[PAS] Probability Abstracts 104
Probability Abstract Service
pas at lists.imstat.org
Tue Jul 8 02:09:34 CDT 2008
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.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_104.shtml
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6994. A DISCRETE CONSTRUCTION FOR GAUSSIAN MARKOV PROCESSES
Thibaud Taillefumier
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
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6995. OPTIMAL ROBUST MEAN-VARIANCE HEDGING IN INCOMPLETE FINANCIAL
MARKETS
N. Lazrieva and T. Toronjadze
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
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6996. COMMUNICATION REQUIREMENTS FOR GENERATING CORRELATED RANDOM
VARIABLES
Paul Cuff (Stanford University)
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
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6997. RANDOM WALKS, ARRANGEMENTS, CELL COMPLEXES, GREEDOIDS, AND SELF-
ORGANIZING LIBRARIES
Anders Bj\"orner
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
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6998. RESONANCES FOR A DIFFUSION WITH SMALL NOISE
Markus Klein and Pierre-Andr\'e Zitt (MODAL'X)
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
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6999. MULTIFRACTAL ANALYSIS IN A MIXED ASYMPTOTIC FRAMEWORK
Emmanuel Bacry and Arnaud Gloter and Marc Hoffmann and Jean-Francois
Muzy
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
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7000. A KHASMINSKII TYPE AVERAGING PRINCIPLE FOR STOCHASTIC REACTION-
DIFFUSION EQUATIONS
Sandra Cerrai
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
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7001. AVERAGING PRINCIPLE FOR A CLASS OF STOCHASTIC REACTION-
DIFFUSION EQUATIONS
Sandra Cerrai and Mark Freidlin
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
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7002. CENTRAL LIMIT THEOREM FOR A CLASS OF LINEAR SYSTEMS
Yukio Nagahata and Nobuo Yoshida
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
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7003. A LIMIT THEOREM FOR PRODUCTS OF FREE UNITARY OPERATORS
Vladislav Kargin
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
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7004. THE PLAYER'S EFFECT
Ronen Gradwohl and Omer Reingold and Ariel Yadin and Amir Yehudayoff
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
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7005. THE MIXING ADVANTAGE IS LESS THAN 2
Kais Hamza and Peter Jagers and Aidan Sudbury and Daniel Tokarev
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
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7006. SPECTRAL GAP FOR THE INTERCHANGE PROCESS IN A BOX
Ben Morris
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
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7007. INTERMITTENCE AND NONLINEAR PARABOLIC STOCHASTIC PARTIAL
DIFFERENTIAL EQUATIONS
Mohammud Foondun and Davar Khoshnevisan
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
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7008. A MARTINGALE APPROACH TO MINIMAL SURFACES
Robert W. Neel
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
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7009. FUNCTIONAL MODERATE DEVIATIONS FOR TRIANGULAR ARRAYS AND
APPLICATIONS
Florence Merlevede and Magda Peligrad
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
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7010. RISK AVERSION AND PORTFOLIO SELECTION IN A CONTINUOUS-TIME MODEL
Jianming Xia
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
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7011. PRINCIPAL EIGENVALUE FOR RANDOM WALK AMONG RANDOM TRAPS ON Z^D
Jean-Christophe Mourrat
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
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7012. RANDOM WALK WEAKLY ATTRACTED TO A WALL
Jo\"el De Coninck and Fran\c{c}ois Dunlop and Thierry Huillet
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
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7013. HYDRODYNAMIC LIMIT FOR A ZERO-RANGE PROCESS IN THE SIERPINSKI
GASKET
M. Jara
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
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7014. LEVY PROCESSES AND SCHROEDINGER EQUATION
Nicola Cufaro Petroni and Modesto Pusterla
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
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7015. LIMIT THEOREMS FOR ADDITIVE C-FREE CONVOLUTION
Jiun-Chau Wang
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
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7016. THE EFFECT OF CLASSICAL NOISE ON A QUANTUM TWO-LEVEL SYSTEM
Jean-Philippe Aguilar (CPT) and Nils Berglund (MAPMO)
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
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7017. CUT POINTS AND DIFFUSIONS IN RANDOM ENVIRONMENT
Ivan del Tenno
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
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7018. BEHAVIOR NEAR THE EXTINCTION TIME IN SELF-SIMILAR FRAGMENTATIONS
I: THE STABLE CASE
Christina Goldschmidt and B\'en\'edicte Haas (CEREMADE)
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
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7019. ON FINE PROPERTIES OF MIXTURES WITH RESPECT TO CONCENTRATION OF
MEASURE AND SOBOLEV TYPE INEQUALITIES
Djalil Chafai (IMT and UPTE) and Florent Malrieu (IRMAR)
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
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7020. THE EFFECTS OF MASS EXTINCTION EVENTS ON THE GENEALOGY OF A
SUBDIVIDED POPULATION
Jesse E. Taylor and Amandine Veber
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
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7021. SOME NEW RANDOM FIELD TOOLS FOR SPATIAL ANALYSIS
Robert J Adler
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
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7022. ON THE EIGENSPACES OF LAMPLIGHTER RANDOM WALKS AND PERCOLATION
CLUSTERS ON GRAPHS
Franz Lehner
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
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7023. CONCENTRATION OF MEASURE VIA APPROXIMATED BRUNN--MINKOWSKI
INEQUALITIES
Masayoshi Watanabe
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
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7024. EXACTNESS OF MARTINGALE APPROXIMATION AND THE CENTRAL LIMIT
THEOREM
Dalibor Voln\'y
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
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7025. THE FINITE HORIZON OPTIMAL MULTI-MODES SWITCHING PROBLEM: THE
VISCOSITY SOLUTION APPROACH
Brahim El Asri and Said Hamadene
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
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7026. HYDRODYNAMIC LIMIT OF PARTICLE SYSTEMS WITH LONG JUMPS
M. Jara
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
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7027. SMALL DEVIATIONS OF GENERAL L\'EVY PROCESSES
Frank Aurzada and Steffen Dereich
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
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7028. A NOTE ON THE ENUMERATION OF DIRECTED ANIMALS VIA GAS
CONSIDERATIONS
Marie Albenque
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
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7029. ADAPTIVE ESTIMATION OF A DISTRIBUTION FUNCTION AND ITS DENSITY
IN SUP-NORM LOSS BY WAVELET AND SPLINE PROJECTIONS
Evarist Gin\'e and Richard Nickl
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
---------------------------------------------------------------
7030. UNIFORM LIMIT THEOREMS FOR WAVELET DENSITY ESTIMATORS
Evarist Gin\'e and Richard Nickl
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
---------------------------------------------------------------
7031. DEGREE-DISTRIBUTION STABILITY OF SCALE-FREE NETWORKS
Zhenting Hou and Xiangxing Kong and Dinghua Shi and Guanrong Chen
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
---------------------------------------------------------------
7032. CONDITIONS FOR STOCHASTIC INTEGRABILITY IN UMD BANACH SPACES
Jan van Neerven and Mark Veraar and Lutz Weis
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
---------------------------------------------------------------
7033. FLUCTUATIONS OF THE PARTITION FUNCTION IN THE GREM WITH EXTERNAL
FIELD
Anton Bovier and Anton Klimovsky
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
---------------------------------------------------------------
7034. THE PROBABILITY OF EXCEEDING A PIECEWISE DETERMINISTIC BARRIER
BY THE HEAVY-TAILED RENEWAL COMPOUND PROCESS
Zbigniew Palmowski and Martijn Pistorius
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
---------------------------------------------------------------
7035. HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS
Bruce Driver and Maria Gordina
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
---------------------------------------------------------------
7036. QUENCHED AND ANNEALED CRITICAL POINTS IN POLYMER PINNING MODELS
Kenneth S. Alexander and Nikos Zygouras
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 $3/2<c<2$ and $c>2$, but only at low temperatures for
$c<3/2$.
For high temperatures and $3/2<c<2$ we establish the exact order of
the gap
between critical points, as a function of temperature. For the
borderline case
$c=3/2$ we show that the gap is positive provided $\phi(n) \to 0$ as
$n \to
\infty$, and for $c >3/2$ 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
---------------------------------------------------------------
7037. ON ISOPERIMETRIC INEQUALITIES FOR LOG-CONVEX MEASURES
Alexander V. Kolesnikov
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
---------------------------------------------------------------
7038. THE COVARIOGRAM DETERMINES THREE-DIMENSIONAL CONVEX POLYTOPES
Gabriele Bianchi
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
---------------------------------------------------------------
7039. LOG-LEVEL COMPARISON PRINCIPLE FOR SMALL BALL PROBABILITIES
A. I. Nazarov
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
---------------------------------------------------------------
7040. CELL CONTAMINATION AND BRANCHING PROCESS IN RANDOM ENVIRONMENT
WITH IMMIGRATION
Vincent Bansaye (PMA)
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
---------------------------------------------------------------
7041. ON THE MARTINGALE PROBEM ASSOCIATED TO THE 2D AND 3D STOCHASTIC
NAVIER-STOKES EQUATIONS
Giuseppe Da Prato (ENS) and Arnaud Debussche (IRMAR)
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
---------------------------------------------------------------
7042. ON ONE PROPERTY OF DISTANCES IN THE INFINITE RANDOM
QUADRANGULATION
Maxim Krikun (IECN)
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
---------------------------------------------------------------
7043. AGGREGATION OF WEAKLY DEPENDENT DOUBLY STOCHASTIC PROCESSES
Lisandro J. Fermin
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
---------------------------------------------------------------
7044. ON A SET OF TRANSFORMATIONS OF GAUSSIAN RANDOM FUNCTIONS
A.I. Nazarov
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
---------------------------------------------------------------
7045. STOCHASTIC CALCULUS FOR CONVOLUTED L\'{E}VY PROCESSES
Christian Bender and Tina Marquardt
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
---------------------------------------------------------------
7046. STOCHASTIC ANALYSIS ON GAUSSIAN SPACE APPLIED TO DRIFT ESTIMATION
Nicolas Privault and Anthony Reveillac
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
---------------------------------------------------------------
7047. THE DBAR STEEPEST DESCENT METHOD FOR ORTHOGONAL POLYNOMIALS ON
THE REAL LINE WITH VARYING WEIGHTS
K. T.-R. McLaughlin and P. D. Miller
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
---------------------------------------------------------------
7048. THE KOLMOGOROV OPERATOR ASSOCIATED TO A BURGERS SPDE IN SPACES
OF CONTINUOUS FUNCTIONS
Luigi Manca
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
---------------------------------------------------------------
7049. UNIVERSAL OPTIMAL STOCHASTIC EXPANSIONS
Simon J.A. Malham and Anke Wiese
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
---------------------------------------------------------------
7050. PROBABILISTIC REPRESENTATION FOR SOLUTIONS OF AN IRREGULAR
POROUS MEDIA TYPE EQUATION
Philippe Blanchard and Michael R\"ockner (SFB 705) and Francesco
Russo (LAGA)
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
---------------------------------------------------------------
7051. N/V-LIMIT FOR LANGEVIN DYNAMICS IN CONTINUUM
Florian Conrad and Martin Grothaus
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
---------------------------------------------------------------
7052. OVERCROWDING AND HOLE PROBABILITIES FOR RANDOM ZEROS ON COMPLEX
MANIFOLDS
Bernard Shiffman and Steve Zelditch and Scott Zrebiec
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
---------------------------------------------------------------
7053. PHASE TRANSITIONS FOR THE GROETH RATE OF LINEAR STOCHASTIC
EVOLUTIONS
Nobuo Yoshida
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
---------------------------------------------------------------
7054. PROBABILITY THEORY AND ITS MODELS
Paul Humphreys
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
---------------------------------------------------------------
7055. DUTCH BOOK IN SIMPLE MULTIVARIATE NORMAL PREDICTION: ANOTHER LOOK
Morris L. Eaton
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
---------------------------------------------------------------
7056. GENERATING UNIFORM RANDOM VECTORS IN $\QTR{BF}{Z}_{P}^{K}$: THE
GENERAL CASE
Claudio Asci
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
---------------------------------------------------------------
7057. MODERATE DEVIATIONS FOR STATIONARY SEQUENCES OF HILBERT VALUED
BOUNDED RANDOM VARIABLES
Sophie Dede (PMA)
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
---------------------------------------------------------------
7058. ON ALMOST RANDOMIZING CHANNELS WITH A SHORT KRAUS DECOMPOSITION
Guillaume Aubrun (ICJ)
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
---------------------------------------------------------------
7059. EVOLUTION EQUATIONS OF THE PROBABILISTIC GENERALIZATION OF THE
VOIGT PROFILE FUNCTION
Gianni Pagnini and Francesco Mainardi
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
---------------------------------------------------------------
7060. NON-MARKOVIAN DIFFUSION EQUATIONS AND PROCESSES: ANALYSIS AND
SIMULATIONS
Antonio Mura and Murad S. Taqqu and Francesco Mainardi
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
---------------------------------------------------------------
7061. DISJOINTNESS OF REPRESENTATIONS ARISING IN HARMONIC ANALYSIS ON
THE INFINITE-DIMENSIONAL UNITARY GROUP
Vadim Gorin
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
---------------------------------------------------------------
7062. ON THE DISTRIBUTION OF THE NODAL SETS OF RANDOM SPHERICAL
HARMONICS
Igor Wigman
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
---------------------------------------------------------------
7063. GENERALIZED BACKWARD STOCHASTIC DIFFERENTIAL EQUATION WITH TWO
REFLECTING BARRIERS AND STOCHASTIC QUADRATIC GROWTH
E. H. Essaky and M. Hassani
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
---------------------------------------------------------------
7064. WIGNER FUNCTIONS AND STOCHASTICALLY PERTURBED LATTICE DYNAMICS
Giada Basile (WIAS) and Stefano Olla (CEREMADE) and Herbert Spohn
(D-Mutu-ZM)
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
---------------------------------------------------------------
7065. ASYMPTOTICS OF CHARACTERISTIC POLYNOMIALS OF WIGNER MATRICES AT
THE EDGE OF THE SPECTRUM
Holger K\"osters
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
---------------------------------------------------------------
7066. CONVERGENCE OF DEPENDENT WALKS IN A RANDOM SCENERY TO FBM-LOCAL
TIME FRACTIONAL STABLE MOTIONS
Serge Cohen (LSProba) and Cl\'ement Dombry (LMA)
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
---------------------------------------------------------------
7067. POISSON-DIRICHLET DISTRIBUTION WITH SMALL MUTATION RATE
Shui Feng
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
---------------------------------------------------------------
7068. HOW T-CELLS USE LARGE DEVIATIONS TO RECOGNIZE FOREIGN ANTIGENS
Natali Zint and Ellen Baake and Frank den Hollander
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
---------------------------------------------------------------
7069. RANDOM MATRICES: A GENERAL APPROACH FOR THE LEAST SINGULAR VALUE
PROBLEM
Terence Tao and Van Vu
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
---------------------------------------------------------------
7070. BROWNIAN ENTROPIC REPULSION
Itai Benjamini and Nathanael Berestycki
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
---------------------------------------------------------------
7071. ON TIGHTNESS OF MUTUAL DEPENDENCE UPPERBOUND FOR SECRET-KEY
CAPACITY OF MULTIPLE TERMINALS
Chung Chan
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
---------------------------------------------------------------
7072. ESTIMATION IN MODELS DRIVEN BY FRACTIONAL BROWNIAN MOTION
Corinne Berzin and Jos\'e R. Le\'on
Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with
parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of
the type
\[X(t)=c+\int_0^t\sigma\bigl(X(u)\bigr)\mathrm {d}b_H(u)+\int
_0^t\mu\bigl(X(u)\bigr)\mathrm {d}u.\] In different particular models
where
$\sigma(x)=\sigma$ or $\sigma(x)=\sigma x$ and $\mu(x)=\mu$ or $\mu(x)=
\mu x$,
we propose a central limit theorem for estimators of $H$ and of $\sigma
$ based
on regression methods. Then we give tests of the hypothesis on $\sigma
$ for
these models. We also consider functional estimation on $\sigma(\cdot)
$ in the
above more general models based in the asymptotic behavior of
functionals of
the 2nd-order increments of the fBm.
http://arxiv.org/abs/0805.3394
---------------------------------------------------------------
7073. THE LEAST SINGULAR VALUE OF A RANDOM SQUARE MATRIX IS O(N^{-1/2})
Mark Rudelson and Roman Vershynin
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
---------------------------------------------------------------
7074. VITESSE DE CONVERGENCE DANS LE TH\'{E}OR\`{E}ME LIMITE CENTRAL
POUR DES CHA\^{I}NES DE MARKOV FORTEMENT ERGODIQUES
Lo\"ic Herv\'e
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
---------------------------------------------------------------
7075. COMPARISON BETWEEN CRITERIA LEADING TO THE WEAK INVARIANCE
PRINCIPLE
Olivier Durieu and Dalibor Voln\'y
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
---------------------------------------------------------------
7076. SINGULAR VALUE DECOMPOSITION OF LARGE RANDOM MATRICES (FOR TWO-
WAY CLASSIFICATION OF MICROARRAYS)
Marianna Bolla and Katalin Friedl and Andras Kramli
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
---------------------------------------------------------------
7077. LINEAR STATISTICS OF POINT PROCESSES VIA ORTHOGONAL POLYNOMIALS
E. Ryckman
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
---------------------------------------------------------------
7078. BERNSTEIN MEASURES ON CONVEX POLYTOPES
Tatsuya Tate
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
---------------------------------------------------------------
7079. RANDOM WALKS IN SPACE TIME MIXING ENVIRONMENTS
Jean Bricmont and Antti Kupiainen
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
---------------------------------------------------------------
7080. PROBABILISTIC STUDY OF THE SPEED OF APPROACH TO EQUILIBRIUM FOR
AN INELASTIC KAC MODEL
Federico Bassetti and Lucia Ladelli and Eugenio Regazzini
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
---------------------------------------------------------------
7081. AVERAGES OF RATIOS OF CHARACTERISTIC POLYNOMIALS IN CIRCULAR
BETA-ENSEMBLES AND SUPER-JACK POLYNOMIALS
Sho Matsumoto
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
---------------------------------------------------------------
7082. ON THE CLUSTER SIZE DISTRIBUTION FOR PERCOLATION ON SOME GENERAL
GRAPHS
Antar Bandyopadhyay and Jeffrey Steif and Adam Timar
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
---------------------------------------------------------------
7083. EXPLICIT ERROR BOUNDS FOR LAZY REVERSIBLE MARKOV CHAIN MONTE CARLO
Daniel Rudolf
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
---------------------------------------------------------------
7084. SUCCESS EXPONENT OF WIRETAPPER: A TRADEOFF BETWEEN SECRECY AND
RELIABILITY
Chung Chan
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
---------------------------------------------------------------
7085. MULTIPLICATIVE FUNCTIONAL FOR REFLECTED BROWNIAN MOTION VIA
DETERMINISTIC ODE
Krzysztof Burdzy and John M. Lee
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
---------------------------------------------------------------
7086. GENERALIZED CHINESE RESTAURANT CONSTRUCTION OF EXCHANGEABLE
GIBBS PARTITIONS AND RELATED RESULTS
Annalisa Cerquetti
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
---------------------------------------------------------------
7087. EXCURSIONS AWAY FROM A REGULAR POINT FOR ONE-DIMENSIONAL
SYMMETRIC LEVY PROCESSES WITHOUT GAUSSIAN PART
Kouji Yano
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
---------------------------------------------------------------
7088. A SET-VALUED FRAMEWORK FOR BIRTH-AND-GROWTH PROCESS
Giacomo Aletti and Enea G. Bongiorno and Vincenzo Capasso
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
---------------------------------------------------------------
7089. OPTIMAL H\"OLDER INDEX FOR DENSITY STATES OF SUPERPROCESSES WITH
(1 + \BETA)-BRANCHING MECHANISM
Klaus Fleischmann and Leonid Mytnik and Vitali Wachtel
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
---------------------------------------------------------------
7090. OPTIMAL INVESTMENT STRATEGY TO MINIMIZE OCCUPATION TIME
Erhan Bayraktar and Virginia R. Young
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
---------------------------------------------------------------
7091. ON THE UNIQUENESS OF THE INFINITE CLUSTER OF THE VACANT SET OF
RANDOM INTERLACEMENTS
A. Q. Teixeira
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
---------------------------------------------------------------
7092. CONVERGENCE OF POINT PROCESSES WITH WEAKLY DEPENDENT POINTS
Raluca Balan and Sana Louhichi
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
---------------------------------------------------------------
7093. ON THE DEPENDENCE STRUCTURE OF WAVELET COEFFICIENTS FOR
SPHERICAL RANDOM FIELDS
Xiaohong Lan and Domenico Marinucci
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
---------------------------------------------------------------
7094. CONTINUOUS TIME RANDOM WALK AND PARAMETRIC SUBORDINATION IN
FRACTIONAL DIFFUSION
Rudolf Gorenflo and Francesco Mainardi and Alessandro Vivoli
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
---------------------------------------------------------------
7095. FRACTIONAL DIFFUSION PROCESSES: PROBABILITY DISTRIBUTIONS AND
CONTINUOUS TIME RANDOM WALK
Rudolf Gorenflo and Francesco Mainardi
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
---------------------------------------------------------------
7096. GENERALIZED STIRLING PERMUTATIONS, FAMILIES OF INCREASING TREES
AND URN MODELS
Svante Janson and Markus Kuba and Alois Panholzer
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
---------------------------------------------------------------
7097. ON THE ENTROPY AND LOG-CONCAVITY OF COMPOUND POISSON MEASURES
Oliver Johnson and Ioannis Kontoyiannis and Mokshay Madiman
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
---------------------------------------------------------------
7098. CHAINS OF DISTRIBUTIONS, HIERARCHICAL BAYESIAN MODELS AND
BENFORD'S LAW
Dennis Jang and Jung Uk Kang and Alex Kruckman and Jun Kudo and
Steven J. Miller
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
---------------------------------------------------------------
7099. EXACT EDGEWORTH EXPANSION FOR A L\'{E}VY PROCESS
Heikki J. Tikanm\"aki
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
---------------------------------------------------------------
7100. ON SUBEXPONENTIALITY OF THE L\'EVY MEASURE OF THE DIFFUSION
INVERSE LOCAL TIME; WITH APPLICATIONS TO PENALIZATIONS
Paavo Salminen and Pierre Vallois
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
---------------------------------------------------------------
7101. CONDITIONING ON AN EXTREME COMPONENT: MODEL CONSISTENCY AND
REGULAR VARIATION ON CONES
Bikramjit Das and Sidney I. Resnick
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
---------------------------------------------------------------
7102. SPARSE POWER-EFFICIENT TOPOLOGIES FOR WIRELESS AD HOC SENSOR
NETWORKS
Amitabha Bagchi
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
---------------------------------------------------------------
7103. QUEUEING SYSTEM WITH PRE-SCHEDULED RANDOM ARRIVALS
G. Guadagni and S. Ndreca and B. Scoppola
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
---------------------------------------------------------------
7104. RANDOM WALKS ON DISCRETE CYLINDERS AND RANDOM INTERLACEMENTS
Alain-Sol Sznitman
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
---------------------------------------------------------------
7105. ON UPPER BOUNDS FOR THE TAIL DISTRIBUTION OF GEOMETRIC SUMS OF
SUBEXPONENTIAL RANDOM VARIABLES
Andrew Richards
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
---------------------------------------------------------------
7106. ASYMPTOTIC PROPERTIES OF AN ESTIMATOR OF THE DRIFT COEFFICIENTS
OF MULTIDIMENSIONAL ORNSTEIN-UHLENBECK PROCESSES THAT ARE NOT
NECESSARILY STABLE
Gopal K. Basak and Philip Lee
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
---------------------------------------------------------------
7107. ON THE FIRST PASSAGE TIME FOR BROWNIAN MOTION SUBORDINATED BY A
LEVY PROCESS
T. R. Hurd and A. Kuznetsov
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
---------------------------------------------------------------
7108. NUMERICAL COMPUTATIONS FOR BACKWARD DOUBLY SDES AND SPDES
Yufeng Shi and Weiqiang Yang and Jing Yuan
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
---------------------------------------------------------------
7109. BACKWARD SDES WITH CONSTRAINED JUMPS AND QUASI-VARIATIONAL
INEQUALITIES
Idris Kharroubi (PMA and CREST) and Jin Ma and Huyen Pham (PMA and
CREST) and Jianfeng Zhang
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
---------------------------------------------------------------
7110. LES DEUX QUADRANGULATIONS INFINIES UNIFORMES ONT M\^EME LOI
Laurent M\'enard
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
---------------------------------------------------------------
7111. DENSENESS OF CERTAIN SMOOTH L\'EVY FUNCTIONALS IN $\DD_{1,2}$
Christel Geiss and Eija Laukkarinen
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
---------------------------------------------------------------
7112. MARKOV CHAIN-BASED STABILITY ANALYSIS OF GROWING NETWORKS
Zhenting Hou and Jinying Tong and Dinghua Shi
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
---------------------------------------------------------------
7113. DIFFERENTIABILITY OF STOCHASTIC FLOW OF REFLECTED BROWNIAN MOTIONS
Krzysztof Burdzy
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
---------------------------------------------------------------
7114. MARKING (1,2) POINTS OF THE BROWNIAN WEB AND APPLICATIONS
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)
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
---------------------------------------------------------------
7115. FROM BLACK-SCHOLES AND DUPIRE FORMULAE TO LAST PASSAGE TIMES OF
LOCAL MARTINGALES. PART A : THE INFINITE TIME HORIZON
Amel Bentata (PMA) and Marc Yor (PMA and Iuf)
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
---------------------------------------------------------------
7116. BACKWARD STOCHASTIC PDES RELATED TO THE UTILITY MAXIMIZATION
PROBLEM
M. Mania and R. Tevzadze
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
---------------------------------------------------------------
7117. SUSCEPTIBILITY IN SUBCRITICAL RANDOM GRAPHS
Svante Janson and Malwina J. Luczak
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
---------------------------------------------------------------
7118. PERTURBATIVE APPROACH ON FINANCIAL MARKETS
Simone Scotti
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
---------------------------------------------------------------
7119. RISK PREMIUM IMPACT IN THE PERTURBATIVE BLACK SCHOLES MODEL
Luca Regis and Simone Scotti
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
---------------------------------------------------------------
7120. THE GROWTH EXPONENT FOR PLANAR LOOP-ERASED RANDOM WALK
Robert Masson
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
---------------------------------------------------------------
7121. DENSITY FLUCTUATIONS FOR A ZERO-RANGE PROCESS ON THE PERCOLATION
CLUSTER
Patricia Goncalves and Milton Jara
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
---------------------------------------------------------------
7122. COARSE GRAINING, FRACTIONAL MOMENTS AND THE CRITICAL SLOPE OF
RANDOM COPOLYMERS
F. Toninelli (Laboratoire de Physique and ENS Lyon and CNRS)
For a much-studied model of random copolymer at a selective interface we
prove that the slope of the critical curve in the weak-disorder limit is
strictly smaller than 1, which is the value given by the annealed
inequality.
The proof is based on a coarse-graining procedure, combined with upper
bounds
on the fractional moments of the partition function.
http://arxiv.org/abs/0806.0365
---------------------------------------------------------------
7123. EVEN WALKS AND ESTIMATES OF HIGH MOMENTS OF LARGE WIGNER RANDOM
MATRICES
O. Khorunzhiy and V. Vengerovsky
We revisit the problem of estimates of moments of random n-dimensional
matrices of Wigner ensemble by using the approach elaborated by Ya.
Sinai and
A. Soshnikov and further developed by A. Ruzmaikina. Our main subject
is given
by the structure of closed even walks and their graphs that arise in
these
studies. We show that the total degree of a vertex of such a graph
depends not
only on the self-intersections degree of but also on the total number
of all
non-closed instants of self-intersections of the walk. This result is
used to
fill the gaps of earlier considerations.
http://arxiv.org/abs/0806.0157
---------------------------------------------------------------
7124. CONDITIONS FOR EXISTENCE AND SMOOTHNESS OF THE DISTRIBUTION
DENSITY FOR AN ORNSTEIN-UHLENBECK PROCESS WITH LEVY NOISE
Semen V.Bodnarchuk and Alexey M.Kulik
Conditions are given, sufficient for the distribution of an
Ornstein-Uhlenbeck process with L\'evy noise to be absolutely
continuous or to
possess a smooth density. For the processes with non-degenerate drift
coefficient, these conditions are a necessary ones. A multidimensional
analogue
for the non-degeneracy condition on the drift coefficient is introduced.
http://arxiv.org/abs/0806.0442
---------------------------------------------------------------
7125. STABILITY OF THE LCD MODEL
Zhenting Hou and Li Tan and Dinghua Shi
In this paper, first-passage probability of Markov chains is used to
get a
strict proof of the existence of degree distribution of the LCD model
presented
by Bollobas (Random Structures and Algorithms 18(2001)). Also, a precise
expression of degree distribution is presented.
http://arxiv.org/abs/0806.0448
---------------------------------------------------------------
7126. ON SUMS OF CONDITIONALLY INDEPENDENT SUBEXPONENTIAL RANDOM
VARIABLES
Serguei Foss and Andrew Richards
The asymptotic tail-behaviour of sums of independent subexponential
random
variables is well understood, one of the main characteristics being
\textit{the
principle of the single big jump}. We study the case of dependent
subexponential random variables, for both deterministic and random
sums, using
a fresh approach, by considering conditional independence structures
on the
random variables. We seek sufficient conditions for the results of the
theory
with independent random variables still to hold. For a subexponential
distribution, we introduce the concept of a boundary class of
functions, which
we hope will be a useful tool in studying many aspects of
subexponential random
variables. The examples we give in the paper demonstrate a variety of
effects
owing to the dependence, and are also interesting in their own right.
http://arxiv.org/abs/0806.0490
---------------------------------------------------------------
7127. A DOUBLE PHASE TRANSITION ARISING FROM BROWNIAN ENTROPIC REPULSION
Itai Benjamini and Nathanael Berestycki
We analyze one-dimensional Brownian motion conditioned on a self-
repelling
behaviour. In the main result of this paper, it is shown that a double
phase
transition occurs when the growth of the local time at the origin is
constrained (in a suitable way) to be slower than the function f(t)=
\sqrt{t}(\log t)^{-c} at every time. In the subcritical phase (c<0), the
process is recurrent and the local time at 0 is diffusive. In the
intermediary
phase (0<c\le 1), the process is recurrent but the local time grows
much slower
than the constraint f. Finally in the supercritical phase (c>1), the
process
becomes transient. The proof exploits the Brownian entropic repulsion
phenomenon.
http://arxiv.org/abs/0806.0597
---------------------------------------------------------------
7128. SUBEQUIVALENCE RELATIONS AND POSITIVE-DEFINITE FUNCTIONS
A. Ioana and A.S. Kechris and and T. Tsankov
We study a positive-definite function associated to a measure-preserving
equivalence relation on a standard probability space and use it to
measure
quantitatively the proximity of subequivalence relations. This is
combined with
a recent co-inducing construction of Epstein to produce new kinds of
mixing
actions of an arbitrary infinite discrete group and it is also used to
show
that orbit equivalence of free, measure preserving, mixing actions of
non-amenable groups is unclassifiable in a strong sense. Finally, in
the case
of property (T) groups we discuss connections with invariant
percolation on
Cayley graphs and the calculation of costs.
http://arxiv.org/abs/0806.0430
---------------------------------------------------------------
7129. BACKWARD UNIQUENESS AND THE EXISTENCE OF THE SPECTRAL LIMIT FOR
SOME PARABOLIC SPDES
Z. Brze\'zniak and M. Neklyudov
The aim of this article is to study the asymptotic behaviour for large
times
of solutions to a certain class of stochastic partial differential
equations of
parabolic type. In particular, we will prove the backward uniqueness
result and
the existence of the spectral limit for abstract SPDEs and then show
how these
results can be applied to some concrete linear and nonlinear SPDEs. For
example, we will consider linear parabolic SPDEs with gradient noise and
stochastic NSEs with multiplicative noise. Our results generalize the
results
proved in Ghidaglia (1986) for deterministic PDEs.
One of the difficulties with extending the results from Ghidaglia
(1986) to
the stochastic case is that the standard It\^o formula is not directly
applicable to the case considered in this article. We use certain
approximations to overcome this problem.
Another difficulty is that conditions 1.3-1.4, p.779 in Ghidaglia
(1986) have
no natural counterpart in the stochastic case. We have only conditions
(1.11),
(1.12). As a result, we require rather strong assumptions on the
regularity of
solutions in the case of stochastic equations with quadratic
nonlinearity. In
the same time, this problem does not appear in the case of linear
stochastic
equations or if nonlinearity has no more than "linear growth".
http://arxiv.org/abs/0806.0616
---------------------------------------------------------------
7130. HOMOGENEOUS NUCLEATION FOR GLAUBER AND KAWASAKI DYNAMICS IN
LARGE VOLUMES AT LOW TEMPERATURES
Anton Bovier and Frank den Hollander and Cristian Spitoni
In this paper we study metastability in large volumes at low
temperatures. We
consider both Ising spins subject to Glauber spin-flip dynamics and
lattice gas
particles subject to Kawasaki hopping dynamics. Let $\b$ denote the
inverse
temperature and let $\L_\b \subset \Z^2$ be a square box with periodic
boundary
conditions such that $\lim_{\b\to\infty}|\L_\b|=\infty$. We run the
dynamics on
$\L_\b$ starting from a random initial configuration where all the
droplets (=
clusters of plus-spins, respectively, clusters of particles)are small.
For
large $\b$, and for interaction parameters that correspond to the
metastable
regime, we investigate how the transition from the metastable state
(with only
small droplets) to the stable state (with one or more large droplets)
takes
place under the dynamics. This transition is triggered by the
appearance of a
single \emph{critical droplet} somewhere in $\L_\b$. Using potential-
theoretic
methods, we compute the \emph{average nucleation time} (= the first
time a
critical droplet appears and starts growing) up to a multiplicative
factor that
tends to one as $\b\to\infty$. It turns out that this time grows as
$Ke^{\Gamma\b}/|\L_\b|$ for Glauber dynamics and $K\b e^{\Gamma\b}/|\L_
\b|$ for
Kawasaki dynamics, where $\Gamma$ is the local canonical, respectively,
grand-canonical energy to create a critical droplet and $K$ is a
constant
reflecting the geometry of the critical droplet, provided these times
tend to
infinity (which puts a growth restriction on $|\L_\b|$). The fact that
the
average nucleation time is inversely proportional to $|\L_\b|$ is
referred to
as \emph{homogeneous nucleation}, because it says that the critical
droplet for
the transition appears essentially independently in small boxes that
partition
$\L_\b$.
http://arxiv.org/abs/0806.0755
---------------------------------------------------------------
7131. ON SLOWDOWN AND SPEEDUP OF TRANSIENT RANDOM WALKS IN RANDOM
ENVIRONMENT
Alexander Fribergh and Nina Gantert and Serguei Popov
We consider one-dimensional random walks in random environment which are
transient to the right. Our main interest is in the study of the sub-
ballistic
regime, where at time n the particle is typically at a distance $O(n^
\kappa)$
from the origin, $\kappa\in(0,1)$. We investigate the probabilities of
moderate
deviations from this behavior. Specifically, we are interested in
quenched and
annealed probabilities of slowdown (at time n, the particle is in
$O(n^{\nu_0})$, $\nu_0\in (0,\kappa)$), and speedup (at time n, the
particle is
around $n^{\nu_1}$, $\nu_1\in (\kappa,1)$), for the current location
of the
particle and for the hitting times. Also, we study probabilities of
backtracking: at time n, the particle is located near $(-n^\nu)$, thus
making
an unusual excursion to the left. For the slowdown, our estimates are
valid in
the ballistic case as well.
http://arxiv.org/abs/0806.0790
---------------------------------------------------------------
7132. CONTINUOUS SPIN MEAN-FIELD MODELS: LIMITING KERNELS AND GIBBS
PROPERTIES OF LOCAL TRANSFORMS
C. Kuelske and A. A. Opoku
We extend the notion of Gibbsianness for mean-field systems to the set-
up of
general (possibly continuous) local state spaces. We investigate the
Gibbs
properties of systems arising from an initial mean-field Gibbs measure
by
application of given local transition kernels. This generalizes previous
case-studies made for spins taking finitely many values to the first
step in
direction to a general theory, containing the following parts: (1) A
formula
for the limiting conditional probability distributions of the
transformed
system. It holds both in the Gibbs and non-Gibbs regime and invokes a
minimization problem for a "constrained rate-function". (2) A
criterion for
Gibbsianness of the transformed system for initial Lipschitz-
Hamiltonians
involving concentration properties of the transition kernels. (3) A
continuity
estimate for the single-site conditional distributions of the
transformed
system. While (2) and (3) have provable lattice-counterparts, the
characterization of (1) is stronger in mean-field. As applications we
show
short-time Gibbsianness of rotator mean-field models on the (q-1)-
dimensional
sphere under diffusive time-evolution and the preservation of
Gibbsianness
under local coarse-graining of the initial local spin space.
http://arxiv.org/abs/0806.0802
---------------------------------------------------------------
7133. A CLASS OF NON HOMOGENEOUS SELF INTERACTING RANDOM PROCESSES
WITH APPLICATIONS TO LEARNING IN GAMES AND VERTEX-REINFORCED RANDOM
WALKS
Michel Benaim (UNINE) and Olivier Raimond (LM-Orsay)
Using an approximation by a set-valued dynamical system, this paper
studies a
class of non Markovian and non homogeneous stochastic processes on a
finite
state space. It provides an unified approach to simulated annealing type
processes. It permits to study new models of vertex reinforced random
walks and
new models of learning in games including Markovian fictitious play.
http://arxiv.org/abs/0806.0806
---------------------------------------------------------------
7134. FLUCTUATION BOUNDS FOR THE ASYMMETRIC SIMPLE EXCLUSION PROCESS
Marton Balazs and Timo Seppalainen
We give a partly new proof of the fluctuation bounds for the second
class
particle and current in the stationary asymmetric simple exclusion
process. One
novelty is a coupling that preserves the ordering of second class
particles in
two systems that are themselves ordered coordinatewise.
http://arxiv.org/abs/0806.0829
---------------------------------------------------------------
7135. THE HEIGHT AND RANGE OF WATERMELONS WITHOUT WALL
Thomas Feierl
We determine the weak limit of the distribution of the random variables
"height" and "range" on the set of p-watermelons without wall
restriction as
the number of steps tends to infinity. Additionally, we provide
asymptotics for
the moments of the random variable "height".
http://arxiv.org/abs/0806.0037
---------------------------------------------------------------
7136. ON RANDOMLY PLACED ARCS ON THE CIRCLE
Arnaud Durand
We completely describe in terms of Hausdorff measures the size of the
set of
points of the circle that are covered infinitely often by a sequence
of random
arcs with given lengths. We also show that this set is a set with large
intersection.
http://arxiv.org/abs/0806.0880
---------------------------------------------------------------
7137. REFLECTED SOLUTIONS OF BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL
EQUATIONS
Weiqiang Yang and Yufeng Shi and Yangling Gu
We study reflected solutions of one-dimensional backward doubly
stochastic
differential equations (BDSDEs in short). The ``reflected'' keeps the
solution
above a given stochastic process. We get the uniqueness and existence by
penalization. For the existence of backward stochastic integral, our
proof is
different from [KKPPQ] slightly. We also obtain a comparison theorem for
reflected BDSDEs.
http://arxiv.org/abs/0806.0917
---------------------------------------------------------------
7138. ASYMPTOTICS OF THE MAXIMAL RADIUS OF AN $L^R$-OPTIMAL SEQUENCE
OF QUANTIZERS
Gilles Pag\`es (PMA) and Abass Sagna (PMA)
Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with
an
Euclidean norm). Let $ r,s > 0 $ and assume $(\alpha_n)_{n \geq 1}$ is
an
(asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. In this
paper we
investigate the asymptotic behavior of the maximal radius sequence
induced by
the sequence $(\alpha_n)_{n \geq 1}$ and defined to be for every $n
\geq 1$, $\
\rho(\alpha_n) = \max \{| a |, a \in \alpha_n \}$. We show that if ${\rm
card(supp}(P))$ is infinite, the maximal radius sequence goes to $\sup
\{| x |,
x \in {\rm supp}(P) \}$ as $n$ goes to infinity. We then give the rate
of
convergence for two classes of distributions with unbounded support :
distributions with exponential tails and distributions with polynomial
tails.
http://arxiv.org/abs/0806.0918
---------------------------------------------------------------
7139. THE STOCHASTIC HAMILTON-JACOBI EQUATION
Joan-Andreu L\'azaro-Cam\'i and Juan-Pablo Ortega
We extend some aspects of the Hamilton-Jacobi theory to the category of
stochastic Hamiltonian dynamical systems. More specifically, we show
that the
stochastic action satisfies the Hamilton-Jacobi equation when, as in the
classical situation, it is written as a function of the configuration
space
using a regular Lagrangian submanifold. Additionally, we will use a
variation
of the Hamilton-Jacobi equation to characterize the generating
functions of
one-parameter groups of symplectomorphisms that allow to rewrite a given
stochastic Hamiltonian system in a form whose solutions are very easy
to find;
this result recovers the classical solutions by reduction to the
equilibrium of
a standard Hamiltonian system.
http://arxiv.org/abs/0806.0993
---------------------------------------------------------------
7140. COMPETITION BETWEEN DISCRETE RANDOM VARIABLES, WITH APPLICATIONS
TO OCCUPANCY PROBLEMS
Julia Eaton and Anant Godbole and Betsy Sinclair
Consider $n$ players whose "scores" are independent and identically
distributed values $\{X_i\}_{i=1}^n$ from some discrete distribution $F
$. We
pay special attention to the cases where (i) $F$ is geometric with
parameter
$p\to0$ and (ii) $F$ is uniform on $\{1,2,...,N\}$; the latter case
clearly
corresponds to the classical occupancy problem. The quantities of
interest to
us are, first, the $U$-statistic $W$ which counts the number of "ties"
between
pairs $i,j$; second, the univariate statistic $Y_r$, which counts the
number of
strict $r$-way ties between contestants, i.e., episodes of the form
${X_i}_1={X_i}_2=...={X_i}_r$; $X_j\ne {X_i}_1;j\ne i_1,i_2,...,i_r$;
and, last
but not least, the multivariate vector $Z_{AB}=(Y_A,Y_{A+1},...,Y_B)$.
We
provide Poisson approximations for the distributions of $W$, $Y_r$ and
$Z_{AB}$
under some general conditions. New results on the joint distribution
of cell
counts in the occupancy problem are derived as a corollary.
http://arxiv.org/abs/0806.1007
---------------------------------------------------------------
7141. SURVIVAL TIME OF RANDOM WALK IN RANDOM ENVIRONMENT AMONG SOFT
OBSTACLES
Nina Gantert and Serguei Popov and Marina Vachkovskaia
We consider a Random Walk in Random Environment (RWRE) moving in an
i.i.d.
random field of obstacles. When the particle hits an obstacle, it
disappears
with a positive probability. We obtain quenched and annealed bounds on
the
tails of the survival time in the general $d$-dimensional case. We then
consider a simplified one-dimensional model (where transition
probabilities and
obstacles are independent and the RWRE only moves to neighbour sites),
and
obtain finer results for the tail of the survival time. In addition,
we study
also the "mixed" probability measures (quenched with respect to the
obstacles
and annealed with respect to the transition probabilities and vice-
versa) and
give results for tails of the survival time with respect to these
probability
measures. Further, we apply the same methods to obtain bounds for the
tails of
the hitting times of Branching Random Walks in Random Environment
(BRWRE).
http://arxiv.org/abs/0806.1030
---------------------------------------------------------------
7142. THE FRACTIONAL LANGEVIN EQUATION: BROWNIAN MOTION REVISITED
Francesco Mainardi and Paolo Pironi
We have revisited the Brownian motion on the basis of the fractional
Langevin
equation which turns out to be a particular case of the generalized
Langevin
equation introduced by Kubo on 1966. The importance of our approach is
to model
the Brownian motion more realistically than the usual one based on the
classical Langevin equation, in that it takes into account also the
retarding
effects due to hydrodynamic backflow, i.e. the added mass and the
Basset memory
drag. On the basis of the two fluctuation-dissipation theorems and of
the
techniques of the Fractional Calculus we have provided the analytical
expressions of the correlation functions (both for the random force
and the
particle velocity) and of the mean squared particle displacement. The
random
force has been shown to be represented by a superposition of the usual
white
noise with a "fractional" noise. The velocity correlation function is
no longer
expressed by a simple exponential but exhibits a slower decay,
proportional to
$t^{-3/2}$ as $t \to \infty$, which indeed is more realistic. Finally,
the mean
squared displacement has been shown to maintain, for sufficiently long
times,
the linear behaviour which is typical of normal diffusion, with the same
diffusion coefficient of the classical case. However, the Basset
history force
induces a retarding effect in the establishing of the linear
behaviour, which
in some cases could appear as a manifestation of anomalous diffusion
to be
correctly interpreted in experimental measurements.
http://arxiv.org/abs/0806.1010
---------------------------------------------------------------
7143. NEGATIVE ENTROPY, PRESSURE AND ZERO TEMPERATURE: A L.D.P. FOR
STATIONARY MARKOV CHAINS ON [0,1]
Artur O. Lopes and Joana Mohr and Rafael R. Souza
We analyze some properties of maximizing stationary Markov
probabilities on
the Bernoulli space $[0,1]^\mathbb{N}$, More precisely, we consider
ergodic
optimization for a continuous potential $A$, where $A: [0,1]^\mathbb{N}
\to
\mathbb{R}$ which depends only on the two first coordinates. We are
interested
in finding stationary Markov probabilities $\mu_\infty$ on $ [0,1]^
\mathbb{N}$
that maximize the value $ \int A d \mu,$ among all stationary Markov
probabilities $\mu$ on $[0,1]^\mathbb{N}$. This problem correspond in
Statistical Mechanics to the zero temperature case for the interaction
described by the potential $A$. The main purpose of this paper is to
show,
under the hypothesis of uniqueness of the maximizing probability, a
Large
Deviation Principle for a family of absolutely continuous Markov
probabilities
$\mu_\beta$ which weakly converges to $\mu_\infty$. The probabilities
$\mu_\beta$ are obtained via an information we get from a Perron
operator and
they satisfy a variational principle similar to the pressure. Under the
hypothesis of $A$ being $C^2$ and the twist condition, that is,
$\frac{\partial^2 A}{\partial_x \partial_y} (x,y) \neq 0$, for all $
(x,y) \in
[0,1]^2$, we show the graph property.
http://arxiv.org/abs/0806.1012
---------------------------------------------------------------
7144. BREAKING THE CHAIN
Michael Allman and Volker Betz
We consider the motion of a Brownian particle in $\mathbb{R}$, moving
between
a particle fixed at the origin and another moving deterministically
away at
slow speed $\epsilon>0$. The middle particle interacts with its
neighbours via
a potential of finite range $b>0$, with a unique minimum at $a>0$, where
$b<2a$. We say that the chain of particles breaks on the left- or
right-hand
side when the middle particle is greater than a distance $b$ from its
left or
right neighbour, respectively. We study the asymptotic location of the
first
break of the chain in the limit of small noise, in the case where $
\epsilon =
\epsilon(\sigma)$ and $\sigma>0$ is the noise intensity.
http://arxiv.org/abs/0806.1163
---------------------------------------------------------------
7145. ASYMPTOTICS OF POSTERIORS FOR BINARY BRANCHING PROCESSES
Didier Piau (IF)
We compute the posterior distributions of the initial population and
parameter of binary branching processes, in the limit of a large
number of
generations. We compare this Bayesian procedure with a more na\"ive
one, based
on hitting times of some random walks. In both cases, central limit
theorems
are available, with explicit variances.
http://arxiv.org/abs/0806.1173
---------------------------------------------------------------
7146. LIMIT THEOREMS FOR SAMPLE EIGENVALUES IN A GENERALIZED SPIKED
POPULATION MODEL
Zhidong Bai (KLASMOE and Dsap) and Jian-Feng Yao (IRMAR)
In the spiked population model introduced by Johnstone (2001),the
population
covariance matrix has all its eigenvalues equal to unit except for a
few fixed
eigenvalues (spikes). The question is to quantify the effect of the
perturbation caused by the spike eigenvalues. Baik and Silverstein
(2006)
establishes the almost sure limits of the extreme sample eigenvalues
associated
to the spike eigenvalues when the population and the sample sizes
become large.
In a recent work (Bai and Yao, 2008), we have provided the limiting
distributions for these extreme sample eigenvalues. In this paper, we
extend
this theory to a {\em generalized} spiked population model where the
base
population covariance matrix is arbitrary, instead of the identity
matrix as in
Johnstone's case. New mathematical tools are introduced for
establishing the
almost sure convergence of the sample eigenvalues generated by the
spikes.
http://arxiv.org/abs/0806.1141
---------------------------------------------------------------
7147. MONOTONE LOOP MODELS AND RATIONAL RESONANCE
Alan Hammond and Richard Kenyon
Let $T_{n,m}=\mathbb Z_n\times\mathbb Z_m$, and define a random mapping
$\phi\colon T_{n,m}\to T_{n,m}$ by $\phi(x,y)=(x+1,y)$ or $(x,y+1)$
independently over $x$ and $y$ and with equal probability. We study
the orbit
structure of such ``quenched random walks'' $\phi$ in the limit $m,n\to
\infty$,
and show how it depends sensitively on the ratio $m/n$. For $m/n$ near a
rational $p/q$, we show that there are likely to be on the order of $
\sqrt{n}$
cycles, each of length O(n), whereas for $m/n$ far from any rational
with small
denominator, there are a bounded number of cycles, and for typical $m/n
$ each
cycle has length on the order of $n^{4/3}$.
http://arxiv.org/abs/0806.1236
---------------------------------------------------------------
7148. SEMIGROUP INEQUALITIES, STOCHASTIC DOMINATION, HARDY'S
INEQUALITY, AND STRONG ERGODICITY
Carl Graham (CMAP)
For the classical Lp spaces of signed measures on the integers, we
devise a
framework in which bounds for a sub-Markovian semigroup of interest
can be
obtained, up to a constant factor, from bounds for another tractable
semigroup
that dominates stochastically the first one. The main tools are the
Hardy
inequality, the definition of related auxiliary Lp spaces suited to take
advantage of the domination, and the proof that the norms are
equivalent to the
classical ones if the reference measure is quasi-geometrically
decreasing. We
illustrate the results using birth-death and single-birth processes.
http://arxiv.org/abs/0806.1263
---------------------------------------------------------------
7149. COMPUTING EXPECTED TRANSITION EVENTS IN REDUCIBLE MARKOV CHAINS
Brian D. Ewald and Jeffrey Humpherys and Jeremy West
We present a method for computing the expected number of times certain
transition events occur during the transient phase of a reducible
Markov chain.
Examples of events include time to absorption, number of visits to a
state,
traversals of a particular transition, loops from a state to itself, and
arrivals to a state from a particular subset of states.
http://arxiv.org/abs/0806.1291
---------------------------------------------------------------
7150. KPZ IN ONE DIMENSIONAL RANDOM GEOMETRY OF MULTIPLICATIVE CASCADES
Itai Benjamini and Oded Schramm
We prove a formula relating the Hausdorff dimension of a subset of the
unit
interval and the Hausdorff dimension of the same set with respect to a
random
path matric on the interval, which is generated using a multiplicative
cascade.
When the random variables generating the cascade are exponentials of
Gaussians,
the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov
from quantum
gravity appears. This note was inspired by the recent work of
Duplantier and
Sheffield proving a somewhat different version of the KPZ formula for
Liouville
gravity. In contrast with the Liouville gravity setting, the one
dimensional
multiplicative cascade framework facilitates the determination of the
Hausdorff
dimension, rather than some expected box count dimension.
http://arxiv.org/abs/0806.1347
---------------------------------------------------------------
7151. THE ALEXANDER-ORBACH CONJECTURE HOLDS IN HIGH DIMENSIONS
Gady Kozma and Asaf Nachmias
We examine the incipient infinite cluster (IIC) of critical
percolation in
regimes where mean-field behavior have been established, namely when the
dimension d is large enough or when d>6 and the lattice is
sufficiently spread
out. We find that random walk on the IIC exhibits anomalous diffusion
with the
spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This
establishes
a conjecture of Alexander and Orbach. En route we calculate the one-arm
exponent with respect to the intrinsic distance.
http://arxiv.org/abs/0806.1442
---------------------------------------------------------------
7152. THE END OF SLEEPING BEAUTY'S NIGHTMARE
Berry Groisman
The way a rational agent changes her belief in certain
propositions/hypotheses in the light of new evidence lies at the heart
of
Bayesian inference. The basic natural assumption, as summarized in van
Fraassen's Reflection Principle ([1984]), would be that in the absence
of new
evidence the belief should not change. Yet, there are examples that
are claimed
to violate this assumption. The apparent paradox presented by such
examples, if
not settled, would demonstrate the inconsistency and/or incompleteness
of the
Bayesian approach and without eliminating this inconsistency, the
approach
cannot be regarded as scientific.
The Sleeping Beauty Problem is just such an example. The existing
attempts to
solve the problem fall into three categories. The first two share the
view that
new evidence is absent, but differ about the conclusion of whether
Sleeping
Beauty should change her belief or not, and why. The third category is
characterized by the view that, after all, new evidence (although
hidden from
the initial view) is involved.
My solution is radically different and does not fall in either of
these
categories. I deflate the paradox by arguing that the two different
degrees of
belief presented in the Sleeping Beauty Problem are in fact beliefs in
two
different propositions, i.e. there is no need to explain the
(un)change of
belief.
http://arxiv.org/abs/0806.1316
---------------------------------------------------------------
7153. ON THE NUMBER OF MATRICES AND A RANDOM MATRIX WITH PRESCRIBED
ROW AND COLUMN SUMS AND 0-1 ENTRIES
Alexander Barvinok
We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries
and
prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ...,
c_n). We
prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the
solution
to a convex optimization problem. We show that if Sigma(R, C) is
sufficiently
large, then a random matrix D in Sigma(R, C) sampled from the uniform
probability measure in Sigma(R,C) is close to a particular matrix
Z=Z(R,C) that
maximizes the entropy among all non-negative matrices with row sums R
and
column sums C. Similar results are obtained for 0-1 matrices with
prescribed
row and column sums and assigned zeros in some positions.
http://arxiv.org/abs/0806.1480
---------------------------------------------------------------
7154. IT\^O'S FORMULA FOR THE $L_{P}$-NORM OF STOCHASTIC $W^{1}_{P}$-
VALUED PROCESSES
N.V. Krylov
We prove It\^o's formula for the $L_{p}$-norm of a stochastic
$W^{1}_{p}$-valued processes appearing in the theory of SPDEs in
divergence
form.
http://arxiv.org/abs/0806.1557
---------------------------------------------------------------
7155. SLOW DECORRELATIONS IN KPZ GROWTH
Patrik L. Ferrari (Weierstrass Institute and WIAS-Berlin)
For stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class in
1+1
dimensions, fluctuations grow as t^{1/3} during time t and the
correlation
length at a fixed time scales as t^{2/3}. In this note we discuss the
scale of
time correlations. It turns out that the space-time is non-trivially
fibred,
having slow directions having decorrelation exponent equal to 1
instead of the
usual 2/3. These directions are the characteristic curves of the PDE
associated
to the surface's slope. As a consequence, previously proven results for
space-like paths will hold in the whole space-time except along the slow
curves.
http://arxiv.org/abs/0806.1350
---------------------------------------------------------------
7156. DISTAL ACTIONS AND SHIFTED CONVOLUTION PROPERTY
C. R. E. Raja and R. Shah
A locally compact group $G$ is said to have shifted convolution property
(abbr. as SCP) if for every regular Borel probability measure $\mu$ on
$G$,
either $\sup_{x\in G} \mu ^n (Cx) \ra 0$ for all compact subsets $C$
of $G$, or
there exist $x\in G$ and a compact subgroup $K$ normalised by $x$ such
that
$\mu^nx^{-n} \ra \omega_K$, the Haar measure on $K$. We first consider
distality of factor actions of distal actions. It is shown that this
holds in
particular for factors under compact groups invariant under the action
and for
factors under the connected component of identity. We then
characterize groups
having SCP in terms of a readily verifiable condition on the
conjugation action
(point-wise distality). This has some interesting corollaries to
distality of
certain actions and Choquet Deny measures which actually motivated SCP
and
point-wise distal groups. We also relate distality of actions on
groups to that
of the extensions on the space of probability measures.
http://arxiv.org/abs/0806.1820
---------------------------------------------------------------
7157. STOCHASTIC EQUATIONS WITH DELAY: OPTIMAL CONTROL VIA BSDES AND
REGULAR SOLUTIONS OF HAMILTON-JACOBI-BELLMAN EQUATIONS
Marco Fuhrman and Federica Masiero and Gianmario Tessitore
We consider an Ito stochastic differential equation with delay, driven
by
brownian motion, whose solution, by an appropriate reformulation,
defines a
Markov process $X$ with values in a space of continuous functions $
\mathbf C$,
with generator $\mathcal L$. We then consider a backward stochastic
differential equation depending on $X$, with unknown processes $(Y,Z)
$, and we
study properties of the resulting system, in particular we identify
the process
$Z$ as a deterministic functional of $X$. We next prove that the
forward-backward system provides a suitable solution to a class of
parabolic
partial differential equations on the space $\mathbf C$ driven by $
\mathcal L$,
and we apply this result to prove a characterization of the fair price
and the
hedging strategy for a financial market with memory effects. We also
include
applications to optimal stochastic control of differential equation
with delay:
in particular we characterize optimal controls as feedback laws in
terms the
process $X$.
http://arxiv.org/abs/0806.1837
---------------------------------------------------------------
7158. THE STOCHASTIC HEAT EQUATION DRIVEN BY A GAUSSIAN NOISE: GERM
MARKOV PROPERTY
Raluca Balan and Doyoon Kim
Let $u=\{u(t,x);t \in [0,T], x \in {\mathbb{R}}^{d}\}$ be the process
solution of the stochastic heat equation $u_{t}=\Delta u+ \dot F,
u(0,\cdot)=0$
driven by a Gaussian noise $\dot F$, which is white in time and has
spatial
covariance induced by the kernel $f$. In this paper we prove that the
process
$u$ is locally germ Markov, if $f$ is the Bessel kernel of order $
\alpha=2k,k
\in \bN_{+}$, or $f$ is the Riesz kernel of order $\alpha=4k,k \in
\bN_{+}$.
http://arxiv.org/abs/0806.1898
---------------------------------------------------------------
7159. THE MIXING TIME EVOLUTION OF GLAUBER DYNAMICS FOR THE MEAN-FIELD
ISING MODEL
Jian Ding and Eyal Lubetzky and Yuval Peres
We consider Glauber dynamics for the Ising model on the complete graph
on $n$
vertices, known as the Curie-Weiss model. It is well-known that the
mixing-time
in the high temperature regime ($\beta < 1$) has order $n\log n$,
whereas the
mixing-time in the case $\beta > 1$ is exponential in $n$. Recently,
Levin,
Luczak and Peres proved that for any fixed $\beta < 1$ there is cutoff
at time
$[2(1-\beta)]^{-1} n\log n$ with a window of order $n$, whereas the
mixing-time
at the critical temperature $\beta=1$ is $\Theta(n^{3/2})$. It is
natural to
ask how the mixing-time transitions from $\Theta(n\log n)$ to $
\Theta(n^{3/2})$
and finally to $\exp(\Theta(n))$. That is, how does the mixing-time
behave when
$\beta=\beta(n)$ is allowed to tend to 1 as $n\to\infty$.
In this work, we obtain a complete characterization of the mixing-
time of the
dynamics as a function of the temperature, as it approaches its
critical point
$\beta_c=1$. In particular, we find a scaling window of order
$1/\sqrt{n}$
around the critical temperature. In the high temperature regime, $
\beta = 1 -
\delta$ for some $0 < \delta < 1$ so that $\delta^2 n \to\infty$ with
$n$, the
mixing-time has order $(n/\delta)\log(\delta^2 n)$, and exhibits
cutoff with
constant 1/2 and window size $n/\delta$. In the critical window, $
\beta = 1\pm
\delta$ where $\delta^2 n$ is O(1), there is no cutoff, and the mixing-
time has
order $n^{3/2}$. At low temperature, $\beta = 1 + \delta$ for $\delta
> 0$ with
$\delta^2 n \to\infty$ and $\delta=o(1)$, there is no cutoff, and the
mixing
time has order $(n/\delta)\exp(({3/4}+o(1))\delta^2 n)$.
http://arxiv.org/abs/0806.1906
---------------------------------------------------------------
7160. QUASI-STATIONARY RANDOM OVERLAP STRUCTURES AND THE CONTINUOUS
CASCADES
Jason Miller
A random overlap structure (ROSt) is a measure on pairs (X,Q) where X
is a
locally finite sequence in the real line with a maximum and Q a positive
semidefinite matrix of overlaps intrinsic to the particles X. Such a
measure is
said to be quasi-stationary provided that the joint law of the gaps of
X and
overlaps Q is stable under a stochastic evolution driven by a Gaussian
sequence
with covariance Q. Aizenman et al. have shown that quasi-stationary
ROSts serve
as an important computational tool in the study of the Sherrington-
Kirkpatrick
(SK) spin-glass model from the perspective of cavity dynamics and the
related
ROSt variational principle for its free energy. In this framework, the
Parisi
solution is reflected in the ansatz that the overlap matrix exhibit a
certain
hierarchical structure. Aizenman et al. have posed the question of
whether the
ansatz could be explained by showing that the only ROSts that are
quasi-stationary in a robust sense are given by a special class of
hierarchical
ROSts known as both the Ruelle Probability Cascades as well the GREM.
Arguin
and Aizenman have given an affirmative answer in the special case that
the set
of values S_Q taken on by the entries of Q is finite. We prove that
this result
holds even when |S_Q| is infinite provided that Q satisfies the
technical
condition that the closure of S_Q has no limit points from below. This
is
relevant to the understanding of the ground states of the SK model, as
they
satisfy |S_Q| = infinity.
http://arxiv.org/abs/0806.1915
---------------------------------------------------------------
7161. ON DIVERGENCE FORM SPDES WITH VMO COEFFICIENTS
N.V. Krylov
We present several results on solvability in Sobolev spaces $W^{1}_{p}
$ of
SPDEs in divergence form in the whole space.
http://arxiv.org/abs/0806.1925
---------------------------------------------------------------
7162. ON DIVERGENCE FORM SPDES WITH VMO COEFFICIENTS IN A HALF SPACE
N.V. Krylov
We extend several known results on solvability in the Sobolev spaces
$W^{1}_{p}$, $p\in[2,\infty)$, of SPDEs in divergence form in $
\bR^{d}_{+}$ to
equations having coefficients which are discontinuous in the space
variable.
http://arxiv.org/abs/0806.1963
---------------------------------------------------------------
7163. STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
Z.-Q. Chen and P. J. Fitzsimmons and K. Kuwae and T.-S. Zhang
Using time-reversal, we introduce a stochastic integral for zero-energy
additive functionals of symmetric Markov processes, extending earlier
work of
S. Nakao. Various properties of such stochastic integrals are
discussed and an
It\^{o} formula for Dirichlet processes is obtained.
http://arxiv.org/abs/0806.2044
---------------------------------------------------------------
7164. SWITCHING GAMES OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
Ying Hu (IRMAR) and Shanjian Tang (SCHOOL of Mathematical Sciences)
In this paper, we study the switching game of one-dimensional backward
stochastic differential equations (BSDEs). This gives rise to a new
type of
multi-dimensional obliquely reflected BSDEs, which is a system of BSDEs
reflected on the boundary of a special unbounded convex domain along
an oblique
direction. The existence of the adapted solution is obtained by the
penalization method, the monotone convergence, and the a priori
estimations.
The uniqueness is obtained by a verification method (the first
component of any
adapted solution is shown to be the vector value of a switching
problem for
Reflected BSDEs). Finally, we show the existence of both the value and
the
saddle point for the switching game. More specifically, we prove that
the value
process of the switching game is given by the first component of the
solution
of the multi-dimensional obliquely reflected BSDEs and the saddle
point can
also be constructed using the latter.
http://arxiv.org/abs/0806.2058
---------------------------------------------------------------
7165. STEIN'S METHOD AND CHARACTERS OF COMPACT LIE GROUPS
Jason Fulman
Stein's method is used to study the trace of a random element from a
compact
Lie group or symmetric space. Central limit theorems are proved using
very
little information: character values on a single element and the
decomposition
of the square of the trace into irreducible components. This is
illustrated for
Lie groups of classical type and Dyson's circular ensembles. The
approach in
this paper will be useful for the study of higher dimensional
characters, where
normal approximations need not hold.
http://arxiv.org/abs/0806.2168
---------------------------------------------------------------
7166. THE ORIENTED SWAP PROCESS
Omer Angel and Alexander Holroyd and Dan Romik
Particles labelled $1,...,n$ are arranged initially in increasing order.
Subsequently, each pair of neighbouring particles that is currently in
increasing order swaps according to a Poisson process of rate 1. We
analyze the
asymptotic behaviour of this process as $n$ goes to infinity. We prove
that the
space-time trajectories of individual particles converge (when
suitably scaled)
to a certain family of random curves with two points of non-
differentiability,
and that the permutation matrix at a given time converges to a certain
deterministic measure with absolutely continuous and singular parts. The
absorbing state (where all particles are in decreasing order) is
reached at
time $(2+o(1))n$. The finishing times of individual particles converge
to
deterministic limits with fluctuations asymptotically governed by the
Tracy-Widom distribution.
http://arxiv.org/abs/0806.2222
---------------------------------------------------------------
7167. TREE-VALUED RESAMPLING DYNAMICS: MARTINGALE PROBLEMS AND
APPLICATIONS
Andreas Greven and Peter Pfaffelhuber and Anita Winter
The measure-valued Fleming-Viot process is a diffusion which models the
evolution of allele frequencies in a multi-type population of constant
size. In
the neutral setting the Kingman coalescent is known to generate the
genealogies
of the "individuals" in the population at a fixed time. The goal of
the present
paper is to replace this static point of view on the genealogies by an
analysis
of the evolving of genealogies. We use well-posed martingale problems to
construct the tree-valued resampling dynamics for both the finite
population
(tree-valued Moran dynamics) and the infinite population (tree-valued
Fleming-Viot dynamics). We show that the tree-valued Moran dynamics
converge in
the limit of large populations to the tree-valued Fleming-Viot
dynamics. In the
long-term behavior we derive the Kingman coalescent measure tree as the
equilibrium. As an application we study the evolution of the
distribution of
the length of the tree spanned by sequentially sampled "individuals".
http://arxiv.org/abs/0806.2224
---------------------------------------------------------------
7168. A CHANGE OF VARIABLE FORMULA FOR THE 2D FRACTIONAL BROWNIAN
MOTION OF HURST INDEX BIGGER OR EQUAL TO 1/4
Ivan Nourdin (PMA)
We prove a change of variable formula for the 2D fractional Brownian
motion
of index H bigger of equal to 1/4. For H strictly bigger than 1/4, our
formula
coincides with that obtained by using the rough paths theory. For
H=1/4 (the
more interesting case), there is an additional term that is a
classical Wiener
integral against an independent standard Brownian motion.
http://arxiv.org/abs/0806.2248
---------------------------------------------------------------
7169. STRONG ASYMMETRIC LIMIT OF THE QUASI-POTENTIAL OF THE BOUNDARY
DRIVEN WEAKLY ASYMMETRIC EXCLUSION PROCESS
Lorenzo Bertini and Davide Gabrielli and Claudio Landim
We consider the weakly asymmetric exclusion process on a bounded
interval
with particles reservoirs at the endpoints. The hydrodynamic limit for
the
empirical density, obtained in the diffusive scaling, is given by the
viscous
Burgers equation with Dirichlet boundary conditions. In the case in
which the
bulk asymmetry is in the same direction as the drift due to the boundary
reservoirs, we prove that the quasi-potential can be expressed in
terms of the
solution to a one-dimensional boundary value problem which has been
introduced
by Enaud and Derrida \cite{de}. We consider the strong asymmetric
limit of the
quasi-potential and recover the functional derived by Derrida,
Lebowitz, and
Speer \cite{DLS3} for the asymmetric exclusion process.
http://arxiv.org/abs/0806.2296
---------------------------------------------------------------
7170. LOCAL BOOTSTRAP PERCOLATION
Janko Gravner and Alexander E. Holroyd
We study a variant of bootstrap percolation in which growth is
restricted to
a single active cluster. Initially there is a single active site at
the origin,
while other sites of Z^2 are independently occupied with small
probability p,
otherwise empty. Subsequently, an empty site becomes active by contact
with 2
or more active neighbors, and an occupied site becomes active if it
has an
active site within distance 2. We prove that the entire lattice
becomes active
with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 +
c sqrt p
and pi^2/9 + C sqrt p (-log p)^3. This corrects previous numerical
predictions
for the scaling of the correction term.
http://arxiv.org/abs/0806.2313
---------------------------------------------------------------
7171. SPECIAL POINTS OF THE BROWNIAN NET
Emmanuel Schertzer and Rongfeng Sun and and Jan M. Swart
The Brownian net, which has recently been introduced by Sun and Swart
[SS08],
and independently by Newman, Ravishankar and Schertzer [NRS08],
generalizes the
Brownian web by allowing branching. In this paper, we study the
structure of
the Brownian net in more detail. In particular, we give an almost sure
classification of each point in $R^2$ according to the configuration
of the
Brownian net paths entering and leaving the point. Along the way, we
establish
various other structural properties of the Brownian net.
http://arxiv.org/abs/0806.2326
---------------------------------------------------------------
7172. A SHARP UNIFORM BOUND FOR THE DISTRIBUTION OF A SUM OF BERNOULLI
RANDOM VARIABLES
Roberto Cominetti and Jose Vaisman
We establish a uniform bound for the distribution of a sum $S^n=X_1+...
+X_n$
of independent non-homogeneous Bernoulli random variables with
$P(X_i=1)=p_i$.
Specifically, we prove that $\sigma^n P(S^n=i)\leq M$ where $\sigma^n$
denotes
the standard deviation of $S^n$ and the constant $M~0.4688$ is the
maximum of
$u\mapsto\sqrt{2u} e^{-2u}\sum_{k=0}^\infty({u^k\over k!})^2$.
http://arxiv.org/abs/0806.2350
---------------------------------------------------------------
7173. RELATIONS BETWEEN INVASION PERCOLATION AND CRITICAL PERCOLATION
IN TWO DIMENSIONS
Michael Damron and Art\"em Sapozhnikov and B\'alint V\'agv\"olgyi
We study invasion percolation in two dimensions. We compare connectivity
properties of the origin's invaded region to those of (a) the critical
percolation cluster of the origin and (b) the incipient infinite
cluster. To
exhibit similarities, we show that for any k > 0 the k-point function
of the
first pond has the same asymptotic behaviour as the probability that k
points
are in the critical cluster of the origin. More prominent, though, are
the
differences. We show that there are infinitely many ponds that contain
many
large disjoint p_c-open clusters. Further, for k > 1, we compute the
exact
decay rate of the distribution of the radius of the kth pond and see
that it is
strictly different than that of the radius of the critical cluster of
the
origin. We finish by showing that the invasion percolation measure and
the
incipient infinite cluster measure are mutually singular.
http://arxiv.org/abs/0806.2425
---------------------------------------------------------------
7174. UNIVERSAL STRUCTURES IN SOME MEAN FIELD SPIN GLASSES, AND AN
APPLICATION
Erwin Bolthausen and Nicola Kistler
We discuss a spin glass reminiscent of the Random Energy Model, which
allows
in particular to recast the Parisi minimization into a more classical
Gibbs
variational principle, thereby shedding some light on the physical
meaning of
the order parameter of the Parisi theory. As an application, we study
the
impact of an extensive cavity field on Derrida's REM: Despite its
simplicity,
this model displays some interesting features such as ultrametricity
and chaos
in temperature.
http://arxiv.org/abs/0806.2446
---------------------------------------------------------------
7175. CENTRAL LIMIT THEOREMS FOR EIGENVALUES IN A SPIKED POPULATION
MODEL
Zhidong Bai and Jian-feng Yao
In a spiked population model, the population covariance matrix has all
its
eigenvalues equal to units except for a few fixed eigenvalues
(spikes). This
model is proposed by Johnstone to cope with empirical findings on
various data
sets. The question is to quantify the effect of the perturbation
caused by the
spike eigenvalues. A recent work by Baik and Silverstein establishes
the almost
sure limits of the extreme sample eigenvalues associated to the spike
eigenvalues when the population and the sample sizes become large.
This paper
establishes the limiting distributions of these extreme sample
eigenvalues. As
another important result of the paper, we provide a central limit
theorem on
random sesquilinear forms.
http://arxiv.org/abs/0806.2503
---------------------------------------------------------------
7176. QUENCHED LAW OF LARGE NUMBERS FOR BRANCHING BROWNIAN MOTION IN A
RANDOM MEDIUM
J\'anos Engl\"ander
We study a spatial branching model, where the underlying motion is
$d$-dimensional ($d\ge1$) Brownian motion and the branching rate is
affected by
a random collection of reproduction suppressing sets dubbed mild
obstacles. The
main result of this paper is the quenched law of large numbers for the
population for all $d\ge1$. We also show that the branching Brownian
motion
with mild obstacles spreads less quickly than ordinary branching
Brownian
motion by giving an upper estimate on its speed. When the underlying
motion is
an arbitrary diffusion process, we obtain a dichotomy for the quenched
local
growth that is independent of the Poissonian intensity. More general
offspring
distributions (beyond the dyadic one considered in the main theorems)
as well
as mild obstacle models for superprocesses are also discussed.
http://arxiv.org/abs/0806.2512
---------------------------------------------------------------
7177. HOMOGENIZATION OF A SINGULAR RANDOM ONE-DIMENSIONAL PDE
Bogdan Iftimie and \'Etienne Pardoux and Andrey Piatnitski
This paper deals with the homogenization problem for a one-dimensional
parabolic PDE with random stationary mixing coefficients in the
presence of a
large zero order term. We show that under a proper choice of the
scaling factor
for the said zero order terms, the family of solutions of the studied
problem
converges in law, and describe the limit process. It should be noted
that the
limit dynamics remain random.
http://arxiv.org/abs/0806.2518
---------------------------------------------------------------
7178. THE QUENCHED INVARIANCE PRINCIPLE FOR RANDOM WALKS IN RANDOM
ENVIRONMENTS ADMITTING A BOUNDED CYCLE REPRESENTATION
Jean-Dominique Deuschel and Holger K\"osters
We derive a quenched invariance principle for random walks in random
environments whose transition probabilities are defined in terms of
weighted
cycles of bounded length. To this end, we adapt the proof for random
walks
among random conductances by Sidoravicius and Sznitman (Probab. Theory
Related
Fields 129 (2004) 219--244) to the non-reversible setting.
http://arxiv.org/abs/0806.2525
---------------------------------------------------------------
7179. ON THE SMALL DEVIATION PROBLEM FOR SOME ITERATED PROCESSES
Frank Aurzada and Mikhail Lifshits
We derive general results on the small deviation behavior for some
classes of
iterated processes. This allows us, in particular, to calculate the
rate of the
small deviations for $n$-iterated Brownian motions and, more
generally, for the
iteration of $n$ fractional Brownian motions. We also give a new and
correct
proof of some results in E. Nane, Laws of the iterated logarithm for
$\alpha$-time Brownian motion, Electron. J. Probab. 11 (2006), no. 18,
434--459.
http://arxiv.org/abs/0806.2559
---------------------------------------------------------------
7180. ON A CLASS OF OPTIMAL STOPPING PROBLEMS FOR DIFFUSIONS WITH
DISCONTINUOUS COEFFICIENTS
Ludger R\"uschendorf and Mikhail A. Urusov
In this paper, we introduce a modification of the free boundary problem
related to optimal stopping problems for diffusion processes. This
modification
allows the application of this PDE method in cases where the usual
regularity
assumptions on the coefficients and on the gain function are not
satisfied. We
apply this method to the optimal stopping of integral functionals with
exponential discount of the form $E_x\int_0^{\tau}e^{-\lambda s}f(X_s)
ds$,
$\lambda\ge0$ for one-dimensional diffusions $X$. We prove a general
verification theorem which justifies the modified version of the free
boundary
problem. In the case of no drift and discount, the free boundary
problem allows
to give a complete and explicit discussion of the stopping problem.
http://arxiv.org/abs/0806.2561
---------------------------------------------------------------
7181. OPTIMAL INVESTMENT AND CONSUMPTION IN A BLACK--SCHOLES MARKET
WITH L\'EVY-DRIVEN STOCHASTIC COEFFICIENTS
{\L}ukasz Delong and Claudia Kl\"uppelberg
In this paper, we investigate an optimal investment and consumption
problem
for an investor who trades in a Black--Scholes financial market with
stochastic
coefficients driven by a non-Gaussian Ornstein--Uhlenbeck process. We
assume
that an agent makes investment and consumption decisions based on a
power
utility function. By applying the usual separation method in the
variables, we
are faced with the problem of solving a nonlinear (semilinear) first-
order
partial integro-differential equation. A candidate solution is derived
via the
Feynman--Kac representation. By using the properties of an operator
defined in
a suitable function space, we prove uniqueness and smoothness of the
solution.
Optimality is verified by applying a classical verification theorem.
http://arxiv.org/abs/0806.2570
---------------------------------------------------------------
7182. A UNIFIED FRAMEWORK FOR UTILITY MAXIMIZATION PROBLEMS: AN ORLICZ
SPACE APPROACH
Sara Biagini and Marco Frittelli
We consider a stochastic financial incomplete market where the price
processes are described by a vector-valued semimartingale that is
possibly
nonlocally bounded. We face the classical problem of utility
maximization from
terminal wealth, with utility functions that are finite-valued over
$(a,\infty)$, $a\in\lbrack-\infty,\infty)$, and satisfy weak regularity
assumptions. We adopt a class of trading strategies that allows for
stochastic
integrals that are not necessarily bounded from below. The embedding
of the
utility maximization problem in Orlicz spaces permits us to formulate
the
problem in a unified way for both the cases $a\in\mathbb{R}$ and $a=-
\infty$.
By duality methods, we prove the existence of solutions to the primal
and dual
problems and show that a singular component in the pricing functionals
may also
occur with utility functions finite on the entire real line.
http://arxiv.org/abs/0806.2582
---------------------------------------------------------------
7183. PALM DISTRIBUTIONS OF WAVE CHARACTERISTICS IN ENCOUNTERING SEAS
Sofia Aberg and Igor Rychlik and M. Ross Leadbetter
Distributions of wave characteristics of ocean waves, such as wave
slope,
waveheight or wavelength, are an important tool in a variety of
oceanographic
applications such as safety of ocean structures or in the study of ship
stability, as will be the focus in this paper. We derive Palm
distributions of
several wave characteristics that can be related to steepness of waves
for two
different cases, namely for waves observed along a line at a fixed
time point
and for waves encountering a ship sailing on the ocean. The relation
between
the distributions obtained in the two cases is also given physical
interpretation in terms of a ``Doppler shift'' that is related to the
velocity
of the ship and the velocities of the individual waves.
http://arxiv.org/abs/0806.2718
---------------------------------------------------------------
7184. CONDITIONALLY IDENTICALLY DISTRIBUTED SPECIES SAMPLING SEQUENCES
Federico Bassetti and Irene Crimaldi and Fabrizio Leisen
Conditional identity in distribution (Berti et al. (2004)) is a new
type of
dependence for random variables, which generalizes the well-known
notion of
exchangeability. In this paper, a class of random sequences, called
Generalized
Species Sampling Sequences, is defined and a condition to have
conditional
identity in distribution is given. Moreover, a class of generalized
species
sampling sequences that are conditionally identically distributed is
introduced
and studied: the Generalized Ottawa sequences (GOS). This class
contains a
'`randomly reinforced'' version of the P\'olya urn and of the
Blackwell-MacQueen urn scheme. For the empirical means and the
predictive means
of a GOS, we prove two convergence results toward suitable mixtures of
Gaussian
distributions. The first one is in the sense of stable convergence and
the
second one in the sense of almost sure conditional convergence. In the
last
part of the paper we study the length of the partition induced by a
GOS at time
$n$, i.e. the random number of distinct values of a GOS until time $n
$. Under
suitable conditions, we prove a strong law of large numbers and a
central limit
theorem in the sense of stable convergence. All the given results in
the paper
are accompanied by some examples.
http://arxiv.org/abs/0806.2724
---------------------------------------------------------------
7185. ${L^P}$-VARIATIONS FOR MULTIFRACTAL FRACTIONAL RANDOM WALKS
Carenne Lude\~na
A multifractal random walk (MRW) is defined by a Brownian motion
subordinated
by a class of continuous multifractal random measures $M[0,t], 0\le t
\le1$. In
this paper we obtain an extension of this process, referred to as
multifractal
fractional random walk (MFRW), by considering the limit in
distribution of a
sequence of conditionally Gaussian processes. These conditional
processes are
defined as integrals with respect to fractional Brownian motion and
convergence
is seen to hold under certain conditions relating the self-similarity
(Hurst)
exponent of the fBm to the parameters defining the multifractal random
measure
$M$. As a result, a larger class of models is obtained, whose fine scale
(scaling) structure is then analyzed in terms of the empirical structure
functions. Implications for the analysis and inference of multifractal
exponents from data, namely, confidence intervals, are also provided.
http://arxiv.org/abs/0806.2731
---------------------------------------------------------------
7186. A MIXED SINGULAR/SWITCHING CONTROL PROBLEM FOR A DIVIDEND POLICY
WITH REVERSIBLE TECHNOLOGY INVESTMENT
Vathana Ly Vath and Huy\^en Pham and St\'ephane Villeneuve
We consider a mixed stochastic control problem that arises in
Mathematical
Finance literature with the study of interactions between dividend
policy and
investment. This problem combines features of both optimal switching and
singular control. We prove that our mixed problem can be decoupled in
two pure
optimal stopping and singular control problems. Furthermore, we
describe the
form of the optimal strategy by means of viscosity solution techniques
and
smooth-fit properties on the corresponding system of variational
inequalities.
Our results are of a quasi-explicit nature. From a financial
viewpoint, we
characterize situations where a firm manager decides optimally to
postpone
dividend distribution in order to invest in a reversible growth
opportunity
corresponding to a modern technology. In this paper a reversible
opportunity
means that the firm may disinvest from the modern technology and
return back to
its old technology by receiving some gain compensation. The results of
our
analysis take qualitatively different forms depending on the
parameters values.
http://arxiv.org/abs/0806.2745
---------------------------------------------------------------
7187. VARIANCE BOUNDING MARKOV CHAINS
Gareth O. Roberts and Jeffrey S. Rosenthal
We introduce a new property of Markov chains, called variance
bounding. We
prove that, for reversible chains at least, variance bounding is
weaker than,
but closely related to, geometric ergodicity. Furthermore, variance
bounding is
equivalent to the existence of usual central limit theorems for all
$L^2$
functionals. Also, variance bounding (unlike geometric ergodicity) is
preserved
under the Peskun order. We close with some applications to Metropolis--
Hastings
algorithms.
http://arxiv.org/abs/0806.2747
---------------------------------------------------------------
7188. STOCHASTIC IMPULSE CONTROL OF NON-MARKOVIAN PROCESSES
Boualem Djehiche and Said Hamadene and Ibtissam Hdhiri
We consider a class of stochastic impulse control problems of general
stochastic processes i.e. not necessarily Markovian. Under fairly
general
conditions we establish existence of an optimal impulse control. We
also prove
existence of combined optimal stochastic and impulse control of a fairly
general class of diffusions with random coefficients. Unlike, in the
Markovian
framework, we cannot apply quasi-variational inequalities techniques.
We rather
derive the main results using techniques involving reflected BSDEs and
the
Snell envelope.
http://arxiv.org/abs/0806.2761
---------------------------------------------------------------
7189. CENTRAL LIMIT THEOREM FOR SIGNAL-TO-INTERFERENCE RATIO OF
REDUCED RANK LINEAR RECEIVER
G. M. Pan and W. Zhou
Let $\mathbf{s}_k=\frac{1}{\sqrt{N}}(v_{1k},...,v_{Nk})^T,$ with
$\{v_{ik},i,k=1,...\}$ independent and identically distributed complex
random
variables. Write $\mathbf{S}_k=(\mathbf{s}_1,...,\mathbf
{s}_{k-1},\mathbf{s}_{k+1},... ,\mathbf{s}_K),$ $\mathbf{P}_k=
\operatorname
{diag}(p_1,...,p_{k-1},p_{k+1},...,p_K)$,
$\mathbf{R}_k=(\mathbf{S}_k\mathbf{P}_k\mathbf{S}_k^*+\sigma
^2\mathbf{I})$ and
$\mathbf{A}_{km}=[\mathbf{s}_k,\mathbf{R}_k\mathbf{s}_k,...
,\mathbf{R}_k^{m-1}\mathbf{s}_k]$. Define
$\beta_{km}=p_k\mathbf{s}_k^*\mathbf{A}_{km}(\mathbf {A}_{km}^*\times\
mathbf{R}_k\mathbf{A}_{km})^{-1}\mathbf{A}_{km}^*\mathbf{s}_k$,
referred to as
the signal-to-interference ratio (SIR) of user $k$ under the
multistage Wiener
(MSW) receiver in a wireless communication system. It is proved that
the output
SIR under the MSW and the mutual information statistic under the
matched filter
(MF) are both asymptotic Gaussian when $N/K\to c>0$. Moreover, we
provide a
central limit theorem for linear spectral statistics of eigenvalues and
eigenvectors of sample covariance matrices, which is a supplement of
Theorem 2
in Bai, Miao and Pan [Ann. Probab. 35 (2007) 1532--1572]. And we also
improve
Theorem 1.1 in Bai and Silverstein [Ann. Probab. 32 (2004) 553--605].
http://arxiv.org/abs/0806.2768
---------------------------------------------------------------
7190. STATIONARY MAX-STABLE FIELDS ASSOCIATED TO NEGATIVE DEFINITE
FUNCTIONS
Zakhar Kabluchko and Martin Schlather and Laurens de Haan
Let $W_i(\cdot)$, $i\in\NN$, be independent copies of a zero mean
gaussian
process $\{W(t), t\in\RR^d\}$ with stationary increments; and denote by
$\sigma^2(t)$ the variance of $W(t)$. Independently from $W_i$, let
$\sum_{i=1}^\infty \delta_{U_i}$ be a Poisson point process on the
real axis
with intensity $e^{-y}dy$. We show that the law of the random family of
functions $\{V_i(\cdot), i\in\NN\}$ defined by
$V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$ is translation invariant. In
particular, the
process $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary max-
stable
process with standard Gumbel margins. The process $\eta$ arises as a
limit of a
suitably normalized and rescaled pointwise maximum of $n$ i.i.d.
stationary
gaussian processes as $n\to\infty$ if and only if $W$ is a (non-
isotropic)
fractional Brownian motion on $\RR^d$. Under suitable conditions on $W
$, the
process $\eta$ has a mixed moving maxima representation.
http://arxiv.org/abs/0806.2780
---------------------------------------------------------------
7191. ANOTHER CORRECTION. ERROR ESTIMATES FOR BINOMIAL APPROXIMATIONS
OF GAME OPTIONS
Yan Dolinsky and Yuri Kifer
The Annals of Applied Probability 16 (2006) 984--1033 [URL:
http://projecteuclid.org/euclid.aoap/1151592257]
http://arxiv.org/abs/0806.2782
---------------------------------------------------------------
7192. ON THE ERGODICITY OF THE ADAPTIVE METROPOLIS ALGORITHM ON
UNBOUNDED DOMAINS
Eero Saksman and Matti Vihola
This paper describes sufficient conditions to ensure the correct
ergodicity
of the Adaptive Metropolis (AM) algorithm of Haario, Saksman, and
Tamminen
(2001) [8], for target distributions with a non-compact support. The
conditions
ensuring a strong law of large numbers and a central limit theorem
require that
the tails of the target density decay super-exponentially, and have
regular
enough convex contours. The result is based on the ergodicity of an
auxiliary
process that is sequentially constrained to feasible adaptation sets,
and
independent estimates of the growth rate of the AM chain and the
corresponding
geometric drift constants. The ergodicity result of the constrained
process is
obtained through a modification of the approach due to Andrieu and
Moulines
(2006) [1].
http://arxiv.org/abs/0806.2933
---------------------------------------------------------------
7193. NEW TECHNIQUES FOR EMPIRICAL PROCESS OF DEPENDENT DATA
Herold Dehling and Olivier Durieu and Dalibor Voln\'y
We present a new technique for proving empirical process invariance
principle
for stationary processes $(X_n)_{n\geq 0}$. The main novelty of our
approach
lies in the fact that we only require the central limit theorem and a
moment
bound for a restricted class of functions $(f(X_n))_{n\geq 0}$, not
containing
the indicator functions. Our approach can be applied to Markov chains
and
dynamical systems, using spectral properties of the transfer operator.
Our
proof consists of a novel application of chaining techniques.
http://arxiv.org/abs/0806.2941
---------------------------------------------------------------
7194. A FOURTH MOMENT INEQUALITY FOR FUNCTIONALS OF STATIONARY PROCESSES
Olivier Durieu
In this paper, a fourth moment bound for partial sums of functional of
strongly ergodic Markov chain is established. This type of inequality
plays an
important role in the study of empirical process invariance principle.
This one
is specially adapted to the technique of Dehling, Durieu and Voln\'y
(2008).
The same moment bound can be proved for dynamical system whose transfer
operator has some spectral properties. Examples of applications are
given.
http://arxiv.org/abs/0806.2980
---------------------------------------------------------------
7195. THE HIGH TEMPERATURE ISING MODEL ON THE TRIANGULAR LATTICE IS A
CRITICAL PERCOLATION MODEL
Andras Balint and Federico Camia and Ronald Meester
The Ising model at inverse temperature $\beta$ and zero external field
can be
obtained via the Fortuin-Kasteleyn (FK) random-cluster model with
$q=2$ and
density of open edges $p=1-e^{-\beta}$ by assigning spin +1 or -1 to
each
vertex in such a way that (1) all the vertices in the same FK cluster
get the
same spin and (2) +1 and -1 have equal probability. We generalize the
above
procedure by assigning spin +1 with probability $r$ and -1 with
probability
$1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $
\beta$, this
generates a dependent (spin) percolation model with parameter $r$. We
show
that, on the triangular lattice and for $\beta<\beta_c$, this model
has a
percolation phase transition at $r=1/2$, corresponding to the Ising
model. This
sheds some light on the conjecture that the high temperature Ising
model on the
triangular lattice is in the percolation universality class and that its
scaling limit can be described in terms of SLE$_6$. We also prove
uniqueness of
the infinite +1 cluster for $r>1/2$, sharpness of the percolation phase
transition (by showing exponential decay of the cluster size
distribution for
$r<1/2$), and continuity of the percolation function for all $r \in
[0,1]$.
http://arxiv.org/abs/0806.3020
---------------------------------------------------------------
7196. A NOTE ON DOMINANT CONTRACTIONS OF JORDAN ALGEBRAS
Farrukh Mukhamedov and Seyit Temir and Hasan Akin
In the paper we consider two positive contractions
$T,S:L^{1}(A,\tau)\longrightarrow L^{1}(A,\tau)$ such that $T\leq S$,
here
$(A,\t)$ is a semi-finite $JBW$-algebra. If there is an $n_{0}\in
\mathbb{N}$
such that $\|S^{n_{0}}-T^{n_{0}}\|<1$. Then we prove that $\|S^{n}-
T^{n}\|<1$
holds for every $n\geq n_{0}.$
http://arxiv.org/abs/0806.2926
---------------------------------------------------------------
7197. STOCHASTIC CONTROL UP TO A HITTING TIME: OPTIMALITY AND ROLLING-
HORIZON IMPLEMENTATION
Debasish Chatterjee and Eugenio Cinquemani and Giorgos Chaloulos
and John Lygeros
We present a dynamic programming-based solution to a stochastic optimal
control problem up to a hitting time for a discrete-time Markov control
process. Firstly, we determine an optimal control policy to steer the
process
toward a compact target set while simultaneously minimizing an expected
discounted cost. We then provide a rolling-horizon strategy for
approximating
the optimal policy, together with quantitative characterization of its
sub-optimality with respect to the optimal policy. Finally, we address
related
issues of asymptotic discount-optimality of the value-iteration
policy. Both
the state and action spaces are assumed to be Polish.
http://arxiv.org/abs/0806.3008
---------------------------------------------------------------
7198. LAWS OF THE ITERATED LOGARITHM FOR A CLASS OF ITERATED PROCESSES
Erkan Nane
Let $X=\{X_t, t\geq 0\}$ be a Brownian motion or a spectrally negative
stable
process of index $1<\a<2$. Let $E=\{E_t,t\geq 0\}$ be the hitting time
of a
stable subordinator of index $0<\beta<1$ independent of $X$. We use a
connection between $X(E_t)$ and the stable subordinator of index $
\beta/\a$ to
derive information on the path behavior of $X(E_t)$. This is an
extension of
the connection of iterated Brownian motion and (1/4)-stable
subordinator due to
Bertoin \cite{bertoin}. Using this connection, we obtain various laws
of the
iterated logarithm for $X(E_t)$. In particular, we establish law of the
iterated logarithm for local time Brownian motion, $X(L_t)$, where $X$
is a
Brownian motion (the case $\a=2$) and $L_t$ is the local time at zero
of a
stable process $Y$ of index $1<\a_2\leq 2$ independent of $X$. In this
case
$E_{\rho t}=L_t$ with $\beta=1-1/\a_2$ for some constant $\rho>0$. This
establishes the lower bound in the law of the iterated logarithm which
we could
not prove with the techniques of our paper \cite{MNX}. We also obtain
exact
small balll probability for $X(E_t)$ using ideas from \cite{aurzada}.
http://arxiv.org/abs/0806.3126
---------------------------------------------------------------
7199. DIRECTIONALLY CONVEX ORDERING OF RANDOM MEASURES, SHOT NOISE
FIELDS AND SOME APPLICATIONS TO WIRELESS COMMUNICATIONS
Bartlomiej Blaszczyszyn (INRIA Rocquencourt) and Dhandapani Yogeshwaran
Directionally convex ($dcx$) ordering is a tool for comparison of
dependence
structure of random vectors that also takes into account the
variability of the
marginal distributions. When extended to random fields it oncerns
comparison of
all finite dimensional distributions. Viewing locally finite measures as
non-negative fields of measure-values indexed by the bounded Borel
subsets of
the space, in this paper we formulate and study the $dcx$ ordering of
random
measures on locally compact spaces. We show that the $dcx$ order is
preserved
under some of the natural operations considered on random measures and
point
processes, such as independent superposition and thinning. Further
operations
such as independent marking and displacement, though do not preserve
the $dcx$
order on all point processes, are shown to preserve the order on Cox
point
processes. We also examine the impact of $dcx$ order on the second
moment
properties, in particular on clustering and on Palm distributions.
Comparisons
of Ripley's functions, pair correlation functions as well as examples
seem to
indicate that p.p. higher in $dcx$ order cluster more. As the main
result, we
show that non-negative integral (shot-noise) fields with respect to
$dcx$
ordered random measures inherit this ordering from the measures.
Numerous
applications of this result are shown, in particular to comparison of
various
Cox processes and some performance measures of wireless networks, in
both of
which shot-noise fields appear as key ingredients. We also mention a few
pertinent open questions.
http://arxiv.org/abs/0806.3180
---------------------------------------------------------------
7200. EXISTENCE OF A CRITICAL POINT FOR THE INFINITE DIVISIBILITY OF
SQUARES OF GAUSSIAN VECTORS IN $R^{2}$ WITH NON--ZERO MEAN
Michael B. Marcus and Jay Rosen
Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with
$EG_{1}G_{2}\neq
0$. Let $c_{1},c_{2}\in R^{1}$. A necessary and sufficient condition for
$G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ to be infinitely
divisible for all $\alpha\in R^{1}$ is that \[ \Ga_{i,i}\geq
\frac{c_{i}}{c_{j}}\Ga_{i,j}>0\qquad\forall 1\le i\ne j\le 2.\] In
this paper
we show that when this does not hold there exists an $0<\alpha_{0}<\ff
$ such
that $G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ is infinitely
divisible for all $|\alpha|\leq \alpha_{0}$ but not for any $|\al|>
\al_{0}$.
http://arxiv.org/abs/0806.3188
---------------------------------------------------------------
7201. RENEWAL SERIES AND SQUARE-ROOT BOUNDARIES FOR BESSEL PROCESSES
Nathanael Enriquez (MODAL'X) and Christophe Sabot (ICJ) and Marc Yor
(PMA and IUF)
We show how a description of Brownian exponential functionals as a
renewal
series gives access to the law of the hitting time of a square-root
boundary by
a Bessel process. This extends classical results by Breiman and Shepp,
concerning Brownian motion, and recovers by different means,
extensions for
Bessel processes, obtained independently by Delong and Yor.
http://arxiv.org/abs/0806.3197
---------------------------------------------------------------
7202. HYDRODYNAMIC LIMIT OF GRADIENT EXCLUSION PROCESSES WITH
CONDUCTANCES
Tertuliano Franco and Claudio Landim
Fix a strictly increasing right continuous with left limits function
$W: \bb
R \to \bb R$ and a smooth function $\Phi : [l,r] \to \bb R$, defined
on some
interval $[l,r]$ of $\bb R$, such that $0<b \le \Phi'\le b^{-1}$. We
prove that
the evolution, on the diffusive scale, of the empirical density of
exclusion
processes, with conductances given by $W$, is described by the weak
solutions
of the non-linear differential equation $\partial_t \rho = (d/dx)(d/dW)
\Phi(\rho)$. We derive some properties of the operator $(d/dx)(d/dW)$
and prove
uniqueness of weak solutions of the previous non-linear differential
equation.
http://arxiv.org/abs/0806.3211
---------------------------------------------------------------
7203. NON-PERTURBATIVE APPROACH TO RANDOM WALK IN MARKOVIAN ENVIRONMENT
Dmitry Dolgopyat and Carlangelo Liverani
We prove an averaged CLT for a random walk in a dynamical environment
where
the states of the environment at different sites are independent
Markov chains.
http://arxiv.org/abs/0806.3236
---------------------------------------------------------------
7204. MAXIMUM LIKELIHOOD DRIFT ESTIMATION FOR MULTISCALE DIFFUSIONS
A.Papavasiliou and G.A. Pavliotis and A.M. Stuart
We study the problem of parameter estimation using maximum likelihood
for
fast/slow systems of stochastic differential equations. Our aim is to
shed
light on the problem of model/data mismatch at small scales. We
consider two
classes of fast/slow problems for which a closed coarse-grained
equation for
the slow variables can be rigorously derived, which we refer to as
averaging
and homogenization problems. We ask whether, given data from the slow
variable
in the fast/slow system, we can correctly estimate parameters in the
drift of
the coarse-grained equation for the slow variable, using maximum
likelihood. We
show that, whereas the maximum likelihood estimator is asymptotically
unbiased
for the averaging problem, for the homogenization problem maximum
likelihood
fails unless we subsample the data at an appropriate rate. An explicit
formula
for the asymptotic error in the log likelihood function is presented.
Our
theory is applied to two simple examples from molecular dynamics.
http://arxiv.org/abs/0806.3248
---------------------------------------------------------------
7205. WELL-POSEDNESS OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR
SOFT POTENTIALS
H\'el\`ene Guerin (IRMAR) and Nicolas Fournier (LAMA)
We consider the spatially homogeneous Landau equation of kinetic
theory, and
provide a differential inequality for the Wasserstein distance with
quadratic
cost between two solutions. We deduce some well-posedness results. The
main
difficulty is that this equation presents a singularity for small
relative
velocities. Our uniqueness result is the first one in the important
case of
soft potentials. Furthermore, it is almost optimal for a class of
moderately
soft potentials, that is for a moderate singularity. Indeed, in such a
case,
our result applies for initial conditions with finite mass, energy, and
entropy. For the other moderatley soft potentials, we assume
additionnally some
moment conditions on the initial data. For very soft potentials, we
obtain only
a local (in time) well-posedness result, under some integrability
conditions.
Our proof is probabilistic, and uses a stochastic version of the Landau
equation, in the spirit of Tanaka.
http://arxiv.org/abs/0806.3379
---------------------------------------------------------------
7206. HETEROGENEOUS CREDIT PORTFOLIOS AND THE DYNAMICS OF THE
AGGREGATE LOSSES
Paolo Dai Pra and Marco Tolotti
We study the impact of contagion in a network of firms facing credit
risk. We
describe an intensity based model where the homogeneity assumption is
broken by
introducing a random environment that makes it possible to take into
account
the idiosyncratic characteristics of the firms. We shall see that our
model
goes behind the identification of groups of firms that can be considered
basically exchangeable. Despite this heterogeneity assumption our
model has the
advantage of being totally tractable. The aim is to quantify the
losses that a
bank may suffer in a large credit portfolio. Relying on a large
deviation
principle on the trajectory space of the process, we state a suitable
law of
large number and a central limit theorem useful to study large portfolio
losses. Simulation results are provided as well as applications to
portfolio
loss distribution analysis.
http://arxiv.org/abs/0806.3399
---------------------------------------------------------------
7207. MINIMAL SUPPORTING SUBTREES FOR THE FREE ENERGY OF POLYMERS ON
DISORDERED TREES
Peter Morters and Marcel Ortgiese
We consider a model of directed polymers on a regular tree with a
disorder
given by independent, identically distributed weights attached to the
vertices.
For suitable weight distributions this model undergoes a phase
transition with
respect to its localization behaviour. We show that, for high
temperatures, the
free energy is supported by a random tree of positive exponential
growth rate,
which is strictly smaller than that of the full tree. The growth rate
of the
minimal supporting subtree is decreasing to zero as the temperature
decreases
to the critical value. At the critical value and all lower
temperatures, a
single polymer suffices to support the free energy. Our proofs rely on
elegant
martingale methods adapted from the theory of branching random walks.
http://arxiv.org/abs/0806.3430
---------------------------------------------------------------
7208. SPECIAL EXAMPLES OF DIFFUSIONS IN RANDOM ENVIRONMENT
Ivan del Tenno
In this note we present some examples of diffusions in random
environment
whose asymptotic behavior is rather surprising. We construct a family of
diffusions that are small perturbations of Brownian motion with non-
vanishing
expected local drift under the static measure of the environment but
where the
ballistic behavior is lost. As slight modifications of this collection
of
diffusions we also provide examples with ballistic behavior where the
non-vanishing limiting velocity points to a direction opposite to the
expected
local drift under the static measure.
http://arxiv.org/abs/0806.3868
---------------------------------------------------------------
7209. HARMONIC MEASURES VERSUS QUASICONFORMAL MEASURES FOR HYPERBOLIC
GROUPS
S\'ebastien Blach\`ere (LATP) and Peter Ha\"issinsky (LATP) and
Pierre Mathieu (LATP)
We establish a dimension formula for the harmonic measure of a finitely
supported and symmetric random walk on a hyperbolic group. We also
characterize
random walks for which this dimension is maximal. Our approach is
based on the
Green metric, a metric which provides a geometric point of view on
random walks
and, in particular, which allows us to interpret harmonic measures as
\qc
measures on the boundary of the group.
http://arxiv.org/abs/0806.3915
---------------------------------------------------------------
7210. WHAT DOES A RANDOM CONTINGENCY TABLE LOOK LIKE?
Alexander Barvinok
Let R=(r_1, ..., r_m) and C=(c_1, ..., c_n) be positive integer
vectors such
that r_1 +... + r_m=c_1 +... + c_n. We consider the set Sigma(R, C) of
non-negative mxn integer matrices (contingency tables) with row sums R
and
column sums C as a finite probability space with the uniform measure.
We prove
that a random table D in Sigma(R,C) with high probability is close to a
particular matrix ("typical table'') Z defined as follows. We let
g(x)=(x+1)
ln(x+1)-x ln x for non-negative x and let g(X)=sum_ij g(x_ij) for a
non-negative matrix X=(x_ij). Then g(X) is strictly concave and
attains its
maximum on the polytope of non-negative mxn matrices X with row sums R
and
column sums C at a unique point, which we call the typical table Z.
http://arxiv.org/abs/0806.3910
---------------------------------------------------------------
7211. STOCHASTIC RELATIONS OF RANDOM VARIABLES AND PROCESSES
Lasse Leskel\"a
This paper generalizes the notion of stochastic order to a relation
between
probability measures over arbitrary measurable spaces. This
generalization is
motivated by the observation that for the stochastic ordering of two
stationary
Markov processes, it suffices that the generators of the processes
preserve
some, not necessarily reflexive or transitive, subrelation of the order
relation. The main contributions of the paper are: a functional
characterization of stochastic relations, necessary and sufficient
conditions
for the preservation of stochastic relations, and an algorithm for
finding
subrelations preserved by probability kernels. The theory is
illustrated with
applications to hidden Markov processes, population processes, and
queueing
systems.
http://arxiv.org/abs/0806.3562
---------------------------------------------------------------
7212. A MATRIX INTERPOLATION BETWEEN CLASSICAL AND FREE MAX
OPERATIONS: I. THE UNIVARIATE CASE
Florent Benaych-Georges (PMA) and Thierry Cabanal-Duvillard (MAP5)
Recently, Ben Arous and Voiculescu considered taking the maximum of
two free
random variables and brought to light a deep analogy with the
operation of
taking the maximum of two independent random variables. We present
here a new
insight on this analogy, based on random matrices giving an
interpolation
between classical and free settings.
http://arxiv.org/abs/0806.3686
---------------------------------------------------------------
7213. DUALITY OF REAL AND QUATERNIONIC RANDOM MATRICES
Wlodzimierz Bryc and Virgil U. Pierce
We show that quaternionic Gaussian random variables satisfy a
generalization
of the Wick formula for computing the expected value of products in
terms of a
family of graphical enumeration problems. When applied to the
quaternionic
Wigner and Wishart families of random matrices the result gives the
duality
between moments of these families and the corresponding real Wigner
and Wishart
families.
http://arxiv.org/abs/0806.3695
---------------------------------------------------------------
7214. INTEGRAL REPRESENTATION OF RENORMALIZED SELF-INTERSECTION LOCAL
TIMES
Yaozhong Hu and David Nualart and Jian Song
In this paper we apply Clark-Ocone formula to deduce an explicit
integral
representation for the renormalized self-intersection local time of
the $d$%
-dimensional fractional Brownian motion with Hurst parameter $H\in
(0,1)$.
As a consequence, we derive the existence of some exponential
moments for
this random variable.
http://arxiv.org/abs/0806.3706
---------------------------------------------------------------
7215. TIME-DEPENDENT SCHR\"ODINGER PERTURBATIONS OF TRANSITION DENSITIES
Krzysztof Bogdan
We construct the fundamental solution of $\partial_t+\Delta_x + q(t,x)
$, for
functions $q$ with a certain integral space-time relative smallness, in
particular for those satisfying a relative negligibility. The resulting
transition density is comparable to the Gaussian kernel in finite
time, and it
is even asymptotically equal to the Gaussian kernel (in small time)
under the
assumption of relative negligibility.
The result is generalized to arbitrary strictly positive and finite
time-inhomogeneous transition densities on measure spaces.
We also discuss specific applications to Schr\"odinger
perturbations of the
fractional Laplacian in view of the fact that the 3P Theorem holds for
the
fundamental solution of the operator.
http://arxiv.org/abs/0806.3549
---------------------------------------------------------------
7216. CARRIES, SHUFFLING AND AN AMAZING MATRIX
Persi Diaconis and Jason Fulman
The number of ``carries'' when $n$ random integers are added forms a
Markov
chain [23]. We show that this Markov chain has the same transition
matrix as
the descent process when a deck of $n$ cards is repeatedly riffle
shuffled.
This gives new results for the statistics of carries and shuffling.
http://arxiv.org/abs/0806.3583
---------------------------------------------------------------
7217. A NONCOMMUTATIVE EXTENDED DE FINETTI THEOREM
Claus K\"ostler
The extended de Finetti theorem characterizes exchangeable infinite
random
sequences as conditionally i.i.d. and shows that the apparently weaker
distributional symmetry of spreadability is equivalent to
exchangeability. Our
main result is a noncommutative version of this theorem.
In contrast to the classical result of Ryll-Nadzewski,
exchangeability turns
out to be stronger than spreadability for infinite noncommutative random
sequences. Out of our investigations emerges noncommutative conditional
independence in terms of a von Neumann algebraic structure closely
related to
Popa's notion of commuting squares and K\"ummerer's generalized
Bernoulli
shifts. Our main result is applicable to classical probability, quantum
probability, in particular free probability, braid group
representations and
Jones subfactors.
http://arxiv.org/abs/0806.3621
---------------------------------------------------------------
7218. WHITE NOISE CALCULUS AND HAMILTONIAN OF A QUANTUM STOCHASTIC
PROCESS
Wilhelm von Waldenfels
A white noise quantum stochastic calculus is developped using classical
measure theory as mathematical tool. Wick's and Ito's theorems have been
established. The simplest quantum stochastic differential equation has
been
solved, unicity and the conditions for unitarity have been proven. The
Hamiltonian of the associated one parameter strongly continuous group
has been
calculated explicitely.
http://arxiv.org/abs/0806.3636
---------------------------------------------------------------
7219. COMPOUND REAL WISHART AND Q-WISHART MATRICES
Wlodek Bryc
We introduce a family of matrices with non-commutative entries that
generalize the classical real Wishart matrices.
With the help of the Brauer product, we derive a non-asymptotic
expression
for the moments of traces of monomials in such matrices; the
expression is
quite similar to the formula derived in our previous work for
independent
complex Wishart matrices. We then analyze the fluctuations about the
Marchenko-Pastur law. We show that after centering by the mean, traces
of real
symmetric polynomials in q-Wishart matrices converge in distribution,
and we
identify the asymptotic law as the normal law when q=1, and as the
semicircle
law when q=0.
http://arxiv.org/abs/0806.4014
---------------------------------------------------------------
7220. AN EXPLICIT SOLUTION FOR AN OPTIMAL STOPPING/OPTIMAL CONTROL
PROBLEM WHICH MODELS AN ASSET SALE
Victoria Henderson and David Hobson
In this article we study an optimal stopping/optimal control problem
which
models the decision facing a risk averse agent over when to sell an
asset. The
market is incomplete so that the asset exposure cannot be hedged. In
addition
to the decision over when to sell, the agent has to choose a control
strategy
which corresponds to a feasible wealth process.
We formulate this problem as one involving the choice of a stopping
time and
a martingale. We conjecture the form of the solution and verify that the
candidate solution is equal to the value function.
The interesting features of the solution are that it is available
in a very
explicit form, that for some parameter values the optimal strategy is
more
sophisticated than might originally be expected, and that although the
set-up
is based on continuous diffusions, the optimal martingale may involve
a jump
process.
One interpretation of the solution is that it is optimal for the
risk averse
agent to gamble.
http://arxiv.org/abs/0806.4061
---------------------------------------------------------------
7221. RUIN MODELS WITH INVESTMENT INCOMES
Jostein Paulsen
This paper is a survey of recent progress in the theory of ruin for risk
processes that earn investment return on invested assets.
http://arxiv.org/abs/0806.4125
---------------------------------------------------------------
7222. ON LOCALIZATION IN KRONECKER'S DIOPHANTINE THEOREM
Michel Weber
We study the localization problem appearing in Kronecker's diophantine
theorem.
We introduce a probabilistic approach allowing to extend for general
$\Q$-linearly independent sequences a result of T\'uran concerning the
sequence
$ (\log p_\ell)$, $p_\ell$ being the $\ell$-th prime.
http://arxiv.org/abs/0806.3990
---------------------------------------------------------------
7223. ON A NON-STANDARD STOCHASTIC CONTROL PROBLEM
Ivar Ekeland and Traian A Pirvu
This paper considers the Merton portfolio management problem. We are
concerned with non-exponential discounting of time and this leads to
time
inconsistencies of the decision maker. Following Ekeland and Pirvu
2006, we
introduce the notion of equilibrium policies and we characterize them
by an
integral equation. The main idea is to come up with the value function
in this
context. If risk preferences are of CRRA type, the integral equation
which
characterizes the value function is shown to have a solution which
leads to an
equilibrium policy. This work is an extension of Ekeland and Pirvu 2006.
http://arxiv.org/abs/0806.4026
---------------------------------------------------------------
7224. THE NOTION OF $\PSI$-WEAK DEPENDENCE AND ITS APPLICATIONS TO
BOOTSTRAPPING TIME SERIES
Paul Doukhan and Michael H. Neumann
We give an introduction to a notion of weak dependence which is more
general
than mixing and allows to treat for example processes driven by discrete
innovations as they appear with time series bootstrap. As a typical
example, we
analyze autoregressive processes and their bootstrap analogues in
detail and
show how weak dependence can be easily derived from a contraction
property of
the process. Furthermore, we provide an overview of classes of processes
possessing the property of weak dependence and describe important
probabilistic
results under such an assumption.
http://arxiv.org/abs/0806.4263
---------------------------------------------------------------
7225. NON-AUTONOMOUS STOCHASTIC EVOLUTION EQUATIONS AND APPLICATIONS
TO STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
Mark Veraar
In this paper we study the following non-autonomous stochastic evolution
equation on a UMD Banach space $E$ with type 2,
\begin{equation}\label{eq:SEab}\tag{SE} {\begin{aligned} dU(t) & =
(A(t)U(t)
+ F(t,U(t))) dt + B(t,U(t)) dW_H(t), \quad t\in [0,T],
U(0) & = u_0. \end{aligned}. \end{equation}
Here $(A(t))_{t\in [0,T]}$ are unbounded operators with domains
$(D(A(t)))_{t\in [0,T]}$ which may be time dependent. We assume that
$(A(t))_{t\in [0,T]}$ satisfies the conditions of Acquistapace and
Terreni. The
functions $F$ and $B$ are nonlinear functions defined on certain
interpolation
spaces and $u_0\in E$ is the initial value. $W_H$ is a cylindrical
Brownian
motion on a separable Hilbert space $H$.
Under Lipschitz and linear growth conditions we show that there
exists a
unique mild solution of \eqref{eq:SEab}. Under assumptions on the
interpolation
spaces we extend the factorization method of Da Prato, Kwapie\'n, and
Zabczyk,
to obtain space-time regularity results for the solution $U$ of
\eqref{eq:SEab}. For Hilbert spaces $E$ we obtain a maximal regularity
result.
The results improve several previous results from the literature.
The theory is applied to a second order stochastic partial
differential
equation which has been studied by Sanz-Sol\'e and Vuillermot. This
leads to
several improvements of their result.
http://arxiv.org/abs/0806.4439
---------------------------------------------------------------
7226. PROBABILITY AND STATISTICS: ESSAYS IN HONOR OF DAVID A. FREEDMAN
Deborah Nolan and Terry Speed
This volume is our tribute to David A. Freedman, whom we regard as one
of the
great statisticians of our time. He received his B.Sc. degree from
McGill
University and his Ph.D. from Princeton, and joined the Department of
Statistics of the University of California, Berkeley, in 1962, where,
apart
from sabbaticals, he has been ever since. In a career of over 45
years, David
has made many fine contributions to probability and statistical
theory, and to
the application of statistics. His early research was on Markov chains
and
martingales, and two topics with which he has had a lifelong
fascination:
exchangeability and De Finetti's theorem, and the consistency of Bayes
estimates. His asymptotic theory for the bootstrap was also highly
influential.
David was elected to the American Academy of Arts and Sciences in
1991, and in
2003 he received the John J. Carty Award for the Advancement of
Science from
the U.S. National Academy of Sciences. In addition to his purely
academic
research, David has extensive experience as a consultant, including
working for
the Carnegie Commission, the City of San Francisco, and the Federal
Reserve, as
well as several Departments of the U.S. Government--Energy, Treasury,
Justice,
and Commerce. He has testified as an expert witness on statistics in a
number
of law cases, including Piva v. Xerox (employment discrimination),
Garza v.
County of Los Angeles (voting rights), and New York v. Department of
Commerce
(census adjustment). Lastly, he is an exceptionally good writer and
teacher,
and his many books and review articles are arguably his most important
contribution to our subject.
http://arxiv.org/abs/0806.4441
---------------------------------------------------------------
7227. NUMERICAL SIMULATION OF BSDES USING EMPIRICAL REGRESSION
METHODS: THEORY AND PRACTICE
Emmanuel Gobet (LJK) and Jean-Philippe Lemor (CMAP)
This article deals with the numerical resolution of backward stochastic
differential equations. Firstly, we consider a rather general case
where the
filtration is generated by a Brownian motion and a Poisson random
measure. We
provide a simulation algorithm based on iterative regressions on
function
bases, which coefficients are evaluated using Monte Carlo simulations.
We state
fully explicit error bounds. Secondly, restricting to the case of a
Brownian
filtration, we consider reflected BSDEs and adapt the previous
algorithm to
that situation. The complexity of the algorithm is very competitive
and allows
us to treat numerical results in dimension 10.
http://arxiv.org/abs/0806.4447
---------------------------------------------------------------
7228. SHARP ASYMPTOTICS FOR METASTABILITY IN THE RANDOM FIELD CURIE-
WEISS MODEL
A. Bianchi and A. Bovier and D. Ioffe
In this paper we study the metastable behavior of one of the simplest
disordered spin system, the random field Curie-Weiss model. We will
show how
the potential theoretic approach can be used to prove sharp estimates on
capacities and metastable exit times also in the case when the
distribution of
the random field is continuous. Previous work was restricted to the
case when
the random field takes only finitely many values, which allowed the
reduction
to a finite dimensional problem using lumping techniques. Here we
produce the
first genuine sharp estimates in a context where entropy is important.
http://arxiv.org/abs/0806.4478
---------------------------------------------------------------
7229. SEMICIRCLE LAW FOR RANDOM MATRICES OF LONG-RANGE PERCOLATION MODEL
Ayadi Slim
We study the normalized eigenvalue counting measure d\sigma of
matrices of
long-range percolation model. These are (2n+1)\times (2n+1) random real
symmetric matrices H=\{H(i,j)\}_{i,j} whose elements are independent
random
variables taking zero value with probability 1-\psi [(i-j)/b], b\in
\mathbb{R}^{+}, where \psi is an even positive function \psi(t)\le{1}
vanishing
at infinity. It is shown that if the third moment of \sqrt{b}H(i,j), i
\leq{j}
is uniformly bounded then the measure d\sigma:=d\sigma_{n,b} weakly
converges
in probability in the limit n,b\to\infty, b=o(n) to the semicircle (or
Wigner)
distribution. The proof uses the resolvent technique combined with the
cumulant
expansions method. We show that the normalized trace of resolvent
g_{n,b}(z)
converges in average and that the variance of g_{n,b}(z) vanishes. In
the
second part of the paper, we estimate the rate of decreasing of the
variance of
g_{n,b}(z), under further conditions on the moments of \sqrt{b}H(i,j), \
i\le{j}.
http://arxiv.org/abs/0806.4497
---------------------------------------------------------------
7230. GEOMETRIC EXTENSION OF PUT-CALL SYMMETRY IN THE MULTIASSET SETTING
Ilya Molchanov and Michael Schmutz
In this paper we show how to relate European call and put options on
multiple
assets to certain convex bodies called lift zonoids. Based on this,
geometric
properties can be translated into economic statements and vice versa.
For
instance, the European call-put parity corresponds to the central
symmetry
property, while the concept of dual markets can be explained by
reflection with
respect to a plane. It is known that the classical univariate log-
normal model
belongs to a large class of distributions with an extra property,
analytically
known as put-call symmetry. The geometric interpretation of this
symmetry
property motivates a natural multivariate extension. The financial
meaning of
this extension is explained, the asset price distributions that have
this
property are characterised and their properties are explored. It is
also shown
how to transform some multivariate asymmetric distributions that
appear after
adjusting for carrying costs to symmetric ones. A particular attention
is
devoted to the case of asset prices driven by L'evy processes. Based
on this,
semi-static hedging techniques for multiasset barrier options are
suggested.
http://arxiv.org/abs/0806.4506
---------------------------------------------------------------
7231. HEAT KERNEL ESTIMATES FOR STRONGLY RECURRENT RANDOM WALK ON
RANDOM MEDIA
Takashi Kumagai and Jun Misumi
We establish general estimates for simple random walk on an arbitrary
infinite random graph, assuming suitable bounds on volume and effective
resistance for the graph. These are generalizations of the results in
\cite[Section 1,2]{BJKS}, and in particular, imply the spectral
dimension of
the random graph. We will also give an application of the results to
random
walk on a long range percolation cluster.
http://arxiv.org/abs/0806.4507
---------------------------------------------------------------
7232. EXIT TIMES FROM CONES FOR NON-HOMOGENEOUS RANDOM WALK WITH
ASYMPTOTICALLY ZERO DRIFT
Iain M. MacPhee and Mikhail V. Menshikov and Andrew R. Wade
We study the first exit time $\tau$ from an arbitrary cone with apex
at the
origin by a non-homogeneous random walk on $\mathbb{Z}^2$ with mean
drift that
is asymptotically zero. Assuming bounded jumps and a form of weak
isotropy, we
give conditions for $\tau$ to be almost surely finite, and for the
existence
and non-existence of moments $\Exp [ \tau^p]$, $p>0$. Specifically, if
the mean
drift at $\bx \in \mathbb{Z}^2$ is of magnitude $O(\| \bx\|^{-1})$, then
$\tau<\infty$ a.s. for any cone, and we give polynomial estimates on
the tail
of $\tau$, while a mean drift of order $\| \bx \|^{-\beta}$, $\beta
\in (0,1)$,
can lead to $\tau=\infty$ with positive probability for any cone. For
mean
drift $o(\|\bx\|^{-1})$ (such as in the driftless case) we essentially
recover,
amongst other things, the random walk analogue of a theorem for
Brownian motion
due to Spitzer.
http://arxiv.org/abs/0806.4561
---------------------------------------------------------------
7233. BOOTSTRAP PERCOLATION IN THREE DIMENSIONS
Jozsef Balogh and Bela Bollobas and Robert Morris
By bootstrap percolation we mean the following deterministic process
on a
graph G. Given a set A of vertices 'infected' at time 0, new vertices
are
subsequently infected, at each time step, if they have at least r \in \N
previously infected neighbours. When the set A is chosen at random,
the main
aim is to determine the critical probability p_c(G,r) at which
percolation
(infection of the entire graph) becomes likely to occur.
This bootstrap process has been extensively studied on the d-
dimensional grid
[n]^d: with 2 \le r \le d fixed, it was proved by Cerf and Cirillo
(for d = r
=3), and by Cerf and Manzo (in general), that p_c([n]^d,r) =
\Theta(\frac{1}{\log_{r-1} n})^{d-r+1}, where \log_r is an r-times
iterated
logarithm. However, the exact threshold function is only known in the
case d =
r = 2, where it was shown by Holroyd to be (1 + o(1))\frac{\pi^2}
{18\log n}. In
this paper we shall determine the exact threshold in the crucial case
d = r =
3, and lay the groundwork for solving the problem for all fixed d and r.
http://arxiv.org/abs/0806.4485
---------------------------------------------------------------
7234. ON THE DEGREE SEQUENCE AND ITS CRITICAL PHENOMENON OF AN
EVOLVING RANDOM GRAPH PROCESS
Xian-Yuan Wu and Zhao Dong and Ke Liu and Kai-Yuan Cai
In this paper we focus on the problem of the degree sequence for the
following random graph process. At any time-step $t$, one of the
following
three substeps is executed: with probability $\alpha_1$, a new vertex
$x_t$ and
$m$ edges incident with $x_t$ are added; or, with probability
$\alpha-\alpha_1$, $m$ edges are added; or finally, with probability
$1-\a$,
$m$ random edges are deleted. Note that in any case edges are added in
the
manner of preferential attachment. we prove that there exists a
critical point
$\alpha_c$ satisfying: 1) if $\alpha_1<\alpha_c$, then the model has
power law
degree sequence; 2) if $\alpha_1>\alpha_c$, then the model has
exponential
degree sequence; and 3) if $\alpha_1=\alpha_c$, then the model has a
degree
sequence lying between the above two cases.
http://arxiv.org/abs/0806.4684
---------------------------------------------------------------
7235. IDLA ON THE SUPERCRITICAL PERCOLATION CLUSTER
Eric Shellef
We consider the internal diffusion limited aggregation (IDLA) process
on the
infinite cluster in supercritical Bernoulli bond percolation on
Euclidean
lattices. It is shown that the process on the cluster behaves like it
does on
the Euclidean lattice, in that the aggregate covers all the vertices
in a
Euclidean ball around the origin, such that the ratio of vertices in
this ball
to the total number of particles sent out approaches one almost surely.
http://arxiv.org/abs/0806.4771
-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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