[PAS] Probability Abstracts 106
Probability Abstract Service
pas at lists.imstat.org
Tue Nov 4 13:39:35 CST 2008
Probability Abstracts 106
This document contains abstracts 7442-7695
from September-1-2008 to October-31-2008.
They have been mailed on Nov 4, 2008.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_106.shtml
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7442. LIMIT THEOREMS FOR ADDITIVE FUNCTIONALS OF A MARKOV CHAIN
Milton Jara (CEREMADE) and Tomasz Komorowski (UMCS) and Stefano
Olla (CEREMADE)
Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability
measure $\pi$. Let $\Psi$ a function on the state space of the chain,
with
$\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find
sufficient
conditions on the probability transition to prove convergence in law of
$N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable law. "Martingale
approximation" approach and "coupling" approach give two different
sets of
conditions. We extend these results to continuous time Markov jump
processes
$X_t$, whose skeleton chain satisfies our assumptions. If waiting time
between
jumps has finite expectation, we prove convergence of $N^{-1/\alpha}
\int_0^{Nt}
V(X_s) ds$ to a stable process. In the case of waiting times with
infinite
average, we prove convergence to a Mittag-Leffler process.
http://arxiv.org/abs/0809.0177
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7443. A NECESSARY AND SUFFICIENT CONDITION FOR THE INVERTIBILITY OF
ADAPTED PERTURBATIONS OF IDENTITY ON THE WIENER SPACE
Ali S\"uleyman \"Ust\"unel
Let $(W,H,\mu)$ be the classical Wiener space, assume that $U=I_W+u$
is an
adapted perturbation of identity satisfying the Girsanov identity.
Then, $U$ is
invertible if and only if the kinetic energy of $u$ is equal to the
relative
entropy of the measure induced with the action of $U$ on the Wiener
measure
$\mu$, in other words $U$ is invertible if and only if $$ \half
\int_W|u|_H^2d\mu=\int_W \frac{dU\mu}{d\mu}\log\frac{dU\mu}{d\mu}d
\mu . $$
http://arxiv.org/abs/0809.0215
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7444. PATHWISE UNIQUENESS FOR STOCHASTIC HEAT EQUATIONS WITH H\"OLDER
CONTINUOUS COEFFICIENTS: THE WHITE NOISE CASE
Leonid Mytnik (Technion) and Edwin Perkins (The University of
British Columbia)
We prove pathwise uniqueness for solutions of parabolic stochastic
pde's with
multiplicative white noise if the coefficient is H\"older continuous
of index
$\gamma>3/4$. The method of proof is an infinite-dimensional version
of the
Yamada-Watanabe argument for ordinary stochastic differential equations.
http://arxiv.org/abs/0809.0248
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7445. REPLICA OVERLAP AND COVERING TIME FOR THE WIENER SAUSAGES AMONG
POISSONIAN OBSTACLES
Ryoki Fukushima
We study two objects concerning the Wiener sausage among Poissonian
obstacles. The first is the asymptotics for the \textit{replica
overlap}, which
is the intersection of two independent Wiener sausages. We show that
it is
asymptotically equal to their union. This result confirms that the
localizing
effect of the media is so strong as to completely determine the
motional range
of particles. The second is an estimate on the \textit{covering time}.
It is
known that the Wiener sausage avoiding Poissonian obstacles up to time
$t$ is
confined in some `clearing' ball near the origin and almost fills it.
We prove
here that the time needed to fill the confinement ball has the same
order as
its volume.
http://arxiv.org/abs/0809.0262
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7446. COMPETING RISKS WITHIN SHOCK MODELS
Antonio Di Crescenzo and Maria Longobardi
We consider a competing risks model, in which system failures are due
to one
out of two mutually exclusive causes, formulated within the framework
of shock
models driven by bivariate Poisson process. We obtain the failure
densities and
the survival functions as well as other related quantities under three
different schemes. Namely, system failures are assumed to occur at the
first
instant in which a random constant threshold is reached by (a) the sum
of
received shocks, (b) the minimum of shocks, (c) the maximum of shocks.
http://arxiv.org/abs/0809.0279
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7447. ESSCHER TRANSFORM AND THE DUALITY PRINCIPLE FOR
MULTIDIMENSIONAL SEMIMARTINGALES
Ernst Eberlein and Antonis Papapantoleon and Albert N. Shiryaev
The duality principle in option pricing aims at simplifying valuation
problems that depend on several variables by associating them to the
corresponding dual option pricing problem. Here we analyze the duality
principle for options that depend on several assets. The asset price
processes
are driven by general semimartingales, and the dual measures are
constructed
via an Esscher transformation. As an application, we can relate swap
and quanto
options to standard call and put options. Explicit calculations for
jump models
are also provided.
http://arxiv.org/abs/0809.0301
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7448. THE T-STABILITY NUMBER OF A RANDOM GRAPH
Nikolaos Fountoulakis and Ross J. Kang and Colin McDiarmid
Given a graph G = (V,E), a vertex subset S is called t-stable (or
t-dependent) if the subgraph G[S] induced on S has maximum degree at
most t.
The t-stability number of G is the maximum order of a t-stable set in
G. We
investigate the typical values that this parameter takes on a random
graph on n
vertices and edge probability equal to p. For any fixed 0 < p < 1 and
fixed
non-negative integer t, we show that, with probability tending to 1 as
n grows,
the t-stability number takes on at most two values which we identify as
functions of t, p and n. The main tool we use is an asymptotic
expression for
the expected number of t-stable sets of order k. We derive this
expression by
performing a precise count of the number of graphs on k vertices that
have
maximum degree at most k. Using the above results, we also obtain
asymptotic
bounds on the t-improper chromatic number of a random graph (this is the
generalisation of the chromatic number, where we partition of the
vertex set of
the graph into t-stable sets).
http://arxiv.org/abs/0809.0141
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7449. THE LONGEST MINIMUM-WEIGHT PATH IN A COMPLETE GRAPH
Louigi Addario-Berry and Nicolas Broutin and Gabor Lugosi
We consider the minimum-weight path between any pair of nodes of the
$n$-vertex complete graph in which the weights of the edges are i.i.d.
exponentially distributed random variables. We show that the longest
of these
minimum-weight paths has about $\alpha^\star \log n$ edges where
$\alpha^\star\approx 3.5911$ is the unique solution of the equation $
\alpha
\log \alpha - \alpha =1$. This answers a question posed by Janson
(1999).
http://arxiv.org/abs/0809.0275
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7450. FLUCTUATIONS OF THE QUENCHED MEAN OF A PLANAR RANDOM WALK IN AN
I.I.D. RANDOM ENVIRONMENT WITH FORBIDDEN DIRECTION
Mathew Joseph
We consider an i.i.d. random environment with a strong form of
transience on
the two dimensional integer lattice. Namely, the walk always moves
forward in
the y-direction. We prove a functional CLT for the quenched expected
position
of the random walk indexed by its level crossing times. We begin with a
variation of the Martingale Central Limit Theorem. The main part of
the paper
checks the conditions of the theorem for our problem.
http://arxiv.org/abs/0809.0320
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7451. ERROR CALCULUS AND REGULARITY OF POISSON FUNCTIONALS : THE LENT
PARTICLE METHOD
Nicolas Bouleau (CIRED and Cermics)
We propose a new method to apply the Lipschitz functional calculus of
local
Dirichlet forms to Poisson random measures.
http://arxiv.org/abs/0809.0382
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7452. AN EXTENDED EXISTENCE RESULT FOR QUADRATIC BSDES WITH JUMPS
WITH APPLICATION TO THE UTILITY MAXIMIZATION PROBLEM
Marie Amelie Morlais
In this study, we consider the exponential utility maximization
problem in
the context of a jump-diffusion model. To solve the problem, we rely
on the
dynamic programming principle and we derive from it a quadratic BSDE
with
jumps. Since this quadratic BSDE is driven both by a Wiener process
and by a
Poisson random measure having a Levy measure with infinite mass, our
main task
consists in establishing a new existence result for the specific BSDE
introduced.
http://arxiv.org/abs/0809.0423
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7453. AVERAGE CONTINUOUS CONTROL OF PIECEWISE DETERMINISTIC MARKOV
PROCESSES
O.L.V. Costa and F. Dufour
This paper deals with the long run average continuous control problem of
piecewise deterministic Markov processes (PDMP's) taking values in a
general
Borel space and with compact action space depending on the state
variable. The
control variable acts on the jump rate and transition measure of the
PDMP, and
the running and boundary costs are assumed to be positive but not
necessarily
bounded. Our first main result is to obtain an optimality equation for
the long
run average cost in terms of a discrete-time optimality equation
related to the
embedded Markov chain given by the post-jump location of the PDMP. Our
second
main result guarantees the existence of a feedback measurable selector
for the
discrete-time optimality equation by establishing a connection between
this
equation and an integro-differential equation. Our final main result
is to
obtain some sufficient conditions for the existence of a solution for a
discrete-time optimality inequality and an ordinary optimal feedback
control
for the long run average cost using the so-called vanishing discount
approach.
http://arxiv.org/abs/0809.0477
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7454. USING RELATIVE ENTROPY TO FIND OPTIMAL APPROXIMATIONS: AN
APPLICATION TO SIMPLE FLUIDS
Chih-Yuan Tseng and Ariel Caticha
We develop a maximum relative entropy formalism to generate optimal
approximations to probability distributions. The central results
consist in (a)
justifying the use of relative entropy as the uniquely natural
criterion to
select a preferred approximation from within a family of trial
parameterized
distributions, and (b) to obtain the optimal approximation by
marginalizing
over parameters using the method of maximum entropy and information
geometry.
As an illustration we apply our method to simple fluids. The "exact"
canonical
distribution is approximated by that of a fluid of hard spheres. The
proposed
method first determines the preferred value of the hard-sphere
diameter, and
then obtains an optimal hard-sphere approximation by a suitably
weighed average
over different hard-sphere diameters. This leads to a considerable
improvement
in accounting for the soft-core nature of the interatomic potential.
As a
numerical demonstration, the radial distribution function and the
equation of
state for a Lennard-Jones fluid (argon) are compared with results from
molecular dynamics simulations.
http://arxiv.org/abs/0808.4160
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7455. GLAUBER DYNAMICS IN HIGH DIMENSIONS
Robert Morris
We study Glauber dynamics on Z^d, which is a dynamic version of the
celebrated Ising model of ferromagnetism. Spins are initially chosen
according
to a Bernoulli distribution with density p, and then the states are
continuously (and randomly) updated according to the majority rule. This
corresponds to the sudden quenching of a ferromagnetic system at high
temperature with an external field, to one at zero temperature with no
external
field. Define p_c(Z^d) to be the infimum over those p \ge 1/2 such
that the
system fixates with probability 1. It is a folklore conjecture that
p_c(Z^d) =
1/2 for every d \ge 2. We prove that p_c(Z^d) \to 1/2 as d \to \infty.
http://arxiv.org/abs/0809.0353
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7456. PROBABILISTIC SOLUTION OF THE AMERICAN OPTIONS
Ali S\"uleyman \"Ust\"unel
The existence and uniqueness of probabilistic solutions of variational
inequalities for the general American options are proved under the
hypothesis
of hypoellipticity of the infinitesimal generator of the underlying
diffusion
process which represents the risky assets of the stock market with
which the
option is created. The main tool is an extension of the It\^o formula
which is
valid for the tempered distributions on $\R^d$ and for nondegenerate It
\^o
processes in the sense of the Malliavin calculus.
http://arxiv.org/abs/0809.0611
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7457. OCCUPATION TIMES OF BRANCHING SYSTEMS WITH INITIAL INHOMOGENEOUS
POISSON STATES AND RELATED SUPERPROCESSES
Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
The $(d,\alpha,\beta,\gamma)$-branching particle system consists of
particles
moving in $R^d$ according to a symmetric $\alpha$-stable L\'evy process
$(0<\alpha\leq 2)$, splitting with a critical $(1+\beta)$-branching law
$(0<\beta\leq 1)$, and starting from an inhomogeneous Poisson random
measure
with intensity measure $\mu_\gamma(dx)=dx/(1+|x|^\gamma), \gamma\geq
0$. By
means of time rescaling $T$ and Poisson intensity measure $H_T\mu_
\gamma$,
occupation time fluctuation limits for the system as $T\to\infty$ have
been
obtained in two special cases: Lebesgue measure ($\gamma=0$, the
homogeneous
case), and finite measures $(\gamma>d)$. In some cases $H_T\equiv 1$
and in
others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit
processes are quite different for Lebesgue and for finite measures.
Therefore
the question arises of what kinds of limits can be obtained for Poisson
intensity measures that are intermediate between Lebesgue measure and
finite
measures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$,
are used
for investigating this question. Occupation time fluctuation limits are
obtained which interpolate in some way between the two previous
extreme cases.
The limit processes depend on different arrangements of the parameters
$d,\alpha,\beta,\gamma$. Related results for the corresponding
$(d,\alpha,\beta,\gamma)$-superprocess are also given.
http://arxiv.org/abs/0809.0665
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7458. A REMARK ON THE INFINITE-VOLUME GIBBS MEASURES OF SPIN GLASSES
Louis-Pierre Arguin
In this note, we point out that infinite-volume Gibbs measures of spin
glass
models on the hypercube can be identified as random probability
measures on the
unit ball of a Hilbert space. This simple observation follows from a
result of
Dovbysh and Sudakov on weakly exchangeable random matrices. Limiting
Gibbs
measures can then be studied as single well-defined objects. This
approach
naturally extends the space of Random Overlap Structures as defined by
Aizenman, Sims and Starr. We discuss the Ruelle Probability Cascades
and the
stochastic stability within this framework. As an application, we use
an idea
of Parisi and Talagrand to prove that if a sequence of finite-volume
Gibbs
measures satisfies the Ghirlanda-Guerra identities, then the infinite-
volume
measure must be singular as a measure on a Hilbert space.
http://arxiv.org/abs/0809.0683
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7459. REFINED ESTIMATES FOR SOME BASIC RANDOM WALKS ON THE SYMMETRIC
AND ALTERNATING GROUPS
L. Saloff-Coste and J. Zuniga
We give refined estimates for the discrete time and continuous time
versions
of some basic random walks on the symmetric and alternating groups $S_n
$ and
$A_n$. We consider the following models: random transposition,
transpose top
with random, random insertion, and walks generated by the uniform
measure on a
conjugacy class. In the case of random walks on $S_n$ and $A_n$
generated by
the uniform measure on a conjugacy class, we show that in continuous
time the
$\ell^2$-cuttoff has a lower bound of $(n/2)\log n$. This result,
along with
the results of M\"uller, Schlage-Puchta and Roichman, demonstrates
that the
continuous time version of these walks may take much longer to reach
stationarity than its discrete time counterpart.
http://arxiv.org/abs/0809.0688
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7460. A DUAL CHARACTERIZATION OF SELF-GENERATION AND LOG-AFFINE
FORWARD PERFORMANCES
Gordan Zitkovic
We propose a mathematical framework for the study of a family of random
fields - called forward performances - which arise as numerical
representation
of certain rational preference relations in mathematical finance.
Their spatial
structure corresponds to that of utility functions, while the temporal
one
reflects a Nisio-type semigroup property, referred to as self-
generation. In
the setting of semimartingale financial markets, we provide a dual
formulation
of self-generation in addition to the original one, and show equivalence
between the two, thus giving a dual characterization of forward
performances.
Then, we focus on random fields with a log-affine structure and provide
necessary and sufficient conditions for self-generation in that case.
Finally,
we illustrate our methods in financial markets driven by It\^o-
processes, where
we obtain an explicit parametrization of all log-affine forward
performances.
http://arxiv.org/abs/0809.0739
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7461. THE NOTION OF CONVEXITY AND CONCAVITY ON WIENER SPACE
D. Feyel and A. S. \"Ust\"unel
We define, in the frame of an abstract Wiener space, the notions of
convexity
and of concavity for the equivalence classes of random variables. As
application we show that some important inequalities of the finite
dimensional
case have their natural counterparts in this setting.
http://arxiv.org/abs/0809.0812
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7462. LARGE DEVIATIONS OF VECTOR-VALUED MARTINGALES IN 2-SMOOTH NORMED
SPACES
Anatoli Juditsky (LJK) and Arkadii S. Nemirovski (ISyE)
We derive exponential bounds on probabilities of large deviations for
"light
tail" martingales taking values in finite-dimensional normed spaces. Our
primary emphasis is on the case where the bounds are dimension-
independent or
nearly so. We demonstrate that this is the case when the norm on the
space can
be approximated, within an absolute constant factor, by a norm which is
differentiable on the unit sphere with a Lipschitz continuous
gradient. We also
present various examples of spaces possessing the latter property.
http://arxiv.org/abs/0809.0813
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7463. SMALL-TIME EXPANSIONS FOR THE TRANSITION DISTRIBUTIONS OF L
\'EVY PROCESSES
Jos\'e E. Figueroa-L\'opez and Christian Houdr\'e
Let $X$ be a L\'evy process with absolutely continuous L\'evy measure $
\nu$.
Small time expansions, polynomial in $t$, are obtained for the tails
$P(X_{t}>y)$ of the process. The conditions imposed on $X$ require for
$X_{t}$
to have a $C^{\infty}$-transition density, whose derivatives remain
uniformly
bounded away from the origin, as $t$ goes to 0. Such conditions are
shown to be
satisfied for symmetric stable L\'evy processes as well as for other
related
L\'evy processes of relevance in mathematical finance. Also, under
very mild
conditions on the L\'evy density of the process and using a different
methodology, a second order power expansion is obtained by identifying
explicitly the limit of $(1/t){P(X_{t}>y)/t -\nu([y,\infty))}$ as $t$
goes to
0. The resulting limit seems to correct a result previously reported
in the
literature and hints at the fact that our higher order expansions
might be
valid under milder conditions.
http://arxiv.org/abs/0809.0849
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7464. SUDDEN EXTINCTION OF A CRITICAL BRANCHING PROCESS IN RANDOM
ENVIRONMENT
V.A. Vatutin V. Wachtel
Let $T$ be the extinction moment of a critical branching process
$Z=(Z_{n},n\geq 0) $ in a random environment specified by iid
probability
generating functions. We study the asymptotic behavior of the
probability of
extinction of the process $Z$ at moment $n\to \infty$, and show that
if the
logarithm of the (random) expectation of the offspring number belongs
to the
domain of attraction of a non-gaussian stable law then the extinction
occurs
owing to very unfavorable environment forcing the process, having at
moment
$T-1$ exponentially large population, to die out. We also give an
interpretation of the obtained results in terms of random walks in
random
environment.
http://arxiv.org/abs/0809.0986
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7465. AN IMPOSSIBILITY RESULT FOR PROCESS DISCRIMINATION
Daniil Ryabko (INRIA Lille - Nord Europe)
Two series of binary observations $x_1,x_1,...$ and $y_1,y_2,...$ are
presented: at each time $n\in\N$ we are given $x_n$ and $y_n$. It is
assumed
that the sequences are generated independently of each other by two
B-processes. We are interested in the question of whether the sequences
represent a typical realization of two different processes or of the
same one.
We demonstrate that this is impossible to decide, in the sense that
every
discrimination procedure is bound to err with non-negligible frequency
when
presented with sequences from some B-processes. This contrasts earlier
positive
results on B-processes, in particular those showing that there are
consistent
$\bar d$-distance estimates for this class of processes.
http://arxiv.org/abs/0809.1053
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7466. A PHASE TRANSITION FOR NON-INTERSECTING BROWNIAN MOTIONS, AND
THE PAINLEVE II EQUATION
Steven Delvaux and Arno B.J.Kuijlaars
We consider n non-intersecting Brownian motions with two fixed starting
positions and two fixed ending positions in the large n limit. We show
that in
case of 'large separation' between the endpoints, the particles are
asymptotically distributed in two separate groups, with no interaction
between
them, as one would intuitively expect. We give a rigorous proof using
the
Riemann-Hilbert formalism. In the case of 'critical separation'
between the
endpoints we are led to a model Riemann-Hilbert problem associated to
the
Hastings-McLeod solution of the Painleve II equation. We show that the
Painleve
II equation also appears in the large n asymptotics of the recurrence
coefficients of the multiple Hermite polynomials that are associated
with the
Riemann-Hilbert problem.
http://arxiv.org/abs/0809.1000
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7467. MEASURABILITY OF OPTIMAL TRANSPORTATION AND STRONG COUPLING OF
MARTINGALE MEASURES
Joaquin Fontbona and Helene Guerin and Sylvie Meleard
We consider the optimal mass transportation problem in $\RR^d$ with
measurably parameterized marginals, for general cost functions and under
conditions ensuring the existence of a unique optimal transport map.
We prove a
joint measurability result for this map, with respect to the space
variable and
to the parameter. The proof needs to establish the measurability of some
set-valued mappings, related to the support of the optimal
transference plans,
which we use to perform a suitable discrete approximation procedure. A
motivation is the construction of a strong coupling between orthogonal
martingale measures. By this we mean that, given a martingale measure,
we
construct in the same probability space a second one with specified
covariance
measure. This is done by pushing forward one martingale measure
through a
predictable version of the optimal transport map between the covariance
measures. This coupling allows us to obtain quantitative estimates in
terms of
the Wasserstein distance between those covariance measures.
http://arxiv.org/abs/0809.1111
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7468. ON ADAPTIVE STRATIFICATION
Pierre Etor\'e (CMAP) and Gersende Fort (LTCI) and Benjamin
Jourdain (CERMICS), Eric Moulines (LTCI)
This paper investigates the use of stratified sampling as a variance
reduction technique for approximating integrals over large dimensional
spaces.
The accuracy of this method critically depends on the choice of the
space
partition, the strata, which should be ideally fitted to thesubsets
where the
functions to integrate is nearly constant, and on the allocation of
the number
of samples within each strata. When the dimension is large and the
function to
integrate is complex, finding such partitions and allocating the
sample is a
highly non-trivial problem. In this work, we investigate a novel
method to
improve the efficiency of the estimator "on the fly", by jointly
sampling and
adapting the strata and the allocation within the strata. The accuracy
of
estimators when this method is used is examined in detail, in the so-
called
asymptotic regime (i.e. when both the number of samples and the number
of
strata are large). We illustrate the use of the method for the
computation of
the price of path-dependent options in models with both constant and
stochastic
volatility. The use of this adaptive technique yields variance
reduction by
factors sometimes larger than 1000 compared to classical Monte Carlo
estimators.
http://arxiv.org/abs/0809.1135
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7469. VERTEX DEGREE OF RANDOM INTERSECTION GRAPH
Bhupendra Gupta
A random intersection graph is constructed by independently assigning a
subset of a given set of objects $W,$ to each vertex of the vertex set
$V$ of a
simple graph $G.$ There is an edge between two vertices of $V,$ iff
their
respective subsets(in $W$,) have at least one common element. The strong
threshold for the connectivity between any two arbitrary vertices of
vertex set
$V,$ is derived. Also we determine the almost sure probability bounds
for the
vertex degree of a typical vertex of graph $G.$
http://arxiv.org/abs/0809.1141
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7470. A STRONG THRESHOLD FOR THE SIZE OF RANDOM CAPS TO COVER A SPHERE
Bhupendra Gupta
In this article, we consider `$N$'spherical caps of area $4\pi p$ were
uniformly distributed over the surface of a unit sphere. We are giving
the
strong threshold function for the size of random caps to cover the
surface of a
unit sphere. We have shown that for large $N,$ if $\frac{Np}{\log\:N}
> 1/2$
the surface of sphere is completely covered by the $N$ caps almost
surely, and
if $\frac{Np}{\log\:N} \leq 1/2$ a partition of the surface of sphere is
remains uncovered by the $N$ caps almost surely.
http://arxiv.org/abs/0809.1142
---------------------------------------------------------------
7471. NUMBER OF EDGES IN RANDOM INTERSECTION GRAPH ON SURFACE OF A
SPHERE
Bhupendra gupta
In this article, we consider `$N$'spherical caps of area $4\pi p$ were
uniformly distributed over the surface of a unit sphere. We study the
random
intersection graph $G_N$ constructed by these caps. We prove that for
$p =
\frac{c}{N^{\al}},\:c >0$ and $\al >2,$ the number of edges in graph
$G_N$
follow the Poisson distribution. Also we derive the strong law results
for the
number of isolated vertices in $G_N$: for $p = \frac{c}{N^{\al}},\:c
>0$ for
$\al < 1,$ there is no isolated vertex in $G_N$ almost surely i.e.,
there are
atleast $N/2$ edges in $G_N$ and for $\al >3,$ every vertex in $G_N$ is
isolated i.e., there is no edge in edge set $\cE_N.$
http://arxiv.org/abs/0809.1143
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7472. LARGE DEVIATIONS FOR RANDOM WALK IN A RANDOM ENVIRONMENT
Atilla Yilmaz
In this work, we study the large deviation properties of random walk
in a
random environment on $\mathbb{Z}^d$ with $d\geq1$.
We start with the quenched case, take the point of view of the
particle, and
prove the large deviation principle (LDP) for the pair empirical
measure of the
environment Markov chain. By an appropriate contraction, we deduce the
quenched
LDP for the mean velocity of the particle and obtain a variational
formula for
the corresponding rate function $I_q$. We propose an Ansatz for the
minimizer
of this formula. This Ansatz is easily verified when $d=1$.
In his 2003 paper, Varadhan proves the averaged LDP for the mean
velocity and
gives a variational formula for the corresponding rate function $I_a$.
Under
the non-nestling assumption (resp. Kalikow's condition), we show that
$I_a$ is
strictly convex and analytic on a non-empty open set $\mathcal{A}$,
and that
the true velocity $\xi_o$ is an element (resp. in the closure) of
$\mathcal{A}$. We then identify the minimizer of Varadhan's
variational formula
at any $\xi\in\mathcal{A}$.
For walks in high dimension, we believe that $I_a$ and $I_q$ agree
on a set
with non-empty interior. We prove this for space-time walks when the
dimension
is at least 3+1. In the latter case, we show that the cheapest way to
condition
the asymptotic mean velocity of the particle to be equal to any $\xi$
close to
$\xi_o$ is to tilt the transition kernel of the environment Markov
chain via a
Doob $h$-transform.
http://arxiv.org/abs/0809.1227
---------------------------------------------------------------
7473. STRONG UNIQUENESS FOR A CLASS OF SINGULAR SDES FOR CATALYTIC
BRANCHING DIFFUSIONS
Hui He
A new result for the strong uniqueness for catalytic branching
diffusions is
established, which improves the work of Dawson, D.A.; Fleischmann, K.;
Xiong,
J.[Strong uniqueness for cyclically symbiotic branching diffusions.
Statist.
Probab. Lett. 73, no. 3, 251--257 (2005)].
http://arxiv.org/abs/0809.1288
---------------------------------------------------------------
7474. RELIABILITY ANALYSIS OF SEMICOHERENT SYSTEMS THROUGH THEIR
LATTICE POLYNOMIAL DESCRIPTIONS
Alexander Dukhovny and Jean-Luc Marichal
A semicoherent system can be described by its structure function or,
equivalently, by a lattice polynomial function expressing the system
lifetime
in terms of the component lifetimes. In this paper we point out the
parallelism
between the two descriptions and use the natural connection of lattice
polynomial functions and relevant random events to collect exact
formulas for
the system reliability. We also discuss the equivalence between
calculating the
reliability of semicoherent systems and calculating the distribution
function
of a lattice polynomial function of random variables.
http://arxiv.org/abs/0809.1332
---------------------------------------------------------------
7475. GENERAL EXISTENCE RESULTS FOR REFLECTED BSDE AND BSDE
E. H. Essaky and M. Hassani
In this paper, we are concerned with the problem of existence of
solutions
for generalized reflected backward stochastic differential equations
(GRBSDEs
for short) and generalized backward stochastic differential equations
(GBSDEs
for short) when the generator $fds + gdA_s$ is continuous with general
growth
with respect to the variable $y$ and stochastic quadratic growth with
respect
to the variable $z$. We deal with the case of a bounded terminal
condition
$\xi$ and a bounded barrier $L$ as well as the case of unbounded ones.
This is
done by using the notion of generalized BSDEs with two reflecting
barriers
studied in \cite{EH}. The work is suggested by the interest the
results might
have in finance, control and game theory.
http://arxiv.org/abs/0809.1353
---------------------------------------------------------------
7476. GRAPHICAL MODELS FOR CORRELATED DEFAULTS
I. Onur Filiz and Xin Guo and Jason Morton and Bernd Sturmfels
A simple graphical model for correlated defaults is proposed, with
explicit
formulas for the loss distribution. Algebraic geometry techniques are
employed
to show that this model is well posed for default dependence: it
represents any
given marginal distribution for single firms and pairwise correlation
matrix.
These techniques also provide a calibration algorithm based on maximum
likelihood estimation. Finally, the model is compared with standard
normal
copula model in terms of tails of the loss distribution and implied
correlation
smile.
http://arxiv.org/abs/0809.1393
---------------------------------------------------------------
7477. A NEW FRAMEWORK OF MULTISTAGE ESTIMATION
Xinjia Chen
In this paper, we have established a new framework of multistage
parametric
estimation. Specially, we have developed sampling schemes for estimating
parameters of common important distributions. Without any information
of the
unknown parameters, our sampling schemes rigorously guarantee
prescribed levels
of precision and confidence, while achieving unprecedented efficiency
in the
sense that the average sampling numbers are virtually the same as that
are
computed as if the exact values of unknown parameters were available.
http://arxiv.org/abs/0809.1241
---------------------------------------------------------------
7478. IRREVERSIBLE MONTE CARLO ALGORITHMS FOR EFFICIENT SAMPLING
Konstantin S. Turitsyn and Michael Chertkov and Marija Vucelja
Equilibrium systems evolve according to Detailed Balance (DB). This
principe
guided development of the Monte-Carlo sampling techniques, of which
Metropolis-Hastings (MH) algorithm is the famous representative. It is
also
known that DB is sufficient but not necessary. We construct irreversible
deformation of a given reversible algorithm capable of dramatic
improvement of
sampling from known distribution. Our transformation modifies
transition rates
keeping the structure of transitions intact. To illustrate the general
scheme
we design an Irreversible version of Metropolis-Hastings (IMH) and
test it on
example of a spin cluster. Standard MH for the model suffers from the
critical
slow down, while IMH is critical slow down free.
http://arxiv.org/abs/0809.0916
---------------------------------------------------------------
7479. WELL-POSEDNESS OF THE TRANSPORT EQUATION BY STOCHASTIC
PERTURBATION
Franco Flandoli and Massimiliano Gubinelli and Enrico Priola
We consider the linear transport equation with a globally Holder
continuous
and bounded vector field. While this deterministic PDE may not be well-
posed,
we prove that a multiplicative stochastic perturbation of Brownian
type is
enough to render the equation well-posed. This seems to be the first
explicit
example of partial differential equation that become well-posed under
the
influece of noise. The key tool is a differentiable stochastic flow
constructed
and analysed by means of a special transformation of the drift of
Tanaka type.
http://arxiv.org/abs/0809.1310
---------------------------------------------------------------
7480. SOME NOTES ON TREES AND PATHS
Ben Hambly and Terry Lyons
These notes cover background material on trees which are used in the
paper
`On uniqueness of the signature of a path of variation and the reduced
path
group'.
http://arxiv.org/abs/0809.1365
---------------------------------------------------------------
7481. MULTIVARIATE JACOBI AND LAGUERRE POLYNOMIALS, INFINITE-
DIMENSIONAL EXTENSIONS, AND THEIR PROBABILISTIC CONNECTIONS WITH
MULTIVARIATE HAHN AND
MEIXNER POLYNOMIALS
Robert C. Griffiths and Dario Span\`o
Multivariate versions of classical orthogonal polynomials such as
Jacobi,
Hahn, Laguerre, Meixner are reviewed and their connection explored by
adopting
a probabilistic approach. Hahn and Meixner polynomials are interpreted
as
posterior mixtures of Jacobi and Laguerre polynomials, respectively.
By using
known properties of Gamma point processes and related transformations,
an
infinite-dimensional version of Jacobi polynomials is constructed with
respect
to the size-biased version of the Poisson-Dirichlet weight measure and
to the
law of the Gamma point process from which it is derived.
http://arxiv.org/abs/0809.1431
---------------------------------------------------------------
7482. CENTRAL LIMIT THEOREM FOR A CLASS OF ONE-DIMENSIONAL KINETIC
EQUATIONS
Federico Bassetti and Lucia Ladelli and Daniel Matthes
We introduce a class of Boltzmann equations on the real line, which
constitute extensions of the classical Kac caricature. The collisional
gain
operators are defined by smoothing transformations with quite general
properties. By establishing a connection to the central limit problem,
we are
able to prove long-time convergence of the equation's solutions
towards a limit
distribution. If the initial condition for the Boltzmann equation
belongs to
the domain of normal attraction of a certain stable law $g_\alpha$,
then the
limit is non-trivial and is a statistical mixture of dilations of $g_
\alpha$.
Under some additional assumptions, explicit exponential rates for the
equilibration in Wasserstein metrics are calculated, and strong
convergence of
the probability densities is shown.
http://arxiv.org/abs/0809.1545
---------------------------------------------------------------
7483. CORRELATED CONTINUOUS TIME RANDOM WALKS
Mark M. Meerschaert and Erkan Nane and Yimin Xiao
Continuous time random walks impose a random waiting time before each
particle jump. Scaling limits of heavy tailed continuous time random
walks are
governed by fractional evolution equations. Space-fractional derivatives
describe heavy tailed jumps, and the time-fractional version codes
heavy tailed
waiting times. This paper develops scaling limits and governing
equations in
the case of correlated jumps. For long-range dependent jumps, this
leads to
fractional Brownian motion or linear fractional stable motion, with
the time
parameter replaced by an inverse stable subordinator in the case of
heavy
tailed waiting times. These scaling limits provide an interesting
class of
non-Markovian, non-Gaussian self-similar processes.
http://arxiv.org/abs/0809.1612
---------------------------------------------------------------
7484. HEAT CONDUCTION AND ENTROPY PRODUCTION IN ANHARMONIC CRYSTALS
WITH SELF-CONSISTENT STOCHASTIC RESERVOIRS
Federico Bonetto and Joel L. Lebowitz and Jani Lukkarinen and
Stefano Olla (CEREMADE)
We investigate a class of anharmonic crystals in $d$ dimensions, $d\ge
1$,
coupled to both external and internal heat baths of the Ornstein-
Uhlenbeck
type. The external heat baths, applied at the boundaries in the 1-
direction,
are at specified, unequal, temperatures $\tlb$ and $\trb$. The
temperatures of
the internal baths are determined in a self-consistent way by the
requirement
that there be no net energy exchange with the system in the non-
equilibrium
stationary state (NESS). We prove the existence of such a stationary
self-consistent profile of temperatures for a finite system and show it
minimizes the entropy production to leading order in $(\tlb -\trb)$.
In the
NESS the heat conductivity $\kappa$ is defined as the heat flux per
unit area
divided by the length of the system and $(\tlb -\trb)$. In the limit
when the
temperatures of the external reservoirs goes to the same temperature $T
$,
$\kappa(T)$ is given by the Green-Kubo formula, evaluated in an
equilibrium
system coupled to reservoirs all having the temperature $T$. This $
\kappa(T)$
remains bounded as the size of the system goes to infinity. We also
show that
the corresponding infinite system Green-Kubo formula yields a finite
result.
Stronger results are obtained under the assumption that the self-
consistent
profile remains bounded.
http://arxiv.org/abs/0809.0953
---------------------------------------------------------------
7485. ON RESOLVENT IDENTITIES IN GAUSSIAN ENSEMBLES AT THE EDGE OF
THE SPECTRUM
Alexander Soshnikov
We obtain the recursive identities for the joint moments of the traces
of the
powers of the resolvent for Gaussian ensembles of random matrices at
the soft
and hard edges of the spectrum. We also discuss the possible ways to
extend
these results to the non-Gaussian case.
http://arxiv.org/abs/0809.1463
---------------------------------------------------------------
7486. ENERGY DIFFUSION AND SUPERDIFFUSION IN OSCILLATORS LATTICE
NETWORKS
Stefano Olla (CEREMADE)
I review here some recent result on the thermal conductivity of chains
of
oscillators whose hamiltonian dynamics is perturbed by a noise
conserving
energy and momentum.
http://arxiv.org/abs/0809.1496
---------------------------------------------------------------
7487. CONVERGENCE OF THE CRITICAL FINITE-RANGE CONTACT PROCESS TO
SUPER-BROWNIAN MOTION ABOVE THE UPPER CRITICAL DIMENSION: I. THE
HIGHER-POINT
FUNCTIONS
Remco van der Hofstad and Akira Sakai
We consider the critical spread-out contact process in Z^d with d\ge1,
whose
infection range is denoted by L\ge1. In this paper, we investigate the
r-point
function \tau_{\vec t}^{(r)}(\vec x) for r\ge3, which is the
probability that,
for all i=1,...,r-1, the individual located at x_i\in Z^d is infected
at time
t_i by the individual at the origin o\in Z^d at time 0. Together with
the
results of the 2-point function in [van der Hofstad and Sakai,
Electron. J.
Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs
crucially
rely, we prove that the r-point functions converge to the moment
measures of
the canonical measure of super-Brownian motion above the upper-critical
dimension 4. We also prove partial results for d\le4 in a local mean-
field
setting.
http://arxiv.org/abs/0809.1712
---------------------------------------------------------------
7488. LOCAL TIME AND THE PRICING OF TIME-DEPENDENT BARRIER OPTIONS
Aleksandar Mijatovic
A time-dependent double-barrier option is a derivative security that
delivers
the terminal value $\phi(S_T)$ at expiry $T$ if neither of the
continuous
time-dependent barriers $b_\pm:[0,T]\to \RR_+$ have been hit during
the time
interval $[0,T]$. Using a probabilistic approach we obtain a
decomposition of
the barrier option price into the corresponding European option price
minus the
barrier premium for a wide class of payoff functions $\phi$, barrier
functions
$b_\pm$ and linear diffusions $(S_t)_{t\in[0,T]}$. We show that the
barrier
premium can be expressed as a sum of integrals along the barriers $b_
\pm$ of
the option's deltas $\Delta_\pm:[0,T]\to\RR$ at the barriers and that
the pair
of functions $(\Delta_+,\Delta_-)$ solves a system of Volterra integral
equations of the first kind. We find a semi-analytic solution for this
system
in the case of constant double barriers and briefly discus a numerical
algorithm for the time-dependent case.
http://arxiv.org/abs/0809.1747
---------------------------------------------------------------
7489. A MARKOV MODEL FOR THE SPREAD OF HEPATITIS C
Laure Coutin (MAP5) and Laurent Decreusefond (LTCI) and Jean-
Stephane Dhersin (MAP5)
We propose a Markov model for the spread of Hepatitis C virus (HCV)
among
drug users who use injections. We then proceed to an asymptotic
analysis (large
initial population) and show that the Markov process is close to the
solution
of a non linear autonomous differential system. We prove both a law of
large
numbers and functional central limit theorem to precise the speed of
convergence towards the limiting system. The deterministic system itself
converges, as time goes to infinity, to an equilibrium point. This
corroborates
the empirical observations about the prevalence of HCV.
http://arxiv.org/abs/0809.1824
---------------------------------------------------------------
7490. THE DISTRIBUTION OF THE ZEROES OF RANDOM TRIGONOMETRIC POLYNOMIALS
Andrew Granville and Igor Wigman
We study the asymptotic distribution of the number $Z_{N}$ of zeros of
random
trigonometric polynomials of degree $N$ as $N\to\infty$. It is known
that as
$N$ grows to infinity, the expected number of the zeros is asymptotic to
$\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was
predicted
by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove
that
$\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian.
In
addition, we find that the analogous result is applicable for the
number of
zeros in short intervals.
http://arxiv.org/abs/0809.1848
---------------------------------------------------------------
7491. ON THE INVARIANT MEASURE OF THE RANDOM DIFFERENCE EQUATION
$X_N=A_N X_{N-1}+ B_N$ IN THE CRITICAL CASE
Sara Brofferio and Dariusz Buraczewski and Ewa Damek
We consider the autoregressive model on $\R^d$ defined by the following
stochastic recursion $X_n = A_n X_{n-1}+B_n$, where $\{(B_n,A_n)\}$
are i.i.d.
random variables valued in $\R^d\times \R^+$. The critical case, when
$\E\big[\log A_1\big]=0$, was studied by Babillot, Bougeorol and Elie,
who
proved that there exists a unique invariant Radon measure $\nu$ for
the Markov
chain $\{X_n \}$. In the present paper we prove that the weak limit of
properly
dilated measure $\nu$ exists and defines a homogeneous measure on
$\R^d\setminus \{0\}$.
http://arxiv.org/abs/0809.1864
---------------------------------------------------------------
7492. INTERSECTING RANDOM GRAPHS AND NETWORKS WITH MULTIPLE ADJACENCY
CONSTRAINTS: A SIMPLE EXAMPLE
N. Prasanth Anthapadmanabhan and Armand M. Makowski
When studying networks using random graph models, one is sometimes
faced with
situations where the notion of adjacency between nodes reflects multiple
constraints. Traditional random graph models are insufficient to
handle such
situations.
A simple idea to account for multiple constraints consists in
taking the
intersection of random graphs. In this paper we initiate the study of
random
graphs so obtained through a simple example. We examine the
intersection of an
Erdos-Renyi graph and of one-dimensional geometric random graphs. We
investigate the zero-one laws for the property that there are no
isolated
nodes. When the geometric component is defined on the unit circle, a
full
zero-one law is established and we determine its critical scaling.
When the
geometric component lies in the unit interval, there is a gap in that
the
obtained zero and one laws are found to express deviations from
different
critical scalings. In particular, the first moment method requires a
larger
critical scaling than in the unit circle case in order to obtain the
one law.
This discrepancy is somewhat surprising given that the zero-one laws
for the
absence of isolated nodes are identical in the geometric random graphs
on both
the unit interval and unit circle.
http://arxiv.org/abs/0809.0918
---------------------------------------------------------------
7493. INCOHERENT DICTIONARIES AND THE STATISTICAL RESTRICTED ISOMETRY
PROPERTY
Shamgar Gurevich (UC Berkeley) and Ronny Hadani (University of Chicago)
In this paper we formulate and prove a statistical version of the
Candes-Tao
restricted isometry property (SRIP for short) which holds in general
for any
incoherent dictionary which is a disjoint union of orthonormal bases. In
addition, we prove that, under appropriate normalization, the
eigenvalues of
the associated Gram matrix fluctuate around \lambda = 1 according to
the Wigner
semicircle distribution. The result is then applied to various
dictionaries
that arise naturally in the setting of finite harmonic analysis,
giving, in
particular, a better understanding on a remark of
Applebaum-Howard-Searle-Calderbank concerning RIP for the Heisenberg
dictionary
of chirp like functions.
http://arxiv.org/abs/0809.1687
---------------------------------------------------------------
7494. AFFINE MODELS
Christa Cuchiero and Damir Filipovic and Josef Teichmann
Affine term structure models have gained a lot of attention in the
finance
literature, which is due to their analytic tractability and statistical
flexibility. The aim of this article is to present both, theoretical
foundations and empirical aspects. Starting from the first short rate
models,
namely the Vasi\v{c}ek and the Cox-Ingersoll-Ross ones, we then give an
overview of some properties of affine processes and explain their
relation to
affine term structure models. Pricing and estimation techniques are
eventually
mentioned, showing how the analytic tractability of affine models can be
exploited for practical purposes.
http://arxiv.org/abs/0809.1985
---------------------------------------------------------------
7495. ROUGH VOLTERRA EQUATIONS 1: THE ALGEBRAIC INTEGRATION SETTING
Aur\'elien Deya (IECN) and Samy Tindel (IECN)
We define and solve Volterra equations driven by an irregular signal, by
means of a variant of the rough path theory called algebraic
integration. In
the Young case, that is for a driving signal with H\"older exponent
greater
than 1/2, we obtain a global solution, and are able to handle the case
of a
singular Volterra coefficient. In case of a driving signal with H\"older
exponent in (1/3,1/2], we get a local existence and uniqueness
theorem. The
results are easily applied to the fractional Brownian motion with Hurst
coefficient H>1/3.
http://arxiv.org/abs/0809.2000
---------------------------------------------------------------
7496. SOME REMARKS ON BETTI NUMBERS OF RANDOM POLYGON SPACES
Cl\'ement Dombry (LMA) and Christian Mazza
Polygon spaces like $M_\ell=\{(u_1,...,u_n)\in S^1\times... S^1 ;\
\sum_{i=1}^n l_iu_i=0\}/SO(2)$ or they three dimensional analogues $N_
\ell$
play an important r\^ole in geometry and topology, and are also of
interest in
robotics where the $l_i$ model the lengths of robot arms. When $n$ is
large,
one can assume that each $l_i$ is a positive real valued random
variable,
leading to a random manifold. The complexity of such manifolds can be
approached by computing Betti numbers, the Euler characteristics, or the
related Poincar\'e polynomial. We study the average values of Betti
numbers of
dimension $p_n$ when $p_n\to\infty$ as $n\to\infty$. We also focus on
the
limiting mean Poincar\'e polynomial, in two and three dimensions. We
show that
in two dimensions, the mean total Betti number behaves as the total
Betti
number associated with the equilateral manifold where $l_i\equiv \bar l
$. In
three dimensions, these two quantities are not any more asymptotically
equivalent. We also provide asymptotics for the Poincar\'e polynomials
http://arxiv.org/abs/0809.2082
---------------------------------------------------------------
7497. LIMIT LAW OF THE LOCAL TIME FOR BROX'S DIFFUSION
Pierre Andreoletti (MAPMO) and Roland Diel (MAPMO)
We consider Brox's model: a one-dimensional diffusion in a Brownian
environment. We show the weak convergence of the normalized local time
process
$(L(x+m_{\log t},t)/t,x\in I \subset \R)$, centered at the coordinate
of the
bottom of the deepest valley $m_{\log t}$ reached by the process
before time
$t$ to a functional of two independent 3-dimensional Bessel processes.
We apply
that result to get the limit law of the supremum of the normalized
local time.
These results are discussed and compared to the discrete time and space
analogous model whose same questions have been solved recently by N.
Ganter, Y.
Peres and Z. Shi.
http://arxiv.org/abs/0809.2195
---------------------------------------------------------------
7498. CONTINUOUS LERW STARTED FROM INTERIOR POINTS
Dapeng Zhan
We use the whole-plane Loewner equation to define a family of
continuous LERW
in finitely connected domains that are started from interior points.
These
continuous LERW satisfy conformal invariance, preserve some continuous
local
martingales, and are the scaling limits of the corresponding discrete
LERW on
the discrete approximation of the domains.
http://arxiv.org/abs/0809.2230
---------------------------------------------------------------
7499. ON INCOMPLETENESS OF BOND MARKETS WITH INFINITE NUMBER OF RANDOM
FACTORS
Michal Baran and Jacek Jakubowski and Jerzy Zabczyk
The completeness of a bond market model with infinite number of
sources of
randomness on a finite time interval in the Heath-Jarrow-Morton
framework is
studied. It is proved that the market in the case of trading
strategies is not
complete. An explicit construction of a bounded contingent claim,
which can not
be replicated, is provided. Moreover, a new concept of generalized
strategies
is introduced and sufficient conditions for the market completeness
with such
strategies are given. An example of a complete model is provided.
http://arxiv.org/abs/0809.2270
---------------------------------------------------------------
7500. RAMSEY-TYPE RESULTS ON RANDOM GRAPHS
A. Berarducci and P. Majer and M. Novaga
We prove some Ramsey-type theorems, where the colours are replaced by
measurable subsets of a probability space. Such results can be also
interpreted
as percolation problems on a countable complete graph, without
requiring any
specific assumption on the probability, such as independency. In
particular, we
provide sharp estimates on the probability of finding an infinite
clique or an
infinite monotone path in a random subgraph.
http://arxiv.org/abs/0809.2335
---------------------------------------------------------------
7501. A NEW PROBABILITY INEQUALITY USING TYPICAL MOMENTS AND
CONCENTRATION RESULTS
Ravi Kannan
A new probability inequality strengthening Azuma's is presented.
Instead of
assuming an absolute bound on the Martingale differences as Azuma's
does, we
use information on conditional moments of each variable. Information
on moments
conditioned on "typical" as well as "worst-case" values are used.
Also, a
weaker assumption than Martingale differences suffices here. The
inequality is
strong enough to yield sub-Gaussian tail bounds in many situations. We
present
several applications. For bin-packing optimal concentration intervals
are
proved settling a question of Talagrand. For chromatic number of
sparse random
graphs, the first sub-Gaussian tail bounds are proved and the first
treatment
of the number of cliques of growing sizes in random graphs is given. For
Lonegest Sub-sequences, a result matching Talagrand's is given. Brief
comparisons to other recent inequalities are made.
http://arxiv.org/abs/0809.2477
---------------------------------------------------------------
7502. EXPONENTIAL FUNCTIONALS OF BROWNIAN MOTION AND CLASS ONE
WHITTAKER FUNCTIONS
Fabrice Baudoin and Neil O'Connell
We consider exponential functionals of a multi-dimensional Brownian
motion
with drift, defined via a collection of linear functionals. We give a
characterisation of the Laplace transform of their joint law as the
unique
bounded solution, up to a constant factor, to a Schrodinger-type partial
differential equation. We derive a similar equation for the probability
density. We then characterise all diffusions which can be interpreted
as having
the law of the Brownian motion with drift conditioned on the law of its
exponential functionals. In the case where the family of linear
functionals is
a set of simple roots, the Laplace transform of the joint law of the
corresponding exponential functionals can be expressed in terms of a
class one
Whittaker function associated with the corresponding root system. In
this
setting, we establish some basic properties of the corresponding
diffusions,
which we call Whittaker processes.
http://arxiv.org/abs/0809.2506
---------------------------------------------------------------
7503. LOOP-ERASED RANDOM WALK AND POISSON KERNEL ON PLANAR GRAPHS
Ariel Yadin and Amir Yehudayoff
Lawler, Schramm and Werner showed that the scaling limit of the loop-
erased
random walk on the square lattice is SLE(2). We consider scaling
limits of the
loop-erasure of random walks on other planar graphs (graphs embedded
into the
complex plane so that edges do not cross one another). We show that if
the
scaling limit of the random walk is planar Brownian motion, then the
scaling
limit of its loop-erasure is SLE(2). Our main contribution is showing
that for
such graphs, the discrete Poisson kernel can be approximated by the
continuous
one.
One example is the infinite component of super-critical percolation
on the
square lattice. Berger and Biskup showed that the scaling limit of the
random
walk on this graph is planar Brownian motion. Our results imply that the
scaling limit of the loop-erased random walk on the super-critical
percolation
cluster is SLE(2).
http://arxiv.org/abs/0809.2643
---------------------------------------------------------------
7504. ANOMALOUS BEHAVIOR OF THE KRAMERS RATE AT BIFURCATIONS IN
CLASSICAL FIELD THEORIES
Nils Berglund (MAPMO) and Barbara Gentz
We consider a Ginzburg-Landau partial differential equation in a bounded
interval, perturbed by weak spatio-temporal noise. As the interval
length
increases, a transition between activation regimes occurs, in which the
classical Kramers rate diverges [R.S. Maier and D.L. Stein, Phys. Rev.
Lett.
87, 270601 (2001)]. We determine a corrected Kramers formula at the
transition
point, yielding a finite, though noise-dependent prefactor, confirming a
conjecture by Maier and Stein [vol. 5114 of SPIE Proceeding (2003)].
For both
periodic and Neumann boundary conditions, we obtain explicit
expressions of the
prefactor in terms of Bessel and error functions.
http://arxiv.org/abs/0809.2652
---------------------------------------------------------------
7505. LOGARITHMIC SOBOLEV INEQUALITIES: REGULARIZING EFFECT OF L\'EVY
OPERATORS AND ASYMPTOTIC CONVERGENCE IN THE L\'EVY-FOKKER-PLANCK
EQUATION
Ivan Gentil (CEREMADE) and Cyril Imbert (CEREMADE)
In this paper we study some applications of the L\'evy logarithmic
Sobolev
inequality to the study of the regularity of the solution of the
fractal heat
equation, i. e. the heat equation where the Laplacian is replaced with
the
fractional Laplacian. It is also used to the study of the asymptotic
behaviour
of the L\'evy-Ornstein-Uhlenbeck process.
http://arxiv.org/abs/0809.2654
---------------------------------------------------------------
7506. PROPORTIONAL FAIRNESS AND ITS RELATIONSHIP WITH MULTI-CLASS
QUEUEING NETWORKS
N.S. Walton
A network of single server queues with routing is considered. These
networks
have a product form stationary distribution. A new limit result proves a
sequence of such networks converges weakly to a stochastic flow level
model.
This stochastic model is insensitive. A large deviation principle for
the
stationary distribution of these queueing networks is found. Its rate
function
has a dual formulation that coincides with proportional fairness. It
is proven
that the stationary throughput of the queueing model converges to a
proportionally fair allocation.
The queueing models considered have no prescribed optimization
structure.
Regardless of this, we find a proportionally fair allocation forms an
entropy
minimizing state of these networks. Proportional fairness occurs as a
consequence of a collapse in the state space of the queueing model.
This work combines classical queueing networks with more recent
work on
stochastic flow level models and proportional fairness. One could view
these
seemingly different models as the same system described at different
levels of
granularity: a microscopic, queueing level description; a macroscopic,
flow
level description and a teleological, optimization description.
http://arxiv.org/abs/0809.2697
---------------------------------------------------------------
7507. CRITERIA FOR STRONG AND WEAK RANDOM ATTRACTORS
Hans Crauel (Goethe-Universit\"at Frankfurt) and Georgi Dimitroff
(ITWM Kaiserslautern), Michael Scheutzow (TU Berlin)
The theory of random attractors has different notions of attraction,
amongst
them pullback attraction and weak attraction. We investigate necessary
and
sufficient conditions for the existence of pullback attractors as well
as of
weak attractors.
http://arxiv.org/abs/0809.2719
---------------------------------------------------------------
7508. DYNAMICS OF VERTEX-REINFORCED RANDOM WALKS
Michel Benaim and Pierre Tarres
We generalize a result from Volkov (2001) and prove that, on an
arbitrary
graph of bounded degree $(G,\sim)$ and for any symmetric reinforcement
matrix
$a=(a_{i,j})_{i\sim j}$, the vertex-reinforced random walk (VRRW)
eventually
localizes with positive probability on subsets which consist of a
complete
$d$-partite subgraph plus its outer boundary.
We first show that, in general, any stable equilibrium of a linear
symmetric
replicator dynamics with positive payoffs on a graph $G$ satisfies the
property
that its support is a complete $d$-partite subgraph of $G$ for some $d
\ge2$.
This result is used here for the study of VRRWs, but also applies to
other
contexts such as evolutionary models in population genetics and game
theory.
Next we generalize the result of Pemantle (1992) and Benaim (1997)
relating
the asymptotic behaviour of the VRRW to replicator dynamics. This
enables us to
conclude that, given any neighbourhood of a strictly stable
equilibrium with
support $S$, the following event occurs with positive probability: the
walk
localizes on $S\cup\partial S$, (where $\partial S$ is the outer
boundary of
$S$) and the density of occupation of the VRRW converges, with
polynomial rate,
to a strictly stable equilibrium in this neighbourhood.
http://arxiv.org/abs/0809.2739
---------------------------------------------------------------
7509. STRONG LAW OF LARGE NUMBERS FOR FRAGMENTATION PROCESSES
S.C. Harris and R. Knobloch and A.E. Kyprianou
In the spirit of a classical results for Crump-Mode-Jagers processes, we
prove a strong law of large numbers for homogenous fragmentation
processes.
Specifically, for self-similar fragmentation processes, including
homogenous
processes, we prove the almost sure convergence of an empirical measure
associated with the stopping line corresponding to first fragments of
size
strictly smaller than $\eta$ for $1\geq \eta >0$.
http://arxiv.org/abs/0809.2958
---------------------------------------------------------------
7510. POISSON LIMIT FOR ASSOCIATED RANDOM FIELDS
Yuri Bakhtin
We prove that under an easily verifiable set of conditions a sequence of
associated random fields converges under rescaling to the Poisson
Point Process
and give a couple of examples.
http://arxiv.org/abs/0809.2971
---------------------------------------------------------------
7511. THERMODYNAMIC LIMIT FOR LARGE RANDOM TREES
Yuri Bakhtin
We consider Gibbs distributions on finite random plane trees with
bounded
branching. We show that as the order of the tree grows to infinity, the
distribution of any finite neighborhood of the root of the tree
converges to a
limit. We compute the limiting distribution explicitly and study its
properties. We introduce an infinite random tree consistent with these
limiting
distributions and show that it satisfies a certain form of the Markov
property.
We also study the growth of this tree and prove several limit theorems
including a diffusion approximation.
http://arxiv.org/abs/0809.2974
---------------------------------------------------------------
7512. UNIFORM ESTIMATES FOR ORDER STATISTICS AND ORLICZ FUNCTIONS
Yehoram Gordon and Alexander Litvak and Carsten Sch\"utt and
Elisabeth Werner
We establish uniform estimates for order statistics of sequences of
independent identically distributed random variables with log-concave
distribution in terms of Orlicz norms associated with the distribution
function
of the random variables.
http://arxiv.org/abs/0809.2989
---------------------------------------------------------------
7513. UNIQUENESS OF THE STATIONARY DISTRIBUTION AND STABILIZABILITY IN
ZHANG'S SANDPILE MODEL
Anne Fey and Haiyan Liu and Ronald Meester
We show that Zhang's sandpile model (N,[a,b]) on N sites and with
uniform
additions on [a,b] has a unique stationary measure for all 0 <= a < b
<= 1.
This generalizes earlier results where this was shown in some special
cases.
We define the infinite volume Zhang's sandpile model in dimension d
>= 1, in
which topplings occur according to a Markov toppling process, and we
study the
stabilizability of initial configurations chosen according to some
measure \mu.
We show that for a stationary ergodic measure \mu with density \rho,
for all
\rho < 1/2, \mu is stabilizable; for all \rho >= 1, \mu is not
stabilizable;
for 1/2 <= \rho < 1, when \rho is near to 1/2 or 1, both possibilities
can
occur.
http://arxiv.org/abs/0809.2913
---------------------------------------------------------------
7514. SOME NOTES ON STANDARD BOREL AND RELATED SPACES
Chris Preston
These notes give an elementary approach to parts of the theory of
standard
Borel and analytic spaces.
http://arxiv.org/abs/0809.3066
---------------------------------------------------------------
7515. STOCHASTIC ANALYSIS OF BERNOULLI PROCESSES
Nicolas Privault
These notes survey some aspects of discrete-time chaotic calculus and
its
applications, based on the chaos representation property for i.i.d.
sequences
of random variables. The topics covered include the Clark formula and
predictable representation, anticipating calculus, covariance
identities and
functional inequalities (such as deviation and logarithmic Sobolev
inequalities), and an application to option hedging in discrete time.
http://arxiv.org/abs/0809.3168
---------------------------------------------------------------
7516. A NEW FRAMEWORK OF MULTISTAGE HYPOTHESIS TESTS
Xinjia Chen
In this paper, we have established a new framework of multistage
hypothesis
tests. Within the new framework, we have developed specific multistage
tests
which guarantee prescribed level of power and are more efficient than
previous
tests in terms of average sampling number and the number of sampling
operations. Without truncation, the maximum sampling numbers of our
testing
plans are absolutely bounded.
http://arxiv.org/abs/0809.3170
---------------------------------------------------------------
7517. ON THE NORM OF THE BEURLING-AHLFORS OPERATOR IN SEVERAL DIMENSIONS
Tuomas Hyt\"onen
The Lp operator norm of the generalized Beurling-Ahlfors
transformation in n
variables is at most (n/2+1)(p-1) for p>2. This improves on earlier
results in
all dimensions n>2. The proof is based on the heat extension and
relies at the
bottom on Burkholder's sharp inequality for martingale transforms.
http://arxiv.org/abs/0809.3127
---------------------------------------------------------------
7518. IMPLIED VOLATILITY EXPLOSIONS: EUROPEAN CALLS AND IMPLIED
VOLATILITIES CLOSE TO EXPIRY IN EXPONENTIAL L\'EVY MODELS
Michael Roper
We examine the small expiry behaviour of European call options in
stock price
models of exponential L\'evy type. In most cases of interest, we are
able to
identify the exact small expiry asymptotics. In "complete generality"
we are
able to show that the time value of the call option has O(\tau) decay
as \tau
(time to expiry) goes to zero. Using our results on the behaviour of
call
options close to expiry we show that implied volatility explodes as
$\tau\to0^+$ in "most" exponential L\'evy models. Attention is
restricted to
calls and implied volatilities that are not at-the-money.
http://arxiv.org/abs/0809.3305
---------------------------------------------------------------
7519. ANALYSIS OF VALUATION FORMULAE AND APPLICATIONS TO EXOTIC
OPTIONS IN L\'EVY MODELS
Ernst Eberlein and Kathrin Glau and Antonis Papapantoleon
We discuss the valuation problem for a broad spectrum of plain vanilla
and
path-dependent options in a general modeling framework, and
specifically for
L\'evy driven models. Among the derivatives which we consider are
digitals,
double digitals, asset-or-nothing options, self-quantos, lookback and
one-touch
options. Extensions to the multivariate case, i.e. basket options and
options
on the minimum or maximum of several assets are considered as well. As
a link
to credit risk we derive a valuation formula for equity default swaps.
The key
idea of this Fourier, resp. Laplace transform based approach is the
clear
separation of two ingredients, namely the payoff function and the
underlying
process. The latter can be the driving process itself, but also the
supremum or
infimum or another process derived from it. For L\'evy processes the
analytically extended characteristic functions of the supremum and the
infimum
process are derived.
http://arxiv.org/abs/0809.3405
---------------------------------------------------------------
7520. SCALING LIMIT FOR A DRAINAGE NETWORK MODEL
C. F. Coletti and E. S. Dias and L. R. G. Fontes
We consider the two dimensional version of a drainage network model
introduced by Gangopadhyay, Roy and Sarkar, and show that the
appropriately
rescaled family of its paths converges in distribution to the Brownian
web. We
do so by verifying the convergence criteria proposed by Fontes, Isopi,
Newman
and Ravishankar.
http://arxiv.org/abs/0809.3454
---------------------------------------------------------------
7521. CONVERGENCE OF SYMMETRIC TRAP MODELS IN THE HYPERCUBE
L. R. G. Fontes and P. H. S. Lima
We consider symmetric trap models in the d-dimensional hypercube whose
ordered mean waiting times, seen as weights of a measure in the natural
numbers, converge to a finite measure as d diverges, and show that the
models
suitably represented converge to a K process as d diverges. We then
apply this
result to get K processes as the scaling limits of the REM-like trap
model and
the Random Hopping Times dynamics for the Random Energy Model in the
hypercube
in time scales corresponding to the ergodic regime for these dynamics.
http://arxiv.org/abs/0809.3463
---------------------------------------------------------------
7522. AVERAGED LARGE DEVIATIONS FOR RANDOM WALK IN A RANDOM ENVIRONMENT
Atilla Yilmaz
In his 2003 paper, Varadhan proves the averaged large deviation
principle
(LDP) for the mean velocity of a particle performing random walk in a
random
environment (RWRE) on $\mathbb{Z}^d$ with $d\geq1$, and gives a
variational
formula for the corresponding rate function $I_a$. Under the non-
nestling
assumption (resp. Kalikow's condition), we show that $I_a$ is strictly
convex
and analytic on a non-empty open set $\mathcal{A}$, and that the true
velocity
$\xi_o$ of the particle is an element (resp. in the closure) of $
\mathcal{A}$.
We then identify the minimizer of Varadhan's variational formula at any
$\xi\in\mathcal{A}$.
http://arxiv.org/abs/0809.3467
---------------------------------------------------------------
7523. PERFECT SIMULATION AND FINITARY CODING FOR MULTICOLOR SYSTEMS
WITH INTERACTIONS OF INFINITE RANGE
A. Galves and N.L. Garcia and E. Loecherbach
We consider a particle system on $Z^d$ with finite state space and
interactions of infinite range. Assuming that the rate of change is
continuous
and decays sufficiently fast, we introduce a perfect simulation
algorithm for
the stationary process. The algorithm follows from a representation of
the
multicolor system as a finitary coding from a sequence of independent
uniform
random variables. This implies that the process is exponentially
ergodic. The
basic tool we use is a representation of the infinite range change
rates as a
mixture of finite range change rates.
http://arxiv.org/abs/0809.3494
---------------------------------------------------------------
7524. OPTIMAL L$^1$-BOUNDS FOR SUBMARTINGALES
Lutz Mattner and Uwe R\"osler
The optimal function $f$ satisfying
$$
\mathbb{E} |\sum_{1}^n X_i |
\ge f(\mathrbb{E}|X_1|,...,\mathbb{E}|X_n|)
$$ for every martingale $(X_1,X_1+X_2, ...,\sum_{i=1}^n X_i)$ is
shown to be
given by $$ f(a) = \max \Big\{a_k-\sum_{i=1}^{k-1} a_i\Big\}_{k=1}^n
\cup
\Big\{\frac {a_k}2\Big\}_{k=3}^n $$ for $a\in{[0,\infty[}^n_{}$. A
similar
result is obtained for submartingales $(0,X_1,X_1+X_2,...,
\sum_{i=1}^n X_i)$.
The optimality proofs use a convex-analytic comparison lemma of
independent
interest.
http://arxiv.org/abs/0809.3522
---------------------------------------------------------------
7525. DYNAMIC TREE ALGORITHMS
Hanene Mohamed (INRIA Rocquencourt) and Philippe Robert
In this paper, a general tree algorithm processing a random flow of
arrivals
is analyzed. Capetanakis-Tsybakov-Mikhailov's protocol in the context of
communication networks with random access is an example of such an
algorithm.
In computer science, this corresponds to a trie structure with a
dynamic input.
Mathematically, it is related to a stopped branching process with
exogeneous
arrivals (immigration). Under quite general assumptions on the
distribution of
the number of arrivals and on the branching procedure, it is shown
that there
exists a {\em positive} constant $\lambda_c$ so that if the arrival
rate is
smaller than $\lambda_c$, then the algorithm is stable under the flow of
requests, i.e. that the total size of an associated tree is
integrable. At the
same time a gap in the earlier proofs of stability of the literature
is fixed.
When the arrivals are Poisson, an explicit characterization of $
\lambda_c$ is
given. Under the stability condition, the asymptotic behavior of the
average
size of a tree starting with a large number of individuals is
analyzed. The
results are obtained with the help of a probabilistic rewriting of the
functional equations describing the dynamic of the system. The proofs
use
extensively this stochastic background throughout the paper. In this
analysis,
two basic limit theorems play a key role: the renewal theorem and the
convergence to equilibrium of an auto-regressive process with moving
average.
http://arxiv.org/abs/0809.3577
---------------------------------------------------------------
7526. SOME LIMIT THEOREMS FOR RESCALED WICK POWERS
Alberto Lanconelli
We establish the strong L2(P)-convergence of properly rescaled Wick
powers as
the power index tends to infinity. The explicit representation of such
limit
will also provide the convergence in distribution to normal and log-
normal
random variables. The proofs rely on some estimates for the L2(P)-norm
of Wick
products and on the properties of second quantization operators.
http://arxiv.org/abs/0809.3702
---------------------------------------------------------------
7527. UNIVERSALITY IN COMPLEX WISHART ENSEMBLES: THE 2 CUT CASE
M. Y. Mo
We studied the universality of Wishart ensembles whose covariance
matrix has
2 distinct eigenvalues. We studied the asymptotic limit when the
number of both
eigenvalues goes to infinity and obtained universality results. In
this case,
the limiting eigenvalue distribution can be supported on 1 or 2 disjoint
intervals. We obtained a necessary and sufficient condition on the
parameters
such that the limiting distribution is supported on 2 disjoint
intervals and
have computed the eigenvalue density in the limit. Furthermore, by using
Riemann-Hilbert analysis, we have shown that under proper rescaling of
the
eigenvalues, the limiting correlation kernel is given by the sine
kernel and
the Airy kernel in the bulk and the edge of the spectrum respectively.
As a
consequence, the behavior of the largest eigenvalue in this model is
described
by the Tracy-Widom distribution.
http://arxiv.org/abs/0809.3750
---------------------------------------------------------------
7528. STOCHASTIC AND DETERMINISTIC APPROACHES FOR POPULATIONS WITH AGE
AND TRAIT-STRUCTURE
Regis Ferriere and Viet Chi Tran (LPP)
This work is a proceeding of the CANUM 2008 conference. Understanding
how
stochastic and non-linear deterministic processes interact is a major
challenge
in population dynamics theory. After a short review, we introduce a
stochastic
individual-centered particle model to describe the evolution in
continuous time
of a population with (continuous) age and trait structures. The
individuals
reproduce asexually, age, interact and die. In a large population
limit, the
random process converges to the solution of a Gurtin-McCamy type PDE.
We show
that the random model has a long time behavior that differs from its
deterministic limit. However, the results on the limiting PDE and large
deviation techniques \textit{\`a la} Freidlin-Wentzell provide
estimates of the
extinction time and a better understanding of the long time behavior
of the
stochastic process. This has applications in the theory of Adaptive
Dynamics.
In a last section, we present on simulations three biological issues
dealing
with the consequences of size plasticity when taking growth into
account, with
growth-reproduction trade-offs and with periodic behavior.
http://arxiv.org/abs/0809.3767
---------------------------------------------------------------
7529. GRASSMANNIAN ESTIMATION
Claude Auderset and Christian Mazza and Ernst Ruh
This paper discusses the family of distributions on the Grassmannian
of the
linear span of r central gaussian vectors parametrized by the covariance
matrix. Our main result is an existence and uniqueness criterion for the
maximum likelihood estimate of a sample.
http://arxiv.org/abs/0809.3697
---------------------------------------------------------------
7530. SPATIAL MARKOV SEMIGROUPS ADMIT HUDSON-PARTHASARATHY DILATIONS
Michael Skeide
For many Markov semigroups dilations in the sense of Hudson and
Parthasarathy, that is a dilation which is a cocycle perturbation of a
noise,
have been constructed with the help of quantum stochastic calculi. In
these
notes we show that every Markov semigroup on the algebra of all bounded
operators on a separable Hilbert space that is spatial in the sense of
Arveson,
admits a Hudson-Parthasarathy dilation. In a sense, the opposite is
also true.
The proof is based on general results on the the relation between
spatial
E_0-semigroups and their product systems.
http://arxiv.org/abs/0809.3538
---------------------------------------------------------------
7531. TIME CONSISTENT DYNAMIC LIMIT ORDER BOOKS CALIBRATED ON OPTIONS
Jocelyne Bion-Nadal
In an incomplete financial market, the axiomatic of Time Consistent
Pricing
Procedure (TCPP), recently introduced, is used to assign to any
financial asset
a dynamic limit order book, taking into account both the dynamics of
basic
assets and the limit order books for options.
Kreps-Yan fundamental theorem is extended to that context. A
characterization
of TCPP calibrated on options is given in terms of their dual
representation.
In case of perfectly liquid options, these options can be used as the
basic
assets to hedge dynamically. A generic family of TCPP calibrated on
option
prices is constructed, from cadlag BMO martingales.
http://arxiv.org/abs/0809.3824
---------------------------------------------------------------
7532. CLUSTERING OF DISCRETELY OBSERVED DIFFUSION PROCESSES
Alessandro De Gregorio and Stefano Maria Iacus
In this paper a new dissimilarity measure to identify groups of assets
dynamics is proposed. The underlying generating process is assumed to
be a
diffusion process solution of stochastic differential equations and
observed at
discrete time. The mesh of observations is not required to shrink to
zero. As
distance between two observed paths, the quadratic distance of the
corresponding estimated Markov operators is considered. Analysis of both
synthetic data and real financial data from NYSE/NASDAQ stocks, give
evidence
that this distance seems capable to catch differences in both the
drift and
diffusion coefficients contrary to other commonly used metrics.
http://arxiv.org/abs/0809.3902
---------------------------------------------------------------
7533. MULTILEVEL DISCRETIZED RANDOM FIELD MODELS WITH "SPIN"
CORRELATIONS FOR THE SIMULATION OF ENVIRONMENTAL SPATIAL DATA
Milan \v{Z}ukovi\v{c} and Dionissios T. Hristopulos
A problem of practical significance is the analysis of large, spatially
distributed data sets. The problem is more challenging for variables
that
follow non-Gaussian distributions. We show that the spatial correlations
between variables can be captured by interactions between "spins". The
spins
represent multilevel discretizations of the initial field with respect
to a
number of pre-defined thresholds. The spatial dependence between the
"spins" is
imposed by means of short-range interactions. We present two approaches,
inspired by the Ising and Potts models, that generate conditional
simulations
from samples with missing data. The simulations of the "spin system"
are forced
to respect locally the sample values and the system statistics
globally. We
compare the two approaches in terms of their ability to reproduce the
sample
statistical properties, to predict data at unsampled locations, as
well as in
terms of their computational complexity. We discuss the impact of
relevant
simulation parameters, such as the domain size, the number of
discretization
levels, and the initial conditions.
http://arxiv.org/abs/0809.3918
---------------------------------------------------------------
7534. FROM PET TO SPLIT
Yuri Kifer
Various forms of the polynomial ergodic theorem (PET) which attracted
substantial attention in ergodic theory study the limits of
expressions having
the form $1/N\sum_{n=1}^NT^{q_1(n)}f_1... T^{q_\ell (n)}f_\ell$ where
$T$ is a
weakly mixing measure preserving transformation, $f_i$'s are bounded
measurable
functions and $q_i$'s are polynomials taking on integer values on the
integers.
Motivated partially by these results we obtain a central limit theorem
for
expressions of the form $1/\sqrt{N}\sum_{n=1}^N
(X_1(q_1(n))X_2(q_2(n))...
X_\ell(q_\ell(n))-a_1a_2... a_\ell)$ (sum-product limit theorem--
SPLIT) where
$X_i$'s are fast $\alpha$-mixing bounded stationary processes,
$a_j=EX_j(0)$
and $q_i$'s are positive functions taking on integer values on
integers with
some growth conditions which are satisfied, for instance, when $q_i$'s
are
polynomials of growing degrees. This result can be applied to the case
when
$X_i(n)=T^nf_i$ where $T$ is a mixing subshift of finite type, a
hyperbolic
diffeomorphism or an expanding transformation taken with a Gibbs
invariant
measure, as well, as to the case when $X_i(n)=f_i(\xi_n)$ where $\xi_n
$ is a
Markov chain satisfying the Doeblin condition considered as a stationary
process with respect to its invariant measure.
http://arxiv.org/abs/0809.4106
---------------------------------------------------------------
7535. A PROBLEM IN ONE-DIMENSIONAL DIFFUSION-LIMITED AGGREGATION (DLA)
AND POSITIVE RECURRENCE OF MARKOV CHAINS
Harry Kesten and Vladas Sidoravicius
We consider the following problem in one-dimensional diffusion-limited
aggregation (DLA). At time $t$, we have an "aggregate" consisting of
$\Bbb{Z}\cap[0,R(t)]$ [with $R(t)$ a positive integer]. We also have
$N(i,t)$
particles at $i$, $i>R(t)$. All these particles perform independent
continuous-time symmetric simple random walks until the first time
$t'>t$ at
which some particle tries to jump from $R(t)+1$ to $R(t)$. The
aggregate is
then increased to the integers in $[0,R(t')]=[0,R(t)+1]$ [so that
$R(t')=R(t)+1$] and all particles which were at $R(t)+1$ at time $t'{-}
$ are
removed from the system. The problem is to determine how fast $R(t)$
grows as a
function of $t$ if we start at time 0 with $R(0)=0$ and the $N(i,0)$
i.i.d.
Poisson variables with mean $\mu>0$. It is shown that if $\mu<1$, then
$R(t)$
is of order $\sqrt{t}$, in a sense which is made precise. It is
conjectured
that $R(t)$ will grow linearly in $t$ if $\mu$ is large enough.
http://arxiv.org/abs/0809.4175
---------------------------------------------------------------
7536. APPROXIMATE ZERO-ONE LAWS AND SHARPNESS OF THE PERCOLATION
TRANSITION IN A CLASS OF MODELS INCLUDING TWO-DIMENSIONAL ISING
PERCOLATION
J. van den Berg
One of the most well-known classical results for site percolation on the
square lattice is the equation $p_c+p_c^*=1$. In words, this equation
means
that for all values $\neq p_c$ of the parameter $p$, the following
holds:
either a.s. there is an infinite open cluster or a.s. there is an
infinite
closed "star" cluster. This result is closely related to the percolation
transition being sharp: below $p_c$, the size of the open cluster of a
given
vertex is not only (a.s.) finite, but has a distribution with an
exponential
tail. The analog of this result has been proven by Higuchi in 1993 for
two-dimensional Ising percolation (at fixed inverse temperature
$\beta<\beta_c$) with external field $h$, the parameter of the model.
Using
sharp-threshold results (approximate zero-one laws) and a modification
of an
RSW-like result by Bollob\'{a}s and Riordan, we show that these
results hold
for a large class of percolation models where the vertex values can be
"nicely"
represented (in a sense which will be defined precisely) by i.i.d.
random
variables. We point out that the ordinary percolation model obviously
belongs
to this class and we also show that the Ising model mentioned above
belongs to
it.
http://arxiv.org/abs/0809.4184
---------------------------------------------------------------
7537. BOUNDED HARMONIC FUNCTIONS FOR THE HECKMAN--OPDAM LAPLACIAN
Bruno Schapira (LM-Orsay)
We describe the set of bounded harmonic functions for the Heckman--Opdam
Laplacian, when the multiplicity function is larger than 1/2. We prove
that
this set is a vector space of dimension the cardinality of the Weyl
group. We
give some consequences in terms of the associated hypergeometric
functions.
http://arxiv.org/abs/0809.4232
---------------------------------------------------------------
7538. EXPECTED COALESCENCE TIME FOR A NONUNIFORM ALLOCATION PROCESS
John K. McSweeney and Boris G. Pittel
We give an asymptotic expression for the expected coalescence time for a
non-uniform balls-into-boxes allocation model. Connections to coalescent
processes in population biology and computer science are discussed.
http://arxiv.org/abs/0809.4233
---------------------------------------------------------------
7539. HEAVY-TRAFFIC LIMITS FOR WAITING TIMES IN MANY-SERVER QUEUES
WITH ABANDONMENT
Rishi Talreja and Ward Whitt
We establish heavy-traffic stochastic-process limits for waiting times
in
many-server queues with customer abandonment. If the system is
asymptotically
critically loaded, as in the quality-and-efficiency-driven (QED)
regime, then a
bounding argument shows that the abandonment does not affect waiting-
time
processes so that the model behaves as if there were no abandonment
assumptions. If instead the system is overloaded, as in the efficiency-
driven
(ED) regime, following Mandelbaum, Massey, Reiman and Stolyar, we treat
customer abandonment by studying the limiting behavior of the queueing
models
with arrivals turned off at some time $t$. Then, the waiting time of an
infinitely patient customer arriving at time $t$ is the additional
time it
takes for the queue to empty. To prove stochastic-process limits for
virtual
waiting times, we establish a two-parameter version of Puhalskii's
invariance
principle for first passage times. That in turn involves proving that
two-parameter versions of the composition and inverse mappings
appropriately
preserve convergence.
http://arxiv.org/abs/0809.4275
---------------------------------------------------------------
7540. TESTING COMPOSITE HYPOTHESES VIA CONVEX DUALITY
Birgit Rudloff and Ioannis Karatzas
We study the problem of testing composite hypotheses versus composite
alternatives, using a convex duality approach. In contrast to
classical results
obtained by Krafft & Witting (1967), where sufficient optimality
conditions are
obtained via Lagrange duality, we obtain necessary and sufficient
optimality
conditions via Fenchel duality under some compactness assumptions. This
approach also differs from the methodology developed in Cvitanic &
Karatzas
(2001).
http://arxiv.org/abs/0809.4297
---------------------------------------------------------------
7541. LONG-RANGE SELF-AVOIDING WALK CONVERGES TO ALPHA-STABLE PROCESSES
Markus Heydenreich
We consider a long-range version of self-avoiding walk in dimension $d >
2(\alpha \wedge 2)$, where $d$ denotes dimension and $\alpha$ the
power-law
decay exponent of the coupling function. Under appropriate scaling we
prove
convergence to Brownian motion for $\alpha \ge 2$, and to $\alpha$-
stable
L\'evy motion for $\alpha < 2$. This complements results by Slade
(1988), who
proves convergence to Brownian motion for nearest-neighbor self-
avoiding walk
in high dimension.
http://arxiv.org/abs/0809.4333
---------------------------------------------------------------
7542. CONVERGENCE TO STABLE LAWS FOR A CLASS OF MULTIDIMENSIONAL
STOCHASTIC RECURSIONS
Dariusz Buraczewski and Ewa Damek and Yves Guivarc'h
We consider a Markov chain $\{X_n\}_{n=0}^\8$ on $\R^d$ defined by the
stochastic recursion $X_{n}=M_n X_{n-1}+Q_n$, where $(Q_n,M_n)$ are
i.i.d.
random variables taking values in the affine group $H=\R^d\rtimes {\rm
GL}(\R^d)$. Assume that $M_n$ takes values in the similarity group of $
\R^d$,
and the Markov chain has a unique stationary measure $\nu$, which has
unbounded
support. We denote by $|M_n|$ the expansion coefficient of $M_n$ and
we assume
$\E |M|^\a=1$ for some positive $\a$. We show that the partial sums
$S_n=\sum_{k=0}^n X_k$, properly normalized, converge to a normal law
($\a\ge
2$) or to an infinitely divisible law, which is stable in a natural
sense
($\a<2$). These laws are fully nondegenerate, if $\nu$ is not
supported on an
affine hyperplane. Under a natural hypothesis, we prove also a local
limit
theorem for the sums $S_n$. If $\a\le 2$, proofs are based on the
homogeneity
at infinity of $\nu$ and on a detailed spectral analysis of a family
of Fourier
operators $P_v$ considered as perturbations of the transition operator
$P$ of
the chain $\{X_n \}$. The characteristic function of the limit law has
a simple
expression in terms of moments of $\nu$ ($\a > 2$) or of the tails of $
\nu$ and
of stationary measure for an associated Markov operator ($\a\le 2$).
We extend
the results to the situation where $M_n$ is a random generalized
similarity.
http://arxiv.org/abs/0809.4349
---------------------------------------------------------------
7543. RUIN PROBABILITIES UNDER GENERAL INVESTMENTS AND HEAVY-TAILED
CLAIMS
Henrik Hult and Filip Lindskog
In this paper we study the asymptotic decay of finite time ruin
probabilities
for an insurance company that faces heavy-tailed claims, uses
predictable
investment strategies and makes investments in risky assets whose
prices evolve
according to quite general semimartingales. We show that the ruin
problem
corresponds to determining hitting probabilities for the solution to a
randomly
perturbed stochastic integral equation. We derive a large deviation
result for
the hitting probabilities that holds uniformly over a family of
semimartingales
and show that this result gives the asymptotic decay of finite time ruin
probabilities under arbitrary investment strategies, including optimal
investment strategies.
http://arxiv.org/abs/0809.4372
---------------------------------------------------------------
7544. LAW OF THE ITERATED LOGARITHM FOR THE RANDOM WALK ON THE
INFINITE PERCOLATION CLUSTER
H. Duminil-Copin
We show that random walks on the infinite supercritical percolation
clusters
in Z^d satisfy the usual Law of the Iterated Logarithm. The proof
combines
Barlow's Gaussian heat kernel estimates and the ergodicity of the
random walk
on the environment viewed from the random walker as derived by Berger
and
Biskup.
http://arxiv.org/abs/0809.4380
---------------------------------------------------------------
7545. SMALL COUNTS IN THE INFINITE OCCUPANCY SCHEME
A.D. Barbour and A.V. Gnedin
The paper is concerned with the classical occupancy scheme with
infinitely
many boxes, in which $n$ balls are thrown independently into boxes
$1,2,...$,
with probability $p_j$ of hitting the box $j$, where $p_1\geq
p_2\geq...>0$ and
$\sum_{j=1}^\infty p_j=1$. We establish joint normal approximation as
$n\to\infty$ for the numbers of boxes containing $r_1,r_2,...,r_m$
balls,
standardized in the natural way, assuming only that the variances of
these
counts all tend to infinity. The proof of this approximation is based
on a
de-Poissonization lemma. We then review sufficient conditions for the
variances
to tend to infinity. Typically, the normal approximation does not mean
convergence. We show that the convergence of the full vector of $r$-
counts only
holds under a condition of regular variation, thus giving a complete
characterization of possible limit correlation structures.
http://arxiv.org/abs/0809.4387
---------------------------------------------------------------
7546. CORRECTION. SDES WITH OBLIQUE REFLECTIONS ON NONSMOOTH DOMAINS
Paul Dupuis and Hitoshi Ishii
Correction to The Annals of Probability 21 (1993) 554--580
[http://projecteuclid.org/euclid.aop/1176989415]
http://arxiv.org/abs/0809.4393
---------------------------------------------------------------
7547. RESCALED LOTKA-VOLTERRA MODELS CONVERGE TO SUPER STABLE PROCESSES
Hui He
Recently, it has been shown that stochastic spatial Lotka-Volterra
models
when suitably rescaled can converge to a super-Brownian motion with
drift. We
show that the limit process could be a super stable process if the
kernel of
the underlying motion is in the domain of attraction of a stable law. As
applications of the convergence theorems, some new results on the
voter model's
asymptotics are obtained.
http://arxiv.org/abs/0809.4520
---------------------------------------------------------------
7548. INFINITE RATE MUTUALLY CATALYTIC BRANCHING
Achim Klenke and Leonid Mytnik
Consider the mutually catalytic branching process with finite
branching rate
$\gamma$. We show that as $\gamma\to\infty$ this process converges in
finite
dimensional distributions (in time) to a certain discontinuous
process. We give
descriptions of this process in terms of its semigroup, the
infinitesimal
generator and as the solution of a martingale problem. We also give a
strong
construction in terms of a planar Brownian motion from which we infer
paths
properties of the process.
This is the first paper of a trilogy where we also construct an
interacting
versions of this process and study its long-time behaviour.
http://arxiv.org/abs/0809.4554
---------------------------------------------------------------
7549. BOUNDARY NON-CROSSINGS OF BROWNIAN PILLOW
Enkelejd Hashorva
Let B_0(s,t) be a Brownian pillow with continuous sample paths, and let
h,u:[0,1]^2\to R be two measurable functions. In this paper we derive
upper and
lower bounds for the boundary non-crossing probability
\psi(u;h):=P{B_0(s,t)+h(s,t) \le u(s,t), \forall s,t\in [0,1]}.
Further we
investigate the asymptotic behaviour of $\psi(u;\gamma h)$ with $\gamma$
tending to infinity, and solve a related minimisation problem.
http://arxiv.org/abs/0809.4560
---------------------------------------------------------------
7550. RANDOM BLOCK MATRICES AND MATRIX ORTHOGONAL POLYNOMIALS
Holger Dette and Bettina Reuther
In this paper we consider random block matrices, which generalize the
general
beta ensembles, which were recently investigated by Dumitriu and
Edelmann
(2002, 2005). We demonstrate that the eigenvalues of these random
matrices can
be uniformly approximated by roots of matrix orthogonal polynomials
which were
investigated independently from the random matrix literature. As a
consequence
we derive the asymptotic spectral distribution of these matrices. The
limit
distribution has a density, which can be represented as the trace of an
integral of densities of matrix measures corresponding to the
Chebyshev matrix
polynomials of the first kind. Our results establish a new relation
between the
theory of random block matrices and the field of matrix orthogonal
polynomials,
which have not been explored so far in the literature.
http://arxiv.org/abs/0809.4601
---------------------------------------------------------------
7551. SENSITIVITY FOR SMOLUCHOWSKI EQUATION
Ismael Bailleul
This article investigates the question of sensitivity of the solutions
of
Smoluchowski equation on R_+^* with respect to parameters \lambda in the
interaction kernel K^\lambda. It is proved that the solution is a C^1
function
of (t,\lambda) with values in a good space of measures under the
hypotheses
K^{\lambda}(x,y)\leq \varphi(x)\varphi(y), for some sub-linear function
\varphi, a (4+epsilon)-moment assumption on the initial condition, and
that the
derivative is a solution, in a suitable sense, of a linearized equation.
http://arxiv.org/abs/0809.4640
---------------------------------------------------------------
7552. BROWNIAN COUPLINGS, CONVEXITY, AND SHY-NESS
Wilfrid S. Kendall
Benjamini, Burdzy and Chen (2007) introduced the notion of a coupling: a
coupling of a Markov process such that, for suitable starting points,
there is
a positive chance of the two component processes of the coupling
staying a
positive distance away from each other for all time. Among other
results, they
showed no shy couplings could exist for reflected Brownian motions in
C^2
bounded convex planar domains whose boundaries contain no line
segments. Here
we use potential-theoretic methods to extend this Benjamini et al.
result (a)
to all bounded convex domains (whether planar and smooth or not) whose
boundaries contain no line segments, (b) to all bounded convex planar
domains
regardless of further conditions on the boundary.
http://arxiv.org/abs/0809.4682
---------------------------------------------------------------
7553. CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF
RANDOM MATRICES WITH INDEPENDENT ENTRIES
Leonid Pastur and Anna Lytova
We consider nxn real symmetric and hermitian Wigner random matrices
n^{-1/2}W
with independent (modulo symmetry condition) entries and the (null)
sample
covariance matrices n^{-1}X^*X with independent entries of mxn matrix X.
Assuming that 4th moments of entries of W and X satisfy a Lindeberg type
condition and the test functions of linear statistics of eigenvalues
of the
above matrices are smooth enough (essentially of the class C^5), we
prove that
linear statistics of eigenvalues satisfy the Central Limit Theorem as
n, m tend
to infinity and m/n tends to finite c>0.
http://arxiv.org/abs/0809.4698
---------------------------------------------------------------
7554. MULTISTAGE ESTIMATION OF BOUNDED-VARIABLE MEANS
Xinjia Chen
In this paper, we develop a multistage approach for estimating the
mean of a
bounded variable. We first focus on the multistage estimation of a
binomial
parameter and then generalize the estimation methods to the case of
general
bounded random variables. A fundamental connection between a binomial
parameter
and the mean of a bounded variable is established. Our multistage
estimation
methods rigorously guarantee prescribed levels of precision and
confidence.
http://arxiv.org/abs/0809.4679
---------------------------------------------------------------
7555. ALMOST ALL ONE-RELATOR GROUPS WITH AT LEAST THREE GENERATORS
ARE RESIDUALLY FINITE
Iva Kozakova and Mark Sapir
We prove that with probability tending to 1, a 1-relator group with at
least
3 generators and relator of length n is residually finite, virtually
residually
(finite p)-group for all sufficiently large p, and coherent. The proof
uses
both combinatorial group theory and non-trivial results about Brownian
motions,
bridges and excursions in R^k.
http://arxiv.org/abs/0809.4693
---------------------------------------------------------------
7556. LARGE DEVIATIONS FOR THE LEAVES IN SOME RANDOM TREES
W. Bryc and D. Minda and S. Sethuraman
Large deviation principles and related results are given for a class of
Markov chains associated to the "leaves" in random recursive trees and
preferential attachment random graphs, as well as the "cherries" in
Yule trees.
In particular, the method of proof, combining analytic and Dupuis-
Ellis type
path arguments, allows for an explicit computation of the large
deviation
pressure.
http://arxiv.org/abs/0809.4741
---------------------------------------------------------------
7557. HIGHER ORDER CAUCHY PROBLEMS IN BOUNDED DOMAINS
Erkan Nane
We study solutions of a class of higher order partial differential
equations
in bounded domains. These partial differential equations appeared
first time in
the papers of Baeumer, Meerschaert and Nane \cite{bmn-07} and
Meerschaert, Nane
and Vellaisamy \cite{MNV}, and Nane \cite{nane-h}. We express the
solutions by
subordinating a killed Markov process by a hitting time of a stable
subordinator of index $0<\beta <1$, or by the absolute value of a
symmetric
$\alpha$-stable process with $0<\alpha\leq 2$, independent of the Markov
process. In some special cases we represent the solutions by running
composition of $k$ independent Brownian motions, called $k$-iterated
Brownian
motion for an integer $k\geq 2$.
http://arxiv.org/abs/0809.4824
---------------------------------------------------------------
7558. CONCENTRATION OF MEASURE AND MIXING FOR MARKOV CHAINS
Malwina J. Luczak
We consider Markovian models on graphs with local dynamics. We show
that,
under suitable conditions, such Markov chains exhibit both rapid
convergence to
equilibrium and strong concentration of measure in the stationary
distribution.
We illustrate our results with applications to some known chains from
computer
science and statistical mechanics.
http://arxiv.org/abs/0809.4856
---------------------------------------------------------------
7559. A NOTE ON RANDOM ORTHOGONAL POLYNOMIALS ON A COMPACT INTERVAL
M. Birke and H. Dette
We consider a uniform distribution on the set $\mathcal{M}_k$ of
moments of
order $k \in \mathbb{N}$ corresponding to probability measures on the
interval
$[0,1]$. To each (random) vector of moments in $\mathcal{M}_{2n-1}$ we
consider
the corresponding uniquely determined monic (random) orthogonal
polynomial of
degree $n$ and study the asymptotic properties of its roots if $n \to
\infty$.
http://arxiv.org/abs/0809.4936
---------------------------------------------------------------
7560. STOCHASTIC VORTEX METHOD FOR FORCED THREE DIMENSIONAL NAVIER-
STOKES EQUATIONS AND PATHWISE CONVERGENCE RATE
Joaquin Fontbona
We develop a McKean-Vlasov interpretation of Navier-Stokes equations
with
external force field in the whole space, by associating with local mild
$L^p-$solutions of the 3d-vortex equation a generalized nonlinear
diffusion
with random space-time birth, that probabilistically describes
creation of
rotation in the fluid due to non-conservativeness of the force. We
establish a
well-posedness result for this process and a stochastic representation
formula
for the vorticity in terms of a vector-weighted version of its law
after its
birth instant. Then, we introduce a stochastic system of 3d vortices
with
mollified interaction and random space-time births, and prove the
propagation
of chaos property, with the nonlinear process as limit, at an explicit
pathwise
convergence rate. Convergence rates for stochastic approximation
schemes of the
velocity and the vorticity fields are also obtained. We thus extend
and refine
previous results concerning a probabilistic interpretation and a
particle
approximation method for the non-forced equation, and generalize a
recently
introduced random space-time-birth particle method for the 2d Navier-
Stokes
equation with force.
http://arxiv.org/abs/0809.4947
---------------------------------------------------------------
7561. SQUARE INTEGRABLE HOLOMORPHIC FUNCTIONS ON INFINITE-DIMENSIONAL
HEISENBERG TYPE GROUPS
Bruce Driver and Maria Gordina
We introduce a class of non-commutative, complex, infinite-dimensional
Heisenberg like Lie groups based on an abstract Wiener space. The
holomorphic
functions which are also square integrable with respect to a heat kernel
measure $\mu$ on these groups are studied. In particular, we establish a
unitary equivalence between the square integrable holomorphic
functions and a
certain completion of the universal enveloping algebra of the "Lie
algebra" of
this class of groups. Using quasi-invariance of the heat kernel
measure, we
also construct a skeleton map which characterizes globally defined
functions
from the $L^{2}(\nu)$-closure of holomorphic polynomials by their
values on the
Cameron-Martin subgroup.
http://arxiv.org/abs/0809.4979
---------------------------------------------------------------
7562. A CLASS OF OPTIMAL STOPPING PROBLEMS FOR MARKOV PROCESSES
Diana Dorobantu (SAF - EA2429)
Our purpose is to study a particular class of optimal stopping
problems for
Markov processes. We justify the value function convexity and we
deduce that
there exists a boundary function such that the smallest optimal
stopping time
is the first time when the Markov process passes over the boundary
depending on
time. Moreover, we propose a method to find the optimal boundary
function.
http://arxiv.org/abs/0809.4990
---------------------------------------------------------------
7563. THE T-IMPROPER CHROMATIC NUMBER OF RANDOM GRAPHS
Ross J. Kang and Colin McDiarmid
We consider the $t$-improper chromatic number of the Erd{\H o}s-
R{\'e}nyi
random graph $G(n,p)$. The t-improper chromatic number $\chi^t(G)$ of
$G$ is
the smallest number of colours needed in a colouring of the vertices
in which
each colour class induces a subgraph of maximum degree at most $t$. If
$t = 0$,
then this is the usual notion of proper colouring. When the edge
probability
$p$ is constant, we provide a detailed description of the asymptotic
behaviour
of $\chi^t(G(n,p))$ over the range of choices for the growth of $t =
t(n)$.
http://arxiv.org/abs/0809.4726
---------------------------------------------------------------
7564. ON CONTINGENT CLAIMS PRICING IN INCOMPLETE MARKETS: A RISK
SHARING APPROACH
Michail Anthropelos and Nikolaos E. Frangos and Stylianos Z.
Xanthopoulos and Athanasios N. Yannacopoulos
In an incomplete market setting, we consider two financial agents, who
wish
to price and trade a non-replicable contingent claim. Assuming that
the agents
are utility maximizers, we propose a transaction price which is a
result of the
minimization of a convex combination of their utility differences. We
call this
price the risk sharing price, we prove its existence for a large
family of
utility functions and we state some of its properties. As an example, we
analyze extensively the case where both agents report exponential
utility.
http://arxiv.org/abs/0809.4781
---------------------------------------------------------------
7565. STATIONARY SOLUTIONS OF SPDES AND INFINITE HORIZON BDSDES WITH
NON-LIPSCHITZ COEFFICIENTS
Qi Zhang and Huaizhong Zhao
We prove a general theorem that the
$L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes
L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of an
infinite
horizon backward doubly stochastic differential equation, if exists,
gives the
stationary solution of the corresponding stochastic partial differential
equation. We prove the existence and uniqueness of the
$L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes
L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solutions for
backward
doubly stochastic differential equations on finite and infinite
horizon with
linear growth without assuming Lipschitz conditions, but under the
monotonicity
condition. Therefore the solution of finite horizon problem gives the
solution
of the initial value problem of the corresponding stochastic partial
differential equations, and the solution of the infinite horizon
problem gives
the stationary solution of the SPDEs according to our general result.
http://arxiv.org/abs/0809.5089
---------------------------------------------------------------
7566. SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS ON
MARKOV CHAINS
Samuel N. Cohen and Robert J. Elliott
We consider backward stochastic differential equations (BSDEs) related
to
finite state, continuous time Markov chains. We show that appropriate
solutions
exist for arbitrary terminal conditions, and are unique up to sets of
measure
zero. We do not require the generating functions to be monotonic,
instead using
only an appropriate Lipschitz continuity condition.
http://arxiv.org/abs/0809.5102
---------------------------------------------------------------
7567. LINEAR UNIVERSAL DECODING FOR COMPOUND CHANNELS: A LOCAL TO
GLOBAL GEOMETRIC APPROACH
Emmanuel Abbe and Lizhong Zheng
Over discrete memoryless channels (DMC), linear decoders (maximizing
additive
metrics) afford several nice properties. In particular, if suitable
encoders
are employed, the use of decoding algorithm with manageable
complexities is
permitted. Maximum likelihood is an example of linear decoder. For a
compound
DMC, decoders that perform well without the channel's knowledge are
required in
order to achieve capacity. Several such decoders have been studied in
the
literature. However, there is no such known decoder which is linear.
Hence, the
problem of finding linear decoders achieving capacity for compound DMC
is
addressed, and it is shown that under minor concessions, such decoders
exist
and can be constructed. This paper also develops a "local geometric
analysis",
which allows in particular, to solve the above problem. By considering
very
noisy channels, the original problem is reduced, in the limit, to an
inner
product space problem, for which insightful solutions can be found.
The local
setting can then provide counterexamples to disproof claims, but also,
it is
shown how in this problem, results proven locally can be "lifted" to
results
proven globally.
http://arxiv.org/abs/0809.5217
---------------------------------------------------------------
7568. CLT VARIANCE ASSOCIATED WITH BAXENDALE'S SDE
Steven R. Finch
Simple analysis of the leftmost eigenvalue of Ince's equation (a
boundary
value problem with periodicity) resolves an open issue surrounding a
stochastic
Lyapunov exponent. Numerical verification is also provided.
http://arxiv.org/abs/0809.5274
---------------------------------------------------------------
7569. ESTIMATING SPEED AND DAMPING IN THE STOCHASTIC WAVE EQUATION
W. Liu and S. V. Lototsky
A parameter estimation problem is considered for a one-dimensional
stochastic
wave equation driven by additive space-time Gaussian white noise.
The estimator is of spectral type and utilizes a finite number of
the spatial
Fourier coefficients of the solution. The asymptotic properties of the
estimator are studied as the number of the Fourier coefficients
increases,
while the observation time and the noise intensity are fixed.
http://arxiv.org/abs/0810.0046
---------------------------------------------------------------
7570. COMPARISONS FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS ON
MARKOV CHAINS AND RELATED NO-ARBITRAGE CONDITIONS
Samuel N. Cohen and Robert J. Elliott
Most previous contributions on BSDEs, and the related theories of
nonlinear
expectation and dynamic risk measures, have been in the framework of
continuous
time diffusions or jump diffusions. Using solutions of BSDEs on spaces
related
to finite state, continuous time Markov Chains, we develop a theory of
nonlinear expectations in the spirit of Peng (2005). We prove basic
properties
of these expectations, and show their applications to dynamic risk
measures on
such spaces. In particular, we prove comparison theorems for scalar
and vector
valued solutions to BSDEs, and discuss arbitrage and risk measures in
the
scalar case.
http://arxiv.org/abs/0810.0055
---------------------------------------------------------------
7571. CONCENTRATION INEQUALITIES FOR MARKOV PROCESSES VIA COUPLING
J.-R. Chazottes and F. Redig
We obtain moment and Gaussian bounds for general Lipschitz functions
evaluated along the sample path of a Markov chain. We treat Markov
chains on
general (possibly unbounded) state spaces via a coupling method. If
the first
moment of the coupling time exists, then we obtain a variance
inequality. If a
moment of order 1+epsilon of the coupling time exists, then depending
on the
behavior of the stationary distribution, we obtain higher moment
bounds. This
immediately implies polynomial concentration inequalities. In the case
that a
moment of order 1+epsilon is finite uniformly in the starting point of
the
coupling, we obtain a Gaussian bound. We illustrate the general
results with
house of cards processes, in which both uniform and non-uniform
behavior of
moments of the coupling time can occur.
http://arxiv.org/abs/0810.0097
---------------------------------------------------------------
7572. HYDRODYNAMIC LIMIT OF ZERO RANGE PROCESSES AMONG RANDOM
CONDUCTANCES ON THE SUPERCRITICAL PERCOLATION CLUSTER
A. Faggionato
We consider i.i.d. random variables {\omega (b):b \in E_d}
parameterized by
the family of bonds in Z^d, d>1. The random variable \omega(b) is
thought of as
the conductance of bond b and it ranges in a finite interval [0,c_0].
Assuming
the probability m of the event {\omega(b)>0} to be supercritical and
denoting
by C(\omega) the unique infinite cluster associated to the bonds with
positive
conductance, we study the zero range process on C(\omega) with
\omega(b)-proportional probability rate of jumps along bond b. For
almost all
realizations of the environment we prove that the hydrodynamic
behavior of the
zero range process is governed by the nonlinear heat equation $
\partial_t \rho=
m \nabla \cdot (D \nabla\phi(\rho/m))$, where the matrix D and the
function
\phi are \omega--independent. We do not require any ellipticity
condition.
http://arxiv.org/abs/0810.0103
---------------------------------------------------------------
7573. PLANAR AGGREGATION AND THE COALESCING BROWNIAN FLOW
James Norris and Amanda Turner
We study a scaling limit associated to a model of planar aggregation.
The
model is obtained by composing certain independent random conformal
maps. The
evolution of harmonic measure on the boundary of the cluster is shown to
converge to the coalescing Brownian flow.
http://arxiv.org/abs/0810.0211
---------------------------------------------------------------
7574. ON A CONTINUOUS TIME GAME WITH INCOMPLETE INFORMATION
Pierre Cardaliaguet (LM-Brest) and Catherine Rainer (LM)
For zero-sum two-player continuous-time games with integral payoff and
incomplete information on one side, one shows that the optimal
strategy of the
informed player can be computed through an auxiliary optimization
problem over
some martingale measures. One also characterizes the optimal martingale
measures and compute it explicitely in several examples.
http://arxiv.org/abs/0810.0220
---------------------------------------------------------------
7575. LIMITING DISTRIBUTIONS AND LARGE DEVIATIONS FOR RANDOM WALKS IN
RANDOM ENVIRONMENTS
Jonathon Peterson
This thesis concerns the study of random walks in random environments
(RWRE).
Since there are two levels of randomness for random walks in random
environments, there are two different distributions for the random
walk that
can be studied. The quenched distribution is the law of the random walk
conditioned on a given environment. The annealed distribution is the
quenched
law averaged over all environments. The main results of the thesis
fall into
two categories: quenched limiting distributions for one-dimensional,
transient
RWRE and annealed large deviations for multidimensional RWRE.
The analysis of the quenched distributions for transient, one-
dimensional
RWRE falls into two separate cases. First, when an annealed central
limit
theorem holds, we prove that a quenched central limit theorem also
holds but
with a random (depending on the environment) centering. In contrast,
when the
annealed limit distribution is not Gaussian, we prove that there is no
quenched
limiting distribution for the RWRE. Moreover, we show that for almost
every
environment, there exist two random (depending on the environment)
sequences of
times, along which random walk has different quenched limiting
distributions.
While an annealed large deviation principle for multidimensional
RWRE was
known previously, very little qualitative information was available
about the
annealed large deviation rate function. We prove that if the law on
environments is non-nestling, then the annealed large deviation rate
function
is analytic in a neighborhood of its unique zero (which is the limiting
velocity of the RWRE).
http://arxiv.org/abs/0810.0257
---------------------------------------------------------------
7576. INTERACTING MULTI-CLASS TRANSMISSIONS IN LARGE STOCHASTIC NETWORKS
Carl Graham (CMAP) and Philippe Robert
The mean-field limit of a Markovian model describing the interaction of
several classes of permanent connections in a network is analyzed. In
the same
way as for the TCP algorithm, each of the connections has a self-
adaptive
behavior in that its transmission rate along its route depends on the
level of
congestion of the nodes of the route. Since several classes of
connections
going through the nodes of the network are considered, an original
mean-field
result in a multi-class context is established. It is shown that, as
the number
of connections goes to infinity, the behavior of the different classes
of
connections can be represented by the solution of an unusual non-linear
stochastic differential equation depending not only on the sample
paths of the
process, but also on its distribution. Existence and uniqueness
results for the
solutions of these equations are derived. Properties of their invariant
distributions are investigated and it is shown that, under some natural
assumptions, they are determined by the solutions of a fixed point
equation in
a finite dimensional space.
http://arxiv.org/abs/0810.0347
---------------------------------------------------------------
7577. PASSAGE-TIME MOMENTS AND HYBRID ZONES FOR THE EXCLUSION-VOTER
MODEL
Iain M. MacPhee and Mikhail V. Menshikov and Stanislav Volkov and
Andrew R. Wade
We study the non-equilibrium dynamics of a one-dimensional interacting
particle system that is a mixture of the voter model and exclusion
process.
With the process started from a finite perturbation of the ground-state
Heaviside configuration consisting of 1s to the left of the origin and
0s
elsewhere, we study the relaxation time $\tau$, that is, the first
hitting time
of the ground-state configuration (up to translation). In particular,
we give
conditions for $\tau$ to be finite and for certain moments of $\tau$
to be
finite or infinite, and prove a result that approaches a conjecture of
Belitsky
{\em et al.} [{\em Bernoulli} {\bf 7} (2001) 119--144]. Ours are the
first
non-existence of moments results for $\tau$ for the mixture model.
Moreover, we
give almost-sure asymptotic results on the long-term evolution of the
size of
the hybrid (disordered) region. Most of our results pertain to the
discrete-time setting, but several transfer to continuous-time. As
well as the
mixture process, some of our results also cover the pure exclusion
case. We
state several significant open problems that remain.
http://arxiv.org/abs/0810.0392
---------------------------------------------------------------
7578. OBSTACLE PROBLEM FOR SPDE WITH NONLINEAR NEUMANN BOUNDARY
CONDITION VIA REFLECTED GENERALIZED BACKWARD DOUBLY SDES
Auguste Aman (LMAI) and Naoul Mrhardy
This paper is intended to give a probabilistic representation for
stochastic
viscosity solution of semi-linear reflected stochastic partial
differential
equations with nonlinear Neumann boundary condition. We use it
connection with
reflected generalized backward doubly stochastic differential equation.
http://arxiv.org/abs/0810.0436
---------------------------------------------------------------
7579. ESTIMATING THE PARAMETERS OF BINOMIAL AND POISSON DISTRIBUTIONS
VIA MULTISTAGE SAMPLING
Xinjia Chen
In this paper, we have developed a new class of sampling schemes for
estimating parameters of binomial and Poisson distributions. Without any
information of the unknown parameters, our sampling schemes rigorously
guarantee prescribed levels of precision and confidence.
http://arxiv.org/abs/0810.0430
---------------------------------------------------------------
7580. INTERSECTION EXPONENTS FOR BIASED RANDOM WALKS ON DISCRETE
CYLINDERS
Brigitta Vermesi
We prove existence of intersection exponents xi(k,lambda) for biased
random
walks on d-dimensional half-infinite discrete cylinders, and show
that, as
functions of lambda, these exponents are real analytic. As part of the
argument, we prove convergence to stationarity of a time-inhomogeneous
Markov
chain on half-infinite random paths. Furthermore, we show this
convergence
takes place at exponential rate, an estimate obtained via a coupling of
weighted half-infinite paths.
http://arxiv.org/abs/0810.0572
---------------------------------------------------------------
7581. INFORMATION INEQUALITIES AND A DEPENDENT CENTRAL LIMIT THEOREM
Oliver Johnson
We adapt arguments concerning information-theoretic convergence in the
Central Limit Theorem to the case of dependent random variables under
Rosenblatt mixing conditions. The key is to work with random variables
perturbed by the addition of a normal random variable, giving us good
control
of the joint density and the mixing coefficient. We strengthen results
of
Takano and of Carlen and Soffer to provide entropy-theoretic, not weak
convergence.
http://arxiv.org/abs/0810.0593
---------------------------------------------------------------
7582. TRANSLATED POISSON APPROXIMATION FOR MARKOV CHAINS
A. D. Barbour and Torgny Lindvall
The paper is concerned with approximating the distribution of a sum W
of n
integer valued random variables Y_i, whose distributions depend on the
state of
an underlying Markov chain X. The approximation is in terms of a
translated
Poisson distribution, with mean and variance chosen to be close to
those of W,
and the error is measured with respect to the total variation norm.
Error
bounds comparable to those found for normal approximation with respect
to the
weaker Kolmogorov distance are established, provided that the
distribution of
the sum of the Y_i's between the successive visits of X to a reference
state is
aperiodic. Without this assumption, approximation in total variation
cannot be
expected to be good.
http://arxiv.org/abs/0810.0599
---------------------------------------------------------------
7583. FROM PERSISTENT RANDOM WALKS TO THE TELEGRAPH NOISE
Samuel Herrmann (IECN) and Pierre Vallois (IECN)
We study a family of memory-based persistent random walks and we prove
weak
convergences after space-time rescaling. The limit processes are not
only
Brownian motions with drift. We have obtained a continuous but non-
Markov
process $(Z_t)$ which can be easely expressed in terms of a counting
process
$(N_t)$. In a particular case the counting process is a Poisson
process, and
$(Z_t)$ permits to represent the solution of the telegraph equation.
We study
in detail the Markov process $((Z_t,N_t); t\ge 0)$.
http://arxiv.org/abs/0810.0650
---------------------------------------------------------------
7584. ON THE PURITY OF THE FREE BOUNDARY CONDITION POTTS MEASURE ON
RANDOM TREES
M. Formentin and C. Kuelske
We consider the free boundary condition Gibbs measure of the Potts
model on a
random tree. We provide an explicit temperature interval below the
ferromagnetic transition temperature for which this measure is extremal,
improving older bounds of Mossel and Peres. In information theoretic
language
extremality of the Gibbs measure corresponds to non-reconstructability
for
symmetric q-ary channels. The bounds are optimal for the Ising model
and appear
to be close to what we conjecture to be the true values up to a factor
of
0.0150 in the case q = 3 and 0.0365 for q = 4. Our proof uses an
iteration of
random boundary entropies from the outside of the tree to the inside,
along
with a symmetrization argument.
http://arxiv.org/abs/0810.0677
---------------------------------------------------------------
7585. BROWNIAN MOTION AND THE PARABOLICITY OF MINIMAL GRAPHS
Robert W. Neel
We prove that minimal graphs (other than planes) are parabolic in the
sense
that any bounded harmonic function is determined by its boundary
values. The
proof relies on using the coupling introduced in the author's earlier
paper "A
martingale approach to minimal surfaces" to show that Brownian motion
on such a
minimal graph almost surely strikes the boundary in finite time.
http://arxiv.org/abs/0810.0669
---------------------------------------------------------------
7586. A CONNECTION BETWEEN GHIRLANDA-GUERRA IDENTITIES AND
ULTRAMETRICITY
Dmitry Panchenko
We consider a symmetric positive definite weakly exchangeable infinite
random
matrix whose elements take a finite number of values and we prove that
if the
distribution of the matrix satisfies the Ghirlanda-Guerra identities
then it is
ultrametric with probability one.
http://arxiv.org/abs/0810.0743
---------------------------------------------------------------
7587. HEDGING AND PRODUCTION DECISIONS UNDER UNCERTAINTY: A SURVEY
Moawia Alghalith
This paper synthesizes and analyzes some important current and recent
contributions to the theory of the firm under uncertainty. In so
doing, it
examines the production and hedging decisions of the competitive firm
under a
single source and multiple sources of uncertainty.
http://arxiv.org/abs/0810.0917
---------------------------------------------------------------
7588. THE COVARIANT MEASURE OF SLE ON THE BOUNDARY
Tom Alberts and Scott Sheffield
We construct a natural measure mu supported on the intersection of a
chordal
SLE(kappa) curve gamma with the real line R, in the range 4 < kappa <
8. The
measure is a function of the SLE path in question. Assuming that
boundary
measures transform in a ``d-dimensional'' way (where d is the Hausdorff
dimension of gamma intersected with R), we show that the measure we
construct
is (up to multiplicative constant) the unique measure-valued function
of the
SLE path that satisfies the Domain Markov property.
http://arxiv.org/abs/0810.0940
---------------------------------------------------------------
7589. SCALING LIMITS OF TWO-DIMENSIONAL PERCOLATION: AN OVERVIEW
Federico Camia
We present a review of the recent progress on percolation scaling
limits in
two dimensions. In particular, we will consider the convergence of
critical
crossing probabilities to Cardy's formula and of the critical
exploration path
to chordal SLE(6), the full scaling limit of critical cluster
boundaries, and
near-critical scaling limits.
http://arxiv.org/abs/0810.1002
---------------------------------------------------------------
7590. ORBIT MEASURES, RANDOM MATRIX THEORY AND INTERLACED
DETERMINANTAL PROCESSES
Manon Defosseux (PMA)
A connection between representation of compact groups and some invariant
ensembles of Hermitian matrices is described. We focus on two types of
invariant ensembles which extend the Gaussian and the Laguerre Unitary
ensembles. We study them using projections and convolutions of invariant
probability measures on adjoint orbits of a compact Lie group. These
measures
are described by semiclassical approximation involving tensor and
restriction
mulltiplicities. We show that a large class of them are determinantal.
http://arxiv.org/abs/0810.1011
---------------------------------------------------------------
7591. MEASURING THE "NON-STOPPING TIMENESS" OF ENDS OF PREVISIBLE SETS
Ju-Yi Yen and Marc Yor
In this paper, we propose several "measurements" of the "non-stopping
timeness" of ends g of previsible sets, such that g avoids stopping
times, in
an ambiant filtration. We then study several explicit examples,
involving last
passage times of some remarkable martingales.
http://arxiv.org/abs/0810.1059
---------------------------------------------------------------
7592. A STRONG LAW OF LARGE NUMBERS WITH APPLICATIONS TO SELF-SIMILAR
STABLE PROCESSES
Erkan Nane and Yimin Xiao and Aklilu Zeleke
Let $p \in (0, \infty)$ be a constant and let $\{\xi_n\} \subset
L^p(\Omega,
{\mathcal F}, \P)$ be a sequence of random variables. For any integers
$m, n
\ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} \xi_k$. It is proved
that, if
there exist a nondecreasing function $\varphi: \R_+\to \R_+$ (which
satisfies a
mild regularity condition) and an appropriately chosen integer $a\ge
2$ such
that $$ \sum_{n=0}^\infty \sup_{k \ge 0} \E\bigg|\frac{S_{k, a^n}}
{\varphi(a^n)} \bigg|^p < \infty,$$ Then $$ \lim_{n \to \infty}
\frac{S_{0, n}}
{\varphi(n)} = 0\qquad \hbox{a.s.} $$ This extends Theorem 1 in
Levental,
Chobanyan and Salehi \cite{chobanyan-l-s} and can be applied
conveniently to a
wide class of self-similar processes with stationary increments
including
stable processes.
http://arxiv.org/abs/0810.1061
---------------------------------------------------------------
7593. ON A SPECULATED RELATION BETWEEN CHV\'ATAL-SANKOFF CONSTANTS OF
SEVERAL SEQUENCES
Marcos Kiwi and Jos\'e Soto
It is well known that, when normalized by n, the expected length of a
longest
common subsequence of d sequences of length n over an alphabet of size
sigma
converges to a constant gamma_{sigma,d}. We disprove a speculation by
Steele
regarding a possible relation between gamma_{2,d} and gamma_{2,2}. In
order to
do that we also obtain new lower bounds for gamma_{sigma,d}, when both
sigma
and d are small integers.
http://arxiv.org/abs/0810.1066
---------------------------------------------------------------
7594. COUNTING NODAL LINES OF RANDOM WAVES ON PLANAR DOMAINS
John A. Toth and Igor Wigman
We compute the asymptotic expectation of the number of {\em open}
nodal lines
for random waves on smooth planar domains. We find that for both the
long
energy window $[0,\lambda]$ and the short one $[\lambda,\lambda+1]$ the
expected number of open nodal lines is proportional to $\lambda$,
asymptotically as $\lambda\to\infty$. Our results are consistent with
the
predictions in the physics literature made by Blum, Gnutzmann and
Smilansky
\cite{BGS}.
http://arxiv.org/abs/0810.1276
---------------------------------------------------------------
7595. THE ROUGH PATH ASSOCIATED TO THE MULTIDIMENSIONAL ANALYTIC FBM
WITH ANY HURST PARAMETER
Samy Tindel (IECN) and J\'er\'emie Unterberger (IECN)
In this paper, we consider a complex-valued d-dimensional fractional
Brownian
motion defined on the closure of the complex upper half-plane, called
analytic
fractional Brownian motion. This process has been introduced by the
second
author of the article, and both its real and imaginary parts,
restricted on the
real axis, are usual fractional Brownian motions. The current note is
devoted
to prove that a rough path based on the analytic fBm can be
constructed for any
value of the Hurst parameter in (0,1/2). This allows in particular to
solve
differential equations driven by this process in a neighborhood of 0
of the
complex upper half-plane, thanks to a variant of the usual rough path
theory
due to Gubinelli.
http://arxiv.org/abs/0810.1408
---------------------------------------------------------------
7596. A NON-MARKOVIAN MODEL OF RILL EROSION
Michael Damron and C.L. Winter
We introduce a new model for rill erosion. We start with a network
similar to
that in the Discrete Web and instantiate a dynamics which makes the
process
highly non-Markovian. The behavior of nodes in the streams is similar
to the
behavior of Polya urns with time-dependent input. In this paper we use a
combination of rigorous arguments and simulation results.
http://arxiv.org/abs/0810.1483
---------------------------------------------------------------
7597. DUALITY AND HIDDEN SYMMETRIES IN INTERACTING PARTICLE SYSTEMS
Cristian Giardina and Jorge Kurchan and Frank Redig and Kiamars
Vafayi
In the context of Markov processes, both in discrete and continuous
setting,
we show a general relation between duality functions and symmetries of
the
generator. If the generator can be written in the form of a
Hamiltonian of a
quantum spin system, then the "hidden" symmetries are easily derived. We
illustrate our approach in processes of symmetric exclusion type, in
which the
symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-
Presutti (KMP)
model for which we unveil its SU(1,1) symmetry. The KMP model is in
turn an
instantaneous thermalization limit of the energy process associated to
a large
family of models of interacting diffusions, which we call Brownian
energy
process (BEP) and which all possess the SU(1,1) symmetry. We treat in
details
the case where the system is in contact with reservoirs and the dual
process
becomes absorbing.
http://arxiv.org/abs/0810.1202
---------------------------------------------------------------
7598. A STRICTLY STATIONARY, N-TUPLEWISE INDEPENDENT COUNTEREXAMPLE TO
THE CENTRAL LIMIT THEOREM
Richard C. Bradley and Alexander R. Pruss
For an arbitrary integer N that is at least 2, this paper gives a
construction of a strictly stationary, N-tuplewise independent
sequence of
(non-degenerate) bounded random variables such that the Central Limit
Theorem
fails to hold. The sequence is in part an adaptation of a non-stationary
example with similar properties constructed by one of the authors
(ARP) in a
paper published in 1998.
http://arxiv.org/abs/0810.1707
---------------------------------------------------------------
7599. ROUGH VOLTERRA EQUATIONS 2: CONVOLUTIONAL GENERALIZED INTEGRALS
Samy Tindel (IECN) and Aur\'elien Deya (IECN)
We define and solve Volterra equations driven by an irregular signal, by
means of a variant of the rough path theory allowing to handle
generalized
integrals weighted by an exponential coefficient. The results are
applied to
the fractional Brownian motion with Hurst coefficient greater than 1/3
http://arxiv.org/abs/0810.1824
---------------------------------------------------------------
7600. THE DISTRIBUTION OF THE DOMINATION NUMBER OF CLASS COVER CATCH
DIGRAPHS FOR NON-UNIFORM ONE-DIMENSIONAL DATA
Elvan Ceyhan
For two or more classes of points in $\R^d$ with $d \ge 1$, the class
cover
catch digraphs (CCCDs) can be constructed using the relative positions
of the
points from one class with respect to the points from the other class.
The
CCCDs were introduced by (Priebe, DeVinney, and Mar-chette, (2001). On
the
distribution of the domination number of random class catch cover di-
graphs.
Statistics and Probability Letters, 55:239-246) who investigated the
case of
two classes, $\X$ and $\Y$. They calculated the exact (finite sample)
distribution of the domination number of the CCCDs based on $\X$ points
relative to $\Y$ points both of which were uniformly distri-buted on a
bounded
interval. We investigate the distribution of the domination number of
the CCCDs
based on data from non-uniform $\X$ points on an interval with end
points from
$\Y$. Then we extend these calculations for multiple $\Y$ points on
bounded
intervals.
http://arxiv.org/abs/0810.1893
---------------------------------------------------------------
7601. STRONG SOLUTIONS OF A CLASS OF SDES WITH JUMPS
Juan Zhao
We study a class of stochastic integral equations with jumps under
non-Lipschitz conditions. We use the method of Euler approximations to
obtain
the existence of the solution and give some criteria for the strong
solution.
http://arxiv.org/abs/0810.1908
---------------------------------------------------------------
7602. CONVOLUTIONS OF LONG-TAILED AND SUBEXPONENTIAL DISTRIBUTIONS
Serguei Foss and Dmitry Korshunov and Stan Zachary
We study convolutions of long-tailed and subexponential distributions
and
measures on the real line, proving some important new results through
a simple
and coherent approach, and showing also that the standard properties
of such
convolutions follow as easy consequences.
http://arxiv.org/abs/0810.1994
---------------------------------------------------------------
7603. HEDGING OF CLAIMS WITH PHYSICAL DELIVERY UNDER CONVEX
TRANSACTION COSTS
Teemu Pennanen and Irina Penner
We study superhedging of contingent claims with physical delivery in a
discrete-time market model with convex transaction costs. Our model
extends
Kabanov's currency market model by allowing for nonlinear illiquidity
effects.
We show that an appropriate generalization of Schachermayer's robust no
arbitrage condition implies that the set of claims hedgeable with zero
cost is
closed in probability. Combined with classical techniques of convex
analysis,
the closedness yields a dual characterization of premium processes
that are
sufficient to superhedge a given claim process. We also extend the
fundamental
theorem of asset pricing for general conical models.
http://arxiv.org/abs/0810.2016
---------------------------------------------------------------
7604. CONSTANT RATE DISTRIBUTIONS ON PARTIALLY ORDERED SETS
Kyle Siegrist
We consider probability distributions with constant rate on partially
ordered
sets, generalizing distributions in the usual reliability setting that
have
constant failure rate. In spite of the minimal algebraic structure,
there is a
surprisingly rich theory, including interesting moment results and an
associated stochastic process with "gamma" distributions. We
concentrate mostly
on discrete posets and characterize constant rate distributions on
trees. We
pose some questions on the existence of constant rate distributions
for general
discrete posets.
http://arxiv.org/abs/0810.2105
---------------------------------------------------------------
7605. ON COLLISIONS OF BROWNIAN PARTICLES
Tomoyuki Ichiba and Ioannis Karatzas
We examine the behavior of n Brownian particles diffusing on the real
line,
with bounded, measurable drift and bounded, piecewise continuous
diffusion
coefficients that depend on the current configuration of particles.
Sufficient
conditions are established for the absence of triple collisions, as
well as for
the presence of (infinitely-many) triple collisions among the
particles. As an
application to the Atlas model of equity markets, we study a special
construction of such systems of diffusing particles using Brownian
motions with
reflection on polyhedral domains.
http://arxiv.org/abs/0810.2149
---------------------------------------------------------------
7606. PHASE TRANSITION FOR THE ISING MODEL ON THE CRITICAL LORENTZIAN
TRIANGULATION
Maxim Krikun and Anatoly Yambartsev
Ising model without external field on an infinite Lorentzian
triangulation
sampled from the uniform distribution is considered. We prove
uniqueness of the
Gibbs measure in the high temperature region and coexistence of at
least two
Gibbs measures at low temperature. The proofs are based on the
disagreement
percolation method and on a variant of Peierls method. The critical
temperature
is shown to be constant a.s.
http://arxiv.org/abs/0810.2182
---------------------------------------------------------------
7607. UPWARD AND DOWNWARD RUNS ON PARTIALLY ORDERED SETS
Kyle Siegrist
We consider Markov chains on partially ordered sets that generalize the
success-runs and remaining life chains in reliability theory. We find
conditions for recurrence and transience and give simple expressions
for the
invariant distributions. We study a number of special cases, including
rooted
trees, uniform posets, and posets associated with positive semigroups.
http://arxiv.org/abs/0810.2289
---------------------------------------------------------------
7608. DISCRETE COMPLEX ANALYSIS ON ISORADIAL GRAPHS
Dmitry Chelkak and Stanislav Smirnov
We study discrete complex analysis and potential theory on a large
family of
planar graphs, the so-called isoradial ones. Along with discrete
analogues of
several classical results, we prove uniform convergence of discrete
harmonic
measures, Green's functions and Poisson kernels to their continuous
counterparts. Among other applications, the results can be used to
establish
universality of the critical Ising and other lattice models.
http://arxiv.org/abs/0810.2188
---------------------------------------------------------------
7609. TRAVELLING WITH/AGAINST THE FLOW. DETERMINISTIC DIFFUSIVE DRIVEN
SYSTEMS
Michael Blank
We introduce and study a deterministic lattice model describing the
motion of
an infinite system of oppositely charged particles under the action of a
constant electric field. As an application this model represents a
traffic flow
of cars moving in opposite directions along a narrow road. Our main
results
concern the Fundamental diagram of the system describing the
dependence of
average particle velocities on their densities and the Phase diagram
describing
the partition of the space of particle configurations into regions
having
different qualitative properties, which we identify with free, jammed
and
hysteresis phases.
http://arxiv.org/abs/0810.2205
---------------------------------------------------------------
7610. RARE EVENTS, ESCAPE RATES AND QUASISTATIONARITY: SOME EXACT
FORMULAE
Gerhard Keller and Carlangelo Liverani
We present a common framework to study decay and exchanges rates in a
wide
class of dynamical systems. Several applications, ranging form the
metric
theory of continuons fractions and the Shannon capacity of contrained
systems
to the decay rate of metastable states, are given.
http://arxiv.org/abs/0810.2229
---------------------------------------------------------------
7611. THE GEOMETRY OF FILTERING
K. D. Elworthy and Y. LeJan and Xue-Mei Li
Geometry arising from two diffusion operators (smooth semi-elliptic,
second
order differential operators) on different spaces but intertwined by a
smooth
map is described. Particular cases arise from Riemannian submersions
when the
operators are Laplace-Beltrami operators, from equivariant operators
on the
total space of a principal bundle, and for the operators on the
diffeomorphism
group arising from stochastic flows. Classical non-linear filtering
problems
also lead to such conffigurations. A basic tool is the, possibly, non-
linear
"semi-connection" induced by this set up, leading to a canonical
decomposition
of the operator on the domain space. Topics discussed include:
generalised
Wietzenbock curvatures arising in the equivariant case, skew -product
decompositions of diffusion processes, conditioned processes, classical
filtering, decomposition of stochastic flows, and connections
determined by
stochastic differential equations.
http://arxiv.org/abs/0810.2253
---------------------------------------------------------------
7612. DSITRIBUTION OF VALUES OF $L$-FUNCTIONS AT THE EDGE OF THE
CRITICAL STRIP
Youness Lamzouri
We prove several results on the distribution of values of $L$-
functions at
the edge of the critical strip, by constructing and studying a large
class of
random Euler products. Among new applications, we provide a precise
estimate
for the distribution of the values at $s=1$ of the family of $k$-th
symmetric
power $L$-functions of primitive cusp forms of weight 2 in the level
aspect
(assuming the automorphy of these $L$-functions), using results of
Cogdell and
Michel on high complex moments of this family. Further we study
families of
$L$-functions of the Selberg Class in the $t$ aspect at Re$(s)=1$, and
of
$L$-functions of quadratic twists of cuspidal automorphic
representations of
$GL(m)/{\Bbb Q}$ at $s=1$, and prove precise estimates for the
corresponding
distribution functions assuming a uniform version of the Sato-Tate
conjecture.
http://arxiv.org/abs/0810.2292
---------------------------------------------------------------
7613. CENTRAL LIMIT THEOREM AND RECURRENCE FOR RANDOM WALKS IN
BISTOCHASTIC RANDOM ENVIRONMENTS
Marco Lenci
We prove the annealed Central Limit Theorem for random walks in
bistochastic
random environments on $Z^d$ with zero local drift. The proof is based
on a
"dynamicist's interpretation" of the system, and requires a much weaker
condition than the customary uniform ellipticity. Moreover, recurrence
is
derived for $d \le 2$.
http://arxiv.org/abs/0810.2324
---------------------------------------------------------------
7614. PARABOLIC HARNACK INEQUALITY AND LOCAL LIMIT THEOREM FOR
PERCOLATION CLUSTERS
Martin Barlow and Ben Hambly
We consider the random walk on supercritical percolation clusters in the
d-dimensional Euclidean lattice. Previous papers have obtained
Gaussian heat
kernel bounds, and a.s. invariance principles for this process. We
show how
this information leads to a parabolic Harnack inequality, a local
limit theorem
and estimates on the Green's function.
http://arxiv.org/abs/0810.2467
---------------------------------------------------------------
7615. PERCOLATION ON CORRELATED NETWORKS
A. V. Goltsev and S. N. Dorogovtsev and J. F. F. Mendes
We reconsider the problem of percolation on an equilibrium random
network
with degree-degree correlations between nearest-neighboring vertices
focusing
on critical singularities at a percolation threshold. We obtain
criteria for
degree-degree correlations to be irrelevant for critical
singularities. We
present examples of networks in which assortative and disassortative
mixing
leads to unusual percolation properties and new critical exponents.
http://arxiv.org/abs/0810.1742
---------------------------------------------------------------
7616. AN INDICATOR FUNCTION LIMIT THEOREM IN DYNAMICAL SYSTEMS
Olivier Durieu and Dalibor Volny
We show by a constructive proof that in all aperiodic dynamical
system, for
all sequences $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty
$ and
$\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a set $A\in\A$
having the
property that the sequence of the distributions of
$(\frac{1}{a_{n}}S_{n}(\ind_A-\mu(A)))_{n\in\N}$ is dense in the space
of all
probability measures on $\R$.
http://arxiv.org/abs/0810.2452
---------------------------------------------------------------
7617. FROM RANDOM WALKS TO ROUGH PATHS
Emmanuel Breuillard and Peter Friz and Martin Huesmann
Donsker's invariance principle is shown to hold for random walks in
rough
path topology. As application, we obtain Donsker-type weak limit
theorems for
stochastic integrals and differential equations.
http://arxiv.org/abs/0810.2681
---------------------------------------------------------------
7618. YET ANOTHER LOOK AT HARRIS' ERGODIC THEOREM FOR MARKOV CHAINS
Martin Hairer and Jonathan C. Mattingly
The aim of this note is to present an elementary proof of a variation of
Harris' ergodic theorem of Markov chains. This theorem, dating back to
the
fifties essentially states that a Markov chain is uniquely ergodic if
it admits
a ``small'' set which is visited infinitely often. This gives an
extension of
the ideas of Doeblin to the unbounded state space setting. Often this is
established by finding a Lyapunov function with ``small'' level sets.
This
topic has been studied by many authors (cf. Harris, Hasminskii,
Nummelin, Meyn
and Tweedie). If the Lyapunov function is strong enough, one has a
spectral gap
in a weighted supremum norm (cf. Meyn and Tweedie).
Traditional proofs of this result rely on the decomposition of the
Markov
chain into excursions away from the small set and a careful analysis
of the
exponential tail of the length of these excursions. There have been
other
variations which have made use of Poisson equations or worked at getting
explicit constants. The present proof is very direct, and relies
instead on
introducing a family of equivalent weighted norms indexed by a parameter
$\beta$ and to make an appropriate choice of this parameter that
allows to
combine in a very elementary way the two ingredients (existence of a
Lyapunov
function and irreducibility) that are crucial in obtaining a spectral
gap.
The original motivation of this proof was the authors' work on
spectral gaps
in Wasserstein metrics. The proof presented in this note is a version
of our
reasoning in the total variation setting which we used to guide the
calculations in arXiv:math/0602479. While we initially produced it for
that
purpose, we hope that it will be of interest in its own right.
http://arxiv.org/abs/0810.2777
---------------------------------------------------------------
7619. ASYMPTOTIC BEHAVIOR OF MAXIMUM LIKELIHOOD ESTIMATOR FOR TIME
INHOMOGENEOUS DIFFUSION PROCESSES
Matyas Barczy and Gyula Pap
We study asymptotic behavior of maximum likelihood estimator for a time
inhomogeneous diffusion process given by a SDE $dX_t=\alpha b(t)X_t dt +
\sigma(t) dB_t$, $t\in[0,T)$, with a parameter $\alpha\in R$, where
$T\in(0,\infty]$ and $(B_t)_{t\in[0,T)}$ is a standard Wiener process.
We
formulate sufficient conditions under which the MLE of $\alpha$
normalized by
Fisher information converges to the limit distribution of Dickey-Fuller
statistics. Next we study a SDE $dY_t=\alpha b(t)a(Y_t) dt + \sigma(t)
dB_t$,
$t\in[0,T)$, with a perturbed drift satisfying $a(x)=x+O(1+|x|^\gamma)
$ with
some $\gamma\in[0,1)$. We give again sufficient conditions under which
the MLE
of $\alpha$ normalized by Fisher information converges to the limit
distribution of Dickey-Fuller statistics.
http://arxiv.org/abs/0810.2688
---------------------------------------------------------------
7620. MULTI-VARIABLE SUBORDINATION DISTRIBUTIONS FOR FREE ADDITIVE
CONVOLUTION
Alexandru Nica
Let k be a positive integer and let D_k denote the space of joint
distributions for k-tuples of selfadjoint elements in C*-probability
space. The
paper studies the concept of "subordination distribution of \mu
\boxplus \nu
with respect to \nu" for \mu, \nu \in D_k, where \boxplus is the
operation of
free additive convolution on D_k. The main tools used in this study are
combinatorial properties of R-transforms for joint distributions and a
related
operator model, with operators acting on the full Fock space
Multi-variable subordination turns out to have nice relations to a
process of
evolution towards \boxplus-infinite divisibility on D_k that was
recently found
by Belinschi and Nica (arXiv:0711.3787). Most notably, one gets better
insight
into a connection which this process was known to have with free
Brownian
motion.
http://arxiv.org/abs/0810.2571
---------------------------------------------------------------
7621. LIMIT THEOREMS FOR INDIVIDUAL-BASED MODELS IN ECONOMICS AND
FINANCE
Daniel Remenik
There is a widespread recent interest in using ideas from statistical
physics
to model certain types of problems in economics and finance. The main
idea is
to derive the macroscopic behavior of the market from the random local
interactions between agents. Our purpose is to present a general
framework that
encompasses a broad range of models, by proving a law of large numbers
and a
central limit theorem for certain interacting particle systems with very
general state spaces. To do this we draw inspiration from some work
done in
mathematical ecology and mathematical physics. The first result is
proved for
the system seen as a measure-valued process, while to prove the second
one we
will need to introduce a chain of embeddings of some abstract Banach and
Hilbert spaces of test functions and prove that the fluctuations
converge to
the solution of a certain generalized Gaussian stochastic differential
equation
taking values in the dual of one of these spaces.
http://arxiv.org/abs/0810.2813
---------------------------------------------------------------
7622. CONCENTRATION INEQUALITIES AND LAWS OF LARGE NUMBERS UNDER
IRRELEVANCE OF LOWER AND UPPER EXPECTATIONS
Fabio Gagliardi Cozman
This paper presents concentration inequalities and laws of large numbers
under weak assumptions of irrelevance, expressed through lower and upper
expectations. The results extend de Cooman and Miranda's recent results
concerning epistemic irrelevance. The proofs indicate connections
between
concepts of irrelevance for lower/upper expectations and the standard
theory of
martingales.
http://arxiv.org/abs/0810.2821
---------------------------------------------------------------
7623. STEIN'S METHOD AND STOCHASTIC ANALYSIS OF RADEMACHER FUNCTIONALS
Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X) and Gesine Reinert
We compute explicit bounds in the Gaussian approximation of
functionals of
infinite Rademacher sequences. Our tools involve Stein's method, as
well as the
use of appropriate discrete Malliavin operators. Although our approach
does not
require the classical use of exchangeable pairs, we employ a chaos
expansion in
order to construct an explicit exchangeable pair vector for any random
variable
which depends on a finite set of Rademacher variables. Among several
examples,
which include random variables which depend on infinitely many
Rademacher
variables, we provide three main applications: (i) to CLTs for
multilinear
forms belonging to a fixed chaos, (ii) to the Gaussian approximation of
weighted infinite 2-runs, and (iii) to the computation of explicit
bounds in
CLTs for multiple integrals over sparse sets. This last application
provides an
alternate proof (and several refinements) of a recent result by Blei and
Janson.
http://arxiv.org/abs/0810.2890
---------------------------------------------------------------
7624. EXPLICIT FORMULAS FOR LAPLACE TRANSFORMS OF CERTAIN FUNCTIONALS
OF SOME TIME INHOMOGENEOUS DIFFUSIONS
Matyas Barczy and Gyula Pap
We consider a process $(X_t)_{t\in[0,T)}$ given by the SDE $dX_t =
\alpha
b(t)X_t dt + \sigma(t) dB_t$, $t\in[0,T)$, with initial condition
$X_0=0$,
where $T\in(0,\infty]$, $\alpha\in R$, $(B_t)_{t\in[0,T)}$ is a
standard Wiener
process, $b:[0,T)\to R\setminus\{0\}$ and $\sigma:[0,T)\to(0,\infty)$
are
continuously differentiable functions. Assuming that $b$ and $\sigma$
satisfy a
certain differential equation we derive an explicit formula for the
joint
Laplace transform of $\int_0^t\frac{b(s)^2}{\sigma(s)^2}(X_s)^2 ds$ and
$(X_t)^2$ for all $t\in[0,T)$. As an application, we study asymptotic
behavior
of the maximum likelihood estimator of $\alpha$ for $\sign(\alpha-K)=
\sign(K)$,
$K\ne0$, and for $\alpha=K$, $K\ne0$. As an example, we examine the so-
called
$\alpha$-Wiener bridges given by SDE $dX_t = -\frac{\alpha}{T-t}X_t dt
+ dB_t$,
$t\in[0,T)$, with initial condition $X_0=0$.
http://arxiv.org/abs/0810.2930
---------------------------------------------------------------
7625. ON THE LIMITING SHAPE OF MARKOVIAN RANDOM YOUNG TABLEAUX
Christian Houdr\'e and Trevis J. Litherland
Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov
chain,
with state space an ordered finite alphabet of size $m$. Using
combinatorial
constructions and weak invariance principles, we obtain the limiting
shape of
the associated Young tableau as a multidimensional Brownian
functional. Since
the length of the top row of the Young tableau is also the length of the
longest (weakly) increasing subsequence of $(X_k)_{1\le k \le n}$, the
corresponding limiting law follows. We relate our results to a
conjecture of
Kuperberg by showing that, under a cyclic condition, a spectral
characterization of the Markov transition matrix delineates precisely
when the
limiting shape is the spectrum of the traceless GUE. For $m=3$, all
cyclic
Markov chains have such a limiting shape, a fact previously known for
$m=2$.
However, this is no longer true for $m \ge 4$.
http://arxiv.org/abs/0810.2982
---------------------------------------------------------------
7626. FROM THE LITTLEWOOD-OFFORD PROBLEM TO THE CIRCULAR LAW:
UNIVERSALITY OF THE SPECTRAL DISTRIBUTION OF RANDOM MATRICES
Terence Tao and Van Vu
The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$
matrix with iid complex entries of mean zero and unit variance, then the
empirical spectral distribution (ESD) of the normalized matrix
$\frac{1}{\sqrt{n}} M_n$ converges almost surely to the uniform
distribution on
the unit disk $\{z \in \C: |z| \leq 1 \}$. After a long sequence of
partial
results that verified this law under additional assumptions on the
distribution
of the entries, the full circular law was recently established in
\cite{TVcir2}. In this survey we describe some of the key ingredients
used in
the establishment of the circular law, in particular recent advances in
understanding the Littlewood-Offord problem and its inverse.
http://arxiv.org/abs/0810.2994
---------------------------------------------------------------
7627. SMALL PROBABILITY EVENTS FOR TWO-LAYER GEOPHYSICAL FLOWS UNDER
UNCERTAINTY
Aijun Du and Jinqiao Duan and Hongjun Gao
The stochastics two-layer quasi-geostrophic flow model is an
intermediate
system between the single-layer two dimensional barotropic flow model
and the
continuously stratified three dimensional baroclinic flow model. This
model is
widely used to investigate basic mechanisms in geophysical flows, such
as
baroclinic effects, the Gulf Stream and subtropical gyres.
A large deviation principle for the two-layer quasi-geostrophic
flow model
under uncertainty is proved. The proof is based on the Laplace
principle and a
variational approach. This approach does not require the exponential
tightness
estimates which are needed in other methods for establishing large
deviation
principles.
http://arxiv.org/abs/0810.2818
---------------------------------------------------------------
7628. TABLEAUX COMBINATORICS FOR THE ASYMMETRIC EXCLUSION PROCESS II
Sylvie Corteel and Lauren Williams
The partially asymmetric exclusion process (PASEP) is an important
model from
statistical mechanics which describes a system of interacting
particles hopping
left and right on a one-dimensional lattice of n sites. It has been
cited as a
model for traffic flow and protein synthesis. In its most general form,
particles may enter and exit at the left with probabilities alpha and
gamma,
and they may exit and enter at the right with probabilities beta and
delta. In
the bulk, the probability of hopping left is q times the probability
of hopping
right.
In previous work we used the matrix ansatz to give a combinatorial
formula
for the steady state probability of each state of the PASEP, when
gamma=delta=0. The formula was the generating function for permutation
tableaux
of a fixed shape, weighted according to three statistics. In this
paper we give
a simple one-parameter generalization of the matrix ansatz, then use
it to
generalize our results about the PASEP to the case of general alpha,
beta,
gamma, delta (and q=1). We replace permutation tableaux by the
slightly more
general bordered permutation tableaux, which we show have cardinality
4^n n!.
We also state our results in terms of alternative tableaux.
http://arxiv.org/abs/0810.2916
---------------------------------------------------------------
7629. ON SUMS OF INDICATOR FUNCTIONS IN DYNAMICAL SYSTEMS
Olivier Durieu and Dalibor Volny
In this paper, we are interested in the limit theorem question for
sums of
indicator functions. We show that in every aperiodic dynamical system,
for
every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that
$a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there
exist a
measurable set $A$ such that the sequence of the partial sums
$\frac{1}{a_n}\sum_{i=0}^{n-1}(\ind_A-\mu(A))\circ T^i$ is dense in
the set of
the probability measures on $\R$. Further, in the ergodic case, we
prove that
there exists a dense $G_\delta$ of such sets.
http://arxiv.org/abs/0810.2917
---------------------------------------------------------------
7630. ALPHA-WIENER BRIDGES: SINGULARITY OF INDUCED MEASURES AND SAMPLE
PATH PROPERTIES
Matyas Barczy and Gyula Pap
Let us consider the process $(X_t^{(\alpha)})_{t\in[0,T)}$ given by
the SDE
$dX_t^{(\alpha)} = -\frac{\alpha}{T-t}X_t^{(\alpha)} dt+ dB_t$, $t
\in[0,T)$,
where $\alpha\in R$, $T\in(0,\infty)$, and $(B_t)_{t\geq 0}$ is a
standard
Wiener process. In case of $\alpha>0$ the process $X^{(\alpha)}$ is
known as an
$\alpha$-Wiener bridge, in case of $\alpha=1$ as the usual Wiener
bridge. We
prove that for all $\alpha,\beta\in R$, $\alpha\ne\beta$, the
probability
measures induced by the processes $X^{(\alpha)}$ and $X^{(\beta)}$ are
singular
on C[0,T). Further, we investigate regularity properties of
$X_t^{(\alpha)}$ as
$t\uparrow T$.
http://arxiv.org/abs/0810.3070
---------------------------------------------------------------
7631. OCCUPANCY SCHEMES ASSOCIATED TO YULE PROCESSES
Philippe Robert and Florian Simatos
An occupancy problem with an infinite number of bins and a random
probability
vector for the locations of the balls is considered. The respective
sizes of
bins are related to the split times of a Yule process. The asymptotic
behavior
of the landscape of first empty bins, i.e., the set of corresponding
indices
represented by point processes, is analyzed and convergences in
distribution to
mixed Poisson processes are established. Additionally, the influence
of the
random environment, the random probability vector, is analyzed. It is
represented by two main components: an i.i.d. sequence and a fixed
random
variable. Each of these components has a specific impact on the
qualitative
behavior of the stochastic model. It is shown in particular that for
some
values of the parameters, some rare events, which are identified, play
an
important role on average values of the number of empty bins in some
regions.
http://arxiv.org/abs/0810.3079
---------------------------------------------------------------
7632. PROBABILISTIC CHARACTERISATION OF BESOV-LIPSCHITZ SPACES ON
METRIC MEASURE SPACES
Katarzyna Pietruska-Pa{\l}uba
We give a probabilistic characterisation of the Besov-Lipschitz spaces
$Lip(\alpha,p,q)(X)$ on domains which support a Markovian kernel with
appropriate exponential bounds. This extends former results of
\cite{Jon,KPP1,KPP2,GHL} which were valid for $\alpha=\frac{d_w}
{2},p=2$,
$q=\infty,$ where $d_w$ is the walk dimension of the space $X.$
http://arxiv.org/abs/0810.3098
---------------------------------------------------------------
7633. ON THE NUMBER OF CUTPOINTS OF THE TRANSIENT NEAREST NEIGHBOR
RANDOM WALK ON THE LINE
Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
We consider transient nearest neighbor random walks on the positive
part of
the real line. We give criteria for the finiteness of the number of
cutpoints
and strong cutpoints. Examples and open problems are presented.
http://arxiv.org/abs/0810.3123
---------------------------------------------------------------
7634. A MODEL FOR INFECTION ON GRAPHS
M.Draief and A. Ganesh
We address the question of understanding the effect of the underlying
network
topology on the spread of a virus and the dissemination of information
when
users are mobile performing independent random walks on a graph. To
this end we
propose a simple model of infection that enables to study the
coincidence time
of two random walkers on an arbitrary graph. By studying the
coincidence time
of a susceptible and an infected individual both moving in the graph
we obtain
estimates of the infection probability. The main result of this paper
is to
pinpoint the impact of the network topology on the infection
probability. More
precisely, we prove that for homogeneous graph including regular
graphs and the
classical Erdos-Renyi model, the coincidence time is inversely
proportional to
the number of nodes in the graph. We then study the model on power-law
graphs,
that exhibit heterogeneous connectivity patterns, and show the
existence of a
phase transition for the coincidence time depending on the parameter
of the
power-law of the degree distribution.
http://arxiv.org/abs/0810.3128
---------------------------------------------------------------
7635. EXPONENTIAL RANDOM GRAPHS AS MODELS OF OVERLAY NETWORKS
M. Draief and A. Ganesh and L. Massoulie
In this paper, we give an analytic solution for graphs with n nodes
and E
edges for which the probability of obtaining a given graph G is
specified in
terms of the degree sequence of G. We describe how this model
naturally appears
in the context of load balancing in communication networks, namely
Peer-to-Peer
overlays. We then analyse the degree distribution of such graphs and
show that
the degrees are concentrated around their mean value. Finally, we derive
asymptotic results on the number of edges crossing a graph cut and use
these
results $(i)$ to compute the graph expansion and conductance, and $(ii)
$ to
analyse the graph resilience to random failures.
http://arxiv.org/abs/0810.3173
---------------------------------------------------------------
7636. OPTIMAL SWITCHING OF ONE-DIMENSIONAL REFLECTED BSDES, AND
ASSOCIATED MULTI-DIMENSIONAL BSDES WITH OBLIQUE REFLECTION
Shanjian Tang and Wei Zhong
In this paper, the optimal switching problem is proposed for one-
dimensional
reflected backward stochastic differential equations (BSDEs, for
short) where
the generators, the terminal values, and the barriers are all switched
with
positive costs. The value process is characterized by a system of
multi-dimensional reflected BSDEs with oblique reflection, whose
existence and
uniqueness is by no means trivial and is therefore carefully examined.
Existence is shown using both methods of the Picard iteration and
penalization,
but under some different conditions. Uniqueness is proved by
representation
either as the equilibrium value process to a stochastic mixed game of
switching
and stopping, or as the value process to our optimal switching problem
for
one-dimensional reflected BSDEs.
http://arxiv.org/abs/0810.3176
---------------------------------------------------------------
7637. SPECTRA OF WINNER-TAKE-ALL STOCHASTIC NEURAL NETWORKS
Tomasz Schreiber
In Piekniewski and Schreiber (2008) we have developed a simple
mathematical
model for information flow structure in a class of recurrent neural
networks
and shown that its asymptotic behaviour is scale-free and admits a
description
in terms of the so-called winner-take-all dynamics. In the present
paper we
establish a limit theorem for spectra of the spike-flow graphs induced
by the
winner-take-all dynamics.
http://arxiv.org/abs/0810.3193
---------------------------------------------------------------
7638. INVARIANCE PRINCIPLE AND RECURRENCE FOR RANDOM WALKS IN RANDOM
ENVIRONMENTS WITH ZERO LOCAL DRIFT
Marco Lenci
We prove the quenched Invariance Principle for random walks in random
environments on $Z^d$ with zero local drift, subject to a transitivity
hypothesis which is much weaker than the customary uniform
ellipticity. We then
obtain recurrence for $d \le 2$. The proofs are based on a
representation of
the system as an "uncountable union" of one-dimensional Markov maps,
and on a
powerful theorem by Schmidt on cocycle recurrence.
http://arxiv.org/abs/0810.3197
---------------------------------------------------------------
7639. WEAK ERROR FOR STABLE DRIVEN SDES: EXPANSION OF THE DENSITIES
Valentin Konakov (CMI RAS) and Stephane Menozzi (PMA)
Consider a multidimensional SDE of the form $X_t = x+\int_{0}^{t}
b(X_{s-})ds+\int{0}^{t} f(X_{s-})dZ_s$ where $(Z_s)_{s\ge 0}$ is a
symmetric
stable process. Under suitable assumptions on the coefficients the
unique
strong solution of the above equation admits a density w.r.t. the
Lebesgue
measure and so does its Euler scheme. Using a parametrix approach, we
derive an
error expansion at order 1 w.r.t. the time step for the difference of
these
densities.
http://arxiv.org/abs/0810.3224
---------------------------------------------------------------
7640. SELF-SIMILAR MARKOV PROCESSES ON CANTOR SET
Yuri Bakhtin
We define analogues of Brownian motion on the triadic Cantor set by
introducing a few natural requirements on the Markov semigroup. We
give a
detailed description of these symmetric self-similar processes and
study their
properties such as mixing and moment asymptotics.
http://arxiv.org/abs/0810.3260
---------------------------------------------------------------
7641. CENTRAL LIMIT THEOREMS FOR THE ENERGY DENSITY IN THE
SHERRINGTON-KIRKPATRICK MODEL
Sourav Chatterjee and Nick Crawford
In this paper we consider central limit theorems for various macroscopic
observables in the high temperature region of the Sherrington-
Kirkpatrick spin
glass model. With a particular focus on obtaining a quenched central
limit
theorem for the energy density of the system with non-zero external
field, we
show how to combine the mean field cavity method with Stein's method
in the
quenched regime. The result for the energy density extends the
corresponding
result of Comets-Neveu.
http://arxiv.org/abs/0810.3279
---------------------------------------------------------------
7642. THE RATE OF CONVERGENCE OF THE WALK ON SPHERES ALGORITHM
Ilia Binder and Mark Braverman
In this paper we examine the rate of convergence of one of the standard
algorithms for emulating exit probabilities of Brownian motion, the
Walk on
Spheres (WoS) algorithm. We obtain the complete characterization of
the rate of
convergence of WoS in terms of the local geomnetry of a domain.
http://arxiv.org/abs/0810.3343
---------------------------------------------------------------
7643. EXCITED BROWNIAN MOTIONS
Olivier Raimond (LM-Orsay and MODAL'X) and Bruno Schapira (LM-Orsay)
We introduce and study a natural continuous time version of excited
random
walks. In the case of nonnegative drift, we obtain a necessary and
sufficient
condition for recurrence. This result is analogous to Zerner's result
\cite{Zer1} for excited (or cookie) random walks. We use similar
arguments.
http://arxiv.org/abs/0810.3538
---------------------------------------------------------------
7644. CENTRAL EXTENSIONS OF THE HEISENBERG ALGEBRA
Luigi Accardi and Andreas Boukas
We study the non-trivial central extensions $CEHeis$ of the Heisenberg
algebra $Heis$ recently constructed in {AccBouCE}. We prove that a
real form of
$CEHeis$ is one the fifteen classified real four--dimensional solvable
Lie
algebras. We also show that $CEHeis$ can be realized (i) as a sub--
Lie--algebra
of the Schroedinger algebra and (ii) in terms of two independent
copies of the
canonical commutation relations (CCR). This gives a natural family of
unitary
representations of $CEHeis$ and allows an explicit determination of the
associated group by exponentiation. In contrast with $Heis$, the group
law for
$CEHeis$ is given by nonlinear (quadratic) functions of the coordinates.
http://arxiv.org/abs/0810.3365
---------------------------------------------------------------
7645. SCALING LIMITS FOR WIDTH TWO PARTIALLY ORDERED SETS: THE
INCOMPARABILITY WINDOW
Nayantara Bhatnagar and Nick Crawford and Elchanan Mossel and Arnab
Sen
We study the structure of a uniformly randomly chosen partial order of
width
2 on n elements. We analyze the local structure by studying the number
of
elements incomparable to a random element in the poset. We show that
under the
appropriate scaling, the number of incomparable elements converges to
the
height of a one dimensional Brownian excursion at a uniformly chosen
random
time in the interval [0,1], which follows the Rayleigh distribution.
http://arxiv.org/abs/0810.3670
---------------------------------------------------------------
7646. DIFFERENCE OPERATORS AND DETERMINANTAL POINT PROCESSES
Grigori Olshanski
We consider a family {P} of determinantal point processes arising in
representation theory and random matrix theory. The processes live on
the
one-dimensional lattice and their correlation kernels correspond to
projection
operators in the l^2 Hilbert space on the lattice. Moreover, these
projections
are spectral projections associated to certain selfadjoint second order
difference operators on the lattice. The aim of the note is to
demonstrate that
the difference operators in question can be efficiently employed in
the study
of limit transitions inside the family {P}.
http://arxiv.org/abs/0810.3751
---------------------------------------------------------------
7647. RANDOM WALKS ON COMPLEX TREES
Andrea Baronchelli and Michele Catanzaro and Romualdo Pastor-Satorras
We study the properties of random walks on complex trees. We observe
that the
absence of loops reflects in physical observables showing large
differences
with respect to their looped counterparts. First, both the vertex
discovery
rate and the mean topological displacement from the origin present a
considerable slowing down in the tree case. Second, the mean first
passage time
(MFPT) displays a logarithmic degree dependence, in contrast to the
inverse
degree shape exhibited in looped networks. This deviation can be
ascribed to
the dominance of source-target topological distance in trees. To show
this, we
study the distance dependence of a symmetrized MFPT and derive its
logarithmic
profile, obtaining good agreement with simulation results. These unique
properties shed light on the recently reported anomalies observed in
diffusive
dynamical systems on trees.
http://arxiv.org/abs/0801.1278
---------------------------------------------------------------
7648. STRUCTURALLY DAMPED PLATE AND WAVE EQUATIONS WITH RANDOM POINT
FORCE IN ARBITRARY SPACE DIMENSIONS
Roland Schnaubelt and Mark Veraar
In this paper we consider structurally damped plate and wave equations
with
point and distributed random forces. In order to treat space
dimensions more
than one, we work in the setting of $L^q$--spaces with (possibly small)
$q\in(1,2)$. We establish existence, uniqueness and regularity of mild
and weak
solutions to the stochastic equations employing recent theory for
stochastic
evolution equations in UMD Banach spaces.
http://arxiv.org/abs/0810.3898
---------------------------------------------------------------
7649. THROWING BALLS ON HOMOGENEOUS SURFACES
Anne Estrade (MAP5) and Jacques Istas (LJK)
Throwing balls on Euclidean spaces have been considered since a while.
With
suitable renormalization, it leads to fractional Brownian motion as
limit
object. We investigate in this paper the throw of balls on spheres and
hyperbolic spaces. This leads to dramatically different behaviors. On
spheres,
we obtain a Gaussian limit that is no more a fractinal Brownian
motion, but is
locally self-similar. On hyperbolic spaces, we prove that there is no
any
limit.
http://arxiv.org/abs/0810.4004
---------------------------------------------------------------
7650. DISCRETE APPROXIMATION OF THE FREE FOCK SPACE
St\'ephane Attal (ICJ) and Ion Nechita (ICJ)
We prove that the free Fock space ${\F}(\R^+;\C)$, which is very
commonly
used in Free Probability Theory, is the continuous free product of
copies of
the space $\C^2$. We describe an explicit embedding and approximation
of this
continuous free product structure by means of a discrete-time
approximation:
the free toy Fock space, a countable free product of copies of $\C^2$.
We show
that the basic creation, annihilation and gauge operators of the free
Fock
space are also limits of elementary operators on the free toy Fock
space. When
applying these constructions and results to the probabilistic
interpretations
of these spaces, we recover some discrete approximations of the semi-
circular
Brownian motion and of the free Poisson process. All these results are
also
extended to the higher multiplicity case, that is, ${\F}(\R^+;\C^N)$
is the
continuous free product of copies of the space $\C^{N+1}$.
http://arxiv.org/abs/0810.4070
---------------------------------------------------------------
7651. MULTISTAGE HYPOTHESIS TESTS FOR THE MEAN OF A NORMAL DISTRIBUTION
Xinjia Chen
In this paper, we have developed new multistage tests which guarantee
prescribed level of power and are more efficient than previous tests
in terms
of average sampling number and the number of sampling operations.
Without
truncation, the maximum sampling numbers of our testing plans are
absolutely
bounded. Based on geometrical arguments, we have derived extremely
tight bounds
for the operating characteristic function.
http://arxiv.org/abs/0810.3946
---------------------------------------------------------------
7652. LOCALIZATION FOR LINEAR STOCHASTIC EVOLUTIONS
Nobuo Yoshida
We consider a discrete-time stochastic growth model on $d$-dimensional
lattice. The growth model describes various interesting examples such as
oriented site/bond percolation, directed polymers in random
environment, time
discretizations of binary contact path process. We show the
equivalence between
the slow population growth and a localization property in terms of
"replica
overlap". This extends a result known for the directed polymers in
random
environment to a large class of models. A new approach, based on the
multiplicative Doob's decomposition, is adopted to overcome the
difficulty that
the total population may get extinct even at finite time.
http://arxiv.org/abs/0810.4218
---------------------------------------------------------------
7653. CHAOS, CONCENTRATION, AND MULTIPLE VALLEYS
Sourav Chatterjee
Disordered systems are an important class of models in statistical
mechanics,
having the defining characteristic that the energy landscape is a fixed
realization of a random field. Examples include various models of
glasses and
polymers. They also arise in other areas, like fitness models in
evolutionary
biology. The ground state of a disordered system is the state with
minimum
energy. The system is said to be chaotic if a small perturbation of
the energy
landscape causes a drastic shift of the ground state. We present a
rigorous
theory of chaos in disordered systems that confirms long-standing
physics
intuition about connections between chaos, anomalous fluctuations of
the ground
state energy, and the existence of multiple valleys in the energy
landscape.
Combining these results with mathematical tools like
hypercontractivity, we
establish the existence of the above phenomena in eigenvectors of GUE
matrices,
the Kauffman-Levin model of evolutionary biology, directed polymers in
random
environment, a subclass of the generalized Sherrington-Kirkpatrick
model of
spin glasses, the discrete Gaussian free field, and continuous
Gaussian fields
on Euclidean spaces. We also list several open questions.
http://arxiv.org/abs/0810.4221
---------------------------------------------------------------
7654. FACILITATED ORIENTED SPIN MODELS:SOME NON EQUILIBRIUM RESULTS
Nicoletta Cancrini and Fabio Martinelli and Roberto H. Schonmann
and Cristina Toninelli
We analyze the relaxation to equilibrium for kinetically constrained
spin
models (KCSM) when the initial distribution $\nu$ is different from the
reversible one, $\mu$. This setting has been intensively studied in
the physics
literature to analyze the slow dynamics which follows a sudden quench
from the
liquid to the glass phase. We concentrate on two basic oriented KCSM:
the East
model on $\bbZ$, for which the constraint requires that the East
neighbor of
the to-be-update vertex is vacant and the model on the binary tree
introduced
in \cite{Aldous:2002p1074}, for which the constraint requires the two
children
to be vacant. While the former model is ergodic at any $p\neq 1$, the
latter
displays an ergodicity breaking transition at $p_c=1/2$. For the East
we prove
exponential convergence to equilibrium with rate depending on the
spectral gap
if $\nu$ is concentrated on any configuration which does not contain a
forever
blocked site or if $\nu$ is a Bernoulli($p'$) product measure for any
$p'\neq
1$. For the model on the binary tree we prove similar results in the
regime
$p,p'<p_c$ and under the (plausible) assumption that the spectral gap is
positive for $p<p_c$. By constructing a proper test function we also
prove that
if $p'>p_c$ and $p\leq p_c$ convergence to equilibrium cannot occur
for all
local functions. Finally we present a very simple argument (different
from the
one in \cite{Aldous:2002p1074}) based on a combination of
combinatorial results
and ``energy barrier'' considerations, which yields the sharp upper
bound for
the spectral gap of East when $p\uparrow 1$.
http://arxiv.org/abs/0810.4237
---------------------------------------------------------------
7655. SYMMETRY AND TIME CHANGED BROWNIAN MOTIONS
Jos\'e Fajardo and Ernesto Mordecki
In this paper we examine which Brownian Subordination with drift
exhibits the
symmetry property introduced by Fajardo and Mordecki (2006). We obtain
that
when the subordination results in a L\'evy process, a necessary and
sufficient
condition for the symmetry to hold is that drift must be equal to -1/2.
http://arxiv.org/abs/0810.4271
---------------------------------------------------------------
7656. CONTINUITY OF THE SLE TRACE IN SIMPLY CONNECTED DOMAINS
Christophe Garban and Steffen Rohde and Oded Schramm
We prove that the $SLE_\kappa$ trace in any simply connected domain $G
$ is
continuous (except possibly near its endpoints) if $\kappa<8$. We also
prove an
SLE analog of Makarov's Theorem about the support of harmonic measure.
http://arxiv.org/abs/0810.4327
---------------------------------------------------------------
7657. SUPERDIFFUSIVITY FOR A BROWNIAN POLYMER IN A CONTINUOUS
GAUSSIAN ENVIRONMENT
S\'ergio Bezerra and Samy Tindel and Frederi Viens
This paper provides information about the asymptotic behavior of a
one-dimensional Brownian polymer in random medium represented by a
Gaussian
field $W$ on ${\mathbb{R}}_+\times{\mathbb{R}}$ which is white noise
in time
and function-valued in space. According to the behavior of the spatial
covariance of $W$, we give a lower bound on the power growth (wandering
exponent) of the polymer when the time parameter goes to infinity: the
polymer
is proved to be superdiffusive, with a wandering exponent exceeding any
$\alpha<3/5$.
http://arxiv.org/abs/0810.4378
---------------------------------------------------------------
7658. KSHIRSAGAR--TAN INDEPENDENCE PROPERTY OF BETA MATRICES AND
RELATED CHARACTERIZATIONS
Konstancja Bobecka and Jacek Weso{\l}owski
A new independence property of univariate beta distributions, related
to the
results of Kshirsagar and Tan for beta matrices, is presented.
Conversely, a
characterization of univariate beta laws through this independence
property is
proved. A related characterization of a family of $2\times2$ random
matrices
including beta matrices is also obtained. The main technical challenge
was a
problem involving the solution of a related functional equation.
http://arxiv.org/abs/0810.4427
---------------------------------------------------------------
7659. CENTRAL LIMIT THEOREMS FOR DOUBLE POISSON INTEGRALS
Giovanni Peccati and Murad S. Taqqu
Motivated by second order asymptotic results, we characterize the
convergence
in law of double integrals, with respect to Poisson random measures,
toward a
standard Gaussian distribution. Our conditions are expressed in terms of
contractions of the kernels. To prove our main results, we use the
theory of
stable convergence of generalized stochastic integrals developed by
Peccati and
Taqqu. One of the advantages of our approach is that the conditions are
expressed directly in terms of the kernel appearing in the multiple
integral
and do not make any explicit use of asymptotic dependence properties
such as
mixing. We illustrate our techniques by an application involving
linear and
quadratic functionals of generalized Ornstein--Uhlenbeck processes, as
well as
examples concerning random hazard rates.
http://arxiv.org/abs/0810.4432
---------------------------------------------------------------
7660. LOCAL TIMES OF MULTIFRACTIONAL BROWNIAN SHEETS
Mark Meerschaert and Dongsheng Wu and Yimin Xiao
Denote by $H(t)=(H_1(t),...,H_N(t))$ a function in $t\in{\mathbb{R}}_
+^N$
with values in $(0,1)^N$. Let
$\{B^{H(t)}(t)\}=\{B^{H(t)}(t),t\in{\mathbb{R}}^N_+\}$ be an
$(N,d)$-multifractional Brownian sheet (mfBs) with Hurst functional
$H(t)$.
Under some regularity conditions on the function $H(t)$, we prove the
existence, joint continuity and the H\"{o}lder regularity of the local
times of
$\{B^{H(t)}(t)\}$. We also determine the Hausdorff dimensions of the
level sets
of $\{B^{H(t)}(t)\}$. Our results extend the corresponding results for
fractional Brownian sheets and multifractional Brownian motion to
multifractional Brownian sheets.
http://arxiv.org/abs/0810.4438
---------------------------------------------------------------
7661. SKEWNESS PREMIUM WITH L\'EVY PROCESSES
Jos\'e Fajardo and Ernesto Mordecki
We study the skewness premium (SK) introduced by Bates (1991) in a
general
context using L\'evy Processes. Under a symmetry condition Fajardo and
Mordecki
(2006) obtain that SK is given by the Bate's $x%$ rule. In this paper
we study
SK under the absence of that symmetry condition. More exactly, we derive
sufficient conditions for SK to be positive, in terms of the
characteristic
triplet of the L\'evy Process under the risk neutral measure.
http://arxiv.org/abs/0810.4485
---------------------------------------------------------------
7662. SHARP LARGE DEVIATIONS FOR THE FRACTIONAL ORNSTEIN-UHLENBECK
PROCESS
Bernard Bercu; Laure Coutin; Nicolas Savy
We investigate the sharp large deviation properties of the energy and
the
maximum likelihood estimator for the Ornstein-Uhlenbeck process driven
by a
fractional Brownian motion with Hurst index greater than one half.
http://arxiv.org/abs/0810.4491
---------------------------------------------------------------
7663. SPONTANEOUS CLUSTERING IN THEORETICAL AND SOME EMPIRICAL
STATIONARY PROCESSES
Tomasz Downarowicz and Yves Lacroix and Didier L\'eandri
In a stationary ergodic process, clustering is defined as the tendency
of
events to appear in series of increased frequency separated by longer
breaks.
Such behavior, contradicting the theoretical "unbiased behavior" with
exponential distribution of the gaps between appearances, is commonly
observed
in experimental processes and often difficult to explain. In the last
section
we relate one such empirical example of clustering, in the area of
marine
technology. In the theoretical part of the paper we prove, using
ergodic theory
and the notion of category, that clustering (even very strong) is in
fact
typical for "rare events" defined as long cylinder sets in processes
generated
by a finite partition of an arbitrary (infinite aperiodic) ergodic
measure
preserving transformation.
http://arxiv.org/abs/0810.4509
---------------------------------------------------------------
7664. CENTRAL LIMIT THEOREMS FOR DOUBLE POISSON INTEGRALS
Giovanni Peccati and Murad S. Taqqu
Motivated by second order asymptotic results, we characterize the
convergence
in law of double integrals, with respect to Poisson random measures,
toward a
standard Gaussian distribution. Our conditions are expressed in terms of
contractions of the kernels. To prove our main results, we use the
theory of
stable convergence of generalized stochastic integrals developed by
Peccati and
Taqqu. One of the advantages of our approach is that the conditions are
expressed directly in terms of the kernel appearing in the multiple
integral
and do not make any explicit use of asymptotic dependence properties
such as
mixing. We illustrate our techniques by an application involving
linear and
quadratic functionals of generalized Ornstein--Uhlenbeck processes, as
well as
examples concerning random hazard rates.
http://www.arxiv.org
---------------------------------------------------------------
7665. THE LAW OF SERIES
Tomasz Downarowicz and Yves Lacroix
We consider an ergodic process on finitely many states, with positive
entropy. Our first main result asserts that the distribution function
of the
normalized waiting time for the first visit to a small (i.e., over a
long
block) cylinder set $B$ is, for majority of such cylinders and up to
epsilon,
dominated by the exponential distribution function $1-e^{-t}$. That
is, the
occurrences of so understood "rare event" $B$ along the time axis can
appear
either with gap sizes of nearly exponential distribution (like in the
independent Bernoulli process), or they "attract" each-other. Our
second main
result states that a {\it typical} ergodic process of positive entropy
has the
following property: the distribution functions of the normalized
hitting times
for the majority of cylinders $B$ of lengths $n'$ converge to zero
along a \sq\
$n'$ whose upper density is 1. The occurrences of such a cylinder $B$
"strongly
attract", i.e., they appear in "series" of many frequent repetitions
separated
by huge gaps of nearly complete absence.
These results, when properly and carefully interpreted, shed some
new light,
in purely statistical terms, independently from physics, on a century
old (and
so far rather avoided by serious science) common-sense phenomenon
known as {\it
the law of series}, asserting that rare events in reality, once
occurred, have
a mysterious tendency for untimely repetitions.
http://arxiv.org/abs/0810.4504
---------------------------------------------------------------
7666. ON ERGODICITY OF SOME MARKOV PROCESSES
T. Komorowski and S. Peszat and and T. Szarek
We formulate a criterion for the existence, uniqueness of an invariant
measure for a Markov process taking values in a Polish phase space. In
addition, the weak star ergodicity, that is, the weak convergence of the
ergodic averages of the laws of the process starting with any initial
distribution, is established. The principal assumptions are the lower
bound of
the ergodic averages of the transition probability function and the e-
property
of the semigroup. The general result is applied to solutions of some
stochastic
evolution equations in Hilbert spaces. As an example we consider an
evolution
equation whose solution describes the Lagrangian observations of the
velocity
field in the passive tracer model. We use the weak star mean
ergodicity of the
respective invariant probability measure to derive the law of large
numbers for
the tra jectory of the passive tracer.
http://arxiv.org/abs/0810.4609
---------------------------------------------------------------
7667. STOCHASTIC FLOWS WITH REFLECTION
Andrey Pilipenko
Some topological properties of stochastic flow $\varphi_t(x)$
generated by
stochastic differential equation in a ${\mathbb R}^d_+$ with normal
reflection
at the boundary are investigated. Sobolev differentiability in initial
condition is received. The absolute continuity of the measure-valued
process
$\mu\circ\varphi_t^{-1}$, where $\mu\ll\lambda^d,$ is studied.
http://arxiv.org/abs/0810.4644
---------------------------------------------------------------
7668. A STOCHASTIC REPRESENTATION FOR BACKWARD INCOMPRESSIBLE NAVIER-
STOKES EQUATIONS
Xicheng Zhang
By reversing the time variable we derive a stochastic representation for
backward incompressible Navier-Stokes equations in terms of stochastic
Lagrangian paths, which is similar to Constantin and Iyer's forward
formulations in {Co-Iy}. Using this representation, a self-contained
proof of
local existence of solutions in Sobolev spaces are provided for the
incompressible Navier-Stokes equation in the whole space. In two
dimensions, an
alternative proof to global existence is also given. Moreover, a large
deviation estimate for stochastic particle trajectories is presented
when the
viscosity tends to zero.
http://arxiv.org/abs/0810.4664
---------------------------------------------------------------
7669. IMAGE AND RECIPROCAL IMAGE OF A MEASURE. COMPATIBILITY THEOREM
Albert Tarantola
It is proposed that to the usual probability theory, three definitions
and a
new theorem are added, the resulting theory allows one to displace the
central
role usually given to the notion of conditional probability. When a
mapping
$\varphi$ is defined between two measurable spaces, to each measure $
\mu$
introduced on the first space, there corresponds an image $\varphi[\mu]
$ on the
second space, and, reciprocally, to each measure $\nu$ defined on the
second
space the corresponds a reciprocal image $\varphi^{-1}[\nu]$ on the
first
space. As the intersection $\cap$ of two measures is easy to
introduce, a
relation like $ \varphi[ \mu \cap \varphi^{-1} [\nu] ] = \varphi[\mu]
\cap \nu
$ makes sense. It is, indeed, a theorem of the theory. This theorem
gives
mathematical consistency to inferences drawn from physical measurements.
http://arxiv.org/abs/0810.4749
---------------------------------------------------------------
7670. A CHARLIER-PARSEVAL APPROACH TO POISSON APPROXIMATION AND ITS
APPLICATIONS
Vytas Zacharovas and Hsien-Kuei Hwang
A new approach to Poisson approximation is proposed. The basic idea is
very
simple and based on properties of the Charlier polynomials and the
Parseval
identity. Such an approach quickly leads to new effective bounds for
several
Poisson approximation problems. A selected survey on diverse Poisson
approximation results is also given.
http://arxiv.org/abs/0810.4756
---------------------------------------------------------------
7671. LIMITS OF BIFRACTIONAL BROWNIAN NOISES
Makoto Maejima and Ciprian Tudor (CES and SAMOS)
Let $B^{H,K}=(B^{H,K}_{t}, t\geq 0)$ be a bifractional Brownian motion
with
two parameters $H\in (0,1)$ and $K\in(0,1]$. The main result of this
paper is
that the increment process generated by the bifractional Brownian motion
$(B^{H,K}_{h+t} -B^{H,K}_{h}, t\geq 0)$ converges when $h\to \infty$ to
$(2^{(1-K)/{2}}B^{HK}_{t}, t\geq 0)$, where $(B^{HK}_{t}, t\geq 0)$ is
the
fractional Brownian motion with Hurst index $HK$. We also study the
behavior of
the noise associated to the bifractional Brownian motion and limit
theorems to
$B^{H,K}$.
http://arxiv.org/abs/0810.4764
---------------------------------------------------------------
7672. A SHARP THRESHOLD FOR MINIMUM BOUNDED-DEPTH AND BOUNDED-
DIAMETER SPANNING TREES AND STEINER TREES IN RANDOM NETWORKS
Omer Angel and Abraham D. Flaxman and and David B. Wilson
In the complete graph on n vertices, when each edge has a weight which
is an
exponential random variable, Frieze proved that the minimum spanning
tree has
weight tending to zeta(3)=1/1^3+1/2^3+1/3^3+... as n goes to infinity.
We
consider spanning trees constrained to have depth bounded by k from a
specified
root. We prove that if k > log_2 log n+omega(1), where omega(1) is any
function
going to infinity with n, then the minimum bounded-depth spanning tree
still
has weight tending to zeta(3) as n -> infinity, and that if k < log_2
log n,
then the weight is doubly-exponentially large in log_2 log n - k. It
is NP-hard
to find the minimum bounded-depth spanning tree, but when k < log_2
log n -
omega(1), a simple greedy algorithm is asymptotically optimal, and
when k >
log_2 log n+omega(1), an algorithm which makes small changes to the
minimum
(unbounded depth) spanning tree is asymptotically optimal. We prove
similar
results for minimum bounded-depth Steiner trees, where the tree must
connect a
specified set of m vertices, and may or may not include other
vertices. In
particular, when m = const * n, if k > log_2 log n+omega(1), the minimum
bounded-depth Steiner tree on the complete graph has asymptotically
the same
weight as the minimum Steiner tree, and if 1 <= k <= log_2 log n-
omega(1), the
weight tends to (1-2^{-k}) sqrt{8m/n} [sqrt{2mn}/2^k]^{1/(2^k-1)} in
both
expectation and probability. The same results hold for minimum bounded-
diameter
Steiner trees when the diameter bound is 2k; when the diameter bound is
increased from 2k to 2k+1, the minimum Steiner tree weight is reduced
by a
factor of 2^{1/(2^k-1)}.
http://arxiv.org/abs/0810.4908
---------------------------------------------------------------
7673. ROBUST ESTIMATION OF MEAN VALUES
Xinjia Chen
In this paper, we develop a computational approach for estimating the
mean
value of a quantity in the presence of uncertainty. We demonstrate
that, under
some mild assumptions, the upper and lower bounds of the mean value are
efficiently computable via a sample reuse technique, of which the
computational
complexity is shown to posses a Poisson distribution.
http://arxiv.org/abs/0810.4727
---------------------------------------------------------------
7674. ON NEARLY RADIAL MARGINALS OF HIGH-DIMENSIONAL PROBABILITY
MEASURES
Bo'az Klartag
We prove that any absolutely continuous probability measure on a
high-dimensional linear space has low-dimensional marginals that are
approximately spherically-symmetric.
http://arxiv.org/abs/0810.4700
---------------------------------------------------------------
7675. ON THE STUDY OF RICHARD TOM ROBERT IDENTITY
Yeong-Shyeong Tsai
In order to estimate the average speed of mosquitoes, a simple
experiment was
designed by Richard (Lu-Hsing Tsai), Tom (Po-Yu Tsai) and Robert (Hung-
Ming
Tsai). The result of the experiment was posted in the science
exhibitions
Taichung Taiwan 1993. The average speed of mosquitoes is inferred by
the simple
relation is obtained In this paper, we will show how to get the data
generated
by computer. Though the rigorous proof is not shown, a sketch proof
will be
shown in this paper
http://arxiv.org/abs/0810.4902
---------------------------------------------------------------
7676. A GENERAL THEORY OF FINITE STATE BACKWARD STOCHASTIC DIFFERENCE
EQUATIONS
Samuel N. Cohen and Robert J. Elliott
By analogy with the theory of Backward Stochastic Differential
Equations, we
define Backward Stochastic Difference Equations on spaces related to
discrete
time, finite state processes. This paper considers properties of these
processes as constructions in their own right, not as approximations
to the
continuous case. We establish the existence and uniqueness of
solutions under
weaker assumptions than are needed in the continuous time setting, and
also
establish a comparison theorem for these solutions. The conditions of
this
theorem are shown to approximate those required in the continuous time
setting.
We also explore the relationship between the driver and the set of
solutions;
in particular, we show under what conditions the driver can be uniquely
determined by the solution. Applications to the theory of nonlinear
expectations are explored, including a representation result in this
context.
http://arxiv.org/abs/0810.4957
---------------------------------------------------------------
7677. LARGE DEVIATIONS FOR BRANCHING PROCESSES IN RANDOM ENVIRONMENT
Vincent Bansaye (PMA) and Julien Berestycki (PMA)
A branching process in random environment $(Z_n, n \in \N)$ is a
generalization of Galton Watson processes where at each generation the
reproduction law is picked randomly. In this paper we give several
results
which belong to the class of {\it large deviations}. By contrast to the
Galton-Watson case, here random environments and the branching process
can
conspire to achieve atypical events such as $Z_n \le e^{cn}$ when $c$ is
smaller than the typical geometric growth rate $\bar L$ and $ Z_n \ge
e^{cn}$
when $c > \bar L$. One way to obtain such an atypical rate of growth
is to have
a typical realization of the branching process in an atypical sequence
of
environments. This gives us a general lower bound for the rate of
decrease of
their probability. When each individual leaves at least one offspring
in the
next generation almost surely, we compute the exact rate function of
these
events and we show that conditionally on the large deviation event, the
trajectory $t \mapsto \frac1n \log Z_{[nt]}, t\in [0,1]$ converges to a
deterministic function $f_c :[0,1] \mapsto \R_+$ in probability in the
sense of
the uniform norm. The most interesting case is when $c < \bar L$ and we
authorize individuals to have only one offspring in the next
generation. In
this situation, conditionally on $Z_n \le e^{cn}$, the population size
stays
fixed at 1 until a time $ \sim n t_c$. After time $n t_c$ an atypical
sequence
of environments let $Z_n$ grow with the appropriate rate ($\neq \bar L
$) to
reach $c.$ The corresponding map $f_c(t)$ is piecewise linear and is 0
on
$[0,t_c]$ and $f_c(t) = c(t-t_c)/(1-t_c)$ on $[t_c,1].$
http://arxiv.org/abs/0810.4991
---------------------------------------------------------------
7678. JUMP-DIFFUSIONS IN HILBERT SPACES: EXISTENCE, STABILITY AND
NUMERICS
Damir Filipovic and Stefan Tappe and Josef Teichmann
By means of an original approach, called "method of the moving frame",
we
establish existence, uniqueness and stability results for mild and weak
solutions of stochastic partial differential equations (SPDEs) with path
dependent coefficients driven by an infinite dimensional Wiener
process and a
compensated Poisson random measure. Our approach is based on a time-
dependent
coordinate transform, which reduces a wide class of SPDEs to a class
of simpler
SDE problems. We try to present the most general results, which we can
obtain
in our setting, within a self-contained framework to demonstrate our
approach
in all details. Also an outlook towards a general theory of numerical
approaches to SPDEs is provided in the spirit of our setting.
http://arxiv.org/abs/0810.5023
---------------------------------------------------------------
7679. ON NOTIONS OF HARMONICITY
Zhen-Qing Chen
In this paper, we address the equivalence of the analytic and
probabilistic
notions of harmonicity in the context of general symmetric Hunt
processes on
locally compact separable metric spaces. Extensions to general
symmetric right
processes on Lusin spaces including infinite dimensional spaces are
mentioned
at the end of this paper.
http://arxiv.org/abs/0810.5089
---------------------------------------------------------------
7680. ON UNIQUE EXTENSION OF TIME CHANGED REFLECTING BROWNIAN MOTIONS
Zhen-Qing Chen and Masatoshi Fukushima
Let $D$ be an unbounded domain in $\RR^d$ with $d\geq 3$. We show that
if $D$
contains an unbounded uniform domain, then the symmetric reflecting
Brownian
motion (RBM) on $\overline D$ is transient. Next assume that RBM $X$ on
$\overline D$ is transient and let $Y$ be its time change by Revuz
measure
${\bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable
function
$m$ on $\overline D$. We further show that if there is some $r>0$ so
that
$D\setminus \overline {B(0, r)}$ is an unbounded uniform domain, then
$Y$
admits one and only one symmetric diffusion that genuinely extends it
and
admits no killings. In other words, in this case $X$ (or equivalently,
$Y$) has
a unique Martin boundary point at infinity.
http://arxiv.org/abs/0810.5096
---------------------------------------------------------------
7681. L2-HOMOGENIZATION OF HEAT EQUATIONS ON TUBULAR NEIGHBORHOODS
O. Wittich
We consider the heat equation with Dirichlet boundary conditions on the
tubular neighborhood of a closed Riemannian submanifold. We show that,
as the
tube radius decreases, the semigroup of a suitably rescaled and
renormalized
generator can be effectively described by a Hamiltonian on the
submanifold with
a potential that depends on the geometry of the submanifold and of the
embedding.
http://arxiv.org/abs/0810.5047
---------------------------------------------------------------
7682. SMOOTH HOMOGENIZATION OF HEAT EQUATIONS ON TUBULAR NEIGHBORHOODS
O. Wittich
We consider the heat equation with Dirichlet boundary conditions on the
tubular neighborhood of a closed Riemannian submanifold. We show that,
as the
tube diameter tends to zero, a suitably rescaled and renormalized
semigroup
converges to a limit semigroup in Sobolev spaces of arbitrarily large
Sobolev
index.
http://arxiv.org/abs/0810.5052
---------------------------------------------------------------
7683. STRUCTURAL PROPERTIES OF SEMILINEAR SPDES DRIVEN BY CYLINDRICAL
STABLE PROCESSES
Enrico Priola and Jerzy Zabczyk
We consider a class of semilinear stochastic evolution equations
driven by an
additive cylindrical stable noise.We investigate structural properties
of the
solutions like Markov, irreducibility, stochastic continuity, Feller
and strong
Feller properties, and study integrability of trajectories. The obtained
results can be applied to semilinear stochastic heat equations with
Dirichlet
boundary conditions and bounded and Lipschitz nonlinearities.
http://arxiv.org/abs/0810.5063
---------------------------------------------------------------
7684. SEMI-STATIC HEDGING FOR CERTAIN MARGRABE TYPE OPTIONS WITH
BARRIERS
Michael Schmutz
It turns out that a slightly generalised Margrabe formula exhibits
symmetry
properties leading to semi-static hedges of rather general options in
the
bivariate Black-Scholes economy. In order to increase the liquidity of
the used
hedging instruments for currency options, the duality principle can be
used to
set up the hedges in a foreign market by using only European vanilla
options
sometimes along with a risk-less bond. Since the semi-static hedges in
the
Black-Scholes economy are exact, a closed form valuation formula of a
certain
weighted barrier swap-option can be easily derived.
http://arxiv.org/abs/0810.5146
---------------------------------------------------------------
7685. ON WEAK GENERALIZED STABILITY AND (C,D)-PSEUDOSTABLE RANDOM
VARIABLES VIA FUNCTIONAL EQUATIONS
W. Jarczyk and J. Misiewicz
In this paper we give a first attempt to define and study stable
distributions with respect to the weak generalized convolution,
focusing our
attention on the symmetric weakly stable distribution. As in the case
of the
classical convolution, characterization of distributions stable in the
sense of
the weak generalized convolution depends on solving some functional
equations
in the class of characteristic functions.
http://arxiv.org/abs/0810.5285
---------------------------------------------------------------
7686. HOW HOT CAN A HEAT BATH GET?
Martin Hairer
We study a model of two interacting Hamiltonian particles subject to a
common
potential in contact with two Langevin heat reservoirs: one at finite
and one
at infinite temperature. This is a toy model for 'extreme' non-
equilibrium
statistical mechanics. We provide a full picture of the long-time
behaviour of
such a system, including the existence / non-existence of a non-
equilibrium
steady state, the precise tail behaviour of the energy in such a
state, as well
as the speed of convergence toward the steady state.
Despite its apparent simplicity, this model exhibits a surprisingly
rich
variety of long time behaviours, depending on the parameter regime: if
the
surrounding potential is 'too stiff', then no stationary state can
exist. In
the softer regimes, the tails of the energy in the stationary state
can be
either algebraic, fractional exponential, or exponential.
Correspondingly, the
speed of convergence to the stationary state can be either algebraic,
stretched
exponential, or exponential. Regarding both types of claims, we obtain
matching
upper and lower bounds.
http://arxiv.org/abs/0810.5431
---------------------------------------------------------------
7687. A NOTE ON TALAGRAND'S TRANSPORTATION INEQUALITY AND LOGARITHMIC
SOBOLEV INEQUALITY
Patrick Cattiaux (LSProba) and Arnaud Guillin and Liming Wu
We give by simple arguments sufficient conditions, so called Lyapunov
conditions, for Talagrand's transportation information inequality and
for the
logarithmic Sobolev inequality. Those sufficient conditions work even
in the
case where the Bakry-Emery curvature is not lower bounded. Several new
examples
are provided.
http://arxiv.org/abs/0810.5435
---------------------------------------------------------------
7688. A THEORY OF TRUNCATED INVERSE SAMPLING
Xinjia Chen
In this paper, we have established a new framework of truncated inverse
sampling for estimating mean values of non-negative random variables
such as
binomial, Poisson, hyper-geometrical, and bounded variables. We have
derived
explicit formulas and computational methods for designing sampling
schemes to
ensure prescribed levels of precision and confidence for point
estimators.
Moreover, we have developed interval estimation methods.
http://arxiv.org/abs/0810.5551
---------------------------------------------------------------
7689. DENSITY OF EIGENVALUES AND ITS PERTURBATION INVARIANCE IN
UNITARY ENSEMBLES OF RANDOM MATRICES
Dang-Zheng Liu and Zheng-Dong Wang and Kui-Hua Yan
We generally study the density of eigenvalues in unitary ensembles of
random
matrices from the recurrence coefficients with regularly varying
conditions for
the orthogonal polynomials. First we calculate directly the moments of
the
density. Then, by studying some deformation of the moments, we get a
family of
differential equations of first order which the densities satisfy (see
Theorem
1.2), and give the densities by solving them. Further, we prove that the
density is invariant after the polynomial perturbation of the weight
function
(see Theorem 1.5).
http://arxiv.org/abs/0810.5425
---------------------------------------------------------------
7690. LIMIT BEHAVIOUR OF SEQUENTIAL EMPIRICAL MEASURE PROCESSES
Omar El-Dakkak (LSTA)
In this paper, we obtain some uniform laws of large numbers and
functional
central limit theorems for sequential empirical measure processes
indexed by
classes of product functions satisfying appropriate Vapnik-Chervonenkis
properties.
http://arxiv.org/abs/0810.5565
---------------------------------------------------------------
7691. AN EXAMPLE OF BRUNET-DERRIDA BEHAVIOR FOR A BRANCHING-SELECTION
PARTICLE SYSTEM ON $\Z$
Jean B\'erard (ICJ)
We consider a branching-selection particle system on $\Z$ with $N \geq
1$
particles. During a branching step, each particle is replaced by two new
particles, whose positions are shifted from that of the original
particle by
independently performing two random walk steps according to the
distribution $p
\delta_{1} + (1-p) \delta_{0}$, from the location of the original
particle.
During the selection step that follows, only the N rightmost particles
are kept
among the $2N$ particles obtained at the branching step, to form a new
population of $N$ particles. After a large number of iterated
branching-selection steps, the displacement of the whole population of
$N$
particles is ballistic, with deterministic asymptotic speed $v_{N}(p)
$. As $N$
goes to infinity, $v_{N}(p)$ converges to a finite limit $v_{\infty}(p)
$. Our
main result is that, for every $0 < p < 1/2$, as $N$ goes to infinity,
the
order of magnitude of the difference $v_{\infty}(p)- v_{N}(p)$ is
$\log(N)^{-2}$. This is called Brunet-Derrida behavior in reference to
the
paper \cite{BruDer1} by E. Brunet and B. Derrida, where such a
behavior is
established for a similar branching-selection particle system, using
both
numerical simulations and heuristic arguments.
http://arxiv.org/abs/0810.5567
---------------------------------------------------------------
7692. A PATHWISE APPROACH TO RELATIVISTIC DIFFUSIONS
Ismael Bailleul
A new class of relativistic diffusions encompassing all the previously
studied examples has recently been introduced by C. Chevalier and F.
Debbasch,
both in a heuristic and analytic way. A pathwise approach of these
processes is
proposed here, in the general framework of Lorentzian geometry. In
considering
the dynamics of the random motion in strongly causal spacetimes, we
are able to
give a simple definition of the one-particle distribution function
associated
with each process of the class and prove its fundamental property.
This result
not only provides a dynamical justification of the analytical approach
developped up to now (enabling us to recover many of the results
obtained so
far), but it provides a new general H-theorem. It also sheds some
light on the
importance of the large scale structure of the manifold in the
asymptotic
behaviour of the Franchi-Le Jan process. This pathwise approach is
also the
source of many interesting questions that have no analytical
counterparts.
http://arxiv.org/abs/0810.5662
---------------------------------------------------------------
7693. THE PACKING MEASURE OF THE RANGE OF SUPER-BROWNIAN MOTION
Thomas Duquesne
We prove that the total range of Super-Brownian motion with quadratic
branching mechanism has an exact packing measure with respect to the
gauge
function $g(r)=r^4 (\log \log1/r)^{-3}$ in super-critical dimensions $d
\geq 5$.
More precisely, we prove that the total occupation measure of Super-
Brownian
motion is equal to the $g$-packing measure restricted to its range, up
to a
deterministic multiplicative constant that only depends on space
dimension $d$.
http://arxiv.org/abs/0810.5673
---------------------------------------------------------------
7694. THE CONTINUOUS TIME NONZERO-SUM DYNKIN GAME PROBLEM AND
APPLICATION IN GAME OPTIONS
Said Hamadene and Jianfeng Zhang
In this paper we study the nonzero-sum Dynkin game in continuous time
which
is a two player non-cooperative game on stopping times. We show that
it has a
Nash equilibrium point for general stochastic processes. As an
application, we
consider the problem of pricing American game contingent claims by the
utility
maximization approach.
http://arxiv.org/abs/0810.5698
---------------------------------------------------------------
7695. CONDITIONAL LIMITS OF L_P SCALE MIXTURE DISTRIBUTIONS
Enkelejd Hashorva
In this paper we introduce the class of L_p random vectors which
includes
beta-independent random vectors and L_p Dirichlet ones. We derive
several
conditional limit results assuming that the associated random radius
of the L_p
random vectors have distribution function in the max-domain of
attraction of a
univariate extreme value distribution function. As an application we
obtain the
joint asymptotic distribution of concomitants of order statics
considering L_p
random samples.
http://arxiv.org/abs/0810.5706
-----------------------------
Stefano Iacus
IMS Groups Editor
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