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Probability Abstracts 93
This document contains abstracts 4255-4513 from
May-1-2006 to Jul-31-2006.
They have been mailed on Aug 1st, 2006.
Author(s): Sourav Chatterjee
Abstract: 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://arXiv.org/abs/math/0604634
http://front.math.ucdavis.edu/math.PR/0604634
(alternate)
Author(s): Mercedes Arriojas and Yaozhong Hu and Salah-Eldin Mohammed and Gyula Pap
Abstract: 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://arXiv.org/abs/math/0604640
http://front.math.ucdavis.edu/math.PR/0604640
(alternate)
Author(s): Mercedes Arriojas and Yaozhong Hu and Salah-Eldin Mohammed and Gyula Pap
Abstract: 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://arXiv.org/abs/math/0604641
http://front.math.ucdavis.edu/math.PR/0604641
(alternate)
Author(s): Bruno Schapira (MAPMO and PMA)
Abstract: 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://arXiv.org/abs/math/0605020
http://front.math.ucdavis.edu/math.PR/0605020
(alternate)
Author(s): Federico Camia and Charles M. Newman
Abstract: 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://arXiv.org/abs/math/0605035
http://front.math.ucdavis.edu/math.PR/0605035
(alternate)
Author(s): Alexei Stepanov
Abstract: In the present note a generalization of Borel-Cantelli Lemma is proposed.
http://arXiv.org/abs/math/0605007
http://front.math.ucdavis.edu/math.ST/0605007
(alternate)
Author(s): Yuval Peres and Oded Schramm and Scott Sheffield and David Wilson
Abstract: 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://arXiv.org/abs/math/0605002
http://front.math.ucdavis.edu/math.AP/0605002
(alternate)
Author(s): Gordon Blower
Abstract: 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://arXiv.org/abs/math/0605010
http://front.math.ucdavis.edu/math.FA/0605010
(alternate)
Author(s): B. Klartag
Abstract: 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://arXiv.org/abs/math/0605014
http://front.math.ucdavis.edu/math.MG/0605014
(alternate)
Author(s): Alexander S. Cherny
Abstract: 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://arXiv.org/abs/math/0605049
http://front.math.ucdavis.edu/math.PR/0605049
(alternate)
Author(s): Itai Benjamini and Roey Izkovsky and Harry Kesten
Abstract: 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://arXiv.org/abs/math/0605050
http://front.math.ucdavis.edu/math.PR/0605050
(alternate)
Author(s): Alexander S. Cherny
Abstract: 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://arXiv.org/abs/math/0605051
http://front.math.ucdavis.edu/math.PR/0605051
(alternate)
Author(s): Samuel Herrmann and Peter Imkeller and Dierk Peithmann
Abstract: 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://arXiv.org/abs/math/0605053
http://front.math.ucdavis.edu/math.PR/0605053
(alternate)
Author(s): Ernesto Mordecki and Paavo Salminen
Abstract: 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://arXiv.org/abs/math/0605054
http://front.math.ucdavis.edu/math.PR/0605054
(alternate)
Author(s): Clement Rau (LATP)
Abstract: 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://arXiv.org/abs/math/0605056
http://front.math.ucdavis.edu/math.PR/0605056
(alternate)
Author(s): Alexander S. Cherny and Dilip B. Madan
Abstract: 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@R and Beta V@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://arXiv.org/abs/math/0605062
http://front.math.ucdavis.edu/math.PR/0605062
(alternate)
Author(s): Alexander S. Cherny and Dilip B. Madan
Abstract: 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://arXiv.org/abs/math/0605064
http://front.math.ucdavis.edu/math.PR/0605064
(alternate)
Author(s): Alexander S. Cherny and Dilip B. Madan
Abstract: 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://arXiv.org/abs/math/0605065
http://front.math.ucdavis.edu/math.PR/0605065
(alternate)
Author(s): K. D. Elworthy and Xue-Mei Li
Abstract: 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://arXiv.org/abs/math/0605089
http://front.math.ucdavis.edu/math.PR/0605089
(alternate)
Author(s): Giacomo Aletti
Abstract: 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://arXiv.org/abs/math/0605099
http://front.math.ucdavis.edu/math.PR/0605099
(alternate)
Author(s): S. Shemehsavar and S. Rezakhah
Abstract: 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://arXiv.org/abs/math/0605116
http://front.math.ucdavis.edu/math.PR/0605116
(alternate)
Author(s): P. Kleban and J. J. H. Simmons and and R. M. Ziff
Abstract: 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://arXiv.org/abs/cond-mat/0605120
http://front.math.ucdavis.edu/cond-mat/0605120
(alternate)
Author(s): Michael J. Kozdron (University of Regina) and Gregory F. Lawler (Cornell University)
Abstract: 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://arXiv.org/abs/math/0605159
http://front.math.ucdavis.edu/math.PR/0605159
(alternate)
Author(s): Alexei Borodin
Abstract: 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://arXiv.org/abs/math/0605168
http://front.math.ucdavis.edu/math.PR/0605168
(alternate)
Author(s): Eric Cator and Sergei Dobrynin
Abstract: 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://arXiv.org/abs/math/0605199
http://front.math.ucdavis.edu/math.PR/0605199
(alternate)
Author(s): Jinho Baik and Toufic Suidan
Abstract: 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://arXiv.org/abs/math/0605212
http://front.math.ucdavis.edu/math.PR/0605212
(alternate)
Author(s): Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
Abstract: 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://arXiv.org/abs/math/0605221
http://front.math.ucdavis.edu/math.PR/0605221
(alternate)
Author(s): Jeremy Quastel and Benedek Valko
Abstract: 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://arXiv.org/abs/math/0605266
http://front.math.ucdavis.edu/math.PR/0605266
(alternate)
Author(s): Erick Herbin and Ely Merzbach
Abstract: 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://arXiv.org/abs/math/0605279
http://front.math.ucdavis.edu/math.PR/0605279
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: 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://arXiv.org/abs/math/0605285
http://front.math.ucdavis.edu/math.PR/0605285
(alternate)
Author(s): Michael Mayer and Ilya Molchanov
Abstract: 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://arXiv.org/abs/math/0605289
http://front.math.ucdavis.edu/math.PR/0605289
(alternate)
Author(s): Oded Schramm and Scott Sheffield
Abstract: 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://arXiv.org/abs/math/0605337
http://front.math.ucdavis.edu/math.PR/0605337
(alternate)
Author(s): Iosif Pinelis
Abstract: 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
http://arXiv.org/abs/math/0605340
http://front.math.ucdavis.edu/math.PR/0605340
(alternate)
Author(s): Mokshay Madiman and Andrew Barron
Abstract: 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://arXiv.org/abs/cs/0605047
http://front.math.ucdavis.edu/cs.IT/0605047
(alternate)
Author(s): Syoiti Ninomiya and Nicolas Victoir
Abstract: 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://arXiv.org/abs/math/0605361
http://front.math.ucdavis.edu/math.PR/0605361
(alternate)
Author(s): P. Chigansky and R. Liptser
Abstract: 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://arXiv.org/abs/math/0605365
http://front.math.ucdavis.edu/math.PR/0605365
(alternate)
Author(s): Tomasz Grzywny and Micha{\l} Ryznar
Abstract: 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 (00, 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://arXiv.org/abs/math/0605370
http://front.math.ucdavis.edu/math.PR/0605370
(alternate)
Author(s): David Coupier
Abstract: 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://arXiv.org/abs/math/0605395
http://front.math.ucdavis.edu/math.PR/0605395
(alternate)
Author(s): Katalin Marton
Abstract: 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://arXiv.org/abs/math/0605397
http://front.math.ucdavis.edu/math.PR/0605397
(alternate)
Author(s): Andr\'{e} Dabrowski and Gail Ivanoof and Rafal Kulik
Abstract: 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://arXiv.org/abs/math/0605400
http://front.math.ucdavis.edu/math.PR/0605400
(alternate)
Author(s): Gastao A. Braga and Leandro M. Cioletti and Remy Sanchis
Abstract: 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://arXiv.org/abs/math-ph/0605047
http://front.math.ucdavis.edu/math-ph/0605047
(alternate)
Author(s): Henri van den Esker and Remco van der Hofstad and Gerard Hooghiemstra
Abstract: 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://arXiv.org/abs/math/0605414
http://front.math.ucdavis.edu/math.PR/0605414
(alternate)
Author(s): Mikhail Lifshits and Werner Linde and Zhan Shi
Abstract: 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://arXiv.org/abs/math/0605417
http://front.math.ucdavis.edu/math.PR/0605417
(alternate)
Author(s): Ted Theodosopoulos and Ming Yuen
Abstract: 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://arXiv.org/abs/math/0605421
http://front.math.ucdavis.edu/math.PR/0605421
(alternate)
Author(s): Panki Kim and Young-Ran Lee
Abstract: 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://arXiv.org/abs/math/0605422
http://front.math.ucdavis.edu/math.PR/0605422
(alternate)
Author(s): Ali Suleyman Ustunel and Moshe Zakai
Abstract: 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://arXiv.org/abs/math/0605433
http://front.math.ucdavis.edu/math.PR/0605433
(alternate)
Author(s): Yves F. Atchade
Abstract: 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://arXiv.org/abs/math/0605452
http://front.math.ucdavis.edu/math.ST/0605452
(alternate)
Author(s): Pierre Patie
Abstract: 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://arXiv.org/abs/math/0605453
http://front.math.ucdavis.edu/math.PR/0605453
(alternate)
Author(s): Ted Theodosopoulos and Alex Trifunovic
Abstract: 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://arXiv.org/abs/math/0605457
http://front.math.ucdavis.edu/math.PR/0605457
(alternate)
Author(s): Alexander M. Chebotarev
Abstract: 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://arXiv.org/abs/math/0605461
http://front.math.ucdavis.edu/math.PR/0605461
(alternate)
Author(s): Jennie Hansen and Cian Reynolds and Stan Zachary
Abstract: 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://arXiv.org/abs/math/0605477
http://front.math.ucdavis.edu/math.PR/0605477
(alternate)
Author(s): J.F. Le Gall and M. Merle
Abstract: 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://arXiv.org/abs/math/0605482
http://front.math.ucdavis.edu/math.PR/0605482
(alternate)
Author(s): J.F. Le Gall
Abstract: 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://arXiv.org/abs/math/0605484
http://front.math.ucdavis.edu/math.PR/0605484
(alternate)
Author(s): Nicolas Pouyanne (LM-Versailles)
Abstract: 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://arXiv.org/abs/math/0605472
http://front.math.ucdavis.edu/math.CO/0605472
(alternate)
Author(s): Adam Hammett and Boris Pittel
Abstract: 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://arXiv.org/abs/math/0605490
http://front.math.ucdavis.edu/math.PR/0605490
(alternate)
Author(s): Myl\`ene Ma\"{\i}da and Jamal Najim and Sandrine P\'ech\'e
Abstract: 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://arXiv.org/abs/math/0605491
http://front.math.ucdavis.edu/math.PR/0605491
(alternate)
Author(s): David Coupier and Paul Doukhan and Bernard Ycart
Abstract: 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://arXiv.org/abs/math/0605502
http://front.math.ucdavis.edu/math.PR/0605502
(alternate)
Author(s): S. Zozor and C. Vignat
Abstract: 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://arXiv.org/abs/math/0605510
http://front.math.ucdavis.edu/math.PR/0605510
(alternate)
Author(s): Jean-Baptiste Bardet (IRMAR) and Gerhard Keller
Abstract: 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://arXiv.org/abs/math/0605501
http://front.math.ucdavis.edu/math.DS/0605501
(alternate)
Author(s): Panki Kim and Renming Song
Abstract: 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://arXiv.org/abs/math/0605533
http://front.math.ucdavis.edu/math.PR/0605533
(alternate)
Author(s): Sourav Chatterjee and Jason Fulman
Abstract: 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://arXiv.org/abs/math/0605552
http://front.math.ucdavis.edu/math.PR/0605552
(alternate)
Author(s): Panki Kim and Renming Song
Abstract: 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://arXiv.org/abs/math/0605556
http://front.math.ucdavis.edu/math.PR/0605556
(alternate)
Author(s): Panki Kim and Renming Song
Abstract: 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://arXiv.org/abs/math/0605557
http://front.math.ucdavis.edu/math.PR/0605557
(alternate)
Author(s): Kalvis M. Jansons
Abstract: 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://arXiv.org/abs/math/0605561
http://front.math.ucdavis.edu/math.PR/0605561
(alternate)
Author(s): B\'eatrice de Tili\`ere
Abstract: 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://arXiv.org/abs/math/0605583
http://front.math.ucdavis.edu/math.PR/0605583
(alternate)
Author(s): Shige Peng
Abstract: 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://arXiv.org/abs/math/0605599
http://front.math.ucdavis.edu/math.PR/0605599
(alternate)
Author(s): Klaus Fleischmann and Vitali Wachtel
Abstract: 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://arXiv.org/abs/math/0605617
http://front.math.ucdavis.edu/math.PR/0605617
(alternate)
Author(s): Nancy L. Garcia and Thomas G. Kurtz
Abstract: 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://arXiv.org/abs/math/0605620
http://front.math.ucdavis.edu/math.PR/0605620
(alternate)
Author(s): Delphine F\'eral and Sandrine P\'ech\'e
Abstract: 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://arXiv.org/abs/math/0605624
http://front.math.ucdavis.edu/math.PR/0605624
(alternate)
Author(s): Malwina J. Luczak and Colin McDiarmid
Abstract: 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://arXiv.org/abs/math/0605639
http://front.math.ucdavis.edu/math.PR/0605639
(alternate)
Author(s): Iva Kozakova and Ronald Meester and Seema Nanda
Abstract: 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://arXiv.org/abs/math/0605640
http://front.math.ucdavis.edu/math.PR/0605640
(alternate)
Author(s): Erik I. Broman and Jeffrey E. Steif
Abstract: 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://arXiv.org/abs/math/0605641
http://front.math.ucdavis.edu/math.PR/0605641
(alternate)
Author(s): Svante Janson
Abstract: 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://arXiv.org/abs/math/0605642
http://front.math.ucdavis.edu/math.PR/0605642
(alternate)
Author(s): N.D. Gagunashvili
Abstract: 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://arXiv.org/abs/physics/0605123
http://front.math.ucdavis.edu/physics/0605123
(alternate)
Author(s): J. Gaertner and F. den Hollander and G. Maillard
Abstract: 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://arXiv.org/abs/math/0605657
http://front.math.ucdavis.edu/math.PR/0605657
(alternate)
Author(s): F. Baudoin and M. Hairer
Abstract: 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://arXiv.org/abs/math/0605658
http://front.math.ucdavis.edu/math.PR/0605658
(alternate)
Author(s): Pablo A. Ferrari and Nevena Maric
Abstract: 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://arXiv.org/abs/math/0605665
http://front.math.ucdavis.edu/math.PR/0605665
(alternate)
Author(s): S. Shemehsavar and S. Rezakhah
Abstract: 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://arXiv.org/abs/math/0605699
http://front.math.ucdavis.edu/math.PR/0605699
(alternate)
Author(s): Daniela Bertacchi
Abstract: 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://arXiv.org/abs/math/0605718
http://front.math.ucdavis.edu/math.PR/0605718
(alternate)
Author(s): Peggy C\'{e}nac (INRIA Rocquencourt) and Brigitte Chauvin (LM-Versailles), St\'{e}phane Ginouillac (LM-Versailles), Nicolas Pouyanne
(LM-Versailles)
Abstract: 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://arXiv.org/abs/math/0605719
http://front.math.ucdavis.edu/math.PR/0605719
(alternate)
Author(s): Lorenzo Zambotti
Abstract: 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://arXiv.org/abs/math/0605720
http://front.math.ucdavis.edu/math.PR/0605720
(alternate)
Author(s): Panki Kim and Renming Song
Abstract: 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://arXiv.org/abs/math/0605757
http://front.math.ucdavis.edu/math.PR/0605757
(alternate)
Author(s): Thomas Bloom and Bernard Shiffman
Abstract: 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://arXiv.org/abs/math/0605739
http://front.math.ucdavis.edu/math.CV/0605739
(alternate)
Author(s): S.Albeverio and O.Baranovskyi and M.Pratsiovytyi and G.Torbin
Abstract: 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://arXiv.org/abs/math/0605747
http://front.math.ucdavis.edu/math.NT/0605747
(alternate)
Author(s): S.Albeverio and M.Pratsiovytyi and G.Torbin
Abstract: 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://arXiv.org/abs/math/0605763
http://front.math.ucdavis.edu/math.NT/0605763
(alternate)
Author(s): I. Ya. Goldsheid
Abstract: 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://arXiv.org/abs/math/0605775
http://front.math.ucdavis.edu/math.PR/0605775
(alternate)
Author(s): Laurent Mazliak (PMA) and Ivan Nourdin (PMA)
Abstract: 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://arXiv.org/abs/math/0606030
http://front.math.ucdavis.edu/math.PR/0606030
(alternate)
Author(s): Mark Conger and D. Viswanath
Abstract: 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://arXiv.org/abs/math/0606031
http://front.math.ucdavis.edu/math.PR/0606031
(alternate)
Author(s): Anatoly Manita
Abstract: 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://arXiv.org/abs/math/0606040
http://front.math.ucdavis.edu/math.PR/0606040
(alternate)
Author(s): Andrew Granville and K. Soundararajan
Abstract: 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://arXiv.org/abs/math/0606039
http://front.math.ucdavis.edu/math.NT/0606039
(alternate)
Author(s): Anders Karlsson and Wolfgang Woess
Abstract: 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://arXiv.org/abs/math/0606046
http://front.math.ucdavis.edu/math.PR/0606046
(alternate)
Author(s): Sebastian M\"uller
Abstract: 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://arXiv.org/abs/math/0606055
http://front.math.ucdavis.edu/math.PR/0606055
(alternate)
Author(s): Peter G. Doyle and Jim Reeds
Abstract: 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://arXiv.org/abs/math/0606068
http://front.math.ucdavis.edu/math.PR/0606068
(alternate)
Author(s): Ida Kruk (LAGA) and Francesco Russo (LAGA) and Ciprian Tudor (SAMOS)
Abstract: 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://arXiv.org/abs/math/0606069
http://front.math.ucdavis.edu/math.PR/0606069
(alternate)
Author(s): Sabir Umarov and Constantino Tsallis and Murray Gell-Mann and Stanly Steinberg
Abstract: 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://arXiv.org/abs/cond-mat/0606038
http://front.math.ucdavis.edu/cond-mat/0606038
(alternate)
Author(s): Sabir Umarov and Constantino Tsallis and Murray Gell-Mann and Stanly Steinberg
Abstract: 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://arXiv.org/abs/cond-mat/0606040
http://front.math.ucdavis.edu/cond-mat/0606040
(alternate)
Author(s): Hitoshi Kondo and Makoto Maejima and Ken-iti Sato
Abstract: 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://arXiv.org/abs/math/0606084
http://front.math.ucdavis.edu/math.PR/0606084
(alternate)
Author(s): L. Decreusefond and D. Nualart
Abstract: 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://arXiv.org/abs/math/0606086
http://front.math.ucdavis.edu/math.PR/0606086
(alternate)
Author(s): W. Bryc and W. Matysiak and R. Szwarc and J. Wesolowski
Abstract: 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://arXiv.org/abs/math/0606092
http://front.math.ucdavis.edu/math.CA/0606092
(alternate)
Author(s): Mark W. Meckes
Abstract: 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://arXiv.org/abs/math/0606073
http://front.math.ucdavis.edu/math.MG/0606073
(alternate)
Author(s): Luc Bouten and Ramon van Handel and and Matthew R. James
Abstract: 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://arXiv.org/abs/math/0606118
http://front.math.ucdavis.edu/math.PR/0606118
(alternate)
Author(s): J.S. Dehesa and B. Olmos & R.J. Yanez
Abstract: 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://arXiv.org/abs/math/0606133
http://front.math.ucdavis.edu/math.CA/0606133
(alternate)
Author(s): Ioana Bena
Abstract: 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://arXiv.org/abs/cond-mat/0606116
http://front.math.ucdavis.edu/cond-mat/0606116
(alternate)
Author(s): Bela Bollobas and Oliver Riordan
Abstract: 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://arXiv.org/abs/math/0606149
http://front.math.ucdavis.edu/math.PR/0606149
(alternate)
Author(s): Ravi Montenegro
Abstract: 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://arXiv.org/abs/math/0606167
http://front.math.ucdavis.edu/math.PR/0606167
(alternate)
Author(s): Iosif Pinelis
Abstract: 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://arXiv.org/abs/math/0606160
http://front.math.ucdavis.edu/math.ST/0606160
(alternate)
Author(s): David Gamarnik and Dmitriy Katz
Abstract: 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://arXiv.org/abs/math/0606143
http://front.math.ucdavis.edu/math.CO/0606143
(alternate)
Author(s): Pau Amengual and Ra\'ul Toral
Abstract: 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://arXiv.org/abs/math/0606181
http://front.math.ucdavis.edu/math.PR/0606181
(alternate)
Author(s): Jir\^o Akahori and Hiroki Aoki and and Yoshihiko Nagata
Abstract: 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://arXiv.org/abs/math/0606183
http://front.math.ucdavis.edu/math.PR/0606183
(alternate)
Author(s): Andrzej Stos
Abstract: 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://arXiv.org/abs/math/0606185
http://front.math.ucdavis.edu/math.PR/0606185
(alternate)
Author(s): Faouzi Chaabane (EASMS) and Hamdi Fathallah (LM-Versailles)
Abstract: 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://arXiv.org/abs/math/0606200
http://front.math.ucdavis.edu/math.PR/0606200
(alternate)
Author(s): L. Decreusefond and D. Nualart
Abstract: 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://arXiv.org/abs/math/0606214
http://front.math.ucdavis.edu/math.PR/0606214
(alternate)
Author(s): Nizar Demni (PMA) and the PMA Collaboration
Abstract: 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://arXiv.org/abs/math/0606218
http://front.math.ucdavis.edu/math.PR/0606218
(alternate)
Author(s): Isabelle Abraham (DCRE) and Romain Abraham (MAPMO) and Agnes Desolneux (MAP5), Sebastien Li-Thiao-Te (CMLA)
Abstract: 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://arXiv.org/abs/math/0606219
http://front.math.ucdavis.edu/math.ST/0606219
(alternate)
Author(s): Jir\^o Akahori
Abstract: 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://arXiv.org/abs/math/0606292
http://front.math.ucdavis.edu/math.PR/0606292
(alternate)
Author(s): John Moriarty and Neil O'Connell
Abstract: 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://arXiv.org/abs/math/0606296
http://front.math.ucdavis.edu/math.PR/0606296
(alternate)
Author(s): Johan Jonasson and Jeffrey Steif
Abstract: 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
http://arXiv.org/abs/math/0606297
http://front.math.ucdavis.edu/math.PR/0606297
(alternate)
Author(s): Andreas Greven and Lea Popovic and and Anita Winter
Abstract: 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://arXiv.org/abs/math/0606313
http://front.math.ucdavis.edu/math.PR/0606313
(alternate)
Author(s): Marcus Hutter
Abstract: 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://arXiv.org/abs/math/0606315
http://front.math.ucdavis.edu/math.ST/0606315
(alternate)
Author(s): Jean-Fran\c{c}ois Marckert (LaBRI)
Abstract: 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://arXiv.org/abs/math/0606338
http://front.math.ucdavis.edu/math.PR/0606338
(alternate)
Author(s): Denis Bell
Abstract: 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://arXiv.org/abs/math/0606365
http://front.math.ucdavis.edu/math.PR/0606365
(alternate)
Author(s): Zhan Shi (PMA) and Olivier Zindy (PMA)
Abstract: 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://arXiv.org/abs/math/0606376
http://front.math.ucdavis.edu/math.PR/0606376
(alternate)
Author(s): Manuel Lladser and Jaime San Martin
Abstract: 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://arXiv.org/abs/math/0606392
http://front.math.ucdavis.edu/math.PR/0606392
(alternate)
Author(s): Arvind Singh (PMA)
Abstract: 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://arXiv.org/abs/math/0606411
http://front.math.ucdavis.edu/math.PR/0606411
(alternate)
Author(s): Kevin P. Costello and Van H. Vu
Abstract: 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
http://arXiv.org/abs/math/0606414
http://front.math.ucdavis.edu/math.PR/0606414
(alternate)
Author(s): Alexey Kulik
Abstract: 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://arXiv.org/abs/math/0606427
http://front.math.ucdavis.edu/math.PR/0606427
(alternate)
Author(s): Irina Ignatiouk-Robert
Abstract: 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://arXiv.org/abs/math/0606439
http://front.math.ucdavis.edu/math.PR/0606439
(alternate)
Author(s): Svante Janson
Abstract: 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://arXiv.org/abs/math/0606454
http://front.math.ucdavis.edu/math.PR/0606454
(alternate)
Author(s): Alessio Sancetta
Abstract: 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://arXiv.org/abs/math/0606462
http://front.math.ucdavis.edu/math.PR/0606462
(alternate)
Author(s): Micha{\l} Rams
Abstract: We estimate from above and below the dimension of invariant measure for
contracting-on-average iterated function systems in $\R^d$.
http://arXiv.org/abs/math/0606420
http://front.math.ucdavis.edu/math.DS/0606420
(alternate)
Author(s): Benoit Collins (Universite Claude Bernard and Lyon 1) and James A. Mingo (Queen's University), Piotr Sniady (Uniwersytet Wroclawski), Roland Speicher
(Queen's University)
Abstract: 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://arXiv.org/abs/math/0606431
http://front.math.ucdavis.edu/math.OA/0606431
(alternate)
Author(s): Le Anh Vinh
Abstract: 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://arXiv.org/abs/math/0606485
http://front.math.ucdavis.edu/math.PR/0606485
(alternate)
Author(s): Istvan Gy\"{o}ngy and Annie Millet (PMA)
Abstract: 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://arXiv.org/abs/math/0606488
http://front.math.ucdavis.edu/math.PR/0606488
(alternate)
Author(s): M. Kwasnicki
Abstract: 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://arXiv.org/abs/math/0606509
http://front.math.ucdavis.edu/math.PR/0606509
(alternate)
Author(s): Jean-Michel Billiot (LABSAD) and Franck Corset (LABSAD) and Eric Fontenas (LABSAD)
Abstract: 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://arXiv.org/abs/cond-mat/0606495
http://front.math.ucdavis.edu/cond-mat/0606495
(alternate)
Author(s): M.Draief; A.Ganesh; L.Massoulie
Abstract: 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://arXiv.org/abs/math/0606514
http://front.math.ucdavis.edu/math.PR/0606514
(alternate)
Author(s): Ignacio Cascos and Ilya Molchanov
Abstract: 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://arXiv.org/abs/math/0606520
http://front.math.ucdavis.edu/math.PR/0606520
(alternate)
Author(s): Remco van der Hofstad and Peter Morters and Nadia Sidorova
Abstract: 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://arXiv.org/abs/math/0606527
http://front.math.ucdavis.edu/math.PR/0606527
(alternate)
Author(s): Ian Melbourne and Matthew Nicol
Abstract: 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://arXiv.org/abs/math/0606535
http://front.math.ucdavis.edu/math.DS/0606535
(alternate)
Author(s): J. Martin Lindsay and Stephen J. Wills
Abstract: 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://arXiv.org/abs/math/0606545
http://front.math.ucdavis.edu/math.FA/0606545
(alternate)
Author(s): Markus Heydenreich and Remco van der Hofstad and Georgi Radulov
Abstract: 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://arXiv.org/abs/math/0606584
http://front.math.ucdavis.edu/math.PR/0606584
(alternate)
Author(s): Nikolai Dokuchaev
Abstract: 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://arXiv.org/abs/math/0606595
http://front.math.ucdavis.edu/math.PR/0606595
(alternate)
Author(s): Zenghu Li
Abstract: 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://arXiv.org/abs/math/0606597
http://front.math.ucdavis.edu/math.PR/0606597
(alternate)
Author(s): Nikolai Dokuchaev
Abstract: 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://arXiv.org/abs/math/0606601
http://front.math.ucdavis.edu/math.PR/0606601
(alternate)
Author(s): Ciprian A. Tudor (SAMOS)
Abstract: 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://arXiv.org/abs/math/0606602
http://front.math.ucdavis.edu/math.PR/0606602
(alternate)
Author(s): Patrizia Berti and Pietro Rigo
Abstract: 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://arXiv.org/abs/math/0606604
http://front.math.ucdavis.edu/math.PR/0606604
(alternate)
Author(s): Marius Junge and Javier Parcet
Abstract: 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://arXiv.org/abs/math/0606596
http://front.math.ucdavis.edu/math.OA/0606596
(alternate)
Author(s): Lorenzo Finesso and Angela Grassi and Peter Spreij
Abstract: 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://arXiv.org/abs/math/0606591
http://front.math.ucdavis.edu/math.OC/0606591
(alternate)
Author(s): Steven N. Evans and Tye Lidman
Abstract: 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://arXiv.org/abs/math/0606609
http://front.math.ucdavis.edu/math.PR/0606609
(alternate)
Author(s): Donald A. Dawson; Zenghu Li; Hao Wang
Abstract: 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://arXiv.org/abs/math/0606615
http://front.math.ucdavis.edu/math.PR/0606615
(alternate)
Author(s): Donald A. Dawson and Luis G. Gorostiza and Zenghu Li
Abstract: 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://arXiv.org/abs/math/0606616
http://front.math.ucdavis.edu/math.PR/0606616
(alternate)
Author(s): Zeng-Hu Li
Abstract: 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://arXiv.org/abs/math/0606617
http://front.math.ucdavis.edu/math.PR/0606617
(alternate)
Author(s): Donald A. Dawson; Zenghu Li
Abstract: 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://arXiv.org/abs/math/0606618
http://front.math.ucdavis.edu/math.PR/0606618
(alternate)
Author(s): Donald A. Dawson and Zenghu Li and Byron Schmuland and Wei Sun
Abstract: 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://arXiv.org/abs/math/0606619
http://front.math.ucdavis.edu/math.PR/0606619
(alternate)
Author(s): Donald A. Dawson; Zenghu Li
Abstract: 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://arXiv.org/abs/math/0606620
http://front.math.ucdavis.edu/math.PR/0606620
(alternate)
Author(s): Donald A. Dawson; Zenghu Li; Xiaowen Zhou
Abstract: 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://arXiv.org/abs/math/0606621
http://front.math.ucdavis.edu/math.PR/0606621
(alternate)
Author(s): Zenghu Li; Hao Wang; Jie Xiong
Abstract: 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://arXiv.org/abs/math/0606622
http://front.math.ucdavis.edu/math.PR/0606622
(alternate)
Author(s): Zenghu Li
Abstract: 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://arXiv.org/abs/math/0606623
http://front.math.ucdavis.edu/math.PR/0606623
(alternate)
Author(s): Charles Bordenave
Abstract: 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://arXiv.org/abs/math/0606624
http://front.math.ucdavis.edu/math.PR/0606624
(alternate)
Author(s): Yuval Peres and Ofer Zeitouni
Abstract: 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<\lambdam$ 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://arXiv.org/abs/math/0606625
http://front.math.ucdavis.edu/math.PR/0606625
(alternate)
Author(s): Rosanna Coviello and Francesco Russo
Abstract: 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://arXiv.org/abs/math/0606642
http://front.math.ucdavis.edu/math.PR/0606642
(alternate)
Author(s): Rami Kanhouche (CMLA)
Abstract: 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://arXiv.org/abs/math/0606643
http://front.math.ucdavis.edu/math.PR/0606643
(alternate)
Author(s): Tomasz Grzywny
Abstract: 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://arXiv.org/abs/math/0606659
http://front.math.ucdavis.edu/math.PR/0606659
(alternate)
Author(s): James Parkinson
Abstract: 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://arXiv.org/abs/math/0606662
http://front.math.ucdavis.edu/math.PR/0606662
(alternate)
Author(s): Brian Rider and Balint Virag
Abstract: 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://arXiv.org/abs/math/0606663
http://front.math.ucdavis.edu/math.PR/0606663
(alternate)
Author(s): Ted Theodosopoulos and Robert Boyer
Abstract: 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://arXiv.org/abs/math/0606667
http://front.math.ucdavis.edu/math.PR/0606667
(alternate)
Author(s): Glenn Merlet
Abstract: 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://arXiv.org/abs/math/0606668
http://front.math.ucdavis.edu/math.PR/0606668
(alternate)
Author(s): Panki Kim and Renming Song
Abstract: 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://arXiv.org/abs/math/0606678
http://front.math.ucdavis.edu/math.PR/0606678
(alternate)
Author(s): Hubert Hennion (Universit\'e de Rennes I)
Abstract: 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://arXiv.org/abs/math/0606680
http://front.math.ucdavis.edu/math.PR/0606680
(alternate)
Author(s): Qing-Yang Guan and Matthias Winkel
Abstract: 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://arXiv.org/abs/math/0606685
http://front.math.ucdavis.edu/math.PR/0606685
(alternate)
Author(s): Panki Kim and Renming Song
Abstract: 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://arXiv.org/abs/math/0606706
http://front.math.ucdavis.edu/math.PR/0606706
(alternate)
Author(s): P. Baldi and D. Marinucci
Abstract: 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://arXiv.org/abs/math/0606709
http://front.math.ucdavis.edu/math.PR/0606709
(alternate)
Author(s): Gerard Ben Arous and Jiri Cerny
Abstract: 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://arXiv.org/abs/math/0606719
http://front.math.ucdavis.edu/math.PR/0606719
(alternate)
Author(s): \'{A}d\'{a}m Tim\'{a}r
Abstract: 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://arXiv.org/abs/math/0606750
http://front.math.ucdavis.edu/math.PR/0606750
(alternate)
Author(s): Andrea Collevecchio
Abstract: 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://arXiv.org/abs/math/0606751
http://front.math.ucdavis.edu/math.PR/0606751
(alternate)
Author(s): C. Houdr\'e and P. Reynaud-Bouret
Abstract: 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://arXiv.org/abs/math/0606752
http://front.math.ucdavis.edu/math.PR/0606752
(alternate)
Author(s): Ciprian Tudor (SAMOS) and Yimin Xiao
Abstract: 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://arXiv.org/abs/math/0606753
http://front.math.ucdavis.edu/math.PR/0606753
(alternate)
Author(s): Peter Buergisser
Abstract: 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://arXiv.org/abs/math/0606755
http://front.math.ucdavis.edu/math.PR/0606755
(alternate)
Author(s): Kurt Johansson and Eric Nordenstam
Abstract: Consider an infinite random matrix $H=(h_{ij})_{0
http://arXiv.org/abs/math/0606760
http://front.math.ucdavis.edu/math.PR/0606760
(alternate)
Author(s): Kijung Lee and Carl Mueller and and Jei Xiong
Abstract: 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://arXiv.org/abs/math/0606761
http://front.math.ucdavis.edu/math.PR/0606761
(alternate)
Author(s): S.N. Lahiri
Abstract: 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://arXiv.org/abs/math/0606739
http://front.math.ucdavis.edu/math.ST/0606739
(alternate)
Author(s): Hanna Jankowski
Abstract: 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://arXiv.org/abs/math/0606778
http://front.math.ucdavis.edu/math.PR/0606778
(alternate)
Author(s): Ivan Nourdin and Thomas Simon
Abstract: 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://arXiv.org/abs/math/0606783
http://front.math.ucdavis.edu/math.PR/0606783
(alternate)
Author(s): Zhen-Qing Chen and Masatoshi Fukushima and Jiangang Ying
Abstract: 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://arXiv.org/abs/math/0606784
http://front.math.ucdavis.edu/math.PR/0606784
(alternate)
Author(s): Ben Goldys and Jan van Neerven
Abstract: 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://arXiv.org/abs/math/0606785
http://front.math.ucdavis.edu/math.PR/0606785
(alternate)
Author(s): Evarist Gin\'{e} and Vladimir Koltchinskii
Abstract: 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://arXiv.org/abs/math/0606788
http://front.math.ucdavis.edu/math.PR/0606788
(alternate)
Author(s): Hariharan Narayanan
Abstract: 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://arXiv.org/abs/math/0607011
http://front.math.ucdavis.edu/math.PR/0607011
(alternate)
Author(s): C. Houdr\'e and P. Marchal and P. Reynaud-Bouret
Abstract: 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://arXiv.org/abs/math/0607019
http://front.math.ucdavis.edu/math.PR/0607019
(alternate)
Author(s): C. Houdr\'e and P. Marchal
Abstract: 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://arXiv.org/abs/math/0607022
http://front.math.ucdavis.edu/math.PR/0607022
(alternate)
Author(s): Ravi Montenegro
Abstract: 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://arXiv.org/abs/math/0607031
http://front.math.ucdavis.edu/math.PR/0607031
(alternate)
Author(s): Jean-Marc Aza\"{\i}s Mario Wschebor
Abstract: 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://arXiv.org/abs/math/0607041
http://front.math.ucdavis.edu/math.PR/0607041
(alternate)
Author(s): Domenico Marinucci and Giovanni Peccati (LSTA)
Abstract: 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://arXiv.org/abs/math/0607044
http://front.math.ucdavis.edu/math.PR/0607044
(alternate)
Author(s): Peter Neal and Gareth Roberts
Abstract: 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://arXiv.org/abs/math/0607054
http://front.math.ucdavis.edu/math.PR/0607054
(alternate)
Author(s): {\L}ukasz Kruk and John Lehoczky and Steven Shreve
Abstract: 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://arXiv.org/abs/math/0607056
http://front.math.ucdavis.edu/math.PR/0607056
(alternate)
Author(s): Donald A. Dawson and Shui Feng
Abstract: 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://arXiv.org/abs/math/0607070
http://front.math.ucdavis.edu/math.PR/0607070
(alternate)
Author(s): Daniel Boivan (LM)
Abstract: 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://arXiv.org/abs/math/0607073
http://front.math.ucdavis.edu/math.PR/0607073
(alternate)
Author(s): S. Koetzer and I. Molchanov
Abstract: 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://arXiv.org/abs/math/0607086
http://front.math.ucdavis.edu/math.PR/0607086
(alternate)
Author(s): S.Ekisheva and C. Houdr\'e
Abstract: 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://arXiv.org/abs/math/0607089
http://front.math.ucdavis.edu/math.PR/0607089
(alternate)
Author(s): Pao-Liu Chow
Abstract: 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://arXiv.org/abs/math/0607097
http://front.math.ucdavis.edu/math.PR/0607097
(alternate)
Author(s): Xianping Guo and Ulrich Rieder
Abstract: 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://arXiv.org/abs/math/0607098
http://front.math.ucdavis.edu/math.PR/0607098
(alternate)
Author(s): Peter Brockwell and Erdenebaatar Chadraa and Alexander Lindner
Abstract: 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://arXiv.org/abs/math/0607109
http://front.math.ucdavis.edu/math.PR/0607109
(alternate)
Author(s): Laurent Denis and Claude Martini
Abstract: 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://arXiv.org/abs/math/0607111
http://front.math.ucdavis.edu/math.PR/0607111
(alternate)
Author(s): Friedrich Hubalek and Jan Kallsen and Leszek Krawczyk
Abstract: 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://arXiv.org/abs/math/0607112
http://front.math.ucdavis.edu/math.PR/0607112
(alternate)
Author(s): Luc Devroye and Hsien-Kuei Hwang
Abstract: 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://arXiv.org/abs/math/0607119
http://front.math.ucdavis.edu/math.PR/0607119
(alternate)
Author(s): Lothar Heinrich and Hendrik Schmidt and Volker Schmidt
Abstract: 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://arXiv.org/abs/math/0607120
http://front.math.ucdavis.edu/math.PR/0607120
(alternate)
Author(s): Jose Blanchet and Peter Glynn
Abstract: 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://arXiv.org/abs/math/0607121
http://front.math.ucdavis.edu/math.PR/0607121
(alternate)
Author(s): Yuri Kifer
Abstract: 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://arXiv.org/abs/math/0607123
http://front.math.ucdavis.edu/math.PR/0607123
(alternate)
Author(s): Johan Jonasson
Abstract: 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://arXiv.org/abs/math/0607124
http://front.math.ucdavis.edu/math.PR/0607124
(alternate)
Author(s): D.A. Dawson and L.G. Gorostiza
Abstract: 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://arXiv.org/abs/math/0607131
http://front.math.ucdavis.edu/math.PR/0607131
(alternate)
Author(s): Pierre-Andr\'{e} Zitt (MODAL'X)
Abstract: 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://arXiv.org/abs/math/0607147
http://front.math.ucdavis.edu/math.PR/0607147
(alternate)
Author(s): Cedric Boutillier
Abstract: 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://arXiv.org/abs/math/0607162
http://front.math.ucdavis.edu/math.PR/0607162
(alternate)
Author(s): Erwin Bolthausen and Ofer Zeitouni
Abstract: 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://arXiv.org/abs/math/0607192
http://front.math.ucdavis.edu/math.PR/0607192
(alternate)
Author(s): F.J. Lopez and G. Sanz and and M. Sobottka
Abstract: 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://arXiv.org/abs/math/0607206
http://front.math.ucdavis.edu/math.PR/0607206
(alternate)
Author(s): H. M. Bui and J. P. Keating
Abstract: 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://arXiv.org/abs/math/0607196
http://front.math.ucdavis.edu/math.NT/0607196
(alternate)
Author(s): Jocelyne Bion-Nadal
Abstract: 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://arXiv.org/abs/math/0607212
http://front.math.ucdavis.edu/math.PR/0607212
(alternate)
Author(s): Jean-Baptiste Gou\'{e}r\'{e} (MAPMO)
Abstract: 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://arXiv.org/abs/math/0607226
http://front.math.ucdavis.edu/math.PR/0607226
(alternate)
Author(s): Julia Dony and Uwe Einmahl
Abstract: 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://arXiv.org/abs/math/0607232
http://front.math.ucdavis.edu/math.ST/0607232
(alternate)
Author(s): Jinqiao Duan and Balasubramanya T. Nadiga
Abstract: 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://arXiv.org/abs/math/0607214
http://front.math.ucdavis.edu/math.AP/0607214
(alternate)
Author(s): Stepanka Bilova and Laurent Mazliak and Pavel Sisma
Abstract: 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://arXiv.org/abs/math/0607217
http://front.math.ucdavis.edu/math.HO/0607217
(alternate)
Author(s): Burgess Davis and Majid Hosseini
Abstract: 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://arXiv.org/abs/math/0607219
http://front.math.ucdavis.edu/math.SP/0607219
(alternate)
Author(s): Marie Th\'eret
Abstract: 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://arXiv.org/abs/math/0607252
http://front.math.ucdavis.edu/math.PR/0607252
(alternate)
Author(s): Marie Th\'eret
Abstract: 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://arXiv.org/abs/math/0607253
http://front.math.ucdavis.edu/math.PR/0607253
(alternate)
Author(s): Harald Luschgy and Gilles Pag\`{e}s (PMA)
Abstract: 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://arXiv.org/abs/math/0607282
http://front.math.ucdavis.edu/math.PR/0607282
(alternate)
Author(s): Ken-iti Sato
Abstract: 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://arXiv.org/abs/math/0607288
http://front.math.ucdavis.edu/math.PR/0607288
(alternate)
Author(s): Tom Schmitz
Abstract: 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://arXiv.org/abs/math/0607293
http://front.math.ucdavis.edu/math.PR/0607293
(alternate)
Author(s): Bergfinnur Durhuus and Thordur Jonsson and John F. Wheater
Abstract: 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://arXiv.org/abs/math-ph/0607020
http://front.math.ucdavis.edu/math-ph/0607020
(alternate)
Author(s): Marta Sanz-Sol\'e and Iv\'an Torrecilla-Tarantino
Abstract: 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://arXiv.org/abs/math/0607310
http://front.math.ucdavis.edu/math.PR/0607310
(alternate)
Author(s): Craig A. Tracy and Harold Widom
Abstract: 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://arXiv.org/abs/math/0607321
http://front.math.ucdavis.edu/math.PR/0607321
(alternate)
Author(s): Vlady Ravelomanana (LIPN) and the Projet PAI Amadeus Collaboration
Abstract: 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://arXiv.org/abs/cs/0607057
http://front.math.ucdavis.edu/cs.DM/0607057
(alternate)
Author(s): Vlady Ravelomanana (LIPN) and Alphonse Laza Rijamame (D.M.I)
Abstract: 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://arXiv.org/abs/cs/0607059
http://front.math.ucdavis.edu/cs.DM/0607059
(alternate)
Author(s): Jose Ramirez and Brian Rider and Balint Virag
Abstract: 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://arXiv.org/abs/math/0607331
http://front.math.ucdavis.edu/math.PR/0607331
(alternate)
Author(s): Ben Hambly and Liza Jones
Abstract: 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://arXiv.org/abs/math/0607345
http://front.math.ucdavis.edu/math.PR/0607345
(alternate)
Author(s): Ronald Meester and Marieke Collins and Richard Gill and Michiel van Lambalgen
Abstract: 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://arXiv.org/abs/math/0607340
http://front.math.ucdavis.edu/math.ST/0607340
(alternate)
Author(s): Cecilia Mancini
Abstract: 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://arXiv.org/abs/math/0607378
http://front.math.ucdavis.edu/math.ST/0607378
(alternate)
Author(s): Dirk Bl\"omker and Jinqiao Duan
Abstract: 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://arXiv.org/abs/math/0607357
http://front.math.ucdavis.edu/math.CA/0607357
(alternate)
Author(s): Aijun Du and Jinqiao Duan
Abstract: 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://arXiv.org/abs/math/0607366
http://front.math.ucdavis.edu/math.DS/0607366
(alternate)
Author(s): David L. Donoho and Jared Tanner
Abstract: 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://arXiv.org/abs/math/0607364
http://front.math.ucdavis.edu/math.MG/0607364
(alternate)
Author(s): Sergi Elizalde and Kevin Woods
Abstract: 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://arXiv.org/abs/math/0607390
http://front.math.ucdavis.edu/math.NT/0607390
(alternate)
Author(s): Pierre Del Moral (JAD) and Fr\'{e}d\'{e}ric Patras (JAD) and Sylvain Rubenthaler (JAD)
Abstract: 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://arXiv.org/abs/math/0607453
http://front.math.ucdavis.edu/math.PR/0607453
(alternate)
Author(s): S\'{e}bastien Blach\`{e}re (LATP) and Peter Ha\"{i}ssinsky (LATP) and Pierre Mathieu (LATP)
Abstract: 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://arXiv.org/abs/math/0607467
http://front.math.ucdavis.edu/math.PR/0607467
(alternate)
Author(s): Jean B. Lasserre
Abstract: 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://arXiv.org/abs/math/0607463
http://front.math.ucdavis.edu/math.FA/0607463
(alternate)
Author(s): Manjunath Krishnapur
Abstract: 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://arXiv.org/abs/math/0607504
http://front.math.ucdavis.edu/math.PR/0607504
(alternate)
Author(s): Hanna Jankowski
Abstract: 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://arXiv.org/abs/math/0607505
http://front.math.ucdavis.edu/math.PR/0607505
(alternate)
Author(s): N. Litvak and W.R.W. Scheinhardt and Y. Volkovich
Abstract: 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://arXiv.org/abs/math/0607507
http://front.math.ucdavis.edu/math.PR/0607507
(alternate)
Author(s): Kwabena Doku-Amponsah and Peter Morters
Abstract: 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://arXiv.org/abs/math/0607545
http://front.math.ucdavis.edu/math.PR/0607545
(alternate)
Author(s): Francis Comets and Jeremy Quastel and Alejandro Ramirez
Abstract: 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://arXiv.org/abs/math/0607549
http://front.math.ucdavis.edu/math.PR/0607549
(alternate)
Author(s): Abhay G Bhatt and Rajeeva L Karandikar and B V Rao
Abstract: 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://arXiv.org/abs/math/0607613
http://front.math.ucdavis.edu/math.PR/0607613
(alternate)
Author(s): Soumik Pal
Abstract: 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://arXiv.org/abs/math/0607617
http://front.math.ucdavis.edu/math.PR/0607617
(alternate)
Author(s): L.V. Bogachev and Z.G. Su
Abstract: 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://arXiv.org/abs/math/0607635
http://front.math.ucdavis.edu/math.PR/0607635
(alternate)
Author(s): Richard F. Bass and Jay Rosen
Abstract: 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://arXiv.org/abs/math/0607636
http://front.math.ucdavis.edu/math.PR/0607636
(alternate)
Author(s): Alessandro De Gregorio and Stefano M. Iacus
Abstract: 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://arXiv.org/abs/math/0607633
http://front.math.ucdavis.edu/math.ST/0607633
(alternate)
Author(s): Codina Cotar and Alexander E. Holroyd and David Revelle
Abstract: 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://arXiv.org/abs/math/0607645
http://front.math.ucdavis.edu/math.PR/0607645
(alternate)
Author(s): Klaus Fleischmann and Carl Mueller and and Pascal Vogt
Abstract: 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://arXiv.org/abs/math/0607667
http://front.math.ucdavis.edu/math.PR/0607667
(alternate)
Author(s): Michael B. Marcus and Jay Rosen
Abstract: 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://arXiv.org/abs/math/0607672
http://front.math.ucdavis.edu/math.PR/0607672
(alternate)
Author(s): Steven J. Miller and Mark J. Nigrini
Abstract: 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://arXiv.org/abs/math/0607686
http://front.math.ucdavis.edu/math.PR/0607686
(alternate)
Author(s): Bernard Bercu and Wlodzimierz Bryc
Abstract: 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://arXiv.org/abs/math/0607687
http://front.math.ucdavis.edu/math.PR/0607687
(alternate)
Author(s): Kalvis M. Jansons
Abstract: 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://arXiv.org/abs/math/0607707
http://front.math.ucdavis.edu/math.PR/0607707
(alternate)
Author(s): Dapeng Zhan
Abstract: 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://arXiv.org/abs/math/0607720
http://front.math.ucdavis.edu/math.PR/0607720
(alternate)
Author(s): Omran Kouba
Abstract: 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://arXiv.org/abs/math/0607694
http://front.math.ucdavis.edu/math.CA/0607694
(alternate)
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