[PAS] Probability Abstracts 100
Probability Abstract Service
pas at lists.imstat.org
Wed Nov 7 18:13:46 CST 2007
Probability Abstracts 100
This document contains abstracts 5997-6227 from
September-1-2007 to October-31-2007.
They have been mailed on November 8th, 2007.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_100.shtml
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5997. A NOTE ON THE CIRCULAR LAW FOR NON-CENTRAL RANDOM MATRICES
Djalil Chafai (IMT and Upte)
Let $(X_{i,j})_{1\leq i,j<\infty}$ be an infinite array of i.i.d.
complex
random variables, with mean $m=0$, variance $\si^2=1$, and say with
finite
fourth moment. The famous circular law theorem states that the empirical
spectral distribution $\frac{1}{n}(\de_{\la_1(\bX)}+...+\de_{\la_n
(\bX)})$ of
$\bX=(n^{-1/2}X_{i,j})_{1\leq i,j\leq n}$ converges almost surely, as
$n\to\infty$, to the uniform law over the unit disc $\{z\in\dC;\ABS{z}
\leq
1\}$. For now, most efforts where focused on the improvement of moments
hypotheses for the centered case $m=0$. Regarding the non-central case
$m\neq0$, Silverstein has already observed that almost surely, the
eigenvalue
of $\bX$ of largest module goes to $+\infty$ as $n\to\infty$, while
the rest of
the spectrum remains bounded. We show in this note that the circular law
theorem remains valid when $m\neq0$, by using logarithmic potentials
and bounds
on extremal singular values.
http://arxiv.org/abs/0709.0036
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5998. ON THE PRECISION OF THE SPECTRAL PROFILE
Gady Kozma
We examine the spectral profile bound of Goel, Montenegro and Tetali
for the
uniform mixing time of continuous-time random walk in reversible
settings. We
find that it is precise up to a log log factor, and that this log log
factor
cannot be improved.
http://arxiv.org/abs/0709.0112
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5999. ON THE SHAPE STABILITY FOR A GROWTH MODEL
M.V. Menshikov and V. Sisko and M. Vachkovskaia
We consider a growth model with $n$ nodes. Let us fix ${\cal K}$ subsets
$S_i\subset \{1, ..., n\}$, the customers arrive to set $S_i$ at rate
$\lambda_i$. A customer arriving to $S_i$ is directed to some node in
$S_i$
accordingly to previously chosen rule (policy). Under some natural
conditions
on $\lambda_i$'s we have found such rules (one of them is Join the
Shortest
Queue routing policy) for which our model is positive recurrent in
shape.
http://arxiv.org/abs/0709.0121
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6000. VALUATIONS AND DYNAMIC CONVEX RISK MEASURES
A. Jobert and L. C. G. Rogers
This paper approaches the definition and properties of dynamic convex
risk
measures through the notion of a family of concave valuation operators
satisfying certain simple and credible axioms. Exploring these in the
simplest
context of a finite time set and finite sample space, we find natural
risk-transfer and time-consistency properties for a firm seeking to
spread its
risk across a group of subsidiaries.
http://arxiv.org/abs/0709.0232
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6001. STRONG LAW OF LARGE NUMBERS FOR BRANCHING DIFFUSIONS
Janos Englander and Simon C. Harris and Andreas E. Kyprianou
Let $X$ be the branching particle diffusion corresponding to the
operator
$Lu+\beta (u^{2}-u)$ on $D\subseteq \mathbb{R}^{d}$
(where $\beta \geq 0$ and $\beta\not\equiv 0$). Let $\lambda_{c}$
denote the
generalized principal eigenvalue for the operator $L+\beta $ on $D$
and assume
that it is finite. When $\lambda_{c}>0$ and $L+\beta-\lambda_{c}$
satisfies
certain spectral theoretical conditions, we prove that the random
measure $\exp
\{-\lambda_{c}t\}X_{t}$ converges almost surely in the vague topology
as $t$
tends to infinity. This result is motivated by a cluster of articles
due to
Asmussen and Hering dating from the mid-seventies as well as the more
recent
work concerning analogous results for superdiffusions of \cite
{ET,EW}. We
extend significantly the results in \cite{AH76,AH77} and include some
key
examples of the branching process literature. As far as the proofs are
concerned, we appeal to modern techniques concerning martingales and
`spine'
decompositions or `immortal particle pictures'.
http://arxiv.org/abs/0709.0272
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6002. ESTIMATING RANDOM VARIABLES FROM RANDOM SPARSE OBSERVATIONS
Andrea Montanari
Let X_1,...., X_n be a collection of iid discrete random variables, and
Y_1,..., Y_m a set of noisy observations of such variables. Assume each
observation Y_a to be a random function of some a random subset of
the X_i's,
and consider the conditional distribution of X_i given the
observations, namely
\mu_i(x_i)\equiv\prob\{X_i=x_i|Y\} (a posteriori probability).
We establish a general relation between the distribution of \mu_i,
and the
fixed points of the associated density evolution operator. Such
relation holds
asymptotically in the large system limit, provided the average number of
variables an observation depends on is bounded. We discuss the
relevance of our
result to a number of applications, ranging from sparse graph codes, to
multi-user detection, to group testing.
http://arxiv.org/abs/0709.0145
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6003. HYDRODYNAMIC BEHAVIOR OF ONE DIMENSIONAL SUBDIFFUSIVE
EXCLUSION PROCESSES WITH RANDOM CONDUCTANCES
A. Faggionato and M. Jara and C. Landim
Consider a system of particles performing nearest neighbor random
walks on
the lattice $\ZZ$ under hard--core interaction. The rate for a jump
over a
given bond is direction--independent and the inverse of the jump
rates are
i.i.d. random variables belonging to the domain of attraction of an
$\a$--stable law, $0<\a<1$. This exclusion process models conduction in
strongly disordered one-dimensional media. We prove that, when
varying over the
disorder and for a suitable slowly varying function $L$, under the
super-diffusive time scaling $N^{1 + 1/\alpha}L(N)$, the density profile
evolves as the solution of the random equation $\partial_t \rho = \mf
L_W
\rho$, where $\mf L_W$ is the generalized second-order differential
operator
$\frac d{du} \frac d{dW}$ in which $W$ is a double sided $\a$--stable
subordinator. This result follows from a quenched hydrodynamic limit
in the
case that the i.i.d. jump rates are replaced by a suitable array $\
{\xi_{N,x} :
x\in\bb Z\}$ having same distribution and fulfilling an a.s. invariance
principle. We also prove a law of large numbers for a tagged particle.
http://arxiv.org/abs/0709.0306
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6004. RANDOM WALK LOCAL TIME APPROXIMATED BY A WIENER SHEET COMBINED
WITH AN INDEPENDENT BROWNIAN MOTION
Endre Cs\'aki and Mikl\'os Cs\"org\H{o} and Ant\'onia F\"oldes and
P\'al R\'ev\'esz
Let $\xi(k,n)$ be the local time of a simple symmetric random walk on
the
line. We give a strong approximation of the centered local time process
$\xi(k,n)-\xi(0,n)$ in terms of a Wiener sheet and an independent Wiener
process, time changed by an independent Brownian local time. Some
related
results and consequences are also established.
http://arxiv.org/abs/0709.0389
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6005. APPROXIMATION VIA REGULARIZATION OF THE LOCAL TIME OF
SEMIMARTINGALES AND BROWNIAN MOTION
Blandine Berard Bergery (IECN) and Pierre Vallois (IECN)
Through a regularization procedure, few approximation schemes of the
local
time of a large class of one dimensional processes are given. We mainly
consider the local time of continuous semimartingales and reversible
diffusions, and the convergence holds in ucp sense. In the case of
standard
Brownian motion, we have been able to determine a rate of convergence
in $L^2$,
and a.s. convergence of some of our schemes.
http://arxiv.org/abs/0709.0402
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6006. DOUBLE CLUSTERING AND GRAPH NAVIGABILITY
Oskar Sandberg
Graphs are called navigable if one can find short paths through them
using
only local knowledge. It has been shown that for a graph to be
navigable, its
construction needs to meet strict criteria. Since such graphs
nevertheless seem
to appear in nature, it is of interest to understand why these
criteria should
be fulfilled.
In this paper we present a simple method for constructing graphs
based on a
model where nodes vertices are ``similar'' in two different ways, and
tend to
connect to those most similar to them - or cluster - with respect to
both. We
prove that this leads to navigable networks for several cases, and
hypothesize
that it also holds in great generality. Enough generality, perhaps,
to explain
the occurrence of navigable networks in nature.
http://arxiv.org/abs/0709.0511
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6007. THE LAGUERRE PROCESS AND GENERALIZED HARTMAN--WATSON LAW
Nizar Demni
In this paper, we study complex Wishart processes or the so-called
Laguerre
processes $(X_t)_{t\geq0}$. We are interested in the behaviour of the
eigenvalue process; we derive some useful stochastic differential
equations and
compute both the infinitesimal generator and the semi-group. We also
give
absolute-continuity relations between different indices. Finally, we
compute
the density function of the so-called generalized Hartman--Watson law
as well
as the law of $T_0:=\inf\{t,\det(X_t)=0\}$ when the size of the
matrix is 2.
demni at ccr.jussieu.fr
http://www.arxiv.org
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6008. ASYMPTOTIC EXPANSION AND CENTRAL LIMIT THEOREM FOR QUADRATIC
VARIATIONS OF GAUSSIAN PROCESSES
Arnaud Begyn
Cohen, Guyon, Perrin and Pontier have given assumptions under which the
second-order quadratic variations of a Gaussian process converge
almost surely
to a deterministic limit. In this paper we present two new
convergence results
about these variations: the first is a deterministic asymptotic
expansion; the
second is a central limit theorem. Next we apply these results to
identify
two-parameter fractional Brownian motion and anisotropic fractional
Brownian
motion.
http://arxiv.org/abs/0709.0598
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6009. ASYMPTOTIC NORMALITY FOR THE COUNTING PROCESS OF WEAK RECORDS
AND \DELTA-RECORDS IN DISCRETE MODELS
Ra\'ul Gouet and F. Javier L\'opez and Gerardo Sanz
Let $\{X_n,n\ge1\}$ be a sequence of independent and identically
distributed
random variables, taking non-negative integer values, and call $X_n$ a
$\delta$-record if $X_n>\max\{X_1,...,X_{n-1}\}+\delta$, where $\delta
$ is an
integer constant. We use martingale arguments to show that the
counting process
of $\delta$-records among the first $n$ observations, suitably
centered and
scaled, is asymptotically normally distributed for $\delta\ne0$. In
particular,
taking $\delta=-1$ we obtain a central limit theorem for the number
of weak
records.
http://arxiv.org/abs/0709.0620
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6010. POISSON-TYPE DEVIATION INEQUALITIES FOR CURVED CONTINUOUS-TIME
MARKOV CHAINS
Ald\'eric Joulin
In this paper, we present new Poisson-type deviation inequalities for
continuous-time Markov chains whose Wasserstein curvature or $\Gamma$-
curvature
is bounded below. Although these two curvatures are equivalent for
Brownian
motion on Riemannian manifolds, they are not comparable in discrete
settings
and yield different deviation bounds. In the case of birth--death
processes, we
provide some conditions on the transition rates of the associated
generator for
such curvatures to be bounded below and we extend the deviation
inequalities
established [An\'{e}, C. and Ledoux, M. On logarithmic Sobolev
inequalities for
continuous time random walks on graphs. Probab. Theory Related Fields
116
(2000) 573--602] for continuous-time random walks, seen as models in
null
curvature. Some applications of these tail estimates are given for
Brownian-driven Ornstein--Uhlenbeck processes and $M/M/1$ queues.
http://arxiv.org/abs/0709.0622
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6011. ON IT\^{O}'S FORMULA FOR ELLIPTIC DIFFUSION PROCESSES
Xavier Bardina and Carles Rovira
Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83--109] prove an
extension of It\^{o}'s formula for $F(X_t,t)$, where $F(x,t)$ has a
locally
square-integrable derivative in $x$ that satisfies a mild continuity
condition
in $t$ and $X$ is a one-dimensional diffusion process such that the
law of
$X_t$ has a density satisfying certain properties. This formula was
expressed
using quadratic covariation. Following the ideas of Eisenbaum
[Potential Anal.
13 (2000) 303--328] concerning Brownian motion, we show that one can
re-express
this formula using integration over space and time with respect to
local times
in place of quadratic covariation. We also show that when the
function $F$ has
a locally integrable derivative in $t$, we can avoid the mild continuity
condition in $t$ for the derivative of $F$ in $x$.
http://arxiv.org/abs/0709.0627
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6012. SAMPLE PATH PROPERTIES OF THE LOCAL TIME OF MULTIFRACTIONAL
BROWNIAN MOTION
Brahim Boufoussi and Marco Dozzi and Raby Guerbaz
We establish estimates for the local and uniform moduli of continuity
of the
local time of multifractional Brownian motion,
$B^H=(B^{H(t)}(t),t\in\mathbb{R}^+)$. An analogue of Chung's law of the
iterated logarithm is studied for $B^H$ and used to obtain the pointwise
H\"{o}lder exponent of the local time. A kind of local asymptotic
self-similarity is proved to be satisfied by the local time of $B^H$.
http://arxiv.org/abs/0709.0637
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6013. LIMIT THEOREMS FOR FUNCTIONALS ON THE FACETS OF STATIONARY
RANDOM TESSELLATIONS
Lothar Heinrich and Hendrik Schmidt and Volker Schmidt
We observe stationary random tessellations $X=\{\Xi_n\}_{n\ge1}$ in
$\mathbb{R}^d$ through a convex sampling window $W$ that expands
unboundedly
and we determine the total $(k-1)$-volume of those $(k-1)$-
dimensional manifold
processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1
$) by their
intersections with the $(d-1)$-facets of independent and identically
distributed motion-invariant tessellations $X_n$ generated within
each cell
$\Xi_n$ of $X$. The cases of $X$ being either a Poisson hyperplane
tessellation
or a random tessellation with weak dependences are treated
separately. In both
cases, however, we obtain that all of the total volumes measured in $W
$ are
approximately normally distributed when $W$ is sufficiently large.
Structural
formulae for mean values and asymptotic variances are derived and
explicit
numerical values are given for planar Poisson--Voronoi tessellations
(PVTs) and
Poisson line tessellations (PLTs).
http://arxiv.org/abs/0709.0650
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6014. ON THE RUIN TIME DISTRIBUTION FOR A SPARRE ANDERSEN PROCESS
WITH EXPONENTIAL CLAIM SIZES
K. A. Borovkov and D. C. M. Dickson
We derive a closed-form (infinite series) representation for the
distribution
of the ruin time for the Sparre Andersen model with exponentially
distributed
claims. This extends a recent result of Dickson et al. (2005) for such
processes with Erlang inter-claim times. We illustrate our result in
the cases
of gamma and mixed exponential inter-claim time distributions.
http://arxiv.org/abs/0709.0764
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6015. SELF-SIMILAR STABLE PROCESSES ARISING FROM HIGH-DENSITY LIMITS
OF OCCUPATION TIMES OF PARTICLE SYSTEMS
Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
We extend results on time-rescaled occupation time fluctuation limits
of the
$(d,\alpha, \beta)$-branching particle system $(0<\alpha \leq 2, 0<
\beta \leq
1)$ with Poisson initial condition. The earlier results in the
homogeneous case
(i.e., with Lebesgue initial intensity measure) were obtained for
dimensions
$d>\alpha / \beta$ only, since the particle system becomes locally
extinct if
$d\le \alpha / \beta$. In this paper we show that by introducing high
density
of the initial Poisson configuration, limits are obtained for all
dimensions,
and they coincide with the previous ones if $d>\alpha/\beta$. We also
give
high-density limits for the systems with finite intensity measures
(without
high density no limits exist in this case due to extinction); the
results are
different and harder to obtain due to the non-invariance of the
measure for the
particle motion. In both cases, i.e., Lebesgue and finite intensity
measures,
for low dimensions ($d<\alpha(1+\beta)/\beta$ and
$d<\alpha(2+\beta)/(1+\beta)$, respectively) the limits are
determined by
non-L\'evy self-similar stable processes. For the corresponding high
dimensions
the limits are qualitatively different: ${\cal S}'(R^d)$-valued L\'evy
processes in the Lebesgue case, stable processes constant in time on
$(0,\infty)$ in the finite measure case. For high dimensions, the
laws of all
limit processes are expressed in terms of Riesz potentials. If $
\beta=1$, the
limits are Gaussian. Limits are also given for particle systems without
branching, which yields in particular weighted fractional Brownian
motions in
low dimensions. The results are obtained in the setup of weak
convergence of
S'(R^d)$-valued processes.
http://arxiv.org/abs/0709.0773
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6016. WEAK APPROXIMATION OF A FRACTIONAL SDE
Ivan Nourdin (PMA) and Samy Tindel (IECN)
In this note, a diffusion approximation result is shown for stochastic
differential equations driven by a fractional Brownian motion B with
Hurst
parameter H>1/3. We shall use a Gaussian regular approximation of B
for sake of
clarity, and our method of proof will rely on the algebraic
integration theory
introduced by Gubinelli.
http://arxiv.org/abs/0709.0805
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6017. RANDOM COLOURINGS OF APERIODIC GRAPHS: ERGODIC AND SPECTRAL
PROPERTIES
Peter M\"uller and Christoph Richard
We study randomly coloured graphs embedded into Euclidean space,
whose vertex
sets are infinite, uniformly discrete subsets of finite local
complexity. We
construct the appropriate ergodic dynamical systems, explicitly
characterise
ergodic measures, and prove an ergodic theorem. For covariant
operators of
finite range defined on those graphs, we show the existence and self-
averaging
of the integrated density of states, as well as the non-randomness of
the
spectrum. Our main result establishes Lifshits tails at the lower
spectral edge
of the graph Laplacian on bond percolation subgraphs, for
sufficiently small
probabilities. Among other assumptions, its proof requires
exponential decay of
the cluster-size distribution for percolation on rather general graphs.
http://arxiv.org/abs/0709.0821
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6018. FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES
Stephen Boyd and Persi Diaconis and Pablo A. Parrilo and Lin Xiao
We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue
modulus of
the transition probability matrix). Exploiting symmetry can lead to
significant
reduction in both the number of variables and the size of matrices in
the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible.
We obtain
analytic or semi-analytic results for particular classes of graphs,
such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection
between
these two approaches.
http://arxiv.org/abs/0709.0955
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6019. FAULT TOLERANCE IN CELLULAR AUTOMATA AT HIGH FAULT RATES
Mark McCann and Nicholas Pippenger
A commonly used model for fault-tolerant computation is that of cellular
automata. The essential difficulty of fault-tolerant computation is
present in
the special case of simply remembering a bit in the presence of
faults, and
that is the case we treat in this paper. We are concerned with the
degree (the
number of neighboring cells on which the state transition function
depends)
needed to achieve fault tolerance when the fault rate is high (nearly
1/2). We
consider both the traditional transient fault model (where faults occur
independently in time and space) and a recently introduced combined
fault model
which also includes manufacturing faults (which occur independently
in space,
but which affect cells for all time). We also consider both a purely
probabilistic fault model (in which the states of cells are perturbed at
exactly the fault rate) and an adversarial model (in which the
occurrence of a
fault gives control of the state to an omniscient adversary). We show
that
there are cellular automata that can tolerate a fault rate $1/2 - \xi
$ (with
$\xi>0$) with degree $O((1/\xi^2)\log(1/\xi))$, even with adversarial
combined
faults. The simplest such automata are based on infinite regular
trees, but our
results also apply to other structures (such as hyperbolic
tessellations) that
contain infinite regular trees. We also obtain a lower bound of
$\Omega(1/\xi^2)$, even with purely probabilistic transient faults only.
http://arxiv.org/abs/0709.0967
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6020. GENERALIZED SOLUTIONS OF THE CAUCHY PROBLEM FOR THE NAVIER-
STOKES SYSTEM AND DIFFUSION PROCESSES
S. Albeverio and Ya. Belopolskaya
We reduce the construction of a weak solution of the Cauchy problem
for the
Navier-Stokes system to the construction of a solution to a
stochastic problem.
Namely, we construct diffusion processes which allow us to obtain a
probabilistic representation of a weak (in distributional sense)
solution to
the Cauchy problem for the Navier- Stokes system.
http://arxiv.org/abs/0709.1008
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6021. CRITICAL SCALING OF STOCHASTIC EPIDEMIC MODELS
Steven P. Lalley
In the simple mean-field SIS and SIR epidemic models, infection is
transmitted from infectious to susceptible members of a finite
population by
independent $p-$coin tosses. Spatial variants of these models are
proposed, in
which finite populations of size $N$ are situated at the sites of a
lattice and
infectious contacts are limited to individuals at neighboring sites.
Scaling
laws for both the mean-field and spatial models are given when the
infection
parameter $p$ is such that the epidemics are critical. It is shown
that in all
cases there is a critical threshold for the numbers initially
infected: below
the threshold, the epidemic evolves in essentially the same manner as
its
branching envelope, but at the threshold evolves like a branching
process with
a size-dependent drift.
http://arxiv.org/abs/0709.1039
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6022. PARAMETER ESTIMATION IN DIAGONALIZABLE BILINEAR STOCHASTIC
PARABOLIC EQUATIONS
Igor Cialenco and Sergey V. Lototsky
A parameter estimation problem is considered for a stochastic parabolic
equation with multiplicative noise under the assumption that the
equation can
be reduced to an infinite system of uncoupled diffusion processes.
From the
point of view of classical statistics, this problem turns out to be
singular
not only for the original infinite-dimensional system but also for most
finite-dimensional projections. This singularity can be exploited to
improve
the rate of convergence of traditional estimators as well as to
construct
completely new closed-form exact estimator.
http://arxiv.org/abs/0709.1135
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6023. CHARACTERIZING HEAVY-TAILED DISTRIBUTIONS INDUCED BY
RETRANSMISSIONS
Predrag R. Jelenkovic and Jian Tan
Consider a generic data unit of random size L that needs to be
transmitted
over a channel of unit capacity. The channel availability dynamics is
modeled
as an i.i.d. sequence {A, A_i},i>0 that is independent of L. During
each period
of time that the channel becomes available, say A_i, we attempt to
transmit the
data unit. If L< A_i, the transmission was considered successful;
otherwise, we
wait for the next available period and attempt to retransmit the data
from the
beginning. We investigate the asymptotic properties of the number of
retransmissions N and the total transmission time T until the data is
successfully transmitted. In the context of studying the completion
times in
systems with failures where jobs restart from the beginning, it was
shown that
this model results in power law and, in general, heavy-tailed delays.
The main
objective of this paper is to uncover the detailed structure of this
class of
heavy-tailed distributions induced by retransmissions. More
precisely, we study
how the functional dependence between P[L>x] and P[A>x] impacts the
distributions of N and T. In the space of this functional dependence, we
discover several functional criticality points that separate classes of
different functional behavior of the distribution of N. We also
discuss the
engineering implications of our results on communication networks since
retransmission strategy is a fundamental component of the existing
network
protocols on all communication layers, from the physical to the
application
one.
http://arxiv.org/abs/0709.1138
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6024. DIVERGENCE OF A STATIONARY RANDOM VECTOR FIELD CAN BE ALWAYS
POSITIVE
Boris Tsirelson
The divergence of a stationary random vector field at a given point is
usually a centered (that is, zero mean) random variable. Strangely
enough, it
can be equal to 1 almost surely.
http://arxiv.org/abs/0709.1270
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6025. EXPONENTIAL CLOGGING TIME FOR A ONE DIMENSIONAL DLA
Itai Benjamini and Christopher Hoffman
When considering DLA on a cylinder it is natural to ask how many
particles it
takes to clog the cylinder, e.g. modeling clogging of arteries. In
this note we
formulate a very simple DLA clogging model and establish an
exponential lower
bound on the number of particles arriving before clogging appears.
http://arxiv.org/abs/0709.1276
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6026. RELATIVE AND DISCRETE UTILITY MAXIMISING ENTROPY
Grzegorz Hara\'nczyk and Wojciech S{\l}omczy\'nski and Tomasz
Zastawniak
The notion of utility maximising entropy (u-entropy) of a probability
density, which was introduced and studied by Slomczynski and
Zastawniak (Ann.
Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), is extended in two
directions. First, the relative u-entropy of two probability measures in
arbitrary probability spaces is defined. Then, specialising to discrete
probability spaces, we also introduce the absolute u-entropy of a
probability
measure. Both notions are based on the idea, borrowed from mathematical
finance, of maximising the expected utility of the terminal wealth of an
investor. Moreover, u-entropy is also relevant in thermodynamics, as
it can
replace the standard Boltzmann-Shannon entropy in the Second Law. If the
utility function is logarithmic or isoelastic (a power function),
then the
well-known notions of the Boltzmann-Shannon and Renyi relative
entropy are
recovered. We establish the principal properties of relative and
discrete
u-entropy and discuss the links with several related approaches in the
literature.
http://arxiv.org/abs/0709.1281
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6027. CORNERS AND RECORDS OF THE POISSON PROCESS IN QUADRANT
Alexander Gnedin
The scale-invariant spacings lemma due to Arratia, Barbour and Tavar
{\'e}
establishes the distributional identity of a self-similar Poisson
process and
the set of spacings between the points of this process. In this note
we connect
this result with properties of a certain set of extreme points of the
unit
Poisson process in the positive quadrant.
http://arxiv.org/abs/0709.1285
---------------------------------------------------------------
6028. A NOTE ON THE COMPONENT STRUCTURE IN RANDOM GRAPHS WITH
TUNABLE CLUSTERING
Andreas Nordvall Lager{\aa}s and Mathias Lindholm
We study the component structure in random intersection graphs with
tunable
clustering, and show that the average degree works as a threshold for
a phase
transition for the size of the largest component. That is, if the
expected
degree is less than one, the size of the largest component is $O_{p}
(\log n)$,
but if the average degree is greater than one, a single large
component of size
$O_{p}(n)$ emerges, and there will only be a finite number of small
vertices of
size less than $O_{p}(\log n)$.
http://arxiv.org/abs/0709.1416
---------------------------------------------------------------
6029. ANALYSIS OF STOCHASTIC FLUID QUEUES DRIVEN BY LOCAL TIME PROCESSES
Takis Konstantopoulos and Andreas Kyprianou and Marina Sirvio and
Paavo Salminen
We consider a stochastic fluid queue served by a constant rate server
and
driven by a process which is the local time of a certain Markov
process. Such a
stochastic system can be used as a model in a priority service system,
especially when the time scales involved are fast. The input (local
time) in
our model is always singular with respect to the Lebesgue measure
which in many
applications is ``close'' to reality. We first discuss how to rigorously
construct the (necessarily) unique stationary version of the system
under some
natural stability conditions. We then consider the distribution of
performance
steady-state characteristics, namely, the buffer content, the idle
period and
the busy period. These derivations are much based on the fact that
the inverse
of the local time of a Markov process is a L\'evy process (a
subordinator)
hence making the theory of L\'evy processes applicable. Another
important
ingredient in our approach is the Palm calculus coming from the point
process
point of view.
http://arxiv.org/abs/0709.1456
---------------------------------------------------------------
6030. ON DIFFERENTIABILITY OF THE PARISI FORMULA
Dmitry Panchenko
It was proved by Michel Talagrand ("Parisi measures") that Parisi
formula is
differentiable with respect to inverse temperature parameter. We
obtain a
simpler proof of this result by using approximate solutions in the
Parisi
formula and give one example of application to replica symmetry
breaking.
http://arxiv.org/abs/0709.1514
---------------------------------------------------------------
6031. AMERICAN OPTIONS UNDER PROPORTIONAL TRANSACTION COSTS: PRICING,
HEDGING AND STOPPING ALGORITHMS FOR LONG AND SHORT POSITIONS
Alet Roux and Tomasz Zastawniak
American options are studied in a general discrete market in the
presence of
proportional transaction costs, modelled as bid-ask spreads. Pricing
algorithms
and constructions of hedging strategies, stopping times and martingale
representations are presented for short (seller's) and long (buyer's)
positions
in an American option with an arbitrary payoff. This general approach
extends
the special cases considered in the literature concerned primarily with
computing the prices of American puts under transaction costs by
relaxing any
restrictions on the form of the payoff, the magnitude of the
transaction costs
or the discrete market model itself. The largely unexplored case of
pricing,
hedging and stopping for the American option buyer under transaction
costs is
also covered. The pricing algorithms are computationally efficient,
growing
only polynomially with the number of time steps in a recombinant tree
model.
The stopping times realising the ask (seller's) and bid (buyer's)
option prices
can differ from one another. The former is generally a so-called mixed
(randomised) stopping time, whereas the latter is always a pure
(ordinary)
stopping time.
http://arxiv.org/abs/0709.1589
---------------------------------------------------------------
6032. DISORDERED PINNING MODELS AND COPOLYMERS: BEYOND ANNEALED BOUNDS
Fabio Toninelli (ENS Lyon and CNRS)
We consider a general model of a disordered copolymer with
adsorption. This
includes, as particular cases, a generalization of the copolymer at a
selective
interface introduced by T. Garel et al., pinning and wetting models
in various
dimensions, and the Poland-Scheraga model of DNA denaturation. We
prove a new
variational upper bound for the free energy via an estimation of non-
integer
moments of the partition function. As an application, we show that
for strong
disorder the quenched critical point differs from the annealed one,
e.g., if
the disorder distribution is Gaussian. In particular, for pinning/
wetting
models with loop exponent 0<alpha<1/2 this implies the existence of a
transition from weak to strong disorder. For the copolymer model,
under some
(restrictive) conditions on the law of the underlying renewal, we
show that the
critical point coincides with the one predicted via renormalization
group
arguments in the theoretical physics literature. A stronger result
holds for a
``reduced wetting model'' introduced by T. Bodineau and G. Giacomin:
under no
restriction on the law of the underlying renewal, the critical point
coincides
with the corresponding renormalization group prediction.
http://arxiv.org/abs/0709.1629
---------------------------------------------------------------
6033. ON THE LOCALIZED PHASE OF A COPOLYMER IN AN EMULSION:
SUPERCRITICAL PERCOLATION REGIME
Frank den Hollander and Nicolas P\'etr\'elis
In this paper we study a two-dimensional directed self-avoiding walk
model of
a random copolymer in a random emulsion. The copolymer is a random
concatenation of monomers of two types, $A$ and $B$, each occurring with
density 1/2. The emulsion is a random mixture of liquids of two
types, $A$ and
$B$, organised in large square blocks occurring with density $p$ and
$1-p$,
respectively, where $p \in (0,1)$. The copolymer in the emulsion has
an energy
that is minus $\alpha$ times the number of $AA$-matches minus $\beta$
times the
number of $BB$-matches, where without loss of generality the interaction
parameters can be taken from the cone $\{(\alpha,\beta)\in\R^2\colon
\alpha\geq
|\beta|\}$. To make the model mathematically tractable, we assume
that the
copolymer is directed and can only enter and exit a pair of
neighbouring blocks
at diagonally opposite corners.
In \cite{dHW06}, it was found that in the supercritical
percolation regime $p
\geq p_c$, with $p_c$ the critical probability for directed bond
percolation on
the square lattice, the free energy has a phase transition along a
curve in the
cone that is independent of $p$. At this critical curve, there is a
transition
from a phase where the copolymer is fully delocalized into the $A$-
blocks to a
phase where it is partially localized near the $AB$-interface. In the
present
paper we prove three theorems that complete the analysis of the phase
diagram :
(1) the critical curve is strictly increasing; (2) the phase
transition is
second order; (3) the free energy is infinitely differentiable
throughout the
partially localized phase.
http://arxiv.org/abs/0709.1659
---------------------------------------------------------------
6034. EXISTENCE, UNIQUENESS AND APPROXIMATION OF STOCHASTIC SCHRODINGER
Clement Pellegrini (ICJ)
Recent developments in quantum physics make heavy use of so-called
``quantum
trajectories''. Mathematically, this theory gives rise to ``stochastic
Schr\"odinger equations'', that is, pertubations of Schrodinger-type
equations
under the form of stochastic differential equations. But such
equations are in
general not of the usual type as considered in the litterature. They
pose a
serious problem in terms of: justifying the existence and uniqueness
of a
solution, justifying the physical pertinence of the equations. In
this article
we concentrate on a particular case: the diffusive case, for a two-level
system. We prove existence and uniqueness of the associated stochastic
Schrodinger equation. We physically justify the equations by proving
that they
are continuous time limit of a concrete physical procedure for
obtainig quantum
trajectory.
http://arxiv.org/abs/0709.1703
---------------------------------------------------------------
6035. ON THE ASYMPTOTIC OF LIKELIHOOD RATIOS FOR SELF-NORMALIZED
LARGE DEVIATIONS
Zhiyi Chi
Motivated by multiple statistical hypothesis testing, we obtain the
limit of
likelihood ratio of large deviations for self-normalized random
variables,
specifically, the ratio of $P(\bar X +d/n \ge x_n V)$ to $P(\bar X
\ge x_n V)$,
as $n\toi$, where $\bar X$ and $V$ are the sample mean and standard
deviation
of iid $X_1, ..., X_n$, respectively, $d>0$ is a constant and $x_n
\toi$. We
show that the limit can have a simple form $e^{d/z_0}$, where $z_0$
is the
unique maximizer of $z f(x)$ with $f$ the density of $X_i$. The
result is
applied to derive the minimum sample size per test in order to
control the
error rate of multiple testing at a target level, when real signals are
different from noise signals only by a small shift.
http://arxiv.org/abs/0709.1506
---------------------------------------------------------------
6036. RANDOM WALKS ON QUASISYMMETRIC FUNCTIONS
Patricia Hersh and Samuel K. Hsiao
Conditions are provided under which an endomorphism on quasisymmetric
functions gives rise to a left random walk on the descent algebra
which is also
a lumping of a left random walk on permutations. Spectral results are
also
obtained. Several well-studied random walks are now realized this way:
Stanley's QS-distribution results from endomorphisms given by
evaluation maps,
a-shuffles result from the a-th convolution power of the universal
character,
and the Tchebyshev operator of the second kind introduced recently by
Ehrenborg
and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg
regarding the spectra for a family of random walks on ab-words is
proven. A
theorem of Stembridge from the theory of enriched P-partitions is also
recovered as a special case.
http://arxiv.org/abs/0709.1477
---------------------------------------------------------------
6037. MEAN-FIELD CONDITIONS FOR PERCOLATION ON FINITE GRAPHS
Asaf Nachmias
Let G_n be a sequence of finite transitive graphs with vertex degree
d=d(n)
and |G_n|=n. Denote by p^t(v,v) the return probability after t steps
of the
non-backtracking random walk on G_n. We show that if p^t(v,v) has
quasi-random
properties, then critical bond-percolation on G_n has a scaling
window of width
n^{-1/3}, as it would on a random graph.
A consequence of our theorems is that if G_n is a transitive
expander family
with girth at least (2/3 + eps) \log_{d-1} n, then the size of the
largest
component in p-bond-percolation with p={1 +O(n^{-1/3}) \over d-1} is
roughly
n^{2/3}. In particular, bond-percolation on the celebrated Ramanujan
graph
constructed by Lubotzky, Phillips and Sarnak has the above scaling
window. This
provides the first examples of quasi-random graphs behaving like
random graphs
with respect to critical bond-percolation.
http://arxiv.org/abs/0709.1719
---------------------------------------------------------------
6038. A HAAR-LIKE CONSTRUCTION FOR THE ORNSTEIN UHLENBECK PROCESS
Thibaud Taillefumier and Marcelo O. Magnasco
The classical Haar construction of Brownian motion uses a binary tree of
triangular wedge-shaped functions. This basis has compactness
properties which
make it especially suited for certain classes of numerical
algorithms. We
present a similar basis for the Ornstein-Uhlenbeck process, in which
the basis
elements approach asymptotically the Haar functions as the index
increases, and
preserve the following properties of the Haar basis: all basis
elements have
compact support on an open interval with dyadic rational endpoints;
these
intervals are nested and become smaller for larger indices of the basis
element, and for any dyadic rational, only a finite number of basis
elements is
nonzero at that number. Thus the expansion in our basis, when
evaluated at a
dyadic rational, terminates in a finite number of steps. We prove the
covariance formulae for our expansion and discuss its statistical
interpretation and connections to asymptotic scale invariance.
http://arxiv.org/abs/0709.1726
---------------------------------------------------------------
6039. ON EXIT TIMES OF LEVY-DRIVEN ORNSTEIN--UHLENBECK PROCESSES
K. Borovkov and A. Novikov
We prove two martingale identities which involve exit times of Levy-
driven
Ornstein--Uhlenbeck processes. Using these identities we find an
explicit
formula for the Laplace transform of the exit time under the
assumption that
positive jumps of the Levy process are exponentially distributed.
http://arxiv.org/abs/0709.1746
---------------------------------------------------------------
6040. ASYMPTOTICS FOR THE WIENER SAUSAGE AMONG POISSONIAN OBSTACLES
Ryoki Fukushima
We consider the Wiener sausage among Poissonian obstacles. The
obstacle is
called hard if Brownian motion entering the obstacle is immediately
killed, and
is called soft if it is killed at certain rate. It is known that
Brownian
motion conditioned to survive among obstacles is confined in a ball
near its
starting point. We show the weak law of large numbers, large deviation
principle in special cases and the moment asymptotics for the volume
of the
corresponding Wiener sausage. One of the consequence of our results
is that the
trajectory of Brownian motion almost fills the confinement ball.
http://arxiv.org/abs/0709.1751
---------------------------------------------------------------
6041. CHAOTICITY FOR MULTI-CLASS SYSTEMS AND EXCHANGEABILITY WITHIN
CLASSES
Carl Graham (CMAP)
Under the natural partial exchangeability assumption for multi-class
interacting particle systems, we prove that these converge to an
independent
system with infinite i.i.d. classes if and only if the empirical
measure of
each class satisfies a weak law of large numbers. This extension of a
classical
result for exchangeable systems (related to the de Finetti Theorem)
is somewhat
surprising, since then convergence of each class to infinite i.i.d.
particles
implies asymptotic independence of particles of different classes.
http://arxiv.org/abs/0709.1918
---------------------------------------------------------------
6042. SYMMETRIC $\ALPHA$-STABLE SUBORDINATORS AND CAUCHY PROBLEMS
Erkan Nane
We survey the results in Nane (E. Nane, Higher order PDE's and iterated
processes, Trans. American Math. Soc. (to appear)) and Baeumer,
Meerschaert,
and Nane (B. Baeumer, M.M. Meerschaert and E. Nane, Brownian
subordinators and
fractional Cauchy problems: Submitted (2007)) which deal with PDE
connection of
some iterated processes, and obtain a new probabilistic proof of the
equivalence of the higher order PDE's and fractional in time PDE's.
http://arxiv.org/abs/0709.1919
---------------------------------------------------------------
6043. A CLARK-OCONE FORMULA IN UMD BANACH SPACES
Jan Maas and Jan van Neerven
Let H be a separable real Hilbert space and let F = (F_t)_{t\in
[0,T]} be the
augmented filtration generated by an H-cylindrical Brownian motion
W_H on
[0,T]. We prove that if E is a UMD Banach space, 1\leq p<\infty, and f
\in
D^{1,p}(E) is F_T-measurable, then f = \E f + \int_0^T P_F(Df) dW_H
where D is
the Malliavin derivative and P_F is the projection onto the F-adapted
elements
in a suitable Banach space of L^p-stochastically integrable L(H,E)-
valued
processes.
http://arxiv.org/abs/0709.2021
---------------------------------------------------------------
6044. NONPARAMETRIC ESTIMATION FOR L\'EVY PROCESSES FROM LOW-
FREQUENCY OBSERVATIONS
Michael H. Neumann and Markus Reiss
We suppose that a L\'evy process is observed at discrete time points. A
rather general construction of minimum-distance estimators is shown
to give
consistent estimators of the L\'evy-Khinchine characteristics as the
number of
observations tends to infinity, keeping the observation distance
fixed. For a
specific $C^2$-criterion this estimator is rate-optimal. The
connection with
deconvolution and inverse problems is explained. A key step in the
proof is a
uniform control on the deviations of the empirical characteristic
function on
the whole real line.
http://arxiv.org/abs/0709.2007
---------------------------------------------------------------
6045. UNIFORM LIMIT LAWS OF THE LOGARITHM FOR NONPARAMETRIC
ESTIMATORS OF THE REGRESSION FUNCTION IN PRESENCE OF CENSORED DATA
Bertrand Maillot and Vivian Viallon
In this paper, we establish uniform-in-bandwidth limit laws of the
logarithm
for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.)
estimators of the multivariate regression function under random
censorship. A
similar result is deduced for estimators of the conditional distribution
function. The uniform-in-bandwidth consistency for estimators of the
conditional density and the conditional hazard rate functions are
also derived
from our main result. Moreover, the logarithm laws we establish are
shown to
yield almost sure simultaneous asymptotic confidence bands for the
functions we
consider. Examples of confidence bands obtained from simulated data are
displayed.
http://arxiv.org/abs/0709.2050
---------------------------------------------------------------
6046. PSEUDO-MAXIMIZATION AND SELF-NORMALIZED PROCESSES
Victor H. de la Pe\~na and Michael J. Klass and Tze Leung Lai
Self-normalized processes are basic to many probabilistic and
statistical
studies. They arise naturally in the the study of stochastic integrals,
martingale inequalities and limit theorems, likelihood-based methods in
hypothesis testing and parameter estimation, and Studentized pivots and
bootstrap-$t$ methods for confidence intervals. In contrast to standard
normalization, large values of the observations play a lesser role as
they
appear both in the numerator and its self-normalized denominator,
thereby
making the process scale invariant and contributing to its
robustness. Herein
we survey a number of results for self-normalized processes in the
case of
dependent variables and describe a key method called ``pseudo-
maximization''
that has been used to derive these results. In the multivariate case,
self-normalization consists of multiplying by the inverse of a positive
definite matrix (instead of dividing by a positive random variable as
in the
scalar case) and is ubiquitous in statistical applications, examples
of which
are given.
http://arxiv.org/abs/0709.2233
---------------------------------------------------------------
6047. ON CONVERGENCE OF GENERATORS OF EQUILIBRIUM DYNAMICS OF
HOPPING PARTICLES TO GENERATOR OF A BIRTH-AND-DEATH PROCESS IN
CONTINUUM
E. Lytvynov and P. T. Polara
We deal with two following classes of equilibrium stochastic dynamics of
infinite particle systems in continuum: hopping particles (also
called Kawasaki
dynamics), i.e., a dynamics where each particle randomly hops over
the space,
and birth-and-death process in continuum (or Glauber dynamics), i.e., a
dynamics where there is no motion of particles, but rather particles
die, or
are born at random. We prove that a wide class of Glauber dynamics
can be
derived as a scaling limit of Kawasaki dynamics. More precisely, we
prove the
convergence of respective generators on a set of cylinder functions,
in the
$L^2$-norm with respect to the invariant measure of the processes.
The latter
measure is supposed to be a Gibbs measure corresponding to a
potential of pair
interaction, in the low activity-high temperature regime. Our result
generalizes that of [Finkelshtein D.L. et al., to appear in Random Oper.
Stochastic Equations], which was proved for a special Glauber (Kawasaki,
respectively) dynamics.
http://arxiv.org/abs/0709.2284
---------------------------------------------------------------
6048. BRANCHED POLYMERS
Richard Kenyon and Peter Winkler
Building on and from the work of Brydges and Imbrie, we give an
elementary
calculation of the volume of the space of branched polymers of order
$n$ in the
plane and in 3-space. Our development reveals some more general
identities, and
allows exact random sampling. In particular we show that a random 3-
dimensional
branched polymer of order $n$ has diameter of order $\sqrt{n}$.
http://arxiv.org/abs/0709.2325
---------------------------------------------------------------
6049. QUEUEING FOR ERGODIC ARRIVALS AND SERVICES
L. Gyorfi and G. Morvai
In this paper we revisit the results of Loynes (1962) on stability of
queues
for ergodic arrivals and services, and show examples when the
arrivals are
bounded and ergodic, the service rate is constant, and under
stability the
limit distribution has larger than exponential tail.
http://arxiv.org/abs/0709.2330
---------------------------------------------------------------
6050. ADJUSTED VITERBI TRAINING FOR HIDDEN MARKOV MODELS
J. Lember and A. Koloydenko
To estimate the emission parameters in hidden Markov models one
commonly uses
the EM algorithm or its variation. Our primary motivation, however,
is the
Philips speech recognition system wherein the EM algorithm is
replaced by the
Viterbi training algorithm. Viterbi training is faster and
computationally less
involved than EM, but it is also biased and need not even be
consistent. We
propose an alternative to the Viterbi training -- adjusted Viterbi
training --
that has the same order of computational complexity as Viterbi
training but
gives more accurate estimators. Elsewhere, we studied the adjusted
Viterbi
training for a special case of mixtures, supporting the theory by
simulations.
This paper proves the adjusted Viterbi training to be also possible
for more
general hidden Markov models.
http://arxiv.org/abs/0709.2317
---------------------------------------------------------------
6051. MUTATION-SELECTION BALANCE WITH RECOMBINATION: CONVERGENCE TO
EQUILIBRIUM FOR POLYNOMIAL SELECTION COSTS
Aubrey Clayton and Steven N. Evans
We study a (possibly infinite-dimensional) dynamical system model for
mutation and selection in the presence of recombination. Some
features of the
model, such as existence and uniqueness of solutions and convergence
to the
dynamical system of an approximating sequence of discrete time
models, were
presented in earlier work by Evans, Steinsaltz, and Wachter for quite
general
selection costs. Here we establish that the phenomenon of mutation-
selection
balance occurs in the special case of ``polynomial'' selection costs
under mild
conditions. That is, we show that the dynamical system has a unique
equilibrium
and that it converges to this equilibrium from all initial conditions.
http://arxiv.org/abs/0709.1750
---------------------------------------------------------------
6052. A PROBABILISTIC PROOF OF WALLIS'S FORMULA FOR PI
Steven J. Miller
Using mostly elementary results and functions from probability, we prove
Wallis's formula for pi: pi/2 = prod_n (2n * 2n) / ((2n-1) * (2n+1)).
The proof
involves normalization constants and the Gamma function, Standard
normal, and
the Student t-Distribution.
http://arxiv.org/abs/0709.2181
---------------------------------------------------------------
6053. RANDOMIZATION IN C*-ALGEBRAS AND THE STABILITY OF QUANTUM FILTERS
Ramon van Handel
The states \rho_i, i=1,2 in the state space S of a C*-algebra A are
absolutely continuous if and only if there exist absolutely continuous
probability measures \mu_i on S such that \rho_i is the barycenter of
\mu_i.
This technique allows one to study the transformation of conditional
expectations under an absolutely continuous change of state using the
classical
Bayes formula, which can be exploited to obtain sufficient conditions
for the
asymptotic stability of quantum Markov filters. In the case that A is
finite
dimensional, explicitly computable observability criteria are obtained.
http://arxiv.org/abs/0709.2216
---------------------------------------------------------------
6054. STOCHASTIC VARIATIONAL PARTITIONED RUNGE-KUTTA INTEGRATORS FOR
CONSTRAINED SYSTEMS
Nawaf Bou-Rabee and Houman Owhadi
Stochastic variational integrators for constrained, stochastic
mechanical
systems are developed in this paper. The main results of the paper
are twofold:
an equivalence is established between a stochastic Hamilton-
Pontryagin (HP)
principle in generalized coordinates and constrained coordinates via
Lagrange
multipliers, and variational partitioned Runge-Kutta (VPRK)
integrators are
extended to this class of systems. Among these integrators are 1/2 and
3/2-order strongly convergent RATTLE-type integrators. We prove order of
accuracy of the methods provided. The paper also reviews the
deterministic
treatment of VPRK integrators from the HP viewpoint.
http://arxiv.org/abs/0709.2222
---------------------------------------------------------------
6055. ON ROUGH ISOMETRIES OF POISSON PROCESSES ON THE LINE
Ron Peled
Benjamini and Szegedy asked (independently) whether two independent
Poisson
point processes on the line with the same intensity are rough isometric
(quasi-isometric) a.s.. Szegedy conjectured the answer is positive.
We prove
that this question is equivalent to the following question, given two
independent Bernoulli percolations $A$ and $B$ on the natural numbers
with 0
adjoined to each of them, there exist constants and probability $p>0$
such that
for any $n$ the first $n$ points of $A$ are rough isometric to an
initial
segment of $B$ with these constants, with 0 mapping to 0 and with
probability
at least $p$. We then make some progress towards the conjecture by
showing that
if the constants of the rough isometry are allowed to grow with $n$ then
constants of order $\sqrt{\log n}$ will suffice (this quantitative
variant was
introduced by Benjamini). It appears that this is the first result to
improve
upon the trivial construction which has constants of order $\log n$.
Furthermore, the rough isometry we construct is (weakly) monotone and we
include a discussion of monotone rough isometries, their properties
and an
interesting lattice structure inherent in them.
http://arxiv.org/abs/0709.2383
---------------------------------------------------------------
6056. A NEW WEAK APPROXIMATION SCHEME OF STOCHASTIC DIFFERENTIAL
EQUATIONS BY USING THE RUNGE-KUTTA METHOD
Shigeo Kusuoka and Mariko Ninomiya and Syoiti Ninomiya
In this paper, authors successfully construct a new algorithm for the
new
higher order scheme of weak approximation of SDEs. The algorithm
presented here
is based on [1][2]. Although this algorithm shares some features with
the
algorithm presented by [3], algorithms themselves are completely
different and
the diversity is not trivial. They apply this new algorithm to the
problem of
pricing Asian options under the Heston stochastic volatility model
and obtain
encouraging results.
[1] Shigeo Kusuoka, ``Approximation of Expectation of Diffusion
Process and
Mathematical Finance,'' Advanced Studies in Pure Mathematics,
Proceedings of
Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp.
147--165. [2] Terry Lyons and Nicolas Victoir, ``Cubature on Wiener
Space,''
Proceedings of the Royal Society of London. Series A. Mathematical
and Physical
Sciences 460 (2004), pp. 169--198. [3] Syoiti Ninomiya, Nicolas
Victoir, ``Weak
approximation of stochastic differential equations and application to
derivative pricing'', arXiv:math/0605361v3
http://arxiv.org/abs/0709.2434
---------------------------------------------------------------
6057. ON THE APPROACH TO EQUILIBRIUM FOR A POLYMER WITH ADSORPTION
AND REPULSION
Pietro Caputo and Fabio Martinelli and Fabio Lucio Toninelli
We consider paths of a one-dimensional simple random walk conditioned
to come
back to the origin after L steps (L an even integer). In the 'pinning
model'
each path \eta has a weight \lambda^{N(\eta)}, where \lambda>0 and N
(\eta) is
the number of zeros in \eta. When the paths are constrained to be non-
negative,
the polymer is said to satisfy a hard-wall constraint. Such models
are well
known to undergo a localization/delocalization transition as the pinning
strength \lambda is varied. In this paper we study a natural 'spin flip'
dynamics for these models and derive several estimates on its
spectral gap and
mixing time. In particular, for the system with the wall we prove that
relaxation to equilibrium is always at least as fast as in the free case
(\lambda=1, no wall), where the gap and the mixing time are known to
scale as
L^{-2} and L^2\log L, respectively. This improves considerably over
previously
known results. For the system without the wall we show that the
equilibrium
phase transition has a clear dynamical manifestation: for \lambda
\geq 1 the
relaxation is again at least as fast as the diffusive free case, but
in the
strictly delocalized phase (\lambda < 1) the gap is shown to be O(L^
{-5/2}), up
to logarithmic corrections. As an application of our bounds, we prove
stretched
exponential relaxation of local functions in the localized regime.
http://arxiv.org/abs/0709.2612
---------------------------------------------------------------
6058. ON HOMOGENEOUS PINNING MODELS AND PENALIZATIONS
Mihai Gradinaru (IRMAR) and Samy Tindel (IECN)
In this note, we show how the penalization method, introduced in
order to
describe some non-trivial changes of the Wiener measure, can be
applied to the
study of some simple polymer models such as the pinning model. The
bulk of the
analysis is then focused on the study of a martingale which has to be
computed
as a Markovian limit.
http://arxiv.org/abs/0709.2656
---------------------------------------------------------------
6059. SHANNON-MACMILLAN THEOREMS FOR DISCRETE RANDOM FIELDS ALONG
CURVES AND LOWER BOUNDS FOR SURFACE-ORDER LARGE DEVIATIONS
Julia Brettschneider
The notion of a surface-order specific entropy h_c(P) of a two-
dimensional
discrete random field P along a curve c is introduced as the limit of
rescaled
entropies along lattice approximations of the blowups of c. Existence
is shown
by proving a corresponding Shannon-MacMillan theorem. We obtain a
representation of h_c(P) as a mixture of specific entropies along the
tangent
lines of c. As an application, the specific entropy along curves is
used to
refine Foellmer and Ort's lower bound for the large deviations of the
empirical
field of an attractive Gibbs measure from its ergodic behavior in the
phase-transition regime.
http://arxiv.org/abs/0709.2662
---------------------------------------------------------------
6060. REGULARITY OF THE DENSITY FOR THE STOCHASTIC HEAT EQUATION
Carl Mueller and David Nualart
We study the smoothness of the density of a semilinear heat equation
with
multiplicative spacetime white noise. Using Malliavin calculus, we
reduce the
problem to a question of negative moments of solutions of a linear heat
equation with multiplicative white noise. Then we settle this
question by
proving that solutions to the linear equation have negative moments
of all
orders.
http://arxiv.org/abs/0709.2663
---------------------------------------------------------------
6061. INTERACTING BROWNIAN MOTIONS AND THE GROSS-PITAEVSKII FORMULA
Stefan Adams and Wolfgang K\"onig
We review probabilistic approaches to the Gross-Pitaevskii theory
describing
interacting dilute systems of particles. The main achievement are large
deviations principles for the mean occupation measure of a large
system of
interacting Brownian motions in a trapping potential. The
corresponding rate
functions are given as variational problems whose solution provide
effective
descriptions of the infinite system.
http://arxiv.org/abs/0709.2771
---------------------------------------------------------------
6062. HOW OFTEN DOES THE RATCHET CLICK? FACTS, HEURISTICS, ASYMPTOTICS
A. Etheridge and P. Pfaffelhuber and A. Wakolbinger
The evolutionary force of recombination is lacking in asexually
reproducing
populations. As a consequence, the population can suffer an irreversible
accumulation of deleterious mutations, a phenomenon known as Muller's
ratchet.
We formulate discrete and continuous time versions of Muller's ratchet.
Inspired by Haigh's (1978) analysis of a dynamical system which
arises in the
limit of large populations, we identify the parameter gamma =
N*lambda/(Ns*log(N*lambda)) as most important for the speed of
accumulation of
deleterious mutations. Here N is population size, s is the selection
coefficient and lambda is the deleterious mutation rate. For large
parts of the
parameter range, measuring time in units of size N, deleterious
mutations
accumulate according to a power law in N*lambda with exponent gamma if
gamma>0.5. For gamma<0.5 mutations cannot accumulate. We obtain
diffusion
approximations for three different parameter regimes, depending on
the speed of
the ratchet. Our approximations shed new light on analyses of Stephan
et al.
(1993) and Gordo & Charlesworth (2000). The heuristics leading to the
approximations are supported by simulations.
http://arxiv.org/abs/0709.2775
---------------------------------------------------------------
6063. ON THE STRUCTURE OF QUASI-STATIONARY COMPETING PARTICLES SYSTEMS
Louis-Pierre Arguin and Michael Aizenman
We study point processes on the real line whose configurations X are
locally
finite, have a maximum, and evolve through increments which are
functions of
correlated gaussian variables. The correlations are intrinsic to the
points and
quantified by a matrix Q={q_ij}. A probability measure on the pair
(X,Q) is
said to be quasi-stationary if the joint law of the gaps of X and of
Q is
invariant under the evolution. A known class of universally quasi-
stationary
processes is given by the Ruelle Probability Cascades (RPC), which
are based on
hierarchally nested Poisson-Dirichlet processes. It was conjectured
that up to
some natural superpositions these processes exhausted the class of
laws which
are robustly quasi-stationary. The main result of this work is a
proof of this
conjecture for the case where q_ij assume only a finite number of
values. The
result is of relevance for mean-field spin glass models, where the
evolution
corresponds to the cavity dynamics, and where the hierarchal
organization of
the Gibbs measure was first proposed as an ansatz.
http://arxiv.org/abs/0709.2901
---------------------------------------------------------------
6064. ON THE NOTION OF CONVEX-COMPACTNESS AND ITS APPLICATIONS
Gordan Zitkovic
The concept of convex-compactness, weaker than the classical notion of
compactness, is introduced and discussed. It is shown that a large
class of
convex subsets of topological vector spaces shares this property and
that is
can be used in lieu of compactness in a variety of cases. In
particular, we
show that bounded-in-probability, convex and closed subsets of the space
$\lzer_+(\Omega,\FF,\PP)$ of finite-valued non-negative random
variables on a
probability space are convex-compact.
Applications in optimization and mathematical economics - versions
of the
Minimax theorem, the fixed-point theorem of Knaster, Kuratowski and
Mazurkiewicz as well as the excess-demand theorem of mathematical
economics -
are provided.
http://arxiv.org/abs/0709.2730
---------------------------------------------------------------
6065. BEHAVIORAL PORTFOLIO SELECTION IN CONTINUOUS TIME
Hanqing Jin and Xunyu Zhou
This paper formulates and studies a general continuous-time behavioral
portfolio selection model under Kahneman and Tversky's (cumulative)
prospect
theory, featuring S-shaped utility (value) functions and probability
distortions. Unlike the conventional expected utility maximization
model, such
a behavioral model could be easily mis-formulated (a.k.a. ill-posed)
if its
different components do not coordinate well with each other. Certain
classes of
an ill-posed model are identified. A systematic approach, which is
fundamentally different from the ones employed for the utility model, is
developed to solve a well-posed model, assuming a complete market and
general
It\^o processes for asset prices. The optimal terminal wealth positions,
derived in fairly explicit forms, possess surprisingly simple structure
reminiscent of a gambling policy betting on a good state of the world
while
accepting a fixed, known loss in case of a bad one. An example with a
two-piece
CRRA utility is presented to illustrate the general results obtained,
and is
solved completely for all admissible parameters. The effect of the
behavioral
criterion on the risky allocations is finally discussed.
http://arxiv.org/abs/0709.2830
---------------------------------------------------------------
6066. SHAPE AND LOCAL GROWTH FOR MULTIDIMENSIONAL BRANCHING RANDOM
WALKS IN RANDOM ENVIRONMENT
Francis Comets and Serguei Popov
We study branching random walks in random environment on the $d$-
dimensional
square lattice, $d \geq 1$. In this model, the environment has finite
range
dependence, and the population size cannot decrease. We prove limit
theorems
(laws of large numbers) for the set of lattice sites which are
visited up to a
large time as well as for the local size of the population. The
limiting shape
of this set is compact and convex, though the local size is given by
a concave
growth exponent. Also, we obtain the law of large numbers for the
logarithm of
the total number of particles in the process.
http://arxiv.org/abs/0709.2926
---------------------------------------------------------------
6067. A UNIFIED APPROACH TO STOCHASTIC EVOLUTION EQUATIONS USING THE
SKOROKHOD INTEGRAL
S. V. Lototsky and B. L. Rozovskii
We study stochastic evolution equations driven by Gaussian noise. The
key
features of the model are that the operators in the deterministic and
stochastic parts can have the same order and the noise can be time-only,
space-only, or space-time. Even the simplest equations of this kind
do not have
a square-integrable solution and must be solved in special weighted
spaces. We
demonstrate that the Cameron-Martin version of the Wiener chaos
decomposition
leads to natural weights and a natural replacement of the square
integrability
condition.
http://arxiv.org/abs/0709.2975
---------------------------------------------------------------
6068. ON SOME PROBABILISTIC PROPERTIES OF PERIODIC GARCH PROCESSES
Abdelouahab Bibi and Abdelhakim Aknouche
This paper examines some probabilistic properties of the class of
periodic
GARCH processes (PGARCH) which feature periodicity in conditional
heteroskedasticity. In these models, the parameters are allowed to
switch
between different regimes, so that their structure shares many
properties with
periodic ARMA process (PARMA). We examine the strict and second order
periodic
stationarities, the existence of higher-order moments, the covariance
structure, the geometric ergodicity and -mixing of the PGARCH(p,q)
process
under general and tractable assumptions. Some examples are proposed to
illustrate the various concepts.
http://arxiv.org/abs/0709.2983
---------------------------------------------------------------
6069. SOME EXTENSIONS OF THE UNCERTAINTY PRINCIPLE
Steeve Zozor and Mariela Portesi and Christophe Vignat
In this paper, we study extensions of entropic inequalities recently
derived
by Bialynicki-Birula [1] and Zozor et al. [2]. These inequalities can be
considered as generalizations of the Heisenberg uncertainty
principle, since
they measure the mutual uncertainty of a wavefunction and its Fourier
transform
through their associated Renyi entropies with conjugated indexes. We
consider
here the case where the entropic indexes are not conjugated, in both
cases
where the state space is discrete and continuous: we discuss the
existence of
an uncertainty inequality depending on the location of the entropic
indexes
$\alpha$ and $\beta$ in the plane $(\alpha, \beta)$. The obtained
results
explain and extend a recent study by Luis [3].
http://arxiv.org/abs/0709.3011
---------------------------------------------------------------
6070. CONSTRUCTION OF A STATIONARY QUEUE WITH IMPATIENT CUSTOMERS
Pascal Moyal
In this paper, we study the stability of queues with impatient
customers.
Under general stationary ergodic assumptions, we first provide some
conditions
for such a queue to be regenerative (i.e. to empty a.s. an infinite
number of
times). In the particular case of a single server operating in First
in, First
out, we prove the existence (in some cases, on an enlarged
probability space)
of a stationary workload. This is done by studying a non-monotonic
stochastic
recursion under the Palm settings, and by stochastic comparison of
stochastic
recursions.
http://arxiv.org/abs/0709.3012
---------------------------------------------------------------
6071. RANDOM EVEN GRAPHS AND THE ISING MODEL
Geoffrey Grimmett and Svante Janson
We explore the relationship between the Ising model with inverse
temperature
$\beta$, the $q=2$ random-cluster model with edge-parameter $p=1-e^{-2
\beta}$,
and the random even subgraph with edge-parameter $\frac 12p$. For a
planar
graph $G$, the boundary edges of the + clusters of the Ising model on
the
planar dual of $G$ forms a random even subgraph of $G$. A coupling of
the
random even subgraph of $G$ and the $q=2$ random-cluster model on $G$ is
presented, thus extending the above observation to general graphs. A
random
even subgraph of a planar lattice undergoes a phase transition at the
parameter-value $\frac 12 \pc$, where $\pc$ is the critical point of
the $q=2$
random-cluster model on the dual lattice. These results are motivated
in part
by an exploration of the so-called random-current method utilised by
Aizenman,
Barsky, Fern\'andez and others to solve the Ising model on the $d$-
dimensional
hypercubic lattice.
http://arxiv.org/abs/0709.3039
---------------------------------------------------------------
6072. THE MARTINGALE PROBLEM FOR A CLASS OF STABLE-LIKE PROCESSES
Richard F. Bass and Huili Tang
Let $\alpha\in (0,2)$ and consider the operator $$L f(x) =\int
[f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+
\alpha}}
dh, $$ where the $\nabla f(x)\cdot h$ term is omitted if $\alpha<1$. We
consider the martingale problem corresponding to the operator $L$ and
under
mild conditions on the function $A$ prove that there exists a unique
solution.
http://arxiv.org/abs/0709.3082
---------------------------------------------------------------
6073. LEAST SQUARES VOLATILITY CHANGE POINT ESTIMATION FOR PARTIALLY
OBSERVED DIFFUSION PROCESSES
A. De Gregorio and S.M. Iacus
A one dimensional diffusion process $X=\{X_t, 0\leq t \leq T\}$, with
drift
$b(x)$ and diffusion coefficient $\sigma(\theta, x)=\sqrt{\theta}
\sigma(x)$
known up to $\theta>0$, is supposed to switch volatility regime at
some point
$t^*\in (0,T)$. On the basis of discrete time observations from $X$, the
problem is the one of estimating the instant of change in the volatility
structure $t^*$ as well as the two values of $\theta$, say $\theta_1$
and
$\theta_2$, before and after the change point. It is assumed that the
sampling
occurs at regularly spaced times intervals of length $\Delta_n$ with
$n\Delta_n=T$. To work out our statistical problem we use a least
squares
approach. Consistency, rates of convergence and distributional
results of the
estimators are presented under an high frequency scheme. We also
study the case
of a diffusion process with unknown drift and unknown volatility but
constant.
http://arxiv.org/abs/0709.2967
---------------------------------------------------------------
6074. QUASI-MAXIMUM LIKELIHOOD ESTIMATION OF PERIODIC GARCH PROCESSES
Abdehakim Aknouche and Abdelouhab Bibi
This paper establishes the strong consistency and asymptotic
normality of the
quasi-maximum likelihood estimator (QMLE) for a GARCH process with
periodically
time-varying parameters. We first give a necessary and sufficient
condition for
the existence of a strictly periodically stationary solution for the
periodic
GARCH (P-GARCH) equation. As a result, it is shown that the moment of
some
positive order of the P-GARCH solution is finite, under which we
prove the
strong consistency and asymptotic normality (CAN) of the QMLE without
any
condition on the moments of the underlying process.
http://arxiv.org/abs/0709.2982
---------------------------------------------------------------
6075. A TAIL INEQUALITY FOR SUPREMA OF UNBOUNDED EMPIRICAL PROCESSES
WITH APPLICATIONS TO MARKOV CHAINS
Rados{\l}aw Adamczak
We present an easy extension of Talagrand's inequality for suprema of
empirical processes to processes generated by variables with finite $
\psi_1$
norm and apply it to some geometrically ergodic Markov chains to derive
versions of Bernstein's and Talagrand's inequalities. We also obtain
a bounded
difference inequality for symmetric statistics of such Markov chains.
http://arxiv.org/abs/0709.3110
---------------------------------------------------------------
6076. LEVEL SETS OF THE STOCHASTIC WAVE EQUATION DRIVEN BY A
SYMMETRIC L\'EVY NOISE
Davar Khoshnevisan and Eulalia Nualart
We consider the solution $\{u(t,x), t \geq 0, x \in \mathbb{R}\}$ of
a system
of $d$ linear stochastic wave equations driven by a $d$ dimensional
symmetric
space-time L\'evy noise. We provide a necessary and sufficient
condition, on
the characteristic exponent of the L\'evy noise, which describes
exactly when
the zero set of $u$ is nonvoid. We also compute the Hausdorff
dimension of that
zero set, when it is nonempty. These results will follow from more
general
potential-theoretic theorems on level sets of L\'evy sheets.
http://arxiv.org/abs/0709.3165
---------------------------------------------------------------
6077. MAXIMAL REGULARITY FOR STOCHASTIC CONVOLUTIONS DRIVEN BY LEVY
NOISE
Zdzislaw Brze\'zniak and Erika Hausenblas
We show that the result from Da Prato and Lunardi is valid for
stochastic
convolutions driven by L\'evy processes.
http://arxiv.org/abs/0709.3179
---------------------------------------------------------------
6078. DISTRIBUTION FUNCTIONS OF LINEAR COMBINATIONS OF LATTICE
POLYNOMIALS FROM THE UNIFORM DISTRIBUTION
Jean-Luc Marichal and Ivan Kojadinovic
We give the distribution functions, the expected values, and the
moments of
linear combinations of lattice polynomials from the uniform
distribution.
Linear combinations of lattice polynomials, which include weighted
sums, linear
combinations of order statistics, and lattice polynomials, are
actually those
continuous functions that reduce to linear functions on each simplex
of the
standard triangulation of the unit cube. They are mainly used in
aggregation
theory, combinatorial optimization, and game theory, where they are
known as
discrete Choquet integrals and Lovasz extensions.
http://arxiv.org/abs/0709.3184
---------------------------------------------------------------
6079. INFINITELY DIVISIBLE DISTRIBUTIONS OVER LOCALLY COMPACT NON-
ARCHIMEDEAN FIELDS
S. V. Ludkovsky
The article is devoted to stochastic processes with values in
finite-dimensional vector spaces over infinite locally compact fields
with
non-trivial non-archimedean valuations. Infinitely divisible
distributions are
investigated. Theorems about their characteristic functionals are
proved.
Particular cases are demonstrated.
http://arxiv.org/abs/0709.3215
---------------------------------------------------------------
6080. INDETERMINACY RELATIONS IN RANDOM DYNAMICS
Piotr Garbaczewski
We analyze various uncertainty measures for spatial diffusion
processes. In
this manifestly non-quantum setting, we focus on the existence issue of
complementary pairs whose joint dispersion measure has strictly
positive lower
bound.
http://arxiv.org/abs/cond-mat/0703204
---------------------------------------------------------------
6081. DISTRIBUTION OF THE TIME AT WHICH THE DEVIATION OF A BROWNIAN
MOTION IS MAXIMUM BEFORE ITS FIRST-PASSAGE TIME
Julien Randon-Furling (LPTMS) and Satya N. Majumdar (LPTMS)
We calculate analytically the probability density $P(t_m)$ of the
time $t_m$
at which a continuous-time Brownian motion (with and without drift)
attains its
maximum before passing through the origin for the first time. We also
compute
the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m
$. In the
driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim
t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m
$. In
presence of a drift towards the origin, $P(t_m)$ decays exponentially
for large
$t_m$. The results from numerical simulations are in excellent
agreement with
our analytical predictions.
http://arxiv.org/abs/0708.2101
---------------------------------------------------------------
6082. ANALYSIS OF EQUILIBRIUM STATES OF MARKOV SOLUTIONS TO THE 3D
NAVIER-STOKES EQUATIONS DRIVEN BY ADDITIVE NOISE
Marco Romito
We prove that every Markov solution to the three dimensional Navier-
Stokes
equation with periodic boundary conditions driven by additive
Gaussian noise is
uniquely ergodic. The convergence to the (unique) invariant measure is
exponentially fast.
Moreover, we give a well-posedness criterion for the equations in
terms of
invariant measures. We also analyse the energy balance and identify
the term
which ensures equality in the balance.
http://arxiv.org/abs/0709.3267
---------------------------------------------------------------
6083. LINE CROSSING PROBLEM FOR BIASED MONOTONIC RANDOM WALKS IN THE
PLANE
Mohammad Javaheri
In this paper, we study the problem of finding the probability that the
two-dimensional (biased) monotonic random walk crosses the line $y=
\alpha x+d$,
where $\alpha,d \geq 0$. A $\beta$-biased monotonic random walk moves
from
$(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(\beta + 1)$ and
$\beta/(\beta + 1)$, respectively. Among our results, we show that if
$\beta
\geq \lceil \alpha \rceil$, then the $\beta$-biased monotonic random
walk,
starting from the origin, crosses the line $y=\alpha x+d$ for all $d
\geq 0$
with probability 1.
http://arxiv.org/abs/0709.3316
---------------------------------------------------------------
6084. A NORMAL DISTRIBUTION FOR THE DISTURBANCE TERM IN REGRESSION
THEORY
Mr. Lambros Iossif
In regression theory, it is stated that the disturbance term follows the
normal distribution when the sample size is large. In Professor
J.Johnston's
words: "In view of the many factors involved, an appeal to the
Central Limit
Theorem would further suggest a normal distribution for u." This
paper includes
an elementary proof that the disturbance term follows the normal
distribution
when n is large.
http://arxiv.org/abs/0709.3414
---------------------------------------------------------------
6085. A NOTE ON "SIGNED VOTER MODELS"
E. Andjel and G. Maillard and T.S. Mountford
We consider some questions raised by the recent paper of Gantert, L
\"owe and
Steif (2005) concerning ``signed'' voter models on locally finite
graphs. These
are voter model like processes with the difference that the edges are
considered to be either positive or negative. If an edge between a
site $x$ and
a site $y$ is negative (respectively positive) the site $y$ will
contribute
towards the flip rate of $x$ if and only if the two current spin
values are
equal (respectively opposed).
http://arxiv.org/abs/0709.3468
---------------------------------------------------------------
6086. SURE WINS, SEPARATING PROBABILITIES AND THE REPRESENTATION OF
LINEAR FUNCTIONALS
Gianluca Cassese
We discuss conditions under which a convex cone $\K\subset \R^{\Omega}$
admits a probability $m$ such that $\sup_{k\in \K} m(k)\leq0$. Based
on these,
we also characterize linear functionals that admit the representation as
finitely additive expectations. A version of Riesz decomposition
based on this
property is obtained as well as a characterisation of positive
functionals on
the space of integrable functions
http://arxiv.org/abs/0709.3411
---------------------------------------------------------------
6087. SINGULARITY SETS OF LEVY PROCESSES
Arnaud Durand
We completely describe the size and large intersection properties of the
Holder singularity sets of Levy processes. We also study the set of
times at
which a given function cannot be a modulus of continuity of a Levy
process. The
Holder singularity sets of the sample paths of certain random wavelet
series
are investigated as well.
http://arxiv.org/abs/0709.3596
---------------------------------------------------------------
6088. RANDOM WAVELET SERIES BASED ON A TREE-INDEXED MARKOV CHAIN
Arnaud Durand
We study the global and local regularity properties of random wavelet
series
whose coefficients exhibit correlations given by a tree-indexed
Markov chain.
We determine the law of the spectrum of singularities of these
series, thereby
performing their multifractal analysis. We also show that almost
every sample
path displays an oscillating singularity at almost every point and
that the
points at which a sample path has at most a given Holder exponent
form a set
with large intersection.
http://arxiv.org/abs/0709.3597
---------------------------------------------------------------
6089. RANDOM FRACTALS AND TREE-INDEXED MARKOV CHAINS
Arnaud Durand
We study the size properties of a general model of fractal sets that are
based on a tree-indexed family of random compacts and a tree-indexed
Markov
chain. These fractals may be regarded as a generalization of those
resulting
from the Moran-like deterministic or random recursive constructions
considered
by various authors. Among other applications, we consider various
extensions of
Mandelbrot's fractal percolation process.
http://arxiv.org/abs/0709.3598
---------------------------------------------------------------
6090. EXISTENCE, UNIQUENESS AND APPROXIMATION FOR STOCHASTIC SCHRODINGER
Clement Pellegrini (ICJ)
In quantum physics, recent investigations deal with the so-called
"quantum
trajectory" theory. Heuristic rules are usually used to give rise to
"stochastic Schrodinger equations" which are stochastic differential
equations
of non-usual type describing the physical models. These equations
pose tedious
problems in terms of mathematical justification: notion of solution,
existence,
uniqueness, justification... In this article, we concentrate on a
particular
case: the Poisson case. Random measure theory is used in order to
give rigorous
sense to such equations. We prove existence and uniqueness of a
solution for
the associated stochastic equation. Furthermore, the stochastic model is
physically justified by proving that the solution can be obtained as
a limit of
a concrete discrete time physical model.
http://arxiv.org/abs/0709.3713
---------------------------------------------------------------
6091. ON THE PATH STRUCTURE OF A SEMIMARTINGALE ARISING FROM
MONOTONE PROBABILITY THEORY
Alexander C. R. Belton
Let X be the unique normal martingale such that X_0 = 0 and d[X]_t =
(1 - t -
X_{t-}) dX_t + dt and let Y_t := X_t + t for all t >= 0; the
semimartingale Y
arises in quantum probability, where it is the monotone-independent
analogue of
the Poisson process. The trajectories of Y are examined and various
probabilistic properties are derived; in particular, the level set {t
>= 0 :
Y_t = 1} is shown to be non-empty, compact, perfect and of zero Lebesgue
measure. The local times of Y are found to be trivial except for that
at level
1; consequently, the jumps of Y are not locally summable.
http://arxiv.org/abs/0709.3788
---------------------------------------------------------------
6092. ON REAL-TIME COMMUNICATION SYSTEMS WITH NOISY FEEDBACK
Aditya Mahajan and Demosthenis Teneketzis
We consider a real-time communication system with noisy feedback
consisting
of a Markov source, a forward and a backward discrete memoryless
channels, and
a receiver with finite memory. The objective is to design an optimal
communication strategy (that is, encoding, decoding, and memory update
strategies) to minimize the total expected distortion over a finite
horizon. We
present a sequential decomposition for the problem, which results in
a set of
nested optimality equations to determine optimal communication
strategies. This
provides a systematic methodology to determine globally optimal joint
source-channel encoding and decoding strategies for real-time
communication
systems with noisy feedback.
http://arxiv.org/abs/0709.3753
---------------------------------------------------------------
6093. MASS TRANSPORT AND VARIANTS OF THE LOGARITHMIC SOBOLEV INEQUALITY
Franck Barthe and Alexander V. Kolesnikov
We develop the optimal transportation approach to modified log-Sobolev
inequalities and to isoperimetric inequalities. Various sufficient
conditions
for such inequalities are given. Some of them are new even in the
classical
log-Sobolev case. The idea behind many of these conditions is that
measures
with a non-convex potential may enjoy such functional inequalities
provided
they have a strong integrability property that balances the lack of
convexity.
In addition, several known criteria are recovered in a simple unified
way by
transportation methods and generalized to the Riemannian setting.
http://arxiv.org/abs/0709.3890
---------------------------------------------------------------
6094. VARIATIONS AND ESTIMATORS FOR THE SELFSIMILARITY ORDER THROUGH
MALLIAVIN CALCULUS
Ciprian Tudor (CES) and Frederi Viens
Using multiple stochastic integrals, we analyze the asymptotic
behavior of
quadratic variations for Gaussian and non-Gaussian selfsimilar
processes. We
apply our results to the study of statistical estimators for the
selfsimilarity
index.
http://arxiv.org/abs/0709.3896
---------------------------------------------------------------
6095. NON-CENTRAL CONVERGENCE OF MULTIPLE INTEGRALS
Ivan Nourdin (PMA) and Giovanni Peccati (LSTA)
Fix $\nu >0$, denote by $G(\nu/2)$ a Gamma random variable with
parameter
$\nu/2$, and let $n\geq2$ be an even integer. Consider a sequence
$\{F_k\}_{k\geq 1}$ of square integrable random variables, belonging
to the
$n$th Wiener chaos of a given Gaussian process and with variance
converging to
$2\nu$. We prove that $\{F_k\}_{k\geq 1}$ converges in distribution to
$2G(\nu/2) - \nu$, if, and only if, $E(F_k^4)-12 E(F_{k}^3)\to 12
\nu^2-48\nu$.
Observe that, if $\nu\geq 1$ is an integer, then $2G(\nu/2) - \nu$ has a
centered $\chi^2$ law with $\nu$ degrees of freedom. Our approach
involves the
techniques of Malliavin calculus recently developed by Nualart and
Ortiz-Latorre (2007). We also obtain some multidimensional non-
central limit
theorems, as well as several equivalent conditions in terms of Malliavin
derivatives and norms of contraction operators. Our results should be
compared
with the main findings by Nualart and Peccati (2005), where it is
shown that a
normalized sequence of multiple Wiener-It\^{o} integrals converges in
law to a
Gaussian random variable if, and only if, the sequence of their
fourth moments
converges to 3.
http://arxiv.org/abs/0709.3903
---------------------------------------------------------------
6096. THE CIRCULAR LAW FOR RANDOM MATRICES
F. G\"otze and A. Tikhomirov
We consider the joint distribution of real and imaginary parts of
eigenvalues
of random matrices with independent real entries with mean zero and unit
variance. We prove the convergence of this distribution to the uniform
distribution on the unit disc without assumptions on the existence of
a density
for the distribution of entries. We assume that the entries have
moment of
order $\E|X_{jk}|^2\phi(x)$, with some positive function $\phi(x)$
which is
growing of order $(\log(1+|x|))^7$, or that they are sparsely non-
zero. The
results are based on and extend previous work of Bai, Rudelson and
the authors.
http://arxiv.org/abs/0709.3995
---------------------------------------------------------------
6097. GEOGRAPHIC GOSSIP: EFFICIENT AVERAGING FOR SENSOR NETWORKS
Alexandros G. Dimakis and Anand D. Sarwate and Martin J. Wainwright
Gossip algorithms for distributed computation are attractive due to
their
simplicity, distributed nature, and robustness in noisy and uncertain
environments. However, using standard gossip algorithms can lead to a
significant waste in energy by repeatedly recirculating redundant
information.
For realistic sensor network model topologies like grids and random
geometric
graphs, the inefficiency of gossip schemes is related to the slow
mixing times
of random walks on the communication graph. We propose and analyze an
alternative gossiping scheme that exploits geographic information. By
utilizing
geographic routing combined with a simple resampling method, we
demonstrate
substantial gains over previously proposed gossip protocols. For
regular graphs
such as the ring or grid, our algorithm improves standard gossip by
factors of
$n$ and $\sqrt{n}$ respectively. For the more challenging case of random
geometric graphs, our algorithm computes the true average to accuracy
$\epsilon$ using $O(\frac{n^{1.5}}{\sqrt{\log n}} \log \epsilon^{-1})
$ radio
transmissions, which yields a $\sqrt{\frac{n}{\log n}}$ factor
improvement over
standard gossip algorithms. We illustrate these theoretical results with
experimental comparisons between our algorithm and standard methods
as applied
to various classes of random fields.
http://arxiv.org/abs/0709.3921
---------------------------------------------------------------
6098. UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS
A. M. Davie
We consider a d-dimensional stochastic differential equation with
additive
noise and a drift coefficient which is assumed only to be a bounded
Borel
function. We show that, for almost all choices of the driving
Brownian path,
the equation has a unique solution.
http://arxiv.org/abs/0709.4147
---------------------------------------------------------------
6099. THRESHOLD PHENOMENA ON PRODUCT SPACES: BKKKL REVISITED (ONCE MORE)
Raphael Rossignol
We revisit the work of Bourgain, Kahn, Kalai, Katznelson and Linial
(1992) --
referred to as ``BKKKL'' in the title -- about influences on Boolean
functions
in order to give a precise statement of threshold phenomenon on the
product
space $\{1,..., r\}^\NN$, generalizing one of the main results of a
paper by
Talagrand (1994).
http://arxiv.org/abs/0709.4178
---------------------------------------------------------------
6100. SPHERICAL MODEL IN A RANDOM FIELD
A.E. Patrick
We investigate the properties of the Gibbs states and thermodynamic
observables of the spherical model in a random field. We show that on
the
low-temperature critical line the magnetization of the model is not a
self-averaging observable, but it self-averages conditionally. We
also show
that an arbitrarily weak homogeneous boundary field dominates over
fluctuations
of the random field once the model transits into a ferromagnetic
phase. As a
result, a homogeneous boundary field restores the conventional self-
averaging
of thermodynamic observables, like the magnetization and the
susceptibility. We
also investigate the effective field created at the sites of the
lattice by the
random field, and show that at the critical temperature of the
spherical model
the effective field undergoes a transition into a phase with long-range
correlations $\sim r^{4-d}$.
http://arxiv.org/abs/0709.3579
---------------------------------------------------------------
6101. WEAK CONVERGENCE OF VERVAAT AND VERVAAT ERROR PROCESSES OF LONG-
RANGE DEPENDENT SEQUENCES
Mikl\'os Cs\"org\Ho and Rafa{\l} Kulik
Following Cs\"{o}rg\H{o}, Szyszkowicz and Wang (Ann. Statist. {\bf 34},
(2006), 1013--1044) we consider a long range dependent linear
sequence. We
prove weak convergence of the uniform Vervaat and the uniform Vervaat
error
processes, extending their results to distributions with unbounded
support and
removing normality assumption.
http://arxiv.org/abs/0709.4285
---------------------------------------------------------------
6102. MODULATED BRANCHING PROCESSES, ORIGINS OF POWER LAWS AND
QUEUEING DUALITY
Predrag R. Jelenkovic and Jian Tan
Power law distributions have been repeatedly observed in a wide
variety of
socioeconomic, biological and technological areas. In the vast
majority of
these observations, e.g., city populations and sizes of living
organisms, the
objects of interest evolve due to the replication of their many
independent
components, e.g., births-deaths of individuals and replications of
cells.
Furthermore, the rates of replication of the many components are often
controlled by exogenous parameters causing periods of expansion and
contraction, e.g., baby booms and busts, economic booms and
recessions, etc. In
addition, the sizes of these objects often either have reflective lower
boundaries, e.g., cities do not fall bellow a certain size, low income
individuals are subsidized by the government, companies are protected by
bankruptcy laws, etc; or have porous/absorbing lower boundaries,
e.g., cities
may degenerate, bankruptcy protections may fail and companies can be
liquidated.
Hence, it is natural to propose reflected modulated branching
processes as
generic models for many of the preceding observations of power laws
that are
typically observed in proportional growth environments. Indeed, our main
results show that the proposed mathematical models result in power law
distributions under quite general polynomial Gartner-Ellis
conditions. The
generality of our results could explain the ubiquitous nature of
power law
distributions. Furthermore, an informal interpretation of our main
results
suggests that alternating periods of expansion and reduction, e.g.,
economic
booms and recessions, are primarily responsible for the appearance of
power law
distributions.
http://arxiv.org/abs/0709.4297
---------------------------------------------------------------
6103. NONCOMMUTATIVE BROWNIAN MOTIONS WITH KESTEN DISTRIBUTIONS AND
RELATED POISSON PROCESSES
Romuald Lenczewski and Rafal Salapata
We introduce and study a noncommutative two-parameter family of
noncommutative Brownian motions in the free Fock space. They are
associated
with Kesten laws and give a continuous interpolation between Brownian
motions
in free probability and monotone probability. The combinatorics of
our model is
based on ordered non-crossing partitions, in which to each such
partition $P$
we assign a weight depending on the numbers of disorders and orders
in $P$
related to the natural partial order on the set of blocks of $P$
implemented by
the relation of being inner or outer. In particular, we obtain a simple
relation between Delaney's numbers (related to inner blocks in non-
crossing
partitions) and generalized Euler's numbers (related to orders and
disorders in
ordered non-crossing partitions). An important feature of our
interpolation is
that the mixed moments of the corresponding creation and annihilation
processes
also reproduce their monotone and free counterparts, which does not
take place
in other interpolations. The same combinatorics is used to construct an
interpolation between free and monotone Poisson processes.
http://arxiv.org/abs/0709.4334
---------------------------------------------------------------
6104. A CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM
PORTFOLIO SELECTION
Hanqing Jin and Zuo Quan Xu and Xun Yu Zhou
A continuous-time financial portfolio selection model with expected
utility
maximization typically boils down to solving a (static) convex
stochastic
optimization problem in terms of the terminal wealth, with a budget
constraint.
In literature the latter is solved by assuming {\it a priori} that
the problem
is well-posed (i.e., the supremum value is finite) and a Lagrange
multiplier
exists (and as a consequence the optimal solution is attainable). In
this paper
it is first shown, via various counter-examples, neither of these two
assumptions needs to hold, and an optimal solution does not
necessarily exist.
These anomalies in turn have important interpretations in and impacts
on the
portfolio selection modeling and solutions. Relations among the non-
existence
of the Lagrange multiplier, the ill-posedness of the problem, and the
non-attainability of an optimal solution are then investigated. Finally,
explicit and easily verifiable conditions are derived which lead to
finding the
unique optimal solution.
http://arxiv.org/abs/0709.4467
---------------------------------------------------------------
6105. GENERALIZED DESCENTS AND NORMALITY
Miklos Bona
We use Janson's dependency criterion to prove that the distribution of
$d$-descents of permutations of length $n$ converge to a normal
distribution as
$n$ goes to infinity. We show that this remains true even if $d$ is
allowed to
grow with $n$, up to a certain degree.
http://arxiv.org/abs/0709.4483
---------------------------------------------------------------
6106. THE DIRICHLET MARKOV ENSEMBLE
Djalil Chafai (IMT and Upte)
We equip the polytope of $n\times n$ Markov matrices with the normalized
trace of the Lebesgue measure of $R^{n^2}$. This probability space
provides
random Markov matrices, with i.i.d. rows following the Dirichlet
distribution
of mean $(1/n,...,1/n)$. We show that if $M$ is such a random matrix,
then the
empirical spectral distribution of $nMM^\top$ tends as $n\to\infty$ to a
Marchenko-Pastur distribution. This phenomenon complements an already
known
result on the sub-dominant eigenvalue of certain random matrices with
independent rows, which suggests that the typical spectral gap of a
uniform
random Markov matrix is of order $1-1/\sqrt{n}$ when $n$ is large.
However,
some computer simulations reveal striking asymptotic spectral
properties of
such random matrices, still waiting for a rigorous mathematical
analysis. In
particular, we conjecture that the empirical distribution of the complex
spectrum of $\sqrt{n}M$ tends as $n\to\infty$ to the uniform
distribution on
the unit disc of the complex plane.
http://arxiv.org/abs/0709.4678
---------------------------------------------------------------
6107. HIGH-ORDER ACCURATE IMPLICIT METHODS FOR THE PRICING OF BARRIER
OPTIONS
J.C. Ndogmo and D. B. Ntwiga
This paper deals with a high-order accurate implicit finite-difference
approach to the pricing of barrier options. In this way various types of
barrier options are priced, including barrier options paying rebates,
and
options on dividend-paying-stocks. Moreover, the barriers may be
monitored
either continuously or discretely. In addition to the high-order
accuracy of
the scheme, and the stretching effect of the coordinate
transformation, the
main feature of this approach lies on a probability-based optimal
determination
of boundary conditions. This leads to much faster and accurate
results when
compared with similar pricing approaches. The strength of the present
scheme is
particularly demonstrated in the valuation of discretely monitored
barrier
options where it yields values closest to those obtained from the only
semi-analytical valuation method available.
http://arxiv.org/abs/0710.0069
---------------------------------------------------------------
6108. APPLICATIONS OF INTEGRAL TRANSFORMS IN FRACTIONAL DIFFUSION
PROCESSES
Francesco Mainardi
The fundamental solution (Green function) for the Cauchy problem of the
space-time fractional diffusion equation is investigated with respect
to its
scaling and similarity properties, starting from its Fourier-Laplace
representation. Then, by using the Mellin transform, a general
representation
of the Green function in terms of Mellin-Barnes integrals in the
complex plane
is derived. This allows us to obtain a suitable computational form of
the Green
function in the space-time domain and to analyse its probability
interpretation.
http://arxiv.org/abs/0710.0145
---------------------------------------------------------------
6109. SELFDECOMPOSABILITY AND SEMI-SELFDECOMPOSABILITY IN
SUBORDINATION OF CONE-PARAMETER CONVOLUTION SEMIGROUPS
Ken-iti Sato
Extension of two known facts concerning subordination is made. The
first fact
is that, in subordination of 1-dimensional Brownian motion with drift,
selfdecomposability is inherited from subordinator to subordinated.
This is
extended to subordination of cone-parameter convolution semigroups.
The second
fact is that, in subordination of strictly stable cone-parameter
convolution
semigroups on $\mathbb{R}^d$, selfdecomposability is inherited from
subordinator to subordinated. This is extended to semi-
selfdecomposability.
http://arxiv.org/abs/0710.0193
---------------------------------------------------------------
6110. BRANCHING DIFFUSIONS, SUPERDIFFUSIONS AND RANDOM MEDIA
J\'anos Engl\"ander
Spatial branching processes became increasingly popular in the past
decades,
not only because of their obvious connection to biology, but also
because
superprocesses are intimately related to nonlinear partial differential
equations. Another hot topic in today's research in probability
theory is
`random media', including the now classical problems on `Brownian
motion among
obstacles' and the more recent `random walks in random environment' and
`catalytic branching' models. These notes aim to give a gentle
introduction
into some topics in spatial branching processes and superprocesses in
deterministic environments (sections 2-6) and in random media
(sections 7-11).
http://arxiv.org/abs/0710.0236
---------------------------------------------------------------
6111. CENTRAL LIMITS AND HOMOGENIZATION IN RANDOM MEDIA
Guillaume Bal
We consider the perturbation of elliptic operators of the form $-
\Delta +
q_0$ by random, rapidly varying, sufficiently mixing, potentials of
the form
$q(\frac{\bx}\eps,\omega)$. We analyze the source and spectral problems
associated to such operators and show that the properly renormalized
difference
between the perturbed and unperturbed solutions may be written
asymptotically
as $\eps\to0$ as explicit Gaussian processes. Such results may be
seen as
central limit corrections to the homogenization (law of large
numbers) process.
Similar results are derived for more general elliptic equations in one
dimension of space. The results are based on the availability of a
rapidly
converging integral formulation for the perturbed solutions and on
the use of
classical central limit results for random processes with appropriate
mixing
conditions.
http://arxiv.org/abs/0710.0363
---------------------------------------------------------------
6112. CONVERGENCE OF SIMPLE RANDOM WALKS ON RANDOM DISCRETE TREES TO
BROWNIAN MOTION ON THE CONTINUUM RANDOM TREE
David Croydon
In this article it is shown that the Brownian motion on the continuum
random
tree is the scaling limit of the simple random walks on any family of
discrete
$n$-vertex ordered graph trees whose search-depth functions converge
to the
Brownian excursion as $n\to\infty$. We prove both a quenched version
(for
typical realisations of the trees) and an annealed version (averaged
over all
realisations of the trees) of our main result. The assumptions of the
article
cover the important example of simple random walks on the trees
generated by
the Galton-Watson branching process, conditioned on the total
population size.
http://arxiv.org/abs/0710.0460
---------------------------------------------------------------
6113. STEADY-STATE ANALYSIS OF A MULTI-SERVER QUEUE IN THE HALFIN-
WHITT REGIME
David Gamarnik and Petar Momcilovic
We examine a multi-server queue in the Halfin-Whitt (Quality- and
Efficiency-Driven) regime: as the number of servers $n$ increases, the
utilization approaches 1 from below at the rate $\Theta(1/\sqrt{n})$.
The
arrival process is renewal and service times have a lattice-valued
distribution
with a finite support. We consider the steady-state distribution of
the queue
length and waiting time in the limit as the number of servers $n$
increases
indefinitely. The queue length distribution, in the limit as $n\to
\infty$, is
characterized in terms of the stationary distribution of an explicitly
constructed Markov chain. As a consequence, the steady-state queue
length and
waiting time scale as $\Theta(\sqrt{n})$ and $\Theta(1/\sqrt{n})$ as
$n\to\infty$, respectively. Moreover, an explicit expression for the
critical
exponent is derived for the moment generating function of a limiting
(scaled)
steady-state queue length. This exponent depends on three parameters:
the
amount of spare capacity and the coefficients of variation of
interarrival and
service times. Interestingly, it matches an analogous exponent
corresponding to
a single-server queue in the conventional heavy-traffic regime. The
results are
derived by analyzing Lyapunov functions.
http://arxiv.org/abs/0710.0654
---------------------------------------------------------------
6114. DIFFERENTIAL EQUATIONS DRIVEN BY ROUGH PATHS: AN APPROACH VIA
DISCRETE APPROXIMATION
A. M. Davie
A theory of differential equations driven by a non-differentiable
path has
recently been developed by Lyons. We develop an alternative approach
to this
theory, using (modified Euler approximations), and investigate its
applicability to stochastic differential equations driven by Brownian
motion.
We also give some other examples showing that the main results are
reasonably
sharp.
http://arxiv.org/abs/0710.0772
---------------------------------------------------------------
6115. MONTE CARLO METHODS AND PATH-GENERATION TECHNIQUES FOR PRICING
MULTI-ASSET PATH-DEPENDENT OPTIONS
Piergiacomo Sabino (Dipartimento di Matematica and Universit\`a
degli Studi di Bari)
We consider the problem of pricing path-dependent options on a basket of
underlying assets using simulations. As an example we develop our
studies using
Asian options. Asian options are derivative contracts in which the
underlying
variable is the average price of given assets sampled over a period
of time.
Due to this structure, Asian options display a lower volatility and are
therefore cheaper than their standard European counterparts. This
paper is a
survey of some recent enhancements to improve efficiency when pricing
Asian
options by Monte Carlo simulation in the Black-Scholes model. We
analyze the
dynamics with constant and time-dependent volatilities of the
underlying asset
returns. We present a comparison between the precision of the
standard Monte
Carlo method (MC) and the stratified Latin Hypercube Sampling (LHS). In
particular, we discuss the use of low-discrepancy sequences, also
known as
Quasi-Monte Carlo method (QMC), and a randomized version of these
sequences,
known as Randomized Quasi Monte Carlo (RQMC). The latter has proven
to be a
useful variance reduction technique for both problems of up to 20
dimensions
and for very high dimensions. Moreover, we present and test a new path
generation approach based on a Kronecker product approximation (KPA)
in the
case of time-dependent volatilities. KPA proves to be a fast generation
technique and reduces the computational cost of the simulation
procedure.
http://arxiv.org/abs/0710.0850
---------------------------------------------------------------
6116. LECTURES ON TWO-DIMENSIONAL CRITICAL PERCOLATION
Wendelin Werner
This is the preliminary version of the notes corresponding to the course
given at the IAS-Park City graduate summer school in July 2007.
http://arxiv.org/abs/0710.0856
---------------------------------------------------------------
6117. DYNAMIC PROGRAMMING OPTIMIZATION OVER RANDOM DATA: THE SCALING
EXPONENT FOR NEAR-OPTIMAL SOLUTIONS
David J. Aldous and Charles Bordenave and Marc Lelarge
A very simple example of an algorithmic problem solvable by dynamic
programming is to maximize, over sets A in {1,2,...,n}, the objective
function
|A| - \sum_i \xi_i 1(i \in A,i+1 \in A) for given \xi_i > 0. This
problem, with
random (\xi_i), provides a test example for studying the relationship
between
optimal and near-optimal solutions of combinatorial optimization
problems. We
show that, amongst solutions differing from the optimal solution in a
small
proportion \delta of places, we can find near-optimal solutions whose
objective
function value differs from the optimum by a factor of order \delta^2
but not
smaller order. We conjecture this relationship holds widely in the
context of
dynamic programming over random data, and Monte Carlo simulations for
the
Kauffman-Levin NK model are consistent with the conjecture. This work
is a
technical contribution to a broad program initiated in Aldous-Percus
(2003) of
relating such scaling exponents to the algorithmic difficulty of
optimization
problems.
http://arxiv.org/abs/0710.0857
---------------------------------------------------------------
6118. A NEW TECHNIQUE FOR PROVING UNIQUENESS FOR MARTINGALE PROBLEMS
Richard F. Bass and Edwin A. Perkins
A new technique for proving uniqueness of martingale problems is
introduced.
The method is illustrated in the context of elliptic diffusions in
$R^d$.
http://arxiv.org/abs/0710.0860
---------------------------------------------------------------
6119. SUBADDITIVITY OF THE ENTROPY AND ITS RELATION TO BRASCAMP-LIEB
TYPE INEQUALITIES
Eric A. Carlen and Dario Cordero-Erausquin
We prove a general duality result showing that a Brascamp--Lieb type
inequality is equivalent to an inequality expressing subadditivity of
the
entropy, with a complete correspondence of best constants and cases of
equality. This open a new approach to the proof of Brascamp--Lieb type
inequalities, via subadditivity of the entropy. We illustrate the
utility of
this approach by proving a general inequality expressing the
subadditivity
property of the entropy on $\R^n$, and fully determining the cases of
equality.
As a consequence of the duality mentioned above, we obtain a simple
new proof
of the classical Brascamp--Lieb inequality, and also a fully explicit
determination of all of the cases of equality. We also deduce several
other
consequences of the general subadditivity inequality, including a
generalization of Hadamard's inequality for determinants. Finally, we
also
prove a second duality theorem relating superadditivity of the Fisher
information and a sharp convolution type inequality for the fundamental
eigenvalues of Schr\"odinger operators. Though we focus mainly on the
case of
random variables in $\R^n$ in this paper, we discuss extensions to other
settings as well.
http://arxiv.org/abs/0710.0870
---------------------------------------------------------------
6120. THE STARTING AND STOPPING PROBLEM UNDER KNIGHTIAN UNCERTAINTY
AND RELATED SYSTEMS OF REFLECTED BSDES
Said Hamadene and Jianfeng Zhang
This article deals with the starting and stopping problem under
Knightian
uncertainty, i.e., roughly speaking, when the probability under which
the
future evolves is not exactly known. We show that the lower price of
a plant
submitted to the decisions of starting and stopping is given by a
solution of a
system of two reflected backward stochastic differential equations
(BSDEs for
short). We solve this latter system and we give the expression of the
optimal
strategy. Further we consider a more general system of $m$ ($m\geq 2$)
reflected BSDEs with interconnected obstacles. Once more we show
existence and
uniqueness of the solution of that system.
http://arxiv.org/abs/0710.0908
---------------------------------------------------------------
6121. STABILIZABILITY AND PERCOLATION IN THE INFINITE VOLUME SANDPILE
MODEL
Anne Fey and Ronald Meester and Frank Redig
We study the sandpile model in infinite volume on $\Zd$. In
particular we are
interested in relations between the density $\rho$ of a stationary
measure
$\mu$ on initial configurations, and the question whether or not initial
configurations are stabilizable. We prove that stabilizability does
not depend
on the particular stabilizability rule we adopt. In $d=1$ and $\mu$ a
product
measure with $\rho=1$ (the known critical value for stabilizability
in $d=1$)
with a positive density of empty sites, we prove that $\mu$ is not
stabilizable.
Furthermore we study, for values of $\rho$ such that $\mu$ is
stabilizable,
percolation of toppled sites. We find that for $\rho>0$ small enough,
there is
a subcritical regime where the distribution of a cluster of toppled
sites has
an exponential tail, as is the case in the subcritical regime for
ordinary
percolation.
http://arxiv.org/abs/0710.0939
---------------------------------------------------------------
6122. SHARP ASYMPTOTICS FOR THE PARTITION FUNCTION OF SOME CONTINUOUS-
TIME DIRECTED POLYMERS
Agnese Cadel (IECN) and Samy Tindel (IECN) and Frederi Viens
This paper is concerned with two related types of directed polymers in a
random medium. The first one is a d-dimensional Brownian motion
living in a
random environment which is Brownian in time and homogeneous in
space. The
second is a continuous-time random walk on the lattice Z^d, in a random
environment with similar properties as in continuous space. The case
of a
space-time white noise environment can be acheived in this second
setting. By
means of some Gaussian tools, we estimate the free energy of these
models at
low temperature, and give some further information on the strong
disorder
regime of the objects under consideration.
http://arxiv.org/abs/0710.0942
---------------------------------------------------------------
6123. SEMANTIC DISTILLATION: A METHOD FOR CLUSTERING OBJECTS BY
THEIR CONTEXTUAL SPECIFICITY
Thomas Sierocinski (IRMAR) and Anthony Le B\'echec and Nathalie Th
\'eret and Dimitri Petritis (IRMAR)
Techniques for data-mining, latent semantic analysis, contextual
search of
databases, etc. have long ago been developed by computer scientists
working on
information retrieval (IR). Experimental scientists, from all
disciplines,
having to analyse large collections of raw experimental data
(astronomical,
physical, biological, etc.) have developed powerful methods for their
statistical analysis and for clustering, categorising, and
classifying objects.
Finally, physicists have developed a theory of quantum measurement,
unifying
the logical, algebraic, and probabilistic aspects of queries into a
single
formalism. The purpose of this paper is twofold: first to show that when
formulated at an abstract level, problems from IR, from statistical data
analysis, and from physical measurement theories are very similar and
hence can
profitably be cross-fertilised, and, secondly, to propose a novel
method of
fuzzy hierarchical clustering, termed \textit{semantic distillation} --
strongly inspired from the theory of quantum measurement --, we
developed to
analyse raw data coming from various types of experiments on DNA
arrays. We
illustrate the method by analysing DNA arrays experiments and
clustering the
genes of the array according to their specificity.
http://arxiv.org/abs/0710.1203
---------------------------------------------------------------
6124. ON FREELY INDECOMPOSABLE MEASURES
Hari Bercovici and Jiun-Chau Wang
We show that a probability measure is not a nontrivial free additive
convolution if it puts no mass in an interval whose endpoints are
atoms. The
analogous results for free multiplicative convolutions are proved as
well. The
proofs use analytic subordination.
http://arxiv.org/abs/0710.1295
---------------------------------------------------------------
6125. ON THE LIMITING EMPIRICAL MEASURE OF THE SUM OF RANK ONE
MATRICES WITH LOG-CONCAVE DISTRIBUTION
Alain Pajor and Leonid Pastur
We consider $n\times n$ real symmetric and hermitian random matrices
$H_{n,m}$ equals the sum of a non-random matrix $H_{n}^{(0)}$ matrix
and the
sum of $m$ rank-one matrices determined by $m$ i.i.d. isotropic
random vectors
with log-concave probability law and i.i.d. random amplitudes $\{\tau_
{\alpha
}\}_{\alpha =1}^{m}$. This is a generalization of the case of vectors
uniformly
distributed over the unit sphere, studied in [Marchenko-Pastur
(1967)]. We
prove that if $n\to \infty, m\to \infty, m/n\to c\in \lbrack 0,\infty)
$ and
that the empirical eigenvalue measure of $H_{n}^{(0)}$ converges
weakly, then
the empirical eigenvalue measure of $H_{n,m}$ converges in
probability to a
non-random limit, found in [Marchenko-Pastur (1967)].
http://arxiv.org/abs/0710.1346
---------------------------------------------------------------
6126. EXACT L_2-SMALL BALL ASYMPTOTICS OF GAUSSIAN PROCESSES AND THE
SPECTRUM OF BOUNDARY VALUE PROBLEMS WITH "NON-SEPARATED" BOUNDARY
CONDITIONS
A. I. Nazarov
We sharpen a classical result on the spectral asymptotics of the
boundary
value problems for self-adjoint ordinary differential operator. Using
this
result we obtain the exact $L_2$-small ball asymptotics for a new
class of zero
mean Gaussian processes. This class includes, in particular, integrated
generalized Slepian process, integrated centered Wiener process and
integrated
centered Brownian bridge.
http://arxiv.org/abs/0710.1408
---------------------------------------------------------------
6127. THE ACCURACY OF MERGING APPROXIMATION IN GENERALIZED ST.
PETERSBURG GAMES
Gyula Pap
Merging asymptotic expansions of arbitrary length are established for
the
distribution functions and for the probabilities of suitably centered
and
normalized cumulative winnings in a full sequence of generalized St.
Petersburg
games, extending the short expansions due to Cs\"org\H{o} [8]. These
expansions
are given in terms of suitably chosen members from the classes of
subsequential
semistable infinitely divisible asymptotic distribution functions and
certain
derivatives of these functions. Depending upon the tail parameter,
uniform or
nonuniform bounds are presented.
http://arxiv.org/abs/0710.1438
---------------------------------------------------------------
6128. A LARGE DEVIATION APPROACH TO OPTIMAL TRANSPORT
Christian L\'eonard (MODAL'x and Cmap)
A probabilistic method for solving the Monge-Kantorovich mass transport
problem on $R^d$ is introduced. A system of empirical measures of
independent
particles is built in such a way that it obeys a doubly indexed large
deviation
principle with an optimal transport cost as its rate function. As a
consequence, new approximation results for the optimal cost function
and the
optimal transport plans are derived. They follow from the Gamma-
convergence of
a sequence of normalized relative entropies toward the optimal
transport cost.
A wide class of cost functions including the standard power cost
functions
$|x-y|^p$ enter this framework.
http://arxiv.org/abs/0710.1461
---------------------------------------------------------------
6129. ASYMMETRY OF NEAR-CRITICAL PERCOLATION INTERFACES
Pierre Nolin (DMA and LM-Orsay) and Wendelin Werner (DMA and LM-
Orsay)
We study the possible scaling limits of percolation interfaces in two
dimensions on the triangular lattice. When one lets the percolation
parameter
p(N) vary with the size N of the box that one is considering, three
possibilities arise in the large-scale limit. It is known that when p
(N) does
not converge to 1/2 fast enough, then the scaling limits are degenerate,
whereas if p(N) - 1 / 2 goes to zero quickly, the scaling limits are
SLE(6) as
when p=1/2. We study some properties of the (non-void) intermediate
regime
where the large scale behavior is neither SLE(6) nor degenerate. We
prove that
in this case, the law of any scaling limit is singular with respect
to that of
SLE(6), even if it is still supported on the set of curves with
Hausdorff
dimension equal to 7/4.
http://arxiv.org/abs/0710.1470
---------------------------------------------------------------
6130. STRONG APPROXIMATIONS OF BSDES IN A DOMAIN
Bruno Bouchard (CEREMADE) and Stephane Menozzi (PMA)
We study the strong approximation of a Backward SDE with finite
stopping time
horizon, namely the first exit time of a forward SDE from a
cylindrical domain.
We use the Euler scheme approach of Bouchard and Touzi, Zhang 04}.
When the
domain is piecewise smooth and under a non-characteristic boundary
condition,
we show that the associated strong error is at most of order $h^
{\frac14-\eps}$
where $h$ denotes the time step and $\eps$ is any positive parameter.
This rate
corresponds to the strong exit time approximation. It is improved to
$h^{\frac12-\eps}$ when the exit time can be exactly simulated or for
a weaker
form of the approximation error. Importantly, these results are obtained
without uniform ellipticity condition.
http://arxiv.org/abs/0710.1519
---------------------------------------------------------------
6131. MULTICOLOR URN MODELS WITH REDUCIBLE REPLACEMENT MATRICES
Arup Bose and Amites Dasgupta and Krishanu Maulik
Consider the multicolored urn model where after every draw, balls of the
different colors are added to the urn in proportion determined by a
given
stochastic replacement matrix. We consider some special replacement
matrices
which are not irreducible. For three and four color urns, we derive the
asymptotic behavior of linear combinations of number of balls. In
particular,
we show that certain linear combinations of the balls of different
colors have
limiting distributions which are variance mixtures of normal
distributions. We
also obtain almost sure limits in certain cases in contrast to the
corresponding irreducible cases, where only weak limits are known.
http://arxiv.org/abs/0710.1520
---------------------------------------------------------------
6132. BALLISTIC TRANSPORT AT UNIFORM TEMPERATURE
Nawaf Bou-Rabee and Houman Owhadi
A paradigm for isothermal, mechanical rectification of stochastic
fluctuations is introduced in this paper. The central idea is to
transform
energy injected by random perturbations into rigid-body rotational
kinetic
energy. The prototype considered in this paper is a mechanical system
consisting of a set of rigid bodies in interaction through magnetic
fields. The
system is stochastically forced by white noise and dissipative through
mechanical friction. The Gibbs-Boltzmann distribution at a specific
temperature
defines the unique invariant measure under the flow of this
stochastic process
and allows us to define ``the temperature'' of the system. This
measure is also
ergodic and weakly mixing. Although the system does not exhibit
global directed
motion, it is shown that global ballistic motion is possible (the
mean-squared
displacement grows like t squared). More precisely, although work
cannot be
extracted from thermal energy by the second law of thermodynamics, it
is shown
that ballistic transport from thermal energy is possible. In
particular, the
dynamics is characterized by a meta-stable state in which the system
exhibits
directed motion over random time scales. This phenomenon is caused by
interaction of three attributes of the system: a non flat (yet bounded)
potential energy landscape, a rigid body effect (coupling translational
momentum and angular momentum through friction) and the degeneracy of
the
noise/friction tensor on the momentums (the fact that noise is not
applied to
all degrees of freedom).
http://arxiv.org/abs/0710.1565
---------------------------------------------------------------
6133. TRANSFORMATIONS OF MARKOV PROCESSES AND CLASSIFICATION SCHEME
FOR SOLVABLE DRIFTLESS DIFFUSIONS
Claudio Albanese and Alexey Kuznetsov
We propose a new classification scheme for diffusion processes for
which the
backward Kolmogorov equation is solvable in analytically closed form by
reduction to hypergeometric equations of the Gaussian or confluent
type. The
construction makes use of transformations of diffusion processes to
eliminate
the drift which combine a measure change given by Doob's h-transform
and a
diffeomorphism. Such transformations have the important property of
preserving
analytic solvability of the process: the transition probability
density for the
driftless process can be expressed through the transition probability
density
of original process. We also make use of tools from the theory of
ordinary
differential equations such as Liouville transformations, canonical
forms and
Bose invariants. Beside recognizing all analytically solvable
diffusion process
known in the previous literature fall into this scheme and we also
discover
rich new families of analytically solvable processes.
http://arxiv.org/abs/0710.1596
---------------------------------------------------------------
6134. LAPLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES
Claudio Albanese and Stephan Lawi
Laplace transforms for integrals of stochastic processes have been
known in
analytically closed form for just a handful of Markov processes:
namely, the
Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the
exponential of
Brownian motion. In virtue of their analytical tractability, these
processes
are extensively used in modelling applications. In this paper, we
construct
broad extensions of these process classes. We show how the known
models fit
into a classification scheme for diffusion processes for which Laplace
transforms for integrals of the diffusion processes and transitional
probability densities can be evaluated as integrals of hypergeometric
functions
against the spectral measure for certain self-adjoint operators. We
also extend
this scheme to a class of finite-state Markov processes related to
hypergeometric polynomials in the discrete series of the Askey
classification
tree.
http://arxiv.org/abs/0710.1599
---------------------------------------------------------------
6135. OPERATOR METHODS, ABELIAN PROCESSES AND DYNAMIC CONDITIONING
Claudio Albanese
A mathematical framework for Continuous Time Finance based on operator
algebraic methods offers a new direct and entirely constructive
perspective on
the field and leads to new numerical analysis techniques. This is
partly a
review paper as it covers and expands on the mathematical framework
underlying
a series of more applied articles. In addition, this article also
presents a
few key new theorems that make the treatment self-contained. Stochastic
processes with continuous time and continuous space variables are
defined
constructively by establishing new convergence estimates for Markov
chains on
simplicial sequences. We emphasize high precision computability by
numerical
linear algebra methods as opposed to the ability of arriving to
analytically
closed form expressions in terms of special functions. Path dependent
processes
adapted to a given Markov filtration are associated to an operator
algebra. If
this algebra is commutative, the corresponding process is named
Abelian, a
concept which provides a far reaching extension of the notion of
stochastic
integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-
Kac-Martin
theorem as a particular case of a broadly general block-diagonalization
algorithm. This technique has many applications ranging from the
problem of
pricing cliquets to target-redemption-notes and volatility derivatives.
Non-Abelian processes are also relevant and appear in several important
applications to for instance snowballs and soft calls. We show that
in these
cases one can effectively use block-factorization algorithms.
Finally, we
discuss the method of dynamic conditioning that allows one to
dynamically
correlate over possibly even hundreds of processes in a numerically
noiseless
framework while preserving marginal distributions.
http://arxiv.org/abs/0710.1606
---------------------------------------------------------------
6136. MINIMIZATION OF ENTROPY FUNCTIONALS
Christian L\'eonard (MODAL'x and Cmap)
Entropy functionals (i.e. convex integral functionals) and extensions of
these functionals are minimized on convex sets. This paper is aimed
at reducing
as much as possible the assumptions on the constraint set. Dual
equalities and
characterizations of the minimizers are obtained with weak constraint
qualifications.
http://arxiv.org/abs/0710.1462
---------------------------------------------------------------
6137. PERCOLATION IN THE SHERRINGTON-KIRKPATRICK SPIN GLASS
J. Machta and C.M. Newman and D.L. Stein
We present extended versions and give detailed proofs of results
concerning
percolation (using various sets of two-replica bond occupation
variables) in
Sherrington-Kirkpatrick spin glasses (with zero external field) that
were first
given in an earlier paper by the same authors. We also explain how
ultrametricity is manifested by the densities of large percolating
clusters.
Our main theorems concern the connection between these densities and
the usual
spin overlap distribution. Their corollaries are that the ordered
spin glass
phase is characterized by a unique percolating cluster of maximal
density
(normally coexisting with a second cluster of nonzero but lower
density). The
proofs involve comparison inequalities between SK multireplica bond
occupation
variables and the independent variables of standard Erdos-Renyi
random graphs.
http://arxiv.org/abs/0710.1399
---------------------------------------------------------------
6138. TRAVELING WAVES IN A ONE-DIMENSIONAL RANDOM MEDIUM
James Nolen and Lenya Ryzhik
We consider solutions of a scalar reaction-diffusion equation of the
ignition
type with a random, stationary and ergodic reaction rate. We show that
solutions of the Cauchy problem spread with a deterministic rate in
the long
time limit. We also establish existence of generalized random
traveling waves
and of transition fronts in general heterogeneous media.
http://arxiv.org/abs/0710.1858
---------------------------------------------------------------
6139. DYNAMICS OF MANDELBROT CASCADES
Julien Barral and Jacques Peyriere and Zhi-Ying Wen
Mandelbrot multiplicative cascades provide a construction of a dynamical
system on a set of probability measures defined by inequalities on
moments. To
be more specific, beyond the first iteration, the trajectories take
values in
the set of fixed points of smoothing transformations (i.e., some
generalized
stable laws).
Studying this system leads to a central limit theorem and to its
functional
version. The limit Gaussian process can also be obtained as limit of an
`additive cascade' of independent normal variables.
http://arxiv.org/abs/0710.1985
---------------------------------------------------------------
6140. A DUAL EIGENVECTOR CONDITION FOR STRONG LUMPABILITY OF MARKOV
CHAINS
Martin Nilsson Jacobi and Olof Goernerup
Necessary and sufficient conditions for identifying strong
lumpability in
Markov chains are presented. We show that the states in a lump
necessarily
correspond to identical elements in eigenvectors of the dual
transition matrix.
If there exist as many dual eigenvectors that respect the necessary
condition
as there are lumps in the aggregation, then the condition is also
sufficient.
The result is demonstrated with two simple examples.
http://arxiv.org/abs/0710.1986
---------------------------------------------------------------
6141. EXPONENTIAL STABILITY OF NON-AUTONOMOUS STOCHASTIC PARTIAL
DIFFERENTIAL EQUATIONS WITH FINITE MEMORY
Li Wan and Jinqiao Duan
The exponential stability, in both mean square and almost sure
senses, for
energy solutions to a class of nonlinear and non-autonomous
stochastic PDEs
with finite memory is investigated. Various criteria for stability are
obtained. An example is presented to demonstrate the main results.
http://arxiv.org/abs/0710.2082
---------------------------------------------------------------
6142. RANDOM WALK DELAYED ON PERCOLATION CLUSTERS
Francis Comets (PMA) and Francois Simenhaus (PMA)
We study a continuous time random walk on the $d$-dimensional lattice,
subject to a drift and an attraction to large clusters of a subcritical
Bernoulli site percolation. We find two distinct regimes: a ballistic
one, and
a subballistic one taking place when the attraction is strong enough. We
identify the speed in the former case, and the algebraic rate of
escape in the
latter case. Finally, we discuss the diffusive behavior in the case
of zero
drift and weak attraction.
http://arxiv.org/abs/0710.2320
---------------------------------------------------------------
6143. STRONG CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR FOR
FINITE MIXTURES OF LOCATION-SCALE DISTRIBUTIONS WHEN PENALTY IS
IMPOSED ON THE
RATIOS OF THE SCALE PARAMETERS
Kentaro Tanaka
In finite mixtures of location-scale distributions, if there is no
constraint
or penalty on the parameters, then the maximum likelihood estimator
does not
exist because the likelihood is unbounded. To avoid this problem, we
consider a
penalized likelihood, where the penalty is a function of the minimum
of the
ratios of the scale parameters and the sample size. It is shown that the
penalized maximum likelihood estimator is strongly consistent. We
also analyze
the consistency of a penalized maximum likelihood estimator where the
penalty
is imposed on the scale parameters themselves.
http://arxiv.org/abs/0710.2183
---------------------------------------------------------------
6144. VERTEX PERCOLATION ON EXPANDER GRAPHS
Sonny Ben-Shimon and Michael Krivelevich
We say that a graph $G=(V,E)$ on $n$ vertices is a $\beta$-expander
for some
constant $\beta>0$ if every $U\subseteq V$ of cardinality $|U|\leq
\frac{n}{2}$
satisfies $|N_G(U)|\geq \beta|U|$ where $N_G(U)$ denotes the
neighborhood of
$U$. We explore the process of uniformly at random deleting vertices
of a
$\beta$-expander with probability $n^{-\alpha}$ for some constant $
\alpha>0$.
Our main result implies that as $n$ tends to infinity, the deletion
process
performed on a $\beta$-expander graph of bounded degree will result
with high
probability in a graph composed of a giant component containing $n-o(n)$
vertices which is itself an expander graph, and small constant size
components.
We proceed by applying the main result to expander graphs with a
positive
spectral gap. In the particular case of $(n,d,\lambda)$-graphs, which
are such
expanders, we compute the values of $\alpha$, under additional
constraints on
the graph, for which with high probability the resulting graph will stay
connected, or will be composed of a giant component and isolated
vertices. As a
graph sampled from the uniform probability space of $d$-regular
graphs with
hight probability meets all of these constraints, this result
strengthens a
recent result due to Greenhill, Holt, and Wormald who prove a similar
theorem
for $\Gnd$. We conclude by showing that performing the deletion
process with
the prescribed deletion probability on expander graphs that expand
sub-linear
sets by an unbounded expansion ratio, with high probability results
in an
expander graph.
http://arxiv.org/abs/0710.2296
---------------------------------------------------------------
6145. A PECULIAR TWO POINT BOUNDARY VALUE PROBLEM
Huadong Pang and Daniel W. Stroock
In this paper we consider a one-dimensional diffusion equation on the
interval $[0,1]$ satisfying non-Feller boundary conditions. As a
consequence,
the initial value Cauchy problem fails to preserve nonnegativity or
boundedness. Nonetheless, probability theory plays an interesting
role in our
analysis and understanding of solutions to this equation.
http://arxiv.org/abs/0710.2396
---------------------------------------------------------------
6146. REGRESSION ESTIMATION FROM AN INDIVIDUAL STABLE SEQUENCE
Gusztav Morvai and Sanjeev R. Kulkarni and Andrew B. Nobel
We consider univariate regression estimation from an individual (non-
random)
sequence $(x_1,y_1),(x_2,y_2), ... \in \real \times \real$, which is
stable in
the sense that for each interval $A \subseteq \real$, (i) the
limiting relative
frequency of $A$ under $x_1, x_2, ...$ is governed by an unknown
probability
distribution $\mu$, and (ii) the limiting average of those $y_i$ with
$x_i \in
A$ is governed by an unknown regression function $m(\cdot)$.
A computationally simple scheme for estimating $m(\cdot)$ is
exhibited, and
is shown to be $L_2$ consistent for stable sequences $\{(x_i,y_i)\}$
such that
$\{y_i\}$ is bounded and there is a known upper bound for the
variation of
$m(\cdot)$ on intervals of the form $(-i,i]$, $i \geq 1$.
Complementing this
positive result, it is shown that there is no consistent estimation
scheme for
the family of stable sequences whose regression functions have finite
variation, even under the restriction that $x_i \in [0,1]$ and $y_i$ is
binary-valued.
http://arxiv.org/abs/0710.2496
---------------------------------------------------------------
6147. DENSITY ESTIMATION FROM AN INDIVIDUAL NUMERICAL SEQUENCE
Andrew B. Nobel and Gusztav Morvai and Sanjeev R. Kulkarni
This paper considers estimation of a univariate density from an
individual
numerical sequence. It is assumed that (i) the limiting relative
frequencies of
the numerical sequence are governed by an unknown density, and (ii)
there is a
known upper bound for the variation of the density on an increasing
sequence of
intervals. A simple estimation scheme is proposed, and is shown to be
$L_1$
consistent when (i) and (ii) apply. In addition it is shown that
there is no
consistent estimation scheme for the set of individual sequences
satisfying
only condition (i).
http://arxiv.org/abs/0710.2500
---------------------------------------------------------------
6148. STOCHASTIC INTEGRALS AND EVOLUTION EQUATIONS WITH GAUSSIAN
RANDOM FIELDS
S. V. Lototsky and K. Stemmann
The paper studies stochastic integration with respect to Gaussian
processes
and fields. It is more convenient to work with a field than a
process: by
definition, a field is a collection of stochastic integrals for a
class of
deterministic integrands. The problem is then to extend the
definition to
random integrands. An orthogonal decomposition of the chaos space of
the random
field, combined with the Wick product, leads to the \Ito-Skorokhod
integral,
and provides an efficient tool to study the integral, both
analytically and
numerically. For a Gaussian process, a natural definition of the
integral
follows from a canonical correspondence between random processes and
a special
class of random fields. Some examples of the corresponding stochastic
differential equations are also considered.
http://arxiv.org/abs/0710.2506
---------------------------------------------------------------
6149. LARGE DEVIATIONS AND ADIABATIC TRANSITIONS FOR DYNAMICAL
SYSTEMS AND MARKOV PROCESSES IN FULLY COUPLED AVERAGING
Yuri Kifer
The work treats systems combining slow and fast motions depending on
each
other where fast motions are perturbations of families of either
dynamical
systems or Markov processes with freezed slow variable. In the first
case we
consider hyperbolic dynamical systems and in the second case we deal
with
random evolutions which are combinations of diffusions and continuous
time
Markov chains. We study first large deviations of the slow motion
from the
averaged one and then use these results together with some Markov
property type
arguments in order to describe very long time behavior of the slow
motion such
as its transitions between attractors of the averaged system.
http://arxiv.org/abs/0710.2405
---------------------------------------------------------------
6150. A NOTE ON MEAN VOLUME AND SURFACE DENSITIES FOR A CLASS OF
BIRTH-AND-GROWTH STOCHASTIC PROCESSES
Elena Villa
Many real phenomena may be modelled as locally finite unions of
$d$-dimensional time dependent random closed sets in $\mathbb{R}^d$,
described
by birth-and-growth stochastic processes, so that their mean volume
and surface
densities, as well as the so called mean extended volume and surface
densities,
may be studied in terms of relevant quantities characterizing the
process. We
extend here known results in the Poissonian case to a wider class of
birth-and-growth processes.
http://arxiv.org/abs/0710.2751
---------------------------------------------------------------
6151. REDUCED BRANCHING PROCESSES WITH VERY HEAVY TAILS
Andreas N. Lager{\aa}s and Serik Sagitov
The reduced Markov branching process is a stochastic model for the
genealogy
of an unstructured biological population. Its limit behavior in the
critical
case is well studied for the Zolotarev-Slack regularity parameter
$\alpha\in(0,1]$. We turn to the case of very heavy tailed reproduction
distribution $\alpha=0$ assuming Zubkov's regularity condition with
parameter
$\beta\in(0,\infty)$. Our main result gives a new asymptotic pattern
for the
reduced branching process conditioned on non-extinction during a long
time
interval.
http://arxiv.org/abs/0710.2755
---------------------------------------------------------------
6152. THE FUNDAMENTAL THEOREM OF ASSET PRICING UNDER PROPORTIONAL
TRANSACTION COSTS
Alet Roux
We extend the fundamental theorem of asset pricing to a model where
the risky
stock is subject to proportional transaction costs in the form of bid-
ask
spreads and the bank account has different interest rates for
borrowing and
lending. We show that such a model is free of arbitrage if and only
if one can
embed in it a friction-free model that is itself free of arbitrage,
in the
sense that there exists an artificial friction-free price for the
stock between
its bid and ask prices and an artificial interest rate between the
borrowing
and lending interest rates such that, if one discounts this stock
price by this
interest rate, then the resulting process is a martingale under some
non-degenerate probability measure. Restricting ourselves to the
simple case of
a finite number of time steps and a finite number of possible
outcomes for the
stock price, the proof follows by combining classical arguments based on
finite-dimensional separation theorems with duality results from linear
optimisation.
http://arxiv.org/abs/0710.2758
---------------------------------------------------------------
6153. DAM RAIN AND CUMULATIVE GAIN
Dorje C. Brody and Lane P. Hughston and Andrea Macrina
We consider a financial contract that delivers a single cash flow
given by
the terminal value of a cumulative gains process. The problem of
modelling and
pricing such an asset and associated derivatives is important, for
example, in
the determination of optimal insurance claims reserve policies, and
in the
pricing of reinsurance contracts. In the insurance setting, the
aggregate
claims play the role of the cumulative gains, and the terminal cash flow
represents the totality of the claims payable for the given
accounting period.
A similar example arises when we consider the accumulation of losses
in a
credit portfolio, and value a contract that pays an amount equal to the
totality of the losses over a given time interval. An explicit
expression for
the value process is obtained. The price of an Arrow-Debreu security
on the
cumulative gains process is determined, and is used to obtain a
closed-form
expression for the price of a European-style option on the value of
the asset.
The results obtained make use of various remarkable properties of the
gamma
bridge process, and are applicable to a wide variety of financial
products
based on cumulative gains processes such as aggregate claims, credit
portfolio
losses, defined-benefit pension schemes, emissions, and rainfall.
http://arxiv.org/abs/0710.2775
---------------------------------------------------------------
6154. MARKET COMPLETION USING OPTIONS
Mark Davis and Jan Obloj
Mathematical models for financial asset prices which include, for
example,
stochastic volatility or jumps are incomplete in that derivative
securities are
generally not replicable by trading in the underlying. In earlier
work (2004)
the first author provided a geometric condition under which trading
in the
underlying and a finite number of vanilla options completes the
market. We
complement this result in several ways. First, we show that the
geometric
condition is not necessary and a weaker, necessary and sufficient,
condition is
presented. While this condition is generally not directly verifiable,
we show
that it simplifies to matrix non-degeneracy in a single point when the
coefficients are real analytic functions. In particular, any stochastic
volatility model is then completed with an arbitrary European type
option.
Further, we show that adding path-dependent options such as a
variance swap to
the set of primary assets, instead of plain vanilla options, also
completes the
market.
http://arxiv.org/abs/0710.2792
---------------------------------------------------------------
6155. A FEYNMAN-KAC-TYPE FORMULA FOR THE DETERMINISTIC AND STOCHASTIC
WAVE EQUATIONS
Robert C. Dalang and Carl Mueller and and Roger Tribe
We establish a probabilistic representation for a wide class of linear
deterministic p.d.e.s with potential term, including the wave
equation in
spatial dimensions 1 to 3. Our representation applies to the heat
equation,
where it is related to the classical Feynman-Kac formula, as well as
to the
telegraph and beam equations. If the potential is a (random) spatially
homogeneous Gaussian noise, then this formula leads to an expression
for the
moments of the solution.
http://arxiv.org/abs/0710.2861
---------------------------------------------------------------
6156. INFORMATION, INFLATION, AND INTEREST
Lane P. Hughston and Andrea Macrina
We propose a class of discrete-time stochastic models for the pricing of
inflation-linked assets. The paper begins with an axiomatic scheme
for asset
pricing and interest rate theory in a discrete-time setting. The
first axiom
introduces a "risk-free" asset, and the second axiom determines the
intertemporal pricing relations that hold for dividend-paying assets.
The
nominal and real pricing kernels, in terms of which the price index
can be
expressed, are then modelled by introducing a Sidrauski-type utility
function
depending on (a) the aggregate rate of consumption, and (b) the
aggregate rate
of real liquidity benefit conferred by the money supply. Consumption
and money
supply policies are chosen such that the expected joint utility
obtained over a
specified time horizon is maximised subject to a budget constraint
that takes
into account the "value" of the liquidity benefit associated with the
money
supply. For any choice of the bivariate utility function, the
resulting model
determines a relation between the rate of consumption, the price
level, and the
money supply. The model also produces explicit expressions for the
real and
nominal pricing kernels, and hence establishes a basis for the
valuation of
inflation-linked securities.
http://arxiv.org/abs/0710.2876
---------------------------------------------------------------
6157. MULTIVARIATE INTEGRATION OF FUNCTIONS DEPENDING EXPLICITLY ON
THE MINIMUM AND THE MAXIMUM OF THE VARIABLES
Jean-Luc Marichal
By using some basic calculus of multiple integration, we provide an
alternative expression of the integral $$ \int_{]a,b[^n} f(\mathbf{x},
\min
x_i,\max x_i) d\mathbf{x}, $$ in which the minimum and the maximum
are replaced
with two single variables. We demonstrate the usefulness of that
expression in
the computation of orness and andness average values of certain
aggregation
functions. By generalizing our result to Riemann-Stieltjes integrals,
we also
provide a method for the calculation of certain expected values and
distribution functions.
http://arxiv.org/abs/0710.2614
---------------------------------------------------------------
6158. AN EXPLICIT FORMULA FOR THE SKOROKHOD MAP ON $[0,A]$
Lukasz Kruk and John Lehoczky and Kavita Ramanan and Steven Shreve
The Skorokhod map is a convenient tool for constructing solutions to
stochastic differential equations with reflecting boundary
conditions. In this
work, an explicit formula for the Skorokhod map $\Gamma_{0,a}$ on
$[0,a]$ for
any $a>0$ is derived. Specifically, it is shown that on the space
$\mathcal{D}[0,\infty)$ of right-continuous functions with left
limits taking
values in $\mathbb{R}$, $\Gamma_{0,a}=\Lambda_a\circ \Gamma_0$, where
$\Lambda_a:\mathcal{D}[0,\infty)\to\mathcal{D}[0,\infty)$ is defined by
\[\Lambda_a(\phi)(t)=\phi(t)-\sup_{s\in[0,t]}\biggl[\bigl(\
phi(s)-a\bigr)^+\wedge\inf_{u\in[s,t]}\phi(u)\biggr]\] and
$\Gamma_0:\mathcal{D}[0,\infty)\to\mathcal{D}[0,\infty)$ is the
Skorokhod map
on $[0,\infty)$, which is given explicitly by
\[\Gamma_0(\psi)(t)=\psi(t)+\sup_{s\in[0,t]}[-\psi(s)]^+.\] In addition,
properties of $\Lambda_a$ are developed and comparison properties of
$\Gamma_{0,a}$ are established.
http://arxiv.org/abs/0710.2977
---------------------------------------------------------------
6159. MOMENT METHODS FOR EXOTIC VOLATILITY DERIVATIVES
Claudio Albanese and Adel Osseiran
The latest generation of volatility derivatives goes beyond variance and
volatility swaps and probes our ability to price realized variance
and sojourn
times along bridges for the underlying stock price process. In this
paper, we
give an operator algebraic treatment of this problem based on Dyson
expansions
and moment methods and discuss applications to exotic volatility
derivatives.
The methods are quite flexible and allow for a specification of the
underlying
process which is semi-parametric or even non-parametric, including
state-dependent local volatility, jumps, stochastic volatility and
regime
switching. We find that volatility derivatives are particularly well
suited to
be treated with moment methods, whereby one extrapolates the
distribution of
the relevant path functionals on the basis of a few moments. We
consider a
number of exotics such as variance knockouts, conditional corridor
variance
swaps, gamma swaps and variance swaptions and give valuation formulas in
detail.
http://arxiv.org/abs/0710.2991
---------------------------------------------------------------
6160. WEAK CONVERGENCE OF MEASURE-VALUED PROCESSES AND $R$-POINT
FUNCTIONS
Mark Holmes and Edwin Perkins
We prove a sufficient set of conditions for a sequence of finite
measures on
the space of cadlag measure-valued paths to converge to the canonical
measure
of super-Brownian motion in the sense of convergence of finite-
dimensional
distributions. The conditions are convergence of the Fourier
transform of the
$r$-point functions and perhaps convergence of the ``survival
probabilities.''
These conditions have recently been shown to hold for a variety of
statistical
mechanical models, including critical oriented percolation, the critical
contact process and lattice trees at criticality, all above their
respective
critical dimensions.
http://arxiv.org/abs/0710.2998
---------------------------------------------------------------
6161. BALLISTIC PHASE OF SELF-INTERACTING RANDOM WALKS
Dmitry Ioffe and Yvan Velenik
We explain a unified approach to a study of ballistic phase for a large
family of self-interacting random walks with a drift and self-
interacting
polymers with an external stretching force. The approach is based on
a recent
version of the Ornstein-Zernike theory developed in earlier works. It
leads to
local limit results for various observables (e.g. displacement of the
end-point
or number of hits of a fixed finite pattern) on paths of n-step walks
(polymers) on all possible deviation scales from CLT to LD. The class of
models, which display ballistic phase in the "universality class"
discussed in
the paper, includes self-avoiding walks, Domb-Joyce model, random
walks in an
annealed random potential, reinforced polymers and weakly reinforced
random
walks.
http://arxiv.org/abs/0710.3095
---------------------------------------------------------------
6162. POISSON CONVERGENCE FOR THE LARGEST EIGENVALUES OF HEAVY TAILED
RANDOM MATRICES
A. Auffinger and G. Ben Arous and S. Peche
We study the statistics of the largest eigenvalues of real symmetric and
sample covariance matrices when the entries are heavy tailed.
Extending the
result obtained by Soshnikov in \cite{Sos1}, we prove that, in the
absence of
the fourth moment, the top eigenvalues behave, in the limit, as the
largest
entries of the matrix.
http://arxiv.org/abs/0710.3132
---------------------------------------------------------------
6163. ON GRADIENT BOUNDS FOR THE HEAT KERNEL ON THE HEISENBERG GROUP
Dominique Bakry (IMT) and Fabrice Baudoin (IMT) and Michel
Bonnefont (IMT) and Djalil Chafai (IMT)
It is known that the couple formed by the two dimensional Brownian
motion and
its L\'evy area leads to the heat kernel on the Heisenberg group,
which is one
of the simplest sub-Riemannian space. The associated diffusion
operator is
hypoelliptic but not elliptic, which makes difficult the derivation of
functional inequalities for the heat kernel. However, Driver and
Melcher and
more recently H.-Q. Li have obtained useful gradient bounds for the
heat kernel
on the Heisenberg group. We provide in this paper simple proofs of these
bounds, and explore their consequences in terms of functional
inequalities,
including Cheeger and Bobkov type isoperimetric inequalities for the
heat
kernel.
http://arxiv.org/abs/0710.3139
---------------------------------------------------------------
6164. $L^1$ BOUNDS IN NORMAL APPROXIMATION
Larry Goldstein
The zero bias distribution $W^*$ of $W$, defined though the
characterizing
equation $\mathit{EW}f(W)=\sigma^2Ef'(W^*)$ for all smooth functions
$f$,
exists for all $W$ with mean zero and finite variance $\sigma^2$. For
$W$ and
$W^*$ defined on the same probability space, the $L^1$ distance
between $F$,
the distribution function of $W$ with $\mathit{EW}=0$ and $Var(W)=1$,
and the
cumulative standard normal $\Phi$ has the simple upper bound \[\Vert
F-\Phi\Vert_1\le2E|W^*-W|.\] This inequality is used to provide
explicit $L^1$
bounds with moderate-sized constants for independent sums,
projections of cone
measure on the sphere $S(\ell_n^p)$, simple random sampling and
combinatorial
central limit theorems.
http://arxiv.org/abs/0710.3262
---------------------------------------------------------------
6165. DIFFERENTIAL EQUATION APPROXIMATIONS FOR MARKOV CHAINS
R.W.R. Darling and J.R. Norris
We formulate some simple conditions under which a Markov chain may be
approximated by the solution to a differential equation, with
quantifiable
error probabilities. The role of a choice of coordinate functions for
the
Markov chain is emphasised. The general theory is illustrated in three
examples: the classical stochastic epidemic, a population process
model with
fast and slow variables, and core-finding algorithms for large random
hypergraphs.
http://arxiv.org/abs/0710.3269
---------------------------------------------------------------
6166. ONE MORE APPROACH TO THE CONVERGENCE OF THE EMPIRICAL PROCESS
TO THE BROWNIAN BRIDGE
Jean-Fran\c{c}ois Marckert (LaBRI)
A theorem of Donsker asserts that the empirical process converges in
distribution to the Brownian bridge. The aim of this paper is to
provide a new
and simple proof of this fact.
http://arxiv.org/abs/0710.3296
---------------------------------------------------------------
6167. TRANSIENT RANDOM WALKS IN RANDOM ENVIRONMENT ON A GALTON-WATSON
TREE
Elie Aidekon (PMA)
We consider a transient random walk $(X_n)$ in random environment on a
Galton--Watson tree. Under fairly general assumptions, we give a
sharp and
explicit criterion for the asymptotic speed to be positive. As a
consequence,
situations with zero speed are revealed to occur. In such cases, we
prove that
$X_n$ is of order of magnitude $n^{\Lambda}$, with $\Lambda \in (0,1)
$. We also
show that the linearly edge reinforced random walk on a regular tree
always has
a positive asymptotic speed, which improves a recent result of
Collevecchio
\cite{Col06}.
http://arxiv.org/abs/0710.3377
---------------------------------------------------------------
6168. DUALITY, VECTOR ADVECTION AND THE NAVIER-STOKES EQUATIONS
Z. Brzezniak and M. Neklyudov
In this article we show that the three dimensional vector advection
equation
is self dual in certain sense defined below. As a consequence, we
infer the
classical result of Serrin on the existence of a strong solution to
the three
dimensional Navier-Stokes equations. Also we deduce a Feynman-Kac
type formula
for solutions of the vector advection equation and show that the
formula is not
unique i.e. there exist flows which are different from the standard
flow, along
which the vorticity is conserved.
http://arxiv.org/abs/0710.3401
---------------------------------------------------------------
6169. MODERATE DEVIATIONS FOR POISSON-DIRICHLET DISTRIBUTION
Shui Feng and Fuqing Gao
Poisson-Dirichlet distribution arises in many different areas. The
parameter
$\theta$ in the distribution is the scaled mutation rate of a
population in the
context of population genetics. The limiting procedure of $\theta$
approaching
infinity is practically motivated and has led to new interesting
mathematical
structures. Results of law of large numbers, fluctuation theorems and
large
deviations have been successfully established. In this paper moderate
deviation
principles are established for Poisson-Dirichlet distribution, GEM
distribution, the homozygosity, and Dirichlet process when parameter $
\theta$
approaches infinity. These results, combined with earlier work, not only
provide a relatively complete picture of the asymptotic behavior of
Poisson-Dirichlet distribution for large $\theta$ but also lead to a
better
understanding of the large deviation problem associated with the scaled
homozygosity. They also reveal some new structures that are not
observed in
existing results of large deviations.
http://arxiv.org/abs/0710.3419
---------------------------------------------------------------
6170. ON AN EXTREME TWO-POINT DISTRIBUTION
V. I. Chebotarev and A. S. Kondrik and K.V. Mikhaylov
A bound for functional $\Delta(F)=\sup_{x\in\mathbb R}|F(x)-\Phi(x)|$ is
obtained, which is uniform for all distribution functions $F$ of random
variables with zero mean-value and unity variance. Moreover, a two-point
distribution is found, for which this bound is reached.
http://arxiv.org/abs/0710.3456
---------------------------------------------------------------
6171. SMALL VALUE PROBABILITIES VIA THE BRANCHING TREE HEURISTIC
Peter Morters (University of Bath) and Marcel Ortgiese (University
of Bath)
In the first part of this paper we give easy and intuitive proofs for
the
small value probabilities of the martingale limit of a supercritical
Galton-Watson process in both the Schr\"oder and the B\"ottcher case.
These
results are well-known, but the most cited proofs rely on generating
function
arguments which are hard to transfer to other settings. In the second
part we
show that the strategy underlying our proofs can be used in the quite
different
context of self-intersections of stochastic processes. Solving a
problem posed
by Wenbo Li, we find the small value probabilities for intersection
local times
of several Brownian motions, as well as for self-intersection local
times of a
single Brownian motion.
http://arxiv.org/abs/0710.3493
---------------------------------------------------------------
6172. CONFORMAL INVARIANCE FOR CERTAIN MODELS OF THE BOND-TRIANGULAR
TYPE
I. Binder and L. Chayes and H. K. Lei
Convergence to SLE_6 of the percolation exploration process for a
correlated
bond--triangular type model studied in [5] is established, which puts
the said
model in the same universality class as the standard site percolation
model on
the triangular lattice [12]. The result is proven for all domains
with boundary
(upper) Minkowski dimension less than 2, following the general
streamlined
approach outlined in [11].
http://arxiv.org/abs/0710.3446
---------------------------------------------------------------
6173. DISTRIBUTIONAL LIMITS FOR THE SYMMETRIC EXCLUSION PROCESS
Thomas M. Liggett
Strong negative dependence properties have recently been proved for the
symmetric exclusion process. In this paper, we apply these results to
prove
convergence to the Poisson and normal distributions for various
functionals of
the process.
http://arxiv.org/abs/0710.3606
---------------------------------------------------------------
6174. A DYNAMIC LOOK-AHEAD MONTE CARLO ALGORITHM FOR PRICING BERMUDAN
OPTIONS
Daniel Egloff and Michael Kohler and Nebojsa Todorovic
Under the assumption of no-arbitrage, the pricing of American and
Bermudan
options can be casted into optimal stopping problems. We propose a
new adaptive
simulation based algorithm for the numerical solution of optimal
stopping
problems in discrete time. Our approach is to recursively compute the
so-called
continuation values. They are defined as regression functions of the
cash flow,
which would occur over a series of subsequent time periods, if the
approximated
optimal exercise strategy is applied. We use nonparametric least squares
regression estimates to approximate the continuation values from a
set of
sample paths which we simulate from the underlying stochastic
process. The
parameters of the regression estimates and the regression problems
are chosen
in a data-dependent manner. We present results concerning the
consistency and
rate of convergence of the new algorithm. Finally, we illustrate its
performance by pricing high-dimensional Bermudan basket options with
strangle-spread payoff based on the average of the underlying assets.
http://arxiv.org/abs/0710.3640
---------------------------------------------------------------
6175. WEAK CONVERGENCE OF METROPOLIS ALGORITHMS FOR NON-I.I.D.
TARGET DISTRIBUTIONS
Myl\`ene B\'edard
In this paper, we shall optimize the efficiency of Metropolis
algorithms for
multidimensional target distributions with scaling terms possibly
depending on
the dimension. We propose a method for determining the appropriate
form for the
scaling of the proposal distribution as a function of the dimension,
which
leads to the proof of an asymptotic diffusion theorem. We show that
when there
does not exist any component with a scaling term significantly
smaller than the
others, the asymptotically optimal acceptance rate is the well-known
0.234.
http://arxiv.org/abs/0710.3684
---------------------------------------------------------------
6176. ON INVARIANT MEASURES OF STOCHASTIC RECURSIONS IN A CRITICAL CASE
Dariusz Buraczewski
We consider an autoregressive model on $\mathbb{R}$ defined by the
recurrence
equation $X_n=A_nX_{n-1}+B_n$, where $\{(B_n,A_n)\}$ are i.i.d. random
variables valued in $\mathbb{R}\times\mathbb{R}^+$ and $\mathbb {E}[\log
A_1]=0$ (critical case). It was proved by Babillot, Bougerol and Elie
that
there exists a unique invariant Radon measure of the process $\{X_n\}
$. The aim
of the paper is to investigate its behavior at infinity. We describe
also
stationary measures of two other stochastic recursions, including one
arising
in queuing theory.
http://arxiv.org/abs/0710.3687
---------------------------------------------------------------
6177. COUPLING A BRANCHING PROCESS TO AN INFINITE DIMENSIONAL
EPIDEMIC PROCESS
A. D. Barbour
Branching process approximation to the initial stages of an epidemic
process
has been used since the 1950's as a technique for providing stochastic
counterparts to deterministic epidemic threshold theorems. One way of
describing the approximation is to construct both branching and epidemic
processes on the same probability space, in such a way that their paths
coincide for as long as possible. In this paper, it is shown, in the
context of
a Markovian model of parasitic infection, that coincidence can be
achieved with
asymptotically high probability until o(N^{2/3}) infections have
occurred,
where N denotes the total number of hosts.
http://arxiv.org/abs/0710.3697
---------------------------------------------------------------
6178. INFERRING THE CONDITIONAL MEAN
Gusztav Morvai and Benjamin Weiss
Consider a stationary real-valued time series $\{X_n\}_{n=0}^{\infty}
$ with a
priori unknown distribution. The goal is to estimate the conditional
expectation $E(X_{n+1}|X_0,..., X_n)$ based on the observations $
(X_0,...,
X_n)$ in a pointwise consistent way. It is well known that this is
not possible
at all values of $n$. We will estimate it along stopping times.
http://arxiv.org/abs/0710.3757
---------------------------------------------------------------
6179. GUESSING THE OUTPUT OF A STATIONARY BINARY TIME SERIES
Gusztav Morvai
The forward prediction problem for a binary time series
$\{X_n\}_{n=0}^{\infty}$ is to estimate the probability that $X_{n+1}
=1$ based
on the observations $X_i$, $0\le i\le n$ without prior knowledge of the
distribution of the process $\{X_n\}$. It is known that this is not
possible if
one estimates at all values of $n$. We present a simple procedure
which will
attempt to make such a prediction infinitely often at carefully selected
stopping times chosen by the algorithm. The growth rate of the
stopping times
is also exhibited.
http://arxiv.org/abs/0710.3760
---------------------------------------------------------------
6180. LIMITATIONS ON INTERMITTENT FORECASTING
Gusztav Morvai and Benjamin Weiss
Bailey showed that the general pointwise forecasting for stationary and
ergodic time series has a negative solution. However, it is known
that for
Markov chains the problem can be solved. Morvai showed that there is
a stopping
time sequence $\{\lambda_n\}$ such that
$P(X_{\lambda_n+1}=1|X_0,...,X_{\lambda_n}) $ can be estimated from
samples
$(X_0,...,X_{\lambda_n})$ such that the difference between the
conditional
probability and the estimate vanishes along these stoppping times for
all
stationary and ergodic binary time series. We will show it is not
possible to
estimate the above conditional probability along a stopping time
sequence for
all stationary and ergodic binary time series in a pointwise sense
such that if
the time series turns out to be a Markov chain, the predictor will
predict
eventually for all $n$.
http://arxiv.org/abs/0710.3773
---------------------------------------------------------------
6181. ON CLASSIFYING PROCESSES
Gusztav Morvai and Benjamin Weiss
We prove several results concerning classifications, based on successive
observations $(X_1,..., X_n)$ of an unknown stationary and ergodic
process, for
membership in a given class of processes, such as the class of all
finite order
Markov chains.
http://arxiv.org/abs/0710.3775
---------------------------------------------------------------
6182. APPROXIMATING CRITICAL PARAMETERS OF BRANCHING RANDOM WALKS
Daniela Bertacchi and Fabio Zucca
Given a branching random walk on a graph, we consider two kinds of
truncations: by inhibiting the reproduction outside a subset of
vertices and by
allowing at most $m$ particles per site. We investigate the
convergence of weak
and strong critical parameters of these truncated branching random
walks to the
analogous parameters of the original branching random walk. As a
corollary, we
apply our results to the study of the strong critical parameter of a
branching
random walk restricted to the cluster of a Bernoulli bond percolation.
http://arxiv.org/abs/0710.3792
---------------------------------------------------------------
6183. THE FRACTIONAL STOCHASTIC HEAT EQUATION ON THE CIRCLE: TIME
REGULARITY AND POTENTIAL THEORY
Eulalia Nualart and Frederi Viens
We consider a system of $d$ linear stochastic heat equations driven
by an
additive infinite-dimensional fractional Brownian noise on the unit
circle
$S^1$. We obtain sharp results on the H\"older continuity in time of
the paths
of the solution $u=\{u(t, x)\}_{t \in \mathbb{R}_+, x \in S^1}$. We then
establish upper and lower bounds on hitting probabilities of $u$, in
terms of
respectively Hausdorff measure and Newtonian capacity.
http://arxiv.org/abs/0710.3952
---------------------------------------------------------------
6184. THE T COPULA WITH MULTIPLE PARAMETERS OF DEGREES OF FREEDOM:
BIVARIATE CHARACTERISTICS AND APPLICATION TO RISK MANAGEMENT
Xiaolin Luo and Pavel V. Shevchenko
The t copula is often used in risk management as it allows for
modelling tail
dependence between risks and it is simple to simulate and calibrate.
However
the use of a standard t copula is often criticized due to its
restriction of
having a single parameter for the degrees of freedom (dof) that may
limit its
capability to model the tail dependence structure in a multivariate
case. To
overcome this problem, grouped t copula was proposed recently where
risks are
grouped a priori in such a way that each group has a standard t
copula with its
specific dof parameter. In this paper we propose the use of a new t
copula,
generalizing grouped t copula to have each group consisting of one
risk only,
so that a priori grouping is not required. The characteristics of
this copula
in the bivariate case are described. We explain simulation and
calibration
procedures and provide examples.
http://arxiv.org/abs/0710.3959
---------------------------------------------------------------
6185. ON THE CAPACITY ACHIEVING COVARIANCE MATRIX FOR RICIAN MIMO
CHANNELS: AN ASYMPTOTIC APPROACH
Julien Dumont (IGM-LabInfo) and W. Hachem (LTCI) and Samson
Lasaulce (LSS) and Philippe Loubaton (IGM-LabInfo), Jamal Najim (LTCI)
The capacity-achieving input covariance matrices for coherent block-
fading
correlated MIMO Rician channels are determined. In this case, no
closed-form
expressions for the eigenvectors of the optimum input covariance
matrix are
available. An approximation of the average mutual information is
evaluated in
this paper in the asymptotic regime where the number of transmit and
receive
antennas converge to $+\infty$. New results related to the accuracy
of the
corresponding large system approximation are provided. An attractive
optimization algorithm of this approximation is proposed and we
establish that
it yields an effective way to compute the capacity achieving
covariance matrix
for the average mutual information. Finally, numerical simulation
results show
that, even for a moderate number of transmit and receive antennas,
the new
approach provides the same results as direct maximization approaches
of the
average mutual information, while being much more computationally
attractive.
http://arxiv.org/abs/0710.4051
---------------------------------------------------------------
6186. CASH SUB-ADDITIVE RISK MEASURES AND INTEREST RATE AMBIGUITY
Nicole El Karoui and Claudia Ravanelli
A new class of risk measures called cash sub-additive risk measures is
introduced to assess the risk of future financial, nonfinancial and
insurance
positions. The debated cash additive axiom is relaxed into the cash sub
additive axiom to preserve the original difference between the
numeraire of the
current reserve amounts and future positions. Consequently, cash sub-
additive
risk measures can model stochastic and/or ambiguous interest rates or
defaultable contingent claims. Practical examples are presented and
in such
contexts cash additive risk measures cannot be used. Several
representations of
the cash sub-additive risk measures are provided. The new risk
measures are
characterized by penalty functions defined on a set of sub-linear
probability
measures and can be represented using penalty functions associated
with cash
additive risk measures defined on some extended spaces. The issue of the
optimal risk transfer is studied in the new framework using inf-
convolution
techniques. Examples of dynamic cash sub-additive risk measures are
provided
via BSDEs where the generator can locally depend on the level of the
cash
sub-additive risk measure.
http://arxiv.org/abs/0710.4106
---------------------------------------------------------------
6187. A DETERMINISTIC APPROACH TO WIRELESS RELAY NETWORKS
A. S. Avestimehr and S. N. Diggavi and D. N. C. Tse
We present a deterministic channel model which captures several key
features
of multiuser wireless communication. We consider a model for a
wireless network
with nodes connected by such deterministic channels, and present an
exact
characterization of the end-to-end capacity when there is a single
source and a
single destination and an arbitrary number of relay nodes. This
result is a
natural generalization of the max-flow min-cut theorem for wireline
networks.
Finally to demonstrate the connections between deterministic model
and Gaussian
model, we look at two examples: the single-relay channel and the diamond
network. We show that in each of these two examples, the capacity-
achieving
scheme in the corresponding deterministic model naturally suggests a
scheme in
the Gaussian model that is within 1 bit and 2 bit respectively from
cut-set
upper bound, for all values of the channel gains. This is the first
part of a
two-part paper; the sequel [1] will focus on the proof of the max-
flow min-cut
theorem of a class of deterministic networks of which our model is a
special
case.
http://arxiv.org/abs/0710.3777
---------------------------------------------------------------
6188. WIRELESS NETWORK INFORMATION FLOW
A. S. Avestimehr and S. N. Diggavi and D. N. C. Tse
We present an achievable rate for general deterministic relay
networks, with
broadcasting at the transmitters and interference at the receivers. In
particular we show that if the optimizing distribution for the
information-theoretic cut-set bound is a product distribution, then
we have a
complete characterization of the achievable rates for such networks.
For linear
deterministic finite-field models discussed in a companion paper [3],
this is
indeed the case, and we have a generalization of the celebrated max-flow
min-cut theorem for such a network.
http://arxiv.org/abs/0710.3781
---------------------------------------------------------------
6189. MATURITY-INDEPENDENT RISK MEASURES
Thaleia Zariphopoulou and Gordan Zitkovic
The new notion of maturity-independent risk measures is introduced and
contrasted with the existing risk measurement concepts. It is shown,
by means
of two examples, one set on a finite probability space and the other
in a
diffusion framework, that, surprisingly, some of the widely utilized
risk
measures cannot be used to build maturity-independent counterparts. We
construct a large class of maturity-independent risk measures and give
representative examples in both continuous- and discrete-time
financial models.
http://arxiv.org/abs/0710.3892
---------------------------------------------------------------
6190. MESSAGE PASSING FOR THE COLORING PROBLEM: GALLAGER MEETS ALON
AND KAHALE
Sonny Ben-Shimon and Dan Vilenchik
Message passing algorithms are popular in many combinatorial
optimization
problems. For example, experimental results show that {\em survey
propagation}
(a certain message passing algorithm) is effective in finding proper
$k$-colorings of random graphs in the near-threshold regime. In 1962
Gallager
introduced the concept of Low Density Parity Check (LDPC) codes, and
suggested
a simple decoding algorithm based on message passing. In 1994 Alon
and Kahale
exhibited a coloring algorithm and proved its usefulness for finding a
$k$-coloring of graphs drawn from a certain planted-solution
distribution over
$k$-colorable graphs. In this work we show an interpretation of Alon and
Kahale's coloring algorithm in light of Gallager's decoding
algorithm, thus
showing a connection between the two problems - coloring and
decoding. This
also provides a rigorous evidence for the usefulness of the message
passing
paradigm for the graph coloring problem. Our techniques can be
applied to
several other combinatorial optimization problems and networking-related
issues.
http://arxiv.org/abs/0710.3928
---------------------------------------------------------------
6191. AREA LIMIT LAWS FOR SYMMETRY CLASSES OF STAIRCASE POLYGONS
Christoph Richard and Uwe Schwerdtfeger and Bhalchandra Thatte
We derive area limit laws for the various symmetry classes of staircase
polygons on the square lattice, in a uniform ensemble where, for fixed
perimeter, each polygon occurs with the same probability. This
complements a
previous study by Leroux and Rassart, where explicit expressions for
the area
and perimeter generating functions of these classes have been derived.
http://arxiv.org/abs/0710.4041
---------------------------------------------------------------
6192. ERGODICITY OF LANGEVIN PROCESSES WITH DEGENERATE DIFFUSION IN
MOMENTUMS
Nawaf Bou-Rabee and Houman Owhadi
This paper introduces a novel method for proving ergodicity of
degenerate
noise driven stochastic processes based on two key conditions: weak
irreducibility and closure under second randomization of the driving
noise. The
paper applies the method to prove ergodicity of a sliding disk
governed by
Langevin-type equations (a simple stochastic rigid body system). The
paper
shows that a key feature of this Langevin process is that even though
the
diffusion and drift matrices associated to the momentums are
degenerate, the
system is still at uniform temperature.
http://arxiv.org/abs/0710.4259
---------------------------------------------------------------
6193. INTEGRAL MEANS SPECTRUM OF RANDOM CONFORMAL SNOWFLAKES
D. Beliaev
In this paper we construct random conformal snowflakes with large
integral
means spectrum at different points. These new estimates are significant
improvement over previously known lower bound of the universal
spectrum. Our
estimates are within 5-10 percent from the conjectured value of the
universal
spectrum.
http://arxiv.org/abs/0710.4175
---------------------------------------------------------------
6194. DYNAMIC IMPORTANCE SAMPLING FOR QUEUEING NETWORKS
Paul Dupuis and Ali Devin Sezer and Hui Wang
Importance sampling is a technique that is commonly used to speed up
Monte
Carlo simulation of rare events. However, little is known regarding
the design
of efficient importance sampling algorithms in the context of queueing
networks. The standard approach, which simulates the system using an
a priori
fixed change of measure suggested by large deviation analysis, has
been shown
to fail in even the simplest network setting (e.g., a two-node tandem
network).
Exploiting connections between importance sampling, differential
games, and
classical subsolutions of the corresponding Isaacs equation, we show
how to
design and analyze simple and efficient dynamic importance sampling
schemes for
general classes of networks. The models used to illustrate the
approach include
$d$-node tandem Jackson networks and a two-node network with
feedback, and the
rare events studied are those of large queueing backlogs, including
total
population overflow and the overflow of individual buffers.
http://arxiv.org/abs/0710.4389
---------------------------------------------------------------
6195. KERNEL ESTIMATION OF GREEK WEIGHTS BY PARAMETER RANDOMIZATION
Romuald Elie and Jean-David Fermanian and Nizar Touzi
A Greek weight associated to a parameterized random variable $Z
(\lambda)$ is
a random variable $\pi$ such that
$\nabla_{\lambda}E[\phi(Z(\lambda))]=E[\phi(Z(\lambda))\pi]$ for any
function
$\phi$. The importance of the set of Greek weights for the purpose of
Monte
Carlo simulations has been highlighted in the recent literature. Our
main
concern in this paper is to devise methods which produce the optimal
weight,
which is well known to be given by the score, in a general context
where the
density of $Z(\lambda)$ is not explicitly known. To do this, we
randomize the
parameter $\lambda$ by introducing an a priori distribution, and we use
classical kernel estimation techniques in order to estimate the score
function.
By an integration by parts argument on the limit of this first kernel
estimator, we define an alternative simpler kernel-based estimator
which turns
out to be closely related to the partial gradient of the kernel-based
estimator
of $\mathbb{E}[\phi(Z(\lambda))]$. Similarly to the finite differences
technique, and unlike the so-called Malliavin method, our estimators are
biased, but their implementation does not require any advanced
mathematical
calculation. We provide an asymptotic analysis of the mean squared
error of
these estimators, as well as their asymptotic distributions. For a
discontinuous payoff function, the kernel estimator outperforms the
classical
finite differences one in terms of the asymptotic rate of
convergence. This
result is confirmed by our numerical experiments.
http://arxiv.org/abs/0710.4392
---------------------------------------------------------------
6196. MARKOVIAN PERTURBATION, RESPONSE AND FLUCTUATION DISSIPATION
THEOREM
Amir Dembo and Jean-Dominique Deuschel
We consider the Fluctuation Dissipation Theorem (FDT) of statistical
physics
from a mathematical perspective. We formalize the concept of ``linear
response
function'' in the general framework of Markov processes. We show that
for
processes out of equilibrium it depends not only on the given Markov
process
X(s) but also on the chosen perturbation of it. We characterize the
set of all
possible response functions for a given Markov process and show that at
equilibrium they all satisfy the FDT. That is, if the initial measure is
invariant for the given Markov semi-group, then for any pair of times
s<t and
nice functions f,g, the dissipation, that is, the derivative in s of the
covariance of g(X(t)) and f(X(s)) equals the infinitesimal response
at time t
and direction g to any Markovian perturbation that alters the
invariant measure
of X(.) in the direction of f at time s. The same applies in the so
called FDT
regime near equilibrium, i.e. in the limit s going to infinity with t-
s fixed,
provided X(s) converges in law to an invariant measure for its
dynamics. We
provide the response function of two generic Markovian perturbations
which we
then compare and contrast for pure jump processes on a discrete
space, for
finite dimensional diffusion processes, and for stochastic spin systems.
http://arxiv.org/abs/0710.4394
---------------------------------------------------------------
6197. ON THE DISCONNECTION OF A DISCRETE CYLINDER BY A BIASED RANDOM
WALK
David Windisch
We consider a random walk on the discrete cylinder (Z/NZ)^d x Z, d >=
3, with
drift N^{-d\alpha} in the Z-direction and investigate the large N-
behavior of
the disconnection time T^{disc}_N, defined as the first time when the
trajectory of the random walk disconnects the cylinder into two infinite
components. We prove that, as long as the drift exponent \alpha is
strictly
greater than 1, the asymptotic behavior of T^{disc}_N remains N^{2d+o
(1)}, as
in the unbiased case considered by Dembo and Sznitman, whereas for
\alpha < 1,
the asymptotic behavior of T^{disc}_N becomes exponential in N.
http://arxiv.org/abs/0710.4427
---------------------------------------------------------------
6198. CORRECTION TO "THE DIVERGENCE OF BANACH SPACE VALUED RANDOM
VARIABLES ON WIENER SPACE", PROB. TH. REL. FIELDS 132, 291-320 (2005)
E. Mayer-Wolf and M. Zakai
As a result of some mistakes discovered in the paper mentioned in the
title,
Corollaries 3.5 and 3.17a) are withdrawn and a new proof is provided for
Proposition 3.14, under the added assumption that the second dual of the
underlying Banach space Y possesses the Radon-Nykodim property.
http://arxiv.org/abs/0710.4483
---------------------------------------------------------------
6199. SOME INFORMATION-THEORETIC COMPUTATIONS RELATED TO THE
DISTRIBUTION OF PRIME NUMBERS
Ioannis Kontoyiannis
We illustrate how elementary information-theoretic ideas may be
employed to
provide proofs for well-known, nontrivial results in number theory.
Specifically, we give an elementary and fairly short proof of the
following
asymptotic result: The sum of (log p)/p, taken over all primes p not
exceeding
n, is asymptotic to log n as n tends to infinity. We also give finite-
n bounds
refining the above limit. This result, originally proved by Chebyshev
in 1852,
is closely related to the celebrated prime number theorem.
http://arxiv.org/abs/0710.4076
---------------------------------------------------------------
6200. FROM THE ENTROPY TO THE STATISTICAL STRUCTURE OF SPIKE TRAINS
Yun Gao and Ioannis Kontoyiannis and Elie Bienenstock
We use statistical estimates of the entropy rate of spike train data
in order
to make inferences about the underlying structure of the spike train
itself. We
first examine a number of different parametric and nonparametric
estimators
(some known and some new), including the ``plug-in'' method, several
versions
of Lempel-Ziv-based compression algorithms, a maximum likelihood
estimator
tailored to renewal processes, and the natural estimator derived from
the
Context-Tree Weighting method (CTW). The theoretical properties of these
estimators are examined, several new theoretical results are
developed, and all
estimators are systematically applied to various types of synthetic
data and
under different conditions.
Our main focus is on the performance of these entropy estimators
on the
(binary) spike trains of 28 neurons recorded simultaneously for a one-
hour
period from the primary motor and dorsal premotor cortices of a
monkey. We show
how the entropy estimates can be used to test for the existence of
long-term
structure in the data, and we construct a hypothesis test for whether
the
renewal process model is appropriate for these spike trains. Further, by
applying the CTW algorithm we derive the maximum a posterior (MAP)
tree model
of our empirical data, and comment on the underlying structure it
reveals.
http://arxiv.org/abs/0710.4117
---------------------------------------------------------------
6201. LIMIT THEOREMS FOR MAXIMUM FLOWS ON A LATTICE
Yu Zhang
We independently assign a non-negative value, as a capacity for the
quantity
of flows per unit time, with a distribution F to each edge on the Z^d
lattice.
We consider the maximum flows through the edges of two disjoint sets,
that is
from a source to a sink, in a large cube. In this paper, we show that
the ratio
of the maximum flow and the size of source is asymptotic to a
constant. This
constant is denoted by the flow constant.
http://arxiv.org/abs/0710.4589
---------------------------------------------------------------
6202. STRONG RECURRENCE FOR BRANCHING MARKOV CHAINS
Sebastian M\"uller
The question of recurrence and transience of branching Markov chains
is more
subtle than for ordinary Markov chains; they can be classified in
transience,
weak recurrence, and strong recurrence. We briefly summarize criteria
for
transience and weak recurrence and give several new conditions for weak
recurrence and strong recurrence. These conditions work well in concrete
examples and provide enough information to distinguish between weak
and strong
recurrence. This represents a step towards a general classification of
branching Markov chains. In particular, we show that in homogeneous
cases weak
recurrence and strong recurrence coincide. Furthermore, we discuss the
generalization of positive and null recurrence to branching Markov
chains.
http://arxiv.org/abs/0710.4651
---------------------------------------------------------------
6203. BAYESIAN SEQUENTIAL CHANGE DIAGNOSIS
Savas Dayanik and Christian Goulding and H. Vincent Poor
Sequential change diagnosis is the joint problem of detection and
identification of a sudden and unobservable change in the
distribution of a
random sequence. In this problem, the common probability law of a
sequence of
i.i.d. random variables suddenly changes at some disorder time to one of
finitely many alternatives. This disorder time marks the start of a
new regime,
whose fingerprint is the new law of observations. Both the disorder
time and
the identity of the new regime are unknown and unobservable. The
objective is
to detect the regime-change as soon as possible, and, at the same
time, to
determine its identity as accurately as possible. Prompt and correct
diagnosis
is crucial for quick execution of the most appropriate measures in
response to
the new regime, as in fault detection and isolation in industrial
processes,
and target detection and identification in national defense. The
problem is
formulated in a Bayesian framework. An optimal sequential decision
strategy is
found, and an accurate numerical scheme is described for its
implementation.
Geometrical properties of the optimal strategy are illustrated via
numerical
examples. The traditional problems of Bayesian change-detection and
Bayesian
sequential multi-hypothesis testing are solved as special cases. In
addition, a
solution is obtained for the problem of detection and identification of
component failure(s) in a system with suspended animation.
http://arxiv.org/abs/0710.4847
---------------------------------------------------------------
6204. FIRST TO MARKET IS NOT EVERYTHING: AN ANALYSIS OF PREFERENTIAL
ATTACHMENT WITH FITNESS
Christian Borgs and Jennifer Chayes and Constantinos Daskalakis
and Sebastien Roch
In this paper, we provide a rigorous analysis of preferential
attachment with
fitness, a random graph model introduced by Bianconi and Barabasi.
Depending on
the shape of the fitness distribution, we observe three distinct
phases: a
first-mover-advantage phase, a fit-get-richer phase and an innovation-
pays-off
phase.
http://arxiv.org/abs/0710.4982
---------------------------------------------------------------
6205. AGE-STRUCTURED TRAIT SUBSTITUTION SEQUENCE PROCESS AND
CANONICAL EQUATION
Sylvie M\'el\'eard (CMAP) and Viet Chi Tran (LPP)
We are interested in a stochastic model of trait and age-structured
population undergoing mutation and selection. We start with a
continuous time,
discrete individual-centered population process. Taking the large
population
and rare mutations limits under a well-chosen time-scale separation
condition,
we obtain a jump process that generalizes the Trait Substitution
Sequence
process describing Adaptive Dynamics for populations without age
structure.
Under the additional assumption of small mutations, we derive an age-
dependent
ordinary differential equation that extends the Canonical Equation.
These
evolutionary approximations have never been introduced to our
knowledge. They
are based on ecological phenomena represented by PDEs that generalize
the
Gurtin-McCamy equation in Demography. Another particularity is that they
involve a fitness function, describing the probability of invasion of
the
resident population by the mutant one, that can not always be computed
explicitly. Examples illustrate how adding an age-structure enrich the
modelling of structured population by including life history features
such as
senescence. In the cases considered, we establish the evolutionary
approximations and study their long time behavior and the nature of
their
evolutionary singularities when computation is tractable. Numerical
procedures
and simulations are carried.
http://arxiv.org/abs/0710.4997
---------------------------------------------------------------
6206. ON FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES
Terhi Kaarakka and Paavo Salminen
In this paper we study Doob's transform of fractional Brownian motion
(FBM).
It is well known that Doob's transform of standard Brownian motion is
identical
in law with the Ornstein-Uhlenbeck diffusion defined as the solution
of the
(stochastic) Langevin equation where the driving process is a
Brownian motion.
It is also known that Doob's transform of FBM and the process
obtained from the
Langevin equation with FBM as the driving process are different.
However, also
the first one of these can be described as a solution of a Langevin
equation
but now with some other driving process than FBM. We are mainly
interested in
the properties of this new driving process denoted Y^{(1)}. We also
study the
solution of the Langevin equation with Y^{(1)} as the driving process.
Moreover, we show that the covariance of Y^{(1)} grows linearly;
hence, in this
respect Y^{(1)} is more like a standard Brownian motion than a FBM.
In fact, it
is proved that a properly scaled version of Y^{(1)} converges weakly to
Brownian motion.
http://arxiv.org/abs/0710.5024
---------------------------------------------------------------
6207. FROM THE PR\'EKOPA-LEINDLER INEQUALITY TO MODIFIED LOGARITHMIC
SOBOLEV INEQUALITY
Ivan Gentil (CEREMADE)
We develop in this paper an improvement of the method given by S.
Bobkov and
M. Ledoux. Using the Pr\'ekopa-Leindler inequality, we prove a modified
logarithmic Sobolev inequality adapted for all measures on $\dR^n$,
with a
strictly convex and super-linear potential. This inequality implies
modified
logarithmic Sobolev inequality for all uniform strictly convex
potential as
well as the Euclidean logarithmic Sobolev inequality.
http://arxiv.org/abs/0710.5025
---------------------------------------------------------------
6208. FORECASTING FOR STATIONARY BINARY TIME SERIES
Gusztav Morvai and Benjamin Weiss
The forecasting problem for a stationary and ergodic binary time series
$\{X_n\}_{n=0}^{\infty}$ is to estimate the probability that $X_{n+1}
=1$ based
on the observations $X_i$, $0\le i\le n$ without prior knowledge of the
distribution of the process $\{X_n\}$. It is known that this is not
possible if
one estimates at all values of $n$. We present a simple procedure
which will
attempt to make such a prediction infinitely often at carefully selected
stopping times chosen by the algorithm. We show that the proposed
procedure is
consistent under certain conditions, and we estimate the growth rate
of the
stopping times.
http://arxiv.org/abs/0710.5144
---------------------------------------------------------------
6209. ERGODICITY AND HYDRODYNAMIC LIMITS FOR AN EPIDEMIC MODEL
Lamia Belhadji
We consider two approaches to study the spread of infectious diseases
within
a spatially structured population distributed in social clusters.
According
whether we consider only the population of infected individuals or both
populations of infected individuals and healthy ones, two models are
given to
study an epidemic phenomenon. Our first approach is at a microscopic
level, its
goal is to determine if an epidemic may occur for those models. The
second one
is the derivation of hydrodynamics limits. By using the relative
entropy method
we prove that the empirical measures of infected and healthy individuals
converge to a deterministic measure absolutely continuous with
respect to the
Lebesgue measure, whose density is the solution of a system of
reaction-diffusion equations.
http://arxiv.org/abs/0710.5185
---------------------------------------------------------------
6210. COEXISTENCE IN LOCALLY REGULATED COMPETING POPULATIONS AND
SURVIVAL OF BRANCHING ANNIHILATING RANDOM WALK
Jochen Blath and Alison Etheridge and Mark Meredith
We propose two models of the evolution of a pair of competing
populations.
Both are lattice based. The first is a compromise between fully
spatial models,
which do not appear amenable to analytic results, and interacting
particle
system models, which do not, at present, incorporate all of the
competitive
strategies that a population might adopt. The second is a
simplification of the
first, in which competition is only supposed to act within lattice
sites and
the total population size within each lattice point is a constant. In
a special
case, this second model is dual to a branching annihilating random
walk. For
each model, using a comparison with oriented percolation, we show
that for
certain parameter values, both populations will coexist for all time
with
positive probability. As a corollary, we deduce survival for all time of
branching annihilating random walk for sufficiently large branching
rates. We
also present a number of conjectures relating to the r\^{o}le of
space in the
survival probabilities for the two populations.
http://arxiv.org/abs/0710.5380
---------------------------------------------------------------
6211. STATIONARY DISTRIBUTION FOR DIOECIOUS BRANCHING PARTICLE
SYSTEMS WITH RAPID STIRRING
Feng Yu
We study dioecious (i.e., two-sex) branching particle system models,
where
there are two types of particles, modeling the male and female
populations, and
where birth of new particles requires the presence of both male and
female
particles. We show that stationary distributions of various dioecious
branching
particle models are nontrivial under certain conditions, for example,
when
there is sufficiently fast stirring.
http://arxiv.org/abs/0710.5404
---------------------------------------------------------------
6212. LIMIT THEOREMS FOR BIFURCATING MARKOV CHAINS. APPLICATION TO
THE DETECTION OF CELLULAR AGING
Julien Guyon
We propose a general method to study dependent data in a binary tree,
where
an individual in one generation gives rise to two different
offspring, one of
type 0 and one of type 1, in the next generation. For any specific
characteristic of these individuals, we assume that the
characteristic is
stochastic and depends on its ancestors' only through the mother's
characteristic. The dependency structure may be described by a
transition
probability $P(x,dy dz)$ which gives the probability that the pair of
daughters' characteristics is around $(y,z)$, given that the mother's
characteristic is $x$. Note that $y$, the characteristic of the
daughter of
type 0, and $z$, that of the daughter of type 1, may be conditionally
dependent
given $x$, and their respective conditional distributions may differ.
We then
speak of bifurcating Markov chains. We derive laws of large numbers
and central
limit theorems for such stochastic processes. We then apply these
results to
detect cellular aging in Escherichia Coli, using the data of Stewart
et al. and
a bifurcating autoregressive model.
http://arxiv.org/abs/0710.5434
---------------------------------------------------------------
6213. MILD SOLUTIONS FOR A CLASS OF FRACTIONAL SPDES AND THEIR SAMPLE
PATHS
Marta Sanz-Sol\'e and Pierre-A. Vuillermot
In this article we introduce and analyze a notion of mild solution for a
class of non-autonomous parabolic stochastic partial differential
equations
defined on a bounded open subset $D\subset\mathbb{R}^{d}$ and driven
by an
infinite-dimensional fractional noise. The noise is derived from an
$L^{2}(D)$-valued fractional Wiener process $W^{H}$ whose covariance
operator
satisfies appropriate restrictions; moreover, the Hurst parameter $H$ is
subjected to constraints formulated in terms of $d$ and the H\"{o}
lder exponent
of the derivative $h^\prime$ of the noise nonlinearity in the
equations. We
prove the existence of such solution, establish its relation with the
variational solution introduced in \cite{nuavu} and also prove the H
\"{o}lder
continuity of its sample paths when we consider it as an $L^{2}(D)$--
valued
stochastic processes. When $h$ is an affine function, we also prove
uniqueness.
The proofs are based on a relation between the notions of mild and
variational
solution established in Sanz-Sol\'e and Vuillermot 2003, and adapted
to our
problem, and on a fine analysis of the singularities of Green's function
associated with the class of parabolic problems we investigate. An
immediate
consequence of our results is the indistinguishability of mild and
variational
solutions in the case of uniqueness.
http://arxiv.org/abs/0710.5485
---------------------------------------------------------------
6214. ANOTHER LOOK AT AR(1)
Steven R. Finch
Given a stationary first-order autoregressive process X_t (with lag-one
correlation rho satisfying |rho|<1), we examine the Central Limit
Theorem for
(1/n)*ln |X_1...X_n| and compute variances to high precision. Given a
nonstationary process X_t (with |rho|>1), we examine instead (1/n)*ln|
X_n| and
study the distribution of ln|X_n|-n*ln|rho|.
http://arxiv.org/abs/0710.5419
---------------------------------------------------------------
6215. LARGE DEVIATIONS ASSOCIATED WITH POISSON--DIRICHLET
DISTRIBUTION AND EWENS SAMPLING FORMULA
Shui Feng
Several results of large deviations are obtained for distributions
that are
associated with the Poisson--Dirichlet distribution and the Ewens
sampling
formula when the parameter $\theta$ approaches infinity. The
motivation for
these results comes from a desire of understanding the exact meaning of
$\theta$ going to infinity. In terms of the law of large numbers and the
central limit theorem, the limiting procedure of $\theta$ going to
infinity in
a Poisson--Dirichlet distribution corresponds to a finite allele
model where
the mutation rate per individual is fixed and the number of alleles
going to
infinity. We call this the finite allele approximation. The first
main result
of this article is concerned with the relation between this finite
allele
approximation and the Poisson--Dirichlet distribution in terms of large
deviations. Large $\theta$ can also be viewed as a limiting procedure
of the
effective population size going to infinity. In the second result a
comparison
is done between the sample size and the effective population size
based on the
Ewens sampling formula.
http://arxiv.org/abs/0710.5577
---------------------------------------------------------------
6216. MINIMAL $F^Q$-MARTINGALE MEASURES FOR EXPONENTIAL L\'EVY PROCESSES
Monique Jeanblanc and Susanne Kl\"oppel and Yoshio Miyahara
Let $L$ be a multidimensional L\'evy process under $P$ in its own
filtration.
The $f^q$-minimal martingale measure $Q_q$ is defined as that
equivalent local
martingale measure for $\mathcal {E}(L)$ which minimizes the $f^q$-
divergence
$E[(dQ/dP)^q]$ for fixed $q\in(-\infty,0)\cup(1,\infty)$. We give
necessary and
sufficient conditions for the existence of $Q_q$ and an explicit
formula for
its density. For $q=2$, we relate the sufficient conditions to the
structure
condition and discuss when the former are also necessary. Moreover,
we show
that $Q_q$ converges for $q\searrow1$ in entropy to the minimal entropy
martingale measure.
http://arxiv.org/abs/0710.5594
---------------------------------------------------------------
6217. THE TWO-TYPE RICHARDSON MODEL WITH UNBOUNDED INITIAL
CONFIGURATIONS
Maria Deijfen and Olle H\"aggstr\"om
The two-type Richardson model describes the growth of two competing
infections on $\mathbb{Z}^d$ and the main question is whether both
infection
types can simultaneously grow to occupy infinite parts of $\mathbb{Z}
^d$. For
bounded initial configurations, this has been thoroughly studied. In
this
paper, an unbounded initial configuration consisting of points
$x=(x_1,...,x_d)$ in the hyperplane $\mathcal{H}=\{x\in\mathbb{Z}
^d:x_1=0\}$ is
considered. It is shown that, starting from a configuration where all
points in
$\mathcal{H} {\mathbf{0}\}$ are type 1 infected and the origin $
\mathbf{0}$ is
type 2 infected, there is a positive probability for the type 2
infection to
grow unboundedly if and only if it has a strictly larger intensity
than the
type 1 infection. If, instead, the initial type 1 infection is
restricted to
the negative $x_1$-axis, it is shown that the type 2 infection at the
origin
can also grow unboundedly when the infection types have the same
intensity.
http://arxiv.org/abs/0710.5602
---------------------------------------------------------------
6218. CENTRAL AND NON-CENTRAL LIMIT THEOREMS FOR WEIGHTED POWER
VARIATIONS OF FRACTIONAL BROWNIAN MOTION
Ivan Nourdin (PMA) and David Nualart and Ciprian Tudor (CES and
SAMOS)
In this paper, we prove some central and non-central limit theorems for
renormalized weighted power variations of order q>1 of the fractional
Brownian
motion with Hurst parameter H in (0,1), where q is an integer. The
central
limit holds for 1/(2q)<H<= 1-1/(2q), the limit being a conditionally
Gaussian
distribution. If H<1/(2q), we show the convergence in L^2 to a limit
which only
depends on the fractional Brownian motion, and if H> 1-1/(2q), we
show the
limit in L^2 to a stochastic integral with respect to the Hermite
process of
order q.
http://arxiv.org/abs/0710.5639
---------------------------------------------------------------
6219. RANDOM WALK ON A SURFACE GROUP: BOUNDARY BEHAVIOR OF THE
GREEN'S FUNCTION AT THE SPECTRAL RADIUS
Steven P. Lalley
It is proved that the Green's function of the simple random walk on a
surface
group of large genus decays exponentially at the spectral radius. It
is also
shown that Ancona's inequalities extend to the spectral radius R, and
therefore
that the Martin boundary for R-potentials coincides with the natural
geometric
boundary S^1.
http://arxiv.org/abs/0710.5745
---------------------------------------------------------------
6220. THE RANDOM CASE OF CONLEY'S THEOREM: II. THE COMPLETE LYAPUNOV
FUNCTION
Zhenxin Liu
Conley in \cite{Con} constructed a complete Lyapunov function for a
flow on
compact metric space which is constant on orbits in the chain
recurrent set and
is strictly decreasing on orbits outside the chain recurrent set. This
indicates that the dynamical complexity focuses on the chain
recurrent set and
the dynamical behavior outside the chain recurrent set is quite
simple. In this
paper, a similar result is obtained for random dynamical systems
under the
assumption that the base space $(\Omega,\mathcal F,\mathbb P)$ is a
separable
metric space endowed with a probability measure. By constructing a
complete
Lyapunov function, which is constant on orbits in the random chain
recurrent
set and is strictly decreasing on orbits outside the random chain
recurrent
set, the random case of Conley's fundamental theorem of dynamical
systems is
obtained. Furthermore, this result for random dynamical systems is
generalized
to noncompact state spaces.
http://arxiv.org/abs/0710.5683
---------------------------------------------------------------
6221. THE RANDOM CASE OF CONLEY'S THEOREM: III. RANDOM SEMIFLOW CASE
AND MORSE DECOMPOSITION
Zhenxin Liu
In the first part of this paper, we generalize the results of the author
\cite{Liu,Liu2} from the random flow case to the random semiflow
case, i.e. we
obtain Conley decomposition theorem for infinite dimensional random
dynamical
systems. In the second part, by introducing the backward orbit for
random
semiflow, we are able to decompose invariant random compact set (e.g.
global
random attractor) into random Morse sets and connecting orbits
between them,
which generalizes the Morse decomposition of invariant sets
originated from
Conley \cite{Con} to the random semiflow setting and gives the
positive answer
to an open problem put forward by Caraballo and Langa \cite{CL}.
http://arxiv.org/abs/0710.5687
---------------------------------------------------------------
6222. ON A RANDOM RECURSION RELATED TO ABSORPTION TIMES OF DEATH
MARKOV CHAINS
Alex Iksanov and Martin M\"ohle
Let $X_1,X_2,...$ be a sequence of random variables satisfying the
distributional recursion $X_1=0$ and $X_n= X_{n-I_n}+1$ for $n=2,3,...
$, where
$I_n$ is a random variable with values in $\{1,...,n-1\}$ which is
independent
of $X_2,...,X_{n-1}$. The random variable $X_n$ can be interpreted as
the
absorption time of a suitable death Markov chain with state space $
{\mathbb
N}:=\{1,2,...\}$ and absorbing state 1, conditioned that the chain
starts in
the initial state $n$.
This paper focuses on the asymptotics of $X_n$ as $n$ tends to
infinity under
the particular but important assumption that the distribution of $I_n$
satisfies ${\mathbb P}\{I_n=k\}=p_k/(p_1+...+p_{n-1})$ for some given
probability distribution $p_k={\mathbb P}\{\xi=k\}$, $k\in{\mathbb N}$.
Depending on the tail behaviour of the distribution of $\xi$, several
scalings
for $X_n$ and corresponding limiting distributions come into play,
among them
stable distributions and distributions of exponential integrals of
subordinators. The methods used in this paper are mainly
probabilistic. The key
tool is a coupling technique which relates the distribution of $X_n$
to a
random walk, which explains, for example, the appearance of the
Mittag-Leffler
distribution in this context. The results are applied to describe the
asymptotics of the number of collisions for certain beta-coalescent
processes.
http://arxiv.org/abs/0710.5826
---------------------------------------------------------------
6223. LINGERING RANDOM WALKS IN RANDOM ENVIRONMENT ON A STRIP
Erwin Bolthausen and Ilya Goldsheid
We consider a recurrent random walk (RW) in random environment (RE) on a
strip. We prove that if the RE is i. i. d. and its distribution is not
supported by an algebraic subsurface in the space of parameters
defining the RE
then the RW exhibits the "(log t)-squared" asymptotic behaviour. The
exceptional algebraic subsurface is described by an explicit system of
algebraic equations.
One-dimensional walks with bounded jumps in a RE are treated as a
particular
case of the strip model. If the one dimensional RE is i. i. d., then our
approach leads to a complete and constructive classification of
possible types
of asymptotic behaviour of recurrent random walks. Namely, the RW
exhibits the
$(\log t)^{2}$ asymptotic behaviour if the distribution of the RE is not
supported by a hyperplane in the space of parameters which shall be
explicitly
described. And if the support of the RE belongs to this hyperplane
then the
corresponding RW is a martingale and its asymptotic behaviour is
governed by
the Central Limit Theorem.
http://arxiv.org/abs/0710.5854
---------------------------------------------------------------
6224. IMPLEMENTING QUASI-MONTE CARLO SIMULATIONS WITH LINEAR
TRANSFORMATIONS
Piergiacomo Sabino (Dipartimento di Matematica Universit\`a degli
Studi di Bari)
Pricing exotic multi-asset path-dependent options requires extensive
Monte
Carlo simulations. In the recent years the interest to the Quasi-
monte Carlo
technique has been renewed and several results have been proposed in
order to
improve its efficiency with the notion of effective dimension. To
this aim,
Imai and Tan introduced a general variance reduction technique in
order to
minimize the nominal dimension of the Monte Carlo method. Taking into
account
these advantages, we investigate this approach in detail in order to
make it
faster from the computational point of view. Indeed, we realize the
linear
transformation decomposition relying on a fast ad hoc QR
decomposition that
considerably reduces the computational burden. This setting makes the
linear
transformation method even more convenient from the computational
point of
view. We implement a high-dimensional (2500) Quasi-Monte Carlo
simulation
combined with the linear transformation in order to price Asian
basket options
with same set of parameters published by Imai and Tan. For the
simulation of
the high-dimensional random sample, we use a 50-dimensional scrambled
Sobol
sequence for the first 50 components, determined by the linear
transformation
method, and pad the remaining ones out by the Latin Hypercube
Sampling. The aim
of this numerical setting is to investigate the accuracy of the
estimation by
giving a higher convergence rate only to those components selected by
the
linear transformation technique. We launch our simulation experiment
also using
the standard Cholesky and the principal component decomposition
methods with
pseudo-random and Latin Hypercube sampling generators. Finally, we
compare our
results and computational times, with those presented in Imai and Tan.
http://arxiv.org/abs/0710.5872
---------------------------------------------------------------
6225. FREE BESSEL LAWS
Teodor Banica and Serban Belinschi and Mireille Capitaine and
Benoit Collins
We introduce and study a remarkable family of real probability measures
$\pi_{st}$, that we call free Bessel laws. These are related to the free
Poisson law $\pi$ via the formulae $\pi_{s1}=\pi^{\boxtimes s}$ and
$\pi_{1t}=\pi^{\boxplus t}$. Our study includes: definition and basic
properties, analytic aspects (supports, atoms, densities), combinatorial
aspects (functional transforms, moments, partitions), and a
discussion of the
relation with random matrices and quantum groups.
http://arxiv.org/abs/0710.5931
---------------------------------------------------------------
6226. SOME ASPECTS OF EXTREME VALUE THEORY UNDER SERIAL DEPENDENCE
Holger Drees
On the occasion of Laurens de Haan's 70th birthday, we discuss two
aspects of
the statistical inference on the extreme value behavior of time
series with a
particular emphasis on his important contributions. First, the
performance of a
direct marginal tail analysis is compared with that of a model-based
approach
using an analysis of residuals. Second, the importance of the
extremal index as
a measure of the serial extremal dependence is discussed by the
example of
solutions of a stochastic recurrence equation.
http://arxiv.org/abs/0710.5879
---------------------------------------------------------------
6227. EXPONENTIAL INEQUALITIES FOR EMPIRICAL UNBOUNDED CONTEXT TREES
Antonio Galves and Florencia G. Leonardi
In this paper we obtain exponential bounds for the rate of
convergence of a
version of the algorithm Context, when the underlying tree is not
necessarily
bounded. The algorithm Context is a well-known tool to estimate the
context
tree of a Variable Length Markov Chain. As a consequence of the
exponential
bounds we obtain a strong consistency result. We generalize in this
way several
previous results in the field.
http://arxiv.org/abs/0710.5900
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