout the paper, a
comparison of the two cases plays a crucial role.
http://front.math.ucdavis.edu/math.PR/0605657
---------------------------------------------------------------
4330. A VERSION OF H\"ORMANDER'S THEOREM FOR THE FRACTIONAL BROWNIAN
MOTION
F. Baudoin and M. Hairer
It is shown that the law of an SDE driven by fractional Brownian
motion with
Hurst parameter greater than 1/2 has a smooth density with respect to
Lebesgue
measure, provided that the driving vector fields satisfy H\"ormander's
condition. The main new ingredient of the proof is an extension of
Norris'
lemma to this situation.
http://front.math.ucdavis.edu/math.PR/0605658
---------------------------------------------------------------
4331. QUASI STATIONARY DISTRIBUTIONS AND FLEMING-VIOT PROCESSES IN
COUNTABLE SPACES
Pablo A. Ferrari and Nevena Maric
We consider an irreducible pure jump Markov process with rates Q=(q
(x,y)) on
\Lambda\cup\{0\} with \Lambda countable and 0 an absorbing state. A
quasi-stationary distribution (qsd) is a probability measure \nu on
\Lambda
that satisfies: starting with \nu, the conditional distribution at
time t,
given that at time t the process has not been absorbed, is still \nu.
That is,
\nu(x) = \nu P_t(x)/(\sum_{y\in\Lambda}\nu P_t(y)), with P_t the
transition
probabilities for the process with rates Q.
A Fleming-Viot (fv) process is a system of N particles moving in
\Lambda.
Each particle moves independently with rates Q until it hits the
absorbing
state 0; but then instantaneously chooses one of the N-1 particles
remaining in
\Lambda and jumps to its position. Between absorptions each particle
moves with
rates Q independently.
Under the condition \alpha:=\sum_x\inf Q(\cdot,x) > \sup Q(\cdot,
0):=C we
prove existence of qsd for Q; uniqueness has been proven by Jacka and
Roberts.
When \alpha>0 the {\fv} process is ergodic for each N. Under \alpha>C
the mean
normalized densities of the fv unique stationary measure converge to
the qsd of
Q, as N \to \infty; in this limit the variances vanish.
http://front.math.ucdavis.edu/math.PR/0605665
---------------------------------------------------------------
4332. ON THE AVERAGE NUMBER OF SHARP CROSSINGS OF CERTAIN GAUSSIAN
RANDOM POLYNOMIALS
S. Shemehsavar and S. Rezakhah
Let $Q_n(x)=\sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraic
polynomial where
the coefficients $A_0,A_1,... $ form a sequence of centered Gaussian
random
variables. Moreover, assume that the increments $\Delta_j=A_j-A_{j-1}$,
$j=0,1,2,...$ are independent, assuming $A_{-1}=0$. The coefficients
can be
considered as $n$ consecutive observations of a Brownian motion. We
obtain the
asymptotic behaviour of the expected number of u-sharp crossings of
polynomial
$Q_n(x)$ . We refer to u-sharp crossings as those zero up-crossings
with slope
greater than $u$, or those down-crossings with slope smaller than $-u
$. We
consider the cases where $u$ is unbounded and is increasing with $n$,
where
$u=o(n^{5/4})$, and $u=o(n^{3/2})$ separately.
http://front.math.ucdavis.edu/math.PR/0605699
---------------------------------------------------------------
4333. ASYMPTOTIC BEHAVIOUR OF THE SIMPLE RANDOM WALK ON THE 2-COMB
Daniela Bertacchi
We analyze the differences between the horizontal and the vertical
component
of the simple random walk on the 2-dimensional comb. In particular we
evaluate
by combinatorial methods the asymptotic behaviour of the expected
value of the
distance from the origin, the maximal deviation and the maximal span
in $n$
steps, proving that for all these quantities the order is $n^{1/4}$
for the
horizontal projection and $n^{1/2}$ for the vertical one (the exact
constants
are determined). Then we rescale the two projections of the random walk
dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical ones,
respectively. The limit process is obtained. As a corollary of the
estimate of
the expected value of the maximal deviation, the walk dimension is
determined,
showing that the Einstein relation between the fractal, spectral and
walk
dimensions does not hold on the comb.
http://front.math.ucdavis.edu/math.PR/0605718
---------------------------------------------------------------
4334. DIGITAL SEARCH TREES AND CHAOS GAME REPRESENTATION
Peggy C\'{e}nac (INRIA Rocquencourt) and Brigitte Chauvin (LM-
Versailles), St\'{e}phane Ginouillac (LM-Versailles), Nicolas Pouyanne
(LM-Versailles)
In this paper, we consider a possible representation of a DNA
sequence in a
quaternary tree, in which on can visualize repetitions of subwords. The
CGR-tree turns a sequence of letters into a digital search tree
(DST), obtained
from the suffixes of the reversed sequence. Several results are known
concerning the height and the insertion depth for DST built from i.i.d.
successive sequences. Here, the successive inserted wors are strongly
dependent. We give the asymptotic behaviour of the insertion depth
and of the
length of branches for the CGR-tree obtained from the suffixes of
reversed
i.i.d. or Markovian sequence. This behaviour turns out to be at first
order the
same one as in the case of independent words. As a by-product,
asymptotic
results on the length of longest runs in a Markovian sequence are
obtained.
http://front.math.ucdavis.edu/math.PR/0605719
---------------------------------------------------------------
4335. ON THE BROWNIAN MEANDER AND EXCURSION CONDITIONED TO HAVE A
FIXED TIME AVERAGE
Lorenzo Zambotti
We study the density of the time average of the Brownian meander/
excursion
over the time interval [0,1]. Moreover we give an expression for the
Brownian
meander/excursion conditioned to have a fixed time average.
http://front.math.ucdavis.edu/math.PR/0605720
---------------------------------------------------------------
4336. INTRINSIC ULTRACONTRACTIVITY OF NON-SYMMETRIC DIFFUSIONS WITH
MEASURE-VALUED DRIFTS AND POTENTIALS
Panki Kim and Renming Song
Recently we extended the concept of intrinsic ultracontractivity to
non-symmetric semigroups. In this paper, we study the intrinsic
ultracontractivity of non-symmetric diffusions with measure-valued
drifts and
measure-valued potentials in bounded domains. We show that scale
invariant
parabolic and elliptic Harnack inequalities are valid for this process.
In this paper, we prove the parabolic boundary Harnack principle
and the
intrinsic ultracontractivity for the killed diffusion with measure-
valued drift
and potential when the domain is one of the following types of
bounded domains:
twisted Holder domains of order (1/3, 1], uniformly Holder domains of
order (0,
2) and domains which can be locally represented as the region above
the graph
of a function. As a consequence of the intrinsic ultracontractivity,
we get
that the supremum of the expected conditional lifetimes finite.
http://front.math.ucdavis.edu/math.PR/0605757
---------------------------------------------------------------
4337. ZEROS OF RANDOM POLYNOMIALS ON C^M
Thomas Bloom and Bernard Shiffman
For a regular compact set $K$ in $C^m$ and a measure $\mu$ on $K$
satisfying
the Bernstein-Markov inequality, we consider the ensemble $P_N$ of
polynomials
of degree $N$, endowed with the Gaussian probability measure induced by
$L^2(\mu)$. We show that for large $N$, the simultaneous zeros of $m$
polynomials in $P_N$ tend to concentrate around the Silov boundary of
$K$; more
precisely, their expected distribution is asymptotic to $N^m \mu_{eq}
$, where
$\mu_{eq}$ is the equilibrium measure of $K$. For the case where $K$
is the
unit ball, we give scaling asymptotics for the expected distribution
of zeros
as $N\to\infty$.
http://front.math.ucdavis.edu/math.CV/0605739
---------------------------------------------------------------
4338. THE OSTROGRADSKY SERIES AND RELATED PROBABILITY MEASURES
S.Albeverio and O.Baranovskyi and M.Pratsiovytyi and G.Torbin
We develop a metric and probabilistic theory for the Ostrogradsky
representation of real numbers, i.e., the expansion of a real number
$x$ in the
following form: \begin{align*} x&= \sum_n\frac{(-1)^{n-1}}{q_1q_2...
q_n}=
&=\sum_n\frac{(-1)^{n-1}}{g_1(g_1+g_2)...(g_1+g_2+...+g_n)}\equiv
\bO1(g_1,g_2,...,g_n,...), \end{align*} where $q_{n+1}>q_n\in\N$,
$g_1=q_1$,
$g_{k+1}=q_{k+1}-q_k$. We compare this representation with the
corresponding
one in terms of continued fractions.
We establish basic metric relations (equalities and inequalities
for ratios
of the length of cylindrical sets). We also compute the Lebesgue
measure of
subsets belonging to some classes of closed nowhere dense sets
defined by
characteristic properties of the $\bO1$-representation. In
particular, the
conditions for the set $\Cset{V}$, consisting of real numbers whose
$\bO1$-symbols take values from the set $V \subset N$, to be of zero
resp.
positive Lebesgue measure are found. For a random variable $\xi$ with
independent $\bO1$-symbols $g_n(\xi)$ we prove the theorem
establishing the
purity of the distribution. In the case of singularity the conditions
for such
distributions to be of Cantor type are also found.
http://front.math.ucdavis.edu/math.NT/0605747
---------------------------------------------------------------
4339. SINGULAR PROBABILITY DISTRIBUTIONS AND FRACTAL PROPERTIES OF
SETS OF REAL NUMBERS DEFINED BY THE ASYMPTOTIC FREQUENCIES OF THEIR
S-ADIC DIGITS
S.Albeverio and M.Pratsiovytyi and G.Torbin
Properties of the set $T_s$ of "particularly non-normal numbers" of
the unit
interval are studied in details ($T_s$ consists of real numbers $x$,
some of
whose s-adic digits have the asymptotic frequencies in the
nonterminating $s-$
adic expansion of $x$, and some do not). It is proven that the set
$T_s$ is
residual in the topological sense (i.e., it is of the first Baire
category) and
it is generic in the sense of fractal geometry ($T_s$ is a
superfractal set,
i.e., its Hausdorff-Besicovitch dimension is equal to~1). A
topological and
fractal classification of sets of real numbers via analysis of
asymptotic
frequencies of digits in their s-adic expansions is presented.
http://front.math.ucdavis.edu/math.NT/0605763
---------------------------------------------------------------
4340. SIMPLE TRANSIENT RANDOM WALKS IN ONE-DIMENSIONAL RANDOM
ENVIRONMENT: THE CENTRAL LIMIT THEOREM
I. Ya. Goldsheid
We consider a simple random walk (dimension one, nearest neighbour
jumps) in
a quenched random environment. The goal of this work is to provide
sufficient
conditions, stated in terms of properties of the environment, under
which the
Central Limit Theorem (CLT) holds for the position of the walk.
Verifying these
conditions leads to a complete solution of the problem in the case of
independent identically distributed environments as well as in the
case of
uniformly ergodic (and thus also weakly mixing) environments.
http://front.math.ucdavis.edu/math.PR/0605775
---------------------------------------------------------------
4341. OPTIMAL CONTROL FOR ROUGH DIFFERENTIAL EQUATIONS
Laurent Mazliak (PMA) and Ivan Nourdin (PMA)
In this note, we consider an optimal control problem associated to a
differential equation driven by a H\"{o}lder continuous function g of
index
greater than 1/2. We split our study in two cases. If the coefficient
of dg\_t
does not depend on the control process, we prove an existence theorem
for a
slightly generalized control problem, that is we obtain a literal
extension of
the corresponding deterministic situation. If the coefficient of dg
\_t depends
on the control process, we also prove an existence theorem but we are
here
obliged to restrict the set of controls to sufficiently regular
functions.
http://front.math.ucdavis.edu/math.PR/0606030
---------------------------------------------------------------
4342. SHUFFLING CARDS FOR BLACKJACK, BRIDGE, AND OTHER CARD GAMES
Mark Conger and D. Viswanath
This paper is about the following question: How many riffle shuffles
mix a
deck of card for games such as blackjack and bridge? An object that
comes up in
answering this question is the descent polynomial associated with
pairs of
decks, where the decks are allowed to have repeated cards. We prove
that the
problem of computing the descent polynomial given a pair of decks is
$#P$-complete. We also prove that the coefficients of these
polynomials can be
approximated using the bell curve. However, as must be expected in
view of the
$#P$-completeness result, approximations using the bell curve are not
good
enough to answer our question. Some of our answers to the main
question are
supported by theorems, and others are based on experiments supported by
heuristic arguments. In the introduction, we carefully discuss the
validity of
our answers.
http://front.math.ucdavis.edu/math.PR/0606031
---------------------------------------------------------------
4343. LONG-TIME BEHAVIOR OF STOCHASTIC MODEL WITH MULTI-PARTICLE
SYNCHRONIZATION
Anatoly Manita
We consider a basic stochastic particle system consisting of $N$
identical
particles with isotropic $k$-particle synchronization, $k\geq 2$. In
the limit
when both number of particles $N$ and time $t=t(N)$ grow to infinity
we study
an asymptotic behavior of a coordinate spread of the particle system. We
describe three time stages of $t(N)$ for which a qualitative behavior
of the
system is completely different. Moreover, we discuss the case when a
spread of
the initial configuration depends on $N$ and increases to infinity as
$N\to
\infty $.
http://front.math.ucdavis.edu/math.PR/0606040
---------------------------------------------------------------
4344. SIEVING AND THE ERD{\H O}S-KAC THEOREM
Andrew Granville and K. Soundararajan
We give a relatively easy proof of the Erd\H os-Kac theorem via
computing
moments. We show how this proof extends naturally in a sieve theory
context,
and how it leads to several related results in the literature.
http://front.math.ucdavis.edu/math.NT/0606039
---------------------------------------------------------------
4345. THE POISSON BOUNDARY OF LAMPLIGHTER RANDOM WALKS ON TREES
Anders Karlsson and Wolfgang Woess
Let T be the homogeneous tree with degree and G a finitely generated
group
whose Cayley graph is T. The associated lamplighter group is the
wreath product
of the cyclic group of order r with G. For a large class of random
walks on
this group, we prove almost sure convergence to a natural geometric
boundary.
If the probability law governing the random walk has finite first
moment, then
the probability space formed by this geometric boundary together with
the limit
distribution of the random walk is proved to be maximal, that is, the
Poisson
boundary. We also prove that the Dirichlet problem at infinity is
solvable for
continuous functions on the active part of the boundary, if the
lamplighter
"operates at bounded range".
http://front.math.ucdavis.edu/math.PR/0606046
---------------------------------------------------------------
4346. RECURRENCE AND TRANSIENCE FOR BRANCHING RANDOM WALKS IN AN IID
RANDOM ENVIRONMENT
Sebastian M\"uller
We give three different criteria for transience of a Branching Markov
Chain.
These conditions enable us to give a classification of Branching
Random Walks
in Random Environment (BRWRE) on Cayley Graphs in recurrence and
transience.
This classification is stated explicitly for BRWRE on $\Z^d.$
Furthermore, we
emphasize the interplay between Branching Markov Chains and the spectral
radius. We prove properties of the spectral radius of the Random Walk
in Random
Environment with the help of appropriate Branching Markov Chains.
http://front.math.ucdavis.edu/math.PR/0606055
---------------------------------------------------------------
4347. THE KNEE-JERK MAPPING
Peter G. Doyle and Jim Reeds
We claim to give the definitive theory of what we call the `knee-jerk
mapping', which is the basis for a class of optimization algorithms
introduced
by Baum, and promoted by Dempster, Laird, and Rubin under the name `EM
algorithm'.
http://front.math.ucdavis.edu/math.PR/0606068
---------------------------------------------------------------
4348. WIENER INTEGRALS, MALLIAVIN CALCULUS AND COVARIANCE MEASURE
STRUCTURE
Ida Kruk (LAGA) and Francesco Russo (LAGA) and Ciprian Tudor (SAMOS)
We introduce the notion of {\em covariance measure structure} for square
integrable stochastic processes. We define Wiener integral, we develop a
suitable formalism for stochastic calculus of variations and we make
Gaussian
assumptions only when necessary. Our main examples are finite quadratric
variation processes with stationary increments and the bifractional
Brownian
motion.
http://front.math.ucdavis.edu/math.PR/0606069
---------------------------------------------------------------
4349. Q-GENERALIZATION OF SYMMETRIC ALPHA-STABLE DISTRIBUTIONS. PART I
Sabir Umarov and Constantino Tsallis and Murray Gell-Mann and
Stanly Steinberg
The classic and the L\'evy-Gnedenko central limit theorems play a key
role in
theory of probabilities, and also in Boltzmann-Gibbs (BG) statistical
mechanics. They both concern the paradigmatic case of probabilistic
independence of the random variables that are being summed. A
generalization of
the BG theory, usually referred to as nonextensive statistical
mechanics and
characterized by the index $q$ ($q=1$ recovers the BG theory),
introduces
global correlations between the random variables, and recovers
independence for
$q=1$. The classic central limit theorem was recently $q$-generalized
by some
of us. In the present paper we $q$-generalize the L\'evy-Gnedenko
central limit
theorem.
http://front.math.ucdavis.edu/cond-mat/0606038
---------------------------------------------------------------
4350. Q-GENERALIZATION OF SYMMETRIC ALPHA-STABLE DISTRIBUTIONS. PART II
Sabir Umarov and Constantino Tsallis and Murray Gell-Mann and
Stanly Steinberg
The classic and the L\'evy-Gnedenko central limit theorems play a key
role in
theory of probabilities, and also in Boltzmann-Gibbs (BG) statistical
mechanics. They both concern the paradigmatic case of probabilistic
independence of the random variables that are being summed. A
generalization of
the BG theory, usually referred to as nonextensive statistical
mechanics and
characterized by the index $q$ ($q=1$ recovers the BG theory),
introduces
global correlations between the random variables, and recovers
independence for
$q=1$. The classic central limit theorem was recently $q$-generalized
by some
of us. In the present paper we $q$-generalize the L\'evy-Gnedenko
central limit
theorem. In Part I we described the $q$-version of the $\alpha$-
stable L\'evy
distributions. In Part II we study the $(q^{\ast},q,q_{\ast})-
$triplet, for
which the mapping $F_{q^{\ast}}: \, \mathcal{G}_{q} \rightarrow
\mathcal{G}_{q_{\ast}}$ holds. This fact allows to study the
corresponding
attractors and to obtain a complete generalization of the $q$-central
limit
theorem for random variables with infinite $(2q-1)$-variance.
http://front.math.ucdavis.edu/cond-mat/0606040
---------------------------------------------------------------
4351. SOME PROPERTIES OF EXPONENTIAL INTEGRALS OF L\'EVY PROCESSES
AND EXAMPLES
Hitoshi Kondo and Makoto Maejima and Ken-iti Sato
The improper stochastic integral $Z=\int_0^{\infty-}\exp(-X_{s-})dY_s
$ is
studied, where $\{(X_t, Y_t), t \geqslant 0 \}$ is a L\'evy process
on $\mathbb
R ^{1+d}$ with $\{X_t \}$ and $\{Y_t \}$ being $\mathbb R$-valued and
$\mathbb
R ^d$-valued, respectively. The condition for existence and
finiteness of $Z$
is given and then the law $\mathcal L(Z)$ of $Z$ is considered. Some
sufficient
conditions for $\mathcal L(Z)$ to be selfdecomposable and some
sufficient
conditions for $\mathcal L(Z)$ to be non-selfdecomposable but
semi-selfdecomposable are given. Attention is paid to the case where
$d=1$,
$\{X_t\}$ is a Poisson process, and $\{X_t\}$ and $\{Y_t\}$ are
independent. An
example of $Z$ of type $G$ with selfdecomposable mixing distribution
is given.
http://front.math.ucdavis.edu/math.PR/0606084
---------------------------------------------------------------
4352. HITTING TIMES FOR GAUSSIAN PROCESSES
L. Decreusefond and D. Nualart
We establish a general formula for the Laplace transform of the
hitting times
of a Gaussian process. Some consequences are derived, and in
particular cases
like the fractional Brownian motion are discussed.
http://front.math.ucdavis.edu/math.PR/0606086
---------------------------------------------------------------
4353. PROJECTION FORMULAS FOR ORTHOGONAL POLYNOMIALS
W. Bryc and W. Matysiak and R. Szwarc and J. Wesolowski
We prove a new projection formula for the four-parameter family of
orthogonal
polynomials outside of the Askey-Wilson class. By carefully analyzing
the
recurrence relations we manage to overcome the lack of explicit
expression for
the orthogonality measure.
http://front.math.ucdavis.edu/math.CA/0606092
---------------------------------------------------------------
4354. GAUSSIAN MARGINALS OF PROBABILITY MEASURES WITH GEOMETRIC
SYMMETRIES
Mark W. Meckes
Motivated by the multivariate version of the central limit problem
for convex
bodies, we prove normal approximation theorems for k-dimensional
marginals of
probability measures on R^n possessing certain geometric symmetries. In
particular, we derive results for uniform measures on 1-unconditional
and
1-symmetric convex bodies and on simplices. We also discuss
connections between
results of E. Meckes and the author for 1-dimensional marginals and a
recent
result of B. Klartag.
http://front.math.ucdavis.edu/math.MG/0606073
---------------------------------------------------------------
4355. A DISCRETE INVITATION TO QUANTUM FILTERING AND FEEDBACK CONTROL
Luc Bouten and Ramon van Handel and and Matthew R. James
The engineering and control of devices at the quantum-mechanical
level--such
as those consisting of small numbers of atoms and photons--is a delicate
business. The fundamental uncertainty that is inherently present at
this scale
manifests itself in the unavoidable presence of noise, making this a
novel
field of application for stochastic estimation and control theory. In
this
expository paper we demonstrate estimation and feedback control of
quantum
mechanical systems in what is essentially a noncommutative version of
the
binomial model that is popular in mathematical finance. The model is
extremely
rich and allows a full development of the theory, while remaining
completely
within the setting of finite-dimensional Hilbert spaces (thus
avoiding the
technical complications of the continuous theory). We introduce
discretized
models of an atom in interaction with the electromagnetic field, obtain
filtering equations for photon counting and homodyne detection, and
solve a
stochastic control problem using dynamic programming and Lyapunov
function
methods.
http://front.math.ucdavis.edu/math.PR/0606118
---------------------------------------------------------------
4356. PARAMETER-BASED FISHER'S INFORMATION OF ORTHOGONAL POLYNOMIALS
J.S. Dehesa and B. Olmos & R.J. Yanez
The Fisher information of the classical orthogonal polynomials with
respect
to a parameter is introduced, its interest justified and its explicit
expression for the Jacobi, Laguerre, Gegenbauer and Grosjean
polynomials found.
http://front.math.ucdavis.edu/math.CA/0606133
---------------------------------------------------------------
4357. DICHOTOMOUS MARKOV NOISE: EXACT RESULTS FOR OUT-OF-EQUILIBRIUM
SYSTEMS (A BRIEF OVERVIEW)
Ioana Bena
Nonequilibrium systems driven by additive or multiplicative dichotomous
Markov noise appear in a wide variety of physical and mathematical
models. We
review here some prototypical examples, with an emphasis on {\em
analytically-solvable} situations. In particular, it has escaped
attention till
recently that the standard results for the long-time properties of
such systems
cannot be applied when unstable fixed points are crossed in the
asymptotic
regime. We show how calculations have to be modified to deal with
these cases
and present a few relevant applications -- the hypersensitive
transport, the
rocking ratchet, and the stochastic Stokes' drift. These results
reinforce the
impression that dichotomous noise can be put on a par with Gaussian
white noise
as far as obtaining analytic results is concerned. They convincingly
illustrate
the interplay between noise and nonlinearity in generating nontrivial
behaviors
of nonequilibrium systems and point to various practical applications.
http://front.math.ucdavis.edu/cond-mat/0606116
---------------------------------------------------------------
4358. PERCOLATION ON DUAL LATTICES WITH K-FOLD SYMMETRY
Bela Bollobas and Oliver Riordan
Zhang found a simple, elegant argument deducing the non-existence of an
infinite open cluster in certain lattice percolation models (for
example, p=1/2
bond percolation on the square lattice) from general results on the
uniqueness
of an infinite open cluster when it exists; this argument requires some
symmetry. Here we show that a simple modification of Zhang's argument
requires
only 2-fold (or 3-fold) symmetry, proving that the critical
probabilities for
percolation on dual planar lattices with such symmetry sum to 1. We
also give a
new proof of a result of Grimmett determining the critical surface for
anisotropic percolation on the triangular lattice.
http://front.math.ucdavis.edu/math.PR/0606149
---------------------------------------------------------------
4359. GENERALIZED CHEEGER INEQUALITIES FOR EIGENVALUES OF NON-
REVERSIBLE MARKOV CHAINS
Ravi Montenegro
We show lower bounds for the smallest non-trivial eigenvalue, and
smallest
real portion of an eigenvalue, of the Laplacian of a non-reversible
Markov
chain in terms of an Evolving set quantity. A myriad of Cheeger-like
inequalities follow for non-reversible chains, which even in the
reversible
case sharpen previously known results. The same argument also
produces a new
Cheeger-like inequality for the smallest eigenvalue of a reversible
chain, and
a Cheeger-like inequality for the second largest magnitude eigenvalue
of a
non-reversible chain.
http://front.math.ucdavis.edu/math.PR/0606167
---------------------------------------------------------------
4360. STUDENT'S T-TEST WITHOUT SYMMETRY CONDITIONS
Iosif Pinelis
An explicit representation of an arbitrary zero-mean distribution as the
mixture of (at-most-)two-point zero-mean distributions is given.
Based in this
representation, tests for (i) asymmetry patterns and (ii) for
location without
symmetry conditions can be constructed. Exact inequalities implying
conservative properties of such tests are presented. These
developments extend
results established earlier by Efron, Eaton, and Pinelis under a
symmetry
condition.
http://front.math.ucdavis.edu/math.ST/0606160
---------------------------------------------------------------
4361. CORRELATION DECAY AND DETERMINISTIC FPTAS FOR COUNTING LIST-
COLORINGS OF A GRAPH
David Gamarnik and Dmitriy Katz
We propose a deterministic algorithm for approximately counting the
number of
list colorings of a graph. Under the assumption that the graph is
triangle
free, the size of every list is at least $\alpha \Delta$, where $
\alpha$ is an
arbitrary constant bigger than $\alpha^{**}=2.8432...$, the solution
of $\alpha
e^{-{1\over \alpha}}=2$, and $\Delta$ is the maximum degree of the
graph, we
obtain the following results. For the case when the size of the each
list is a
large constant, we show the existence of a \emph{deterministic} FPTAS
for
computing the total number of list colorings. The same deterministic
algorithm
has complexity $2^{O(\log^2 n)}$, without any assumptions on the
sizes of the
lists, where $n$ is the size of the instance.
Our results are not based on the most powerful existing counting
technique --
rapidly mixing Markov chain method. Rather we build upon concepts from
statistical physics, in particular, the decay of correlation
phenomena and its
implication for the uniqueness of Gibbs measures in infinite graphs.
This
approach was proposed in two recent papers \cite
{BandyopadhyayGamarnikCounting}
and \cite{weitzCounting}. The principle insight of the present work
is that the
correlation decay property can be established with respect to certain
\emph{computation tree}, as opposed to the conventional correlation
decay
property which is typically established with respect to graph theoretic
neighborhoods of a given node. This allows truncation of computation
at a
logarithmic depth in order to obtain polynomial accuracy in
polynomial time.
While the analysis conducted in this paper is limited to the problem of
counting list colorings, the proposed algorithm can be extended to an
arbitrary
constraint satisfaction problem in a straightforward way.
http://front.math.ucdavis.edu/math.CO/0606143
---------------------------------------------------------------
4362. TRUELS, OR THE SURVIVAL OF THE WEAKEST
Pau Amengual and Ra\'ul Toral
In this paper we review some of the main results obtained in the
field of
truels. A "truel" is a generalization of a duel involving three players.
Depending on the rules used for chosing the players, we may
distinguish between
the random, sequential and simultaneous truel. A paradoxical result
appears in
these games, as the player with the highest marksmanship does not
necessarily
possess the highest survival (or winning) probability. In this work
we limit
ourselves to the random and sequential truels in which players use
their best
possible strategy with no coalitions. Furthermore, we have modified
the random
truel and converted it into an opinion model. In this version each of
the three
players holds a different opinion on a given topic. We address next the
question of who wins a "truel league". We will see that, despite the
paradoxical result mentioned above, still the distribution of winners
is peaked
around the players with the higher marksmanship for the random and
opinion
versions. In the sequential truel, however, the paradoxical result
remains
partially since the distribution of winners is peaked around the
intermediate
players.
If the rules of truels are extended from three to $N$ players, the
paradoxical results shows up even more clearly since as $N$ increases
it is
more difficult for the player with the highest marksmanship to win
the game.
Finally, we consider the dynamics of the games in a spatial
distribution in a
given network of interactions.
http://front.math.ucdavis.edu/math.PR/0606181
---------------------------------------------------------------
4363. GENERALIZATIONS OF HO-LEE'S BINOMIAL INTEREST RATE MODEL I:
FROM ONE- TO MULTI-FACTOR
Jir\^o Akahori and Hiroki Aoki and and Yoshihiko Nagata
In this paper a multi-factor generalization of Ho-Lee model is
proposed. In
sharp contrast to the classical Ho-Lee, this generalization allows
for those
movements other than parallel shifts, while it still is described by a
recombining tree, and is stationary to be compatible with principal
component
analysis. Based on the model, generalizations of duration-based
hedging are
proposed. A continuous-time limit of the model is also discussed.
http://front.math.ucdavis.edu/math.PR/0606183
---------------------------------------------------------------
4364. STABLE SEMIGROUPS ON HOMOGENEOUS TREES AND HYPERBOLIC SPACES
Andrzej Stos
We prove the kernel estimates related to subordinated semigroups on
homogeneous trees. We study the long time propagation problem. We
exploit this
to show exit time estimates for (large) balls. We use an abstract
setting of
metric measure spaces. This enables us to give these results for
trees end
hyperbolic spaces as well. Finally, we show some estimates for the
Poisson
kernel of a ball.
http://front.math.ucdavis.edu/math.PR/0606185
---------------------------------------------------------------
4365. IDENTIFICATION D'UN PROCESSUS AUTOR\'{E}GRESSIF GAUSSIEN STABLE
PAR LA M\'{E}THODE DE MOYENNISATION LOGARITHMIQUE DANS LE CAS R\'{E}EL
Faouzi Chaabane (EASMS) and Hamdi Fathallah (LM-Versailles)
In the present work, we consider a stable one-dimensional gaussian
autoregressive model in continous time. Using the limit theorems with
logarithmic averaging obtained for continous local martingales, we
construct
then an estimator of the noise covariance $\sigma^{2}$ and an
estimator of
$\theta$ different of the one of the least squares estimator. By
exploiting the
weighting method we ameliorate the convergence rates of these new
estimators.
http://front.math.ucdavis.edu/math.PR/0606200
---------------------------------------------------------------
4366. FLOW PROPERTIES OF DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL
BROWNIAN MOTION
L. Decreusefond and D. Nualart
We prove that solutions of stochastic differential equations driven by
fractional Brownian motion for $H>1/2$ define flows of homeomorphisms on
$\mathbb{R}^{d}$.
http://front.math.ucdavis.edu/math.PR/0606214
---------------------------------------------------------------
4367. FREE JACOBI PROCESS
Nizar Demni (PMA) and the PMA Collaboration
Using a matrix approach, we define the free Jacobi process as the
limit of
the complex Jacobi matrix process. The we derive a free SDE which is
analogous
to its classical counterpart. To proceed, we prove that fro suitable
parameters
the process remains injective if it is initially injective and then
use the
polar decomposition. In the stationnary case, this will be easily
deduced from
the explicit expression of the spectral measure. In the general
setting we
derive a recurrence formula for the moments. Moreover, a p. d. e. for
the
Cauchy transform of the law is given.
http://front.math.ucdavis.edu/math.PR/0606218
---------------------------------------------------------------
4368. SIGNIFICANT EDGES IN THE CASE OF A NON-STATIONARY GAUSSIAN NOISE
Isabelle Abraham (DCRE) and Romain Abraham (MAPMO) and Agnes
Desolneux (MAP5), Sebastien Li-Thiao-Te (CMLA)
In this paper, we propose an edge detection technique based on some
local
smoothing of the image followed by a statistical hypothesis testing
on the
gradient. An edge point being defined as a zero-crossing of the
Laplacian, it
is said to be a significant edge point if the gradient at this point
is larger
than a threshold $s(\eps)$ defined by: if the image $I$ is pure
noise, then
$\P(\norm{\nabla I}\geq s(\eps) \bigm| \Delta I = 0) \leq\eps$. In
other words,
a significant edge is an edge which has a very low probability to be
there
because of noise. We will show that the threshold $s(\eps)$ can be
explicitly
computed in the case of a stationary Gaussian noise. In images we are
interested in, which are obtained by tomographic reconstruction from a
radiograph, this method fails since the Gaussian noise is not stationary
anymore. But in this case again, we will be able to give the law of the
gradient conditionally on the zero-crossing of the Laplacian, and
thus compute
the threshold $s(\eps)$. We will end this paper with some experiments
and
compare the results with the ones obtained with some other methods of
edge
detection.
http://front.math.ucdavis.edu/math.ST/0606219
---------------------------------------------------------------
4369. A DISCRETE IT\^O CALCULUS APPROACH TO HE'S FRAMEWORK FOR MULTI-
FACTOR DISCRETE MARKETS
Jir\^o Akahori
In the present paper, a discrete version of It\^o's formula for a
class of
multi-dimensional random walk is introduced and applied to the study
of a
discrete-time complete market model which we call He's framework. The
formula
unifies continuous-time and discrete-time settings and by regarding
the latter
as the finite difference scheme of the former, the order of
convergence is
obtained. The result shows that He's framework cannot be of order 1
scheme
except for the one dimensional case.
http://front.math.ucdavis.edu/math.PR/0606292
---------------------------------------------------------------
4370. ON THE FREE ENERGY OF A DIRECTED POLYMER IN A BROWNIAN ENVIRONMENT
John Moriarty and Neil O'Connell
We prove a formula conjectured in O'Connell and Yor (2001) for the free
energy density of a directed polymer in a Brownian environment in 1+1
dimensions.
http://front.math.ucdavis.edu/math.PR/0606296
---------------------------------------------------------------
4371. DYNAMICAL MODELS FOR CIRCLE COVERING: BROWNIAN MOTION AND
POISSON UPDATING
Johan Jonasson and Jeffrey Steif
We consider two dynamical variants of the classical problem of random
interval coverings of the unit circle, the latter having been
completely solved
by L. Shepp. In the first model, the centers of the intervals perform
independent Brownian motions and in the second model, the positions
of the
intervals are updated according to independent Poisson processes
where an
interval of length l is updated at rate l^{-alpha} where alpha is a
parameter.
For the model with Brownian motions, a special case of our results is
that if
the length of the nth interval is c/n, then there are times at which
a fixed
point is not covered if and only if c <2 and there are times at which
the
circle is not fully covered if and only if c <3. For the Poisson
updating
model, we obtain analogous results with c <alpha and c <alpha +1
instead. We
also compute the Hausdorff dimension of the set of exceptional times
for some
of these questions.
http://front.math.ucdavis.edu/math.PR/0606297
---------------------------------------------------------------
4372. GENEALOGY OF CATALYTIC BRANCHING MODELS
Andreas Greven and Lea Popovic and and Anita Winter
We consider catalytic branching populations. They consist of a catalyst
population evolving according to a critical binary branching process in
continuous time with a constant branching rate, and of a reactant
population
with a branching rate proportional to the number of catalyst
individuals alive.
The reactant forms a process in random medium.
We describe asymptotically the genealogy of catalytic branching
populations
coded as the induced forest of $\R$-trees using the many individuals
-- rapid
branching continuum limit. The limiting continuum genealogical
forests are then
studied in detail from both the quenched and annealed point of view.
The result
is obtained by constructing a contour process and analyzing the
appropriately
rescaled version and its limit. The genealogy of the limiting forest is
described by a point-process. We compare geometric properties and
statistics of
the reactant limit forest with those of the ``classical'' forest.
http://front.math.ucdavis.edu/math.PR/0606313
---------------------------------------------------------------
4373. BAYESIAN REGRESSION OF PIECEWISE CONSTANT FUNCTIONS
Marcus Hutter
We derive an exact and efficient Bayesian regression algorithm for
piecewise
constant functions of unknown segment number, boundary location, and
levels. It
works for any noise and segment level prior, e.g. Cauchy which can
handle
outliers. We derive simple but good estimates for the in-segment
variance. We
also propose a Bayesian regression curve as a better way of smoothing
data
without blurring boundaries. The Bayesian approach also allows
straightforward
determination of the evidence, break probabilities and error
estimates, useful
for model selection and significance and robustness studies. We
discuss the
performance on synthetic and real-world examples. Many possible
extensions will
be discussed.
http://front.math.ucdavis.edu/math.ST/0606315
---------------------------------------------------------------
4374. GLOBALLY CENTERED DISCRETE SNAKES
Jean-Fran\c{c}ois Marckert (LaBRI)
We consider branching random walks built on Galton-Watson trees with
offspring distribution having a bounded support, conditioned to have
$n$ nodes,
and their rescaled convergences to the Brownian snake. We exhibit a
notion of
"globally centered discrete snake'' that extends the usual settings
in which
the displacements are supposed centered. We show that under some
additional
moment conditions, when $n$ goes to $+\infty$, "globally centered
discrete
snakes'' converge to the Brownian snake. The proof relies on a
precise study of
the "lineage'' of the nodes in a Galton-Watson tree conditioned by
the size,
and their links with a multinomial process. Some consequences concerning
Galton-Watson trees conditioned by the size are also derived.
http://front.math.ucdavis.edu/math.PR/0606338
---------------------------------------------------------------
4375. QUASI-INVARIANT MEASURES ON THE PATH SPACE OF A DIFFUSION
Denis Bell
The author has previously constructed a class of admissible vector
fields on
the path space of an elliptic diffusion process $x$ taking values in
a closed
compact manifold. In this Note the existence of flows for this class
of vector
fields is established and it is shown that the law of $x$ is quasi-
invariant
under these flows.
http://front.math.ucdavis.edu/math.PR/0606365
---------------------------------------------------------------
4376. A WEAKNESS IN STRONG LOCALIZATION FOR SINAI'S WALK
Zhan Shi (PMA) and Olivier Zindy (PMA)
Sinai's walk is a recurrent one-dimensional nearest-neighbour random
walk in
random environment. It is known for a phenomenon of strong localization,
namely, the walk spends almost all time at or near the bottom of deep
valleys
of the potential. Our main result shows a weakness of this localization
phenomenon: with probability one, the zones where the walk stays for
the most
time can be far away from the sites where the walk spends the most
time. In
particular, this gives a negative answer to a problem of Erd\H os and
R\'ev\'esz \cite{erdos-revesz}, originally formulated for the usual
homogeneous
random walk.
http://front.math.ucdavis.edu/math.PR/0606376
---------------------------------------------------------------
4377. DOMAIN OF ATTRACTION OF THE QUASI-STATIONARY DISTRIBUTIONS FOR
THE ORNSTEIN-UHLENBECK PROCESS
Manuel Lladser and Jaime San Martin
Let $X=(X_t)$ be a one-dimensional Ornstein-Uhlenbeck process with an
initial
density function $f$ supported on the positive real-line that is a
regularly
varying function with exponent $-(1+\eta)$, with $\eta\in (0,1)$. We
prove the
existence of a probability measure $\nu$ with a Lebesgue density,
depending on
$\eta$, such that for every Borel set $A$ of the positive real-line:
$\lim_{t\to\infty} P_f(X_t\in A | T_0^X>t)=\nu(A)$, where $T_0^X$ is the
hitting time of 0 of $X$.
http://front.math.ucdavis.edu/math.PR/0606392
---------------------------------------------------------------
4378. RATES OF CONVERGENCE OF A TRANSIENT DIFFUSION IN A SPECTRALLY
NEGATIVE L\'{E}VY POTENTIAL
Arvind Singh (PMA)
We consider a diffusion process $X$ in a random L\'{e}vy potential $V
$. We
study the rates of convergence when the diffusion is transient under the
assumption that the L\'{e}vy process does not possess positive jumps. We
generalize the previous results of Hu-Shi-Yor (1999) for drifted
Brownian
potentials. In particular, we prove a conjecture of Carmona: provided
that
there exists $0<\kappa<1$ such that $E[e^{\kappa V\_1}]=1$, then
$X\_t/t^\kappa$ converges to some non-degenerate distribution. These
results
are in a way analogous to those obtained by Kesten-Kozlov-Spitzer
(1975) for
the random walk in a random environment.
http://front.math.ucdavis.edu/math.PR/0606411
---------------------------------------------------------------
4379. THE RANK OF RANDOM GRAPHS
Kevin P. Costello and Van H. Vu
We show that almost surely the rank of the adjacency matrix of the
Erd\"os-R\'enyi random graph $G(n,p)$ equals the number of non-isolated
vertices for any $c\ln n/n<p<1/2$, where $c$ is an arbitrary positive
constant
larger than 1/2. In particular, the giant component (a.s.) has full
rank in
this range.
http://front.math.ucdavis.edu/math.PR/0606414
---------------------------------------------------------------
4380. STOCHASTIC CALCULUS OF VARIATIONS FOR GENERAL L\'EVY PROCESSES
AND ITS APPLICATIONS TO JUMP-TYPE SDE'S WITH NON-DEGENERATED DRIFT
Alexey Kulik
We consider an SDE in R^m of the type dX(t)=a(X(t))dt+dU(t) with a L
\'evy
process U and study the problem for the distribution of a solution to be
regular in various senses. We do not impose any specific conditions
on the
L\'evy measure of the noise, and this is the main difference between
our method
and the known methods by J.Bismut or J.Picard. The main tool in our
approach is
the stochastic calculus of variations for a L\'evy process, based on the
time-stretching transformations of the trajectories. Three problems
are solved
in this framework. First, we prove that if the drift coefficient a is
non-degenerated in an appropriate sense, then the law of the solution
to the
Cauchy problem for the initial equation is absolutely continuous, as
soon as
the L\'evy measure of the noise satisfies one of the rather weak
intensity
conditions, for instance the so-called wide cone condition. Secondly, we
provide the sufficient conditions for the density of the distribution
of the
solution to the Cauchy problem to be smooth in the terms of the
family of the
so-called order indices of the L\'evy measure of the noise (the drift
again is
supposed to be non-degenerated). At last, we show that an invariant
distribution to the initial equation, if exists, possesses a C^\infty-
density
provided the drift is non-degenerated and the L\'evy measure of the
noise
satisfies the wide cone condition.
http://front.math.ucdavis.edu/math.PR/0606427
---------------------------------------------------------------
4381. MARTIN BOUNDARY OF A KILLED RANDOM WALK ON A HALF-SPACE
Irina Ignatiouk-Robert
A complete representation of the Martin boundary of killed random
walks on a
half-space $\Z^{d-1}\times\N^*$ is obtained. In particular, it is
proved that
the corresponding Martin boundary is homemorphic to the half-sphere $
{\cal
S}^d_+ = \{z\in\R^{d-1}\times\R_+ : |z|=1\}$. The method is based on a
combination of ratio limits theorems and large deviation techniques.
http://front.math.ucdavis.edu/math.PR/0606439
---------------------------------------------------------------
4382. ON A RANDOM GRAPH RELATED TO QUANTUM THEORY
Svante Janson
We show that a random graph studied by Ioffe and Levit is an example
of an
inhomogeneous random graph of the type studied by Bollobas, Janson
and Riordan,
which enables us to give a new, simple, proof of their result on a phase
transition.
http://front.math.ucdavis.edu/math.PR/0606454
---------------------------------------------------------------
4383. WEAK CONVERGENCE OF LAWS ON R^{K} WITH COMMON MARGINALS
Alessio Sancetta
We present a result on topologically equivalent integral metrics
(Rachev,
1991, Muller, 1997) that metrize weak convergence of laws with common
marginals. This result is relevant for applications, as shown in a
few simple
examples.
http://front.math.ucdavis.edu/math.PR/0606462
---------------------------------------------------------------
4384. DIMENSION ESTIMATES FOR INVARIANT MEASURES OF CONTRACTING-ON-
AVERAGE ITERATED FUNCTION SYSTEMS
Micha{\l} Rams
We estimate from above and below the dimension of invariant measure for
contracting-on-average iterated function systems in $\R^d$.
http://front.math.ucdavis.edu/math.DS/0606420
---------------------------------------------------------------
4385. SECOND ORDER FREENESS AND FLUCTUATIONS OF RANDOM MATRICES, III.
HIGHER ORDER FREENESS AND FREE CUMULANTS
Benoit Collins (Universite Claude Bernard and Lyon 1) and James A.
Mingo (Queen's University), Piotr Sniady (Uniwersytet Wroclawski),
Roland Speicher
(Queen's University)
We extend the relation between random matrices and free probability
theory
from the level of expectations to the level of all correlation
functions (which
are classical cumulants of traces of products of the matrices). We
introduce
the notion of "higher order freeness" and develop a theory of
corresponding
free cumulants. We show that two independent random matrix ensembles
are free
of arbitrary order if one of them is unitarily invariant. We prove R-
transform
formulas for second order freeness. Much of the presented theory
relies on a
detailed study of the properties of "partitioned permutations".
http://front.math.ucdavis.edu/math.OA/0606431
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4386. RANDOM WALKS ON HYPERGROUP OF CONICS IN FINITE FIELDS
Le Anh Vinh
In this paper we study random walks on the hypergroup of conics in
finite
fields. We investigate the behavior of random walks on this
hypergroup, the
equilibrium distribution and the mixing times. We use the coupling
method to
show that the mixing time of random walks on hypergroup of conics is
only
linear.
http://front.math.ucdavis.edu/math.PR/0606485
---------------------------------------------------------------
4387. RATE OF CONVERGENCE OF IMPLICIT APPROXIMATIONS FOR STOCHASTIC
EVOLUTION EQUATIONS
Istvan Gy\"{o}ngy and Annie Millet (PMA)
Stochastic evolution equations in Banach spaces with unbounded nonlinear
drift and diffusion operators are considered. Under some regularity
condition
assumed for the solution, the rate of convergence of implicit Euler
approximations is estimated under strong monotonicity and Lipschitz
conditions.
The results are applied to a class of quasilinear stochastic PDEs of
parabolic
type.
http://front.math.ucdavis.edu/math.PR/0606488
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4388. SPECTRAL GAP ESTIMATE FOR FRACTIONAL LAPLACIAN
M. Kwasnicki
A lower bound estimate \lambda_2 - \lambda_1 \ge c \lambda_1^{-d /
\alpha}
(\diam D)^{-d - \alpha} for the spectral gap of the Dirichlet fractional
Laplacian on arbitrary bounded domain D is proved. This follows from a
variational formula for the spectral gap and an upper bound estimate
for the
supremum norm of the ground state eigenfunction.
http://front.math.ucdavis.edu/math.PR/0606509
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4389. ON THE NUMBER OF CLUSTERS FOR PLANAR GRAPHS
Jean-Michel Billiot (LABSAD) and Franck Corset (LABSAD) and Eric
Fontenas (LABSAD)
The Tutte polynomial is a powerfull analytic tool to study the
structure of
planar graphs. In this paper, we establish some relations between the
number of
clusters per bond for planar graph and its dual : these relations
bring into
play the coordination number of the graphs. The factorial moment
measure of the
number of clusters per bond are given using the derivative of the Tutte
polynomial. Examples are presented for simple planar graph. The cases of
square, triangular, honeycomb, Archimedean and Laves lattices are
discussed.
http://front.math.ucdavis.edu/cond-mat/0606495
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4390. THRESHOLDS FOR VIRUS SPREAD ON NETWORKS
M.Draief; A.Ganesh; L.Massoulie
We study how the spread of computer viruses, worms, and other
self-replicating malware is affected by the logical topology of the
network
over which they propagate. We consider a model in which each host can
be in one
of 3 possible states - susceptible, infected or removed (cured, and
no longer
susceptible to infection). We characterise how the size of the
population that
eventually becomes infected depends on the network topology.
Specifically, we
show that if the ratio of cure to infection rates is larger than the
spectral
radius of the graph, and the initial infected population is small,
then the
final infected population is also small in a sense that can be made
precise.
Conversely, if this ratio is smaller than the spectral radius, then
we show in
some graph models of practical interest (including power law random
graphs)
that the final infected population is large. These results yield
insights into
what the critical parameters are in determining virus spread in
networks.
http://front.math.ucdavis.edu/math.PR/0606514
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4391. MULTIVARIATE RISKS AND DEPTH-TRIMMED REGIONS
Ignacio Cascos and Ilya Molchanov
We describe a general framework for measuring risks, where the risk
measure
takes values in an abstract cone. It is shown that this approach
naturally
includes the classical risk measures and set-valued risk measures and
yields a
natural definition of vector-valued risk measures. Several main
constructions
of risk measures are described in this abstract axiomatic framework.
It is shown that the concept of depth-trimmed (or central) regions
from the
multivariate statistics is closely related to the definition of risk
measures.
In particular, the halfspace trimming corresponds to the Value-at-
Risk, while
the zonoid trimming yields the expected shortfall. In the abstract
framework,
it is shown how to establish a both-ways correspondence between risk
measures
and depth-trimmed regions. It is also demonstrated how the lattice
structure of
the space of risk values influences this relationship.
http://front.math.ucdavis.edu/math.PR/0606520
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4392. WEAK AND ALMOST SURE LIMITS FOR THE PARABOLIC ANDERSON MODEL
WITH HEAVY TAILED POTENTIALS
Remco van der Hofstad and Peter Morters and Nadia Sidorova
We study the parabolic Anderson problem, i.e., the heat equation with
independent identically distributed random potential and localised
initial
condition. Our interest is in the long-term behaviour of the random
total mass
of the unique non-negative solution in the case that the distribution
of the
potential at one site is heavy tailed. For this, we study two
paradigm cases of
fields with infinite moment generating functions: the case of
polynomial or
Frechet tails, and the case of stretched exponential or Weibull
tails. For
potentials with either polynomial or stretched exponential right
tails, we find
asymptotic expansions for the logarithm of the total mass up to the
first
random term, which we describe in terms of weak limit theorems. In
the case of
polynomial tails, already the leading term in the expansion is
random. For
stretched exponential tails, we observe random fluctuations in the
almost sure
asymptotics of the second term of the expansion, but in the weak
sense the
fourth term is the first random term of the expansion. The main tool
in our
proofs is extreme value theory.
http://front.math.ucdavis.edu/math.PR/0606527
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4393. A VECTOR-VALUED ALMOST SURE INVARIANCE PRINCIPLE FOR
HYPERBOLIC DYNAMICAL SYSTEMS
Ian Melbourne and Matthew Nicol
We prove an almost sure invariance principle (approximation by d-
dimensional
Brownian motion) for vector-valued Holder observables of large
classes of
nonuniformly hyperbolic dynamical systems. These systems include Axiom~A
diffeomorphisms and flows as well as systems modelled by Young towers
with
moderate tail decay rates. In particular, the position variable of
the planar
periodic Lorentz gas with finite horizon approximates a 2-dimensional
Brownian
motion.
http://front.math.ucdavis.edu/math.DS/0606535
---------------------------------------------------------------
4394. CONSTRUCTION OF SOME QUANTUM STOCHASTIC OPERATOR COCYCLES BY
THE SEMIGROUP METHOD
J. Martin Lindsay and Stephen J. Wills
A new method for the construction of Fock-adapted operator Markovian
cocycles
is outlined, and its use is illustrated by application to a number of
examples
arising in physics and probability. The construction uses the Trotter-
Kato
Theorem and a recent characterisation of such cocycles in terms of an
associated family of contraction semigroups.
http://front.math.ucdavis.edu/math.FA/0606545
---------------------------------------------------------------
4395. FUNCTIONALS OF BROWNIAN BRIDGES ARISING IN THE CURRENT MISMATCH
IN D/A-CONVERTERS
Markus Heydenreich and Remco van der Hofstad and Georgi Radulov
Digital-to-analog converters (DAC) transform signals from the abstract
digital domain to the real analog world. In many applications, DAC's
play a
crucial role.
Due to variability in the production, various errors arise that
influence the
performance of the DAC. We focus on the current errors, which
describe the
fluctuations in the currents of the various unit current elements in
the DAC. A
key performance measure of the DAC is the Integrated Non-linearity
(INL), which
we study in this paper.
There are several DAC architectures. The most widely used
architectures are
the thermometer, the binary and the segmented architectures. We study
the two
extreme architectures, namely, the thermometer and the binary
architectures. We
assume that the current errors are i.i.d. normally distributed, and
reformulate
the INL as a functional of a Brownian bridge. We then proceed by
investigating
these functionals. For the thermometer case, the functional is the
maximal
absolute value of the Brownian bridge, which has been investigated in
the
literature. For the binary case, we investigate properties of the
functional,
such as its mean, variance and density.
http://front.math.ucdavis.edu/math.PR/0606584
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4396. BACKWARD PARABOLIC ITO EQUATIONS AND SECOND FUNDAMENTAL INEQUALITY
Nikolai Dokuchaev
Existence, uniqueness, and a priori estimates for solutions are
studied for
stochastic parabolic Ito equations. An analog of the second fundamental
inequality and the related existence theorem are obtained for backward
stochastic parabolic Ito equation.
http://front.math.ucdavis.edu/math.PR/0606595
---------------------------------------------------------------
4397. A LIMIT THEOREM OF DISCRETE GALTON-WATSON BRANCHING PROCESSES
WITH IMMIGRATION
Zenghu Li
We provide a simple set of sufficient conditions for the weak
convergence of
discrete Galton-Watson branching processes with immigration to
continuous time
and continuous state branching processes with immigration.
http://front.math.ucdavis.edu/math.PR/0606597
---------------------------------------------------------------
4398. REPRESENTATION OF FUNCTIONALS OF ITO PROCESSES IN BOUNDED
DOMAINS VIA PARABOLIC ITO EQUATIONS
Nikolai Dokuchaev
Representation of functionals of non-Markov processes is studied for
bounded
and unbounded domains. These functionals are represented via
solutions of
backward parabolic Ito equations. This results is based on an analog
of the
second fundamental inequality and the related existence theorem are
obtained
for backward parabolic Ito equations.
http://front.math.ucdavis.edu/math.PR/0606601
---------------------------------------------------------------
4399. ANALYSIS OF THE ROSENBLATT PROCESS
Ciprian A. Tudor (SAMOS)
We analyze {\em the Rosenblatt process} which is a selfsimilar
process with
stationary increments and which appears as limit in the so-called
{\em Non
Central Limit Theorem} (Dobrushin and Major (1979), Taqqu (1979)).
This process
is non-Gaussian and it lives in the second Wiener chaos. We give its
representation as a Wiener-It\^o multiple integral with respect to
the Brownian
motion on a finite interval and we develop a stochastic calculus with
respect
to it by using both pathwise type calculus and Malliavin calculus.
http://front.math.ucdavis.edu/math.PR/0606602
---------------------------------------------------------------
4400. 0-1 LAWS FOR REGULAR CONDITIONAL DISTRIBUTIONS
Patrizia Berti and Pietro Rigo
Let $(\Omega,\mathcal{B},P)$ be a probability space,
$\mathcal{A}\subset\mathcal{B}$ a sub-$\sigma$-field, and $\mu$ a
regular
conditional distribution for $P$ given $\mathcal{A}$. Necessary and
sufficient
conditions for $\mu(\omega)(A)$ to be 0-1, for all $A\in\mathcal{A}$ and
$\omega\in A_0$, where $A_0\in\mathcal{A}$ and $P(A_0)=1$, are given.
Such
conditions apply, in particular, when $\mathcal{A}$ is a tail
sub-$\sigma$-field. Let $H(\omega)$ denote the $\mathcal{A}$-atom
including the
point $\omega\in\Omega$. Necessary and sufficient conditions for
$\mu(\omega)(H(\omega))$ to be 0-1, for all $\omega\in A_0$, are also
given. If
$(\Omega,\mathcal{B})$ is a standard space, the latter 0-1 law is
true for
various classically interesting sub-$\sigma$-fields $\mathcal{A}$,
including
tail, symmetric, invariant, as well as some sub-$\sigma$-fields
connected with
continuous time processes.
http://front.math.ucdavis.edu/math.PR/0606604
---------------------------------------------------------------
4401. OPERATOR SPACE EMBEDDING OF LQ INTO LP
Marius Junge and Javier Parcet
Let 1 \le p < q \le 2 and let M be any von Neumann algebra. We use
recent
techniques from free harmonic analysis to construct a completely
isomorphic
embedding of Lq(M) (equipped with its natural operator space
structure) into
Lp(A) for some sufficiently large von Neumann algebra A. We show that
hyperfiniteness and the QWEP are preserved in our construction.
http://front.math.ucdavis.edu/math.OA/0606596
---------------------------------------------------------------
4402. APPROXIMATION OF STATIONARY PROCESSES BY HIDDEN MARKOV MODELS
Lorenzo Finesso and Angela Grassi and Peter Spreij
We propose an algorithm for the construction of a Hidden Markov Model
(HMM)
of assigned complexity (number of states of the underlying Markov
chain) which
best approximates, in Kullback-Leibler divergence rate, a given
stationary
process. We establish, under mild conditions, the existence of the
divergence
rate between a stationary process and an HMM, and approximate it with a
properly defined divergence between their Hankel matrices. The proposed
three-step algorithm, based on the Nonnegative Matrix Factorization
technique,
realizes an HMM optimal with respect to the Hankel approximated
criterion. A
full theoretical analysis of the algorithm is given in the special
case of
Markov approximation.
http://front.math.ucdavis.edu/math.OC/0606591
---------------------------------------------------------------
4403. EXPECTATION, CONDITIONAL EXPECTATION AND MARTINGALES IN LOCAL
FIELDS
Steven N. Evans and Tye Lidman
We investigate a possible definition of expectation and conditional
expectation for random variables with values in a local field such as
the
$p$-adic numbers. We define the expectation by analogy with the
observation
that for real-valued random variables in $L^2$ the expected value is the
orthogonal projection onto the constants. Previous work has shown
that the
local field version of $L^\infty$ is the appropriate counterpart of
$L^2$, and
so the expected value of a local field-valued random variable is
defined to be
its ``projection'' in $L^\infty$ onto the constants. Unlike the real
case, the
resulting projection is not typically a single constant, but rather a
ball in
the metric on the local field. However, many properties of this
expectation
operation and the corresponding conditional expectation mirror those
familiar
from the real-valued case; for example, conditional expectation is, in a
suitable sense, a contraction on $L^\infty$ and the tower property
holds. We
also define the corresponding notion of martingale, show that several
standard
examples of martingales (for example, sums or products of suitable
independent
random variables or ``harmonic'' functions composed with Markov
chains) have
local field analogues, and obtain versions of the optional sampling and
martingale convergence theorems.
http://front.math.ucdavis.edu/math.PR/0606609
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4404. SUPERPROCESSES WITH DEPENDENT SPATIAL MOTION AND GENERAL
BRANCHING DENSITIES
Donald A. Dawson; Zenghu Li; Hao Wang
We construct a class of superprocesses by taking the high density
limit of a
sequence of interacting-branching particle systems. The spatial
motion of the
superprocess is determined by a system of interacting diffusions, the
branching
density is given by an arbitrary bounded non-negative Borel function,
and the
superprocess is characterized by a martingale problem as a diffusion
process
with state space $M(\IR)$, improving and extending considerably the
construction of Wang (1997, 1998). It is then proved in a special
case that a
suitable rescaled process of the superprocess converges to the usual
super
Brownian motion. An extension to measure-valued branching catalysts
is also
discussed.
http://front.math.ucdavis.edu/math.PR/0606615
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4405. NON-LOCAL BRANCHING SUPERPROCESSES AND SOME RELATED MODELS
Donald A. Dawson and Luis G. Gorostiza and Zenghu Li
A new formulation of non-local branching superprocesses is given from
which
we derive as special cases the rebirth, the multitype, the mass-
structured, the
multilevel and the age-reproduction-structured superprocesses and the
superprocess-controlled immigration process. This unified treatment
simplifies
considerably the proof of existence of the old classes of
superprocesses and
also gives rise to some new ones.
http://front.math.ucdavis.edu/math.PR/0606616
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4406. SKEW CONVOLUTION SEMIGROUPS AND RELATED IMMIGRATION PROCESSES
Zeng-Hu Li
A special type of immigration associated with measure-valued branching
processes is formulated by using skew convolution semigroups. We give
characterization for a general inhomogeneous skew convolution
semigroup in
terms of probability entrance laws. The related immigration process is
constructed by summing up measure-valued paths in the Kuznetsov process
determined by an entrance rule. The behavior of the Kuznetsov process
is then
studied, which provides insights into trajectory structures of the
immigration
process. Some well-known results on excessive measures are formulated
in terms
of stationary immigration processes.
http://front.math.ucdavis.edu/math.PR/0606617
---------------------------------------------------------------
4407. CONSTRUCTION OF IMMIGRATION SUPERPROCESSES WITH DEPENDENT
SPATIAL MOTION FROM ONE-DIMENSIONAL EXCURSIONS
Donald A. Dawson; Zenghu Li
A superprocess with dependent spatial motion and interactive
immigration is
constructed as the pathwise unique solution of a stochastic integral
equation
carried by a stochastic flow and driven by Poisson processes of one-
dimensional
excursions.
http://front.math.ucdavis.edu/math.PR/0606618
---------------------------------------------------------------
4408. GENERALIZED MEHLER SEMIGROUPS AND CATALYTIC BRANCHING PROCESSES
WITH IMMIGRATION
Donald A. Dawson and Zenghu Li and Byron Schmuland and Wei Sun
Skew convolution semigroups play an important role in the study of
generalized Mehler semigroups and Ornstein-Uhlenbeck processes. We
give a
characterization for a general skew convolution semigroup on real
separable
Hilbert space whose characteristic functional is not necessarily
differentiable
at the initial time. A connection between this subject and catalytic
branching
superprocesses is established through fluctuation limits, providing a
rich
class of non-differentiable skew convolution semigroups. Path
regularity of the
corresponding generalized Ornstein-Uhlenbeck processes in different
topologies
is also discussed.
http://front.math.ucdavis.edu/math.PR/0606619
---------------------------------------------------------------
4409. NON-DIFFERENTIABLE SKEW CONVOLUTION SEMIGROUPS AND RELATED
ORNSTEIN-UHLENBECK PROCESSES
Donald A. Dawson; Zenghu Li
It is proved that a general non-differentiable skew convolution
semigroup
associated with a strongly continuous semigroup of linear operators
on a real
separable Hilbert space can be extended to a differentiable one on
the entrance
space of the linear semigroup. A cadlag strong Markov process on an
enlargement
of the entrance space is constructed from which we obtain a
realization of the
corresponding Ornstein-Uhlenbeck process. Some explicit
characterizations of
the entrance spaces for special linear semigroups are given.
http://front.math.ucdavis.edu/math.PR/0606620
---------------------------------------------------------------
4410. SUPERPROCESSES WITH COALESCING BROWNIAN SPATIAL MOTION AS LARGE-
SCALE LIMITS
Donald A. Dawson; Zenghu Li; Xiaowen Zhou
A superprocess with coalescing spatial motion is constructed in terms of
one-dimensional excursions. Based on this construction, it is proved
that the
superprocess is purely atomic and arises as scaling limit of a
special form of
the superprocess with dependent spatial motion studied in Dawson {\it
et al.}
(2001) and Wang (1997, 1998).
http://front.math.ucdavis.edu/math.PR/0606621
---------------------------------------------------------------
4411. CONDITIONAL LOG-LAPLACE FUNCTIONALS OF IMMIGRATION
SUPERPROCESSES WITH DEPENDENT SPATIAL MOTION
Zenghu Li; Hao Wang; Jie Xiong
A non-critical branching immigration superprocess with dependent spatial
motion is constructed and characterized as the solution of a stochastic
equation driven by a time-space white noise and an orthogonal martingale
measure. A representation of its conditional log-Laplace functionals is
established, which gives the uniqueness of the solution and hence its
Markov
property. Some properties of the superprocess including an ergodic
theorem are
also obtained.
http://front.math.ucdavis.edu/math.PR/0606622
---------------------------------------------------------------
4412. BRANCHING PROCESSES WITH IMMIGRATION AND RELATED TOPICS
Zenghu Li
This is a survey on recent progresses in the study of branching
processes
with immigration, generalized Ornstein-Uhlenbeck processes and affine
Markov
processes. We mainly focus on the applications of skew convolution
semigroups
and the connections in those processes.
http://front.math.ucdavis.edu/math.PR/0606623
---------------------------------------------------------------
4413. EIGENVALUES OF EUCLIDEAN RANDOM MATRICES
Charles Bordenave
We study the spectral measure of large Euclidean random matrices. The
entries
of these matrices are determined by the relative position of $n$
random points
in a compact set $\Omega_n$ of $\R^d$. Under various assumptions we
establish
the almost sure convergence of the limiting spectral measure as the
number of
points goes to infinity. The moments of the limiting distribution are
computed,
and we prove that the limit of this limiting distribution as the
density of
points goes to infinity has a nice expression. We apply our results
to the
adjacency matrix of the geometric graph.
http://front.math.ucdavis.edu/math.PR/0606624
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4414. A CENTRAL LIMIT THEOREM FOR BIASED RANDOM WALKS ON GALTON-
WATSON TREES
Yuval Peres and Ofer Zeitouni
Let ${\cal T}$ be a rooted Galton-Watson tree with offspring
distribution
$\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails.
Consider the $\lambda$-biased random walk $\{X_n\}_{n\geq 0}$ on $
{\cal T}$;
this is the nearest neighbor random walk which, when at a vertex $v$
with $d_v$
offspring, moves closer to the root with probability $\lambda/(\lambda
+d_v)$,
and moves to each of the offspring with probability $1/(\lambda+d_v)
$. It is
known that this walk has an a.s. constant speed $\v=\lim_n |X_n|/n$
(where $|X_n|$ is the distance of $X_n$ from the root), with $\v>0
$ for $
0<\lambda<m$ and $\v=0$ for $\lambda \ge m$. For all $\lambda \le m$,
we prove
a quenched CLT for $|X_n|-n\v$. (For $\lambda>m$ the walk is positive
recurrent, and there is no CLT.) The most interesting case by far is
$\lambda=m$, where the CLT has the following form: for almost every $
{\cal T}$,
the ratio $|X_{[nt]}|/\sqrt{n}$ converges in law as $n \to \infty$ to a
deterministic multiple of the absolute value of a Brownian motion.
Our approach
to this case is based on an explicit description of an invariant
measure for
the walk from the point of view of the particle (previously, such a
measure was
explicitly known only for $\lambda=1$) and the construction of
appropriate
harmonic coordinates.
http://front.math.ucdavis.edu/math.PR/0606625
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4415. MODELING FINANCIAL ASSETS WITHOUT SEMIMARTINGALES
Rosanna Coviello and Francesco Russo
This paper does not suppose a priori that the evolution of the price
of a
financial asset is a semimartingale. Since possible strategies of
investors are
self-financing, previous prices are forced to be finite quadratic
variation
processes. The non-arbitrage property is not excluded if the class $
{\cal A}$
of admissible strategies is restricted. The classical notion of
martingale is
replaced with the notion of ${\cal A}$-martingale. A calculus related
to ${\cal
A}$-martingales with some examples is developed. Some applications to
the
maximization of the utility of an insider are expanded.
http://front.math.ucdavis.edu/math.PR/0606642
---------------------------------------------------------------
4416. ENTROPY AND VISION
Rami Kanhouche (CMLA)
In vector quantization the number of vectors used to construct the
codebook
is always an undefined problem, there is always a compromise between
the number
of vectors and the quantity of information lost during the
compression. In this
text we present a minimum of Entropy principle that gives solution to
this
compromise and represents an Entropy point of view of signal
compression in
general. Also we present a new adaptive Object Quantization technique
that is
the same for the compression and the perception.
http://front.math.ucdavis.edu/math.PR/0606643
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4417. INTRINSIC ULTRACONTRACTIVITY FOR LEVY PROCESSES
Tomasz Grzywny
We prove the intrinsic ultracontractivity for the semigroup generated
by a
large class of symmetric Levy processes such that the Levy measure
satisfies
some conditions in the neighborhood of 0, killed on exiting a bounded
and
connected Lipschitz domain.
http://front.math.ucdavis.edu/math.PR/0606659
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4418. ISOTROPIC RANDOM WALKS ON AFFINE BUILDINGS
James Parkinson
In this paper we apply techniques of spherical harmonic analysis to
prove a
local limit theorem, a rate of escape theorem, and a central limit
theorem for
isotropic random walks on arbitrary thick regular affine buildings of
irreducible type.
http://front.math.ucdavis.edu/math.PR/0606662
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4419. THE NOISE IN THE CIRCULAR LAW AND THE GAUSSIAN FREE FIELD
Brian Rider and Balint Virag
Fill an n x n matrix with independent complex Gaussians of variance 1/
n. As n
approaches infinity, the eigenvalues {z_k} converge to a sum of an
H^1-noise on
the unit disk and an independent H^{1/2}-noise on the unit circle. More
precisely, for C^1 functions of suitable growth, the distribution of
sum_{k=1}^n (f(z_k)-E f(z_k)) converges to that of a mean-zero
Gaussian with
variance given by the sum of the squares of the disk H^1 and the
circle H^{1/2}
norms of f. Moreover, with p_n the characteristic polynomial, log|
p_n|- E
log|p_n| tends to the planar Gaussian free field conditioned to be
harmonic
outside the unit disk. Finally, for polynomial test functions f, we
prove that
the limiting covariance structure is universal for a class of models
including
Haar distributed unitary matrices.
http://front.math.ucdavis.edu/math.PR/0606663
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4420. PERIODIC ATTRACTORS OF RANDOM TRUNCATOR MAPS
Ted Theodosopoulos and Robert Boyer
This paper introduces the \textit{truncator} map as a dynamical
system on the
space of configurations of an interacting particle system. We
represent the
symbolic dynamics generated by this system as a non-commutative
algebra and
classify its periodic orbits using properties of endomorphisms of the
resulting
algebraic structure. A stochastic model is constructed on these
endomorphisms,
which leads to the classification of the distribution of periodic
orbits for
random truncator maps. This framework is applied to investigate the
periodic
transitions of Bornholdt's spin market model.
http://front.math.ucdavis.edu/math.PR/0606667
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4421. A CENTRAL LIMIT THEOREM FOR STOCHASTIC RECURSIVE SEQUENCES OF
TOPICAL OPERATORS
Glenn Merlet
Let $(A\_n)\_{n\in\N}$ be a sequence of stationary topical (i.e.
isotone and
additively homogeneous) operators. Let $x(n,x\_0)$ be defined by
$x(0,x\_0)=x\_0$ and $x(n+1,x\_0)=A\_nx(n,x\_0)$. This can modelize a
wide
range of systems including, train or queuing networks, job-shop,
timed digital
circuits or parallel processing systems. When $(A\_n)\_{n\in\N}$ has
the memory
loss property, $(x(n,x\_0))\_{n\in\N}$ satisfy a strong law of large
numbers.
We show that it also satisfy the CLT if $\sAn$ satisfy the same
mixing and
integrability assumptions that ensure the CLT for a sum of real
variables in
the results by P. Billingsley and I. Ibragimov. This article is based
on the
work by H. Ishitani for products of random positive matrices.
http://front.math.ucdavis.edu/math.PR/0606668
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4422. INTRINSIC ULTRACONTRACTIVITY FOR NON-SYMMETRIC LEVY PROCESSES
Panki Kim and Renming Song
Recently we extended the concept of intrinsic ultracontractivity to
non-symmetric semigroups and proved that for a large class of non-
symmetric
diffusions Z with measure-valued drift and potential, the semigroup
of Z^D (the
process obtained by killing Z upon exiting D) in a bounded domain is
intrinsic
ultracontractive under very mild assumptions.
In this paper, we study the intrinsic ultracontractivity for non-
symmetric
discontinuous Levy processes. We prove that, for a large class of non-
symmetric
discontinuous Levy processes X such that the Lebesgue measure is
absolutely
continuous with respect to the Levy measure of X, the semigroup of
X^D in any
bounded open set D is intrinsic ultracontractive. In particular, for the
non-symmetric stable process X, the semigroup of X^D is intrinsic
ultracontractive for any bounded set D. Using the intrinsic
ultracontractivity,
we show that the parabolic boundary Harnack principle is true for those
processes. Moreover, we get that the supremum of the expected
conditional
lifetimes in a bounded open set is finite. We also have results of
the same
nature when the Levy measure is compactly supported.
http://front.math.ucdavis.edu/math.PR/0606678
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4423. QUASI-COMPACTNESS AND ABSOLUTELY CONTINUOUS KERNELS,
APPLICATIONS TO MARKOV CHAINS
Hubert Hennion (Universit\'e de Rennes I)
We show how the essential spectral radius of a bounded positive kernel,
acting on bounded functions, is linked to its lower approximation by
certain
absolutely continuous kernels. The standart Doeblin's condition can be
interpreted in this context, and, when suitably reformulated, it
leads to a
formula for the essential spectral radius. This results may be used to
characterize the Markov kernels having a quasi-compact action on a
space of
measurable functions bounded with respect to some test function, when no
irreducibilty and aperiodicity are assumed.
http://front.math.ucdavis.edu/math.PR/0606680
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4424. SLE AND ALPHA-SLE DRIVEN BY LEVY PROCESSES
Qing-Yang Guan and Matthias Winkel
Stochastic Loewner Evolutions (SLE) with a multiple sqrt(kappa)B of
Brownian
motion B as driving process are random planar curves (if kappa<=4) or
growing
compact sets generated by a curve (if kappa>4). We consider here more
general
Levy processes as driving processes and obtain evolutions expected to
look like
random trees or compact sets generated by trees, respectively. We
show that
when the driving force is of the form sqrt(kappa)B+theta^(1/alpha)S
for a
symmetric alpha-stable Levy process S, the cluster has zero or positive
Lebesgue measure according to whether kappa<=4 or kappa>4. We also give
mathematical evidence that a further phase transition at alpha=1 is
attributable to the recurrence/transience dychotomy of the driving Levy
process. We introduce a new class of evolutions that we call alpha-
SLE. They
have alpha-self-similarity properties for alpha-stable Levy driving
processes.
We show the phase transition at a critical coefficient theta=theta_0
(alpha)
analogous to the kappa=4 phase transition.
http://front.math.ucdavis.edu/math.PR/0606685
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4425. BOUNDARY BEHAVIOR OF HARMONIC FUNCTIONS FOR TRUNCATED STABLE
PROCESSES
Panki Kim and Renming Song
For any \alpha in (0, 2), a truncated symmetric \alpha-stable process
is a
symmetric Levy process with no diffusion part and with a Levy density
given by
c|x|^{-d-\alpha} 1_{|x|< 1} for some constant c. In previous paper we
have
studied the potential theory of truncated symmetric stable processes.
Among
other things, we proved that the boundary Harnack principle is valid
for the
positive harmonic functions of a truncated symmetric stable process
in any
bounded convex domain and showed that the Martin boundary of any
bounded convex
domain with respect to this process is the same as the Euclidean
boundary.
However, for truncated symmetric stable processes, the boundary Harnack
principle is not valid in non-convex domains. In this paper, we show
that, for
a large class of not necessarily convex bounded open sets called bounded
roughly connected \kappa-fat open sets (including bounded non-convex
\kappa-fat
domains), the Martin boundary with respect to any truncated symmetric
stable
process is still the same as the Euclidean boundary. We also show
that, for
truncated symmetric stable processes a relative Fatou type theorem is
true in
bounded roughly connected \kappa-fat open sets.
http://front.math.ucdavis.edu/math.PR/0606706
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4426. SOME CHARACTERIZATIONS OF THE SPHERICAL HARMONICS COEFFICIENTS
FOR ISOTROPIC RANDOM FIELDS
P. Baldi and D. Marinucci
In this paper we provide some simple characterizations for the spherical
harmonics coefficients of an isotropic random field on the sphere.
The main
result is a characterization of isotropic gaussian fields through
independence
of the coefficients of their development in spherical harmonics.
http://front.math.ucdavis.edu/math.PR/0606709
---------------------------------------------------------------
4427. SCALING LIMIT FOR TRAP MODELS ON Z^D
Gerard Ben Arous and Jiri Cerny
We give the ``quenched'' scaling limit of Bouchaud's trap model in
dimension
d larger or equal to two. This scaling limit is the Fractional-Kinetics
process, that is the time change of a d-dimensional Brownian motion
by the
inverse of an independent stable subordinator.
http://front.math.ucdavis.edu/math.PR/0606719
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4428. ENDS IN FREE MINIMAL SPANNING FORESTS
\'{A}d\'{a}m Tim\'{a}r
We show that for a transitive unimodular graph, the number of ends is
the
same for every tree of the free minimal spanning forest. This answers a
question of Lyons, Peres and Schramm.
http://front.math.ucdavis.edu/math.PR/0606750
---------------------------------------------------------------
4429. ON THE TRANSIENCE OF PROCESSES DEFINED ON GALTON--WATSON TREES
Andrea Collevecchio
We introduce a simple technique for proving the transience of certain
processes defined on the random tree $\mathcal{G}$ generated by a
supercritical
branching process. We prove the transience for once-reinforced random
walks on
$\mathcal{G}$, that is, a generalization of a result of Durrett,
Kesten and
Limic [Probab. Theory Related Fields 122 (2002) 567--592]. Moreover,
we give a
new proof for the transience of a family of biased random walks
defined on
$\mathcal{G}$. Other proofs of this fact can be found in [Ann.
Probab. 16
(1988) 1229--1241] and [Ann. Probab. 18 (1990) 931--958] as part of more
general results. A similar technique is applied to a vertex-
reinforced jump
process. A by-product of our result is that this process is transient
on the
3-ary tree. Davis and Volkov [Probab. Theory Related Fields 128
(2004) 42--62]
proved that a vertex-reinforced jump process defined on the $b$-ary
tree is
transient if $b\ge 4$ and recurrent if $b=1$. The case $b=2$ is still
open.
http://front.math.ucdavis.edu/math.PR/0606751
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4430. CONCENTRATION FOR INFINITELY DIVISIBLE VECTORS WITH
INDEPENDENT COMPONENTS
C. Houdr\'e and P. Reynaud-Bouret
For various classes of Lipschitz functions we provide dimension free
concentration inequalities for infinitely divisible random vectors with
independent components and finite exponential moments.
http://front.math.ucdavis.edu/math.PR/0606752
---------------------------------------------------------------
4431. SAMPLE PATH PROPERTIES OF BIFRACTIONAL BROWNIAN MOTION
Ciprian Tudor (SAMOS) and Yimin Xiao
Let $B^{H, K}= \big\{B^{H, K}(t), t \in \R\_+ \big\}$ be a bifractional
Brownian motion in $\R^d$. We prove that $B^{H, K}$ is strongly locally
nondeterministic. Applying this property and a stochastic integral
representation of $B^{H, K}$, we establish Chung's law of the iterated
logarithm for $B^{H, K}$, as well as sharp H\"older conditions and tail
probability estimates for the local times of $B^{H, K}$. We also
consider the
existence and the regularity of the local times of multiparameter
bifractional
Brownian motion $B^{\bar{H}, \bar{K}}= \big\{B^{\bar{H}, \bar{K}}(t),
t \in
\R^N\_+ \big\}$ in $\R^d$ using Wiener-It\^o chaos expansion.
http://front.math.ucdavis.edu/math.PR/0606753
---------------------------------------------------------------
4432. AVERAGE VOLUME, CURVATURES, AND EULER CHARACTERISTIC OF RANDOM
REAL ALGEBRAIC VARIETIES
Peter Buergisser
We determine the expected curvature polynomial of random real projective
varieties given as the zero set of independent random polynomials
with Gaussian
distribution, whose distribution is invariant under the action of the
orthogonal group. In particular, the expected Euler characteristic of
such
random real projective varieties is found. This considerably extends
previously
known results on the number of roots, the volume, and the Euler
characteristic
of the solution set of random polynomial equations
http://front.math.ucdavis.edu/math.PR/0606755
---------------------------------------------------------------
4433. EIGENVALUES OF GUE MINORS
Kurt Johansson and Eric Nordenstam
Consider an infinite random matrix $H=(h_{ij})_{0<i,j}$ picked from the
Gaussian Unitary Ensemble (GUE). Denote its main minors by $H_i=(h_
{rs})_{1\leq
r,s\leq i}$ and let the $j$:th largest eigenvalue of $H_i$ be $\mu^i_j
$. We
show that the configuration of all these eigenvalues $(i,\mu_j^i)$
form a
determinantal point process on $\mathbb{N}\times\mathbb{R}$.
Furthermore we show that this process can be obtained as the
scaling limit in
random tilings of the Aztec diamond close to the boundary. We also
discuss the
corresponding limit for random lozenge tilings of a hexagon.
http://front.math.ucdavis.edu/math.PR/0606760
---------------------------------------------------------------
4434. SOME PROPERTIES FOR SUPERPROCESS UNDER A STOCHASTIC FLOW
Kijung Lee and Carl Mueller and and Jei Xiong
For a superprocess under a stochastic flow, we prove that it has a
density
with respect to the Lebesgue measure for d=1 and is singular for d>1.
For d=1,
a stochastic partial differential equation is derived for the
density. The
regularity of the solution is then proved by using Krylov's L_p-
theory for
linear SPDE. A snake representation for this superprocess is
established. As
applications of this representation, we prove the compact support
property for
general d and singularity of the process when d>1.
http://front.math.ucdavis.edu/math.PR/0606761
---------------------------------------------------------------
4435. ASYMPTOTIC EXPANSIONS FOR SUMS OF BLOCK-VARIABLES UNDER WEAK
DEPENDENCE
S.N. Lahiri
Let $\{X_i\}\sipmi$ be a sequence of random vectors and $Y_{in}=f_
{in} ({\cal
X}_{i,\ell})$ be zero mean block-variables where ${\cal
X}_{i,\ell}=(X_i,...,X_{i+\ell-1}), ~i\geq 1$ are overlapping blocks
of length
$\ell$ and where $f_{in}$ are Borel measurable functions. This paper
establishes valid joint asymptotic expansions of general orders for
the joint
distribution of the sums $\sum_{i=1}^n X_i$ and $\sum_{i=1}^n Y_{in}$
under
weak dependence conditions on the sequence $\{X_i\}\sipmi$ when the
block
length $\ell$ grows to infinity. Similar expansions are also derived
for sums
of block variables based on non-overlapping blocks. In contrast to the
classical Edgeworth expansion results where the terms in the
expansions are
given by powers of $n^{-1/2}$, the expansions derived here are
mixtures of two
series, one in powers of $n^{-1/2}$ while the other in powers of
$[\frac{n}{\ell}]^{-1/2}$. Applications of the main results to
expansions for
studentized statistics of time series data and to second order
correctness of
the blocks of blocks bootstrap method for studentized statistics are
given.
http://front.math.ucdavis.edu/math.ST/0606739
---------------------------------------------------------------
4436. LOGARITHMIC SOBOLEV INEQUALITY FOR THE INHOMOGENEOUS ZERO RANGE
PROCESS
Hanna Jankowski
We prove that the logarithmic Sobolev constant for the inhomogeneous
symmetric nearest neighbour zero range process on a cube of size N^d
grows as
N^2. We apply this result to the inhomogeneous process which arises
in the
study of the homogeneous version of the zero range interacting
particle system
with colours.
http://front.math.ucdavis.edu/math.PR/0606778
---------------------------------------------------------------
4437. ON THE ABSOLUTE CONTINUITY OF L\'{E}VY PROCESSES WITH DRIFT
Ivan Nourdin and Thomas Simon
We consider the problem of absolute continuity for the one-
dimensional SDE
\[X_t=x+\int_0^ta(X_s) ds+Z_t,\] where $Z$ is a real L\'{e}vy process
without
Brownian part and $a$ a function of class $\mathcal{C}^1$ with bounded
derivative. Using an elementary stratification method, we show that
if the
drift $a$ is monotonous at the initial point $x$, then $X_t$ is
absolutely
continuous for every $t>0$ if and only if $Z$ jumps infinitely often.
This
means that the drift term has a regularizing effect, since $Z_t$
itself may not
have a density. We also prove that when $Z_t$ is absolutely
continuous, then
the same holds for $X_t$, in full generality on $a$ and at every
fixed time
$t$. These results are then extended to a larger class of elliptic jump
processes, yielding an optimal criterion on the driving Poisson
measure for
their absolute continuity.
http://front.math.ucdavis.edu/math.PR/0606783
---------------------------------------------------------------
4438. TRACES OF SYMMETRIC MARKOV PROCESSES AND THEIR CHARACTERIZATIONS
Zhen-Qing Chen and Masatoshi Fukushima and Jiangang Ying
Time change is one of the most basic and very useful transformations for
Markov processes. The time changed process can also be regarded as
the trace of
the original process on the support of the Revuz measure used in the
time
change. In this paper we give a complete characterization of time
changed
processes of an arbitrary symmetric Markov process, in terms of the
Beurling--Deny decomposition of their associated Dirichlet forms and
of Feller
measures of the process. In particular, we determine the jumping and
killing
measure (or, equivalently, the L\'{e}vy system) for the time-changed
process.
We further discuss when the trace Dirichlet form for the time changed
process
can be characterized as the space of finite Douglas integrals defined
by Feller
measures. Finally, we give a probabilistic characterization of Feller
measures
in terms of the excursions of the base process.
http://front.math.ucdavis.edu/math.PR/0606784
---------------------------------------------------------------
4439. TRANSITION SEMIGROUPS OF BANACH SPACE VALUED ORNSTEIN-
UHLENBECK PROCESSES
Ben Goldys and Jan van Neerven
We investigate the transition semigroup of the solution to a stochastic
evolution equation $dX(t) = AX(t)dt +dW_H(t)$, $t\ge 0,$ where $A$ is
the
generator of a $C_0$-semigroup $S$ on a separable real Banach space $E
$ and
$W_H$ is cylindrical white noise with values in a real Hilbert space
$H$ which
is continuously embedded in $E$. Various properties of these
semigroups, such
as the strong Feller property, the spectral gap property, and
analyticity, are
characterized in terms of the behaviour of $S$ in $H$. In particular we
investigate the interplay between analyticity of the transition
semigroup,
$S$-invariance of $H$, and analyticity of the restricted semigroup
$S_H$.
http://front.math.ucdavis.edu/math.PR/0606785
---------------------------------------------------------------
4440. CONCENTRATION INEQUALITIES AND ASYMPTOTIC RESULTS FOR RATIO
TYPE EMPIRICAL PROCESSES
Evarist Gin\'{e} and Vladimir Koltchinskii
Let $\mathcal{F}$ be a class of measurable functions on a measurable
space
$(S,\mathcal{S})$ with values in $[0,1]$ and let
\[P_n=n^{-1}\sum_{i=1}^n\delta_{X_i}\] be the empirical measure based
on an
i.i.d. sample $(X_1,...,X_n)$ from a probability distribution $P$ on
$(S,\mathcal{S})$. We study the behavior of suprema of the following
type:
\[\sup_{r_n<\sigma_Pf\leq \delta_n}\frac{|P_nf-Pf|}{\phi(\sigma_Pf)},
\] where
$\sigma_Pf\ge\operatorname {Var}^{1/2}_Pf$ and $\phi$ is a
continuous, strictly
increasing function with $\phi(0)=0$. Using Talagrand's concentration
inequality for empirical processes, we establish concentration
inequalities for
such suprema and use them to derive several results about their
asymptotic
behavior, expressing the conditions in terms of expectations of
localized
suprema of empirical processes. We also prove new bounds for expected
values of
sup-norms of empirical processes in terms of the largest $\sigma_Pf$
and the
$L_2(P)$ norm of the envelope of the function class, which are
especially
suited for estimating localized suprema. With this technique, we
extend to
function classes most of the known results on ratio type suprema of
empirical
processes, including some of Alexander's results for VC classes of
sets. We
also consider applications of these results to several important
problems in
nonparametric statistics and in learning theory (including general
excess risk
bounds in empirical risk minimization and their versions for $L_2$-
regression
and classification and ratio type bounds for margin distributions in
classification).
http://front.math.ucdavis.edu/math.PR/0606788
---------------------------------------------------------------
4441. RANDOM TREES IN ELECTRICAL NETWORKS
Hariharan Narayanan
This paper contains results relating currents and voltages in resistive
networks to appropriate random trees or forests in those networks.
Since each
resistive network has a reversible Markov chain equivalent, we obtain
equivalent results for the latter as well. We describe a way of
obtaining a
harmonic function on a weighted graph given the boundary values, by
choosing
random forests of the graph. As applications of the theorems
discussed, (which
give formulae of the Kirchhoff tree kind), we obtain an expression
for the
expected transit time from one state to another in a reversible
Markov chain in
terms of its arborescences. The methods of this paper can also be
used to give
alternative proofs of the Kirchhoff tree formula.
http://front.math.ucdavis.edu/math.PR/0607011
---------------------------------------------------------------
4442. CONCENTRATION FOR NORMS OF INFINITELY DIVISIBLE VECTORS WITH
INDEPENDENT COMPONENTS
C. Houdr\'e and P. Marchal and P. Reynaud-Bouret
We obtain dimension free concentration inequalities for $L^p$, $p\ge 2$,
norms of infinitely divisible random vectors with independent
coordinates. The
methods and results extend to some other classes of Lipschitz functions.
http://front.math.ucdavis.edu/math.PR/0607019
---------------------------------------------------------------
4443. MEDIAN, CONCENTRATION AND FLUCTUATION FOR L\'EVY PROCESSES
C. Houdr\'e and P. Marchal
We estimate a median of $f(X_t)$ where $f$ is a Lipschitz function, $X
$ is a
L\'evy process and $t$ an arbitrary time. This leads to concentration
inequalities for $f(X_t)$. In turn, corresponding fluctuation
estimates are
obtained under assumptions typically satisfied if the process has a
regular
behavior in small time and a, possibly different, regular behavior in
large
time.
http://front.math.ucdavis.edu/math.PR/0607022
---------------------------------------------------------------
4444. DUALITY AND EVOLVING SET BOUNDS ON MIXING TIMES
Ravi Montenegro
We sharpen the Evolving set methodology of Morris and Peres and
extend it to
study convergence in total variation, relative entropy, $L^2$ and other
distances. Bounds in terms of a modified form of conductance are
given which
apply even for walks with no holding probability. These bounds are
found to be
strictly better than earlier Evolving set bounds, may be
substantially better
than conductance profile results derived via Spectral profile, and
drastically
sharpen Blocking Conductance bounds if there are no bottlenecks at
small sets.
http://front.math.ucdavis.edu/math.PR/0607031
---------------------------------------------------------------
4445. A GENERAL FORMULA FOR THE DISTRIBUTION OF THE MAXIMUM OF A
GAUSSIAN FIELD AND THE APPROXIMATION OF THE TAIL
Jean-Marc Aza\"{\i}s Mario Wschebor
We study the probability distribution $F(u)$ of the maximum of smooth
Gaussian fields defined on compact subsets of $\R^d$ having some
geometric
regularity.
Our main result is a general formula for the density of $F$. Even
though this
is an implicit formula, one can deduce from it explicit bounds for
the density,
hence for the distribution, as well as improved expansions for $ 1-F
(u)$ for
large values of $u$.
The main tool is the Rice formula for the moments of the number of
roots of a
random system of equation over the reals, of which we give a new
simplified
proof.
This method enables also to study second order properties of the so-
called
expected Euler Characteristic approximation using only elementary
arguments and
to extend these kind of results to some interesting classes of
Gaussian fields.
We obtain more precise results for the "direct method" to compute the
distribution of the maximum, using spectral theory of GOE random
matrices.
http://front.math.ucdavis.edu/math.PR/0607041
---------------------------------------------------------------
4446. HIGH-FREQUENCY ASYMPTOTICS FOR SUBORDINATED ISOTROPIC FIELDS ON
AN ABELIAN COMPACT GROUP
Domenico Marinucci and Giovanni Peccati (LSTA)
Let T* be a random field indexed by an Abelian compact group G, and
suppose
that T* has the form T* = F(T(g)), where T is Gaussian and isotropic.
The aim
of this paper is to establish high-frequency central limit theorems
for the
Fourier coefficients associated to T*. The proofs of our main results
involve
recently established criteria for the weak convergence of multiple
Wiener-It\^{o} integrals. Our research is motivated by physical
applications,
mainly related to the probabilistic modelization of the Cosmic Microwave
Background radiation. In this connection, the case of the n-
dimensional torus
is analyzed in detail.
http://front.math.ucdavis.edu/math.PR/0607044
---------------------------------------------------------------
4447. OPTIMAL SCALING FOR PARTIALLY UPDATING MCMC ALGORITHMS
Peter Neal and Gareth Roberts
In this paper we shall consider optimal scaling problems for high-
dimensional
Metropolis--Hastings algorithms where updates can be chosen to be lower
dimensional than the target density itself. We find that the optimal
scaling
rule for the Metropolis algorithm, which tunes the overall algorithm
acceptance
rate to be 0.234, holds for the so-called Metropolis-within-Gibbs
algorithm as
well. Furthermore, the optimal efficiency obtainable is independent
of the
dimensionality of the update rule. This has important implications
for the MCMC
practitioner since high-dimensional updates are generally
computationally more
demanding, so that lower-dimensional updates are therefore to be
preferred.
Similar results with rather different conclusions are given for so-
called
Langevin updates. In this case, it is found that high-dimensional
updates are
frequently most efficient, even taking into account computing costs.
http://front.math.ucdavis.edu/math.PR/0607054
---------------------------------------------------------------
4448. ACCURACY OF STATE SPACE COLLAPSE FOR EARLIEST-DEADLINE-FIRST
QUEUES
{\L}ukasz Kruk and John Lehoczky and Steven Shreve
This paper presents a second-order heavy traffic analysis of a single
server
queue that processes customers having deadlines using the
earliest-deadline-first scheduling policy. For such systems, referred
to as
real-time queueing systems, performance is measured by the fraction of
customers who meet their deadline, rather than more traditional
performance
measures, such as customer delay, queue length or server utilization.
To model
such systems, one must keep track of customer lead times (the time
remaining
until a customer deadline elapses) or equivalent information. This paper
reviews the earlier heavy traffic analysis of such systems that provided
approximations to the system's behavior. The main result of this
paper is the
development of a second-order analysis that gives the accuracy of the
approximations and the rate of convergence of the sequence of real-time
queueing systems to its heavy traffic limit.
http://front.math.ucdavis.edu/math.PR/0607056
---------------------------------------------------------------
4449. ASYMPTOTIC BEHAVIOR OF THE POISSON--DIRICHLET DISTRIBUTION FOR
LARGE MUTATION RATE
Donald A. Dawson and Shui Feng
The large deviation principle is established for the Poisson--Dirichlet
distribution when the parameter $\theta$ approaches infinity. The
result is
then used to study the asymptotic behavior of the homozygosity and the
Poisson--Dirichlet distribution with selection. A phase transition
occurs
depending on the growth rate of the selection intensity. If the
selection
intensity grows sublinearly in $\theta$, then the large deviation
rate function
is the same as the neutral model; if the selection intensity grows at
a linear
or greater rate in $\theta$, then the large deviation rate function
includes an
additional term coming from selection. The application of these
results to the
heterozygote advantage model provides an alternate proof of one of
Gillespie's
conjectures in [Theoret. Popul. Biol. 55 145--156].
http://front.math.ucdavis.edu/math.PR/0607070
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4450. TAIL ESTIMATES FOR HOMOGENIZATION THEOREMS IN RANDOM MEDIA
Daniel Boivan (LM)
It is known that a random walk on $\Z^d$ among i.i.d. uniformly elliptic
random bond conductances verifies a central limit theorem. It is also
known
that approximations of the covariance matrix can be obtained by
considering
periodic environments. Here we estimate the speed of convergence of this
homogenization result. We obtain similar estimates for finite volume
approximations of the effective conductance and of the lowest Dirichlet
eigenvalue. A lower bound is also given for the variance of the Green
function
of a random walk in a random non-negative potential.
http://front.math.ucdavis.edu/math.PR/0607073
---------------------------------------------------------------
4451. ON THE DOMAIN OF ATTRACTION FOR THE LOWER TAIL IN WICKSELL'S
CORPUSCLE PROBLEM
S. Koetzer and I. Molchanov
We consider the classical Wicksell corpuscle problem with spherical
particles
in R^n and investigate the shapes of lower tails of distributions of
`sphere
radii' in R^n and `sphere radii' in a k-dimensional section plane. We
show in
which way the domains of attraction are related to each other.
http://front.math.ucdavis.edu/math.PR/0607086
---------------------------------------------------------------
4452. TRANSPORTATION DISTANCE AND THE CENTRAL LIMIT THEOREM
S.Ekisheva and C. Houdr\'e
For probability measures on a complete separable metric space, we
present
sufficient conditions for the existence of a solution to the Kantorovich
transportation problem. We also obtain sufficient conditions (which
sometimes
also become necessary) for the convergence, in transportation, of
probability
measures when the cost function is continuous, non-decreasing and
depends on
the distance. As an application, the CLT in the transportation
distance is
proved for independent and some dependent stationary sequences.
http://front.math.ucdavis.edu/math.PR/0607089
---------------------------------------------------------------
4453. ASYMPTOTICS OF SOLUTIONS TO SEMILINEAR STOCHASTIC WAVE EQUATIONS
Pao-Liu Chow
Large-time asymptotic properties of solutions to a class of semilinear
stochastic wave equations with damping in a bounded domain are
considered.
First an energy inequality and the exponential bound for a linear
stochastic
equation are established. Under appropriate conditions, the existence
theorem
for a unique global solution is given. Next the questions of bounded
solutions
and the exponential stability of an equilibrium solution, in mean-
square and
the almost sure sense, are studied. Then, under some sufficient
conditions, the
existence of a unique invariant measure is proved. Two examples are
presented
to illustrate some applications of the theorems.
http://front.math.ucdavis.edu/math.PR/0607097
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4454. AVERAGE OPTIMALITY FOR CONTINUOUS-TIME MARKOV DECISION
PROCESSES IN POLISH SPACES
Xianping Guo and Ulrich Rieder
This paper is devoted to studying the average optimality in
continuous-time
Markov decision processes with fairly general state and action
spaces. The
criterion to be maximized is expected average rewards. The transition
rates of
underlying continuous-time jump Markov processes are allowed to be
unbounded,
and the reward rates may have neither upper nor lower bounds. We
first provide
two optimality inequalities with opposed directions, and also give
suitable
conditions under which the existence of solutions to the two optimality
inequalities is ensured. Then, from the two optimality inequalities
we prove
the existence of optimal (deterministic) stationary policies by using
the
Dynkin formula. Moreover, we present a ``semimartingale
characterization'' of
an optimal stationary policy. Finally, we use a generalized Potlach
process
with control to illustrate the difference between our conditions and
those in
the previous literature, and then further apply our results to
average optimal
control problems of generalized birth--death systems, upwardly skip-free
processes and two queueing systems. The approach developed in this
paper is
slightly different from the ``optimality inequality approach'' widely
used in
the previous literature.
http://front.math.ucdavis.edu/math.PR/0607098
---------------------------------------------------------------
4455. CONTINUOUS-TIME GARCH PROCESSES
Peter Brockwell and Erdenebaatar Chadraa and Alexander Lindner
A family of continuous-time generalized autoregressive conditionally
heteroscedastic processes, generalizing the $\operatorname {COGARCH}
(1,1)$
process of Kl\"{u}ppelberg, Lindner and Maller [J. Appl. Probab. 41
(2004)
601--622], is introduced and studied. The resulting $\operatorname
{COGARCH}(p,q)$ processes, $q\ge p\ge 1$, exhibit many of the
characteristic
features of observed financial time series, while their corresponding
volatility and squared increment processes display a broader range of
autocorrelation structures than those of the $\operatorname {COGARCH}
(1,1)$
process. We establish sufficient conditions for the existence of a
strictly
stationary nonnegative solution of the equations for the volatility
process
and, under conditions which ensure the finiteness of the required
moments,
determine the autocorrelation functions of both the volatility and
the squared
increment processes. The volatility process is found to have the
autocorrelation function of a continuous-time autoregressive moving
average
process.
http://front.math.ucdavis.edu/math.PR/0607109
---------------------------------------------------------------
4456. A THEORETICAL FRAMEWORK FOR THE PRICING OF CONTINGENT CLAIMS IN
THE PRESENCE OF MODEL UNCERTAINTY
Laurent Denis and Claude Martini
The aim of this work is to evaluate the cheapest superreplication
price of a
general (possibly path-dependent) European contingent claim in a
context where
the model is uncertain. This setting is a generalization of the
uncertain
volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The
uncertainty is specified by a family of martingale probability
measures which
may not be dominated. We obtain a partial characterization result and
a full
characterization which extends Avellaneda, Levy and Paras results in
the UVM
case.
http://front.math.ucdavis.edu/math.PR/0607111
---------------------------------------------------------------
4457. VARIANCE-OPTIMAL HEDGING FOR PROCESSES WITH STATIONARY
INDEPENDENT INCREMENTS
Friedrich Hubalek and Jan Kallsen and Leszek Krawczyk
We determine the variance-optimal hedge when the logarithm of the
underlying
price follows a process with stationary independent increments in
discrete or
continuous time. Although the general solution to this problem is
known as
backward recursion or backward stochastic differential equation, we
show that
for this class of processes the optimal endowment and strategy can be
expressed
more explicitly. The corresponding formulas involve the moment,
respectively,
cumulant generating function of the underlying process and a Laplace- or
Fourier-type representation of the contingent claim. An example
illustrates
that our formulas are fast and easy to evaluate numerically.
http://front.math.ucdavis.edu/math.PR/0607112
---------------------------------------------------------------
4458. WIDTH AND MODE OF THE PROFILE FOR SOME RANDOM TREES OF
LOGARITHMIC HEIGHT
Luc Devroye and Hsien-Kuei Hwang
We propose a new, direct, correlation-free approach based on central
moments
of profiles to the asymptotics of width (size of the most abundant
level) in
some random trees of logarithmic height. The approach is simple but
gives
precise estimates for expected width, central moments of the width
and almost
sure convergence. It is widely applicable to random trees of logarithmic
height, including recursive trees, binary search trees, quad trees,
plane-oriented ordered trees and other varieties of increasing trees.
http://front.math.ucdavis.edu/math.PR/0607119
---------------------------------------------------------------
4459. CENTRAL LIMIT THEOREMS FOR POISSON HYPERPLANE TESSELLATIONS
Lothar Heinrich and Hendrik Schmidt and Volker Schmidt
We derive a central limit theorem for the number of vertices of convex
polytopes induced by stationary Poisson hyperplane processes in $
\mathbb{R}^d$.
This result generalizes an earlier one proved by Paroux [Adv. in
Appl. Probab.
30 (1998) 640--656] for intersection points of motion-invariant
Poisson line
processes in $\mathbb{R}^2$. Our proof is based on Hoeffding's
decomposition of
$U$-statistics which seems to be more efficient and adequate to
tackle the
higher-dimensional case than the ``method of moments'' used in [Adv.
in Appl.
Probab. 30 (1998) 640--656] to treat the case $d=2$. Moreover, we
extend our
central limit theorem in several directions. First we consider $k$-flat
processes induced by Poisson hyperplane processes in $\mathbb{R}^d$
for $0\le
k\le d-1$. Second we derive (asymptotic) confidence intervals for the
intensities of these $k$-flat processes and, third, we prove
multivariate
central limit theorems for the $d$-dimensional joint vectors of
numbers of
$k$-flats and their $k$-volumes, respectively, in an increasing
spherical
region.
http://front.math.ucdavis.edu/math.PR/0607120
---------------------------------------------------------------
4460. COMPLETE CORRECTED DIFFUSION APPROXIMATIONS FOR THE MAXIMUM OF
A RANDOM WALK
Jose Blanchet and Peter Glynn
Consider a random walk $(S_n:n\geq0)$ with drift $-\mu$ and $S_0=0$.
Assuming
that the increments have exponential moments, negative mean, and are
strongly
nonlattice, we provide a complete asymptotic expansion (in powers of $
\mu>0$)
that corrects the diffusion approximation of the all time maximum
$M=\max_{n\geq0}S_n$. Our results extend both the first-order
correction of
Siegmund [Adv. in Appl. Probab. 11 (1979) 701--719] and the full
asymptotic
expansion provided in the Gaussian case by Chang and Peres [Ann.
Probab. 25
(1997) 787--802]. We also show that the Cram\'{e}r--Lundberg constant
(as a
function of $\mu$) admits an analytic extension throughout a
neighborhood of
the origin in the complex plane $\mathbb{C}$. Finally, when the
increments of
the random walk have nonnegative mean $\mu$, we show that the Laplace
transform, $E_{\mu}\exp(-bR(\infty))$, of the limiting overshoot, $R
(\infty)$,
can be analytically extended throughout a disc centered at the origin in
$\mathbb{C\times C}$ (jointly for both $b$ and $\mu$). In addition,
when the
distribution of the increments is continuous and appropriately
symmetric, we
show that $E_{\mu}S_{\tau}$ [where $\tau$ is the first (strict)
ascending
ladder epoch] can be analytically extended to a disc centered at the
origin in
$\mathbb{C}$, generalizing the main result in [Ann. Probab. 25 (1997)
787--802]
and extending a related result of Chang [Ann. Appl. Probab. 2 (1992)
714--738].
http://front.math.ucdavis.edu/math.PR/0607121
---------------------------------------------------------------
4461. ERROR ESTIMATES FOR BINOMIAL APPROXIMATIONS OF GAME OPTIONS
Yuri Kifer
We justify and give error estimates for binomial approximations of game
(Israeli) options in the Black--Scholes market with Lipschitz
continuous path
dependent payoffs which are new also for usual American style
options. We show
also that rational (optimal) exercise times and hedging self-financing
portfolios of binomial approximations yield for game options in the
Black--Scholes market ``nearly'' rational exercise times and
``nearly'' hedging
self-financing portfolios with small average shortfalls and initial
capitals
close to fair prices of the options. The estimates rely on strong
invariance
principle type approximations via the Skorokhod embedding.
http://front.math.ucdavis.edu/math.PR/0607123
---------------------------------------------------------------
4462. BIASED RANDOM-TO-TOP SHUFFLING
Johan Jonasson
Recently Wilson [Ann. Appl. Probab. 14 (2004) 274--325] introduced an
important new technique for lower bounding the mixing time of a
Markov chain.
In this paper we extend Wilson's technique to find lower bounds of
the correct
order for card shuffling Markov chains where at each time step a
random card is
picked and put at the top of the deck. Two classes of such shuffles are
addressed, one where the probability that a given card is picked at a
given
time step depends on its identity, the so-called move-to-front
scheme, and one
where it depends on its position. For the move-to-front scheme, a
test function
that is a combination of several different eigenvectors of the
transition
matrix is used. A general method for finding and using such a test
function,
under a natural negative dependence condition, is introduced. It is
shown that
the correct order of the mixing time is given by the biased coupon
collector's
problem corresponding to the move-to-front scheme at hand. For the
second
class, a version of Wilson's technique for complex-valued
eigenvalues/eigenvectors is used. Such variants were presented in
[Random Walks
and Geometry (2004) 515--532] and [Electron. Comm. Probab. 8 (2003)
77--85].
Here we present another such variant which seems to be the most
natural one for
this particular class of problems. To find the eigenvalues for the
general case
of the second class of problems is difficult, so we restrict
attention to two
special cases. In the first case the card that is moved to the top is
picked
uniformly at random from the bottom $k=k(n)=o(n)$ cards, and we find
the lower
bound $(n^3/(4\pi^2k(k-1)))\log n$. Via a coupling, an upper bound
exceeding
this by only a factor 4 is found. This generalizes Wilson's
[Electron. Comm.
Probab. 8 (2003) 77--85] result on the Rudvalis shuffle and Goel's
[Ann. Appl.
Probab. 16 (2006) 30--55] result on top-to-bottom shuffles. In the
second case
the card moved to the top is, with probability 1/2, the bottom card
and with
probability 1/2, the card at position $n-k$. Here the lower bound is
again of
order $(n^3/k^2)\log n$, but in this case this does not seem to be
tight unless
$k=O(1)$. What the correct order of mixing is in this case is an open
question.
We show that when $k=n/2$, it is at least $\Theta(n^2)$.
http://front.math.ucdavis.edu/math.PR/0607124
---------------------------------------------------------------
4463. PERCOLATION IN A HIERARCHICAL RANDOM GRAPH
D.A. Dawson and L.G. Gorostiza
We study asymptotic percolation as $N\to \infty$ in an infinite
random graph
${\cal G}_N$ embedded in the hierarchical group of order $N$, with
connection
probabilities depending on an ultrametric distance between vertices. $
{\cal
G}_N$ is structured as a cascade of finite random subgraphs of
(approximate)
Erd\"os-Renyi type. We give a criterion for percolation, and show that
percolation takes place along giant components of giant components at
the
previous level in the cascade of subgraphs for all consecutive
hierarchical
distances. The proof involves a hierarchy of random graphs with
vertices having
an internal structure and random connection probabilities.
http://front.math.ucdavis.edu/math.PR/0607131
---------------------------------------------------------------
4464. ANNEALING DIFFUSIONS IN A SLOWLY GROWING POTENTIAL
Pierre-Andr\'{e} Zitt (MODAL'X)
We consider a continuous analogue of the simulated annealing
algorithm in
$R^d$. We prove a convergence result, under hypotheses weaker than
the usual
ones. In particular, we cover cases where the gradient of the
potential goes to
zero at infinity. The proof follows an idea of L. Miclo, but we
replace the
Poincar\'{e} and log-Sobolev inequalities (which do not hold in our
setting) by
weak Poincar\'{e} inequalities. We estimate the latter with measure-
capacity
criteria. We show that, despite the absence of a spectral gap, the
convergence
still holds for the "classical" schedule t = c/ ln(t), if c is bigger
than a
constant related to the potential.
http://front.math.ucdavis.edu/math.PR/0607147
---------------------------------------------------------------
4465. THE BEAD MODEL & LIMIT BEHAVIORS OF DIMER MODELS
Cedric Boutillier
In this paper, we study the bead model: beads are threaded on a set
of wires
on the plane represented by parallel straight lines. We add the
constraint that
between two consecutive beads on a wire, there must be exactly one
bead on each
neighboring wire. We construct a one-parameter family of Gibbs
measures on the
bead configurations that are uniform in a certain sense. When endowed
with one
of these measures, this model is shown to be a determinantal point
process,
whose marginal on each wire is the sine process (given by eigenvalues
of large
hermitian random matrices). We prove then that this process appears
as a limit
of any dimer model on a planar bipartite graph when some weights
degenerate.
http://front.math.ucdavis.edu/math.PR/0607162
---------------------------------------------------------------
4466. MULTISCALE ANALYSIS OF EXIT DISTRIBUTIONS FOR RANDOM WALKS IN
RANDOM ENVIRONMENTS
Erwin Bolthausen and Ofer Zeitouni
We present a multiscale analysis for the exit measures from large
balls in
Z^d, d\geq 3, of random walks in certain i.i.d. random environments
which are
small perturbations of the fixed environment corresponding to simple
random
walk. Our main assumption is an isotropy assumption on the law of the
environment, introduced by Bricmont and Kupianien. The analysis is
based on
propagating estimates on the variational distance between the exit
measure and
that of simple random walk, in addition to estimates on the variational
distance between smoothed versions of these quantities.
http://front.math.ucdavis.edu/math.PR/0607192
---------------------------------------------------------------
4467. DUALITIES FOR MULTI-STATE PROBABILISTIC CELLULAR AUTOMATA
F.J. Lopez and G. Sanz and and M. Sobottka
The present work treats dualities for probabilistic cellular automata
(PCA).
A general result of duality is presented and it is used to study two
models of
PCA: a multi-opinion noisy general voter model; and a multi-state
attractive
biased model.
http://front.math.ucdavis.edu/math.PR/0607206
---------------------------------------------------------------
4468. ON TWIN PRIMES ASSOCIATED WITH THE HAWKINS RANDOM SIEVE
H. M. Bui and J. P. Keating
We establish an asymptotic formula for the number of $k$-difference twin
primes associated with the Hawkins random sieve, which is a
probabilistic model
of the Eratosthenes sieve. The formula for $k = 1$ was obtained by
Wunderlich
[Acta Arith. \textbf{26} (1974), 59 - 81]. We here extend this to $k
\geq 2$
and generalize it to all $l$-tuples of Hawkins primes.
http://front.math.ucdavis.edu/math.NT/0607196
---------------------------------------------------------------
4469. TIME CONSISTENT DYNAMIC RISK PROCESSES, CADLAG MODIFICATION
Jocelyne Bion-Nadal
Working in a continuous time setting, we extend to the general case of
dynamic risk measures continuous from above the characterization of time
consistency in terms of ``cocycle condition'' of the minimal penalty
function.
We prove also the supermartingale property for general time
consistent dynamic
risk measures. When the time consistent dynamic risk measure
(continuous from
above) is normalized and non degenerate, we prove, under a mild
condition, that
the dynamic risk process of any financial instrument has a cadlag
modification.
This condition is always satisfied in case of continuity from below.
http://front.math.ucdavis.edu/math.PR/0607212
---------------------------------------------------------------
4470. SHAPE OF TERRITORIES IN SOME COMPETING GROWTH MODELS
Jean-Baptiste Gou\'{e}r\'{e} (MAPMO)
We study two competing growth models. Each of these models describes the
spread of a finite number of infections on a graph. Each infection
evolves like
an (oriented or unoriented) first passage percolation process except
that once
a vertex is infected by type $i$ infection, it remains of type $i$
forever. We
give results about the shape of the area ultimately infected by the
different
infections.
http://front.math.ucdavis.edu/math.PR/0607226
---------------------------------------------------------------
4471. WEIGHTED UNIFORM CONSISTENCY OF KERNEL DENSITY ESTIMATORS WITH
GENERAL BANDWIDTH SEQUENCES
Julia Dony and Uwe Einmahl
We are interested in the rate of consistency of kernel density
estimators
with respect to the weighted sup-norm determined by some unbounded
weight
function. This problem has been considered by Gine, Koltchinskii and
Zinn
(2004) for a deterministic bandwidth sequence. We provide "uniform in h"
versions of some of their results, allowing us to determine the
corresponding
rates of consistency for kernel density estimators where the bandwidth
sequences may depend on the data and/or the location.
http://front.math.ucdavis.edu/math.ST/0607232
---------------------------------------------------------------
4472. STOCHASTIC PARAMETERIZATION FOR LARGE EDDY SIMULATION OF
GEOPHYSICAL FLOWS
Jinqiao Duan and Balasubramanya T. Nadiga
Recently, stochastic, as opposed to deterministic, parameterizations are
being investigated to model the effects of unresolved subgrid scales
(SGS) in
large eddy simulations (LES) of geophysical flows. We analyse such a
stochastic
approach in the barotropic vorticity equation to show that (i) if the
stochastic parameterization approximates the actual SGS stresses,
then the
solution of the stochastic LES approximates the "true" solution at
appropriate
scale sizes; and that (ii) when the filter scale size approaches
zero, the
solution of the stochastic LES approaches the true solution.
http://front.math.ucdavis.edu/math.AP/0607214
---------------------------------------------------------------
4473. THE AXIOMATIC MELTING POT. TEACHING PROBABILITY THEORY IN
PRAGUE DURING THE 1930'S
Stepanka Bilova and Laurent Mazliak and Pavel Sisma
In this paper, we are interested in the teaching of probability
theory in
Prague and Czechoslovakia, in particular during the 1930's. We focus
specially
on a textbook, published in Prague by Karel Rychlik in 1938, which uses
Kolmogorov's axiomatization, a very exceptional fact before World War
II.
http://front.math.ucdavis.edu/math.HO/0607217
---------------------------------------------------------------
4474. ON THE SPECTRAL GAP FOR CONVEX DOMAINS
Burgess Davis and Majid Hosseini
We prove the following for a bounded convex planar domain that is
symmetric
with respect to both coordinate axes. Consider a centered rectangle
with sides
parallel to the axes that strictly contains the domain. If the domain
is not a
certain kind of rectangle, the spectral gap of the domain is larger
than the
spectral gap of the rectangle. We also provide explicit lower bounds
for the
differnce between the gaps.
http://front.math.ucdavis.edu/math.SP/0607219
---------------------------------------------------------------
4475. ON THE SMALL MAXIMAL FLOWS IN FIRST PASSAGE PERCOLATION
Marie Th\'eret
We consider the standard first passage percolation on $\mathbb{Z}^{d}
$: with
each edge of the lattice we associate a random capacity. We are
interested in
the maximal flow through a cylinder in this graph. Under some
assumptions
Kesten proved in 1987 a law of large number for the rescaled flow. We
give here
a partial answer to one of his questions: the large deviations far
away below
its typical value are of surface order.
http://front.math.ucdavis.edu/math.PR/0607252
---------------------------------------------------------------
4476. UPPER LARGE DEVIATIONS FOR THE MAXIMAL FLOW IN FIRST PASSAGE
PERCOLATION
Marie Th\'eret
We consider the standard first passage percolation in $\mathbb{Z}^{d}
$ for
$d\geq 2$ and we denote by $\phi_{n^{d-1},h(n)}$ the maximal flow
through the
cylinder $]0,n]^{d-1} \times ]0,h(n)]$ from its bottom to its top.
Kesten
proved a law of large numbers for the maximal flow in dimension
three: under
some assumptions, $\phi_{n^{d-1},h(n)} / n^{d-1}$ converges towards a
constant
$\nu$. We look now at the probability that $\phi_{n^{d-1},h(n)} / n^
{d-1}$ is
greater than $\nu + \epsilon$ for some $\epsilon >0$, and we show
under some
assumptions that this probability decays exponentially fast with the
volume of
the cylinder. Moreover, we prove a large deviations principle for the
sequence
$(\phi_{n^{d-1},h(n)} / n^{d-1}, n\in \mathbb{N})$.
http://front.math.ucdavis.edu/math.PR/0607253
---------------------------------------------------------------
4477. MOMENT ESTIMATES FOR L\'{E}VY PROCESSES
Harald Luschgy and Gilles Pag\`{e}s (PMA)
For real L\'{e}vy processes $(X\_t)\_{t \geq 0}$ having no Brownian
component
with Blumenthal-Getoor index $\beta$, the estimate $\E \sup\_{s \leq
t} | X\_s
- a\_p s |^p \leq C\_p t$ for every $t \in [0,1]$ and suitable $a\_p
\in \R$
has been established by Millar \cite{MILL} for $\beta < p \leq 2$
provided
$X\_1 \in L^p$. We derive extensions of these estimates to the cases
$p > 2$
and $p \leq\beta$.
http://front.math.ucdavis.edu/math.PR/0607282
---------------------------------------------------------------
4478. MONOTONICITY AND NON-MONOTONICITY OF DOMAINS OF STOCHASTIC
INTEGRAL OPERATORS
Ken-iti Sato
A L\'evy process on $R^d$ with distribution $\mu$ at time 1 is
denoted by
$X^{(\mu)}=\{X_t^{(\mu)}\}$. If the improper stochastic integral
$\int_0^{\infty-} f(s)dX_s^{(\mu)}$ of $f$ with respect to $X^{(\mu)}
$ is
definable, its distribution is denoted by $\Phi_f(\mu)$. The class of
all
infinitely divisible distributions $\mu$ on $R^d$ such that $\Phi_f
(\mu)$ is
definable is denoted by $D(\Phi_f)$. The class $D(\Phi_f)$, its two
extensions
$D_c(\Phi_f)$ and $D_e(\Phi_f)$ (compensated and essential), and its
restriction $D^0(\Phi_f)$ (absolutely definable) are studied. It is
shown that
$D_e(\Phi_f)$ is monotonic with respect to $f$, which means that $|
f_2|\leq
|f_1|$ implies $D_e(\Phi_{f_1})\subset D_e(\Phi_{f_2})$. Further, $D^0
(\Phi_f)$
is monotonic with respect to $f$ but neither $D(\Phi_f)$ nor $D_c
(\Phi_f)$ is
monotonic with respect to $f$. Furthermore, there exist $\mu$, $f_1$,
and $f_2$
such that $0\leq f_2\leq f_1$, $\mu\in D(\Phi_{f_1})$, and $\mu\not\in
D(\Phi_{f_2})$. An explicit example for this is related to some
properties of a
class of martingale L\'evy processes.
http://front.math.ucdavis.edu/math.PR/0607288
---------------------------------------------------------------
4479. EXAMPLES OF CONDITION (T) FOR DIFFUSIONS IN A RANDOM ENVIRONMENT
Tom Schmitz
With the help of the methods developed in our previous article
[Schmitz, to
appear in "Annales de l'I.H.P. Prob. & Stat.], we highlight condition
(T) as a
source of new examples of 'ballistic' diffusions in a random
environment when
d>1 ('ballistic' means that a strong law of large numbers with non-
vanishing
limiting velocity holds). In particular we are able to treat the case of
non-constant diffusion coefficients, a feature that causes problems.
Further we
recover the ballistic character of two important classes of
diffusions in a
random environment by simply checking condition (T). This not only
points out
to the broad range of examples where condition (T) can be checked,
but also
fortifies our belief that condition (T) is a natural contender for the
characterisation of ballistic diffusions in a random environment when
d>1.
http://front.math.ucdavis.edu/math.PR/0607293
---------------------------------------------------------------
4480. THE SPECTRAL DIMENSION OF GENERIC TREES
Bergfinnur Durhuus and Thordur Jonsson and John F. Wheater
We define generic ensembles of infinite trees. These are limits as
$N\to\infty$ of ensembles of finite trees of fixed size $N$, defined
in terms
of a set of branching weights. Among these ensembles are those
supported on
trees with vertices of a uniformly bounded order. The associated
probability
measures are supported on trees with a single spine and Hausdorff
dimension
$d_h =2$. Our main result is that their spectral dimension is $d_s=4/3
$, and
that the critical exponent of the mass, defined as the exponential
decay rate
of the two-point function along the spine, is 1/3.
http://front.math.ucdavis.edu/math-ph/0607020
---------------------------------------------------------------
4481. PROBABILITY DENSITY FOR A HYPERBOLIC SPDE WITH TIME DEPENDENT
COEFFICIENTS
Marta Sanz-Sol\'e and Iv\'an Torrecilla-Tarantino
We prove the existence and smoothness of density for the solution of a
hyperbolic SPDE with free term coefficients depending on time, under
hypoelliptic non degeneracy conditions. The result extends those
proved in
Cattiaux and Mesnager, PTRF 2002, to an infinite dimensional setting.
http://front.math.ucdavis.edu/math.PR/0607310
---------------------------------------------------------------
4482. NONINTERSECTING BROWNIAN EXCURSIONS
Craig A. Tracy and Harold Widom
We consider the process of n Brownian excursions conditioned to be
nonintersecting. We show the distribution functions for the top curve
and the
bottom curve are equal to Fredholm determinants whose kernel we give
explicity.
In the simplest case, these determinants are expressible in terms of
Painlev\'e
V functions. We prove that as n tends to infinity the distributional
limit of
the bottom curve is the Bessel process with parameter 1/2. We apply
these
results to study the expected area under the bottom and top curves.
http://front.math.ucdavis.edu/math.PR/0607321
---------------------------------------------------------------
4483. THE AVERAGE SIZE OF GIANT COMPONENTS BETWEEN THE DOUBLE-JUMP
Vlady Ravelomanana (LIPN) and the Projet PAI Amadeus Collaboration
We study the sizes of connected components according to their
excesses during
a random graph process built with $n$ vertices. The considered model
is the
continuous one defined in Janson 2000. An ${\ell}$-component is a
connected
component with ${\ell}$ edges more than vertices. $\ell$ is also
called the
\textit{excess} of such component. As our main result, we show that
when $\ell$
and ${n \over \ell}$ are both large, the expected number of vertices
that ever
belong to an $\ell$-component is about ${12}^{1/3} {\ell}^{1/3} n^
{2/3}$. We
also obtain limit theorems for the number of creations of $\ell$-
components.
http://front.math.ucdavis.edu/cs.DM/0607057
---------------------------------------------------------------
4484. CREATION AND GROWTH OF COMPONENTS IN A RANDOM HYPERGRAPH PROCESS
Vlady Ravelomanana (LIPN) and Alphonse Laza Rijamame (D.M.I)
Denote by an $\ell$-component a connected $b$-uniform hypergraph with
$k$
edges and $k(b-1) - \ell$ vertices. We prove that the expected number of
creations of $\ell$-component during a random hypergraph process
tends to 1 as
$\ell$ and $b$ tend to $\infty$ with the total number of vertices $n$
such that
$\ell = o(\sqrt[3]{\frac{n}{b}})$. Under the same conditions, we also
show that
the expected number of vertices that ever belong to an $\ell$-
component is
approximately $12^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}$. As an immediate
consequence, it follows that with high probability the largest $\ell$-
component
during the process is of size $O((b-1)^{1/3} \ell^{1/3} n^{2/3})$.
Our results
give insight about the size of giant components inside the phase
transition of
random hypergraphs.
http://front.math.ucdavis.edu/cs.DM/0607059
---------------------------------------------------------------
4485. BETA ENSEMBLES, STOCHASTIC AIRY SPECTRUM, AND A DIFFUSION
Jose Ramirez and Brian Rider and Balint Virag
Building on earlier work of A. Edelman, I. Dumitriu, and B. Sutton we
prove
that the largest eigenvalues of the general beta-ensemble of Random
Matrix
Theory, properly centered and scaled, converge in distribution to the
law of
the low lying eigenvalues of a random operator of Schroedinger type.
The latter
is $ -\frac{d^2}{dx^2} + x + \frac{2}{\sqrt{\beta}} b^{\prime}(x)$
acting on
$L^2(R_+)$ with Dirichlet boundary condition at $x=0$. Here $b^
{\prime}(x)$
denotes a standard White Noise and the $\beta > 0$ is that of the
original
ensemble. Based on this convergence, we provide a new
characterization of the
Tracy-Widom type laws (for all $\beta$) in terms of the explosion/non-
explosion
a one-dimensional diffusion.
http://front.math.ucdavis.edu/math.PR/0607331
---------------------------------------------------------------
4486. NUMBER VARIANCE FROM A PROBABILISTIC PERSPECTIVE: INFINITE
SYSTEMS OF INDEPENDENT BROWNIAN MOTIONS AND SYMMETRIC ALPHA-STABLE
PROCESSES
Ben Hambly and Liza Jones
Some probabilistic aspects of the number variance statistic are
investigated.
Infinite systems of independent Brownian motions and symmetric alpha-
stable
processes are used to construct new examples of processes which
exhibit both
divergent and saturating number variance behaviour. We derive a general
expression for the number variance for the spatial particle
configurations
arising from these systems and this enables us to deduce various
limiting
distribution results for the fluctuations of the associated counting
functions.
In particular, knowledge of the number variance allows us to
introduce and
characterize a novel family of centered, long memory Gaussian
processes. We
obtain fractional Brownian motion as a weak limit of these constructed
processes.
http://front.math.ucdavis.edu/math.PR/0607345
---------------------------------------------------------------
4487. ON THE (AB)USE OF STATISTICS IN THE LEGAL CASE AGAINST THE
NURSE LUCIA DE B
Ronald Meester and Marieke Collins and Richard Gill and Michiel
van Lambalgen
We discuss the statistics involved in the legal case of the nurse
Lucia de B.
in The Netherlands, 2003-2004. Lucia de B. witnessed an unusually
high number
of incidents during her shifts, and the question arose as to whether
this could
be attributed to chance. We discuss and criticise the statistical
analysis of
Henk Elffers, a statistician who was asked by the prosecutor to write a
statistical report on the issue. We discuss several other
possibilities for
statistical analysis. Our main point is that several statistical
models exist,
leading to very different predictions, or perhaps different answers to
different questions. There is no such thing as a `best' statistical
analysis.
http://front.math.ucdavis.edu/math.ST/0607340
---------------------------------------------------------------
4488. NON PARAMETRIC THRESHOLD ESTIMATION FOR MODELS WITH STOCHASTIC
DIFFUSION COEFFICIENTS AND JUMPS
Cecilia Mancini
We consider a stochastic process driven by a diffusion and jumps. We
devise a
technique, which is based on a discrete record of observations, for
identifying
the times when jumps larger than a suitably defined threshold
occurred. The
technique allows also jump size estimation. We prove the consistency
of a
nonparametric estimator of the integrated infinitesimal variance of
the process
continuous part when the jump component with infinite activity is
Levy. Central
limit results are proved in the case where the jump component has finite
activity. Some simulations illustrate the reliability of the
methodology in
finite samples.
http://front.math.ucdavis.edu/math.ST/0607378
---------------------------------------------------------------
4489. PREDICTABILITY OF THE BURGERS DYNAMICS UNDER MODEL UNCERTAINTY
Dirk Bl\"omker and Jinqiao Duan
Complex systems may be subject to various uncertainties. A great
effort has
been concentrated on predicting the dynamics under uncertainty in
initial
conditions. In the present work, we consider the well-known Burgers
equation
with random boundary forcing or with random body forcing. Our goal is to
attempt to understand the stochastic Burgers dynamics by predicting or
estimating the solution processes in various diagnostic metrics, such
as mean
length scale, correlation function and mean energy. First, for the
linearized
model, we observe that the important statistical quantities like mean
energy or
correlation functions are the same for the two types of random
forcing, even
though the solutions behave very differently. Second, for the full
nonlinear
model, we estimate the mean energy for various types of random body
forcing,
highlighting the different impact on the overall dynamics of space-
time white
noises, trace class white-in-time and colored-in-space noises, point
noises,
additive noises or multiplicative noises.
http://front.math.ucdavis.edu/math.CA/0607357
---------------------------------------------------------------
4490. INVARIANT MANIFOLD REDUCTION FOR STOCHASTIC DYNAMICAL SYSTEMS
Aijun Du and Jinqiao Duan
Invariant manifolds facilitate the understanding of nonlinear stochastic
dynamics. When an invariant manifold is represented approximately by
a graph
for example, the whole stochastic dynamical system may be reduced or
restricted
to this manifold. This reduced system may provide valuable dynamical
information for the original system. The authors have derived an
invariant
manifold reduction or restriction principle for systems of
Stratonovich or Ito
stochastic differential equations.
Two concepts of invariance are considered for invariant manifolds.
The first invariance concept is in the framework of cocycles -- an
invariant
manifold being a random set. The dynamical reduction is achieved by
investigating random center manifolds.
The second invariance concept is in the sense of almost sure -- an
invariant
manifold being a deterministic set which is not necessarily
attracting. The
restriction of the original stochastic system on this deterministic
local
invariant manifold is still a stochastic system but with reduced
dimension.
http://front.math.ucdavis.edu/math.DS/0607366
---------------------------------------------------------------
4491. COUNTING FACES OF RANDOMLY-PROJECTED POLYTOPES WHEN THE
PROJECTION RADICALLY LOWERS DIMENSION
David L. Donoho and Jared Tanner
This paper develops asymptotic methods to count faces of random
high-dimensional polytopes. Beyond its intrinsic interest, our
conclusions have
surprising implications - in statistics, probability, information
theory, and
signal processing - with potential impacts in practical subjects like
medical
imaging and digital communications. Three such implications concern:
convex
hulls of Gaussian point clouds, signal recovery from random
projections, and
how many gross errors can be efficiently corrected from Gaussian error
correcting codes.
http://front.math.ucdavis.edu/math.MG/0607364
---------------------------------------------------------------
4492. THE PROBABILITY OF CHOOSING PRIMITIVE SETS
Sergi Elizalde and Kevin Woods
We generalize a theorem of Nymann that the density of points in Z^d
that are
visible from the origin is 1/zeta(d), where zeta(a) is the Riemann zeta
function 1/1^a + 1/2^a + 1/3^a + ...
A subset S of Z^d is called primitive if it is a Z-basis for the
lattice
composed of the integer points in the R-span of S, or, equivalently,
if S can
be completed to a Z-basis of Z^d. We prove that if m points in Z^d
are chosen
uniformly and independently at random from a large box, then as the
size of the
box goes to infinity, the probability that the points form a
primitive set
approaches 1/[\zeta(d)\zeta(d-1)...zeta(d-m+1)].
http://front.math.ucdavis.edu/math.NT/0607390
---------------------------------------------------------------
4493. COALESCENT TREE BASED FUNCTIONAL REPRESENTATIONS FOR SOME
FEYNMAN-KAC PARTICLE MODELS
Pierre Del Moral (JAD) and Fr\'{e}d\'{e}ric Patras (JAD) and
Sylvain Rubenthaler (JAD)
We design a theoretic tree-based functional representation of a class of
Feynman-Kac particle distributions, including an extension of the
Wick product
formula to interacting particle systems. These weak expansions rely
on an
original combinatorial, and permutation group analysis of a special
class of
forests. They provide refined non asymptotic propagation of chaos type
properties, as well as sharp $\LL\_p$-mean error bounds, and laws of
large
numbers for $U$-statistics. Applications to particle interpretations
of the top
eigenvalues, and the ground states of Schr\"{o}dinger semigroups are
also
discussed.
http://front.math.ucdavis.edu/math.PR/0607453
---------------------------------------------------------------
4494. ASYMPTOTIC ENTROPY AND GREEN SPEED FOR RANDOM WALKS ON GROUPS
S\'{e}bastien Blach\`{e}re (LATP) and Peter Ha\"{i}ssinsky (LATP)
and Pierre Mathieu (LATP)
We study asymptotic properties of the Green metric associated to
random walks
on discrete transient groups. We prove that the rate of escape of the
random
walk computed in the Green metric equals its asymptotic entropy. Two
proofs are
given. One relies on integral representations of both quantities with
the
extended Martin kernel. The other proof (valid only when the volume
growth of
the group is superpolynomial) relies on a version of the so called
fundamental
inequality (relating the rate of escape, the entropy and the
logarithmic volume
growth) extended to random walk with unbounded support.
http://front.math.ucdavis.edu/math.PR/0607467
---------------------------------------------------------------
4495. THE MOMENT PROBLEM WITH BOUNDED DENSITY
Jean B. Lasserre
Let $\mu$ be a given Borel measure on $\K\subseteq\R^n$ and let
$y=(y_\alpha)$, $\alpha\in\N^n$, be a given sequence. We provide several
conditions linking $y$ and the moment sequence $z=(z_\alpha)$ of $\mu
$, for $y$
to be the moment sequence of a Borel measure $\nu$ on $\K$ which is
absolutely
continuous with respect to $\mu$ and such that its density is in
$L_\infty(\K,\mu)$. The conditions are necessary and sufficient if $\K
$ is a
compact basic semi-algebraic set, and sufficient if $\K\equiv\R^n$.
Moreover,
arbitrary finitely many of these conditions can be checked by solving
either a
semidefinite program or a linear program with a single variable
http://front.math.ucdavis.edu/math.FA/0607463
---------------------------------------------------------------
4496. ZEROS OF RANDOM ANALYTIC FUNCTIONS
Manjunath Krishnapur
The dominant theme of this thesis is that random matrix valued analytic
functions, generalizing both random matrices and random analytic
functions, for
many purposes can (and perhaps should) be effectively studied in that
level of
generality. We study zeros of random analytic functions in one complex
variable. It is known that there is a one parameter family of
Gaussian analytic
functions with zero sets that are stationary in each of the three
symmetric
spaces, namely the plane, the sphere and the unit disk, under the
corresponding
group of isometries. We show a way to generate non Gaussian random
analytic
functions whose zero sets are also stationary in the same domains.
There are
particular cases where the exact distribution of the zero set turns
out to
belong to an important class of point processes known as
determinantal point
processes. Apart from questions regarding the exact distribution of
zero sets,
we also study certain asymptotic properties. We show asymptotic
normality for
smooth statistics applied to zeros of these random analytic
functions. Lastly,
we present some results on certain large deviation problems for the
zeros of
the planar and hyperbolic Gaussian analytic functions.
http://front.math.ucdavis.edu/math.PR/0607504
---------------------------------------------------------------
4497. NONEQUILIBRIUM DENSITY FLUCTUATIONS FOR THE ZERO RANGE PROCESS
WITH COLOUR
Hanna Jankowski
We examine the fluctuations of the empirical density measure for the
colour
version of the symmetric nearest neighbour zero range particle
systems in
dimension one. We show that the weak limit of these fluctuations is the
solution of a system of coupled generalized Ornstein-Uhlenbeck
processes. We
also discuss how this result may be used to prove a central limit
theorem for
the tagged particle on the level of finite dimensional distributions,
and
identify the limiting variance. This is the central limit theorem
associated to
propagation of chaos for this interacting particle system.
http://front.math.ucdavis.edu/math.PR/0607505
---------------------------------------------------------------
4498. IN-DEGREE AND PAGERANK OF WEB PAGES: WHY DO THEY FOLLOW SIMILAR
POWER LAWS?
N. Litvak and W.R.W. Scheinhardt and Y. Volkovich
The PageRank is a popularity measure designed by Google to rank Web
pages.
Experiments confirm that the PageRank obeys a `power law' with the same
exponent as the In-Degree. This paper presents a novel mathematical
model that
explains this phenomenon. The relation between the PageRank and In-
Degree is
modelled through a stochastic equation, which is inspired by the
original
definition of the PageRank, and is analogous to the well-known
distributional
identity for the busy period in the M/G/1 queue. Further, we employ
the theory
of regular variation and Tauberian theorems to analytically prove
that the tail
behavior of the PageRank and the In-Degree differ only by a
multiplicative
factor, for which we derive a closed-form expression. Our analytical
results
are in good agreement with experimental data.
http://front.math.ucdavis.edu/math.PR/0607507
---------------------------------------------------------------
4499. LARGE DEVIATION PRINCIPLES FOR EMPIRICAL MEASURES OF COLOURED
RANDOM GRAPHS
Kwabena Doku-Amponsah and Peter Morters
For any finite coloured graph we define the empirical neighbourhood
measure,
which counts the number of vertices of a given colour connected to a
given
number of vertices of each colour, and the empirical pair measure,
which counts
the number of edges connecting each pair of colours. For a class of
sparse
coloured random graphs, we prove large deviation principles for these
empirical
measures in the weak topology. The rate functions governing our large
deviation
principles can be expressed explicitly in terms of relative
entropies. We
derive a large deviation principle for the degree distribution of
Erdos-Renyi
graphs near criticality.
http://front.math.ucdavis.edu/math.PR/0607545
---------------------------------------------------------------
4500. FLUCTUATIONS OF THE FRONT IN A ONE DIMENSIONAL MODEL OF X+Y-->2X
Francis Comets and Jeremy Quastel and Alejandro Ramirez
We consider a model of the reaction $X+Y\to 2X$ on the integer
lattice in
which $Y$ particles do not move while $X$ particles move as independent
continuous time, simple symmetric random walks. $Y$ particles are
transformed
instantaneously to $X$ particles upon contact. We start with a fixed
number
$a\ge 1$ of $Y$ particles at each site to the right of the origin,
and define a
class of configurations of the $X$ particles to the left of the
origin having a
finite $l^1$ norm with a specified exponential weight. Starting from any
configuration of $X$ particles to the left of the origin within such
a class,
we prove a central limit theorem for the position of the rightmost
visited site
of the $X$ particles.
http://front.math.ucdavis.edu/math.PR/0607549
---------------------------------------------------------------
4501. ON CHARACTERISATION OF MARKOV PROCESSES VIA MARTINGALE PROBLEMS
Abhay G Bhatt and Rajeeva L Karandikar and B V Rao
It is well-known that well-posedness of a martingale problem in the
class of
continuous (or r.c.l.l.) solutions enables one to construct the
associated
transition probability functions. We extend this result to the case
when the
martingale problem is well-posed in the class of solutions which are
continuous
in probability. This extension is used to improve on a criterion for a
probability measure to be invariant for the semigroup associated with
the
Markov process. We also give examples of martingale problems that are
well-posed in the class of solutions which are continuous in
probability but
for which no r.c.l.l. solution exists.
http://front.math.ucdavis.edu/math.PR/0607613
---------------------------------------------------------------
4502. COMPUTING STRATEGIES FOR ACHIEVING ACCEPTABILITY
Soumik Pal
We consider a trader who wants to direct his portfolio towards a set of
acceptable wealths given by a convex risk measure. We propose a black-
box
algorithm, whose inputs are the joint law of stock prices and the
convex risk
measure, and whose outputs are the numerical values of initial capital
requirement and the functional form of a trading strategy to achieve
acceptability. We also prove optimality of the obtained capital.
http://front.math.ucdavis.edu/math.PR/0607617
---------------------------------------------------------------
4503. CENTRAL LIMIT THEOREM FOR RANDOM PARTITIONS UNDER THE
PLANCHEREL MEASURE
L.V. Bogachev and Z.G. Su
In this work, we obtain the central limit theorem for fluctuations of
Young
diagrams around their limit shape in the bulk of the "spectrum" of
partitions
of a large integer n (under the Plancherel measure). More
specifically, we show
that, under the suitable normalization (growing as the square root of
log n),
the corresponding random process converges, in the sense of finite
dimensional
distributions, to a Gaussian process with independent values. The
proof uses
heavily the determinantal structure of the correlation functions and
is based
on the application of the Costin-Lebowitz-Soshnikov central limit
theorem. At
the spectrum edges, the fluctuation asymptotics is expressed in terms
of the
largest members of the Airy ensemble; in particular, at the upper
edge the
limit distribution appears to be discrete (without any
normalization). These
results admit an elegant symmetric reformulation under the rotation
of Young
diagrams by 45 degrees, where the normalization no longer depends on the
location of the spectrum point. We also discuss the link of our
central limit
theorem with an earlier result by S.V. Kerov on the convergence to a
generalized Gaussian process.
http://front.math.ucdavis.edu/math.PR/0607635
---------------------------------------------------------------
4504. FREQUENT POINTS FOR RANDOM WALKS IN TWO DIMENSIONS
Richard F. Bass and Jay Rosen
For a symmetric random walk in $Z^2$ which does not necessarily have
bounded
jumps we study those points which are visited an unusually large
number of
times. We prove the analogue of the Erd\H{o}s-Taylor conjecture and
obtain the
asymptotics for the number of visits to the most visited site. We
also obtain
the asymptotics for the number of points which are visited very
frequently by
time $n$. Among the tools we use are Harnack inequalities and Green's
function
estimates for random walks with unbounded jumps; some of these are of
independent interest.
http://front.math.ucdavis.edu/math.PR/0607636
---------------------------------------------------------------
4505. PARAMETRIC ESTIMATION FOR THE STANDARD AND GEOMETRIC TELEGRAPH
PROCESS OBSERVED AT DISCRETE TIMES
Alessandro De Gregorio and Stefano M. Iacus
The telegraph process $X(t)$, $t>0$, (Goldstein, 1951) and the geometric
telegraph process $S(t) = s_0 \exp\{(\mu -\frac12\sigma^2)t + \sigma X
(t)\}$
with $\mu$ a known constant and $\sigma>0$ a parameter are supposed
to be
observed at $n+1$ equidistant time points $t_i=i\Delta_n,i=0,1,..., n
$. For
both models $\lambda$, the underlying rate of the Poisson process, is a
parameter to be estimated. In the geometric case, also $\sigma>0$ has
to be
estimated. We propose different estimators of the parameters and we
investigate
their performance under the high frequency asymptotics, i.e. $
\Delta_n \to 0$,
$n\Delta = T<\infty$ as $n \to \infty$, with $T>0$ fixed. The process
$X(t)$ in
non markovian, non stationary and not ergodic thus we use approximation
arguments to derive estimators. Given the complexity of the equations
involved
only estimators on the first model can be studied analytically.
Therefore, we
run an extensive Monte Carlo analysis to study the performance of the
proposed
estimators also for small sample size $n$.
http://front.math.ucdavis.edu/math.ST/0607633
---------------------------------------------------------------
4506. A PERCOLATING HARD SPHERE MODEL
Codina Cotar and Alexander E. Holroyd and David Revelle
Given a homogeneous Poisson point process in R^d, Haggstrom and
Meester asked
whether it is possible to place spheres (of differing radii) centred
at the
points, in a translation-invariant way, so that the spheres do not
overlap but
there is an unbounded component of touching spheres. We prove that
the answer
is yes in sufficiently high dimension.
http://front.math.ucdavis.edu/math.PR/0607645
---------------------------------------------------------------
4507. ON THE LARGE SCALE BEHAVIOR OF SUPER-BROWNIAN MOTION IN THREE
DIMENSIONS WITH A SINGLE POINT SOURCE
Klaus Fleischmann and Carl Mueller and and Pascal Vogt
In a recent work, Fleischmann and Mueller (2004) showed the existence
of a
super-Brownian motion in R^d, d=2,3, with extra birth at the origin.
Their
construction made use of an analytical approach based on the fundamental
solution of the heat equation with a one point potential worked out by
Albeverio et al. (1995). The present note addresses two properties of
this
measure-valued process in the three-dimensional case, namely the
scaling of the
process and the large scale behavior of its mean.
http://front.math.ucdavis.edu/math.PR/0607667
---------------------------------------------------------------
4508. $L^P$ MODULI OF CONTINUITY OF GAUSSIAN PROCESSES AND LOCAL
TIMES OF SYMMETRIC L\'EVY PROCESSES
Michael B. Marcus and Jay Rosen
Let $X=\{X(t), t\in R_+\}$ be a real valued symmetric L\'evy process
with
continuous local times $\{L^x_t,(t,x)\in R_+\times R\}$ and
characteristic
function $E e^{i\lambda X(t)} = e^{-t\psi(\lambda)} $. Let \sigma^2_0
(x-y) =
(4/\pi)\int_0^\infty \sin^2(\lambda (x-y)/2) / \psi(\lambda) d
\lambda . If
$\sigma^2_0(h)$ is concave, and satisfies some addtional very weak
regularity
conditions, then for any $ p\ge 1$, and all $t\in R_+$ \[ \lim_{h
\downarrow 0}
\int_{a}^{b} \bigg|{L^{x+h}_{t} -L^{x}_{t}\over\sigma_0(h)}\bigg|^p dx
=2^pE|\eta|^p \int_a^b |L^{x}_{t}|^{p/2} dx \] for all $a,b $ in the
extended
real line almost surely, and also in $L^m$, $m\ge 1$. (Here $\eta$ is
a normal
random variable with mean zero and variance one.) This result is
obtained via
the Eisenbaum Isomorphism Theorem and depends on the related result for
Gaussian processes with stationary increments, $\{G(x),x\in R^1\}$,
for which
$E(G(x)-G(y))^2=\sigma_0^2(x-y)$; \[ \lim_{h\to 0}
\int_a^b\bigg|\frac{G(x+h)-G(x)}{\sigma_0(h)}\bigg|^p dx =E|\eta |^p
(b-a) \]
for all $a,b\in R^1$, almost surely.
http://front.math.ucdavis.edu/math.PR/0607672
---------------------------------------------------------------
4509. THE MODULO 1 CENTRAL LIMIT THEOREM AND BENFORD'S LAW FOR PRODUCTS
Steven J. Miller and Mark J. Nigrini
We derive a necessary and sufficient condition for the sum of M
independent
continuous random variables modulo 1 to converge to the uniform
distribution in
L^1([0,1]), and discuss generalizations to discrete random variables. A
consequence is that if X_1, ..., X_M are independent continuous random
variables with densities f_1, ..., f_M, for any base B as M \to
\infty for many
choices of the densities the distribution of the digits of X_1 * ...
* X_M
converges to Benford's law base B. The rate of convergence can be
quantified in
terms of the Fourier coefficients of the densities, and provides an
explanation
for the prevalence of Benford behavior in many diverse systems.
http://front.math.ucdavis.edu/math.PR/0607686
---------------------------------------------------------------
4510. ASYMPTOTIC RESULTS FOR EMPIRICAL MEASURES OF WEIGHTED SUMS OF
INDEPENDENT RANDOM VARIABLES
Bernard Bercu and Wlodzimierz Bryc
We prove that if a rectangular matrix with uniformly small entries and
approximately orthogonal rows is applied to the independent
standardized random
variables with uniformly bounded third moments, then the empirical
CDF of the
resulting partial sums converges to the normal CDF with probability
one. This
implies almost sure convergence of empirical periodograms, almost sure
convergence of spectra of circulant and reverse circulant matrices,
and almost
sure convergence of the CDF's generated from independent random
variables by
independent random orthogonal matrices.
For special trigonometric matrices, the speed of the almost sure
convergence
is described by the normal approximation and by the large deviation
principle.
http://front.math.ucdavis.edu/math.PR/0607687
---------------------------------------------------------------
4511. STOCHASTIC STOKES' DRIFT WITH INERTIA
Kalvis M. Jansons
We consider both the effect of particle inertia on stochastic Stokes'
drift,
and also a related process which could be considered as a crude model of
stochastic Stokes' drift driven by an eddy diffusivity. In the
latter, the
stochastic forcing is a stable OU process rather than Brownian
motion. We show
that the eddy Stokes' drift velocity has a peak at a non-zero value
of the
correlation time-scale for particles that have the same (limiting)
diffusivity.
For both of the models considered, this study shows that not only can
stochastic Stokes' drift be used to sort particles with different
diffusivities, but also it can be used to sort particles of the same
diffusivities but with different particle masses or correlation time-
scales.
This effect may be important in particle sorting applications.
http://front.math.ucdavis.edu/math.PR/0607707
---------------------------------------------------------------
4512. SOME PROPERTIES OF ANNULUS SLE
Dapeng Zhan
An annulus SLE$_\kappa$ trace tends to a single point on the target
circle,
and the density function of the end point satisfies some differential
equation.
Some martingales or local martingales are found for annulus SLE$_4$,
SLE$_8$
and SLE$_{8/3}$. From the local martingale for annulus SLE$_4$ we find a
candidate of discrete lattice model that may have annulus SLE$_4$ as its
scaling limit. The local martingale for annulus SLE$_{8/3}$ is
similar to those
for chordal and radial SLE$_{8/3}$. But it seems that annulus SLE$_
{8/3}$ does
not satisfy the restriction property.
http://front.math.ucdavis.edu/math.PR/0607720
---------------------------------------------------------------
4513. INEQUALITIES RELATED TO THE ERROR FUNCTION
Omran Kouba
In this note we consider inequalities involving the error function $
\phi$.
Our methodes give new proofs of some known inequalities of Komatsu,
and of
Szarek and Werner, and also produce two families of inequalities that
give
upper and lower bounds for $\phi$. Moreover the continued fractions
expansion
of $\phi$ is obtained.
http://front.math.ucdavis.edu/math.CA/0607694
-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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