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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)
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