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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 annea | |
