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Probability Abstracts 104
This document contains abstracts 6994-7235
from May-1-2008 to June-30-2008.
They have been mailed on July 8th, 2008.
Author(s): Thibaud Taillefumier
Abstract: In the L\'evy construction of Brownian motion, a Haar-derived basis of
functions is used to form a finite-dimensional process $W^{N}$ and to define
the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends
to infinity. We generalize such a construction to the class of centered
Gaussian Markov processes $X$ which can be written $X_{t} = g(t) \cdot
\int_{0}^{t} f(t) dW_{t}$ with $f$ and $g$ being continuous functions. We build
the finite-dimensional process $X^{N}$ so that it gives an exact representation
of the conditional expectation of $X$ with respect to the filtration generated
by ${\lbrace X_{k/2^{N}}\rbrace}$ for $0 \leq k \leq 2^{N}$. Moreover, we prove
that the process $X^{N}$ converges in distribution toward $X$.
http://arxiv.org/abs/0805.0048
Author(s): N. Lazrieva and T. Toronjadze
Abstract: Optimal B-robust estimate is constructed for multidimensional parameter in
drift coefficient of diffusion type process with small noise. Optimal
mean-variance robust (optimal V -robust) trading strategy is find to hedge in
mean-variance sense the contingent claim in incomplete financial market with
arbitrary information structure and misspecified volatility of asset price,
which is modelled by multidimensional continuous semimartingale. Obtained
results are applied to stochastic volatility model, where the model of latent
volatility process contains unknown multidimensional parameter in drift
coefficient and small parameter in diffusion term.
http://arxiv.org/abs/0805.0122
Author(s): Paul Cuff (Stanford University)
Abstract: Two familiar notions of correlation are rediscovered as extreme operating
points for simulating a discrete memoryless channel, in which a channel output
is generated based only on a description of the channel input. Wyner's "common
information" coincides with the minimum description rate needed. However, when
common randomness independent of the input is available, the necessary
description rate reduces to Shannon's mutual information. This work
characterizes the optimal tradeoff between the amount of common randomness used
and the required rate of description.
http://arxiv.org/abs/0805.0065
Author(s): Anders Bj\"orner
Abstract: The starting point is the known fact that some much-studied random walks on
permutations, such as the Tsetlin library, arise from walks on real hyperplane
arrangements. This paper explores similar walks on complex hyperplane
arrangements. This is achieved by involving certain cell complexes naturally
associated with the arrangement. In a particular case this leads to walks on
libraries with several shelves.
We also show that interval greedoids give rise to random walks belonging to
the same general family. Members of this family of Markov chains, based on
certain semigroups, have the property that all eigenvalues of the transition
matrices are non-negative real and given by a simple combinatorial formula.
Background material needed for understanding the walks is reviewed in rather
great detail.
http://arxiv.org/abs/0805.0083
Author(s): Markus Klein and Pierre-Andr\'e Zitt (MODAL'X)
Abstract: We study resonances for the generator of a diffusion with small noise in
$R^d$ :$ L_\epsilon = -\epsilon\Delta + \nabla F \cdot \nabla$, when the
potential F grows slowly at infinity (typically as a square root of the norm).
The case when F grows fast is well known, and under suitable conditions one can
show that there exists a family of exponentially small eigenvalues, related to
the wells of F . We show that, for an F with a slow growth, the spectrum is R+,
but we can find a family of resonances whose real parts behave as the
eigenvalues of the "quick growth" case, and whose imaginary parts are small.
http://arxiv.org/abs/0805.0106
Author(s): Emmanuel Bacry and Arnaud Gloter and Marc Hoffmann and Jean-Francois Muzy
Abstract: Multifractal analysis of multiplicative random cascades is revisited within
the framework of {\em mixed asymptotics}. In this new framework, statistics are
estimated over a sample which size increases as the resolution scale (or the
sampling period) becomes finer. This allows one to continuously interpolate
between the situation where one studies a single cascade sample at arbitrary
fine scales and where at fixed scale, the sample length (number of cascades
realizations) becomes infinite. We show that scaling exponents of ''mixed''
partitions functions i.e., the estimator of the cumulant generating function of
the cascade generator distribution, depends on some ``mixed asymptotic''
exponent $\chi$ respectively above and beyond two critical value $p_\chi^-$ and
$p_\chi^+$. We study the convergence properties of partition functions in mixed
asymtotics regime and establish a central limit theorem. These results are
shown to remain valid within a general wavelet analysis framework. Their
interpretation in terms of Besov frontier are discussed. Moreover, within the
mixed asymptotic framework, we establish a ``box-counting'' multifractal
formalism that can be seen as a rigorous formulation of Mandelbrot's negative
dimension theory. Numerical illustrations of our purpose on specific examples
are also provided.
http://arxiv.org/abs/0805.0194
Author(s): Sandra Cerrai
Abstract: We prove that an averaging principle holds for a general class of stochastic
reaction-diffusion systems, having unbounded multiplicative noise, in any space
dimension. We show that the classical Khasminskii approach for systems with a
finite number of degrees of freedom can be extended to infinite dimensional
systems.
http://arxiv.org/abs/0805.0294
Author(s): Sandra Cerrai and Mark Freidlin
Abstract: We consider the averaging principle for stochastic reaction-diffusion
equations. Under some assumptions providing existence of a unique invariant
measure of the fast motion with the frozen slow component, we calculate
limiting slow motion. The study of solvability of Kolmogorov equations in
Hilbert spaces and the analysis of regularity properties of solutions, allow to
generalize the classical approach to finite-dimensional problems of this type
in the case of SPDE's.
http://arxiv.org/abs/0805.0297
Author(s): Yukio Nagahata and Nobuo Yoshida
Abstract: We consider a class of interacting particle systems with values in
$[0,\8)^{\zd}$, of which the binary contact path process is an example. For $d
\ge 3$ and under a certain square integrability condition on the total number
of the particles, we prove a central limit theorem for the density of the
particles, together with upper bounds for the density of the most populated
site and the replica overlap.
http://arxiv.org/abs/0805.0342
Author(s): Vladislav Kargin
Abstract: This paper establishes necessary and sufficient conditions for the products
of freely independent unitary operators to converge in distribution to the
uniform law on the unit circle.
http://arxiv.org/abs/0805.0374
Author(s): Ronen Gradwohl and Omer Reingold and Ariel Yadin and Amir Yehudayoff
Abstract: In a function that takes its inputs from various players, the effect of a
player measures the variation he can cause in the expectation of that function.
In this paper we prove a tight upper bound on the number of players with large
effect, a bound that holds even when the players' inputs are only known to be
pairwise independent. We also study the effect of a set of players, and show
that there always exists a "small" set that, when eliminated, leaves every set
with little effect. Finally, we ask whether there always exists a player with
positive effect. We answer this question differently in various scenarios,
depending on the properties of the function and the distribution of players'
inputs. More specifically, we show that if the function is non-monotone or the
distribution is only known to be pairwise independent, then it is possible that
all players have 0 effect. If the distribution is pairwise independent with
minimal support, on the other hand, then there must exist a player with "large"
effect.
http://arxiv.org/abs/0805.0400
Author(s): Kais Hamza and Peter Jagers and Aidan Sudbury and Daniel Tokarev
Abstract: Corresponding to $n$ independent non-negative random variables $X_1,...,X_n$,
are values $M_1,...,M_n$, where each $M_i$ is the expected value of the maximum
of $n$ independent copies of $X_i$. We obtain an upper bound to the expected
value of the maximum of $X_1,...,X_n$ in terms of $M_1,...,M_n$. This
inequality is sharp in the sense that the quantity and its bound can be made as
close to each other as we want. We also present related comparison results.
http://arxiv.org/abs/0805.0447
Author(s): Ben Morris
Abstract: We show that the spectral gap for the interchange process (and the symmetric
exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to
$\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in
any graph the spectral gap for the interchange process is the same as the
spectral gap for a corresponding continuous-time random walk. Our proof uses a
technique that is similar to that used by Handjani and Jungreis, who proved
that Aldous's conjecture holds when the graph is a tree.
http://arxiv.org/abs/0805.0480
Author(s): Mohammud Foondun and Davar Khoshnevisan
Abstract: We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u +
\sigma(u)\dot w$, where $\dot w$ denotes space-time white noise,
$\sigma:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the
$L^2$-generator of a L\'evy process. We present precise criteria for existence
as well as uniqueness of solutions. More significantly, we prove that these
solutions grow in time with at most a precise exponential rate. We establish
also that when $\sigma$ is globally Lipschitz and asymptotically sublinear, the
solution to the nonlinear heat equation is ``weakly intermittent,'' provided
that the symmetrization of $\sL$ is recurrent and the initial data is
sufficiently large.
Among other things, our results lead to general formulas for the upper
second-moment Liapounov exponent of the parabolic Anderson model for $\sL$ in
dimension $(1+1)$. When $\sL=\kappa\partial_{xx}$ for $\kappa>0$, these
formulas agree with the earlier results of statistical physics
\cite{Kardar,KrugSpohn,LL63}, and also probability theory \cite{BC,CM94} in the
two exactly-solvable cases where $u_0=\delta_0$ and $u_0\equiv 1$.
http://arxiv.org/abs/0805.0557
Author(s): Robert W. Neel
Abstract: We provide a probabilistic approach to studying minimal surfaces in
three-dimensional Euclidean space. Following a discussion of the basic
relationship between Brownian motion on a surface and minimality of the
surface, we introduce a way of coupling Brownian motions on two minimal
surfaces. This coupling is then used to study two classes of results in the
theory of minimal surfaces, maximum principle-type results, such as weak and
strong halfspace theorems and the maximum principle at infinity, and Liouville
theorems.
http://arxiv.org/abs/0805.0556
Author(s): Florence Merlevede and Magda Peligrad
Abstract: Motivated by the study of dependent random variables by coupling with
independent blocks of variables, we obtain first sufficient conditions for the
moderate deviation principle in its functional form for triangular arrays of
independent random variables. Under some regularity assumptions our conditions
are also necessary in the stationary case. The results are then applied to
derive moderate deviation principles for linear processes, kernel estimators of
a density and some classes of dependent random variables.
http://arxiv.org/abs/0805.0617
Author(s): Jianming Xia
Abstract: The comparative statics of the optimal portfolios across individuals is
carried out for a continuous-time complete market model, where the risky assets
price process follows a joint geometric Brownian motion with time-dependent and
deterministic coefficients. It turns out that the indirect utility functions
inherit the order of risk aversion (in the Arrow-Pratt sense) from the von
Neumann-Morgenstern utility functions, and therefore, a more risk-averse agent
would invest less wealth (in absolute value) in the risky assets.
http://arxiv.org/abs/0805.0618
Author(s): Jean-Christophe Mourrat
Abstract: Let $(\tau_x)_{x \in \Z^d}$ be i.i.d. random variables with heavy
(polynomial) tails. Given $a \in [0,1]$, we consider the Markov process defined
by the jump rates $\omega_{x \to y} = {\tau_x}^{-(1-a)} {\tau_y}^a$ between two
neighbours $x$ and $y$ in $\Z^d$. We give the asymptotic behaviour of the
principal eigenvalue of the generator of this process, with Dirichlet boundary
condition. The prominent feature is a phase transition that occurs at some
threshold depending on the dimension. Our method relies mainly on results
proved in the Appendix, which are of independent interest. They consist of a
Gaussian-like upper bound on the transition kernel of any symmetric
nearest-neighbour continuous-time random walk on $\Z^d$, provided its jump
rates are uniformly bounded from below, together with an upper bound on the
Green function when $d \ge 3$.
http://arxiv.org/abs/0805.0706
Author(s): Jo\"el De Coninck and Fran\c{c}ois Dunlop and Thierry Huillet
Abstract: We consider a random walk $X_n$ in $\Ze_+$, starting at $X_0=x\ge0$, with
transition probabilities
$$\Pe(X_{n+1}=X_n\pm1|X_n=y\ge1)={1\over2}\mp{\del\over4y+2\del}$$ and
$X_{n+1}=1$ whenever $X_n=0$. We prove $\Ee X_n\sim{\rm const.}
n^{1-{\del\over2}}$ as $n\nea\infty$ when $\del\in(1,2)$. The proof is based
upon the Karlin-McGregor spectral representation, which is made explicit for
this random walk.
http://arxiv.org/abs/0805.0729
Author(s): M. Jara
Abstract: We prove that the hydrodynamic limit of a zero-range process evolving in
graphs approximating the Sierpinski gasket is given by a nonlinear heat
equation. We also prove existence and uniqueness of the hydrodynamic equation
by considering a finite-difference scheme.
http://arxiv.org/abs/0805.0380
Author(s): Nicola Cufaro Petroni and Modesto Pusterla
Abstract: We analyze the extension of the well known relation between Brownian motion
and Schroedinger equation to the family of Levy processes. We propose a
Levy-Schroedinger equation where the usual kinetic energy operator - the
Laplacian - is generalized by means of a pseudodifferential operator whose
symbol is the logarithmic characteristic of an infinitely divisible law. The
Levy-Khintchin formula shows then how to write down this operator in an
integro--differential form. When the underlying Levy process is stable we
recover as a particular case the recently proposed fractional Schroedinger
equation. A few examples are finally given and we find that there are
physically relevant models (such as a form of the relativistic Schroedinger
equation) that are in the domain of the possible Levy-Schroedinger equations.
http://arxiv.org/abs/0805.0503
Author(s): Jiun-Chau Wang
Abstract: In this paper we find necessary and sufficient conditions for the weak
convergence of c-free convolution of pairs of measures, where the measures are
assumed to be infinitesimal and their support may be unbounded. These results
are obtained by complex analytic methods.
http://arxiv.org/abs/0805.0607
Author(s): Jean-Philippe Aguilar (CPT) and Nils Berglund (MAPMO)
Abstract: We consider a quantum two-level system perturbed by classical noise. The
noise is implemented as a stationary diffusion process in the off-diagonal
matrix elements of the Hamiltonian, representing a transverse magnetic field.
We determine the invariant measure of the system and prove its uniqueness. In
the case of Ornstein-Uhlenbeck noise, we determine the speed of convergence to
the invariant measure. Finally, we determine an approximate one-dimensional
diffusion equation for the transition probabilities. The proofs use both
spectral-theoretic and probabilistic methods.
http://arxiv.org/abs/0805.0869
Author(s): Ivan del Tenno
Abstract: In this article we investigate the asymptotic behavior of a new class of
multi-dimensional diffusions in random environment. We introduce cut times in
the spirit of the work done by Bolthausen, Sznitman and Zeitouni, see [4], in
the discrete setting providing a decoupling effect in the process. This allows
us to take advantage of an ergodic structure to derive a strong law of large
numbers with possibly vanishing limiting velocity and a central limit theorem
under the quenched measure.
http://arxiv.org/abs/0805.0886
Author(s): Christina Goldschmidt and B\'en\'edicte Haas (CEREMADE)
Abstract: The stable fragmentation with index of self-similarity $\alpha \in [-1/2,0)$
is derived by looking at the masses of the subtrees formed by discarding the
parts of a $(1 + \alpha)^{-1}$--stable continuum random tree below height $t$,
for $t \geq 0$. We give a detailed limiting description of the distribution of
such a fragmentation, $(F(t), t \geq 0)$, as it approaches its time of
extinction, $\zeta$. In particular, we show that $t^{1/\alpha}F((\zeta - t)^+)$
converges in distribution as $t \to 0$ to a non-trivial limit. In order to
prove this, we go further and describe the limiting behavior of (a) an
excursion of the stable height process (conditioned to have length 1) as it
approaches its maximum; (b) the collection of open intervals where the
excursion is above a certain level and (c) the ranked sequence of lengths of
these intervals. Our principal tool is excursion theory. We also consider the
last fragment to disappear and show that, with the same time and space
scalings, it has a limiting distribution given in terms of a certain
size-biased version of the law of $\zeta$.
http://arxiv.org/abs/0805.0967
Author(s): Djalil Chafai (IMT and UPTE) and Florent Malrieu (IRMAR)
Abstract: Mixtures are convex combinations of laws. Despite this simple definition, a
mixture can be far more subtle than its mixed components. For instance, mixing
Gaussian laws may produce a wild potential with multiple wells. We study in the
present work fine properties of mixtures with respect to concentration of
measure and Gross type functional inequalities. We provide sharp Laplace bounds
for Lipschitz functions in the case of generic mixtures, involving a
transportation cost diameter of the mixed family. We also provide precise upper
bounds for two-components mixtures. Additionally, our analysis of Gross type
inequalities for two-components mixtures reveals natural relations with some
kind of band isoperimetry and support constrained interpolation via mass
transportation. We show that the Poincar\'e constant of a two-components
mixture may remain bounded as the mixture proportion goes to 0 or 1 while the
Gross constant may surprisingly blow up. Additionally, this counter-intuitive
result is not reducible to support disconnections. As far as mixture of
distributions are concerned, the Gross inequality is less stable than the
sub-Gaussian concentration for Lipschitz functions. We illustrate our results
on a gallery of concrete two-components mixtures.
http://arxiv.org/abs/0805.0987
Author(s): Jesse E. Taylor and Amandine Veber
Abstract: We investigate the infinitely many demes limit of the genealogy of a sample
of individuals from a subdivided population subject to sporadic mass extinction
events. By exploiting a separation of timescales property of Wright's island
model, we show that as the number of demes tends to infinity the limiting form
of the genealogy can be described in terms of the alternation of instantaneous
'scattering' phases dominated by local demographic processes, and extended
'collecting' phases dominated by global processes. When extinction and
recolonization events are local, this genealogy is given by Kingman's
coalescent and the scattering phase influences only the overall rate of the
process. In contrast, if the vacant demes left by a mass extinction event can
be recolonized by individuals emerging from a small number of demes, then the
limiting genealogy is a colaescent with simultaneous multiple mergers. In this
case, the details of the within-deme population dynamics influence not only the
overall rate of the coalescent process, but also the statistics of the complex
mergers that can occur within sample genealogies. This study gives some insight
into the genealogical consequences of mass extinction in structured
populations.
http://arxiv.org/abs/0805.1010
Author(s): Robert J Adler
Abstract: This is a brief review, in relatively non-technical terms, of recent advances
in the theory of random field geometry. These advances have provided a
collection of explicit new formulae describing mean values of a variety of
geometric characteristics of excursion sets of random fields. As well as a
review of the theory, we provide brief descriptions of some of the more
interesting applications.
http://arxiv.org/abs/0805.1031
Author(s): Franz Lehner
Abstract: We show that the Plancherel measure of the lamplighter random walk on a graph
coincides with the expected spectral measure of the absorbing random walk on
the Bernoulli percolation clusters. In the subcritical regime the spectrum is
pure point and we construct a complete set of finitely supported
eigenfunctions.
http://arxiv.org/abs/0805.0867
Author(s): Masayoshi Watanabe
Abstract: We prove that an approximated version of the Brunn--Minkowski inequality with
volume distortion coefficient implies a Gaussian concentration-of-measure
phenomenon. Our main theorem is applicable to discrete spaces.
http://arxiv.org/abs/0805.0902
Author(s): Dalibor Voln\'y
Abstract: The article is showing sharpness of central limit theorems of Kipnis and
Varadhan, Derriennic and Lin, Maxwell and Woodroofe. In the case of the CLT of
Derriennic and Lin (for Markov chains with a normal operator) it is shown that
the assumption of normality cannot be relaxed. In the case of the CLT of
Maxwell and Woodroofe, the example of Peligrad and Utev is improved in the
sense of getting a convergence to different laws.
http://arxiv.org/abs/0805.1198
Author(s): Brahim El Asri and Said Hamadene
Abstract: In this paper we show existence and uniqueness of a solution for a system of
m variational partial differential inequalities with inter-connected obstacles.
This system is the deterministic version of the Verification Theorem of the
Markovian optimal m-states switching problem. The switching cost functions are
arbitrary. This problem is in relation with the valuation of firms in a
financial market.
http://arxiv.org/abs/0805.1306
Author(s): M. Jara
Abstract: We consider some interacting particle processes with long-range dynamics: the
zero-range and exclusion processes with long jumps. We prove that the
hydrodynamic limit of these processes corresponds to a (possibly non-linear)
fractional heat equation. The scaling in this case is superdiffusive. In
addition, we discuss a central limit theorem for a tagged particle on the
zero-range process and existence and uniqueness of solutions of the Cauchy
problem for the fractional heat equation.
http://arxiv.org/abs/0805.1326
Author(s): Frank Aurzada and Steffen Dereich
Abstract: We study the small deviation problem $\log \mathbb{P}(\sup_{t\in[0,1]} |X_t|
\leq \epsilon)$, as $\epsilon\to 0$, for general L\'evy processes $X$. The
techniques enable us to determine the asymptotic rate for general real-valued
L\'evy processes, which we demonstrate with many examples.
As a particular consequence, we show that a L\'evy process with non-vanishing
Gaussian component has the same (strong) asymptotic small deviation rate as the
corresponding Brownian motion.
http://arxiv.org/abs/0805.1330
Author(s): Marie Albenque
Abstract: In the literature, most of the results about the enumeration of directed
animals on lattices via gas considerations are obtained by a formal passage to
the limit of enumeration of directed animals on cyclical versions of the
lattice.
We provide here a new point of view on this phenomenon. Using the gas
construction given introduced by Le Borgne and Marckert, we represent the gas
process on the cyclical versions of the lattices as a cyclical Markov chain
(roughly speaking, Markov chains conditioned to come back to their starting
point). Then we provide a notion of convergence of graphs, such that if $(G_n)$
converges to $G$ then the gas process built on $G_n$ converges in distribution
to the gas process on $G$. This gives a general tool to show that gas processes
related to animals enumeration are often Markovian on some extracted line of
the lattice.
We provide examples and computations of new generating functions for directed
animals with various sources on some families of lattices.
http://arxiv.org/abs/0805.1349
Author(s): Evarist Gin\'e and Richard Nickl
Abstract: Given an i.i.d. sample from a distribution $F$ on $\mathbb R$ with uniformly
continuous density $p_0$, purely-data driven estimators are constructed that
efficiently estimate $F$ in sup-norm loss, and simultaneously estimate $p_0$ at
the best possible rate of convergence over H\"{o}lder balls, also in sup-norm
loss. The estimators are obtained from applying a model selection procedure
close to Lepski's method with random thresholds to projections of the empirical
measure onto spaces spanned by wavelets or $B$-splines. Explicit constants in
the asymptotic risk of the estimator are obtained, as well as oracle-type
inequalities in sup-norm loss. The random thresholds are based on suprema of
Rademacher processes indexed by wavelet or spline projection kernels. This
requires Bernstein-analogues of the inequalities in Koltchinskii (2006) for the
deviation of suprema of empirical processes from their Rademacher
symmetrizations.
http://arxiv.org/abs/0805.1404
Author(s): Evarist Gin\'e and Richard Nickl
Abstract: Let $p_n (y)=\sum_k \hat \alpha_{k} \phi(y-k) + \sum_{l=0}^{j_n-1} \sum_k
\hat \beta_{lk} 2^{l/2} \psi(2^ly-k)$ be the wavelet density estimator, where
$\phi$, $\psi$ are a father and a mother wavelet (with compact support), $\hat
\alpha_k$, $\hat \beta_{lk}$ are the empirical wavelet coefficients based on an
i.i.d. sample of random variables distributed according to a density $p_0$ on
$\mathbb R$, and $j_n \in \mathbb Z$, $j_n \nearrow \infty$. Several uniform
limit theorems are proved: First, the almost sure rate of convergence of
$\sup_{y \in \mathbb R} |p_n(y)-Ep_n(y)|$ is obtained, and a law of the
logarithm for a suitably scaled version of this quantity is established. This
implies that $\sup_{y \in \mathbb R} |p_n(y)-p_0(y)|$ attains the optimal
almost sure rate of convergence for estimating $p_0$, if $j_n$ is suitably
chosen. Second, a uniform central limit theorem as well as strong invariance
principles for the distribution function of $p_n$, that is, for the stochastic
processes $\sqrt n (F_n^W(s) - F(s))= \sqrt n \int_{-\infty}^s (p_n-p_0), s \in
\mathbb R$, are proved; and more generally, uniform central limit theorems for
the processes $\sqrt n \int (p_n-p_0)f; f \in \mathcal F$, for other Donsker
classes $\mathcal F$ of interest are considered. As a statistical application,
it is shown that essentially the same limit theorems can be obtained for the
hard thresholding wavelet estimator introduced by Donoho, Johnstone,
Kerkyacharian and Picard (1996).
http://arxiv.org/abs/0805.1406
Author(s): Zhenting Hou and Xiangxing Kong and Dinghua Shi and Guanrong Chen
Abstract: Based on the concept and techniques of first-passage probability in Markov
chain theory, this letter provides a rigorous proof for the existence of the
steady-state degree distribution of the scale-free network generated by the
Barabasi-Albert (BA) model, and mathematically re-derives the exact analytic
formulas of the distribution. The approach developed here is quite general,
applicable to many other scale-free types of complex networks.
http://arxiv.org/abs/0805.1434
Author(s): Jan van Neerven and Mark Veraar and Lutz Weis
Abstract: A detailed theory of stochastic integration in UMD Banach spaces has been
developed recently by the authors. The present paper is aimed at giving various
sufficient conditions for stochastic integrability.
http://arxiv.org/abs/0805.1458
Author(s): Anton Bovier and Anton Klimovsky
Abstract: We study Derrida's generalized random energy model in the presence of uniform
external field. We compute the fluctuations of the ground state and of the
partition function in the thermodynamic limit for all admissible values of
parameters. We find that the fluctuations are described by a hierarchical
structure which is obtained by a certain coarse-graining of the initial
hierarchical structure of the GREM with external field. We provide an explicit
formula for the free energy of the model. We also derive some large deviation
results providing an expression for the free energy in a class of models with
Gaussian Hamiltonians and external field. Finally, we prove that the
coarse-grained parts of the system emerging in the thermodynamic limit tend to
have a certain optimal magnetization, as prescribed by strength of external
field and by parameters of the GREM.
http://arxiv.org/abs/0805.1478
Author(s): Zbigniew Palmowski and Martijn Pistorius
Abstract: We analyze the asymptotics of crossing a high piecewise linear barriers by a
renewal compound process with the subexponential jumps. The study is motivated
by ruin probabilities of two insurance companies (or two branches of the same
company) that divide between them both claims and premia in some specified
proportions when the initial reserves of both companies tend to infinity.
http://arxiv.org/abs/0805.1631
Author(s): Bruce Driver and Maria Gordina
Abstract: We introduce a class of non-commutative Heisenberg like infinite dimensional
Lie groups based on an abstract Wiener space. The Ricci curvature tensor for
these groups is computed and shown to be bounded. Brownian motion and the
corresponding heat kernel measures, $\{\nu_t\}_{t>0},$ are also studied. We
show that these heat kernel measures admit: 1) Gaussian like upper bounds, 2)
Cameron-Martin type quasi-invariance results, 3) good $L^p$ -- bounds on the
corresponding Radon-Nykodim derivatives, 4) integration by parts formulas, and
5) logarithmic Sobolev inequalities. The last three results heavily rely on the
boundedness of the Ricci tensor.
http://arxiv.org/abs/0805.1650
Author(s): Kenneth S. Alexander and Nikos Zygouras
Abstract: We consider a polymer with configuration modeled by the path of a Markov
chain, interacting with a potential $u+V_n$ which the chain encounters when it
visits a special state 0 at time $n$. The disorder $(V_n)$ is a fixed
realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends
a positive fraction of its time at state 0, when $u$ exceeds a critical value.
We assume that for the Markov chain in the absence of the potential, the
probability of an excursion from 0 of length $n$ has the form $n^{-c}\phi(n)$
with $c \geq 1$ and $\phi$ slowly varying. Comparing to the corresponding
annealed system, in which the $V_n$ are effectively replaced by a constant, it
is known that the quenched and annealed critical points differ at all
temperatures for $3/22$, but only at low temperatures for $c<3/2$.
For high temperatures and $3/23/2$ with arbitrary temperature we provide a new proof
that the gap is positive, and extend it to $c=2$.
http://arxiv.org/abs/0805.1708
Author(s): Alexander V. Kolesnikov
Abstract: We study isoperimetric inequalities for measures of the type $\mu=e^{V} dx$,
where $V$ is convex. Using optimal transportation techniques we estimate
isoperimetric profiles for a broad class of such measures. We consider many
examples and reviel some relations to the hyperbolic geometry and curvature
flows.
http://arxiv.org/abs/0805.1584
Author(s): Gabriele Bianchi
Abstract: The cross covariogram g_{K,L} of two convex sets K, L in R^n is the function
which associates to each x in R^n the volume of the intersection of K with L+x.
The problem of determining the sets from their covariogram is relevant in
stochastic geometry, in probability and it is equivalent to a particular case
of the phase retrieval problem in Fourier analysis. It is also relevant for the
inverse problem of determining the atomic structure of a quasicrystal from its
X-ray diffraction image.
The two main results of this paper are that g_{K,K} determines
three-dimensional convex polytopes K and that g_{K,L} determines both K and L
when K and L are convex polyhedral cones satisfying certain assumptions. These
results settle a conjecture of G. Matheron in the class of convex polytopes.
Further results regard the known counterexamples in dimension n>=4. We also
introduce and study the notion of synisothetic polytopes. This concept is
related to the rearrangement of the faces of a convex polytope.
http://arxiv.org/abs/0805.1605
Author(s): A. I. Nazarov
Abstract: We prove a new variant of comparison principle for logarithmic $L_2$-small
ball probabilities of Gaussian processes. As an application, we obtain
logarithmic small ball asymptotics for some well-known processes with smooth
covariances.
http://arxiv.org/abs/0805.1773
Author(s): Vincent Bansaye (PMA)
Abstract: We consider a branching model for a population of dividing cells infected by
parasites. Each cell receives parasites by inheritance from its mother cell and
independent contamination from outside the population. Parasites multiply
randomly inside the cell and are shared randomly between the two daughter cells
when the cell divides. The law of the number of parasites which contaminate a
given cell depends only on whether the cell is already infected or not. We
determine the asymptotic behavior of the number of parasites in a cell line,
which follows a branching process in random environment with state dependent
immigration. We then derive a law of large numbers for the asymptotic
proportions of cells with a given number of parasites. The main tools are
branching processes in random environment and laws of large numbers for Markov
tree.
http://arxiv.org/abs/0805.1863
Author(s): Giuseppe Da Prato (ENS) and Arnaud Debussche (IRMAR)
Abstract: We consider the martingale problem associated to the Navier-Stokes in
dimension 2 or 3. Existence is well known and it has been recently shown that
markovian transition semi group associated to these equations can be
constructed. We study the Kolmogorov operator associated to these equations. It
can be defined formally as a differential operator on an infinite dimensional
Hilbert space. It can be also defined in an abstract way as the infinitesimal
generator of the transition semi group. We explicit cores for these abstract
operators and identify them with the concrete differential operators on these
cores. In dimension 2, the core is explicit and we can use a classical argument
to prove uniqueness for the martingale problem. In dimension 3, we are only
able to exhibit a core which is defined abstractly and does not allow to prove
uniqueness for the martingale problem. Instead, we exhibit a core for a
modified Kolmogorov operator which enables us to prove uniqueness for the
martingale problem up to the time the solutions are regular.
http://arxiv.org/abs/0805.1906
Author(s): Maxim Krikun (IECN)
Abstract: We show that the Schaeffer's tree for an infinite quadrangulation only
changes locally when changing the root of the quadrangulation. This follows
from one property of distances in the infinite uniform random quadrangulation.
http://arxiv.org/abs/0805.1907
Author(s): Lisandro J. Fermin
Abstract: The aim of this paper is to extend the aggregation convergence results given
in (Dacunha-Castelle and Fermin 2005, Dacunha-Castelle and Fermin 2008) to
doubly stochastic linear and nonlinear processes with weakly dependent
innovations. First, we introduce a weak dependence notion for doubly stochastic
processes, based in the weak dependence definition given in (Doukhan and
Louhichi 1999), and we exhibe several models satisfying this notion, such as:
doubly stochastic Volterra processes and doubly stochastic Bernoulli scheme
with weakly dependent innovations. Afterwards we derive a central limit theorem
for the partial aggregation sequence considering weakly dependent doubly
stochastic processes. Finally, show a new SLLN for the covariance function of
the partial aggregation process in the case of doubly stochastic Volterra
processes with interactive innovations.
Keywords: Aggregation, weak dependence, doubly stochastic processes, Volterra
processes, Bernoulli shift, TCL, SLLN.
http://arxiv.org/abs/0805.1949
Author(s): A.I. Nazarov
Abstract: We consider a set of one-dimensional transformations of Gaussian random
functions. Under natural assumptions we obtain a connection between $L_2$-small
ball asymptotics of the transformed function and of the original one. Also the
explicit Karhunen -- Lo\'eve expansion is obtained for a proper class of
Gaussian processes.
http://arxiv.org/abs/0805.1967
Author(s): Christian Bender and Tina Marquardt
Abstract: We develop a stochastic calculus for processes which are built by convoluting
a pure jump, zero expectation L\'{e}vy process with a Volterra-type kernel.
This class of processes contains, for example, fractional L\'{e}vy processes as
studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we
introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities
from Malliavin calculus and white noise analysis and give an elementary
definition based on expectations under change of measure. As a main result, we
derive an It\^{o} formula which separates the different contributions from the
memory due to the convolution and from the jumps.
http://arxiv.org/abs/0805.2084
Author(s): Nicolas Privault and Anthony Reveillac
Abstract: In this paper we consider the nonparametric functional estimation of the
drift of Gaussian processes using Paley-Wiener and Karhunen-Lo\`eve expansions.
We construct efficient estimators for the drift of such processes, and prove
their minimaxity using Bayes estimators. We also construct superefficient
estimators of Stein type for such drifts using the Malliavin integration by
parts formula and stochastic analysis on Gaussian space, in which superharmonic
functionals of the process paths play a particular role. Our results are
illustrated by numerical simulations and extend the construction of James-Stein
type estimators for Gaussian processes by Berger and Wolper.
http://arxiv.org/abs/0805.2002
Author(s): K. T.-R. McLaughlin and P. D. Miller
Abstract: We obtain Plancherel-Rotach type asymptotics valid in all regions of the
complex plane for orthogonal polynomials with varying weights of the form
$e^{-NV(x)}$ on the real line, assuming that $V$ has only two Lipschitz
continuous derivatives and that the corresponding equilibrium measure has
typical support properties. As an application we extend the universality class
for bulk and edge asymptotics of eigenvalue statistics in unitary invariant
Hermitian random matrix theory. Our methodology involves developing a new
technique of asymptotic analysis for matrix Riemann-Hilbert problems with
nonanalytic jump matrices suitable for analyzing such problems even near
transition points where the solution changes from oscillatory to exponential
behavior.
http://arxiv.org/abs/0805.1980
Author(s): Luigi Manca
Abstract: We are concerned with a viscous Burgers equation forced by a perturbation of
white noise type. We study the corresponding transition semigroup in a space of
continuous functions weighted by a proper potential, and we show that the
infinitesimal generator is the closure (with respect to a suitable topology) of
the Kolmogorov operator associated to the stochastic equation. In the last part
of the paper we use this result to solve the corresponding Fokker-Planck
equation.
http://arxiv.org/abs/0805.2011
Author(s): Simon J.A. Malham and Anke Wiese
Abstract: We study solutions to nonlinear stochastic differential systems driven by a
multi-dimensional Wiener process with non-commuting diffusion vector fields,
and no drift. We construct universal optimal solution expansions. They are
optimal because the solution series truncated at any order is at least as
accurate as the corresponding stochastic Taylor truncation in the mean-square
sense. They are universal because this property is independent of the vector
fields concerned. This series is the hyperbolic sine of the logarithm of the
stochastic Taylor flow. Our proof utilizes the underlying Hopf algebra
structure of these series, and a two-alphabet associative algebra of shuffle
and concatenation operations that distinguish the coefficients of each term in
the series.
http://arxiv.org/abs/0805.2340
Author(s): Philippe Blanchard and Michael R\"ockner (SFB 705) and Francesco Russo (LAGA)
Abstract: We consider a porous media type equation over all of $\R^d$ with $d = 1$,
with monotone discontinuous coefficients with linear growth and prove a
probabilistic representation of its solution in terms of an associated
microscopic diffusion. This equation is motivated by some singular behaviour
arising in complex self-organized critical systems. One of the main analytic
ingredients of the proof, is a new result on uniqueness of distributional
solutions of a linear PDE on $\R^1$ with non-continuous coefficients.
http://arxiv.org/abs/0805.2383
Author(s): Florian Conrad and Martin Grothaus
Abstract: We construct an infinite particle/infinite volume Langevin dynamics on the
space of configurations in $\R^d$ having velocities as marks. The construction
is done via a limiting procedure using $N$-particle dynamics in cubes
$(-\lambda,\lambda]^d$ with periodic boundary conditions. A main step to this
result is to derive an (improved) Ruelle bound for the canonical correlation
functions of $N$-particle systems in $(-\lambda,\lambda]^d$ with periodic
boundary conditions. After proving tightness of the laws of finite particle
dynamics, the identification of accumulation points as martingale solutions of
the Langevin equation is based on a general study of properties of measures on
configuration space (and their weak limit) fulfilling a uniform Ruelle bound.
Additionally, we prove that the initial/invariant distribution of the
constructed dynamics is a tempered grand canonical Gibbs measure. All proofs
work for general repulsive interaction potentials $\phi$ of Ruelle type (e.g.
the Lennard-Jones potential) and all temperatures, densities and dimensions
$d\geq 1$.
http://arxiv.org/abs/0805.2518
Author(s): Bernard Shiffman and Steve Zelditch and Scott Zrebiec
Abstract: We give asymptotic large deviations estimates for the volume inside a domain
U of the zero set of a random holomorphic section of the N-th power of a
positive line bundle on a compact Kaehler manifold. In particular, we show that
for all $\delta>0$, the probability that this volume differs by more than
$\delta N$ from its average value is less than $\exp(-C_{\delta,U}N^{m+1})$,
for some constant $C_{\delta,U}>0$. As a consequence, the "hole probability"
that a random section does not vanish in U has an upper bound of the form
$\exp(-C_{U}N^{m+1})$.
http://arxiv.org/abs/0805.2598
Author(s): Nobuo Yoshida
Abstract: We consider a simple discrete-time Markov chain with values in
$[0,\infty)^{Z^d}$. The Markov chain describes various interesting examples
such as oriented percolation, directed polymers in random environment, time
discretizations of binary contact path process and the voter model. We study
the phase transition for the growth rate of the "total number of particles" in
this framework. The main results are roughly as follows: If $d \ge 3$ and the
Markov chain is "not too random", then, with positive probability, the growth
rate of the total number of particles is of the same order as its expectation.
If on the other hand, $d=1,2$, or the Markov chain is "random enough", then the
growth rate is slower than its expectation. We also discuss the above phase
transition for the dual processes and its connection to the structure of
invariant measures for the Markov chain with proper normalization.
http://arxiv.org/abs/0805.2652
Author(s): Paul Humphreys
Abstract: This paper argues for the status of formal probability theory as a
mathematical, rather than a scientific, theory. David Freedman and Philip
Stark's concept of model based probabilities is examined and is used as a
bridge between the formal theory and applications.
http://arxiv.org/abs/0805.2801
Author(s): Morris L. Eaton
Abstract: In this expository paper we describe a relatively elementary method of
establishing the existence of a Dutch book in a simple multivariate normal
prediction setting. The method involves deriving a nonstandard predictive
distribution that is motivated by invariance. This predictive distribution
satisfies an interesting identity which in turn yields an elementary
demonstration of the existence of a Dutch book for a variety of possible
predictive distributions.
http://arxiv.org/abs/0805.2808
Author(s): Claudio Asci
Abstract: This paper is about the rate of convergence of the Markov chain
$X_{n+1}=AX_{n}+B_{n}$ (mod $p$), where $A$ is an integer matrix with nonzero
eigenvalues and ${B_{n}}_{n}$ is a sequence of independent and identically
distributed integer vectors, with support not parallel to a proper subspace of
$Q^{k}$ invariant under $A$. If $|\lambda_{i}|\not=1$ for all eigenvalues
$\lambda_{i}$ of $A$, then $n=O((\ln p)^{2}) $ steps are sufficient and
$n=O(\ln p)$ steps are necessary to have $X_{n}$ sampling from a nearly uniform
distribution. Conversely, if $A$ has the eigenvalues $\lambda_{i}$ that are
roots of positive integer numbers, $|\lambda_{1}|=1$ and $|\lambda_{i}|>1$ for
all $i\not=1$, then $O(p^{2}) $ steps are necessary and sufficient.
http://arxiv.org/abs/0805.2830
Author(s): Sophie Dede (PMA)
Abstract: In this paper, we derive the moderate deviation principle for stationary
sequences of bounded random variables with values in a Hilbert space. The
conditions obtained are expressed in terms of martingale-type conditions. The
main tools are martingale approximations and a new Hoeffding inequality for non
adpated sequences of Hilbert-valued random variables. Applications to
Cramer-Von Mises statistics, functions of linear processes and stable Markov
chains are given.
http://arxiv.org/abs/0805.2899
Author(s): Guillaume Aubrun (ICJ)
Abstract: For large $d$, we study quantum channels on $\C^d$ obtained by selecting
randomly $N$ independent Kraus operators according to a probability measure
$\mu$ on the unitary group $\mU(d)$. When $\mu$ is the Haar measure, we show
that for $N \succcurlyeq d/\e^2$, such a channel is $\e$-randomizing with high
probability, which means that it maps every state within distance $\e/d$ (in
operator norm) of the maximally mixed state. This slightly improves on a result
by Hayden, Leung, Shor and Winter by optimizing their discretization argument.
Moreover, for general $\mu$, we obtain a $\e$-randomizing channel provided $N
\succcurlyeq d (\log d)^6/\e^2$. For $d=2^k$ ($k$ qubits), this includes Kraus
operators obtained by tensoring $k$ random Pauli matrices. The proof uses
recent results on empirical processes in Banach spaces.
http://arxiv.org/abs/0805.2900
Author(s): Gianni Pagnini and Francesco Mainardi
Abstract: The spectrum profile that emerges in molecular spectroscopy and atmospheric
radiative transfer as the combined effect of Doppler and pressure broadenings
is known as the Voigt profile function. Because of its convolution integral
representation, the Voigt profile can be interpreted as the probability density
function of the sum of two independent random variables with Gaussian density
(due to the Doppler effect) and Lorentzian density (due to the pressure
effect). Since these densities belong to the class of symmetric L\'evy stable
distributions, a probabilistic generalization is proposed as the convolution of
two arbitrary symmetric L\'evy densities. We study the case when the widths of
the considered distributions depend on a scale-factor $\tau$ that is
representative of spatial inhomogeneity or temporal non-stationarity. The
evolution equations for this probabilistic generalization of the Voigt function
are here introduced and interpreted as generalized diffusion equations
containing two Riesz space-fractional derivatives, thus classified as
space-fractional diffusion equations of double order.
http://arxiv.org/abs/0711.4246
Author(s): Antonio Mura and Murad S. Taqqu and Francesco Mainardi
Abstract: In this paper we introduce and analyze a class of diffusion type equations
related to certain non-Markovian stochastic processes. We start from the
forward drift equation which is made non-local in time by the introduction of a
suitable chosen memory kernel K(t). The resulting non-Markovian equation can be
interpreted in a natural way as the evolution equation of the marginal density
function of a random time process l(t). We then consider the subordinated
process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding
time-evolution of the marginal density function of Y(t) is governed by a
non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We
develop several applications and derive the exact solutions. We consider
different stochastic models for the given equations providing path simulations.
http://arxiv.org/abs/0712.0240
Author(s): Vadim Gorin
Abstract: We prove pairwise disjointness of representations T_{z,w} of the
infinite-dimensional unitary group. These representations provide a natural
generalization of the regular representation for the case of "big" group
U(\infty). They were introduced and studied by G.Olshanski and A.Borodin.
Disjointness of the representations can be reduced to disjointness of certain
probability measures on the space of paths in the Gelfand-Tsetlin graph. We
prove the latter disjointness using probabilistic and combinatorial methods.
http://arxiv.org/abs/0805.2660
Author(s): Igor Wigman
Abstract: We study the length of the nodal set of eigenfunctions of the Laplacian on
the $\spheredim$-dimensional sphere. It is well known that the eigenspaces
corresponding to $\eigval=n(n+\spheredim-1)$ are the spaces $\eigspc$ of
spherical harmonics of degree $n$, of dimension $\eigspcdim$. We use the
multiplicity of the eigenvalues to endow $\eigspc$ with the Gaussian
probability measure and study the distribution of the $\spheredim$-dimensional
volume of the nodal sets of a randomly chosen function. The expected volume is
proportional to $\sqrt{\eigval}$. One of our main results is bounding the
variance of the volume to be $O(\frac{\eigval}{\sqrt{\eigspcdim}})$.
In addition to the volume of the nodal set, we study its Leray measure. For
every $n$, the expected value of the Leray measure is $\frac{1}{\sqrt{2\pi}}$.
We are able to determine that the asymptotic form of the variance is
$\frac{const}{\eigspcdim}$.
http://arxiv.org/abs/0805.2768
Author(s): E. H. Essaky and M. Hassani
Abstract: In this paper we study one-dimensional generalized reflected backward
stochastic differential equation with two barriers and stochastic quadratic
growth. We prove the existence of a maximal solution when there exists a
semimartingale between the barriers L and U, the generator f is continuous with
general growth with respect to the variable y and stochastic quadratic growth
with respect to the variable z and without assuming any P-integrability
conditions on the data. The proof of our result is based on the use of a
comparison theorem, an exponential transformation and an approximation
technique. Our result is applied to the Dynkin game problem as well as to the
American game option.
http://arxiv.org/abs/0805.2979
Author(s): Giada Basile (WIAS) and Stefano Olla (CEREMADE) and Herbert Spohn (D-Mutu-ZM)
Abstract: We consider lattice dynamics with a small stochastic perturbation of order
\epsilon and prove that for a space-time scale of order \epsilon-1 the Wigner
function evolves according to a linear transport equation describing inelastic
collisions. For an energy and momentum conserving chain the transport equation
predicts a slow decay, as 1/\sqrt{t}, for the energy current correlation in
equilibrium. This is in agreement with previous studies using a different
method.
http://arxiv.org/abs/0805.3012
Author(s): Holger K\"osters
Abstract: We investigate the asymptotic behaviour of the second-order correlation
function of the characteristic polynomial of a Hermitian Wigner matrix at the
edge of the spectrum. We show that the suitably rescaled second-order
correlation function is asymptotically given by the Airy kernel, thereby
generalizing the well-known result for the Gaussian Unitary Ensemble (GUE).
Moreover, we obtain similar results for real-symmetric Wigner matrices.
http://arxiv.org/abs/0805.3044
Author(s): Serge Cohen (LSProba) and Cl\'ement Dombry (LMA)
Abstract: It is classical to approximate the distribution of fractional Brownian motion
by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this
paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $
for $ j \in \mathbb Z,$ when the ceiling of the walk $ S_n $ is located at $
j.$ The random reward (or scenery) $ \xi_j $ is independent of the walk and
with heavy tail. We show the convergence of the sum of independent copies of $
Z_n$ suitably renormalized to a stable motion with integral representation,
whose kernel is the local time of a fractional Brownian motion (fBm). This work
extends a previous work where the random walk $ S_n$ had independent increments
limits.
http://arxiv.org/abs/0805.3054
Author(s): Shui Feng
Abstract: The behavior of the Poisson-Dirichlet distribution with small mutation rate
is studied through large deviations. The structure of the rate function
indicates that the number of alleles is finite at the instant when mutation
appears. The large deviation results are then used to study the asymptotic
behavior of the homozygosity, and the Poisson-Dirichlet distribution with
symmetric selection. The latter shows that several alleles can coexist when
selection intensity goes to infinity in a particular way as the mutation rate
approaches zero.
http://arxiv.org/abs/0805.3113
Author(s): Natali Zint and Ellen Baake and Frank den Hollander
Abstract: A stochastic model for the activation of T-cells is analysed. T-cells are
part of the immune system and recognize foreign antigens against a background
of the body's own molecules. The model under consideration is a slight
generalization of a model introduced by Van den Berg, Rand and Burroughs in
2001, and is capable of explaining how this recognition works on the basis of
rare stochastic events. With the help of a refined large deviation theorem and
numerical evaluation it is shown that, for a wide range of parameters, T-cells
can distinguish reliably between foreign antigens and self-antigens.
http://arxiv.org/abs/q-bio/0605016
Author(s): Terence Tao and Van Vu
Abstract: Let $x$ be a complex random variable with mean zero and bounded variance. Let
$N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $x$
and $M$ be an arbitrary matrix. We give a general estimate for the least
singular value of the matrix $M_{n}:=M + N_{n}$. In various special cases, our
estimate extends or refines previous known results.
http://arxiv.org/abs/0805.3167
Author(s): Itai Benjamini and Nathanael Berestycki
Abstract: We consider one-dimensional Brownian motion conditioned (in a suitable sense)
to have a local time at every point and at every moment bounded by some fixed
constant. Our main result shows that a phenomenon of entropic repulsion occurs:
that is, this process is ballistic and has an asymptotic velocity approximately
4.58... as high as required by the conditioning (the exact value of this
constant involves the first zero of a Bessel function). We also study the
random walk case and show that the process is asymptotically ballistic but with
an unknown speed.
http://arxiv.org/abs/0805.3326
Author(s): Chung Chan
Abstract: Csiszar and Narayan[3] show that the secret-key capacity with unlimited
public discussion and the smallest achievable rate of communication for
omniscience of a group of at least two active users sum up to the entropy rate
of the discrete multiple memoryless sources for all terminals. They then derive
a heuristically appealing upperbound[3,(26)] on the secret-key capacity, which
is in the form of the information divergence from joint to product probability
measure commonly interpreted as the mutual dependence of a set of random
variables. Tightness of this bound would confirm its heuristic interpretation
with the operational meaning of the secret-key capacity, i.e. the maximum
mutual consensus among the active users that need not be explicitly described
in public. While one can easily check that the bound is tight for any system
with three or less users, testing the case with more users quickly becomes
unmanageable. Yet, there is no apparent reason, other than its heuristic
interpretation, that the bound is tight, nor is there a counter-example that
suggests otherwise.
This paper proves that the bound is indeed tight when all users are active,
as a consequence of the polymatroidal structure[6] underlying the source coding
problem. This already confirms the heuristic interpretation of the bound as a
measure of mutual dependence of random variables. For the other case when some
users are helpers, there is a counter-example with three active users and three
helpers for which the bound is loose.
http://arxiv.org/abs/0805.3200
Author(s): Corinne Berzin and Jos\'e R. Le\'on
Abstract: Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with
parameter $0
http://arxiv.org/abs/0805.3394
Author(s): Mark Rudelson and Roman Vershynin
Abstract: Let A be a matrix whose entries are real i.i.d. centered random variables
with unit variance and suitable moment assumptions. Then the smallest singular
value of A is of order n^{-1/2} with high probability. The lower estimate of
this type was proved recently by the authors; in this note we establish the
matching upper estimate.
http://arxiv.org/abs/0805.3407
Author(s): Lo\"ic Herv\'e
Abstract: Let $Q$ be a transition probability on a measurable space $E$ which admits an
invariant probability measure, let $(X_n)_n$ be a Markov chain associated to
$Q$, and let $\xi$ be a real-valued measurable function on $E$, and $S_n=\sum
_{k=1}^n\xi(X_k)$. Under functional hypotheses on the action of $Q$ and the
Fourier kernels $Q(t)$, we investigate the rate of convergence in the central
limit theorem for the sequence $(\frac{S_n}{\sqrt{n}})_n$. According to the
hypotheses, we prove that the rate is, either $\mathrm{O}(n^{-{\tau}/{2}})$ for
all $\tau<1$, or $\mathrm{O}(n^{-{1}/{2}})$. We apply the spectral Nagaev's
method which is improved by using a perturbation theorem of Keller and
Liverani, and a majoration of $|\mathbb{E}[\mathrm{e}^{\mat
hrm{i}t{S_n}/{\sqrt{n}}}]-\mathrm{e}^{{-t^2}/{2}}|$ obtained by a method of
martingale difference reduction. When $E$ is not compact or $\xi$ is not
bounded, the conditions required here on $Q(t)$ (in substance, some moment
conditions on $\xi$) are weaker than the ones usually imposed when the standard
perturbation theorem is used in the spectral method. For example, in the case
of $V$-geometric ergodic chains or Lipschitz iterative models, the rate of
convergence in the c.l.t. is $\mathrm{O}(n^{-{1}/{2}})$ under a third moment
condition on $\xi$.
http://arxiv.org/abs/0805.3418
Author(s): Olivier Durieu and Dalibor Voln\'y
Abstract: The aim of this paper is to compare various criteria leading to the central
limit theorem and the weak invariance principle. These criteria are the
martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk
SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab.
Theory Related Fields 110 (1998), which was subsequently improved by Dedecker
and Rio in Ann. Inst. H. Poincar\'{e} Probab. Statist. 36 (2000) and the
condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later
improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in
every ergodic dynamical system with positive entropy, if we consider two of
these criteria, we can find a function in $\mathbb{L}^2$ satisfying the first
but not the second.
http://arxiv.org/abs/0805.3450
Author(s): Marianna Bolla and Katalin Friedl and Andras Kramli
Abstract: Asymptotic behavior of the singular value decomposition (SVD) of blown up
matrices and normalized blown up contingency tables exposed to Wigner-noise is
investigated.It is proved that such an m\times n matrix almost surely has a
constant number of large singular values (of order \sqrt{mn}), while the rest
of the singular values are of order \sqrt{m+n} as m,n\to\infty. Concentration
results of Alon et al. for the eigenvalues of large symmetric random matrices
are adapted to the rectangular case, and on this basis, almost sure results for
the singular values as well as for the corresponding isotropic subspaces are
proved. An algorithm, applicable to two-way classification of microarrays, is
also given that finds the underlying block structure.
http://arxiv.org/abs/0805.3476
Author(s): E. Ryckman
Abstract: For arbitrary $\beta > 0$, we use the orthogonal polynomials techniques
developed by R. Killip and I. Nenciu to study certain linear statistics
associated with the circular and Jacobi $\beta$ ensembles. We identify the
distribution of these statistics then prove a joint central limit theorem. In
the circular case, similar statements have been proved using different methods
by a number of authors. In the Jacobi case these results are new.
http://arxiv.org/abs/0805.3516
Author(s): Tatsuya Tate
Abstract: We define the notion of Bernstein measures and Bernstein approximations over
general convex polytopes. This generalizes well-known Bernstein polynomials
which are used to prove the Weierstrass approximation theorem on one
dimensional intervals. We discuss some properties of Bernstein measures and
approximations, and prove an asymptotic expansion of the Bernstein
approximations for smooth functions which is a generalization of the asymptotic
expansion of the Bernstein polynomials on the standard $m$-simplex obtained by
Abel-Ivan and H\"{o}rmander. These are different from the Bergman-Bernstein
approximations over Delzant polytopes recently introduced by Zelditch. We
discuss relations between Bernstein approximations defined in this paper and
Zelditch's Bergman-Bernstein approximations.
http://arxiv.org/abs/0805.3379
Author(s): Jean Bricmont and Antti Kupiainen
Abstract: We prove that random walks in random environments, that are exponentially
mixing in space and time, are almost surely diffusive, in the sense that their
scaling limit is given by the Wiener measure.
http://arxiv.org/abs/0805.3455
Author(s): Federico Bassetti and Lucia Ladelli and Eugenio Regazzini
Abstract: This paper deals with a one--dimensional model for granular materials, which
boils down to an inelastic version of the Kac kinetic equation, with
inelasticity parameter $p>0$. In particular, the paper provides bounds for
certain distances -- such as specific weighted $\chi$--distances and the
Kolmogorov distance -- between the solution of that equation and the limit. It
is assumed that the even part of the initial datum (which determines the
asymptotic properties of the solution) belongs to the domain of normal
attraction of a symmetric stable distribution with characteristic exponent
$\a=2/(1+p)$. With such initial data, it turns out that the limit exists and is
just the aforementioned stable distribution. A necessary condition for the
relaxation to equilibrium is also proved. Some bounds are obtained without
introducing any extra--condition. Sharper bounds, of an exponential type, are
exhibited in the presence of additional assumptions concerning either the
behaviour, near to the origin, of the initial characteristic function, or the
behaviour, at infinity, of the initial probability distribution function.
http://arxiv.org/abs/0805.3508
Author(s): Sho Matsumoto
Abstract: We study the averages of ratios of characteristic polynomials over circular
$\beta$-ensembles, where $\beta$ is a positive real number. Using Jack
polynomial theory, we obtain three expressions for ratio averages. Two of them
are given as sums of super-Jack polynomials and another one is given by a
hyperdeterminant. As applications, we give duality relations for ratio averages
between $\beta$ and $4/\beta$.
http://arxiv.org/abs/0805.3573
Author(s): Antar Bandyopadhyay and Jeffrey Steif and Adam Timar
Abstract: We show that for any Cayley graph, the probability (at any $p$) that the
cluster of the origin has size n decays at a well-defined exponential rate
(possibly 0). For general graphs, we relate this rate being positive in the
supercritical regime with the amenability/nonamenability of the underlying
graph.
http://arxiv.org/abs/0805.3620
Author(s): Daniel Rudolf
Abstract: We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte
Carlo methods, such as the Metropolis algorithm. The problem is to compute the
expectation (or integral) of f with respect to a measure which can be given by
a density with respect to another measure. A straight simulation of the desired
distribution by a random number generator is in general not possible. Thus it
is reasonable to use Markov chain sampling with a burn-in. We study such an
algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an
explicit error bound.
http://arxiv.org/abs/0805.3587
Author(s): Chung Chan
Abstract: Equivocation rate has been widely used as an information-theoretic measure of
security after Shannon[10]. It simplifies problems by removing the effect of
atypical behavior from the system. In [9], however, Merhav and Arikan
considered the alternative of using guessing exponent to analyze the Shannon's
cipher system. Because guessing exponent captures the atypical behavior, the
strongest expressible notion of secrecy requires the more stringent condition
that the size of the key, instead of its entropy rate, to be equal to the size
of the message. The relationship between equivocation and guessing exponent are
also investigated in [6][7] but it is unclear which is a better measure, and
whether there is a unifying measure of security.
Instead of using equivocation rate or guessing exponent, we study the wiretap
channel in [2] using the success exponent, defined as the exponent of a
wiretapper successfully learn the secret after making an exponential number of
guesses to a sequential verifier that gives yes/no answer to each guess. By
extending the coding scheme in [2][5] and the converse proof in [4] with the
new Overlap Lemma 5.2, we obtain a tradeoff between secrecy and reliability
expressed in terms of lower bounds on the error and success exponents of
authorized and respectively unauthorized decoding of the transmitted messages.
From this, we obtain an inner bound to the strongly achievable public,
private and guessing rate triple for which the exponents are strictly positive.
The closure of this region contains the region in Theorem 1 of [2] when we
treat equivocation rate as the guessing rate. It would be surprising if one can
show that the subset relationship is strict, the region is tight, or a better
coding scheme exists to improve it. These problems remain open.
http://arxiv.org/abs/0805.3605
Author(s): Krzysztof Burdzy and John M. Lee
Abstract: We prove that a sequence of semi-discrete approximations converges to a
multiplicative functional for reflected Brownian motion, which intuitively
represents the Lyapunov exponent for the corresponding stochastic flow. The
method of proof is based on a study of the deterministic version of the problem
and the excursion theory.
http://arxiv.org/abs/0805.3740
Author(s): Annalisa Cerquetti
Abstract: By resorting to sequential constructions of exchangeable random partitions
(Pitman, 2006), and exploiting some known facts about generalized Stirling
numbers, we derive a generalized Chinese restaurant process construction of
exchangeable Gibbs partitions of type $\alpha$ (Gnedin and Pitman, 2006). Our
construction represents the natural theoretical probabilistic framework in
which to embed some recent results about a Bayesian nonparametric treatment of
estimation problems arising in genetic experiment under Gibbs, species
sampling, models priors.
http://arxiv.org/abs/0805.3853
Author(s): Kouji Yano
Abstract: The characteristic measure of excursions away from a regular point is studied
for a class of symmetric Levy processes without Gaussian part. It is proved
that the harmonic transform of the killed process enjoys Feller property. The
result is applied to prove extremeness and oscillatory entrance properties of
the excursion measure.
http://arxiv.org/abs/0805.3881
Author(s): Giacomo Aletti and Enea G. Bongiorno and Vincenzo Capasso
Abstract: We propose a set-valued framework for the well-posedness of birth-and-growth
process. Our birth-and-growth model is rigorously defined as a suitable
combination, involving Minkowski sum and Aumann integral, of two very general
set-valued processes representing nucleation and growth respectively. The
simplicity of the used geometrical approach leads us to avoid problems arising
by an analytical definition of the front growth such as boundary regularities.
In this framework, growth is generally anisotropic and, according to a
mesoscale point of view, it is not local, i.e. for a fixed time instant, growth
is the same at each space point.
http://arxiv.org/abs/0805.3912
Author(s): Klaus Fleischmann and Leonid Mytnik and Vitali Wachtel
Abstract: For 0 < \alpha \leq 2, a super-\alpha-stable motion X in R^d with branching
of index 1 + \beta in (1,2) is considered. If d < \alpha / \beta, a dichotomy
for the density of states X_t at fixed times t > 0 holds: the density function
is locally H\"older continuous if d = 1 and \alpha > 1 + \beta, but locally
unbounded otherwise. Moreover, in the case of continuity, we determine the
optimal H\"older index.
http://arxiv.org/abs/0805.3914
Author(s): Erhan Bayraktar and Virginia R. Young
Abstract: We find the optimal investment strategy to minimize the expected time that an
individual's wealth stays below zero, the so-called {\it occupation time}. The
individual consumes at a constant rate and invests in a Black-Scholes financial
market consisting of one riskless and one risky asset, with the risky asset's
price process following a geometric Brownian motion. We also consider an
extension of this problem by penalizing the occupation time for the degree to
which wealth is negative.
http://arxiv.org/abs/0805.3981
Author(s): A. Q. Teixeira
Abstract: We consider the model of random interlacements on Z^d introduced in [8]. For
this model, we prove the uniqueness of the infinite component of the vacant
set. As a consequence, we derive the continuity in u of the probability that
the origin belongs to the infinite component of the vacant set at level u in
the supercritical phase u < u_*.
http://arxiv.org/abs/0805.4106
Author(s): Raluca Balan and Sana Louhichi
Abstract: For each $n \geq 1$, let $\{X_{j,n}\}_{1 \leq j \leq n}$ be a sequence of
strictly stationary random variables. In this article, we give some asymptotic
weak dependence conditions for the convergence in distribution of the point
process $N_n=\sum_{j=1}^{n}\delta_{X_{j,n}}$ to an infinitely divisible point
process. From the point process convergence, we obtain the convergence in
distribution of the partial sum sequence $S_n=\sum_{j=1}^{n}X_{j,n}$ to an
infinitely divisible random variable, whose L\'{e}vy measure is related to the
canonical measure of the limiting point process. As examples, we discuss the
case of triangular arrays which possess known (row-wise) dependence structures,
like the strong mixing property, the association, or the dependence structure
of a stochastic volatility model.
http://arxiv.org/abs/0805.4128
Author(s): Xiaohong Lan and Domenico Marinucci
Abstract: We consider the correlation structure of the random coefficients for a wide
class of wavelet systems on the sphere which was recently introduced in the
literature. We provide necessary and sufficient conditions for these
coefficients to be asymptotic uncorrelated in the real and in the frequency
domain. Here, the asymptotic theory is developed in the high resolution sense.
Statistical applications are also discussed, in particular with reference to
the analysis of cosmological data.
http://arxiv.org/abs/0805.4154
Author(s): Rudolf Gorenflo and Francesco Mainardi and Alessandro Vivoli
Abstract: The well-scaled transition to the diffusion limit in the framework of the
theory of continuous-time random walk (CTRW)is presented starting from its
representation as an infinite series that points out the subordinated character
of the CTRW itself. We treat the CTRW as a combination of a random walk on the
axis of physical time with a random walk in space, both walks happening in
discrete operational time. In the continuum limit we obtain a generally
non-Markovian diffusion process governed by a space-time fractional diffusion
equation. The essential assumption is that the probabilities for waiting times
and jump-widths behave asymptotically like powers with negative exponents
related to the orders of the fractional derivatives. By what we call parametric
subordination, applied to a combination of a Markov process with a positively
oriented L\'evy process, we generate and display sample paths for some special
cases.
http://arxiv.org/abs/cond-mat/0701126
Author(s): Rudolf Gorenflo and Francesco Mainardi
Abstract: A physical-mathematical approach to anomalous diffusion may be based on
fractional diffusion equations and related random walk models. The fundamental
solutions of these equations can be interpreted as probability densities
evolving in time of peculiar self-similar stochastic processes: an integral
representation of these solutions is here presented. A more general approach to
anomalous diffusion is known to be provided by the master equation for a
continuous time random walk (CTRW). We show how this equation reduces to our
fractional diffusion equation by a properly scaled passage to the limit of
compressed waiting times and jump widths. Finally, we describe a method of
simulation and display (via graphics) results of a few numerical case studies.
http://arxiv.org/abs/0709.3990
Author(s): Svante Janson and Markus Kuba and Alois Panholzer
Abstract: Bona [2007+] studied the distribution of ascents, plateaux and descents in
the class of Stirling permutations, introduced by Gessel and Stanley [1978].
Recently, Janson [2008+] showed the connection between Stirling permutations
and plane recursive trees and proved a joint normal law for the parameters
considered by Bona. Here we will consider generalized Stirling permutations
extending the earlier results of Bona and Janson, and relate them with certain
families of generalized plane recursive trees, and also $(k+1)$-ary increasing
trees. We also give two different bijections between certain families of
increasing trees, which both give as a special case a bijection between ternary
increasing trees and plane recursive trees. In order to describe the
(asymptotic) behaviour of the parameters of interests, we study three
(generalized) Polya urn models using various methods.
http://arxiv.org/abs/0805.4084
Author(s): Oliver Johnson and Ioannis Kontoyiannis and Mokshay Madiman
Abstract: Motivated, in part, by the desire to develop an information-theoretic
foundation for compound Poisson approximation limit theorems (analogous to the
corresponding developments for the central limit theorem and for simple Poisson
approximation), this work examines sufficient conditions under which the
compound Poisson distribution has maximal entropy within a natural class of
probability measures on the nonnegative integers. We show that the natural
analog of the Poisson maximum entropy property remains valid if the measures
under consideration are log-concave, but that it fails in general. A parallel
maximum entropy result is established for the family of compound binomial
measures. The proofs are largely based on ideas related to the semigroup
approach introduced in recent work by Johnson for the Poisson family.
Sufficient conditions are given for compound distributions to be log-concave,
and specific examples are presented illustrating all the above results.
http://arxiv.org/abs/0805.4112
Author(s): Dennis Jang and Jung Uk Kang and Alex Kruckman and Jun Kudo and Steven J. Miller
Abstract: Alex Ely Kossovsky recently conjectured that the distribution of leading
digits of a chain of probability distributions converges to Benford's law as
the length of the chain grows. We prove his conjecture in many cases, and
provide an interpretation in terms of products of independent random variables
and a central limit theorem. An important consequence is that in hierarchical
Bayesian models priors tend to satisfy Benford's Law as the number of levels of
the hyper-parameters increases. We give explicit formulas for the error terms
as sums of Mellin transforms, which converges extremely rapidly as the number
of terms in the chain grows.
http://arxiv.org/abs/0805.4226
Author(s): Heikki J. Tikanm\"aki
Abstract: The one dimensional distribution of a L\'{e}vy process is not known in
general even though its characteristic function is given by the famous
L\'{e}vy-Khinchine theorem. This article gives an exact series representation
for the one dimensional distribution of a L\'{e}vy process satisfying certain
moment conditions. Moreover, this work clarifies an old result by Cram\'{e}r on
Edgeworth expansions for the distribution function of a L\'{e}vy process.
http://arxiv.org/abs/0805.4332
Author(s): Paavo Salminen and Pierre Vallois
Abstract: For a recurrent linear diffusion on $\R_+$ we study the asymptotics of the
distribution of its local time at 0 as the time parameter tends to infinity.
Under the assumption that the L\'evy measure of the inverse local time is
subexponential this distribution behaves asymtotically as a multiple of the
L\'evy measure. Using spectral representations we find the exact value of the
multiple. For this we also need a result on the asymptotic behavior of the
convolution of a subexponential distribution and an arbitrary distribution on
$\R_+.$ The exact knowledge of the asymptotic behavior of the distribution of
the local time allows us to analyze the process derived via a penalization
procedure with the local time. This result generalizes the penalizations
obtained in Roynette, Vallois and Yor \cite{rvyV} for Bessel processes.
http://arxiv.org/abs/0805.4353
Author(s): Bikramjit Das and Sidney I. Resnick
Abstract: Multivariate extreme value theory assumes a multivariate domain of attraction
condition for the distribution of a random vector necessitating that each
component satisfy a marginal domain of attraction condition.
\cite{heffernan:tawn:2004} and \cite{heffernan:resnick:2007} developed an
approximation to the joint distribution of the random vector by conditioning
that one of the components be extreme. The prior papers left unresolved the
consistency of different models obtained by conditioning on {different}
components being extreme and we provide understanding of this issue. We also
clarify the relationship between the conditional distributions and multivariate
extreme value theory. We discuss conditions under which the two models are the
same and when one can extend the conditional model to the extreme value model.
We also discuss the relationship between the conditional extreme value model
and standard regular variation on cones of the form
$[0,\infty]\times(0,\infty]$ or $(0,\infty]\times[0,\infty]$.
http://arxiv.org/abs/0805.4373
Author(s): Amitabha Bagchi
Abstract: We study the problem of power-efficient routing for multihop wireless ad hoc
sensor networks. The guiding insight of our work is that unlike an ad hoc
wireless network, a wireless ad hoc sensor network does not require full
connectivity among the nodes. As long as the sensing region is well covered by
connected nodes, the network can perform its task. We consider two kinds of
geometric random graphs as base interconnection structures: unit disk graphs
$\UDG(2,\lambda)$ and $k$-nearest-neighbor graphs $\NN(2,k)$ built on points
generated by a Poisson point process of density $\lambda$ in $\RR^2$. We
provide subgraph constructions for these two models $\US(2,\lambda)$ and
$\NS(2,k)$ and show that there are values $\lambda_s$ and $k_s$ above which
these constructions have the following good properties: (i) they are sparse;
(ii) they are power-efficient in the sense that the graph distance is no more
than a constant times the Euclidean distance between any pair of points; (iii)
they cover the space well; (iv) the subgraphs can be set up easily using local
information at each node. We also describe a simple local algorithm for routing
packets on these subgraphs.
http://arxiv.org/abs/0805.4060
Author(s): G. Guadagni and S. Ndreca and B. Scoppola
Abstract: We consider a point process obtained summing to each point $i$ of the set of
the integer $\mathbb{Z}$ an i.i.d random variable $\xi_i$ having a variance
that can be also much larger than 1. We compare the process obtained with this
construction with the standard Poisson process, and we show that in some sense
our process tends to converge for large variance of $\xi$ to the Poisson
process in total variation. We then consider analytically and numerically a
simple queueing system having our process as arrival process. This model is
motivated by the study of air traffic systems.
http://arxiv.org/abs/0805.4472
Author(s): Alain-Sol Sznitman
Abstract: We explore some of the connections between the local picture left by the
trace of simple random walk on a discrete cylinder with base a d-dimensional
torus, d at least 2, of side-length N running for times of order N^{2d} and the
model of random interlacements recently introduced in arXiv:0704.2560. In
particular we show that when the base becomes large, in the neighborhood of a
point of the cylinder with a vertical component of order N^d, the complement of
the set of points visited by the walk up to times of order N^{2d}, is close in
distribution to the law of the vacant set of random interlacements at a level
which is determined by an independent Brownian local time. The limit of the
local pictures in the neighborhood of finitely many points is also derived.
http://arxiv.org/abs/0805.4516
Author(s): Andrew Richards
Abstract: The approach used by Kalashnikov and Tsitsiashvili for constructing upper
bounds for the tail distribution of a geometric sum with subexponential
summands is reconsidered. By expressing the problem in a more probabilistic
light, several improvements and one correction are made, which enables the
constructed bound to be significantly tighter. Several examples are given,
showing how to implement the theoretical result.
http://arxiv.org/abs/0805.4548
Author(s): Gopal K. Basak and Philip Lee
Abstract: In this paper, we investigate the consistency and asymptotic efficiency of an
estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are
not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$
are in the right half space (i.e., eigenvalues with positive real parts). In
this case the process grows exponentially fast. (2) The eigenvalues of $F$ are
on the left half space (i.e., the eigenvalues with negative or zero real
parts). The process where all eigenvalues of $F$ have negative real parts is
called a stable process and has a unique invariant (i.e., stationary)
distribution. In this case the process does not grow. When the eigenvalues of
$F$ have zero real parts (i.e., the case of zero eigenvalues and purely
imaginary eigenvalues) the process grows polynomially fast. Considering (1) and
(2) separately, we first show that an estimator, $\hat{F}$, of $F$ is
consistent. We then combine them to present results for the general
Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic
efficiency of the estimator.
http://arxiv.org/abs/0805.4535
Author(s): T. R. Hurd and A. Kuznetsov
Abstract: This paper considers the class of L\'evy processes that can be written as a
Brownian motion time changed by an independent L\'evy subordinator. Examples in
this class include the variance gamma model, the normal inverse Gaussian model,
and other processes popular in financial modeling. The question addressed is
the precise relation between the standard first passage time and an alternative
notion, which we call first passage of the second kind, as suggested by Hurd
(2007) and others. We are able to prove that standard first passage time is the
almost sure limit of iterations of first passage of the second kind. Many
different problems arising in financial mathematics are posed as first passage
problems, and motivated by this fact, we are lead to consider the implications
of the approximation scheme for fast numerical methods for computing first
passage. We find that the generic form of the iteration can be competitive with
other numerical techniques. In the particular case of the VG model, the scheme
can be further refined to give very fast algorithms.
http://arxiv.org/abs/0805.4618
Author(s): Yufeng Shi and Weiqiang Yang and Jing Yuan
Abstract: In this paper we present two numerical schemes of approximating solutions of
backward doubly stochastic differential equations (BDSDEs for short). We give a
method to discretize a BDSDE. And we also give the proof of the convergence of
these two kinds of solutions for BDSDEs respectively. We give a sample of
computation of BDSDEs.
http://arxiv.org/abs/0805.4662
Author(s): Idris Kharroubi (PMA and CREST) and Jin Ma and Huyen Pham (PMA and CREST) and Jianfeng Zhang
Abstract: We consider a class of backward stochastic differential equations (BSDEs)
driven by Brownian motion and Poisson random measure, and subject to
constraints on the jump component. We prove the existence and uniqueness of the
minimal solution for the BSDEs by using a penalization approach. Moreover, we
show that under mild conditions the minimal solutions to these constrained
BSDEs can be characterized as the unique viscosity solution of
quasi-variational inequalities (QVIs), which leads to a probabilistic
representation for solutions to QVIs. Such a representation in particular gives
a new stochastic formula for value functions of a class of impulse control
problems. As a direct consequence we obtain a numerical scheme for the solution
of such QVIs via the simulation of the penalized BSDEs.
http://arxiv.org/abs/0805.4676
Author(s): Laurent M\'enard
Abstract: We prove that the uniform infinite random quadrangulations introduced
respectively by Chassaing-Durhuus and Krikun have the same distribution.
http://arxiv.org/abs/0805.4687
Author(s): Christel Geiss and Eija Laukkarinen
Abstract: The Malliavin derivative for a L\'evy process $(X_t)$ can be defined on the
space $\DD_{1,2}$ using a chaos expansion or in the case of a pure jump process
also via an increment quotient operator \cite{sole-utzet-vives}. In this paper
we define the Malliavin derivative operator $\D$ on the class $\mathcal{S}$ of
smooth random variables $f(X_{t_1}, ..., X_{t_n}),$ where $f$ is a smooth
function with compact support. We show that the closure of $L_2(\Om) \supseteq
\mathcal{S} \stackrel{\D}{\to} L_2(\m\otimes \mass)$ yields to the space
$\DD_{1,2}.$ As an application we conclude that Lipschitz functions map from
$\DD_{1,2}$ into $\DD_{1,2}.$
http://arxiv.org/abs/0805.4704
Author(s): Zhenting Hou and Jinying Tong and Dinghua Shi
Abstract: From the perspective of probability, the stability of growing network is
studied in the present paper. Using the DMS model as an example, we establish a
relation between the growing network and
Markov process. Based on the concept and technique of first-passage
probability in Markov theory, we provide a rigorous proof for existence of the
steady-state degree distribution, mathematically re-deriving the exact formula
of the distribution. The approach based on Markov chain theory is universal and
performs well in a large class of growing networks.
http://arxiv.org/abs/0805.4765
Author(s): Krzysztof Burdzy
Abstract: We prove that a stochastic flow of reflected Brownian motions in a smooth
multidimensional domain is differentiable with respect to its initial position.
The derivative is a linear map represented by a multiplicative functional for
reflected Brownian motion. The method of proof is based on excursion theory and
analysis of the deterministic Skorokhod equation.
http://arxiv.org/abs/0806.0119
Author(s): C. M. Newman (1) and K. Ravishankar (2) and E. Schertzer (1) ((1) Courant Inst. of Mathematical Sciences, NYU, (2) Dept. of Mathematics, SUNY College
at New Paltz)
Abstract: The Brownian web (BW), which developed from the work of Arratia and then
T\'{o}th and Werner, is a random collection of paths (with specified starting
points) in one plus one dimensional space-time that arises as the scaling limit
of the discrete web (DW) of coalescing simple random walks. Two recently
introduced extensions of the BW, the Brownian net (BN) constructed by Sun and
Swart, and the dynamical Brownian web (DyBW) proposed by Howitt and Warren, are
(or should be) scaling limits of corresponding discrete extensions of the DW --
the discrete net (DN) and the dynamical discrete web (DyDW). These discrete
extensions have a natural geometric structure in which the underlying Bernoulli
left or right "arrow" structure of the DW is extended by means of branching
(i.e., allowing left and right simultaneously) to construct the DN or by means
of switching (i.e., from left to right and vice-versa) to construct the DyDW.
In this paper we show that there is a similar structure in the continuum where
arrow direction is replaced by the left or right parity of the (1,2) space-time
points of the BW (points with one incoming path from the past and two outgoing
paths to the future, only one of which is a continuation of the incoming path).
We then provide a complete construction of the DyBW and an alternate
construction of the BN to that of Sun and Swart by proving that the switching
or branching can be implemented by a Poissonian marking of the (1,2) points.
http://arxiv.org/abs/0806.0158
Author(s): Amel Bentata (PMA) and Marc Yor (PMA and Iuf)
Abstract: These notes are the first half of the contents of the course given by the
second author at the Bachelier Seminar (February 8-15-22 2008) at IHP. They
also correspond to topics studied by the first author for her Ph.D.thesis.
http://arxiv.org/abs/0806.0239
Author(s): M. Mania and R. Tevzadze
Abstract: We study utility maximization problem for general utility functions using
dynamic programming approach. We consider an incomplete financial market model,
where the dynamics of asset prices are described by an $R^d$-valued continuous
semimartingale. Under some regularity assumptions we derive backward stochastic
partial differential equation (BSPDE) related directly to the primal problem
and show that the strategy is optimal if and only if the corresponding wealth
process satisfies a certain forward-SDE. As examples the cases of power,
exponential and logarithmic utilities are considered.
http://arxiv.org/abs/0806.0240
Author(s): Svante Janson and Malwina J. Luczak
Abstract: We study the evolution of the susceptibility in the subcritical random graph
$G(n,p)$ as $n$ tends to infinity. We obtain precise asymptotics of its
expectation and variance, and show it obeys a law of large numbers. We also
prove that the scaled fluctuations of the susceptibility around its
deterministic limit converge to a Gaussian law. We further extend our results
to higher moments of the component size of a random vertex, and prove that they
are jointly asymptotically normal.
http://arxiv.org/abs/0806.0252
Author(s): Simone Scotti
Abstract: We study the point of transition between complete and incomplete financial
models thanks to Dirichlet Forms methods. We apply recent techniques,
developped by Bouleau, to hedging procedures in order to perturbate parameters
and stochastic processes, in the case of a volatility parameter fixed but
uncertain for traders; we call this model Perturbed Black Scholes (PBS) Model.
We show that this model can reproduce at the same time a smile effect and a
bid-ask spread; we exhibit the volatility function associated to the
local-volatility model equivalent to PBS model when vanilla options are
concerned.
Lastly, we present a connection between Error Theory using Dirichlet Forms
and Utility Function Theory.
http://arxiv.org/abs/0806.0287
Author(s): Luca Regis and Simone Scotti
Abstract: We study the risk premium impact in the Perturbative Black Scholes model. The
Perturbative Black Scholes model, developed by Scotti, is a subjective
volatility model based on the classical Black Scholes one, where the volatility
used by the trader is an estimation of the market one and contains measurement
errors. In this article we analyze the correction to the pricing formulas due
to the presence of an underlying drift different from the risk free return. We
prove that, under some hypothesis on the parameters, if the asset price is a
sub-martingale under historical probability, then the implied volatility
presents a skewed structure, and the position of the minimum depends on the
risk premium $\lambda$.
http://arxiv.org/abs/0806.0307
Author(s): Robert Masson
Abstract: We give a new proof of a result of Kenyon that the growth exponent for
loop-erased random walks in two dimensions is 5/4. The proof uses the
convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid
for irreducible bounded symmetric random walks on any two-dimensional discrete
lattice.
http://arxiv.org/abs/0806.0357
Author(s): Patricia Goncalves and Milton Jara
Abstract: We prove that the density fluctuations for a zero-range process evolving on
the supercritical percolation cluster are given by a generalized
Ornstein-Uhlenbeck process in the space of distributions $\mc S'(\bb R^d)$.
http://arxiv.org/abs/0806.0362
Author(s): F. Toninelli (Laboratoire de Physique and ENS Lyon and CNRS)
Abstract: For a much-studied model of random copolymer at a selective interface we
prove that the slope of the critical curve in the weak-disorder limit is
strictly smaller than 1, which is the value given by the annealed inequality.
The proof is based on a coarse-graining procedure, combined with upper bounds
on the fractional moments of the partition function.
http://arxiv.org/abs/0806.0365
Author(s): O. Khorunzhiy and V. Vengerovsky
Abstract: We revisit the problem of estimates of moments of random n-dimensional
matrices of Wigner ensemble by using the approach elaborated by Ya. Sinai and
A. Soshnikov and further developed by A. Ruzmaikina. Our main subject is given
by the structure of closed even walks and their graphs that arise in these
studies. We show that the total degree of a vertex of such a graph depends not
only on the self-intersections degree of but also on the total number of all
non-closed instants of self-intersections of the walk. This result is used to
fill the gaps of earlier considerations.
http://arxiv.org/abs/0806.0157
Author(s): Semen V.Bodnarchuk and Alexey M.Kulik
Abstract: Conditions are given, sufficient for the distribution of an
Ornstein-Uhlenbeck process with L\'evy noise to be absolutely continuous or to
possess a smooth density. For the processes with non-degenerate drift
coefficient, these conditions are a necessary ones. A multidimensional analogue
for the non-degeneracy condition on the drift coefficient is introduced.
http://arxiv.org/abs/0806.0442
Author(s): Zhenting Hou and Li Tan and Dinghua Shi
Abstract: In this paper, first-passage probability of Markov chains is used to get a
strict proof of the existence of degree distribution of the LCD model presented
by Bollobas (Random Structures and Algorithms 18(2001)). Also, a precise
expression of degree distribution is presented.
http://arxiv.org/abs/0806.0448
Author(s): Serguei Foss and Andrew Richards
Abstract: The asymptotic tail-behaviour of sums of independent subexponential random
variables is well understood, one of the main characteristics being \textit{the
principle of the single big jump}. We study the case of dependent
subexponential random variables, for both deterministic and random sums, using
a fresh approach, by considering conditional independence structures on the
random variables. We seek sufficient conditions for the results of the theory
with independent random variables still to hold. For a subexponential
distribution, we introduce the concept of a boundary class of functions, which
we hope will be a useful tool in studying many aspects of subexponential random
variables. The examples we give in the paper demonstrate a variety of effects
owing to the dependence, and are also interesting in their own right.
http://arxiv.org/abs/0806.0490
Author(s): Itai Benjamini and Nathanael Berestycki
Abstract: We analyze one-dimensional Brownian motion conditioned on a self-repelling
behaviour. In the main result of this paper, it is shown that a double phase
transition occurs when the growth of the local time at the origin is
constrained (in a suitable way) to be slower than the function f(t)=
\sqrt{t}(\log t)^{-c} at every time. In the subcritical phase (c<0), the
process is recurrent and the local time at 0 is diffusive. In the intermediary
phase (01), the process
becomes transient. The proof exploits the Brownian entropic repulsion
phenomenon.
http://arxiv.org/abs/0806.0597
Author(s): A. Ioana and A.S. Kechris and and T. Tsankov
Abstract: We study a positive-definite function associated to a measure-preserving
equivalence relation on a standard probability space and use it to measure
quantitatively the proximity of subequivalence relations. This is combined with
a recent co-inducing construction of Epstein to produce new kinds of mixing
actions of an arbitrary infinite discrete group and it is also used to show
that orbit equivalence of free, measure preserving, mixing actions of
non-amenable groups is unclassifiable in a strong sense. Finally, in the case
of property (T) groups we discuss connections with invariant percolation on
Cayley graphs and the calculation of costs.
http://arxiv.org/abs/0806.0430
Author(s): Z. Brze\'zniak and M. Neklyudov
Abstract: The aim of this article is to study the asymptotic behaviour for large times
of solutions to a certain class of stochastic partial differential equations of
parabolic type. In particular, we will prove the backward uniqueness result and
the existence of the spectral limit for abstract SPDEs and then show how these
results can be applied to some concrete linear and nonlinear SPDEs. For
example, we will consider linear parabolic SPDEs with gradient noise and
stochastic NSEs with multiplicative noise. Our results generalize the results
proved in Ghidaglia (1986) for deterministic PDEs.
One of the difficulties with extending the results from Ghidaglia (1986) to
the stochastic case is that the standard It\^o formula is not directly
applicable to the case considered in this article. We use certain
approximations to overcome this problem.
Another difficulty is that conditions 1.3-1.4, p.779 in Ghidaglia (1986) have
no natural counterpart in the stochastic case. We have only conditions (1.11),
(1.12). As a result, we require rather strong assumptions on the regularity of
solutions in the case of stochastic equations with quadratic nonlinearity. In
the same time, this problem does not appear in the case of linear stochastic
equations or if nonlinearity has no more than "linear growth".
http://arxiv.org/abs/0806.0616
Author(s): Anton Bovier and Frank den Hollander and Cristian Spitoni
Abstract: In this paper we study metastability in large volumes at low temperatures. We
consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas
particles subject to Kawasaki hopping dynamics. Let $\b$ denote the inverse
temperature and let $\L_\b \subset \Z^2$ be a square box with periodic boundary
conditions such that $\lim_{\b\to\infty}|\L_\b|=\infty$. We run the dynamics on
$\L_\b$ starting from a random initial configuration where all the droplets (=
clusters of plus-spins, respectively, clusters of particles)are small. For
large $\b$, and for interaction parameters that correspond to the metastable
regime, we investigate how the transition from the metastable state (with only
small droplets) to the stable state (with one or more large droplets) takes
place under the dynamics. This transition is triggered by the appearance of a
single \emph{critical droplet} somewhere in $\L_\b$. Using potential-theoretic
methods, we compute the \emph{average nucleation time} (= the first time a
critical droplet appears and starts growing) up to a multiplicative factor that
tends to one as $\b\to\infty$. It turns out that this time grows as
$Ke^{\Gamma\b}/|\L_\b|$ for Glauber dynamics and $K\b e^{\Gamma\b}/|\L_\b|$ for
Kawasaki dynamics, where $\Gamma$ is the local canonical, respectively,
grand-canonical energy to create a critical droplet and $K$ is a constant
reflecting the geometry of the critical droplet, provided these times tend to
infinity (which puts a growth restriction on $|\L_\b|$). The fact that the
average nucleation time is inversely proportional to $|\L_\b|$ is referred to
as \emph{homogeneous nucleation}, because it says that the critical droplet for
the transition appears essentially independently in small boxes that partition
$\L_\b$.
http://arxiv.org/abs/0806.0755
Author(s): Alexander Fribergh and Nina Gantert and Serguei Popov
Abstract: We consider one-dimensional random walks in random environment which are
transient to the right. Our main interest is in the study of the sub-ballistic
regime, where at time n the particle is typically at a distance $O(n^\kappa)$
from the origin, $\kappa\in(0,1)$. We investigate the probabilities of moderate
deviations from this behavior. Specifically, we are interested in quenched and
annealed probabilities of slowdown (at time n, the particle is in
$O(n^{\nu_0})$, $\nu_0\in (0,\kappa)$), and speedup (at time n, the particle is
around $n^{\nu_1}$, $\nu_1\in (\kappa,1)$), for the current location of the
particle and for the hitting times. Also, we study probabilities of
backtracking: at time n, the particle is located near $(-n^\nu)$, thus making
an unusual excursion to the left. For the slowdown, our estimates are valid in
the ballistic case as well.
http://arxiv.org/abs/0806.0790
Author(s): C. Kuelske and A. A. Opoku
Abstract: We extend the notion of Gibbsianness for mean-field systems to the set-up of
general (possibly continuous) local state spaces. We investigate the Gibbs
properties of systems arising from an initial mean-field Gibbs measure by
application of given local transition kernels. This generalizes previous
case-studies made for spins taking finitely many values to the first step in
direction to a general theory, containing the following parts: (1) A formula
for the limiting conditional probability distributions of the transformed
system. It holds both in the Gibbs and non-Gibbs regime and invokes a
minimization problem for a "constrained rate-function". (2) A criterion for
Gibbsianness of the transformed system for initial Lipschitz-Hamiltonians
involving concentration properties of the transition kernels. (3) A continuity
estimate for the single-site conditional distributions of the transformed
system. While (2) and (3) have provable lattice-counterparts, the
characterization of (1) is stronger in mean-field. As applications we show
short-time Gibbsianness of rotator mean-field models on the (q-1)-dimensional
sphere under diffusive time-evolution and the preservation of Gibbsianness
under local coarse-graining of the initial local spin space.
http://arxiv.org/abs/0806.0802
Author(s): Michel Benaim (UNINE) and Olivier Raimond (LM-Orsay)
Abstract: Using an approximation by a set-valued dynamical system, this paper studies a
class of non Markovian and non homogeneous stochastic processes on a finite
state space. It provides an unified approach to simulated annealing type
processes. It permits to study new models of vertex reinforced random walks and
new models of learning in games including Markovian fictitious play.
http://arxiv.org/abs/0806.0806
Author(s): Marton Balazs and Timo Seppalainen
Abstract: We give a partly new proof of the fluctuation bounds for the second class
particle and current in the stationary asymmetric simple exclusion process. One
novelty is a coupling that preserves the ordering of second class particles in
two systems that are themselves ordered coordinatewise.
http://arxiv.org/abs/0806.0829
Author(s): Thomas Feierl
Abstract: We determine the weak limit of the distribution of the random variables
"height" and "range" on the set of p-watermelons without wall restriction as
the number of steps tends to infinity. Additionally, we provide asymptotics for
the moments of the random variable "height".
http://arxiv.org/abs/0806.0037
Author(s): Arnaud Durand
Abstract: We completely describe in terms of Hausdorff measures the size of the set of
points of the circle that are covered infinitely often by a sequence of random
arcs with given lengths. We also show that this set is a set with large
intersection.
http://arxiv.org/abs/0806.0880
Author(s): Weiqiang Yang and Yufeng Shi and Yangling Gu
Abstract: We study reflected solutions of one-dimensional backward doubly stochastic
differential equations (BDSDEs in short). The ``reflected'' keeps the solution
above a given stochastic process. We get the uniqueness and existence by
penalization. For the existence of backward stochastic integral, our proof is
different from [KKPPQ] slightly. We also obtain a comparison theorem for
reflected BDSDEs.
http://arxiv.org/abs/0806.0917
Author(s): Gilles Pag\`es (PMA) and Abass Sagna (PMA)
Abstract: Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an
Euclidean norm). Let $ r,s > 0 $ and assume $(\alpha_n)_{n \geq 1}$ is an
(asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. In this paper we
investigate the asymptotic behavior of the maximal radius sequence induced by
the sequence $(\alpha_n)_{n \geq 1}$ and defined to be for every $n \geq 1$, $\
\rho(\alpha_n) = \max \{| a |, a \in \alpha_n \}$. We show that if ${\rm
card(supp}(P))$ is infinite, the maximal radius sequence goes to $\sup \{| x |,
x \in {\rm supp}(P) \}$ as $n$ goes to infinity. We then give the rate of
convergence for two classes of distributions with unbounded support :
distributions with exponential tails and distributions with polynomial tails.
http://arxiv.org/abs/0806.0918
Author(s): Joan-Andreu L\'azaro-Cam\'i and Juan-Pablo Ortega
Abstract: We extend some aspects of the Hamilton-Jacobi theory to the category of
stochastic Hamiltonian dynamical systems. More specifically, we show that the
stochastic action satisfies the Hamilton-Jacobi equation when, as in the
classical situation, it is written as a function of the configuration space
using a regular Lagrangian submanifold. Additionally, we will use a variation
of the Hamilton-Jacobi equation to characterize the generating functions of
one-parameter groups of symplectomorphisms that allow to rewrite a given
stochastic Hamiltonian system in a form whose solutions are very easy to find;
this result recovers the classical solutions by reduction to the equilibrium of
a standard Hamiltonian system.
http://arxiv.org/abs/0806.0993
Author(s): Julia Eaton and Anant Godbole and Betsy Sinclair
Abstract: Consider $n$ players whose "scores" are independent and identically
distributed values $\{X_i\}_{i=1}^n$ from some discrete distribution $F$. We
pay special attention to the cases where (i) $F$ is geometric with parameter
$p\to0$ and (ii) $F$ is uniform on $\{1,2,...,N\}$; the latter case clearly
corresponds to the classical occupancy problem. The quantities of interest to
us are, first, the $U$-statistic $W$ which counts the number of "ties" between
pairs $i,j$; second, the univariate statistic $Y_r$, which counts the number of
strict $r$-way ties between contestants, i.e., episodes of the form
${X_i}_1={X_i}_2=...={X_i}_r$; $X_j\ne {X_i}_1;j\ne i_1,i_2,...,i_r$; and, last
but not least, the multivariate vector $Z_{AB}=(Y_A,Y_{A+1},...,Y_B)$. We
provide Poisson approximations for the distributions of $W$, $Y_r$ and $Z_{AB}$
under some general conditions. New results on the joint distribution of cell
counts in the occupancy problem are derived as a corollary.
http://arxiv.org/abs/0806.1007
Author(s): Nina Gantert and Serguei Popov and Marina Vachkovskaia
Abstract: We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d.
random field of obstacles. When the particle hits an obstacle, it disappears
with a positive probability. We obtain quenched and annealed bounds on the
tails of the survival time in the general $d$-dimensional case. We then
consider a simplified one-dimensional model (where transition probabilities and
obstacles are independent and the RWRE only moves to neighbour sites), and
obtain finer results for the tail of the survival time. In addition, we study
also the "mixed" probability measures (quenched with respect to the obstacles
and annealed with respect to the transition probabilities and vice-versa) and
give results for tails of the survival time with respect to these probability
measures. Further, we apply the same methods to obtain bounds for the tails of
the hitting times of Branching Random Walks in Random Environment (BRWRE).
http://arxiv.org/abs/0806.1030
Author(s): Francesco Mainardi and Paolo Pironi
Abstract: We have revisited the Brownian motion on the basis of the fractional Langevin
equation which turns out to be a particular case of the generalized Langevin
equation introduced by Kubo on 1966. The importance of our approach is to model
the Brownian motion more realistically than the usual one based on the
classical Langevin equation, in that it takes into account also the retarding
effects due to hydrodynamic backflow, i.e. the added mass and the Basset memory
drag. On the basis of the two fluctuation-dissipation theorems and of the
techniques of the Fractional Calculus we have provided the analytical
expressions of the correlation functions (both for the random force and the
particle velocity) and of the mean squared particle displacement. The random
force has been shown to be represented by a superposition of the usual white
noise with a "fractional" noise. The velocity correlation function is no longer
expressed by a simple exponential but exhibits a slower decay, proportional to
$t^{-3/2}$ as $t \to \infty$, which indeed is more realistic. Finally, the mean
squared displacement has been shown to maintain, for sufficiently long times,
the linear behaviour which is typical of normal diffusion, with the same
diffusion coefficient of the classical case. However, the Basset history force
induces a retarding effect in the establishing of the linear behaviour, which
in some cases could appear as a manifestation of anomalous diffusion to be
correctly interpreted in experimental measurements.
http://arxiv.org/abs/0806.1010
Author(s): Artur O. Lopes and Joana Mohr and Rafael R. Souza
Abstract: We analyze some properties of maximizing stationary Markov probabilities on
the Bernoulli space $[0,1]^\mathbb{N}$, More precisely, we consider ergodic
optimization for a continuous potential $A$, where $A: [0,1]^\mathbb{N}\to
\mathbb{R}$ which depends only on the two first coordinates. We are interested
in finding stationary Markov probabilities $\mu_\infty$ on $ [0,1]^\mathbb{N}$
that maximize the value $ \int A d \mu,$ among all stationary Markov
probabilities $\mu$ on $[0,1]^\mathbb{N}$. This problem correspond in
Statistical Mechanics to the zero temperature case for the interaction
described by the potential $A$. The main purpose of this paper is to show,
under the hypothesis of uniqueness of the maximizing probability, a Large
Deviation Principle for a family of absolutely continuous Markov probabilities
$\mu_\beta$ which weakly converges to $\mu_\infty$. The probabilities
$\mu_\beta$ are obtained via an information we get from a Perron operator and
they satisfy a variational principle similar to the pressure. Under the
hypothesis of $A$ being $C^2$ and the twist condition, that is,
$\frac{\partial^2 A}{\partial_x \partial_y} (x,y) \neq 0$, for all $(x,y) \in
[0,1]^2$, we show the graph property.
http://arxiv.org/abs/0806.1012
Author(s): Michael Allman and Volker Betz
Abstract: We consider the motion of a Brownian particle in $\mathbb{R}$, moving between
a particle fixed at the origin and another moving deterministically away at
slow speed $\epsilon>0$. The middle particle interacts with its neighbours via
a potential of finite range $b>0$, with a unique minimum at $a>0$, where
$b<2a$. We say that the chain of particles breaks on the left- or right-hand
side when the middle particle is greater than a distance $b$ from its left or
right neighbour, respectively. We study the asymptotic location of the first
break of the chain in the limit of small noise, in the case where $\epsilon =
\epsilon(\sigma)$ and $\sigma>0$ is the noise intensity.
http://arxiv.org/abs/0806.1163
Author(s): Didier Piau (IF)
Abstract: We compute the posterior distributions of the initial population and
parameter of binary branching processes, in the limit of a large number of
generations. We compare this Bayesian procedure with a more na\"ive one, based
on hitting times of some random walks. In both cases, central limit theorems
are available, with explicit variances.
http://arxiv.org/abs/0806.1173
Author(s): Zhidong Bai (KLASMOE and Dsap) and Jian-Feng Yao (IRMAR)
Abstract: In the spiked population model introduced by Johnstone (2001),the population
covariance matrix has all its eigenvalues equal to unit except for a few fixed
eigenvalues (spikes). The question is to quantify the effect of the
perturbation caused by the spike eigenvalues. Baik and Silverstein (2006)
establishes the almost sure limits of the extreme sample eigenvalues associated
to the spike eigenvalues when the population and the sample sizes become large.
In a recent work (Bai and Yao, 2008), we have provided the limiting
distributions for these extreme sample eigenvalues. In this paper, we extend
this theory to a {\em generalized} spiked population model where the base
population covariance matrix is arbitrary, instead of the identity matrix as in
Johnstone's case. New mathematical tools are introduced for establishing the
almost sure convergence of the sample eigenvalues generated by the spikes.
http://arxiv.org/abs/0806.1141
Author(s): Alan Hammond and Richard Kenyon
Abstract: Let $T_{n,m}=\mathbb Z_n\times\mathbb Z_m$, and define a random mapping
$\phi\colon T_{n,m}\to T_{n,m}$ by $\phi(x,y)=(x+1,y)$ or $(x,y+1)$
independently over $x$ and $y$ and with equal probability. We study the orbit
structure of such ``quenched random walks'' $\phi$ in the limit $m,n\to\infty$,
and show how it depends sensitively on the ratio $m/n$. For $m/n$ near a
rational $p/q$, we show that there are likely to be on the order of $\sqrt{n}$
cycles, each of length O(n), whereas for $m/n$ far from any rational with small
denominator, there are a bounded number of cycles, and for typical $m/n$ each
cycle has length on the order of $n^{4/3}$.
http://arxiv.org/abs/0806.1236
Author(s): Carl Graham (CMAP)
Abstract: For the classical Lp spaces of signed measures on the integers, we devise a
framework in which bounds for a sub-Markovian semigroup of interest can be
obtained, up to a constant factor, from bounds for another tractable semigroup
that dominates stochastically the first one. The main tools are the Hardy
inequality, the definition of related auxiliary Lp spaces suited to take
advantage of the domination, and the proof that the norms are equivalent to the
classical ones if the reference measure is quasi-geometrically decreasing. We
illustrate the results using birth-death and single-birth processes.
http://arxiv.org/abs/0806.1263
Author(s): Brian D. Ewald and Jeffrey Humpherys and Jeremy West
Abstract: We present a method for computing the expected number of times certain
transition events occur during the transient phase of a reducible Markov chain.
Examples of events include time to absorption, number of visits to a state,
traversals of a particular transition, loops from a state to itself, and
arrivals to a state from a particular subset of states.
http://arxiv.org/abs/0806.1291
Author(s): Itai Benjamini and Oded Schramm
Abstract: We prove a formula relating the Hausdorff dimension of a subset of the unit
interval and the Hausdorff dimension of the same set with respect to a random
path matric on the interval, which is generated using a multiplicative cascade.
When the random variables generating the cascade are exponentials of Gaussians,
the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum
gravity appears. This note was inspired by the recent work of Duplantier and
Sheffield proving a somewhat different version of the KPZ formula for Liouville
gravity. In contrast with the Liouville gravity setting, the one dimensional
multiplicative cascade framework facilitates the determination of the Hausdorff
dimension, rather than some expected box count dimension.
http://arxiv.org/abs/0806.1347
Author(s): Gady Kozma and Asaf Nachmias
Abstract: We examine the incipient infinite cluster (IIC) of critical percolation in
regimes where mean-field behavior have been established, namely when the
dimension d is large enough or when d>6 and the lattice is sufficiently spread
out. We find that random walk on the IIC exhibits anomalous diffusion with the
spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes
a conjecture of Alexander and Orbach. En route we calculate the one-arm
exponent with respect to the intrinsic distance.
http://arxiv.org/abs/0806.1442
Author(s): Berry Groisman
Abstract: The way a rational agent changes her belief in certain
propositions/hypotheses in the light of new evidence lies at the heart of
Bayesian inference. The basic natural assumption, as summarized in van
Fraassen's Reflection Principle ([1984]), would be that in the absence of new
evidence the belief should not change. Yet, there are examples that are claimed
to violate this assumption. The apparent paradox presented by such examples, if
not settled, would demonstrate the inconsistency and/or incompleteness of the
Bayesian approach and without eliminating this inconsistency, the approach
cannot be regarded as scientific.
The Sleeping Beauty Problem is just such an example. The existing attempts to
solve the problem fall into three categories. The first two share the view that
new evidence is absent, but differ about the conclusion of whether Sleeping
Beauty should change her belief or not, and why. The third category is
characterized by the view that, after all, new evidence (although hidden from
the initial view) is involved.
My solution is radically different and does not fall in either of these
categories. I deflate the paradox by arguing that the two different degrees of
belief presented in the Sleeping Beauty Problem are in fact beliefs in two
different propositions, i.e. there is no need to explain the (un)change of
belief.
http://arxiv.org/abs/0806.1316
Author(s): Alexander Barvinok
Abstract: We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and
prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We
prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution
to a convex optimization problem. We show that if Sigma(R, C) is sufficiently
large, then a random matrix D in Sigma(R, C) sampled from the uniform
probability measure in Sigma(R,C) is close to a particular matrix Z=Z(R,C) that
maximizes the entropy among all non-negative matrices with row sums R and
column sums C. Similar results are obtained for 0-1 matrices with prescribed
row and column sums and assigned zeros in some positions.
http://arxiv.org/abs/0806.1480
Author(s): N.V. Krylov
Abstract: We prove It\^o's formula for the $L_{p}$-norm of a stochastic
$W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence
form.
http://arxiv.org/abs/0806.1557
Author(s): Patrik L. Ferrari (Weierstrass Institute and WIAS-Berlin)
Abstract: For stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class in 1+1
dimensions, fluctuations grow as t^{1/3} during time t and the correlation
length at a fixed time scales as t^{2/3}. In this note we discuss the scale of
time correlations. It turns out that the space-time is non-trivially fibred,
having slow directions having decorrelation exponent equal to 1 instead of the
usual 2/3. These directions are the characteristic curves of the PDE associated
to the surface's slope. As a consequence, previously proven results for
space-like paths will hold in the whole space-time except along the slow
curves.
http://arxiv.org/abs/0806.1350
Author(s): C. R. E. Raja and R. Shah
Abstract: A locally compact group $G$ is said to have shifted convolution property
(abbr. as SCP) if for every regular Borel probability measure $\mu$ on $G$,
either $\sup_{x\in G} \mu ^n (Cx) \ra 0$ for all compact subsets $C$ of $G$, or
there exist $x\in G$ and a compact subgroup $K$ normalised by $x$ such that
$\mu^nx^{-n} \ra \omega_K$, the Haar measure on $K$. We first consider
distality of factor actions of distal actions. It is shown that this holds in
particular for factors under compact groups invariant under the action and for
factors under the connected component of identity. We then characterize groups
having SCP in terms of a readily verifiable condition on the conjugation action
(point-wise distality). This has some interesting corollaries to distality of
certain actions and Choquet Deny measures which actually motivated SCP and
point-wise distal groups. We also relate distality of actions on groups to that
of the extensions on the space of probability measures.
http://arxiv.org/abs/0806.1820
Author(s): Marco Fuhrman and Federica Masiero and Gianmario Tessitore
Abstract: We consider an Ito stochastic differential equation with delay, driven by
brownian motion, whose solution, by an appropriate reformulation, defines a
Markov process $X$ with values in a space of continuous functions $\mathbf C$,
with generator $\mathcal L$. We then consider a backward stochastic
differential equation depending on $X$, with unknown processes $(Y,Z)$, and we
study properties of the resulting system, in particular we identify the process
$Z$ as a deterministic functional of $X$. We next prove that the
forward-backward system provides a suitable solution to a class of parabolic
partial differential equations on the space $\mathbf C$ driven by $\mathcal L$,
and we apply this result to prove a characterization of the fair price and the
hedging strategy for a financial market with memory effects. We also include
applications to optimal stochastic control of differential equation with delay:
in particular we characterize optimal controls as feedback laws in terms the
process $X$.
http://arxiv.org/abs/0806.1837
Author(s): Raluca Balan and Doyoon Kim
Abstract: Let $u=\{u(t,x);t \in [0,T], x \in {\mathbb{R}}^{d}\}$ be the process
solution of the stochastic heat equation $u_{t}=\Delta u+ \dot F, u(0,\cdot)=0$
driven by a Gaussian noise $\dot F$, which is white in time and has spatial
covariance induced by the kernel $f$. In this paper we prove that the process
$u$ is locally germ Markov, if $f$ is the Bessel kernel of order $\alpha=2k,k
\in \bN_{+}$, or $f$ is the Riesz kernel of order $\alpha=4k,k \in \bN_{+}$.
http://arxiv.org/abs/0806.1898
Author(s): Jian Ding and Eyal Lubetzky and Yuval Peres
Abstract: We consider Glauber dynamics for the Ising model on the complete graph on $n$
vertices, known as the Curie-Weiss model. It is well-known that the mixing-time
in the high temperature regime ($\beta < 1$) has order $n\log n$, whereas the
mixing-time in the case $\beta > 1$ is exponential in $n$. Recently, Levin,
Luczak and Peres proved that for any fixed $\beta < 1$ there is cutoff at time
$[2(1-\beta)]^{-1} n\log n$ with a window of order $n$, whereas the mixing-time
at the critical temperature $\beta=1$ is $\Theta(n^{3/2})$. It is natural to
ask how the mixing-time transitions from $\Theta(n\log n)$ to $\Theta(n^{3/2})$
and finally to $\exp(\Theta(n))$. That is, how does the mixing-time behave when
$\beta=\beta(n)$ is allowed to tend to 1 as $n\to\infty$.
In this work, we obtain a complete characterization of the mixing-time of the
dynamics as a function of the temperature, as it approaches its critical point
$\beta_c=1$. In particular, we find a scaling window of order $1/\sqrt{n}$
around the critical temperature. In the high temperature regime, $\beta = 1 -
\delta$ for some $0 < \delta < 1$ so that $\delta^2 n \to\infty$ with $n$, the
mixing-time has order $(n/\delta)\log(\delta^2 n)$, and exhibits cutoff with
constant 1/2 and window size $n/\delta$. In the critical window, $\beta = 1\pm
\delta$ where $\delta^2 n$ is O(1), there is no cutoff, and the mixing-time has
order $n^{3/2}$. At low temperature, $\beta = 1 + \delta$ for $\delta > 0$ with
$\delta^2 n \to\infty$ and $\delta=o(1)$, there is no cutoff, and the mixing
time has order $(n/\delta)\exp(({3/4}+o(1))\delta^2 n)$.
http://arxiv.org/abs/0806.1906
Author(s): Jason Miller
Abstract: A random overlap structure (ROSt) is a measure on pairs (X,Q) where X is a
locally finite sequence in the real line with a maximum and Q a positive
semidefinite matrix of overlaps intrinsic to the particles X. Such a measure is
said to be quasi-stationary provided that the joint law of the gaps of X and
overlaps Q is stable under a stochastic evolution driven by a Gaussian sequence
with covariance Q. Aizenman et al. have shown that quasi-stationary ROSts serve
as an important computational tool in the study of the Sherrington-Kirkpatrick
(SK) spin-glass model from the perspective of cavity dynamics and the related
ROSt variational principle for its free energy. In this framework, the Parisi
solution is reflected in the ansatz that the overlap matrix exhibit a certain
hierarchical structure. Aizenman et al. have posed the question of whether the
ansatz could be explained by showing that the only ROSts that are
quasi-stationary in a robust sense are given by a special class of hierarchical
ROSts known as both the Ruelle Probability Cascades as well the GREM. Arguin
and Aizenman have given an affirmative answer in the special case that the set
of values S_Q taken on by the entries of Q is finite. We prove that this result
holds even when |S_Q| is infinite provided that Q satisfies the technical
condition that the closure of S_Q has no limit points from below. This is
relevant to the understanding of the ground states of the SK model, as they
satisfy |S_Q| = infinity.
http://arxiv.org/abs/0806.1915
Author(s): N.V. Krylov
Abstract: We present several results on solvability in Sobolev spaces $W^{1}_{p}$ of
SPDEs in divergence form in the whole space.
http://arxiv.org/abs/0806.1925
Author(s): N.V. Krylov
Abstract: We extend several known results on solvability in the Sobolev spaces
$W^{1}_{p}$, $p\in[2,\infty)$, of SPDEs in divergence form in $\bR^{d}_{+}$ to
equations having coefficients which are discontinuous in the space variable.
http://arxiv.org/abs/0806.1963
Author(s): Z.-Q. Chen and P. J. Fitzsimmons and K. Kuwae and T.-S. Zhang
Abstract: Using time-reversal, we introduce a stochastic integral for zero-energy
additive functionals of symmetric Markov processes, extending earlier work of
S. Nakao. Various properties of such stochastic integrals are discussed and an
It\^{o} formula for Dirichlet processes is obtained.
http://arxiv.org/abs/0806.2044
Author(s): Ying Hu (IRMAR) and Shanjian Tang (SCHOOL of Mathematical Sciences)
Abstract: In this paper, we study the switching game of one-dimensional backward
stochastic differential equations (BSDEs). This gives rise to a new type of
multi-dimensional obliquely reflected BSDEs, which is a system of BSDEs
reflected on the boundary of a special unbounded convex domain along an oblique
direction. The existence of the adapted solution is obtained by the
penalization method, the monotone convergence, and the a priori estimations.
The uniqueness is obtained by a verification method (the first component of any
adapted solution is shown to be the vector value of a switching problem for
Reflected BSDEs). Finally, we show the existence of both the value and the
saddle point for the switching game. More specifically, we prove that the value
process of the switching game is given by the first component of the solution
of the multi-dimensional obliquely reflected BSDEs and the saddle point can
also be constructed using the latter.
http://arxiv.org/abs/0806.2058
Author(s): Jason Fulman
Abstract: Stein's method is used to study the trace of a random element from a compact
Lie group or symmetric space. Central limit theorems are proved using very
little information: character values on a single element and the decomposition
of the square of the trace into irreducible components. This is illustrated for
Lie groups of classical type and Dyson's circular ensembles. The approach in
this paper will be useful for the study of higher dimensional characters, where
normal approximations need not hold.
http://arxiv.org/abs/0806.2168
Author(s): Omer Angel and Alexander Holroyd and Dan Romik
Abstract: Particles labelled $1,...,n$ are arranged initially in increasing order.
Subsequently, each pair of neighbouring particles that is currently in
increasing order swaps according to a Poisson process of rate 1. We analyze the
asymptotic behaviour of this process as $n$ goes to infinity. We prove that the
space-time trajectories of individual particles converge (when suitably scaled)
to a certain family of random curves with two points of non-differentiability,
and that the permutation matrix at a given time converges to a certain
deterministic measure with absolutely continuous and singular parts. The
absorbing state (where all particles are in decreasing order) is reached at
time $(2+o(1))n$. The finishing times of individual particles converge to
deterministic limits with fluctuations asymptotically governed by the
Tracy-Widom distribution.
http://arxiv.org/abs/0806.2222
Author(s): Andreas Greven and Peter Pfaffelhuber and Anita Winter
Abstract: The measure-valued Fleming-Viot process is a diffusion which models the
evolution of allele frequencies in a multi-type population of constant size. In
the neutral setting the Kingman coalescent is known to generate the genealogies
of the "individuals" in the population at a fixed time. The goal of the present
paper is to replace this static point of view on the genealogies by an analysis
of the evolving of genealogies. We use well-posed martingale problems to
construct the tree-valued resampling dynamics for both the finite population
(tree-valued Moran dynamics) and the infinite population (tree-valued
Fleming-Viot dynamics). We show that the tree-valued Moran dynamics converge in
the limit of large populations to the tree-valued Fleming-Viot dynamics. In the
long-term behavior we derive the Kingman coalescent measure tree as the
equilibrium. As an application we study the evolution of the distribution of
the length of the tree spanned by sequentially sampled "individuals".
http://arxiv.org/abs/0806.2224
Author(s): Ivan Nourdin (PMA)
Abstract: We prove a change of variable formula for the 2D fractional Brownian motion
of index H bigger of equal to 1/4. For H strictly bigger than 1/4, our formula
coincides with that obtained by using the rough paths theory. For H=1/4 (the
more interesting case), there is an additional term that is a classical Wiener
integral against an independent standard Brownian motion.
http://arxiv.org/abs/0806.2248
Author(s): Lorenzo Bertini and Davide Gabrielli and Claudio Landim
Abstract: We consider the weakly asymmetric exclusion process on a bounded interval
with particles reservoirs at the endpoints. The hydrodynamic limit for the
empirical density, obtained in the diffusive scaling, is given by the viscous
Burgers equation with Dirichlet boundary conditions. In the case in which the
bulk asymmetry is in the same direction as the drift due to the boundary
reservoirs, we prove that the quasi-potential can be expressed in terms of the
solution to a one-dimensional boundary value problem which has been introduced
by Enaud and Derrida \cite{de}. We consider the strong asymmetric limit of the
quasi-potential and recover the functional derived by Derrida, Lebowitz, and
Speer \cite{DLS3} for the asymmetric exclusion process.
http://arxiv.org/abs/0806.2296
Author(s): Janko Gravner and Alexander E. Holroyd
Abstract: We study a variant of bootstrap percolation in which growth is restricted to
a single active cluster. Initially there is a single active site at the origin,
while other sites of Z^2 are independently occupied with small probability p,
otherwise empty. Subsequently, an empty site becomes active by contact with 2
or more active neighbors, and an occupied site becomes active if it has an
active site within distance 2. We prove that the entire lattice becomes active
with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p
and pi^2/9 + C sqrt p (-log p)^3. This corrects previous numerical predictions
for the scaling of the correction term.
http://arxiv.org/abs/0806.2313
Author(s): Emmanuel Schertzer and Rongfeng Sun and and Jan M. Swart
Abstract: The Brownian net, which has recently been introduced by Sun and Swart [SS08],
and independently by Newman, Ravishankar and Schertzer [NRS08], generalizes the
Brownian web by allowing branching. In this paper, we study the structure of
the Brownian net in more detail. In particular, we give an almost sure
classification of each point in $R^2$ according to the configuration of the
Brownian net paths entering and leaving the point. Along the way, we establish
various other structural properties of the Brownian net.
http://arxiv.org/abs/0806.2326
Author(s): Roberto Cominetti and Jose Vaisman
Abstract: We establish a uniform bound for the distribution of a sum $S^n=X_1+...+X_n$
of independent non-homogeneous Bernoulli random variables with $P(X_i=1)=p_i$.
Specifically, we prove that $\sigma^n P(S^n=i)\leq M$ where $\sigma^n$ denotes
the standard deviation of $S^n$ and the constant $M~0.4688$ is the maximum of
$u\mapsto\sqrt{2u} e^{-2u}\sum_{k=0}^\infty({u^k\over k!})^2$.
http://arxiv.org/abs/0806.2350
Author(s): Michael Damron and Art\"em Sapozhnikov and B\'alint V\'agv\"olgyi
Abstract: We study invasion percolation in two dimensions. We compare connectivity
properties of the origin's invaded region to those of (a) the critical
percolation cluster of the origin and (b) the incipient infinite cluster. To
exhibit similarities, we show that for any k > 0 the k-point function of the
first pond has the same asymptotic behaviour as the probability that k points
are in the critical cluster of the origin. More prominent, though, are the
differences. We show that there are infinitely many ponds that contain many
large disjoint p_c-open clusters. Further, for k > 1, we compute the exact
decay rate of the distribution of the radius of the kth pond and see that it is
strictly different than that of the radius of the critical cluster of the
origin. We finish by showing that the invasion percolation measure and the
incipient infinite cluster measure are mutually singular.
http://arxiv.org/abs/0806.2425
Author(s): Erwin Bolthausen and Nicola Kistler
Abstract: We discuss a spin glass reminiscent of the Random Energy Model, which allows
in particular to recast the Parisi minimization into a more classical Gibbs
variational principle, thereby shedding some light on the physical meaning of
the order parameter of the Parisi theory. As an application, we study the
impact of an extensive cavity field on Derrida's REM: Despite its simplicity,
this model displays some interesting features such as ultrametricity and chaos
in temperature.
http://arxiv.org/abs/0806.2446
Author(s): Zhidong Bai and Jian-feng Yao
Abstract: In a spiked population model, the population covariance matrix has all its
eigenvalues equal to units except for a few fixed eigenvalues (spikes). This
model is proposed by Johnstone to cope with empirical findings on various data
sets. The question is to quantify the effect of the perturbation caused by the
spike eigenvalues. A recent work by Baik and Silverstein establishes the almost
sure limits of the extreme sample eigenvalues associated to the spike
eigenvalues when the population and the sample sizes become large. This paper
establishes the limiting distributions of these extreme sample eigenvalues. As
another important result of the paper, we provide a central limit theorem on
random sesquilinear forms.
http://arxiv.org/abs/0806.2503
Author(s): J\'anos Engl\"ander
Abstract: We study a spatial branching model, where the underlying motion is
$d$-dimensional ($d\ge1$) Brownian motion and the branching rate is affected by
a random collection of reproduction suppressing sets dubbed mild obstacles. The
main result of this paper is the quenched law of large numbers for the
population for all $d\ge1$. We also show that the branching Brownian motion
with mild obstacles spreads less quickly than ordinary branching Brownian
motion by giving an upper estimate on its speed. When the underlying motion is
an arbitrary diffusion process, we obtain a dichotomy for the quenched local
growth that is independent of the Poissonian intensity. More general offspring
distributions (beyond the dyadic one considered in the main theorems) as well
as mild obstacle models for superprocesses are also discussed.
http://arxiv.org/abs/0806.2512
Author(s): Bogdan Iftimie and \'Etienne Pardoux and Andrey Piatnitski
Abstract: This paper deals with the homogenization problem for a one-dimensional
parabolic PDE with random stationary mixing coefficients in the presence of a
large zero order term. We show that under a proper choice of the scaling factor
for the said zero order terms, the family of solutions of the studied problem
converges in law, and describe the limit process. It should be noted that the
limit dynamics remain random.
http://arxiv.org/abs/0806.2518
Author(s): Jean-Dominique Deuschel and Holger K\"osters
Abstract: We derive a quenched invariance principle for random walks in random
environments whose transition probabilities are defined in terms of weighted
cycles of bounded length. To this end, we adapt the proof for random walks
among random conductances by Sidoravicius and Sznitman (Probab. Theory Related
Fields 129 (2004) 219--244) to the non-reversible setting.
http://arxiv.org/abs/0806.2525
Author(s): Frank Aurzada and Mikhail Lifshits
Abstract: We derive general results on the small deviation behavior for some classes of
iterated processes. This allows us, in particular, to calculate the rate of the
small deviations for $n$-iterated Brownian motions and, more generally, for the
iteration of $n$ fractional Brownian motions. We also give a new and correct
proof of some results in E. Nane, Laws of the iterated logarithm for
$\alpha$-time Brownian motion, Electron. J. Probab. 11 (2006), no. 18,
434--459.
http://arxiv.org/abs/0806.2559
Author(s): Ludger R\"uschendorf and Mikhail A. Urusov
Abstract: In this paper, we introduce a modification of the free boundary problem
related to optimal stopping problems for diffusion processes. This modification
allows the application of this PDE method in cases where the usual regularity
assumptions on the coefficients and on the gain function are not satisfied. We
apply this method to the optimal stopping of integral functionals with
exponential discount of the form $E_x\int_0^{\tau}e^{-\lambda s}f(X_s) ds$,
$\lambda\ge0$ for one-dimensional diffusions $X$. We prove a general
verification theorem which justifies the modified version of the free boundary
problem. In the case of no drift and discount, the free boundary problem allows
to give a complete and explicit discussion of the stopping problem.
http://arxiv.org/abs/0806.2561
Author(s): {\L}ukasz Delong and Claudia Kl\"uppelberg
Abstract: In this paper, we investigate an optimal investment and consumption problem
for an investor who trades in a Black--Scholes financial market with stochastic
coefficients driven by a non-Gaussian Ornstein--Uhlenbeck process. We assume
that an agent makes investment and consumption decisions based on a power
utility function. By applying the usual separation method in the variables, we
are faced with the problem of solving a nonlinear (semilinear) first-order
partial integro-differential equation. A candidate solution is derived via the
Feynman--Kac representation. By using the properties of an operator defined in
a suitable function space, we prove uniqueness and smoothness of the solution.
Optimality is verified by applying a classical verification theorem.
http://arxiv.org/abs/0806.2570
Author(s): Sara Biagini and Marco Frittelli
Abstract: We consider a stochastic financial incomplete market where the price
processes are described by a vector-valued semimartingale that is possibly
nonlocally bounded. We face the classical problem of utility maximization from
terminal wealth, with utility functions that are finite-valued over
$(a,\infty)$, $a\in\lbrack-\infty,\infty)$, and satisfy weak regularity
assumptions. We adopt a class of trading strategies that allows for stochastic
integrals that are not necessarily bounded from below. The embedding of the
utility maximization problem in Orlicz spaces permits us to formulate the
problem in a unified way for both the cases $a\in\mathbb{R}$ and $a=-\infty$.
By duality methods, we prove the existence of solutions to the primal and dual
problems and show that a singular component in the pricing functionals may also
occur with utility functions finite on the entire real line.
http://arxiv.org/abs/0806.2582
Author(s): Sofia Aberg and Igor Rychlik and M. Ross Leadbetter
Abstract: Distributions of wave characteristics of ocean waves, such as wave slope,
waveheight or wavelength, are an important tool in a variety of oceanographic
applications such as safety of ocean structures or in the study of ship
stability, as will be the focus in this paper. We derive Palm distributions of
several wave characteristics that can be related to steepness of waves for two
different cases, namely for waves observed along a line at a fixed time point
and for waves encountering a ship sailing on the ocean. The relation between
the distributions obtained in the two cases is also given physical
interpretation in terms of a ``Doppler shift'' that is related to the velocity
of the ship and the velocities of the individual waves.
http://arxiv.org/abs/0806.2718
Author(s): Federico Bassetti and Irene Crimaldi and Fabrizio Leisen
Abstract: Conditional identity in distribution (Berti et al. (2004)) is a new type of
dependence for random variables, which generalizes the well-known notion of
exchangeability. In this paper, a class of random sequences, called Generalized
Species Sampling Sequences, is defined and a condition to have conditional
identity in distribution is given. Moreover, a class of generalized species
sampling sequences that are conditionally identically distributed is introduced
and studied: the Generalized Ottawa sequences (GOS). This class contains a
'`randomly reinforced'' version of the P\'olya urn and of the
Blackwell-MacQueen urn scheme. For the empirical means and the predictive means
of a GOS, we prove two convergence results toward suitable mixtures of Gaussian
distributions. The first one is in the sense of stable convergence and the
second one in the sense of almost sure conditional convergence. In the last
part of the paper we study the length of the partition induced by a GOS at time
$n$, i.e. the random number of distinct values of a GOS until time $n$. Under
suitable conditions, we prove a strong law of large numbers and a central limit
theorem in the sense of stable convergence. All the given results in the paper
are accompanied by some examples.
http://arxiv.org/abs/0806.2724
Author(s): Carenne Lude\~na
Abstract: A multifractal random walk (MRW) is defined by a Brownian motion subordinated
by a class of continuous multifractal random measures $M[0,t], 0\le t\le1$. In
this paper we obtain an extension of this process, referred to as multifractal
fractional random walk (MFRW), by considering the limit in distribution of a
sequence of conditionally Gaussian processes. These conditional processes are
defined as integrals with respect to fractional Brownian motion and convergence
is seen to hold under certain conditions relating the self-similarity (Hurst)
exponent of the fBm to the parameters defining the multifractal random measure
$M$. As a result, a larger class of models is obtained, whose fine scale
(scaling) structure is then analyzed in terms of the empirical structure
functions. Implications for the analysis and inference of multifractal
exponents from data, namely, confidence intervals, are also provided.
http://arxiv.org/abs/0806.2731
Author(s): Vathana Ly Vath and Huy\^en Pham and St\'ephane Villeneuve
Abstract: We consider a mixed stochastic control problem that arises in Mathematical
Finance literature with the study of interactions between dividend policy and
investment. This problem combines features of both optimal switching and
singular control. We prove that our mixed problem can be decoupled in two pure
optimal stopping and singular control problems. Furthermore, we describe the
form of the optimal strategy by means of viscosity solution techniques and
smooth-fit properties on the corresponding system of variational inequalities.
Our results are of a quasi-explicit nature. From a financial viewpoint, we
characterize situations where a firm manager decides optimally to postpone
dividend distribution in order to invest in a reversible growth opportunity
corresponding to a modern technology. In this paper a reversible opportunity
means that the firm may disinvest from the modern technology and return back to
its old technology by receiving some gain compensation. The results of our
analysis take qualitatively different forms depending on the parameters values.
http://arxiv.org/abs/0806.2745
Author(s): Gareth O. Roberts and Jeffrey S. Rosenthal
Abstract: We introduce a new property of Markov chains, called variance bounding. We
prove that, for reversible chains at least, variance bounding is weaker than,
but closely related to, geometric ergodicity. Furthermore, variance bounding is
equivalent to the existence of usual central limit theorems for all $L^2$
functionals. Also, variance bounding (unlike geometric ergodicity) is preserved
under the Peskun order. We close with some applications to Metropolis--Hastings
algorithms.
http://arxiv.org/abs/0806.2747
Author(s): Boualem Djehiche and Said Hamadene and Ibtissam Hdhiri
Abstract: We consider a class of stochastic impulse control problems of general
stochastic processes i.e. not necessarily Markovian. Under fairly general
conditions we establish existence of an optimal impulse control. We also prove
existence of combined optimal stochastic and impulse control of a fairly
general class of diffusions with random coefficients. Unlike, in the Markovian
framework, we cannot apply quasi-variational inequalities techniques. We rather
derive the main results using techniques involving reflected BSDEs and the
Snell envelope.
http://arxiv.org/abs/0806.2761
Author(s): G. M. Pan and W. Zhou
Abstract: Let $\mathbf{s}_k=\frac{1}{\sqrt{N}}(v_{1k},...,v_{Nk})^T,$ with
$\{v_{ik},i,k=1,...\}$ independent and identically distributed complex random
variables. Write $\mathbf{S}_k=(\mathbf{s}_1,...,\mathbf
{s}_{k-1},\mathbf{s}_{k+1},... ,\mathbf{s}_K),$ $\mathbf{P}_k=\operatorname
{diag}(p_1,...,p_{k-1},p_{k+1},...,p_K)$,
$\mathbf{R}_k=(\mathbf{S}_k\mathbf{P}_k\mathbf{S}_k^*+\sigma ^2\mathbf{I})$ and
$\mathbf{A}_{km}=[\mathbf{s}_k,\mathbf{R}_k\mathbf{s}_k,...
,\mathbf{R}_k^{m-1}\mathbf{s}_k]$. Define
$\beta_{km}=p_k\mathbf{s}_k^*\mathbf{A}_{km}(\mathbf {A}_{km}^*\times\
mathbf{R}_k\mathbf{A}_{km})^{-1}\mathbf{A}_{km}^*\mathbf{s}_k$, referred to as
the signal-to-interference ratio (SIR) of user $k$ under the multistage Wiener
(MSW) receiver in a wireless communication system. It is proved that the output
SIR under the MSW and the mutual information statistic under the matched filter
(MF) are both asymptotic Gaussian when $N/K\to c>0$. Moreover, we provide a
central limit theorem for linear spectral statistics of eigenvalues and
eigenvectors of sample covariance matrices, which is a supplement of Theorem 2
in Bai, Miao and Pan [Ann. Probab. 35 (2007) 1532--1572]. And we also improve
Theorem 1.1 in Bai and Silverstein [Ann. Probab. 32 (2004) 553--605].
http://arxiv.org/abs/0806.2768
Author(s): Zakhar Kabluchko and Martin Schlather and Laurens de Haan
Abstract: Let $W_i(\cdot)$, $i\in\NN$, be independent copies of a zero mean gaussian
process $\{W(t), t\in\RR^d\}$ with stationary increments; and denote by
$\sigma^2(t)$ the variance of $W(t)$. Independently from $W_i$, let
$\sum_{i=1}^\infty \delta_{U_i}$ be a Poisson point process on the real axis
with intensity $e^{-y}dy$. We show that the law of the random family of
functions $\{V_i(\cdot), i\in\NN\}$ defined by
$V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$ is translation invariant. In particular, the
process $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary max-stable
process with standard Gumbel margins. The process $\eta$ arises as a limit of a
suitably normalized and rescaled pointwise maximum of $n$ i.i.d. stationary
gaussian processes as $n\to\infty$ if and only if $W$ is a (non-isotropic)
fractional Brownian motion on $\RR^d$. Under suitable conditions on $W$, the
process $\eta$ has a mixed moving maxima representation.
http://arxiv.org/abs/0806.2780
Author(s): Yan Dolinsky and Yuri Kifer
Abstract: The Annals of Applied Probability 16 (2006) 984--1033 [URL:
http://projecteuclid.org/euclid.aoap/1151592257]
http://arxiv.org/abs/0806.2782
Author(s): Eero Saksman and Matti Vihola
Abstract: This paper describes sufficient conditions to ensure the correct ergodicity
of the Adaptive Metropolis (AM) algorithm of Haario, Saksman, and Tamminen
(2001) [8], for target distributions with a non-compact support. The conditions
ensuring a strong law of large numbers and a central limit theorem require that
the tails of the target density decay super-exponentially, and have regular
enough convex contours. The result is based on the ergodicity of an auxiliary
process that is sequentially constrained to feasible adaptation sets, and
independent estimates of the growth rate of the AM chain and the corresponding
geometric drift constants. The ergodicity result of the constrained process is
obtained through a modification of the approach due to Andrieu and Moulines
(2006) [1].
http://arxiv.org/abs/0806.2933
Author(s): Herold Dehling and Olivier Durieu and Dalibor Voln\'y
Abstract: We present a new technique for proving empirical process invariance principle
for stationary processes $(X_n)_{n\geq 0}$. The main novelty of our approach
lies in the fact that we only require the central limit theorem and a moment
bound for a restricted class of functions $(f(X_n))_{n\geq 0}$, not containing
the indicator functions. Our approach can be applied to Markov chains and
dynamical systems, using spectral properties of the transfer operator. Our
proof consists of a novel application of chaining techniques.
http://arxiv.org/abs/0806.2941
Author(s): Olivier Durieu
Abstract: In this paper, a fourth moment bound for partial sums of functional of
strongly ergodic Markov chain is established. This type of inequality plays an
important role in the study of empirical process invariance principle. This one
is specially adapted to the technique of Dehling, Durieu and Voln\'y (2008).
The same moment bound can be proved for dynamical system whose transfer
operator has some spectral properties. Examples of applications are given.
http://arxiv.org/abs/0806.2980
Author(s): Andras Balint and Federico Camia and Ronald Meester
Abstract: The Ising model at inverse temperature $\beta$ and zero external field can be
obtained via the Fortuin-Kasteleyn (FK) random-cluster model with $q=2$ and
density of open edges $p=1-e^{-\beta}$ by assigning spin +1 or -1 to each
vertex in such a way that (1) all the vertices in the same FK cluster get the
same spin and (2) +1 and -1 have equal probability. We generalize the above
procedure by assigning spin +1 with probability $r$ and -1 with probability
$1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $\beta$, this
generates a dependent (spin) percolation model with parameter $r$. We show
that, on the triangular lattice and for $\beta<\beta_c$, this model has a
percolation phase transition at $r=1/2$, corresponding to the Ising model. This
sheds some light on the conjecture that the high temperature Ising model on the
triangular lattice is in the percolation universality class and that its
scaling limit can be described in terms of SLE$_6$. We also prove uniqueness of
the infinite +1 cluster for $r>1/2$, sharpness of the percolation phase
transition (by showing exponential decay of the cluster size distribution for
$r<1/2$), and continuity of the percolation function for all $r \in [0,1]$.
http://arxiv.org/abs/0806.3020
Author(s): Farrukh Mukhamedov and Seyit Temir and Hasan Akin
Abstract: In the paper we consider two positive contractions
$T,S:L^{1}(A,\tau)\longrightarrow L^{1}(A,\tau)$ such that $T\leq S$, here
$(A,\t)$ is a semi-finite $JBW$-algebra. If there is an $n_{0}\in\mathbb{N}$
such that $\|S^{n_{0}}-T^{n_{0}}\|<1$. Then we prove that $\|S^{n}-T^{n}\|<1$
holds for every $n\geq n_{0}.$
http://arxiv.org/abs/0806.2926
Author(s): Debasish Chatterjee and Eugenio Cinquemani and Giorgos Chaloulos and John Lygeros
Abstract: We present a dynamic programming-based solution to a stochastic optimal
control problem up to a hitting time for a discrete-time Markov control
process. Firstly, we determine an optimal control policy to steer the process
toward a compact target set while simultaneously minimizing an expected
discounted cost. We then provide a rolling-horizon strategy for approximating
the optimal policy, together with quantitative characterization of its
sub-optimality with respect to the optimal policy. Finally, we address related
issues of asymptotic discount-optimality of the value-iteration policy. Both
the state and action spaces are assumed to be Polish.
http://arxiv.org/abs/0806.3008
Author(s): Erkan Nane
Abstract: Let $X=\{X_t, t\geq 0\}$ be a Brownian motion or a spectrally negative stable
process of index $1<\a<2$. Let $E=\{E_t,t\geq 0\}$ be the hitting time of a
stable subordinator of index $0<\beta<1$ independent of $X$. We use a
connection between $X(E_t)$ and the stable subordinator of index $\beta/\a$ to
derive information on the path behavior of $X(E_t)$. This is an extension of
the connection of iterated Brownian motion and (1/4)-stable subordinator due to
Bertoin \cite{bertoin}. Using this connection, we obtain various laws of the
iterated logarithm for $X(E_t)$. In particular, we establish law of the
iterated logarithm for local time Brownian motion, $X(L_t)$, where $X$ is a
Brownian motion (the case $\a=2$) and $L_t$ is the local time at zero of a
stable process $Y$ of index $1<\a_2\leq 2$ independent of $X$. In this case
$E_{\rho t}=L_t$ with $\beta=1-1/\a_2$ for some constant $\rho>0$. This
establishes the lower bound in the law of the iterated logarithm which we could
not prove with the techniques of our paper \cite{MNX}. We also obtain exact
small balll probability for $X(E_t)$ using ideas from \cite{aurzada}.
http://arxiv.org/abs/0806.3126
Author(s): Bartlomiej Blaszczyszyn (INRIA Rocquencourt) and Dhandapani Yogeshwaran
Abstract: Directionally convex ($dcx$) ordering is a tool for comparison of dependence
structure of random vectors that also takes into account the variability of the
marginal distributions. When extended to random fields it oncerns comparison of
all finite dimensional distributions. Viewing locally finite measures as
non-negative fields of measure-values indexed by the bounded Borel subsets of
the space, in this paper we formulate and study the $dcx$ ordering of random
measures on locally compact spaces. We show that the $dcx$ order is preserved
under some of the natural operations considered on random measures and point
processes, such as independent superposition and thinning. Further operations
such as independent marking and displacement, though do not preserve the $dcx$
order on all point processes, are shown to preserve the order on Cox point
processes. We also examine the impact of $dcx$ order on the second moment
properties, in particular on clustering and on Palm distributions. Comparisons
of Ripley's functions, pair correlation functions as well as examples seem to
indicate that p.p. higher in $dcx$ order cluster more. As the main result, we
show that non-negative integral (shot-noise) fields with respect to $dcx$
ordered random measures inherit this ordering from the measures. Numerous
applications of this result are shown, in particular to comparison of various
Cox processes and some performance measures of wireless networks, in both of
which shot-noise fields appear as key ingredients. We also mention a few
pertinent open questions.
http://arxiv.org/abs/0806.3180
Author(s): Michael B. Marcus and Jay Rosen
Abstract: Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $EG_{1}G_{2}\neq
0$. Let $c_{1},c_{2}\in R^{1}$. A necessary and sufficient condition for
$G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ to be infinitely
divisible for all $\alpha\in R^{1}$ is that \[ \Ga_{i,i}\geq
\frac{c_{i}}{c_{j}}\Ga_{i,j}>0\qquad\forall 1\le i\ne j\le 2.\] In this paper
we show that when this does not hold there exists an $0<\alpha_{0}<\ff $ such
that $G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ is infinitely
divisible for all $|\alpha|\leq \alpha_{0}$ but not for any $|\al|>\al_{0}$.
http://arxiv.org/abs/0806.3188
Author(s): Nathanael Enriquez (MODAL'X) and Christophe Sabot (ICJ) and Marc Yor (PMA and IUF)
Abstract: We show how a description of Brownian exponential functionals as a renewal
series gives access to the law of the hitting time of a square-root boundary by
a Bessel process. This extends classical results by Breiman and Shepp,
concerning Brownian motion, and recovers by different means, extensions for
Bessel processes, obtained independently by Delong and Yor.
http://arxiv.org/abs/0806.3197
Author(s): Tertuliano Franco and Claudio Landim
Abstract: Fix a strictly increasing right continuous with left limits function $W: \bb
R \to \bb R$ and a smooth function $\Phi : [l,r] \to \bb R$, defined on some
interval $[l,r]$ of $\bb R$, such that $0
http://arxiv.org/abs/0806.3211
Author(s): Dmitry Dolgopyat and Carlangelo Liverani
Abstract: We prove an averaged CLT for a random walk in a dynamical environment where
the states of the environment at different sites are independent Markov chains.
http://arxiv.org/abs/0806.3236
Author(s): A.Papavasiliou and G.A. Pavliotis and A.M. Stuart
Abstract: We study the problem of parameter estimation using maximum likelihood for
fast/slow systems of stochastic differential equations. Our aim is to shed
light on the problem of model/data mismatch at small scales. We consider two
classes of fast/slow problems for which a closed coarse-grained equation for
the slow variables can be rigorously derived, which we refer to as averaging
and homogenization problems. We ask whether, given data from the slow variable
in the fast/slow system, we can correctly estimate parameters in the drift of
the coarse-grained equation for the slow variable, using maximum likelihood. We
show that, whereas the maximum likelihood estimator is asymptotically unbiased
for the averaging problem, for the homogenization problem maximum likelihood
fails unless we subsample the data at an appropriate rate. An explicit formula
for the asymptotic error in the log likelihood function is presented. Our
theory is applied to two simple examples from molecular dynamics.
http://arxiv.org/abs/0806.3248
Author(s): H\'el\`ene Guerin (IRMAR) and Nicolas Fournier (LAMA)
Abstract: We consider the spatially homogeneous Landau equation of kinetic theory, and
provide a differential inequality for the Wasserstein distance with quadratic
cost between two solutions. We deduce some well-posedness results. The main
difficulty is that this equation presents a singularity for small relative
velocities. Our uniqueness result is the first one in the important case of
soft potentials. Furthermore, it is almost optimal for a class of moderately
soft potentials, that is for a moderate singularity. Indeed, in such a case,
our result applies for initial conditions with finite mass, energy, and
entropy. For the other moderatley soft potentials, we assume additionnally some
moment conditions on the initial data. For very soft potentials, we obtain only
a local (in time) well-posedness result, under some integrability conditions.
Our proof is probabilistic, and uses a stochastic version of the Landau
equation, in the spirit of Tanaka.
http://arxiv.org/abs/0806.3379
Author(s): Paolo Dai Pra and Marco Tolotti
Abstract: We study the impact of contagion in a network of firms facing credit risk. We
describe an intensity based model where the homogeneity assumption is broken by
introducing a random environment that makes it possible to take into account
the idiosyncratic characteristics of the firms. We shall see that our model
goes behind the identification of groups of firms that can be considered
basically exchangeable. Despite this heterogeneity assumption our model has the
advantage of being totally tractable. The aim is to quantify the losses that a
bank may suffer in a large credit portfolio. Relying on a large deviation
principle on the trajectory space of the process, we state a suitable law of
large number and a central limit theorem useful to study large portfolio
losses. Simulation results are provided as well as applications to portfolio
loss distribution analysis.
http://arxiv.org/abs/0806.3399
Author(s): Peter Morters and Marcel Ortgiese
Abstract: We consider a model of directed polymers on a regular tree with a disorder
given by independent, identically distributed weights attached to the vertices.
For suitable weight distributions this model undergoes a phase transition with
respect to its localization behaviour. We show that, for high temperatures, the
free energy is supported by a random tree of positive exponential growth rate,
which is strictly smaller than that of the full tree. The growth rate of the
minimal supporting subtree is decreasing to zero as the temperature decreases
to the critical value. At the critical value and all lower temperatures, a
single polymer suffices to support the free energy. Our proofs rely on elegant
martingale methods adapted from the theory of branching random walks.
http://arxiv.org/abs/0806.3430
Author(s): Ivan del Tenno
Abstract: In this note we present some examples of diffusions in random environment
whose asymptotic behavior is rather surprising. We construct a family of
diffusions that are small perturbations of Brownian motion with non-vanishing
expected local drift under the static measure of the environment but where the
ballistic behavior is lost. As slight modifications of this collection of
diffusions we also provide examples with ballistic behavior where the
non-vanishing limiting velocity points to a direction opposite to the expected
local drift under the static measure.
http://arxiv.org/abs/0806.3868
Author(s): S\'ebastien Blach\`ere (LATP) and Peter Ha\"issinsky (LATP) and Pierre Mathieu (LATP)
Abstract: We establish a dimension formula for the harmonic measure of a finitely
supported and symmetric random walk on a hyperbolic group. We also characterize
random walks for which this dimension is maximal. Our approach is based on the
Green metric, a metric which provides a geometric point of view on random walks
and, in particular, which allows us to interpret harmonic measures as \qc
measures on the boundary of the group.
http://arxiv.org/abs/0806.3915
Author(s): Alexander Barvinok
Abstract: Let R=(r_1, ..., r_m) and C=(c_1, ..., c_n) be positive integer vectors such
that r_1 +... + r_m=c_1 +... + c_n. We consider the set Sigma(R, C) of
non-negative mxn integer matrices (contingency tables) with row sums R and
column sums C as a finite probability space with the uniform measure. We prove
that a random table D in Sigma(R,C) with high probability is close to a
particular matrix ("typical table'') Z defined as follows. We let g(x)=(x+1)
ln(x+1)-x ln x for non-negative x and let g(X)=sum_ij g(x_ij) for a
non-negative matrix X=(x_ij). Then g(X) is strictly concave and attains its
maximum on the polytope of non-negative mxn matrices X with row sums R and
column sums C at a unique point, which we call the typical table Z.
http://arxiv.org/abs/0806.3910
Author(s): Lasse Leskel\"a
Abstract: This paper generalizes the notion of stochastic order to a relation between
probability measures over arbitrary measurable spaces. This generalization is
motivated by the observation that for the stochastic ordering of two stationary
Markov processes, it suffices that the generators of the processes preserve
some, not necessarily reflexive or transitive, subrelation of the order
relation. The main contributions of the paper are: a functional
characterization of stochastic relations, necessary and sufficient conditions
for the preservation of stochastic relations, and an algorithm for finding
subrelations preserved by probability kernels. The theory is illustrated with
applications to hidden Markov processes, population processes, and queueing
systems.
http://arxiv.org/abs/0806.3562
Author(s): Florent Benaych-Georges (PMA) and Thierry Cabanal-Duvillard (MAP5)
Abstract: Recently, Ben Arous and Voiculescu considered taking the maximum of two free
random variables and brought to light a deep analogy with the operation of
taking the maximum of two independent random variables. We present here a new
insight on this analogy, based on random matrices giving an interpolation
between classical and free settings.
http://arxiv.org/abs/0806.3686
Author(s): Wlodzimierz Bryc and Virgil U. Pierce
Abstract: We show that quaternionic Gaussian random variables satisfy a generalization
of the Wick formula for computing the expected value of products in terms of a
family of graphical enumeration problems. When applied to the quaternionic
Wigner and Wishart families of random matrices the result gives the duality
between moments of these families and the corresponding real Wigner and Wishart
families.
http://arxiv.org/abs/0806.3695
Author(s): Yaozhong Hu and David Nualart and Jian Song
Abstract: In this paper we apply Clark-Ocone formula to deduce an explicit integral
representation for the renormalized self-intersection local time of the $d$%
-dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$.
As a consequence, we derive the existence of some exponential moments for
this random variable.
http://arxiv.org/abs/0806.3706
Author(s): Krzysztof Bogdan
Abstract: We construct the fundamental solution of $\partial_t+\Delta_x + q(t,x)$, for
functions $q$ with a certain integral space-time relative smallness, in
particular for those satisfying a relative negligibility. The resulting
transition density is comparable to the Gaussian kernel in finite time, and it
is even asymptotically equal to the Gaussian kernel (in small time) under the
assumption of relative negligibility.
The result is generalized to arbitrary strictly positive and finite
time-inhomogeneous transition densities on measure spaces.
We also discuss specific applications to Schr\"odinger perturbations of the
fractional Laplacian in view of the fact that the 3P Theorem holds for the
fundamental solution of the operator.
http://arxiv.org/abs/0806.3549
Author(s): Persi Diaconis and Jason Fulman
Abstract: The number of ``carries'' when $n$ random integers are added forms a Markov
chain [23]. We show that this Markov chain has the same transition matrix as
the descent process when a deck of $n$ cards is repeatedly riffle shuffled.
This gives new results for the statistics of carries and shuffling.
http://arxiv.org/abs/0806.3583
Author(s): Claus K\"ostler
Abstract: The extended de Finetti theorem characterizes exchangeable infinite random
sequences as conditionally i.i.d. and shows that the apparently weaker
distributional symmetry of spreadability is equivalent to exchangeability. Our
main result is a noncommutative version of this theorem.
In contrast to the classical result of Ryll-Nadzewski, exchangeability turns
out to be stronger than spreadability for infinite noncommutative random
sequences. Out of our investigations emerges noncommutative conditional
independence in terms of a von Neumann algebraic structure closely related to
Popa's notion of commuting squares and K\"ummerer's generalized Bernoulli
shifts. Our main result is applicable to classical probability, quantum
probability, in particular free probability, braid group representations and
Jones subfactors.
http://arxiv.org/abs/0806.3621
Author(s): Wilhelm von Waldenfels
Abstract: A white noise quantum stochastic calculus is developped using classical
measure theory as mathematical tool. Wick's and Ito's theorems have been
established. The simplest quantum stochastic differential equation has been
solved, unicity and the conditions for unitarity have been proven. The
Hamiltonian of the associated one parameter strongly continuous group has been
calculated explicitely.
http://arxiv.org/abs/0806.3636
Author(s): Wlodek Bryc
Abstract: We introduce a family of matrices with non-commutative entries that
generalize the classical real Wishart matrices.
With the help of the Brauer product, we derive a non-asymptotic expression
for the moments of traces of monomials in such matrices; the expression is
quite similar to the formula derived in our previous work for independent
complex Wishart matrices. We then analyze the fluctuations about the
Marchenko-Pastur law. We show that after centering by the mean, traces of real
symmetric polynomials in q-Wishart matrices converge in distribution, and we
identify the asymptotic law as the normal law when q=1, and as the semicircle
law when q=0.
http://arxiv.org/abs/0806.4014
Author(s): Victoria Henderson and David Hobson
Abstract: In this article we study an optimal stopping/optimal control problem which
models the decision facing a risk averse agent over when to sell an asset. The
market is incomplete so that the asset exposure cannot be hedged. In addition
to the decision over when to sell, the agent has to choose a control strategy
which corresponds to a feasible wealth process.
We formulate this problem as one involving the choice of a stopping time and
a martingale. We conjecture the form of the solution and verify that the
candidate solution is equal to the value function.
The interesting features of the solution are that it is available in a very
explicit form, that for some parameter values the optimal strategy is more
sophisticated than might originally be expected, and that although the set-up
is based on continuous diffusions, the optimal martingale may involve a jump
process.
One interpretation of the solution is that it is optimal for the risk averse
agent to gamble.
http://arxiv.org/abs/0806.4061
Author(s): Jostein Paulsen
Abstract: This paper is a survey of recent progress in the theory of ruin for risk
processes that earn investment return on invested assets.
http://arxiv.org/abs/0806.4125
Author(s): Michel Weber
Abstract: We study the localization problem appearing in Kronecker's diophantine
theorem.
We introduce a probabilistic approach allowing to extend for general
$\Q$-linearly independent sequences a result of T\'uran concerning the sequence
$ (\log p_\ell)$, $p_\ell$ being the $\ell$-th prime.
http://arxiv.org/abs/0806.3990
Author(s): Ivar Ekeland and Traian A Pirvu
Abstract: This paper considers the Merton portfolio management problem. We are
concerned with non-exponential discounting of time and this leads to time
inconsistencies of the decision maker. Following Ekeland and Pirvu 2006, we
introduce the notion of equilibrium policies and we characterize them by an
integral equation. The main idea is to come up with the value function in this
context. If risk preferences are of CRRA type, the integral equation which
characterizes the value function is shown to have a solution which leads to an
equilibrium policy. This work is an extension of Ekeland and Pirvu 2006.
http://arxiv.org/abs/0806.4026
Author(s): Paul Doukhan and Michael H. Neumann
Abstract: We give an introduction to a notion of weak dependence which is more general
than mixing and allows to treat for example processes driven by discrete
innovations as they appear with time series bootstrap. As a typical example, we
analyze autoregressive processes and their bootstrap analogues in detail and
show how weak dependence can be easily derived from a contraction property of
the process. Furthermore, we provide an overview of classes of processes
possessing the property of weak dependence and describe important probabilistic
results under such an assumption.
http://arxiv.org/abs/0806.4263
Author(s): Mark Veraar
Abstract: In this paper we study the following non-autonomous stochastic evolution
equation on a UMD Banach space $E$ with type 2,
\begin{equation}\label{eq:SEab}\tag{SE} {\begin{aligned} dU(t) & = (A(t)U(t)
+ F(t,U(t))) dt + B(t,U(t)) dW_H(t), \quad t\in [0,T],
U(0) & = u_0. \end{aligned}. \end{equation}
Here $(A(t))_{t\in [0,T]}$ are unbounded operators with domains
$(D(A(t)))_{t\in [0,T]}$ which may be time dependent. We assume that
$(A(t))_{t\in [0,T]}$ satisfies the conditions of Acquistapace and Terreni. The
functions $F$ and $B$ are nonlinear functions defined on certain interpolation
spaces and $u_0\in E$ is the initial value. $W_H$ is a cylindrical Brownian
motion on a separable Hilbert space $H$.
Under Lipschitz and linear growth conditions we show that there exists a
unique mild solution of \eqref{eq:SEab}. Under assumptions on the interpolation
spaces we extend the factorization method of Da Prato, Kwapie\'n, and Zabczyk,
to obtain space-time regularity results for the solution $U$ of
\eqref{eq:SEab}. For Hilbert spaces $E$ we obtain a maximal regularity result.
The results improve several previous results from the literature.
The theory is applied to a second order stochastic partial differential
equation which has been studied by Sanz-Sol\'e and Vuillermot. This leads to
several improvements of their result.
http://arxiv.org/abs/0806.4439
Author(s): Deborah Nolan and Terry Speed
Abstract: This volume is our tribute to David A. Freedman, whom we regard as one of the
great statisticians of our time. He received his B.Sc. degree from McGill
University and his Ph.D. from Princeton, and joined the Department of
Statistics of the University of California, Berkeley, in 1962, where, apart
from sabbaticals, he has been ever since. In a career of over 45 years, David
has made many fine contributions to probability and statistical theory, and to
the application of statistics. His early research was on Markov chains and
martingales, and two topics with which he has had a lifelong fascination:
exchangeability and De Finetti's theorem, and the consistency of Bayes
estimates. His asymptotic theory for the bootstrap was also highly influential.
David was elected to the American Academy of Arts and Sciences in 1991, and in
2003 he received the John J. Carty Award for the Advancement of Science from
the U.S. National Academy of Sciences. In addition to his purely academic
research, David has extensive experience as a consultant, including working for
the Carnegie Commission, the City of San Francisco, and the Federal Reserve, as
well as several Departments of the U.S. Government--Energy, Treasury, Justice,
and Commerce. He has testified as an expert witness on statistics in a number
of law cases, including Piva v. Xerox (employment discrimination), Garza v.
County of Los Angeles (voting rights), and New York v. Department of Commerce
(census adjustment). Lastly, he is an exceptionally good writer and teacher,
and his many books and review articles are arguably his most important
contribution to our subject.
http://arxiv.org/abs/0806.4441
Author(s): Emmanuel Gobet (LJK) and Jean-Philippe Lemor (CMAP)
Abstract: This article deals with the numerical resolution of backward stochastic
differential equations. Firstly, we consider a rather general case where the
filtration is generated by a Brownian motion and a Poisson random measure. We
provide a simulation algorithm based on iterative regressions on function
bases, which coefficients are evaluated using Monte Carlo simulations. We state
fully explicit error bounds. Secondly, restricting to the case of a Brownian
filtration, we consider reflected BSDEs and adapt the previous algorithm to
that situation. The complexity of the algorithm is very competitive and allows
us to treat numerical results in dimension 10.
http://arxiv.org/abs/0806.4447
Author(s): A. Bianchi and A. Bovier and D. Ioffe
Abstract: In this paper we study the metastable behavior of one of the simplest
disordered spin system, the random field Curie-Weiss model. We will show how
the potential theoretic approach can be used to prove sharp estimates on
capacities and metastable exit times also in the case when the distribution of
the random field is continuous. Previous work was restricted to the case when
the random field takes only finitely many values, which allowed the reduction
to a finite dimensional problem using lumping techniques. Here we produce the
first genuine sharp estimates in a context where entropy is important.
http://arxiv.org/abs/0806.4478
Author(s): Ayadi Slim
Abstract: We study the normalized eigenvalue counting measure d\sigma of matrices of
long-range percolation model. These are (2n+1)\times (2n+1) random real
symmetric matrices H=\{H(i,j)\}_{i,j} whose elements are independent random
variables taking zero value with probability 1-\psi [(i-j)/b], b\in
\mathbb{R}^{+}, where \psi is an even positive function \psi(t)\le{1} vanishing
at infinity. It is shown that if the third moment of \sqrt{b}H(i,j), i\leq{j}
is uniformly bounded then the measure d\sigma:=d\sigma_{n,b} weakly converges
in probability in the limit n,b\to\infty, b=o(n) to the semicircle (or Wigner)
distribution. The proof uses the resolvent technique combined with the cumulant
expansions method. We show that the normalized trace of resolvent g_{n,b}(z)
converges in average and that the variance of g_{n,b}(z) vanishes. In the
second part of the paper, we estimate the rate of decreasing of the variance of
g_{n,b}(z), under further conditions on the moments of \sqrt{b}H(i,j), \
i\le{j}.
http://arxiv.org/abs/0806.4497
Author(s): Ilya Molchanov and Michael Schmutz
Abstract: In this paper we show how to relate European call and put options on multiple
assets to certain convex bodies called lift zonoids. Based on this, geometric
properties can be translated into economic statements and vice versa. For
instance, the European call-put parity corresponds to the central symmetry
property, while the concept of dual markets can be explained by reflection with
respect to a plane. It is known that the classical univariate log-normal model
belongs to a large class of distributions with an extra property, analytically
known as put-call symmetry. The geometric interpretation of this symmetry
property motivates a natural multivariate extension. The financial meaning of
this extension is explained, the asset price distributions that have this
property are characterised and their properties are explored. It is also shown
how to transform some multivariate asymmetric distributions that appear after
adjusting for carrying costs to symmetric ones. A particular attention is
devoted to the case of asset prices driven by L'evy processes. Based on this,
semi-static hedging techniques for multiasset barrier options are suggested.
http://arxiv.org/abs/0806.4506
Author(s): Takashi Kumagai and Jun Misumi
Abstract: We establish general estimates for simple random walk on an arbitrary
infinite random graph, assuming suitable bounds on volume and effective
resistance for the graph. These are generalizations of the results in
\cite[Section 1,2]{BJKS}, and in particular, imply the spectral dimension of
the random graph. We will also give an application of the results to random
walk on a long range percolation cluster.
http://arxiv.org/abs/0806.4507
Author(s): Iain M. MacPhee and Mikhail V. Menshikov and Andrew R. Wade
Abstract: We study the first exit time $\tau$ from an arbitrary cone with apex at the
origin by a non-homogeneous random walk on $\mathbb{Z}^2$ with mean drift that
is asymptotically zero. Assuming bounded jumps and a form of weak isotropy, we
give conditions for $\tau$ to be almost surely finite, and for the existence
and non-existence of moments $\Exp [ \tau^p]$, $p>0$. Specifically, if the mean
drift at $\bx \in \mathbb{Z}^2$ is of magnitude $O(\| \bx\|^{-1})$, then
$\tau<\infty$ a.s. for any cone, and we give polynomial estimates on the tail
of $\tau$, while a mean drift of order $\| \bx \|^{-\beta}$, $\beta \in (0,1)$,
can lead to $\tau=\infty$ with positive probability for any cone. For mean
drift $o(\|\bx\|^{-1})$ (such as in the driftless case) we essentially recover,
amongst other things, the random walk analogue of a theorem for Brownian motion
due to Spitzer.
http://arxiv.org/abs/0806.4561
Author(s): Jozsef Balogh and Bela Bollobas and Robert Morris
Abstract: By bootstrap percolation we mean the following deterministic process on a
graph G. Given a set A of vertices 'infected' at time 0, new vertices are
subsequently infected, at each time step, if they have at least r \in \N
previously infected neighbours. When the set A is chosen at random, the main
aim is to determine the critical probability p_c(G,r) at which percolation
(infection of the entire graph) becomes likely to occur.
This bootstrap process has been extensively studied on the d-dimensional grid
[n]^d: with 2 \le r \le d fixed, it was proved by Cerf and Cirillo (for d = r
=3), and by Cerf and Manzo (in general), that p_c([n]^d,r) =
\Theta(\frac{1}{\log_{r-1} n})^{d-r+1}, where \log_r is an r-times iterated
logarithm. However, the exact threshold function is only known in the case d =
r = 2, where it was shown by Holroyd to be (1 + o(1))\frac{\pi^2}{18\log n}. In
this paper we shall determine the exact threshold in the crucial case d = r =
3, and lay the groundwork for solving the problem for all fixed d and r.
http://arxiv.org/abs/0806.4485
Author(s): Xian-Yuan Wu and Zhao Dong and Ke Liu and Kai-Yuan Cai
Abstract: In this paper we focus on the problem of the degree sequence for the
following random graph process. At any time-step $t$, one of the following
three substeps is executed: with probability $\alpha_1$, a new vertex $x_t$ and
$m$ edges incident with $x_t$ are added; or, with probability
$\alpha-\alpha_1$, $m$ edges are added; or finally, with probability $1-\a$,
$m$ random edges are deleted. Note that in any case edges are added in the
manner of preferential attachment. we prove that there exists a critical point
$\alpha_c$ satisfying: 1) if $\alpha_1<\alpha_c$, then the model has power law
degree sequence; 2) if $\alpha_1>\alpha_c$, then the model has exponential
degree sequence; and 3) if $\alpha_1=\alpha_c$, then the model has a degree
sequence lying between the above two cases.
http://arxiv.org/abs/0806.4684
Author(s): Eric Shellef
Abstract: We consider the internal diffusion limited aggregation (IDLA) process on the
infinite cluster in supercritical Bernoulli bond percolation on Euclidean
lattices. It is shown that the process on the cluster behaves like it does on
the Euclidean lattice, in that the aggregate covers all the vertices in a
Euclidean ball around the origin, such that the ratio of vertices in this ball
to the total number of particles sent out approaches one almost surely.
http://arxiv.org/abs/0806.4771
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