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Probability Abstracts 105
This document contains abstracts 7236-7441
from July-1-2008 to August-31-2008.
They have been mailed on Sept 2nd, 2008.
Author(s): S. C.Lim and L. P. Teo
Abstract: Two types of Gaussian processes, namely the Gaussian field with generalized
Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy
covariance (GSGCC) are considered. Some of the basic properties and the
asymptotic properties of the spectral densities of these random fields are
studied. The associated self-similar random fields obtained by applying the
Lamperti transformation to GFGCC and GSGCC are studied.
http://arxiv.org/abs/0807.0022
Author(s): Hui He
Abstract: We construct a class of discontinuous superprocesses with dependent spatial
motion and general branching mechanism. The process arises as the weak limit of
critical interacting-branching particle systems where the spatial motions of
the particles are not independent. The main work is to solve the martingale
problem. When we turn to the uniqueness of the process, we generalize the
localization method introduced by [D.W. Stroock, Diffusion processes associated
with Levy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32(1975)
209--244] to the measure-valued context. As for existence, we use particle
system approximation and a perturbation method. This work generalizes the model
introduced in [D.A. Dawson, Z. Li, H. Wang, Superprocesses with dependent
spatial motion and general branching densities, Electron. J. Probab. 6(2001),
no.25, 33 pp. (electronic)] where quadratic branching mechanism was considered.
We also investigate some properties of the process.
http://arxiv.org/abs/0807.0054
Author(s): Matthieu Fradelizi
Abstract: We prove a sharp inequality conjectured by Bobkov on the measure of dilations
of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives
a common generalization of an inequality of Nazarov, Sodin and Volberg and a
concentration inequality of Gu\'edon. Applying our inequality to the level sets
of functions satisfying a Remez type inequality, we deduce, as it is classical,
that these functions enjoy dimension free distribution inequalities and
Kahane-Khintchine type inequalities with positive and negative exponent, with
respect to an arbitrary $s$-concave probability.
http://arxiv.org/abs/0807.0080
Author(s): Diego Armentano and Mario Wschebor
Abstract: We consider random systems of equations over the reals, with $m$ equations
and $m$ unknowns $ P_i(t)+X_i(t)=0$, $t \in \R^m$, $i=1,...,m$, where the
$P_i's$ are non-random polynomials having degrees $d_i's$ (the "signal") and
the $X_i's$ (the "noise") are independent real-valued Gaussian centered random
polynomial fields defined on $\R^m$, with a probability law satisfying some
invariance properties.
For each $i$, $P_i$ and $X_i$ have degree $d_i$.
The problem is the behavior of the number of roots for large $m$. We prove
that under specified conditions on the relation signal over noise, which imply
that in a certain sense this relation is neither too large nor too small, it
follows that the quotient between the expected value of the number of roots of
the perturbed system and the expected value corresponding to the centered
system (i.e. $P_i$ identically zero for all $i=1,...,m$), tends to zero
geometrically fast as $m$ tends to infinity. In particular, this means that the
behavior of this expected value, is governed by the noise part.
http://arxiv.org/abs/0807.0262
Author(s): Christophette Blanchet-Scalliet (ICJ) and Fr\'ed\'eric Patras (JAD)
Abstract: The valuation of counterparty risk for single name credit derivatives
requires the computa- tion of joint distributions of default times of two
default-prone entities. For a Merton-type model, we derive some formulas for
these joint distribu- tions. As an application, closed formulas for
counterparty risk on a CDS or for a first-to-default swap on two underlyings
are obtained.
http://arxiv.org/abs/0807.0309
Author(s): Alessandra Cretarola and Fausto Gozzi and Huy\^en Pham (PMA and CREST) and Peter Tankov (PMA)
Abstract: We investigate optimal consumption policies in the liquidity risk model
introduced in Pham and Tankov (2007). Our main result is to derive smoothness
results for the value functions of the portfolio/consumption choice problem. As
an important consequence, we can prove the existence of the optimal control
(portfolio/consumption strategy) which we characterize both in feedback form in
terms of the derivatives of the value functions and as the solution of a
second-order ODE. Finally, numerical illustrations of the behavior of optimal
consumption strategies between two trading dates are given.
http://arxiv.org/abs/0807.0326
Author(s): Martin R. Evans and Pablo A. Ferrari and Kirone Mallick
Abstract: In this work we construct the stationary measure of the N species totally
asymmetric simple exclusion process in a matrix product formulation. We make
the connection between the matrix product formulation and the queueing theory
picture of Ferrari and Martin. In particular, in the standard representation,
the matrices act on the space of queue lengths. For N >2 the matrices in fact
become tensor products of elements of quadratic algebras. This enables us to
give a purely algebraic proof of the stationary measure which we present for N
=3.
http://arxiv.org/abs/0807.0327
Author(s): Yacin Ameur and Haakan hedenmalm and Nikolai Makarov
Abstract: In this note, we prove Gaussian field convergence of fluctuations of
eigenvalues of random normal matrices in the interior of a quantum droplet.
http://arxiv.org/abs/0807.0375
Author(s): A. B. Cruzeiro and E. Shamarova
Abstract: We establish a connection between the strong solution to the Cauchy problem
for the 2D Navier-Stokes equations and the solution to the system of
forward-backward stochastic differential equations (FBSDEs) on the group of
volume-preserving diffeomorphisms of the 2D torus. We construct a
representation of the strong solution to the Navier-Stokes equations in terms
of diffusion processes.
http://arxiv.org/abs/0807.0421
Author(s): Olivier Garet (IECN) and R\'egine Marchand (IECN)
Abstract: The aim of this article is to prove asymptotic shape theorems for the contact
pr ocess in stationary random environment. These theorems generalize known
results for the classical contact process. In particular, if $H_t$ denotes the
set of al ready occupied sites at time $t$, we show that for almost every
environment, whe n the contact process survives, the set $H_t/t$ almost surely
converges to a co mpact set that only depends on the law of the environment. We
introduce new obje cts which also simplify the proof of the shape theorem in a
deterministic environment.
http://arxiv.org/abs/0807.0426
Author(s): Antonio Mura and Gianni Pagnini
Abstract: In this paper we study a parametric class of stochastic processes to model
both fast and slow anomalous diffusion. This class, called generalized grey
Brownian motion (ggBm), is made up off self-similar with stationary increments
processes (H-sssi) and depends on two real parameters alpha in (0,2) and beta
in (0,1]. It includes fractional Brownian motion when alpha in (0,2) and
beta=1, and time-fractional diffusion stochastic processes when alpha=beta in
(0,1). The latters have marginal probability density function governed by
time-fractional diffusion equations of order beta. The ggBm is defined through
the explicit construction of the underline probability space. However, in this
paper we show that it is possible to define it in an unspecified probability
space. For this purpose, we write down explicitly all the finite dimensional
probability density functions. Moreover, we provide different ggBm
characterizations. The role of the M-Wright function, which is related to the
fundamental solution of the time-fractional diffusion equation, emerges as a
natural generalization of the Gaussian distribution. Furthermore, we show that
ggBm can be represented in terms of the product of a random variable, which is
related to the M-Wright function, and an independent fractional Brownian
motion. This representation highlights the $H$-{\bf sssi} nature of the ggBm
and provides a way to study and simulate the trajectories. For this purpose, we
developed a random walk model based on a finite difference approximation of a
partial integro-differenital equation of fractional type.
http://arxiv.org/abs/0801.4879
Author(s): Florent Benaych-Georges (PMA)
Abstract: Debbah and Ryan have recently proved a result about the limit empirical
singular distribution of the sum of two rectangular random matrices whose
dimensions tend to infinity. In this paper, we reformulate it in terms of the
rectangular free convolution introduced in a previous paper and then we give a
new, shorter, proof of this result under weaker hypothesis: we do not suppose
the \pro measure in question in this result to be compactly supported anymore.
At last, we discuss the inclusion of this result in the family of relations
between rectangular and square random matrices.
http://arxiv.org/abs/0807.0505
Author(s): Bernard Bercu and Benoite de Saporta and Anne Gegout-Petit
Abstract: We study the asymptotic behavior of the least squares estimators of the
unknown parameters of bifurcating autoregressive processes. Under very weak
assumptions on the driven noise of the process, namely conditional pair-wise
independence and suitable moment conditions, we establish the almost sure
convergence of our estimators together with the quadratic strong law and the
central limit theorem. All our analysis relies on non-standard asymptotic
results for martingales.
http://arxiv.org/abs/0807.0528
Author(s): Bo Chen and Daniel Ford and Matthias Winkel
Abstract: We introduce a simple tree growth process that gives rise to a new
two-parameter family of discrete fragmentation trees that extends Ford's alpha
model to multifurcating trees and includes the trees obtained by uniform
sampling from Duquesne and Le Gall's stable continuum random tree. We call
these new trees the alpha-gamma trees. In this paper, we obtain their splitting
rules, dislocation measures both in ranked order and in sized-biased order, and
we study their limiting behaviour.
http://arxiv.org/abs/0807.0554
Author(s): Domenico Marinucci and Giovanni Peccati (LSTA)
Abstract: We characterize the angular polyspectra, of arbitrary order, associated with
isotropic fields defined on the sphere S^2. Our techniques rely heavily on
group representation theory, and specifically on the properties of Wigner
matrices and Clebsch-Gordan coefficients. The findings of the present paper
constitute a basis upon which one can build formal procedures for the
statistical analysis and the probabilistic modelization of the Cosmic Microwave
Background radiation, which is currently a crucial topic of investigation in
cosmology. We also outline an application to random data compression and
"simulation" of Clebsch-Gordan coefficients.
http://arxiv.org/abs/0807.0687
Author(s): Jean-Fran\c{c}ois Delmas (CERMICS) and Laurence Marsalle (LPP)
Abstract: We consider the bifurcating Markov chain model introduced by Guyon to detect
cellular aging from cell lineage. To take into account the possibility for a
cell to die, we use an underlying Galton-Watson process to describe the
evolution of the cell lineage. We give in this more general framework a weak
law of large number, an invariance principle and thus fluctuation results for
the average over one generation or up to one generation. We also prove the
fluctuations over each generation are independent. Then we present the natural
modifications of the tests given by Guyon in cellular aging detection within
the particular case of the auto-regressive model.
http://arxiv.org/abs/0807.0749
Author(s): Vincent Lemaire (PMA) and Gilles Pag\`es (PMA)
Abstract: We propose an unconstrained stochastic approximation method of finding the
optimal measure change (in an a priori parametric family) for Monte Carlo
simulations. We consider different parametric families based on the Girsanov
theorem and the Esscher transform (or exponential-tilting). In a
multidimensional Gaussian framework, Arouna uses a projected Robbins-Monro
procedure to select the parameter minimizing the variance. In our approach, the
parameter (scalar or process) is selected by a classical Robbins-Monro
procedure without projection or truncation. To obtain this unconstrained
algorithm we intensively use the regularity of the density of the law without
assume smoothness of the payoff. We prove the convergence for a large class of
multidimensional distributions and diffusion processes. We illustrate the
effectiveness of our algorithm via pricing a Basket payoff under a
multidimensional NIG distribution, and pricing a barrier options in different
markets.
http://arxiv.org/abs/0807.0762
Author(s): Kenneth Falconer and Ronan Le Gu\'evel (LMJL) and Jacques L\'evy-V\'ehel (INRIA Saclay - Ile de France)
Abstract: We study a particular class of moving average processes which possess a
property called localisability. This means that, at any given point, they admit
a ``tangent process'', in a suitable sense. We give general conditions on the
kernel g defining the moving average which ensures that the process is
localisable and we characterize the nature of the associated tangent processes.
Examples include the reverse Ornstein-Uhlenbeck process and the multistable
reverse Ornstein-Uhlenbeck process. In the latter case, the tangent process is,
at each time t, a L\'evy stable motion with stability index possibly varying
with t. We also consider the problem of path synthesis, for which we give both
theoretical results and numerical simulations.
http://arxiv.org/abs/0807.0764
Author(s): Amel Bentata (PMA) and Marc Yor (PMA and Iuf)
Abstract: These notes are the second half of the contents of the course given by the
second author at the Bachelier Seminar (8-15-22 February 2008) at IHP. They
also correspond to topics studied by the first author for her Ph.D.thesis.
http://arxiv.org/abs/0807.0788
Author(s): Xian-Yuan Wu and Ping Feng
Abstract: We consider the Bernoulli first-passage percolation on $\mathbb Z^d (d\ge
2)$. That is, the edge passage time is taken independently to be 1 with
probability $1-p$ and 0 otherwise. Let ${\mu(p)}$ be the time constant. We
prove in this paper that \[ \mu(p_1)-\mu({p_2})\ge
\frac{\mu(p_2)}{1-p_2}(p_2-p_1)\] for all $ 0\leq p_1
http://arxiv.org/abs/0807.0839
Author(s): Vadim A. Kaimanovich and Vincent Le Prince
Abstract: Any Zariski dense countable subgroup of $SL(d,R)$ is shown to carry a
non-degenerate finitely supported symmetric random walk such that its harmonic
measure on the flag space is singular. The main ingredients of the proof are:
(1) a new upper estimate for the Hausdorff dimension of the projections of the
harmonic measure onto Grassmannians in $R^d$ in terms of the associated
differential entropies and differences between the Lyapunov exponents; (2) an
explicit construction of random walks with uniformly bounded entropy and
Lyapunov exponents going to infinity.
http://arxiv.org/abs/0807.1015
Author(s): Raoul Robert (IF) and Vincent Vargas (CEREMADE)
Abstract: In this article, we extend the theory of multiplicative chaos for positive
definite functions in Rd of the form f(x) = 2 ln+ T|x|+ g(x) where g is a
continuous and bounded function. The construction is simpler and more general
than the one defined by Kahane in 1985. As main application, we give a rigorous
mathematical meaning to the Kolmogorov-Obukhov model of energy dissipation in a
turbulent flow.
http://arxiv.org/abs/0807.1030
Author(s): R\'emi Rhodes (CEREMADE) and Vincent Vargas (CEREMADE)
Abstract: We consider the continuous model of log-infinitely divisible multifractal
random mea- sures (MRM) introduced in [1]. If M is a non degenerate
multifractal measure with associated metric rho(x, y) = M ([x, y]) and
structure function zeta, we show that we have the following relation between
the (Euclidian) Hausdorff dimension dimH of a measurable set K and the
Hausdorff dimension dim_H with respect to rho of the same set: $\zeta({\rm
dim}_H^{\rho}(K))={\r m dim}_H(K)$.
http://arxiv.org/abs/0807.1036
Author(s): Raoul Robert (IF) and Vincent Vargas (CEREMADE)
Abstract: In this article, we construct two families of nonsymmetrical multifractal
fields. One of these families is used for the modelization of the velocity
field of turbulent flows.
http://arxiv.org/abs/0807.1042
Author(s): Ramon van Handel
Abstract: The nonlinear filter associated with the discrete time signal-observation
model $(X_k,Y_k)$ is known to forget its initial condition as $k\to\infty$
regardless of the observation structure when the signal possesses sufficiently
strong ergodic properties. Conversely, it stands to reason that if the
observations are sufficiently informative, then the nonlinear filter should
forget its initial condition regardless of the properties of the signal. We
show that the latter is indeed the case for the additive noise model
$Y_k=h(X_k)+\xi_k$ with invertible observation function $h$ under mild
regularity assumptions on $h$ and on the distribution of the noise variables
$\xi_k$.
http://arxiv.org/abs/0807.1072
Author(s): Gregory Schehr and Satya N. Majumdar and Alain Comtet and Julien Randon-Furling
Abstract: We study p non intersecting one-dimensional Brownian walks, either excursions
(p-watermelons with a wall) or bridges (p-watermelons without wall). We focus
on the maximal height H_p of these p-watermelons configurations on the unit
time interval. Using path integral techniques associated to corresponding
models of free Fermions, we compute exactly the distribution of H_p for generic
integer p. For large p, one obtains < H_p > \sim \sqrt{2p} for p-watermelons
with a wall whereas < H_p > \sim \sqrt{p} for p-watermelons without wall. We
point out and solve a discrepancy between these exact asymptotic behaviors and
numerical experiments, which recently attracted much attention, and we show
that only the pre-asymptotic behaviors of these averages were actually
measured. In addition, our method, using tools of many-body physics, provides a
simpler physical derivation of the connection between vicious walkers and
random matrix theory.
http://arxiv.org/abs/0807.0522
Author(s): Claus K\"ostler and Roland Speicher
Abstract: We show that the classical de Finetti theorem has a canonical noncommutative
counterpart if we strengthen `exchangeability' (i.e., invariance of the joint
distribution of the random variables under the action of the permutation group)
to invariance under the action of the quantum permutation group. More
precisely, for an infinite sequence of noncommutative random variables, we
prove that invariance of their joint distribution under quantum permutations is
equivalent to the fact that the random variables are identically distributed
and free with respect to the conditional expectation onto their tail algebra.
http://arxiv.org/abs/0807.0677
Author(s): Dennis Cheung and Felipe Cucker
Abstract: We prove an O(log n) bound for the expected value of the logarithm of the
componentwise (and, a fortiori, the mixed) condition number of a random sparse
n x n matrix. As a consequence, small bounds on the average loss of accuracy
for triangular linear systems follow.
http://arxiv.org/abs/0807.0956
Author(s): Florian Simatos and Danielle Tibi (PMA and INRIA Rocquencourt)
Abstract: A stochastic model for a mobile network is studied. Users enter the network,
and then perform independent Markovian routes between nodes, where they receive
service according to the Processor-Sharing policy. Once their service
requirement is satisfied, they leave the system. The stability region is
identified via a fluid limit approach, and strongly relies on a ``spatial
homogenization'' property: At the fluid level, customers are instantaneously
distributed across the network according to the stationary distribution of
their Markovian dynamics and stay distributed as such as long as the network is
not empty. In the unstable regime, spatial homogenization almost surely holds
asymptotically as time goes to infinity (on the normal scale), telling how the
system fills up. One of the technical achievements of the paper is the
construction of a family of martingales associated to the multidimensional
process of interest, which makes it possible to get crucial estimates for
certain exit times.
http://arxiv.org/abs/0807.1205
Author(s): Alexandra Chronopoulou and Ciprian Tudor (CES and SAMOS) and Frederi Viens
Abstract: We consider the class of all the Hermite processes $(Z_{t}^{(q,H)})_{t\in
\lbrack 0,1]}$ of order $q\in \mathbf{N}^{\ast}$ and with Hurst parameter $%
H\in ({1/2},1)$. The process $Z^{(q,H)}$ is $H$-selfsimilar, it has stationary
increments and it exhibits long-range dependence identical to that of
fractional Brownian motion (fBm). For $q=1$, $Z^{(1,H)}$ is fBm, which is
Gaussian; for $q=2$, $Z^{(2,H)}$ is the Rosenblatt process, which lives in the
second Wiener chaos; for any $q>2$, $Z^{(q,H)}$ is a process in the $q$th
Wiener chaos. We study the variations of $Z^{(q,H)}$ for any $q$, by using
multiple Wiener -It\^{o} stochastic integrals and Malliavin calculus. We prove
a reproduction property for this class of processes in the sense that the terms
appearing in the chaotic decomposition of their variations give rise to other
Hermite processes of different orders and with different Hurst parameters. We
apply our results to construct a strongly consistent estimator for the
self-similarity parameter $H$ from discrete observations of $Z^{(q,H)}$; the
asymptotics of this estimator, after appropriate normalization, are proved to
be distributed like a Rosenblatt random variable (value at time 1 of a
Rosenblatt process).with self-similarity parameter $1+2(H-1)/q$.
http://arxiv.org/abs/0807.1208
Author(s): Peter Spreij (University of Amsterdam) and Enno Veerman (University of Amsterdam)
Abstract: In affine term structure models the short rate is modelled as an affine
transformation of a multi-dimensional square root process. Sufficient
conditions to avoid negative volatility factors are the multivariate Feller
conditions. We will prove their necessity for a 2-dimensional square root SDE
with one volatility factor by presenting a methodology based on measure
transformations and solving linear systems of ordinary differential equations.
http://arxiv.org/abs/0807.1224
Author(s): Friedrich Hubalek (Technical University of Vienna) Carlo Sgarra (Technical University of Milan)
Abstract: We compute and discuss the Esscher martingale transform for exponential
processes, the Esscher martingale transform for linear processes, the minimal
martingale measure, the class of structure preserving martingale measures, and
the minimum entropy martingale measure for stochastic volatility models of
Ornstein-Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We
show, that in the model with leverage, with jumps both in the volatility and in
the returns, all those measures are different, whereas in the model without
leverage, with jumps in the volatility only and a continuous return process,
several measures coincide, some simplifications can be made and the results are
more explicit. We illustrate our results with parametric examples used in the
literature.
http://arxiv.org/abs/0807.1227
Author(s): Dorje C. Brody and Mark H. A. Davis and Robyn L. Friedman and Lane P. Hughston
Abstract: A model is introduced in which there is a small agent who is more susceptible
to the flow of information in the market than the general market participant,
and who tries to implement strategies based on the additional information. In
this model market participants have access to a stream of noisy information
concerning the future return of an asset, whereas the informed trader has
access to a further information source which is obscured by an additional noise
that may be correlated with the market noise. The informed trader uses the
extraneous information source to seek statistical arbitrage opportunities,
while at the same time accommodating the additional risk. The amount of
information available to the general market participant concerning the asset
return is measured by the mutual information of the asset price and the
associated cash flow. The worth of the additional information source is then
measured in terms of the difference of mutual information between the general
market participant and the informed trader. This difference is shown to be
nonnegative when the signal-to-noise ratio of the information flow is known in
advance. Explicit trading strategies leading to statistical arbitrage
opportunities, taking advantage of the additional information, are constructed,
illustrating how excess information can be translated into profit.
http://arxiv.org/abs/0807.1253
Author(s): Federico Bassetti
Abstract: The main object of Bayesian statistical inference is the determination of
posterior distributions. Sometimes these laws are given for quantities devoid
of empirical value. This serious drawback vanishes when one confines oneself to
considering a finite horizon framework. However, assuming infinite
exchangeability gives rise to fairly tractable {\it a posteriori} quantities,
which is very attractive in applications. Hence, with a view to a
reconciliation between these two aspects of the Bayesian way of reasoning, in
this paper we provide quantitative comparisons between posterior distributions
of finitary parameters and posterior distributions of allied parameters
appearing in usual statistical models.
http://arxiv.org/abs/0807.1201
Author(s): Joerg Kampen and Anastasia Kolodko and and John Schoenmakers
Abstract: In this paper we introduce efficient Monte Carlo estimators for the valuation
of high-dimensional derivatives and their sensitivities (''Greeks''). These
estimators are based on an analytical, usually approximative representation of
the underlying density. We study approximative densities obtained by the WKB
method. The results are applied in the context of a Libor market model.
http://arxiv.org/abs/0807.1213
Author(s): Bernard Bercu and Francois Dufour and G. George Yin
Abstract: This work is devoted to the almost sure stabilization of adaptive control
systems that involve an unknown Markov chain. The control system displays
continuous dynamics represented by differential equations and discrete events
given by a hidden Markov chain. Different from previous work on stabilization
of adaptive controlled systems with a hidden Markov chain, where average
criteria were considered, this work focuses on the almost sure stabilization or
sample path stabilization of the underlying processes. Under simple conditions,
it is shown that as long as the feedback controls have linear growth in the
continuous component, the resulting process is regular. Moreover, by
appropriate choice of the Lyapunov functions, it is shown that the adaptive
system is stabilizable almost surely. As a by-product, it is also established
that the controlled process is positive recurrent.
http://arxiv.org/abs/0807.1413
Author(s): Adrien Richou (IRMAR)
Abstract: We study a new class of ergodic backward stochastic differential equations
(EBSDEs for short) which is linked with semi-linear Neumann type boundary value
problems related to ergodic phenomenas. The particularity of these problems is
that the ergodic constant appears in Neumann boundary conditions. We study the
existence and uniqueness of solutions to EBSDEs and the link with partial
differential equations. Then we apply these results to optimal ergodic control
problems.
http://arxiv.org/abs/0807.1521
Author(s): Sonny Ben-Shimon and Michael Krivelevich
Abstract: In this work we analyze the distribution of the number of edges spanned by a
single set and between two disjoint sets of vertices in the random regular
graph model $\mathcal{G}_{n,d}$ in the range $d=o(\sqrt n)$. We show it to be
very similar to that of the binomial random graph model, $\mathcal{G}(n,p)$
with $p=\frac{d}{n}$. Some graph properties are known to exist solely based on
the edge distribution of the graph. We demonstrate how our analysis can be used
to prove the high probability of existence of such graph properties in the
$\mathcal{G}_{n,d}$ probability space, specifically spectral gap and
Hamiltonicity. Next, using our results on the distribution of edges spanned by
a single set of vertices, we show that for every fixed $\epsilon>0$ and
$d=o(n^{1/5})$, the chromatic number of $\mathcal{G}_{n,d}$ is concentrated in
two consecutive values with probability at least $1-\epsilon$ for large enough
values of $n$.
http://arxiv.org/abs/math/0511343
Author(s): David Gamarnik and Dmitriy Katz
Abstract: We propose a new method for the problems of computing free energy and surface
pressure for various statistical mechanics models on a lattice $\Z^d$. Our
method is based on representing the free energy and surface pressure in terms
of certain marginal probabilities in a suitably modified sublattice of $\Z^d$.
Then recent deterministic algorithms for computing marginal probabilities are
used to obtain numerical estimates of the quantities of interest. The method
works under the assumption of Strong Spatial Mixing (SSP), which is a form of a
correlation decay.
We illustrate our method for the hard-core and monomer-dimer models, and
improve several earlier estimates. For example we show that the exponent of the
monomer-dimer coverings of $\Z^3$ belongs to the interval $[0.78595,0.78599]$,
improving best previously known estimate of (approximately) $[0.7850,0.7862]$
obtained in \cite{FriedlandPeled},\cite{FriedlandKropLundowMarkstrom}.
Moreover, we show that given a target additive error $\epsilon>0$, the
computational effort of our method for these two models is
$(1/\epsilon)^{O(1)}$ \emph{both} for free energy and surface pressure. In
contrast, prior methods, such as transfer matrix method, require
$\exp\big((1/\epsilon)^{O(1)}\big)$ computation effort.
http://arxiv.org/abs/0807.1551
Author(s): Geoffrey R. Grimmett and Thomas M. Liggett and Thomas Richthammer
Abstract: We consider a type of long-range percolation problem on the positive
integers, motivated by earlier work of others on the appearance of (in)finite
words within a site percolation model. The main issue is whether a given
infinite binary word appears within an iid Bernoulli sequence at locations that
satisfy certain constraints. We settle the issue in some cases, and provide
partial results in others.
http://arxiv.org/abs/0807.1676
Author(s): Nils Berglund (MAPMO) and Barbara Gentz
Abstract: The Eyring-Kramers law describes the mean transition time of an overdamped
Brownian particle between local minima in a potential landscape. In the
weak-noise limit, the transition time is to leading order exponential in the
potential difference to overcome. This exponential is corrected by a prefactor
which depends on the principal curvatures of the potential at the starting
minimum and at the highest saddle crossed by an optimal transition path. The
Eyring-Kramers law, however, does not hold whenever one of these principal
curvatures vanishes, since it would predict a vanishing or infinite transition
time. We derive the correct prefactor up to multiplicative errors that tend to
one in the zero-noise limit. As an illustration, we discuss the case of a
symmetric pitchfork bifurcation, in which the prefactor can be expressed in
terms of modified Bessel functions. The results extend work by Bovier, Eckhoff,
Gayrard and Klein, who rigorously analysed the case of quadratic saddles, using
methods from potential theory.
http://arxiv.org/abs/0807.1681
Author(s): Gregorio Moreno Flores (PMA)
Abstract: We consider the model of Directed Polymers in an i.i.d. gaussian or bounded
Environment in the $L^2$ region. We prove the convergence of the law of the
environment seen by the particle. As a main technical step, we establish a
lower tail concentration inequality for the partition function for bounded
environments. Our proof is based on arguments developed by Talagrand in the
context of the Hopfield Model. This improves in some sense a concentration
inequality obtained by Carmona and Hu for gaussian environments. We use this
and a Local Limit Theorem to prove the $L^1$ convergence of the density of the
law of the environment seen by the particle with respect to the product
measure.
http://arxiv.org/abs/0807.1685
Author(s): Craig A. Tracy and Harold Widom
Abstract: In previous work the authors considered the asymmetric simple exclusion
process on the integer lattice in the case of step initial condition, particles
beginning at the positive integers. There it was shown that the probability
distribution for the position of an individual particle is given by an integral
whose integrand involves a Fredholm determinant. Here we use this formula to
obtain two asymptotic results for the positions of these particles.
http://arxiv.org/abs/0807.1713
Author(s): Igor Chueshov and Annie Millet (PMA and Matisse and Samos)
Abstract: We deal with a class of abstract nonlinear stochastic models, which covers
many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD
models and 2D magnetic B\'enard problem and also some shell models of
turbulence. We first prove the existence and uniqueness theorem for the class
considered. Our main result is a Wentzell-Freidlin type large deviation
principle for small multiplicative noise which we prove by weak convergence
method.
http://arxiv.org/abs/0807.1810
Author(s): Said Hamadene and Alexandre Popier
Abstract: This paper deals with the problem of existence and uniqueness of a solution
for a backward stochastic differential equation (BSDE for short) with one
reflecting barrier in the case when the terminal value, the generator and the
obstacle process are Lp-integrable with p in ]1,2[. To construct the solution
we use two methods: penalization and Snell envelope. As an application we
broaden the class of functions for which the related obstacle partial
differential equation problem has a unique viscosity solution.
http://arxiv.org/abs/0807.1846
Author(s): Rafael Diaz and Eddy Pariguan
Abstract: We present study of the Gaussian q-measure introduced by Diaz and Teruel from
a probabilistic and a categorical viewpoint. We show that the Gaussian
q-measure interpolates between that uniform measure on the interval [-1,1] and
the Gaussian measure on the real line.
http://arxiv.org/abs/0807.1918
Author(s): Krzysztof Bogdan and Bart{\l}omiej Dyda
Abstract: We prove an optimal Hardy inequality for the fractional Laplacian on the
half-space.
http://arxiv.org/abs/0807.1825
Author(s): Nicolas Bouleau (CIRED and Cermics) and Laurent Denis (DP)
Abstract: We introduce a new approach to absolute continuity of laws of Poisson
functionals. It is based on the {\it energy image density} property for
Dirichlet forms and on what we call {\it the lent particle method} which
consists in adding a particle and taking it back after some calculation.
http://arxiv.org/abs/0807.1963
Author(s): Bela Bollobas and Svante Janson and Oliver Riordan
Abstract: In 2007 we introduced a general model of sparse random graphs with
independence between the edges. The aim of this paper is to present an
extension of this model in which the edges are far from independent, and to
prove several results about this extension. The basic idea is to construct the
random graph by adding not only edges but also other small graphs. In other
words, we first construct an inhomogeneous random hypergraph with independent
hyperedges, and then replace each hyperedge by a (perhaps complete) graph.
Although flexible enough to produce graphs with significant dependence between
edges, this model is nonetheless mathematically tractable. Indeed, we find the
critical point where a giant component emerges in full generality, in terms of
the norm of a certain integral operator, and relate the size of the giant
component to the survival probability of a certain (non-Poisson) multi-type
branching process. While our main focus is the phase transition, we also study
the degree distribution and the numbers of small subgraphs. We illustrate the
model with a simple special case that produces graphs with power-law degree
sequences with a wide range of degree exponents and clustering coefficients.
http://arxiv.org/abs/0807.2040
Author(s): Shaolin Ji and Zhen Wu and Li Zhou
Abstract: In this paper we study reflected backward stochastic differential equations
with a continuous, linear growth coefficient and two barriers which belong to
L^2. We prove that there exists at least by penalization method.
http://arxiv.org/abs/0807.2075
Author(s): Yong Ren and Xiliang Fan
Abstract: In this paper, we deal with a class of reflected backward stochastic
differential equations associated to the subdifferential operator of a lower
semi-continuous convex function driven by Teugels martingales associated with
L\'{e}vy process. We obtain the existence and uniqueness of solutions to these
equations by means of the penalization method. As its application, we give a
probabilistic interpretation for the solutions of a class of partial
differential-integral inclusions.
http://arxiv.org/abs/0807.2076
Author(s): Kotaro Endo and Muneya Matsui
Abstract: We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU)
process where the integrand is replaced by the exponential of an independent
L\'evy process. We call the process the generalized fractional
Ornstein-Uhlenbeck (GFOU) process. Alternatively, the process can be
constructed from a generalized Ornstein-Uhlenbeck (GOU) process using an
independent fractional Brownian motion (FBM) as integrator. We show that the
GFOU process is well-defined by checking the existence of the integral included
in the process, and investigate its properties. It is proved that the process
has a stationary version and exhibits long memory. We also find that the
process satisfies a certain stochastic differential equation. Our underlying
intention is to introduce long memory into the GOU process which has short
memory without losing the possibility of jumps. Note that both FOU and GOU
processes have found application in a variety of fields as useful alternatives
to the Ornstein-Uhlenbeck (OU) process.
http://arxiv.org/abs/0807.2110
Author(s): Andrea Macrina
Abstract: A new framework for asset pricing based on modelling the information
available to market participants is presented. Each asset is characterised by
the cash flows it generates. Each cash flow is expressed as a function of one
or more independent random variables called market factors or "X-factors". Each
X-factor is associated with a "market information process", the values of which
become available to market participants. In addition to true information about
the X-factor, the information process contains an independent "noise" term
modelled here by a Brownian bridge. The information process thus gives partial
information about the X-factor, and the value of the market factor is only
revealed at the termination of the process. The market filtration is assumed to
be generated by the information processes associated with the X-factors. The
price of an asset is given by the risk-neutral expectation of the sum of the
discounted cash flows, conditional on the information available from the
filtration. The theory is developed in some detail, with a variety of
applications to credit risk management, share prices, interest rates, and
inflation. A number of new exactly solvable models are obtained for the price
processes of various types of assets and derivative securities; and a novel
mechanism is proposed to account for the dynamics of stochastic volatility and
dynamic correlation. A discrete-time version of the information-based framework
is also developed, and is used to construct a new class of models for the real
and nominal interest rate term structures, and the dynamics of the associated
price index.
http://arxiv.org/abs/0807.2124
Author(s): Bruno Schapira (LM-Orsay) and Robert Young (IHES)
Abstract: We prove a sharp estimate on the expected value of the integral of the index
of a simple random walk on the square or triangular lattice. This gives new
lower bounds on the averaged Dehn function, which measures the expected area
needed to fill a random curve with a disc.
http://arxiv.org/abs/0807.2192
Author(s): Nasir Ganikhodjaev
Abstract: Applying non-ergodic quadratic stochastic operator the continual family of
weak ergodic non-homogeneous Markov chains is constructed.
http://arxiv.org/abs/0807.2012
Author(s): Lorenz A. Gilch
Abstract: We consider random walks on the set of all words over a finite alphabet such
that in each step only the last two letters of the current word may be modified
and only one letter may be adjoined or deleted. We assume that the transition
probabilities depend only on the last two letters of the current word.
Furthermore, we consider also the special case of random walks on free products
by amalgamation of finite groups which arise in a natural way from random walks
on the single factors. The aim of this paper is to compute several equivalent
formulas for the rate of escape with respect to natural length functions for
these random walks using different techniques.
http://arxiv.org/abs/0807.2264
Author(s): Jean B\'erard (ICJ) and Alejandro Ram\'irez
Abstract: We investigate the probabilities of large deviations for the position of the
front in a stochastic model of the reaction $X+Y \to 2X$ on the integer lattice
in which $Y$ particles do not move while $X$ particles move as independent
simple continuous time random walks of total jump rate $2$. For a wide class of
initial conditions, we prove that a large deviations principle holds and we
show that the zero set of the rate function is the interval $[0,v]$, where $v$
is the velocity of the front given by the law of large numbers. We also give
more precise estimates for the rate of decay of the slowdown probabilities. Our
results indicate a gapless property of the generator of the process as seen
from the front, as it happens in the context of nonlinear diffusion equations
describing the propagation of a pulled front into an unstable state.
http://arxiv.org/abs/0807.2349
Author(s): Nicolas Broutin and Philippe Flajolet
Abstract: This extended abstract is dedicated to the analysis of the height of
non-plane unlabelled rooted binary trees. The height of such a tree chosen
uniformly among those of size $n$ is proved to have a limiting theta
distribution, both in a central and local sense. Moderate as well as large
deviations estimates are also derived. The proofs rely on the analysis (in the
complex plane) of generating functions associated with trees of bounded height.
http://arxiv.org/abs/0807.2365
Author(s): Ryoki Fukushima
Abstract: We consider the annealed asymptotics for the survival probability of Brownian
motion among randomly distributed traps. The configuration of traps is given by
independent displacements of the lattice points. We determined the asymptotics
for the logarithm of the survival probability up to multiplicative constant. As
applications, we show the Lifshitz tail effect of the density of states of
associated random Schr\"{o}dinger operator and intermittency for the parabolic
Anderson problem.
http://arxiv.org/abs/0807.2486
Author(s): Teemu Pennanen
Abstract: This paper presents a stochastic model for discrete-time trading in financial
markets where trading costs are given by convex cost functions and portfolios
are constrained by convex sets. The model does not assume the existence of a
cash account/numeraire. In addition to classical frictionless markets and
markets with transaction costs or bid-ask spreads, our framework covers markets
with nonlinear illiquidity effects for large instantaneous trades. In the
presence of nonlinearities, the classical notion of arbitrage turns out to have
two equally meaningful generalizations, a marginal and a scalable one. We study
their relations to state price deflators by analyzing two auxiliary market
models describing the local and global behavior of the cost functions and
constraints.
http://arxiv.org/abs/0807.2526
Author(s): Matthias Birkner and Andreas Greven and Frank den Hollander
Abstract: When we cut an i.i.d. sequence of letters into words according to an
independent renewal process, we obtain an i.i.d. sequence of words. In the
annealed large deviation principle (LDP) for the empirical process of words,
the rate function is the specific relative entropy of the observed law of words
w.r.t. the reference law of words. In the present paper we consider the
quenched LDP, i.e., we condition on a typical letter sequence. We focus on the
case where the renewal process has an algebraic tail. The rate function turns
out to be a sum of two terms, one being the annealed rate function, the other
being proportional to the specific relative entropy of the observed law of
letters w.r.t. the reference law of letters, with the former being obtained by
concatenating the words and randomising the location of the origin. The
proportionality constant equals the tail exponent of the renewal process.
Earlier work by Birkner considered the case where the renewal process has an
exponential tail, in which case the rate function turns out to be the first
term on the set where the second term vanishes and to be infinite elsewhere.
We apply our LDP to prove that the radius of convergence of the moment
generating function of the collision local time of two strongly transient
random walks on \Z^d, d \geq 1, strictly increases when we condition on one of
the random walks, both in discrete time and in continuous time. The presence of
these gaps implies the existence of an intermediate phase for the long-time
behaviour of a class of coupled branching processes, interacting diffusions,
respectively, directed polymers in random environments.
http://arxiv.org/abs/0807.2611
Author(s): Codina Cotar and Jean-Dominique Deuschel
Abstract: We consider a gradient interface model on the lattice with interaction
potential which is a nonconvex perturbation of a convex potential. Using a
technique which decouples the neighboring vertices sites into even and odd
vertices, we show for a class of non-convex potentials: the uniqueness of
ergodic component for \nabla\phi-Gibbs measures, the decay of covariances, the
scaling limit and the strict convexity of the surface tension.
http://arxiv.org/abs/0807.2621
Author(s): Morris L. Eaton and Robb J. Muirhead
Abstract: Let H be a Haar distributed random matrix on the group of pxp real orthogonal
matrices. Partition H into four blocks: (1) the (1,1) element, (2)the rest of
the first row, (3) the rest of the first column, and (4)the remaining
(p-1)x(p-1) matrix. The marginal distribution of (1) is well known. In this
paper, we give the conditional distribution of (2) and (3) given (1), and the
conditional distribution of (4) given (1), (2), (3). This conditional
specification uniquely determines the Haar distribution. The two conditional
distributions involve well known probability distributions namely, the uniform
distribution on the unit sphere in p-1 dimensional space and the Haar
distribution on (p-2)x(p-2) orthogonal matrices. Our results show how to
construct the Haar distribution on pxp orthogonal matrices from the Haar
distribution on (p-2)x(p-2) orthogonal matrices coupled with the uniform
distribution on the unit sphere in p-1 dimensions.
http://arxiv.org/abs/0807.2598
Author(s): Alexander Gnedin and Alex Iksanov and Martin M\"ohle
Abstract: We study the number of collisions $X_n$ of an exchangeable coalescent with
multiple collisions ($\Lambda$-coalescent) which starts with $n$ particles and
is driven by rates determined by a finite characteristic measure $\nu({\rm
d}x)=x^{-2}\Lambda({\rm d}x)$. Via a coupling technique we derive limiting laws
of $X_n$, using previous results on regenerative compositions derived from
stick-breaking partitions of the unit interval. The possible limiting laws of
$X_n$ include normal, stable with index $1\le\alpha<2$ and Mittag-Leffler
distributions. The results apply, in particular, to the case when $\nu$ is a
beta$(a-2,b)$ distribution with parameters $a>2$ and $b>0$. The approach taken
allows to derive asymptotics of three other functionals of the coalescent, the
absorption time, the length of an external branch chosen at random from the $n$
external branches, and the number of collision events that occur before the
randomly selected external branch coalesces with one of its neighbours.
http://arxiv.org/abs/0807.2742
Author(s): Matthias Birkner and Rongfeng Sun
Abstract: We study a random walk pinning model, where conditioned on a simple random
walk Y on Z^d acting as a random medium, the path measure of a second
independent simple random walk X up to time t is Gibbs transformed with
Hamiltonian -L_t(X,Y), where L_t(X,Y) is the collision local time between X and
Y up to time t. This model arises naturally in various contexts, including the
study of the parabolic Anderson model with moving catalysts, the parabolic
Anderson model with Brownian noise, and the directed polymer model. It falls in
the same framework as the pinning and copolymer models, and exhibits a
localization-delocalization transition as the inverse temperature \beta varies.
We show that in dimensions d=1,2, the annealed and quenched critical values of
\beta are both 0, while in dimensions d\geq 4, the quenched critical value of
\beta is strictly larger than the annealed critical value (which is positive).
This implies the existence of certain intermediate regimes for the parabolic
Anderson model with Brownian noise and the directed polymer model. For d\geq 5,
the same result has recently been established by Birkner, Greven and den
Hollander via a quenched large deviation principle. Our proof is based on a
fractional moment method used recently by Derrida, Giacomin, Lacoin and
Toninelli to establish the non-coincidence of annealed and quenched critical
points for the pinning model in the disorder-relevant regime. The critical case
d=3 remains open.
http://arxiv.org/abs/0807.2752
Author(s): Xian-Yuan Wu and Zhao Dong and Ke Liu and Kai-Yuan Cai
Abstract: This paper focuses on the problem of the degree sequence for a mixed random
graph process which continuously combines the {\it classical} model and the BA
model. Note that the number of step added edges for the mixed model are random
and unbounded. By developing a comparing argument, phase transition on the
degree distributions of the mixed model is revealed: while the {\it pure
classical model possesses a {\it exponential} degree sequence, the {\it pure}
BA model and the mixed model possess {\it power law} degree sequences. We point
out that the intermediate mixed model can be looked as a BA model with {\it
sublinear preferential attachment}.
http://arxiv.org/abs/0807.2811
Author(s): Gao-Rong Ning and Xian-Yuan Wu and Kai-Yuan Cai
Abstract: We consider a simple random graph process with {\it hard} copying as
following: At any Time-Step $t$, with probability $0<\alpha\leq 1$ a new vertex
$v_t$ is added and then $m$ edges incident with $v_t$ are added in the manner
of {\it preferential attachment}; or with probability $1-\alpha$ a existing
vertex is copied uniformly at random. We prove in the paper that, when $\alpha$
large enough, the model possesses a mean degree sequence as $ d_{k}\thicksim
Ck^{-(1+2\alpha)}$, where $d_k$ be the limit mean proportion of vertices of
degree $k$. Note that in the present model, while a vertex with large degree is
copied, the number of added edges is just its degree, so the number of added
edges is not upper bounded.
http://arxiv.org/abs/0807.2819
Author(s): Konstantinos Spiliopoulos
Abstract: Ornstein-Uhlenbeck processes driven by general L\'{e}vy process are
considered in this paper. We derive strongly consistent estimators for the
moments of the underlying L\'{e}vy process and for the mean reverting parameter
of the Ornstein-Uhlenbeck process. Moreover, we prove that the estimators are
asymptotically normal. Finally, we test the empirical performance of our
estimators in a simulation study and we fit the model to real data.
http://arxiv.org/abs/0807.2832
Author(s): Irene Valsecchi
Abstract: The survey is concerned with the issue of information transmission from
experts to non-experts. Two main approaches to the use of experts can be
traced. According to the game-theoretic approach expertise is a case of
asymmetric information between the expert, who is the better informed agent,
and the non-expert, who is either a decision-maker or an evaluator of the
expert's performance. According to the Bayesian decision-theoretic approach the
expert is the agent who announces his probabilistic opinion, and the non-expert
has to incorporate that opinion into his beliefs in a consistent way, despite
his poor understanding of the expert's substantive knowledge. The two
approaches ground the relationships between experts and non-experts on so
different premises that their results are very poorly connected.
http://arxiv.org/abs/0807.2931
Author(s): Yves Atchad\'e and Gersende Fort (LTCI)
Abstract: This paper deals with the ergodicity and the existence of a strong law of
large numbers for adaptive Markov Chain Monte Carlo. We show that a diminishing
adaptation assumption together with a drift condition for positive recurrence
is enough to imply ergodicity. Strengthening the drift condition to a
polynomial drift condition yields a strong law of large numbers for possibly
unbounded functions. These results broaden considerably the class of adaptive
MCMC algorithms for which rigorous analysis is now possible. As an example, we
give a detailed analysis of the Adaptive Metropolis Algorithm of Haario et al.
(2001) when the target distribution is sub-exponential in the tails.
http://arxiv.org/abs/0807.2952
Author(s): D. Coupier and P. Heinrich
Abstract: A three types competition model governed by directed last passage percolation
on $\mathbb{N}^{2}$ is considered. We prove that coexistence of the three
types, i.e. the sets of vertices of the three types are simultaneously
unbounded, occurs with positive probability. Moreover, the asymptotic angles
formed by the two competition interfaces with the horizontal axis are
determined and their probability of being different is positive. As a key step,
a stochastic domination between subtrees of the last passage percolation tree
is obtained.
http://arxiv.org/abs/0807.2987
Author(s): Teemu Pennanen
Abstract: We study contingent claims in a discrete-time market model where trading
costs are given by convex functions and portfolios are constrained by convex
sets. In addition to classical frictionless markets and markets with
transaction costs or bid-ask spreads, our framework covers markets with
nonlinear illiquidity effects for large instantaneous trades. We derive dual
characterizations of superhedging conditions for contingent claim processes in
a market without a cash account. The characterizations are given in terms of
stochastic discount factors that correspond to martingale densities in a market
with a cash account. The dual representations are valid under a topological
condition and a weak consistency condition reminiscent of the ``law of one
price'', both of which are implied by the no arbitrage condition in the case of
classical perfectly liquid market models. We give alternative sufficient
conditions that apply to market models with nonlinear cost functions and
portfolio constraints.
http://arxiv.org/abs/0807.2962
Author(s): Giuseppina Guatteri
Abstract: In this paper we prove necessary conditions for optimality of a stochastic
control problem for a class of stochastic partial differential equations that
is controlled through the boundary. This kind of problems can be interpreted as
a stochastic control problem for an evolution system in an Hilbert space. The
regularity of the solution of the adjoint equation, that is a backward
stochastic equation in infinite dimension, plays a crucial role in the
formulation of the maximum principle.
http://arxiv.org/abs/0807.3096
Author(s): Patrick Cattiaux (LSProba) and Nathael Gozlan (LAMA) and Arnaud Guillin (LATP), Cyril Roberto (LAMA)
Abstract: This paper is devoted to the study of probability measures with heavy tails.
Using the Lyapunov function approach we prove that such measures satisfy
different kind of functional inequalities such as weak Poincar\'e and weak
Cheeger, weighted Poincar\'e and weighted Cheeger inequalities and their dual
forms. Proofs are short and we cover very large situations. For product
measures on $\R^n$ we obtain the optimal dimension dependence using the mass
transportation method. Then we derive (optimal) isoperimetric inequalities.
Finally we deal with spherically symmetric measures. We recover and improve
many previous results.
http://arxiv.org/abs/0807.3112
Author(s): Joachim Hermisson and Peter Pfaffelhuber
Abstract: Genetic hitchhiking describes evolution at a neutral locus that is linked to
a selected locus. If a beneficial allele rises to fixation at the selected
locus, a characteristic polymorphism pattern (so-called selective sweep)
emerges at the neutral locus. The classical model assumes that fixation of the
beneficial allele occurs from a single copy of this allele that arises by
mutation. However, recent theory (Pennings and Hermisson, 2006a; Pennings and
Hermisson, 2006b) has shown that recurrent beneficial mutation at biologically
realistic rates can lead to markedly different polymorphism patterns, so called
soft selective sweeps. We extend an approach that has recently been developed
for the classical hitchhiking model (Schweinsbergand Durrett, 2005; Etheridge,
Pfaffelhuber, Wakolbinger, 2006) to study the recurrent mutation scenario. We
show that the genealogy at the neutral locus can be approximated (to leading
orders in the selection strength) by a marked Yule process with immigration.
Using this formalism, we derive an improved analytical approximation for the
expected heterozygosity at the neutral locus at the time of fixation of the
beneficial allele.
http://arxiv.org/abs/0807.3167
Author(s): Frank den Hollander and Nicolas Petrelis
Abstract: The present paper is a continuation of \cite{dHP07b}. The object of interest
is a two-dimensional model of a directed copolymer, consisting of a random
concatenation of hydrophobic and hydrophilic monomers, immersed in an emulsion,
consisting of large blocks of oil and water arranged in a percolation-type
fashion. The copolymer interacts with the emulsion through an interaction
Hamiltonian that favors matches and disfavors mismatches between the monomers
and the solvents, in such a way that the interaction with the oil is stronger
than with the water.
The model has two regimes, supercritical and subcritical, depending on
whether the oil blocks percolate or not. In \cite{dHP07b} we focussed on the
supercritical regime and obtained a complete description of the phase diagram,
which consists of two phases separated by a single critical curve. In the
present paper we focus on the subcritical regime and show that the phase
diagram consists of four phases separated by three critical curves meeting in
two tricritical points.
http://arxiv.org/abs/0807.3190
Author(s): Dapeng Zhan
Abstract: We prove that, for $\kappa\in(0,4)$ and $\rho\ge (\kappa-4)/2$, the chordal
SLE$(\kappa;\rho)$ trace started from $(0;0^+)$ or $(0;0^-)$ satisfies the
reversibility property. And we obtain the equation for the reversal of the
chordal SLE$(\kappa;\rho)$ trace started from $(0;b_0)$, where $b_0>0$.
http://arxiv.org/abs/0807.3265
Author(s): R.F. Bass and T. Kumagai and and T. Uemura
Abstract: For each $n$ let $Y^n_t$ be a continuous time symmetric Markov chain with
state space $n^{-1} \Z^d$. A condition in terms of the conductances is given
for the convergence of the $Y^n_t$ to a symmetric Markov process $Y_t$ on
$\R^d$. We have weak convergence of $\{Y^n_t: t\leq t_0\}$ for every $t_0$ and
every starting point. The limit process $Y$ has a continuous part and may also
have jumps.
http://arxiv.org/abs/0807.3268
Author(s): Itai Benjamini and Johan Jonasson and Oded Schramm and Johan Tykesson
Abstract: Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$,
whose law is invariant under isometries of $\H^2$. We prove that if the
probability that $Z$ contains a fixed ball of radius 1 is larger than some
universal constant $p<1$, then there is positive probability that $Z$ contains
(bi-infinite) lines.
We then consider a family of random sets in $\H^2$ that satisfy some
additional natural assumptions. An example of such a set is the covered region
in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment
of length $r$ is contained in such a set $Z$. We show that if $f(r)$ decays
fast enough, then there are almost surely no lines in $Z$. We also show that if
the decay of $f(r)$ is not too fast, then there are almost surely lines in $Z$.
In the case of the Poisson Boolean model with balls of fixed radius $R$ we
characterize the critical intensity for the almost sure existence of lines in
the covered region by an integral equation.
We also determine when there are lines in the complement of a Poisson process
on the Grassmannian of lines in $\H^2$.
http://arxiv.org/abs/0807.3308
Author(s): Rohini Kumar
Abstract: In a system made up of independent random walks, fluctuations of order
$n^{1/4}$ from the hydrodynamic limit come from particle current across
characteristics. We show that a two-parameter space-time particle current
process converges to a two-parameter Gaussian process. These Gaussian processes
also appear as the limit for the one-dimensional random average process. The
final section of this paper looks at large deviations of the current process.
http://arxiv.org/abs/0807.3313
Author(s): Theodore P. Hill
Abstract: Without additional hypotheses, Proposition 7.1 in Brams and Taylor's book
"Fair Division" (Cambridge University Press, 1996) is false, as are several
related Pareto-optimality theorems of Brams, Jones and Klamler in their 2006
cake-cutting paper.
http://arxiv.org/abs/0807.3117
Author(s): Pietro Caputo
Abstract: We give a new and elementary computation of the spectral gap of the Kac walk
on the N-sphere. The result is obtained as a by-product of a more general
observation which allows to reduce the analysis of the spectral gap of an
N-component system to that of the same system for N=3. The method applies to a
number of random 'binary collision' processes with complete-graph structure,
including non-homogeneous examples such as exclusion and colored exclusion
processes with site disorder.
http://arxiv.org/abs/0807.3415
Author(s): Nadine Guillotin-Plantard (ICJ) and Cl\'ementine Prieur (LSProba)
Abstract: Let $S=(S_k)_{k\geq 0}$ be a random walk on $\mathbb{Z}$ and
$\xi=(\xi_{i})_{i\in\mathbb{Z}}$ a stationary random sequence of centered
random variables, independent of $S$. We consider a random walk in random
scenery that is the sequence of random variables $(\Sigma_n)_{n\geq 0}$ where
$$\Sigma_n=\sum_{k=0}^n \xi_{S_k}, n\in\mathbb{N}.$$ Under a weak dependence
assumption on the scenery $\xi$ we prove a functional limit theorem
generalizing Kesten and Spitzer's theorem (1979).
http://arxiv.org/abs/0807.3441
Author(s): St\'ephan Cl\'emen\c{c}on (LTCI and METaRISK) and Viet Chi Tran (LPP) and Hector De Arazoza (MATCOM)
Abstract: A stochastic epidemic model accounting for the effect of contact-tracing on
the spread of an infectious disease is studied. Precisely, individuals
identified as infected may contribute to detecting other infectious individuals
by providing information related to persons with whom they have had possibly
infectious contacts. The population evolves through demographic, infection and
detection processes, in a way that its temporal evolution is described by a
stochastic Markov process, of which the component accounting for the
contact-tracing feature is assumed to be valued in a space of point measures.
For adequate scalings of the demographic, infection and detection rates, it is
shown to converge to the weak deterministic solution of a PDE system, as a
parameter n, interpreted as the population size roughly speaking, becomes
large. From the perspective of the analysis of infectious disease data, this
approximation result may serve as a key tool for exploring the asymptotic
properties of standard inference methods such as maximum likelihood estimation.
We state preliminary statistical results in this context. Eventually, relation
of the model to the available data of the HIV epidemic in Cuba, in which
country a contact-tracing detection system has been set up since 1986, is
investigated and numerical applications are carried out.
http://arxiv.org/abs/0807.3462
Author(s): Friedrich Hubalek and Petra Posedel
Abstract: We introduce a variant of the Barndorff-Nielsen and Shephard stochastic
volatility model where the non Gaussian Ornstein-Uhlenbeck process describes
some measure of trading intensity like trading volume or number of trades
instead of unobservable instantaneous variance. We develop an explicit
estimator based on martingale estimating functions in a bivariate model that is
not a diffusion, but admits jumps. It is assumed that both the quantities are
observed on a discrete grid of fixed width, and the observation horizon tends
to infinity. We show that the estimator is consistent and asymptotically normal
and give explicit expressions of the asymptotic covariance matrix. Our method
is illustrated by a finite sample experiment and a statistical analysis on the
International Business Machines Corporation (IBM) stock from the New York Stock
Exchange (NYSE) and the Microsoft Corporation (MSFT) stock from Nasdaq during a
history of five years.
http://arxiv.org/abs/0807.3464
Author(s): Friedrich Hubalek and Petra Posedel
Abstract: We provide a simple explicit estimator for discretely observed
Barndorff-Nielsen and Shephard models, prove rigorously consistency and
asymptotic normality based on the single assumption that all moments of the
stationary distribution of the variance process are finite, and give explicit
expressions for the asymptotic covariance matrix.
We develop in detail the martingale estimating function approach for a
bivariate model, that is not a diffusion, but admits jumps. We do not use
ergodicity arguments.
We assume that both, logarithmic returns and instantaneous variance are
observed on a discrete grid of fixed width, and the observation horizon tends
to infinity. As the instantaneous variance is not observable in practice, our
results cannot be applied immediately. Our purpose is to provide a theoretical
analysis as a starting point and benchmark for further developments concerning
optimal martingale estimating functions, and for theoretical and empirical
investigations, that replace the variance process with a substitute, such as
number or volume of trades or implied variance from option data.
http://arxiv.org/abs/0807.3479
Author(s): Isaac Meilijson
Abstract: Consider a random walk whose (light-tailed) increments have positive mean.
Lower and upper bounds are provided for the expected maximal value of the
random walk until it experiences a given drawdown d. These bounds, related to
the Calmar ratio in Finance, are of the form (exp{alpha d}-1)/alpha and (K
exp{alpha d}-1)/alpha for some K>1, in terms of the adjustment coefficient
alpha (E[exp{-alpha X}]=1) of the insurance risk literature. Its inverse
1/alpha has been recently derived by Aumann and Serrano as an index of
riskiness of the random variable X. This article also complements the Lundberg
exponential stochastic upper bound and the Cramer-Lundberg approximation for
the expected minimum of the random walk, with an exponential stochastic lower
bound. The tail probability bounds are of the form C exp{-alpha x} and
exp{-alpha x} respectively, for some 1/K < C < 1. Our treatment of the problem
involves Skorokhod embeddings of random walks in Martingales, especially via
the Azema-Yor and Dubins stopping times, adapted from standard Brownian Motion
to exponential Martingales.
http://arxiv.org/abs/0807.3506
Author(s): Rinaldo B. Schinazi
Abstract: We propose two simple probability models to compute the probability of an
influenza pandemic. Under a random walk model the probability that all
pandemics between times 0 and 300 occur by time 150 is 1/2. Under a Poisson
model with mean inter arrival time of 30 years the probability that no pandemic
occurs during at least 60 years is 14%. These probabilities are much higher
than generally perceived. So yes the next influenza pandemic will happen but
maybe much later than generally thought.
http://arxiv.org/abs/0807.3524
Author(s): P. Albini and E. De Vito and A. Toigo
Abstract: We define a positive operator valued measure $E$ on $[0,2\pi]\times R$
describing the measurement of randomly sampled quadratures in quantum homodyne
tomography, and we study its probabilistic properties. Moreover, we give a
mathematical analysis of the relation between the description of a state in
terms of $E$ and the description provided by its Wigner transform.
http://arxiv.org/abs/0807.3437
Author(s): Lester E. Dubins and David Gilat and Isaac Meilijson
Abstract: It is shown that the ratio between the expected diameter of an L2-bounded
martingale and the standard deviation of its last term cannot exceed sqrt(3).
Moreover, a one-parameter family of stopping times on standard Brownian Motion
is exhibited, for which the sqrt(3) upper bound is attained. These stopping
times, one for each cost-rate c, are optimal when the payoff for stopping at
time t is the diameter D(t) obtained up to time t minus the hitherto
accumulated cost c t. A quantity related to diameter, maximal drawdown (or
rise), is introduced and its expectation is shown to be bounded by sqrt(2)
times the standard deviation of the last term of the martingale. These results
complement the Dubins and Schwarz respective bounds 1 and sqrt(2) for the
ratios between the expected maximum and maximal absolute value of the
martingale and the standard deviation of its last term. Dynamic programming
(gambling theory) methods are used for the proof of optimality.
http://arxiv.org/abs/0807.3571
Author(s): Yael Dekel and James R. Lee and Nathan Linial
Abstract: We initiate a systematic study of eigenvectors of random graphs. Whereas much
is known about eigenvalues of graphs and how they reflect properties of the
underlying graph, relatively little is known about the corresponding
eigenvectors. Our main focus in this paper is on the nodal domains associated
with the different eigenfunctions. In the analogous realm of Laplacians of
Riemannian manifolds, nodal domains have been the subject of intensive research
for well over a hundred years. Graphical nodal domains turn out to have
interesting and unexpected properties. Our main theorem asserts that there is a
constant c such that for almost every graph G, each eigenfunction of G has at
most two large nodal domains, and in addition at most c exceptional vertices
outside these primary domains. We also discuss variations of these questions
and briefly report on some numerical experiments which, in particular, suggest
that almost surely there are just two nodal domains and no exceptional
vertices.
http://arxiv.org/abs/0807.3675
Author(s): Leonid V. Bogachev and Sakhavat M. Zarbaliev
Abstract: Let $\Pi_n$ be the set of planar convex lattice polygons $\Gamma$ (i.e., with
vertices on $\mathbb{Z}_+^2$ and non-negative inclination of all edges) with
fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the limit
shape of a typical polygon $\Gamma\in\Pi_n$ as $n\to\infty$ with respect to a
certain parametric family of probability measures $\{P_n^r\}$ ($0
http://arxiv.org/abs/0807.3682
Author(s): David L. Donoho and Jared Tanner
Abstract: Let $A$ be an $n$ by $N$ real valued random matrix, and $\h$ denote the
$N$-dimensional hypercube. For numerous random matrix ensembles, the expected
number of $k$-dimensional faces of the random $n$-dimensional zonotope $A\h$
obeys the formula $E f_k(A\h) /f_k(\h) = 1-P_{N-n,N-k}$, where $P_{N-n,N-k}$ is
a fair-coin-tossing probability. The formula applies, for example, where the
columns of $A$ are drawn i.i.d. from an absolutely continuous symmetric
distribution. The formula exploits Wendel's Theorem\cite{We62}.
Let $\po$ denote the positive orthant; the expected number of $k$-faces of
the random cone$A \po$ obeys $ {\cal E} f_k(A\po) /f_k(\po) = 1 - P_{N-n,N-k}$.
The formula applies to numerous matrix ensembles, including those with iid
random columns from an absolutely continuous, centrally symmetric distribution.
There is an asymptotically sharp threshold in the behavior of face counts of
the projected hypercube; thresholds known for projecting the simplex and the
cross-polytope, occur at very different locations. We briefly consider face
counts of the projected orthant when $A$ does not have mean zero; these do
behave similarly to those for the projected simplex. We consider non-random
projectors of the orthant; the 'best possible' $A$ is the one associated with
the first $n$ rows of the Fourier matrix.
These geometric face-counting results have implications for signal
processing, information theory, inverse problems, and optimization. Most of
these flow in some way from the fact that face counting is related to
conditions for uniqueness of solutions of underdetermined systems of linear
equations.
http://arxiv.org/abs/0807.3590
Author(s): Lo\"ic Chaumont (LAREMA) and L. Vostrikova (LAREMA)
Abstract: Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic process. We
prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real numbers which
converges to 0 and such that $M$ satisfies the reflection property at all
levels $a_n$ and $2a_n$ with $n\geq 1$, then $M$ is an Ocone local martingale
with respect to its natural filtration. We state the subsequent open question:
is this result still true when the property only holds at levels $a_n$? Then we
prove that the later question is equivalent to the fact that for Brownian
motion, the $\sigma$-field of the invariant events by all reflections at levels
$a_n$, $n\ge1$ is trivial. We establish similar results for skip free
$\mathbb{Z}$-valued processes and use them for the proof in continuous time,
via a discretisation in space.
http://arxiv.org/abs/0807.3816
Author(s): Salim Bouzebda (LSTA) and Nabil Nessigha (LSTA)
Abstract: In this paper we establish strong approximations of the uniform
non-overlapping m-spacings process extending the preceding results. Our methods
rely on the Mason and Van Zwet's invariance principle.
http://arxiv.org/abs/0807.3868
Author(s): Martin Hutzenthaler
Abstract: The BKR-inequality (a.k.a. BK-conjecture) asserts that the probability of two
events occurring on disjoint subsets of a finite family of independent
variables is dominated by the product of the probabilities of the two events.
This correlation inequality has been conjectured and partially established by
van den Berg and Kesten (1985) and has been proved by Reimer (2000). We give a
short proof of the BKR-inequality and generalize it to arbitrary families of
independent variables.
http://arxiv.org/abs/0807.4038
Author(s): Stefano Bonaccorsi and Delio Mugnolo
Abstract: Our investigation is specially motivated by the stochastic version of a
common model of potential spread in a dendritic tree. We do not assume the
noise in the junction points to be Markovian. In fact, we allow for long-range
dependence in time of the stochastic perturbation. This leads to an abstract
formulation in terms of a stochastic diffusion with dynamic boundary
conditions, featuring fractional Brownian motion. We prove results on
existence, uniqueness and asymptotics of weak and strong solutions to such a
stochastic differential equation.
http://arxiv.org/abs/0807.4057
Author(s): Abhimanyu Mitra and Sidney I. Resnick
Abstract: We study the tail behavior of the distribution of the sum of asymptotically
independent risks whose marginal distributions belong to the maximal domain of
attraction of the Gumbel distribution. We impose conditions on the distribution
of the risks $(X,Y)$ such that $P(X + Y > x) \sim (const)P (X > x)$. With the
further assumption of non-negativity of the risks, the result is extended to
more than two risks. We note a sufficient condition for a distribution to
belong to both the maximal domain of attraction of the Gumbel distribution and
the subexponential class. We provide examples of distributions which satisfy
our assumptions. The examples include cases where the marginal distributions of
$X$ and $Y$ are subexponential and also cases where they are not. In addition,
the asymptotic behavior of linear combinations of such risks with positive
coefficients is explored leading to an approximate solution of an optimization
problem which is applied to portfolio design.
http://arxiv.org/abs/0807.4200
Author(s): Francesco Caravenna and Giambattista Giacomin and Massimiliano Gubinelli
Abstract: We consider a general discrete model for heterogeneous semiflexible polymer
chains. Both the thermal noise and the inhomogeneous character of the chain
(the disorder) are modeled in terms of random rotations. We focus on the
quenched regime, i.e., the analysis is performed for a given realization of the
disorder. Semiflexible models differ substantially from random walks on short
scales, but on large scales a Brownian behavior emerges. By exploiting
techniques from tensor analysis and non-commutative Fourier analysis, we
establish the Brownian character of the model on large scale and we obtain an
expression for the diffusion constant. We moreover give conditions yielding
quantitative mixing properties.
http://arxiv.org/abs/0807.4232
Author(s): Julien Berestycki and Nathanael Berestycki and and Vlada Limic
Abstract: Consider a $\Lambda$-coalescent that comes down from infinity (meaning that
it starts from a configuration containing infinitely many blocks at time 0, yet
it has a finite number $N_t$ of blocks at any positive time $t>0$). We exhibit
a deterministic function $v:(0,\infty)\to (0,\infty)$, such that $N_t/v(t)\to
1$, almost surely and in $L^p$ for any $p\geq 1$, as $t\to 0$. Our approach
relies on a novel martingale technique.
http://arxiv.org/abs/0807.4278
Author(s): Kouji Yano and Yuko Yano and Marc Yor
Abstract: Limit theorems for the normalized laws with respect to two kinds of weight
functionals are studied for any symmetric stable L\'evy process of index $ 1 <
\alpha \le 2 $. The first kind is a function of the local time at the origin,
and the second kind is the exponential of an occupation time integral. Special
emphasis is put on the role played by a stable L\'evy counterpart of the
universal $ \sigma $-finite measure, found in [9] and [10], which unifies the
corresponding limit theorems in the Brownian setup for which $ \alpha =2 $.
http://arxiv.org/abs/0807.4336
Author(s): Seid Bahlali
Abstract: We consider a stochastic control problem where the set of strict (classical)
controls is not necessarily convex, and the system is governed by a nonlinear
backward stochastic differential equation. By introducing a new approach, we
establish necessary as well as sufficient conditions of optimality for two
models. The first concerns the relaxed controls, who are measure-valued
processes. The second is a particular case of the first and relates to strict
control problems.
http://arxiv.org/abs/0807.4297
Author(s): Michael Krivelevich and Benny Sudakov and Dan Vilenchik
Abstract: In this work we suggest a new model for generating random satisfiable k-CNF
formulas. To generate such formulas -- randomly permute all 2^k\binom{n}{k}
possible clauses over the variables x_1, ..., x_n, and starting from the empty
formula, go over the clauses one by one, including each new clause as you go
along if after its addition the formula remains satisfiable. We study the
evolution of this process, namely the distribution over formulas obtained after
scanning through the first m clauses (in the random permutation's order).
Random processes with conditioning on a certain property being respected are
widely studied in the context of graph properties. This study was pioneered by
Ruci\'nski and Wormald in 1992 for graphs with a fixed degree sequence, and
also by Erd\H{o}s, Suen, and Winkler in 1995 for triangle-free and bipartite
graphs. Since then many other graph properties were studied such as planarity
and H-freeness. Thus our model is a natural extension of this approach to the
satisfiability setting.
Our main contribution is as follows. For m \geq cn, c=c(k) a sufficiently
large constant, we are able to characterize the structure of the solution space
of a typical formula in this distribution. Specifically, we show that typically
all satisfying assignments are essentially clustered in one cluster, and all
but e^{-\Omega(m/n)} n of the variables take the same value in all satisfying
assignments. We also describe a polynomial time algorithm that finds with high
probability a satisfying assignment for such formulas.
http://arxiv.org/abs/0807.4326
Author(s): Andrew N. Downes
Abstract: The paper presents new simple sharp bounds for transition density functions
for time-homogeneous diffusions processes. The bounds are obtained under mild
conditions on the drift and diffusion coefficients, extending and substantially
improving previous results in the literature which were limited to drifts
satisfying a linear growth condition. They lead to an asymptotic expression for
the transition density as the transition time approaches zero. While the focus
is on the one-dimensional case, an extension to multiple dimensions is
discussed. Results are illustrated by numerical examples.
http://arxiv.org/abs/0807.4586
Author(s): Anatolii Puhalskii
Abstract: We prove a heavy traffic limit theorem for a non-time homogeneous Markovian
many server queue with customer abandonment.
http://arxiv.org/abs/0807.4621
Author(s): Aryeh Kontorovich and Anthony Brockwell
Abstract: We prove a strong law of large numbers for a class of strongly mixing
processes. Our result rests on recent advances in understanding of
concentration of measure. It is simple to apply and gives finite-sample (as
opposed to asymptotic) bounds, with readily computable rate constants. In
particular, this makes it suitable for analysis of inhomogeneous Markov
processes. We demonstrate how it can be applied to establish an almost-sure
convergence result for a class of models that includes as a special case a
class of adaptive Markov chain Monte Carlo algorithms.
http://arxiv.org/abs/0807.4665
Author(s): Jean-Christophe Breton (MIA) and Cl\'ement Dombry (LMA)
Abstract: We consider weighted random balls in $\real^d$ distributed according to a
random Poisson measure with heavy-tailed intensity and study the asymptotic
behaviour of the total weight of some configurations in $\real^d$. This
procedure amounts to be very rich and several regimes appear in the limit,
depending on the intensity of the balls, the zooming factor, the tail
parameters of the radii and of the weights. Statistical properties of the limit
fields are also evidenced, such as isotropy, self-similarity or dependence. One
regime is of particular interest and yields $\alpha$-stable stationary
isotropic self-similar generalized random fields which recovers Takenaka
fields, Telecom process or fractional Brownian motion.
http://arxiv.org/abs/0807.4700
Author(s): Mihai N. Pascu and Maria E. Gageonea
Abstract: A well known conjecture of R. Laugesen and C. Morpurgo asserts that the
diagonal element of the Neumann heat kernel of the unit ball in
$\mathbb{R}^{n}$ ($n\geq 1$) is a radially increasing function. In this paper,
we use probabilistic arguments to settle this conjecture, and, as an
application, we derive a new proof of the Hot Spots conjecture of J. Rauch in
the case of the unit disk.
http://arxiv.org/abs/0807.4726
Author(s): Hubert Lacoin
Abstract: We study a hierarchical disordered pinning model with site disorder for
which, like in the bond disordered case [5, 8], there exists a value of a
parameter b (enters in the definition of the hierarchical lattice) that
separates an irrelevant disorder regime and a relevant disorder regime. We show
that for such a value of b the critical point of the disordered system is
different from the critical point of the annealed version of the model. The
proof goes beyond the technique used in [8] and it takes explicitly advantage
of the inhomogeneous character of the Green function of the model.
http://arxiv.org/abs/0807.4864
Author(s): Terence Tao and Van Vu
Abstract: Given an $n \times n$ complex matrix $A$, let
$$\mu_{A}(x,y):= \frac{1}{n} |\{1\le i \le n, \Re \lambda_i \le x, \Im
\lambda_i \le y\}|$$ be the empirical spectral distribution (ESD) of its
eigenvalues
$\lambda_i \in \BBC, i=1, ... n$.
We consider the limiting distribution (both in probability and in the almost
sure convergence sense) of the normalized ESD $\mu_{\frac{1}{\sqrt{n}} A_n}$ of
a random matrix $A_n = (a_{ij})_{1 \leq i,j \leq n}$ where the random variables
$a_{ij} - \E(a_{ij})$ are iid copies of a fixed random variable $x$ with unit
variance. We prove a \emph{universality principle} for such ensembles, namely
that the limit distribution in question is {\it independent} of the actual
choice of $x$. In particular, in order to compute this distribution, one can
assume that $x$ is real of complex gaussian. As a related result, we show how
laws for this ESD follow from laws for the \emph{singular} value distribution
of $\frac{1}{\sqrt{n}} A_n - zI$ for complex $z$. As a corollary we establish
the Circular Law conjecture (in both strong and weak forms), that asserts that
$\mu_{\frac{1}{\sqrt{n}} A_n}$ converges to the uniform measure on the unit
disk when the $a_{ij}$ have zero mean.
http://arxiv.org/abs/0807.4898
Author(s): Steffen Dereich and Peter Morters
Abstract: We define a dynamic model of random networks, where new vertices are
connected to old ones with a probability proportional to a sublinear function
of their degree. We first give a strong limit law for the empirical degree
distribution, and then have a closer look at the temporal evolution of the
degrees of individual vertices, which we describe in terms of large and
moderate deviation principles. Using these results, we expose an interesting
phase transition: in cases of strong preference of large degrees, eventually a
single vertex emerges forever as vertex of maximal degree, whereas in cases of
weak preference, the vertex of maximal degree is changing infinitely often.
Loosely speaking, the transition between the two phases occurs in the case when
a new edge is attached to an existing vertex with a probability proportional to
the root of its current degree.
http://arxiv.org/abs/0807.4904
Author(s): Zhiyi Chi
Abstract: Multiple hypothesis testing often involves composite nulls, i.e., nulls that
are associated with two or more distributions. In many cases, it is reasonable
to assume that there is a prior distribution on the distributions despite it is
unknown. When the number of distributions under true nulls is finite, we show
that under the above assumption, the false discover rate (FDR) can be
controlled using $p$-values computed under constraints imposed by the empirical
distribution of the observations. Comparing to FDR control using $p$-values
defined as maximum significance level over all null distributions, the proposed
FDR control can have substantially more power.
http://arxiv.org/abs/0807.4879
Author(s): Jean Jacod and Emmanuel Kowalski and Ashkan Nikeghbali
Abstract: We introduce a new type of convergence in probability theory, which we call
``mod-Gaussian convergence''. It is directly inspired by theorems and
conjectures, in random matrix theory and number theory, concerning moments of
values of characteristic polynomials or zeta functions. We study this type of
convergence in detail in the framework of infinitely divisible distributions,
and exhibit some unconditional occurrences in number theory, in particular for
families of $L$-functions over function fields in the Katz-Sarnak framework. A
similar phenomenon of ``mod-Poisson convergence'' turns out to also appear in
the classical Erd\H{o}s-K\'ac Theorem.
http://arxiv.org/abs/0807.4739
Author(s): Delia Coculescu and Ashkan Nikeghbali
Abstract: In this paper, we provide a solution to two problems which have been open in
default time modeling in credit risk. We first show that if $\tau$ is an
arbitrary random (default) time such that its Az\'ema's supermartingale
$Z_t^\tau=\P(\tau>t|\F_t)$ is continuous, then $\tau$ avoids stopping times. We
then disprove a conjecture about the equality between the hazard process and
the martingale hazard process, which first appeared in \cite{jenbrutk1}, and we
show how it should be modified to become a theorem. The pseudo-stopping times,
introduced in \cite{AshkanYor}, appear as the most general class of random
times for which these two processes are equal. We also show that these two
processes always differ when $\tau$ is an honest time.
http://arxiv.org/abs/0807.4958
Author(s): Raouf Ghomrasni and Olivier Menoukeu Pamen
Abstract: In a recent work \cite{BG}, given a collection of continuous semimartingales,
authors derive a semimartingale decomposition from the corresponding ranked
processes in the case that the ranked processes can meet more than two original
processes at the same time. This has led to a more general decomposition of
ranked processes. In this paper, we derive a more general result for
semimartingales (not necessarily continuous) using a simpler approach.
Furthermore, we also give a generalization of Ouknine \cite{O1, O2} and Yan's
\cite{Y1} formula for local times of ranked processes
http://arxiv.org/abs/0807.5001
Author(s): Giovanni Peccati (LSTA) and Josep Llu\'is Sol\'e (Universitat Aut\'Onoma De Barcelona), Murad S. Taqqu (Boston University), Frederic Utzet
(Universitat Aut\'Onoma De Barcelona)
Abstract: We combine Stein's method with a version of Malliavin calculus on the Poisson
space. As a result, we obtain explicit Berry-Ess\'een bounds in Central Limit
Theorems (CLTs) involving multiple Wiener-It\^o integrals with respect to a
general Poisson measure. We provide several applications to CLTs related to
Ornstein-Uhlenbeck L\'evy processes.
http://arxiv.org/abs/0807.5035
Author(s): Anna De Masi and Immacolata Merola and Errico Presutti and Yvon Vignaud
Abstract: This is the second of two papers on a continuum version of the Potts model,
where particles are points in $\mathbb R^d$, $d\ge 2$, with a spin which may
take $S\ge 3$ possible values. Particles with different spins repel each other
via a Kac pair potential of range $\ga^{-1}$, $\ga>0$. In this paper we prove
phase transition, namely we prove that if the scaling parameter of the Kac
potential is suitably small, given any temperature there is a value of the
chemical potential such that at the given temperature and chemical potential
there exist $S+1$ mutually distinct DLR measures.
http://arxiv.org/abs/0807.5080
Author(s): Elisa Al\`os and Jorge A. Le\'on and Josep Vives
Abstract: In this paper, we use the Malliavin calculus techniques to obtain an
anticipative version of the change of variable formula for L\'evy processes.
Here the coefficients are in the domain of the anihilation (gradient) operator
in the "future sense", which includes the family of all adapted and
square-integrable processes. This domain was introduced on the Wiener space by
Al\`os and Nualart.
http://arxiv.org/abs/0808.0035
Author(s): Peter Friz and Harald Oberhauser
Abstract: The Wong-Zakai theorem asserts that ODEs driven by "reasonable" (e.g.
piecewise linear) approximations of Brownian motion converge to the
corresponding Stratonovich stochastic differential equation. With the aid of
rough path analysis, we study "non-reasonable" approximations and go beyond a
well-known criterion of [Ikeda--Watanabe, North Holland 1989] in the sense that
our result applies to perturbations on all levels, exhibiting additional drift
terms involving any iterated Lie brackets of the driving vector fields. In
particular, this applies to the approximations by McShane ('72) and Sussmann
('91). Our approach is not restricted to Brownian driving signals. At last,
these ideas can be used to prove optimality of certain rough path estimates.
http://arxiv.org/abs/0808.0337
Author(s): Matthias Birkner and Jochen Blath and Martin Moehle and Matthias Steinruecken and Johanna Tams
Abstract: Let $\Lambda$ be a finite measure on the unit interval. A
$\Lambda$-Fleming-Viot process is a probability measure valued Markov process
which is dual to a coalescent with multiple collisions ($\Lambda$-coalescent)
in analogy to the duality known for the classical Fleming Viot process and
Kingman's coalescent, where $\Lambda$ is the Dirac measure in 0.
We explicitly construct a dual process of the coalescent with simultaneous
multiple collisions ($\Xi$-coalescent) with mutation, the $\Xi$-Fleming-Viot
process with mutation, and provide a representation based on the empirical
measure of an exchangeable particle system along the lines of Donnelly and
Kurtz (1999). We establish pathwise convergence of the approximating systems to
the limiting $\Xi$-Fleming-Viot process with mutation. An alternative
construction of the semigroup based on the Hille-Yosida theorem is provided and
various types of duality of the processes are discussed.
In the last part of the paper a population is considered which undergoes
recurrent bottlenecks. In this scenario, non-trivial $\Xi$-Fleming-Viot
processes naturally arise as limiting models.
http://arxiv.org/abs/0808.0412
Author(s): Richard Kiefer and Peter Morters
Abstract: The frontier of a planar Brownian motion is the boundary of the unbounded
component of the complement of its range. In this paper we find the Hausdorff
dimension of the set of double points on the frontier.
http://arxiv.org/abs/0808.0425
Author(s): Arup Bose and Amites Dasgupta and Krishanu Maulik
Abstract: Consider an urn model whose replacement matrix is triangular, has all entries
nonnegative and the row sums are all equal to one. We obtain the strong laws
for the counts of balls corresponding to each color. The scalings for these
laws depend on the diagonal elements of a rearranged replacement matrix. We use
the strong laws obtained to study further behavior of certain three color urn
models.
http://arxiv.org/abs/0808.0426
Author(s): A.B. Dieker and J. Warren
Abstract: We show that Markov transition probabilities for the queue length process in
series Jackson networks can be written as a finite sum of non-crossing
probabilities. Using this, we prove a conjecture of Blanc that the relaxation
time (i.e., the reciprocal of the `spectral gap') of a positive recurrent
system equals the relaxation time of an M/M/1 queue with the same arrival and
service rates as the network's bottleneck station.
http://arxiv.org/abs/0808.0513
Author(s): Sreekar Vadlamani and Robert J. Adler
Abstract: We consider global geometric properties of a codimension one manifold
embedded in Euclidean space, as it evolves under an isotropic and volume
preserving Brownian flow of diffeomorphisms. In particular, we obtain
expressions describing the expected rate of growth of the Lipschitz-Killing
curvatures, or intrinsic volumes, of the manifold under the flow.
These results shed new light on some of the intriguing growth properties of
flows from a global perspective, rather than the local perspective, on which
there is a much larger literature.
http://arxiv.org/abs/0808.0720
Author(s): Lucian Maticiuc and Aurel Rascanu and Adrian Zalinescu
Abstract: The aim of this paper is to study the existence and uniqueness of the
solution of the backward stochastic differential equations involving the
subdifferential operator $\partial\varphi$ (also called backward stochastic
variational inequalities): \[ {\begin{array} [c]{l}%
-dY_{t}+\partial\varphi(Y_{t}) dt\ni F(t,Y_{t}%, Z_{t}) dt-Z_{t}dB_{t}, 0\leq
t
http://arxiv.org/abs/0808.0801
Author(s): de gregorio alessandro and stefano iacus
Abstract: In this paper we propose the use of $\phi$-divergences as test statistics to
verify simple hypotheses about a one-dimensional parametric diffusion process
$\de X_t = b(X_t, \theta)\de t + \sigma(X_t, \theta)\de W_t$, from discrete
observations $\{X_{t_i}, i=0, ..., n\}$ with $t_i = i\Delta_n$, $i=0, 1, >...,
n$, under the asymptotic scheme $\Delta_n\to0$, $n\Delta_n\to\infty$ and
$n\Delta_n^2\to 0$. The class of $\phi$-divergences is wide and includes
several special members like Kullback-Leibler, R\'enyi, power and
$\alpha$-divergences. We derive the asymptotic distribution of the test
statistics based on $\phi$-divergences. The limiting law takes different forms
depending on the regularity of $\phi$. These convergence differ from the
classical results for independent and identically distributed random variables.
Numerical analysis is used to show the small sample properties of the test
statistics in terms of estimated level and power of the test.
http://arxiv.org/abs/0808.0853
Author(s): Cedric Bernardin (UMPA-Ensl)
Abstract: We consider a $d$-dimensional disordered harmonic chain (DHC) perturbed by an
energy conservative noise. We obtain uniform in the volume upper and lower
bounds for the thermal conductivity defined through the Green-Kubo formula.
These bounds indicate a positive finite conductivity. We prove also that the
infinite volume homogenized Green-Kubo formula converges.
http://arxiv.org/abs/0808.0660
Author(s): Cedric Bernardin (UMPA-Ensl)
Abstract: We discuss an asymmetric energy model (AEM) introduced by Giardina et al. in
\cite{7}. This model is expected to belong to the KPZ class. We obtain lower
bounds for the diffusion coefficient. In particular, the diffusion coefficient
is diverging in dimension one and two as it is expected in the KPZ picture.
http://arxiv.org/abs/0808.0661
Author(s): Cedric Bernardin (UMPA-Ensl)
Abstract: We consider a stochastic heat conduction model for solids composed by N
interacting atoms. The system is in contact with two heat baths at different
temperature $T_\ell$ and $T_r$. The bulk dynamics conserve two quantities: the
energy and the deformation between atoms. If $T_\ell \neq T_r$, a heat flux
takes place in the system. For large $N$, the system adopts a linear
temperature profile between $T_\ell$ and $T_r$. We establish the hydrodynamic
limit for the two conserved quantities. We introduce the fluctuations field of
the energy and of the deformation in the non-equilibrium steady state. As $N$
goes to infinity, we show that this field converges to a Gaussian field and we
compute the limiting covariance matrix. The main contribution of the paper is
the study of large deviations for the temperature profile in the
non-equilibrium stationary state. A variational formula for the rate function
is derived following the recent macroscopic fluctuation theory of Bertini et
al.
http://arxiv.org/abs/0808.0662
Author(s): Zhihui Yang and Jinqiao Duan
Abstract: A dynamical system driven by non-Gaussian L\'evy noises of small intensity is
considered. The first exit time of solution orbits from a bounded neighborhood
of an attracting equilibrium state is estimated. For a class of non-Gaussian
L\'evy noises, it is shown that the mean exit time is asymptotically faster
than exponential (the well-known Gaussian Brownian noise case) but slower than
polynomial (the stable L\'evy noise case), in terms of the reciprocal of the
small noise intensity.
http://arxiv.org/abs/0808.1085
Author(s): Jason Morton
Abstract: We describe a Groebner basis of relations among conditional probabilities in
a discrete probability space, with any set of conditioned-upon events. They may
be specialized to the partially-observed random variable case, the purely
conditional case, and other special cases. We also investigate the connection
to generalized permutohedra and describe a conditional probability simplex.
http://arxiv.org/abs/0808.1149
Author(s): M. Bal\'azs and J. Komj\'athy and T. Sepp\"al\"ainen
Abstract: This paper develops a general approach to proving order of magnitude t^{1/3}
for fluctuations in characteristic directions for asymmetric deposition models,
or asymmetric conservative particle systems, in the case of a strictly concave
flux function. For hypothesis the argument requires control of second class
particles in a manner that deserves to be called microscopic concavity by
analogy with the effects of concavity of the flux. This hypothesis has been
verified for the asymmetric simple exclusion process and is verified in the
present paper for totally asymmetric zero range processes with jump rates that
increase with exponentially decaying slope.
http://arxiv.org/abs/0808.1177
Author(s): Jinqiao Duan and Jia-an Yan
Abstract: The expressions of solutions for general $n\times m$ matrix-valued
inhomogeneous linear stochastic differential equations are derived. This
generalizes a result of Jaschke (2003) for scalar inhomogeneous linear
stochastic differential equations. As an application, some $\R^n$ vector-valued
inhomogeneous nonlinear stochastic differential equations are reduced to random
differential equations, facilitating pathwise study of the solutions.
http://arxiv.org/abs/0808.1112
Author(s): Gordon Blower
Abstract: Tracy and Widom showed that fundamentally important kernels in random matrix
theory arise from differential equations with rational coefficients. More
generally, this paper considers symmetric Hamiltonian systems abd determines
the properties of kernels that arise from them. The inverse spectral problem
for self-adjoint Hankel operators gives a sufficient condition for a
self-adjoint operator to be the Hankel operator on $L^2(0, \infty)$ from a
linear system in continuous time; thus this paper expresses certain kernels as
squares of Hankel operators. For a suitable linear system $(-A,B,C)$ with one
dimensional input and output spaces, there exists a Hankel operator $\Gamma$
with kernel $\phi_{(x)}(s+t)=Ce^{-(2x+s+t)A}B$ such that $\det
(I+(z-1)\Gamma\Gamma^\dagger)$ is the generating function of a determinantal
random point field.
http://arxiv.org/abs/0808.1276
Author(s): Arka P. Ghosh and Alexander Roitershtein and Ananda Weerasinghe
Abstract: We consider a stochastic control model driven by a fractional Brownian
motion. This model is a formal approximation to a queueing network with an
on-off input process. We study stochastic control problems associated with the
long-run average cost, the infinite horizon discounted cost, and the finite
horizon cost. In addition, we find a solution to a constrained minimization
problem as an application of our solution to the long-run average cost problem.
We also establish Abelian limit relationships among the value functions of the
above control problems.
http://arxiv.org/abs/0808.1299
Author(s): Martin Hairer and Jonathan C. Mattingly
Abstract: We present a theory of hypoellipticity and unique ergodicity for semilinear
parabolic stochastic PDEs with ``polynomial'' nonlinearities and additive
noise, considered as abstract evolution equations in some Hilbert space. It is
shown that if Hormander's bracket condition holds at every point of this
Hilbert space, then a lower bound on the Malliavin covariance operator $M_t$
can be obtained. Informally, this bound can be read as ``Fix any
finite-dimensional projection $\Pi$ on a subspace of sufficiently regular
functions. Then the eigenfunctions of $M_t$ with small eigenvalues have only a
very small component in the image of $\Pi$.''
We also show how to use a priori bounds on the solutions to the equation to
obtain good control on the dependency of the bounds on the Malliavin matrix on
the initial condition. These bounds are sufficient in many cases to obtain the
asymptotic strong Feller property introduced in [HairerMattingly06].
One of the main novel technical tools is an almost sure bound from below on
the size of ``Wiener polynomials,'' where the coefficients are possibly
non-adapted stochastic processes satisfying a Lipschitz condition. By
exploiting the polynomial structure of the equations, this result can be used
to replace Norris' lemma, which is unavailable in the present context.
We conclude by showing that the two-dimensional stochastic Navier-Stokes
equations and a large class of reaction-diffusion equations fit the framework
of our theory.
http://arxiv.org/abs/0808.1361
Author(s): Rami Atar and Amarjit Budhiraja
Abstract: Given a bounded $\calC^2$ domain $G\subset\R^m$ and functions $g\in\calC(\pl
G,\R)$ and $h\in\calC(\bar G,\R\setminus\{0\})$, let $u$ denote the unique
viscosity solution to the equation $-2\Del_\infty u=h$ in $G$ with boundary
data $g$. We provide a representation for $u$ as the value of a two-player
zero-sum stochastic differential game.
http://arxiv.org/abs/0808.1457
Author(s): Djalil Chafai (IMT and Upte)
Abstract: Let $(X_{i,j})$ be an infinite array of i.i.d. non negative real random
variables with unit mean, finite positive variance $\sigma^2$, and finite
fourth moment. Let $M$ be the $n\times n$ random Markov matrix with i.i.d. rows
defined by $M_{i,j}=X_{i,j}/(X_{i,1}+...+X_{i,n})$. It belongs to the Dirichlet
Markov Ensemble when $X_{1,1}$ follows an exponential law. We show that with
probability one, the empirical spectral distribution
$\frac{1}{n}(\delta_{\lambda_1}+...+\delta_{\lambda_n})$ of $\sqrt{n}M$
converges weakly as $n\to\infty$ to the uniform law on the disc
$\{z\in\mathbb{C};|z|\leq \sigma\}$ and moreover, the spectral gap of $M$ is of
order $n^{-1/2}$ for large enough $n$. There is for now a gap in the proof of
Theorem 1.3 (convergence to the circular law).
http://arxiv.org/abs/0808.1502
Author(s): Carlo Marinelli and Claudia Pr\'ev\^ot and Michael R\"ockner
Abstract: We prove existence, uniqueness and Lipschitz dependence on the initial datum
for mild solutions of stochastic partial differential equations with Lipschitz
coefficients driven by Wiener and Poisson noise. Under additional assumptions,
we prove Gateaux and Frechet differentiability of solutions with respect to the
initial datum. As an application, we obtain gradient estimates for the
resolvent associated to the mild solution. Finally, we prove the strong Feller
property of the associated semigroup.
http://arxiv.org/abs/0808.1509
Author(s): Bertrand Duplantier and Scott Sheffield
Abstract: Consider a bounded planar domain D, an instance h of the Gaussian free field
on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma <
2. The Liouville quantum gravity measure on D is the weak limit as \epsilon
tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz,
where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h
on the circle of radius \epsilon centered at z. Given a random (or
deterministic) subset X of D one can define the scaling dimension of X using
either Lebesgue measure or this random measure. We derive a general quadratic
relation between these two dimensions, which we view as a probabilistic
formulation of the KPZ relation from conformal field theory. We also present a
boundary analog of KPZ (for subsets of the boundary of D). We illustrate (via
heuristics and announced results) the connection between discrete and continuum
quantum gravity and provide a framework for understanding Euclidean scaling
exponents via quantum gravity.
http://arxiv.org/abs/0808.1560
Author(s): Thomas Gilbert and Raphael Lefevere
Abstract: We discuss the transport properties of a class of Hamiltonian dynamics with
local confinement, in which interactions between neighboring particles occur
through hard core elastic collisions. Such dynamics may be described as
high-dimensional billiards. We consider the case where the collisions are rare
and, for large systems, derive a Boltzmann-like equation for the evolution of
the probability densities. We solve this equation in the linear regime and
compute the heat conductivity in the approximate stationary state and with the
help of the Green-Kubo formula. We demonstrate the validity of the molecular
chaos hypothesis by comparing our theoretical predictions to the results of
numerical simulations performed on a new class of models, which are defocusing
chaotic billiards, likened to higher-dimensional stadia.
http://arxiv.org/abs/0808.1179
Author(s): Ludolf E. Meester
Abstract: Two old conjectures from problem sections, one of which from SIAM Review,
concern the question of finding distributions that maximize P(Sn <= t), where
Sn is the sum of i.i.d. random variables X1, ..., Xn on the interval [0,1],
satisfying E[X1]=m. In this paper a Lagrange multiplier technique is applied to
this problem, yielding necessary conditions for distributions to be extremal,
for arbitrary n. For n=2, a complete solution is derived from them: extremal
distributions are discrete and have one of the following supports, depending on
m and t: {0,t}, {t-1,1}, {t/2,1}, or {0,t,1}. These results suffice to refute
both conjectures. However, acquired insight naturally leads to a revised
conjecture: that extremal distributions always have at most three support
points and belong to a (for each n, specified) finite collection of two and
three point distributions.
http://arxiv.org/abs/0808.1669
Author(s): Jeong Han Kim
Abstract: For the random 2-SAT formula $F(n,p)$, let $F_C (n,p)$ be the formula left
after the pure literal algorithm applied to $F(n,p)$ stops. Using the recently
developed Poisson cloning model together with the cut-off line algorithm
(COLA), we completely analyze the structure of $F_{C} (n,p)$. In particular, it
is shown that, for $\gl:= p(2n-1) = 1+\gs $ with $\gs\gg n^{-1/3}$, the core of
$F(n,p)$ has $\thl^2 n +O((\thl n)^{1/2})$ variables and $\thl^2 \gl n+O((\thl
n))^{1/2}$ clauses, with high probability, where $\thl$ is the larger solution
of the equation $\th- (1-e^{-\thl \gl})=0$. We also estimate the probability of
$F(n,p)$ being satisfiable to obtain $$ \pr[ F_2(n, \sfrac{\gl}{2n-1}) is
satisfiable ] = \caseth{1-\frac{1+o(1)}{16\gs^3 n}}{if $\gl= 1-\gs$ with
$\gs\gg n^{-1/3}$}{}{}{e^{-\Theta(\gs^3n)}}{if $\gl=1+\gs$ with $\gs\gg
n^{-1/3}$,} $$ where $o(1)$ goes to 0 as $\gs$ goes to 0. This improves the
bounds of Bollob\'as et al. \cite{BBCKW}.
http://arxiv.org/abs/0808.1599
Author(s): Uwe Franz and Adam Skalski
Abstract: Idempotent states on a compact quantum group are shown to yield group-like
projections in the multiplier algebra of the dual discrete quantum group. This
allows to deduce that every idempotent state on a finite quantum group arises
as the Haar state on a finite quantum hypergroup. A natural order structure on
the set of idempotent states is also studied and some examples discussed.
http://arxiv.org/abs/0808.1683
Author(s): Fabian Freund and Martin M\"ohle
Abstract: Let $K_n$ denote the number of types of a sample of size $n$ taken from an
exchangeable coalescent process ($\Xi$-coalescent) with mutation. A
distributional recursion for the sequence $(K_n)_{n\in{\mathbb N}}$ is derived.
If the coalescent does not have proper frequencies, i.e., if the characterizing
measure $\Xi$ on the infinite simplex $\Delta$ does not have mass at zero and
satisfies $\int_\Delta |x|\Xi(dx)/(x,x)<\infty$, where $|x|:=\sum_{i=1}^\infty
x_i$ and $(x,x):=\sum_{i=1}^\infty x_i^2$ for $x=(x_1,x_2,...)\in\Delta$, then
$K_n/n$ converges weakly as $n\to\infty$ to a limiting variable $K$ which is
characterized by an exponential integral of the subordinator associated with
the coalescent process. For so-called simple measures $\Xi$ satisfying
$\int_\Delta\Xi(dx)/(x,x)<\infty$ we characterize the distribution of $K$ via a
fixed-point equation.
http://arxiv.org/abs/0808.1792
Author(s): Theodore P. Hill
Abstract: The conflation of a finite number of probability distributions P_1,...,P_n is
the probability distribution that minimizes the loss of Shannon Information in
consolidating the combined information from P_1,...,P_n into a single
distribution Q, and is also the minimax likelihood ratio consolidation of the
distributions. Intuitively, the conflation is the conditional distribution of
independent random variables, given that they are all equal, so in large
classes of distributions the conflation is the distribution determined by the
normalized product of the probability density or probability mass functions.
When P_1,...,P_n are Gaussian, Q is Gaussian with mean the classical
weighted-mean-squares reciprocal of variances. For unbiased estimators, the
mean of the conflation thus yields a BLUE (best linear unbiased estimator) and
if the underlying data is Gaussian, the conflation mean is also an MLE (maximum
likelihood estimator). A version of the classical convolution theorem holds for
conflations of a large class of a.c. measures.
http://arxiv.org/abs/0808.1808
Author(s): T. Antal and D. ben-Avraham and E. Ben-Naim and P.L. Krapivsky
Abstract: We study a directed flipping process that underlies the performance of the
random edge simplex algorithm. In this stochastic process, which takes place on
a one-dimensional lattice whose sites may be either occupied or vacant,
occupied sites become vacant at a constant rate and simultaneously cause all
sites to the right to change their state. This random process exhibits rich
phenomenology. First, there is a front, defined by the position of the
left-most occupied site, that propagates at a nontrivial velocity. Second, the
front involves a depletion zone with an excess of vacant sites. The total
excess D_k increases logarithmically, D_k ~ ln k, with the distance k from the
front. Third, the front exhibits rejuvenation -- young fronts are vigorous but
old fronts are sluggish. We investigate these phenomena using a quasi-static
approximation, direct solutions of small systems, and numerical simulations.
http://arxiv.org/abs/0808.0159
Author(s): Shamgar Gurevich and Ronny Hadani and Nir Sochen
Abstract: A system of functions (signals) on the finite line, called the oscillator
system, is described and studied. Applications of this system for discrete
radar and digital communication theory are explained.
Keywords: Weil representation, commutative subgroups, eigenfunctions, random
behavior, deterministic construction
http://arxiv.org/abs/0808.1417
Author(s): A. Faggionato and D. Gabrielli and M. Ribezzi Crivellari
Abstract: We consider Piecewise Deterministic Markov Processes (PDMPs) with a finite
set of discrete states. In the regime of fast jumps between discrete states, we
prove a law of large number and a large deviation principle. In the regime of
fast and slow jumps, we analyze a coarse-grained process associated to the
original one and prove its convergence to a new PDMP with effective force
fields and jump rates. In all the above cases, the continuous variables evolve
slowly according to ODEs.
Finally, we discuss some applications related to the mechanochemical cycle of
macromolecules, including strained--dependent power--stroke molecular motors.
Our analysis covers the case of fully--coupled slow and fast motions.
http://arxiv.org/abs/0808.1910
Author(s): Greg Martin and Erick B. Wong
Abstract: Random matrices arise in many mathematical contexts, and it is natural to ask
about the properties that such matrices satisfy. If we choose a matrix with
integer entries at random, for example, what is the probability that it will
have a particular integer as an eigenvalue, or an integer eigenvalue at all? If
we choose a matrix with real entries at random, what is the probability that it
will have a real eigenvalue in a particular interval? The purpose of this paper
is to resolve these questions, once they are made suitably precise, in the
setting of 2x2 matrices.
http://arxiv.org/abs/0808.1922
Author(s): Olga Holtz and Fedor Nazarov and Yuval Peres
Abstract: Given a (known) function $f:[0,1] \to (0,1)$, we consider the problem of
simulating a coin with probability of heads $f(p)$ by tossing $N$ times a coin
with unknown heads probability $p$, where $N$ may be random. Keane and O'Brien
(1994) showed that such a simulation scheme with $\P_p(N<\infty)=1$ exists iff
$f$ is continuous. Nacu and Peres (2005) proved that $f$ is real analytic iff
such a simulation scheme exists with $\P_p(N>n)$ decaying exponentially. We
prove that for $\alpha>0$ noninteger, $f$ is in the space $C^\alpha [0,1]$ if
and only if a simulation scheme as above exists with $\P_p(N>n) \le
C\Delta_n(p)$,where $\Delta_n(x)\eqbd \max \{\sqrt{x(1-x)/n},1/n \}$. The key
to the proof is a new result in approximation theory:
Let $H_n^+$ be the space of homogenous polynomials of degree $n$ with
non-negative coefficients in two variables. We show that a function $f:[0,1]
\to (0,1)$ is in $C^\alpha [0,1]$ if and only if $f$ has a series
representation $\sum_{n=1}^\infty F_n(x,1-x)$ with $F_n(x,y) \in H_n^+$ and
$\sum_{k>n} F_k(x,1-x) \le C\Delta_n(x)$ for all $ x \in [0,1]$ and $n \ge 1$.
We also provide a counterexample to a theorem stated without proof by Lorentz
(1963), who claimed that if some $\phi_n(x,y) \in H_n^+$ satisfy
$|f(x)-\phi_n(x,1-x)| \le C \Delta_n(x)$ for all $ x \in [0,1]$ and $n \ge 1$,
then $f \in C^\alpha [0,1]$.
http://arxiv.org/abs/0808.1936
Author(s): Michael Krivelevich and Benny Sudakov and Nicholas Wormald
Abstract: An old problem of Erd\H{o}s, Fajtlowicz and Staton asks for the order of a
largest induced regular subgraph that can be found in every graph on n
vertices. Motivated by this problem, we consider the order of such a subgraph
in a typical graph on n vertices, i.e., in a binomial random graph G(n,1/2). We
prove that with high probability a largest induced regular subgraph of G(n,1/2)
has about n^{2/3} vertices.
http://arxiv.org/abs/0808.2023
Author(s): Ivan Nourdin (University of Paris 6) and Frederi G. Viens (Purdue University)
Abstract: We show how to use the Malliavin calculus to obtain density estimates of the
law of general centered random variables. In particular, under a non-degeneracy
condition, we prove and use a new formula for the density of a random variable
which is measurable and differentiable with respect to a given isonormal
Gaussian process. Among other results, we apply our techniques to bound the
density of the maximum of a general Gaussian process from above and below;
several new results ensue, including improvements on the so-called
Borell-Sudakov inequality. We then explain what can be done when one is only
interested in or capable of deriving concentration inequalities, i.e. tail
bounds from above or below but not necessarily both simultaneously.
http://arxiv.org/abs/0808.2088
Author(s): Balazs Rath (TU Budapest and Institute of Mathematics) and Balint Toth (TU Budapest, Institute of Mathematics)
Abstract: We modify the usual Erdos-Renyi random graph evolution by letting connected
clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a
Poisson flow of lightnings. In a range of the intensity of rate of lightnings
the system sticks to a permanent critical state.
http://arxiv.org/abs/0808.2116
Author(s): Tim Austin (UCLA)
Abstract: In his 1985 survey of notions of exchangeability, Aldous introduced a form of
exchangeability corresponding to the symmetries of the infinite discrete cube,
and asked whether these exchangeable probability measures enjoy a
representation theorem similar to those for exchangeable sequences, arrays and
set-indexed families. In this note we to prove that, whereas the known
representation theorems for different classes of partially exchangeable
probability measure imply that the compact convex set of such measures is a
Bauer simplex (that is, its subset of extreme points is closed), in the case of
cube-exchangeability it is a copy of the Poulsen simplex (in which the extreme
points are dense). This follows from the arguments used by Glasner and Weiss'
for their characterization of property (T) in terms of the geometry of the
simplex of invariant measures for associated generalized Bernoulli actions.
The emergence of this Poulsen simplex suggests that, if a representation
theorem for these processes is available at all, it must take a very different
form from the case of set-indexed exchangeable families.
http://arxiv.org/abs/0808.2268
Author(s): Yves Le Jan (LM-Orsay)
Abstract: This is a preliminary version of the notes of a series of lectures given in
St Flour. It includes a discussion of relations between the occupation field of
Markov loops with the corresponding free field and new developments on
reflection positivity.
http://arxiv.org/abs/0808.2303
Author(s): George Lowther
Abstract: We consider the problem of finding a real valued martingale fitting specified
marginal distributions. For this to be possible, the marginals must be
increasing in the convex order and have constant mean. We show that, under the
extra condition that they are weakly continuous, the marginals can always be
fitted in a unique way by a martingale which lies in a particular class of
strong Markov processes.
It is also shown that the map that this gives from the sets of marginal
distributions to the martingale measures is continuous. Furthermore, we prove
that it is the unique continuous method of fitting martingale measures to the
marginal distributions.
http://arxiv.org/abs/0808.2319
Author(s): Rinaldo B. Schinazi
Abstract: We propose a one mutation model for cancer with a mutation rate that
increases with time. Under rather general hypotheses the number of mutations is
necessarily a (non homogeneous) Poisson process with the prescribed mutation
rate. We show that the cumulative probability of cancer up to time $t$ is, up
to a multiplicative constant, an antiderivative of the mutation rate.
http://arxiv.org/abs/0808.2424
Author(s): Tim Austin (UCLA)
Abstract: We offer a new proof of the Furstenberg-Katznelson multiple recurrence
theorem for several commuting probability-preserving transformations
$T_1,T_2,...,T_d:\bbZ\curvearrowright (X,\S,\mu)$, and so, via the Furstenberg
correspondence principle, a new proof of the multi-dimensional Szemer\'edi
Theorem. We bypass the detailed analysis of certain towers of factors of a
probability-preserving systems that underlies the Furstenberg-Katznelson
analysis, instead modifying an approach recently developed for the analysis of
nonconventional ergodic averages to pass to a large extension of our original
system in which this analysis greatly simplifies. In particular, we find that
this simplifies our setting quite quickly to data that can be analyzed using
the infinitary version of the hypergraph removal lemma studied by Tao, and we
complete the proof by a simple application of that lemma. This addresses the
difficulty, highlighted by Tao, of establishing a direct connection between his
infinitary, probabilistic approach to the hypergraph removal lemma and the
infinitary, ergodic-theoretic approach to Szemer\'edi's Theorem set in motion
by Furstenberg.
http://arxiv.org/abs/0808.2267
Author(s): Sourav Chatterjee and Michel Ledoux
Abstract: Let M be an arbitrary Hermitian matrix of order n, and k be a positive
integer less than or equal to n. We show that if k is large, the distribution
of eigenvalues on the real line is almost the same for almost all principal
submatrices of M of order k. The proof uses results about random walks on
symmetric groups and concentration of measure. In a similar way, we also show
that almost all k x n submatrices of M have almost the same distribution of
singular values.
http://arxiv.org/abs/0808.2521
Author(s): David Applebaum
Abstract: We review and extend Lindsay's work on abstract gradient and divergence
operators in Fock space over a general complex Hilbert space. Precise
expressions for the domains are given, the $L^2$-equivalence of norms is proved
and an abstract version of the It\^{o}-Skorohod isometry is established. We
then outline a new proof of It\^{o}'s chaos expansion of complex
L\'{e}vy-It\^{o} space in terms of multiple Wiener-L\'{e}vy integrals based on
Brownian motion and a compensated Poisson random measure. The duality transform
now identifies L\'{e}vy-It\^{o} space as a Fock space. We can then easily
obtain key properties of the gradient and divergence of a general L\'{e}vy
process. In particular we establish maximal domains of these operators and
obtain the It\^{o}-Skorohod isometry on its maximal domain.
http://arxiv.org/abs/0808.2593
Author(s): Martin Bender
Abstract: A family of random matrix ensembles interpolating between the GUE and the
Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian
entries is considered. The asymptotic spectral distribution in these models is
uniform in an ellipse in the complex plane, which collapses to an interval of
the real line as the degree of non-Hermiticity diminishes. Scaling limit
theorems are proven for the eigenvalue point process at the rightmost edge of
the spectrum, and it is shown that a non-trivial transition occurs between
Poisson and Airy point process statistics when the ratio of the axes of the
supporting ellipse is of order $n^{-1/3}$. In this regime, the family of
limiting probability distributions of the maximum of the real parts of the
eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.
http://arxiv.org/abs/0808.2608
Author(s): Carl Mueller and Zhixin Wu
Abstract: We give a new representation of fractional Brownian motion with Hurst
parameter H<=1/2 using stochastic partial differential equations. This
representation allows us to use the Markov property and time reversal, tools
which are not usually available for fractional Brownian motion. We then give
simple proofs that fractional Brownian motion does not hit points in the
critical dimension, and that it does not have double points in the critical
dimension. These facts were already known, but our proofs are quite simple and
use some ideas of Levy.
http://arxiv.org/abs/0808.2634
Author(s): Jose Blanchet and Peter Glynn
Abstract: Let $(X_n:n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set
$S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling
algorithm to estimate the tail of $M=\max \{S_n:n\geq 0\}$ that is strongly
efficient for both light and heavy-tailed increment distributions. Moreover, in
the case of heavy-tailed increments and under additional technical assumptions,
our estimator can be shown to have asymptotically vanishing relative variance
in the sense that its coefficient of variation vanishes as the tail parameter
increases. A key feature of our algorithm is that it is state-dependent. In the
presence of light tails, our procedure leads to Siegmund's (1979) algorithm.
The rigorous analysis of efficiency requires new Lyapunov-type inequalities
that can be useful in the study of more general importance sampling algorithms.
http://arxiv.org/abs/0808.2731
Author(s): Sebastien Destercke and Didier Dubois and Eric Chojnacki
Abstract: There exist several simple representations of uncertainty that are easier to
handle than more general ones. Among them are random sets, possibility
distributions, probability intervals, and more recently Ferson's p-boxes and
Neumaier's clouds. Both for theoretical and practical considerations, it is
very useful to know whether one representation is equivalent to or can be
approximated by other ones. In this paper, we define a generalized form of
usual p-boxes. These generalized p-boxes have interesting connections with
other previously known representations. In particular, we show that they are
equivalent to pairs of possibility distributions, and that they are special
kinds of random sets. They are also the missing link between p-boxes and
clouds, which are the topic of the second part of this study.
http://arxiv.org/abs/0808.2747
Author(s): Sebastien Destercke and Didier Dubois and Eric Chojnacki
Abstract: There exist many simple tools for jointly capturing variability and
incomplete information by means of uncertainty representations. Among them are
random sets, possibility distributions, probability intervals, and the more
recent Ferson's p-boxes and Neumaier's clouds, both defined by pairs of
possibility distributions. In the companion paper, we have extensively studied
a generalized form of p-box and situated it with respect to other models . This
paper focuses on the links between clouds and other representations.
Generalized p-boxes are shown to be clouds with comonotonic distributions. In
general, clouds cannot always be represented by random sets, in fact not even
by 2-monotone (convex) capacities.
http://arxiv.org/abs/0808.2779
Author(s): Antonio Lijoi and Igor Pr\"unster and Stephen G. Walker
Abstract: We consider discrete nonparametric priors which induce Gibbs-type
exchangeable random partitions and investigate their posterior behavior in
detail. In particular, we deduce conditional distributions and the
corresponding Bayesian nonparametric estimators, which can be readily exploited
for predicting various features of additional samples. The results provide
useful tools for genomic applications where prediction of future outcomes is
required.
http://arxiv.org/abs/0808.2863
Author(s): Rami Atar
Abstract: Given a random variable $N$ with values in ${\mathbb{N}}$, and $N$ i.i.d.
positive random variables $\{\mu_k\}$, we consider a queue with renewal
arrivals and $N$ exponential servers, where server $k$ serves at rate $\mu_k$,
under two work conserving routing schemes. In the first, the service rates
$\{\mu_k\}$ need not be known to the router, and each customer to arrive at a
time when some servers are idle is routed to the server that has been idle for
the longest time (or otherwise it is queued). In the second, the service rates
are known to the router, and a customer that arrives to find idle servers is
routed to the one whose service rate is greatest. In the many-server heavy
traffic regime of Halfin and Whitt, the process that represents the number of
customers in the system is shown to converge to a one-dimensional diffusion
with a random drift coefficient, where the law of the drift depends on the
routing scheme. A related result is also provided for nonrandom environments.
http://arxiv.org/abs/0808.2865
Author(s): Peter Eichelsbacher and Gesine Reinert
Abstract: Stein's method provides a way of bounding the distance of a probability
distribution to a target distribution $\mu$. Here we develop Stein's method for
the class of discrete Gibbs measures with a density $e^V$, where $V$ is the
energy function. Using size bias couplings, we treat an example of Gibbs
convergence for strongly correlated random variables due to Chayes and Klein
[Helv. Phys. Acta 67 (1994) 30--42]. We obtain estimates of the approximation
to a grand-canonical Gibbs ensemble. As side results, we slightly improve on
the Barbour, Holst and Janson [Poisson Approximation (1992)] bounds for Poisson
approximation to the sum of independent indicators, and in the case of the
geometric distribution we derive better nonuniform Stein bounds than Brown and
Xia [Ann. Probab. 29 (2001) 1373--1403].
http://arxiv.org/abs/0808.2877
Author(s): Ashkan Nikeghbali and Eckhard Platen
Abstract: This paper demonstrates the usefulness and importance of the concept of
honest times to financial modeling. It studies a financial market with asset
prices that follow jump-diffusions with negative jumps. The central building
block of the market model is its growth optimal portfolio (GOP), which
maximizes the growth rate of strictly positive portfolios. Primary security
account prices, when expressed in units of the GOP, turn out to be nonnegative
local martingales. In the proposed framework an equivalent risk neutral
probability measure need not exist. Derivative prices are obtained as
conditional expectations of corresponding future payoffs, with the GOP as
numeraire and the real world probability as pricing measure. The time when the
global maximum of a portfolio with no positive jumps, when expressed in units
of the GOP, is reached, is shown to be a generic representation of an honest
time. We provide a general formula for the law of such honest times and compute
the conditional distributions of the global maximum of a portfolio in this
framework. Moreover, we provide a stochastic integral representation for
uniformly integrable martingales whose terminal values are functions of the
global maximum of a portfolio. These formulae are model independent and
universal. We also specialize our results to some examples where we hedge a
payoff that arrives at an honest time.
http://arxiv.org/abs/0808.2892
Author(s): B. G. Pittel
Abstract: A uniformly random graph on $n$ vertices with a fixed degree sequence,
obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$,
in a subcritical phase with high probability the largest component size does
not exceed $n^{1/\gamma+\varepsilon_n}$, $\varepsilon_n=O(\ln\ln n/\ln n)$,
$1/\gamma$ being the best power for this random graph. This is similar to the
best possible $n^{1/(\gamma-1)}$ bound for a different model of the random
graph, one with independent vertex degrees, conjectured by Durrett, and proved
recently by Janson.
http://arxiv.org/abs/0808.2907
Author(s): Yu Zhang
Abstract: We consider the first passage percolation model on the ${\bf Z}^d$ lattice.
In this model, we assign independently to each edge $e$ a non-negative passage
time $t(e)$ with a common distribution $F$. Let $a_{0,n}$ be the passage time
from the origin to $(n,0,..., 0)$. Under the exponential tail assumption,
Kesten (1993) and Talagrand (1995) investigated the concentration of
$a_{0,n}$ from its mean using different methods. With this concentration and
the exponential tail assumption, Alexander gave an estimate for the convergence
rate for ${\bf E} a_{0,n}$. In this paper, focusing on a moment condition, we
reinvestigate the concentration and the convergence rate for $a_{0,n}$ using a
special martingale structure.
http://arxiv.org/abs/0808.3021
Author(s): Oleksandr Chybiryakov
Abstract: In this paper we obtain skew-product representations of the multidimensional
Dunkl processes which generalize the skew-product decomposition in dimension 1
obtained in L. Gallardo and M. Yor. Some remarkable properties of the Dunkl
martingales. S\'{e}minaire de Probabilit\'{e}s XXXIX, 2006. We also study the
radial part of the Dunkl process, i.e. the projection of the Dunkl process on a
Weyl chamber.
http://arxiv.org/abs/0808.3033
Author(s): Xia Chen
Abstract: In this paper we obtain the central limit theorems, moderate deviations and
the laws of the iterated logarithm for the energy \[H_n=\sum_{1\le j
http://arxiv.org/abs/0808.3037
Author(s): J\'er\^ome Dedecker and Emmanuel Rio
Abstract: In this paper, we give estimates of the minimal ${\mathbb{L}}^1$ distance
between the distribution of the normalized partial sum and the limiting
Gaussian distribution for stationary sequences satisfying projective criteria
in the style of Gordin or weak dependence conditions.
http://arxiv.org/abs/0808.3048
Author(s): Antoine Ayache and Dongsheng Wu and Yimin Xiao
Abstract: Let $B^H=\{B^H(t),t\in{{\mathbb{R}}_+^N}\}$ be an $(N,d)$-fractional Brownian
sheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by
$B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where
$B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian
sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, then the
local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao
and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp
local and global H\"{o}lder conditions for the local times of $B^H$. These
results are applied to study analytic and geometric properties of the sample
paths of $B^H$.
http://arxiv.org/abs/0808.3054
Author(s): D. Loukianova and O. Loukianov
Abstract: Usually the problem of drift estimation for a diffusion process is considered
under the hypothesis of ergodicity. It is less often considered under the
hypothesis of null-recurrence, simply because there are fewer limit theorems
and existing ones do not apply to the whole null-recurrent class. The aim of
this paper is to provide some limit theorems for additive functionals and
martingales of a general (ergodic or null) recurrent diffusion which would
allow us to have a somewhat unified approach to the problem of non-parametric
kernel drift estimation in the one-dimensional recurrent case. As a particular
example we obtain the rate of convergence of the Nadaraya--Watson estimator in
the case of a locally H\"{o}lder-continuous drift.
http://arxiv.org/abs/0808.3069
Author(s): M. D. Taylor
Abstract: We explore the consequences of a set of axioms which extend Scarsini's axioms
for bivariate measures of concordance to the multivariate case and exhibit the
following results: (1) A method of extending measures of concordance from the
bivariate case to arbitrarily high dimensions. (2) A formula expressing the
measure of concordance of the random vectors $(\pm X_1,...,\pm X_n)$ in terms
of the measures of concordance of the "marginal" random vectors
$(X_{i_1},...,X_{i_k})$. (3) A method of expressing the measure of concordance
of an odd-dimensional copula in terms of the measures of concordance of its
even-dimensional marginals. (4) A family of relations which exist between the
measures of concordance of the marginals of a given copula.
http://arxiv.org/abs/0808.3105
Author(s): Ross G. Pinsky
Abstract: The diffusion operator $$ H_D=-\frac12\frac d{dx}a\frac d{dx}-b\frac
d{dx}=-\frac12\exp(-2B)\frac d{dx}a\exp(2B)\frac d{dx}, $$ where
$B(x)=\int_0^x\frac ba(y)dy$, defined either on $R^+=(0,\infty)$ with the
Dirichlet boundary condition at $x=0$, or on $R$, can be realized as a
self-adjoint operator with respect to the density $\exp(2Q(x))dx$. The operator
is unitarily equivalent to the Schr\"odinger-type operator $H_S=-\frac12\frac
d{dx}a\frac d{dx}+V_{b,a}$, where $V_{b,a}=\frac12(\frac{b^2}a+b')$. We obtain
an explicit criterion for the existence of a compact resolvent and explicit
formulas up to the multiplicative constant 4 for the infimum of the spectrum
and for the infimum of the essential spectrum for these operators. We give some
applications which show in particular how $\inf\sigma(H_D)$ scales when $a=\nu
a_0$ and $b=\gamma b_0$, where $\nu$ and $\gamma$ are parameters, and $a_0$ and
$b_0$ are chosen from certain classes of functions. We also give applications
to self-adjoint, multi-dimensional diffusion operators.
http://arxiv.org/abs/0808.3044
Author(s): Eugen J. Ionascu and Alin A. Stancu
Abstract: We are interested in constructing concrete independent events in purely
atomic probability spaces with geometric distribution. Among other facts we
prove that there are uncountable many sequences of independent events.
http://arxiv.org/abs/0808.3155
Author(s): E.A. Carlen and M.C. Carvalho and J. Le Roux and M. Loss and C. Villani
Abstract: We investigate the behavior in $N$ of the $N$--particle entropy functional
for Kac's stochastic model of Boltzmann dynamics, and its relation to the
entropy function for solutions of Kac's one dimensional nonlinear model
Boltzmann equation. We prove a number of results that bring together the notion
of propagation of chaos, which Kac introduced in the context of this model,
with the problem of estimating the rate of equilibration in the model in
entropic terms, and obtain a bound showing that the entropic rate of
convergence can be arbitrarily slow. Results proved here show that one can in
fact use entropy production bounds in Kac's stochastic model to obtain entropic
convergence bounds for his non linear model Boltzmann equation, though the
problem of obtaining optimal lower bounds of this sort for the original Kac
model remains open, and the upper bounds obtained here show that this problem
is somewhat subtle.
http://arxiv.org/abs/0808.3192
Author(s): R\'emi Rhodes
Abstract: We study the long time behavior (homogenization) of a diffusion in random
medium with time and space dependent coefficients. The diffusion coefficient
may degenerate. In Stochastic Process. Appl. (2007) (to appear), an invariance
principle is proved for the critical rescaling of the diffusion. Here, we
generalize this approach to diffusions whose space-time scaling differs from
the critical one.
http://arxiv.org/abs/0808.3301
Author(s): Vladas Sidoravicius and Alain-Sol Sznitman
Abstract: We investigate random interlacements on Z^d, d bigger or equal to 3. This
model recently introduced in arXiv:0704.2560 corresponds to a Poisson cloud on
the space of doubly infinite trajectories modulo time-shift tending to infinity
at positive and negative infinite times. A non-negative parameter u measures
how many trajectories enter the picture. Our main interest lies in the
percolative properties of the vacant set left by random interlacements at level
u. We show that for all d bigger or equal to 3 the vacant set at level u
percolates when u is small. This solves an open problem of arXiv:0704.2560,
where this fact has only been established when d is bigger or equal to 7. It
also completes the proof of the non-degeneracy in all dimensions d bigger or
equal to 3 of the critical parameter introduced in arXiv:0704.2560.
http://arxiv.org/abs/0808.3344
Author(s): E. Ostrovsky and E. Rogover
Abstract: In this paper non-asymptotic exact rearrangement invariant norm estimates are
derived for the maximum distribution of the family elements of some
rearrangement invariant (r.i.) space over unbounded measure in the entropy
terms and in the terms of generic chaining.
We consider some applications in the martingale theory and in the theory of
Fourier series.
http://arxiv.org/abs/0808.3247
Author(s): E. Ostrovsky
Abstract: We prove in this article that every Borelian measure, for example, the
distribution of a random variable, in separable Banach space has a support
which is compact embedded Banach subspace; and prove that if the norm of the
random variable belongs to some exponential Orlicz space, then the new subspace
can be choose such that the norm of this variable in the new space also belongs
to other exponential Orlicz space.
http://arxiv.org/abs/0808.3248
Author(s): Ju-Yi Yen and Marc Yor
Abstract: As a complement to some recent work by Pal and Protter, "Strict local
martingales, bubbles, and no early exercise", we show that the call option
prices associated with the Bessel strict local martingales are integrable over
time, and we discuss the probability densities obtained thus.
http://arxiv.org/abs/0808.3402
Author(s): Jeremie Unterberger
Abstract: Let $B=(B^{(1)},B^{(2)})$ be a two-dimensional fractional Brownian motion
with Hurst index $\alpha\in (0,1/4)$. Using an analytic approximation $B(\eta)$
of $B$ introduced in \cite{Unt08}, we prove that the rescaled L\'evy area
process $(s,t)\to \eta^{\half(1-4\alpha)}\int_s^t dB_{t_1}^{(1)}(\eta)
\int_s^{t_1} dB_{t_2}^{(2)}(\eta)$ converges in law to $W_t-W_s$ where $W$ is a
Brownian motion independent from $B$. The method relies on a very general
scheme of analysis of singularities of analytic functions, applied to the
moments of finite-dimensional distributions of the L\'evy area.
http://arxiv.org/abs/0808.3458
Author(s): L. R. G. Fontes and P. Mathieu
Abstract: We study K-processes, which are Markov processes in a denumerable state
space, all of whose elements are stable, with the exception of a single state,
starting from which the process enters finite sets of stable states with
uniform distribution. We show how these processes arise, in a particular
instance, as scaling limits of the trap model in the complete graph, and
subsequently derive aging results for those models in this context.
http://arxiv.org/abs/0808.3494
Author(s): Maria Vlasiou and Zbigniew Palmowski
Abstract: We investigate the tail behaviour of the steady state distribution of a
stochastic recursion that generalises Lindley's recursion. This recursion
arises in queuing systems with dependent interarrival and service times, and
includes alternating service systems and carousel storage systems as special
cases. We obtain precise tail asymptotics in three qualitatively different
cases, and compare these with existing results for Lindley's recursion and for
alternating service systems.
http://arxiv.org/abs/0808.3495
Author(s): Boris Pittel
Abstract: Consider a uniformly random regular graph of a fixed degree $d\ge3$, with $n$
vertices. Suppose that each edge is open (closed), with probability $p(q=1-p)$,
respectively. In 2004 Alon, Benjamini and Stacey proved that $p^*=(d-1)^{-1}$
is the threshold probability for emergence of a giant component in the subgraph
formed by the open edges. In this paper we show that the transition window
around $p^*$ has width roughly of order $n^{-1/3}$. More precisely, suppose
that $p=p(n)$ is such that $\omega:=n^{1/3}|p-p^*|\to\infty$. If $pp^*$, and $\log\omega\gg\log\log n$, then whp the largest
component has about $n(1-(p\pi+q)^d)\asymp n(p-p^*)$ vertices, and the second
largest component is of size $(p-p^*)^{-2}(\log n)^{1+o(1)}$, at most, where
$\pi=(p\pi+q)^{d-1},\pi\in(0,1)$. If $\omega$ is merely polylogarithmic in $n$,
then whp the largest component contains $n^{2/3+o(1)}$ vertices.
http://arxiv.org/abs/0808.3516
Author(s): L. R. G. Fontes and C. M. Newman and K. Ravishankar and E. Schertzer
Abstract: The dynamical discrete web (DyDW),introduced in recent work of Howitt and
Warren, is a system of coalescing simple symmetric one-dimensional random walks
which evolve in an extra continuous dynamical time parameter \tau. The
evolution is by independent updating of the underlying Bernoulli variables
indexed by discrete space-time that define the discrete web at any fixed \tau.
In this paper, we study the existence of exceptional (random) values of \tau
where the paths of the web do not behave like usual random walks and the
Hausdorff dimension of the set of exceptional such \tau. Our results are
motivated by those about exceptional times for dynamical percolation in high
dimension by H\"{a}ggstrom, Peres and Steif, and in dimension two by Schramm
and Steif. The exceptional behavior of the walks in the DyDW is rather
different from the situation for the dynamical random walks of Benjamini,
H\"{a}ggstrom, Peres and Steif. For example, we prove that the walk from the
origin S^\tau_0 violates the law of the iterated logarithm (LIL) on a set of
\tau of Hausdorff dimension one. We also discuss how these and other results
extend to the dynamical Brownian web, the natural scaling limit of the DyDW.
http://arxiv.org/abs/0808.3599
Author(s): Amarjit Budhiraja and Paul Dupuis and Vasileios Maroulas
Abstract: The large deviations analysis of solutions to stochastic differential
equations and related processes is often based on approximation. The
construction and justification of the approximations can be onerous, especially
in the case where the process state is infinite dimensional. In this paper we
show how such approximations can be avoided for a variety of infinite
dimensional models driven by some form of Brownian noise. The approach is based
on a variational representation for functionals of Brownian motion. Proofs of
large deviations properties are reduced to demonstrating basic qualitative
properties (existence, uniqueness and tightness) of certain perturbations of
the original process.
http://arxiv.org/abs/0808.3631
Author(s): Dapeng Zhan
Abstract: We prove that the chordal SLE$_{\kappa}$ trace is reversible for
$\kappa\in(0,4]$.
http://arxiv.org/abs/0808.3649
Author(s): Ioannis Karatzas and Ingrid-Mona Zamfirescu
Abstract: We develop a martingale approach for studying continuous-time stochastic
differential games of control and stopping, in a non-Markovian framework and
with the control affecting only the drift term of the state-process. Under
appropriate conditions, we show that the game has a value and construct a
saddle pair of optimal control and stopping strategies. Crucial in this
construction is a characterization of saddle pairs in terms of pathwise and
martingale properties of suitable quantities.
http://arxiv.org/abs/0808.3656
Author(s): Denis Denisov and Sergey Foss and Dmitry Korshunov
Abstract: We study conditions under which $P(S_\tau>x)\sim P(M_\tau>x)\sim E\tau
P(\xi_1>x)$ as $x\to\infty$, where $S_\tau$ is a sum $\xi_1+...+\xi_\tau$ of
random size $\tau$ and $M_\tau$ is a maximum of partial sums
$M_\tau=\max_{n\le\tau}S_n$. Here $\xi_n$, $n=1$, 2, ..., are independent
identically distributed random variables whose common distribution is assumed
to be subexponential. We consider mostly the case where $\tau$ is independent
of the summands; also, in a particular situation, we deal with a stopping time.
Also we consider the case where $E\xi>0$ and where the tail of $\tau$ is
comparable with or heavier than that of $\xi$, and obtain the asymptotics
$P(S_\tau>x) \sim E\tau P(\xi_1>x)+P(\tau>x/E\xi)$ as $x\to\infty$. This case
is of a primary interest in the branching processes.
In addition, we obtain new uniform (in all $x$ and $n$) upper bounds for the
ratio $P(S_n>x)/P(\xi_1>x)$ which substantially improve Kesten's bound in the
subclass ${\mathcal S}^*$ of subexponential distributions.
http://arxiv.org/abs/0808.3697
Author(s): Siva R. Athreya and Jan M. Swart
Abstract: We consider contact processes on the hierarchical group $\Omega_N$ with
freedom $N$, where sites infect other sites at hierarchical distance $k$ with
rate $\alpha_kN^{-k}$, and sites become healthy with recovery rate $\delta$. We
show that the critical recovery rate $\delta_{\rm c}$ is zero (i.e., the
process dies out for any $\delta>0$) if
$\liminf_{k\to\infty}N^{-k}\log(\beta_k)=-\infty$, where
$\beta_k:=\sum_{n=k}^\infty\alpha_n$. On the other hand, in the special case
that $N$ is a power of two, we show that $\delta_{\rm c}>0$ provided that
$\sum_kN^{-k}\log(\alpha_k)>-\infty$. The proof of this latter fact is based on
a coupling argument that compares contact processes on $\Omega_2$ with contact
processes on a renormalized lattice.
http://arxiv.org/abs/0808.3732
Author(s): Sotirios Sabanis
Abstract: This paper presents the general form and essential properties of the
q-optimal measure following the approach of Delbaen and Schachermayer (1996)
and proves its existence under mild conditions. Most importantly, it states a
necessary and sufficient condition for a candidate measure to be the q-optimal
measure in the case even of signed measures. Finally, an updated
characterization of the q-optimal measure for continuous asset price processes
is presented in the light of the counterexample appearing in Cerny and Kallsen
(2006) concerning Hobson's (2004) approach.
http://arxiv.org/abs/0808.3751
Author(s): Piotr Milos
Abstract: We consider a branching system consisting of particles moving according to a
Markov family in $\Rd$ and undergoing subcritical branching with a constant
rate $V>0$. New particles immigrate to the system according to homogeneous
space-time Poisson random field. The process of the fluctuations of the
rescaled occupation time is studied with very mild assumptions on the Markov
family. In this general setting a functional central limit theorem is proved.
The subcriticality of the branching law is crucial for the limit behaviour and
in a sense overwhelms the properties of the particles' motion. It is for this
reason that the limit is the same for all dimensions and can be obtained for a
wide class of Markov processes. Another consequence is the form of the limit -
$\SP$-valued Wiener process with a simple temporal structure and a complicated
spatial one. This behaviour contrasts sharply with the case of critical
branching systems.
http://arxiv.org/abs/0808.3755
Author(s): Zhenting Hou; Xiangxing Kong; Qinggui Zhao
Abstract: In this paper we abstract a kind of stochastic processes from evolving
processes of growing networks, which are called as growing network Markov
chains, threrefore the existence of the steady degree distribution and the
formulas of the degree distribution are transformed to the corresponding
problems of growing network Markov chains. We divide growing network markov
chains into two classes: non-multiple and multiple, and then, obtain the
condition in which the steady degree distribution exists and the exact formulas
respectively, and then applied it to the various growing networks. So we have
rigorous, exact and united solution of the steady degree distribution of the
growing networks.
http://arxiv.org/abs/0808.3661
Author(s): Michel Fliess (LIX and INRIA Saclay - Ile de France)
Abstract: The signal to noise ratio, which plays such an important r\^ole in
information theory, is shown to become pointless for digital communications
where the demodulation is achieved via new fast estimation techniques.
Operational calculus, differential algebra, noncommutative algebra and
nonstandard analysis are the main mathematical tools.
http://arxiv.org/abs/0808.3712
Author(s): Philippe Carmona (LMJL)
Abstract: The sequence of random probability measures $\nu_n$ that gives a path of
length $n$, $\unsur{n}$ times the sum of the random weights collected along the
paths, is shown to satisfy a large deviations principle with good rate function
the Legendre transform of the free energy of the associated directed polymer in
a random environment. Consequences on the asymptotics of the typical number of
paths whose collected weight is above a fixed proportion are then drawn.
http://arxiv.org/abs/0808.3842
Author(s): Florent Benaych-Georges (PMA)
Abstract: In this paper, we prove a result linking the square and the rectangular
R-transforms, which consequence is a surprising relation between the square and
rectangular free convolutions, involving the Marchenko-Pastur law. Consequences
on infinite divisibility and on the arithmetics of Voiculescu's free additive
and multiplicative convolutions are given.
http://arxiv.org/abs/0808.3938
Author(s): Zhen-Qing Chen and Takashi Kumagai
Abstract: In this paper, we consider the following type of non-local
(pseudo-differential) operators $\LL $ on $\R^d$:
$$
\LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x)
\frac{\partial}{\partial x_j}) +
\lim_{\eps \downarrow 0} \int_{\{y\in \R^d: |y-x|>\eps\}}
(u(y)-u(x)) J(x, y) dy,
$$ where $A(x)=(a_{ij}(x))_{1\leq i, j\leq d}$ is a measurable $d\times d$
matrix-valued function on $\R^d$ that is uniform elliptic and bounded and $J$
is a symmetric measurable non-trivial non-negative kernel on $\R^d\times \R^d$
satisfying certain conditions. Corresponding to $\LL$ is a symmetric strong
Markov process $X$ on $\R^d$ that has both the diffusion component and pure
jump component. We establish a priori H\"older estimate for bounded parabolic
functions of $\LL$ and parabolic Harnack principle for positive parabolic
functions of $\LL$. Moreover, two-sided sharp heat kernel estimates are derived
for such operator $\LL$ and jump-diffusion $X$. In particular, our results
apply to the mixture of symmetric diffusion of uniformly elliptic divergence
form operator and mixed stable-like processes on $\R^d$. To establish these
results, we employ methods from both probability theory and analysis.
http://arxiv.org/abs/0808.4010
Author(s): Alexander M. G. Cox and Jan K. Ob{\l}\'oj
Abstract: We consider model-free pricing of digital options, which pay out if the
underlying asset has crossed both upper and lower barriers. We make only weak
assumptions about the underlying process (typically continuity), but assume
that the initial prices of call options with the same maturity and all strikes
are known. Under such circumstances, we are able to give upper and lower bounds
on the arbitrage-free prices of the relevant options, and further, using
techniques from the theory of Skorokhod embeddings, to show that these bounds
are tight. Additionally, martingale inequalities are derived, which provide the
trading strategies with which we are able to realise any potential arbitrages.
We show that, depending of the risk aversion of the investor, the resulting
hedging strategies can outperform significantly the standard delta/vega-hedging
in presence of market frictions and/or model misspecification.
http://arxiv.org/abs/0808.4012
Author(s): Janos Englander
Abstract: In this paper we prove that the center of mass of a supercritical
branching-Brownian motion, or that of a supercritical super-Brownian motion
tends to a limiting position almost surely, which, in a sense complements a
result of Tribe on the final behavior of a critical super-Brownian motion. This
is shown to be true also for a model where branching Brownian motion is
modified by attraction/repulsion between particles.
We then put this observation together with the description of the interacting
system as viewed from its center of mass, and get the following asymptotic
behavior: the system asymptotically becomes a branching Ornstein Uhlenbeck
process (inward for attraction and outward for repulsion), but the origin is
shifted to a random point which has normal distribution, and the Ornstein
Uhlenbeck particles are not independent but constitute a system with a degree
of freedom which is less by their number by precisely one.
http://arxiv.org/abs/0808.4024
Author(s): Oliver Riordan and Nicholas Wormald
Abstract: In this paper we study the diameter of the random graph $G(n,p)$, i.e., the
largest distance between two vertices in the same component, for a wide range
of functions $p=p(n)$. For $p=\la/n$ with $\la>1$ constant, we give a simple
proof of an essentially best possible result, with an $\Op(1)$ additive
correction term. For $p=(1+\eps)/n$ with $\eps\to 0$, we obtain a corresponding
result that applies almost all the way down to the scaling window of the phase
transition, with an $\Op(1/\eps)$ additive correction term whose (appropriately
scaled) limiting distribution we describe. Throughout we use branching process
methods, rather than the more common approach of separate analysis of the
2-core.
http://arxiv.org/abs/0808.4067
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