[PAS] Probability Abstracts 105
Probability Abstract Service
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Tue Sep 2 02:10:40 CDT 2008
Probability Abstracts 105
This document contains abstracts 7236-7441
from July-1-2008 to August-31-2008.
They have been mailed on Sept 2nd, 2008.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_105.shtml
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7236. GAUSSIAN FIELDS AND GAUSSIAN SHEETS WITH GENERALIZED CAUCHY
COVARIANCE STRUCTURE
S. C.Lim and L. P. Teo
Two types of Gaussian processes, namely the Gaussian field with
generalized
Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy
covariance (GSGCC) are considered. Some of the basic properties and the
asymptotic properties of the spectral densities of these random fields
are
studied. The associated self-similar random fields obtained by
applying the
Lamperti transformation to GFGCC and GSGCC are studied.
http://arxiv.org/abs/0807.0022
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7237. DISCONTINUOUS SUPERPROCESSES WITH DEPENDENT SPATIAL MOTION
Hui He
We construct a class of discontinuous superprocesses with dependent
spatial
motion and general branching mechanism. The process arises as the weak
limit of
critical interacting-branching particle systems where the spatial
motions of
the particles are not independent. The main work is to solve the
martingale
problem. When we turn to the uniqueness of the process, we generalize
the
localization method introduced by [D.W. Stroock, Diffusion processes
associated
with Levy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete,
32(1975)
209--244] to the measure-valued context. As for existence, we use
particle
system approximation and a perturbation method. This work generalizes
the model
introduced in [D.A. Dawson, Z. Li, H. Wang, Superprocesses with
dependent
spatial motion and general branching densities, Electron. J. Probab.
6(2001),
no.25, 33 pp. (electronic)] where quadratic branching mechanism was
considered.
We also investigate some properties of the process.
http://arxiv.org/abs/0807.0054
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7238. CONCENTRATION INEQUALITIES FOR $S$-CONCAVE MEASURES OF DILATIONS
OF BOREL SETS AND APPLICATIONS
Matthieu Fradelizi
We prove a sharp inequality conjectured by Bobkov on the measure of
dilations
of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our
result gives
a common generalization of an inequality of Nazarov, Sodin and Volberg
and a
concentration inequality of Gu\'edon. Applying our inequality to the
level sets
of functions satisfying a Remez type inequality, we deduce, as it is
classical,
that these functions enjoy dimension free distribution inequalities and
Kahane-Khintchine type inequalities with positive and negative
exponent, with
respect to an arbitrary $s$-concave probability.
http://arxiv.org/abs/0807.0080
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7239. RANDOM SYSTEMS OF POLYNOMIAL EQUATIONS. THE EXPECTED NUMBER OF
ROOTS UNDER SMOOTH ANALYSIS
Diego Armentano and Mario Wschebor
We consider random systems of equations over the reals, with $m$
equations
and $m$ unknowns $ P_i(t)+X_i(t)=0$, $t \in \R^m$, $i=1,...,m$, where
the
$P_i's$ are non-random polynomials having degrees $d_i's$ (the
"signal") and
the $X_i's$ (the "noise") are independent real-valued Gaussian
centered random
polynomial fields defined on $\R^m$, with a probability law satisfying
some
invariance properties.
For each $i$, $P_i$ and $X_i$ have degree $d_i$.
The problem is the behavior of the number of roots for large $m$.
We prove
that under specified conditions on the relation signal over noise,
which imply
that in a certain sense this relation is neither too large nor too
small, it
follows that the quotient between the expected value of the number of
roots of
the perturbed system and the expected value corresponding to the
centered
system (i.e. $P_i$ identically zero for all $i=1,...,m$), tends to zero
geometrically fast as $m$ tends to infinity. In particular, this means
that the
behavior of this expected value, is governed by the noise part.
http://arxiv.org/abs/0807.0262
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7240. COUNTERPARTY RISK VALUATION FOR CDS
Christophette Blanchet-Scalliet (ICJ) and Fr\'ed\'eric Patras (JAD)
The valuation of counterparty risk for single name credit derivatives
requires the computa- tion of joint distributions of default times of
two
default-prone entities. For a Merton-type model, we derive some
formulas for
these joint distribu- tions. As an application, closed formulas for
counterparty risk on a CDS or for a first-to-default swap on two
underlyings
are obtained.
http://arxiv.org/abs/0807.0309
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7241. OPTIMAL CONSUMPTION POLICIES IN ILLIQUID MARKETS
Alessandra Cretarola and Fausto Gozzi and Huy\^en Pham (PMA and
CREST) and Peter Tankov (PMA)
We investigate optimal consumption policies in the liquidity risk model
introduced in Pham and Tankov (2007). Our main result is to derive
smoothness
results for the value functions of the portfolio/consumption choice
problem. As
an important consequence, we can prove the existence of the optimal
control
(portfolio/consumption strategy) which we characterize both in
feedback form in
terms of the derivatives of the value functions and as the solution of a
second-order ODE. Finally, numerical illustrations of the behavior of
optimal
consumption strategies between two trading dates are given.
http://arxiv.org/abs/0807.0326
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7242. MATRIX REPRESENTATION OF THE STATIONARY MEASURE FOR THE
MULTISPECIES TASEP
Martin R. Evans and Pablo A. Ferrari and Kirone Mallick
In this work we construct the stationary measure of the N species
totally
asymmetric simple exclusion process in a matrix product formulation.
We make
the connection between the matrix product formulation and the queueing
theory
picture of Ferrari and Martin. In particular, in the standard
representation,
the matrices act on the space of queue lengths. For N >2 the matrices
in fact
become tensor products of elements of quadratic algebras. This enables
us to
give a purely algebraic proof of the stationary measure which we
present for N
=3.
http://arxiv.org/abs/0807.0327
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7243. FLUCTUATIONS OF EIGENVALUES OF RANDOM NORMAL MATRICES
Yacin Ameur and Haakan hedenmalm and Nikolai Makarov
In this note, we prove Gaussian field convergence of fluctuations of
eigenvalues of random normal matrices in the interior of a quantum
droplet.
http://arxiv.org/abs/0807.0375
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7244. NAVIER-STOKES EQUATIONS AND FORWARD-BACKWARD SDES ON THE GROUP
OF VOLUME-PRESERVING DIFFEOMORPHISMS
A. B. Cruzeiro and E. Shamarova
We establish a connection between the strong solution to the Cauchy
problem
for the 2D Navier-Stokes equations and the solution to the system of
forward-backward stochastic differential equations (FBSDEs) on the
group of
volume-preserving diffeomorphisms of the 2D torus. We construct a
representation of the strong solution to the Navier-Stokes equations
in terms
of diffusion processes.
http://arxiv.org/abs/0807.0421
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7245. LA FORME ASYMPTOTIQUE DU PROCESSUS DE CONTACT EN ENVIRONNEMENT
AL\'EATOIRE
Olivier Garet (IECN) and R\'egine Marchand (IECN)
The aim of this article is to prove asymptotic shape theorems for the
contact
pr ocess in stationary random environment. These theorems generalize
known
results for the classical contact process. In particular, if $H_t$
denotes the
set of al ready occupied sites at time $t$, we show that for almost
every
environment, whe n the contact process survives, the set $H_t/t$
almost surely
converges to a co mpact set that only depends on the law of the
environment. We
introduce new obje cts which also simplify the proof of the shape
theorem in a
deterministic environment.
http://arxiv.org/abs/0807.0426
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7246. CHARACTERIZATIONS AND SIMULATIONS OF A CLASS OF STOCHASTIC
PROCESSES TO MODEL ANOMALOUS DIFFUSION
Antonio Mura and Gianni Pagnini
In this paper we study a parametric class of stochastic processes to
model
both fast and slow anomalous diffusion. This class, called generalized
grey
Brownian motion (ggBm), is made up off self-similar with stationary
increments
processes (H-sssi) and depends on two real parameters alpha in (0,2)
and beta
in (0,1]. It includes fractional Brownian motion when alpha in (0,2) and
beta=1, and time-fractional diffusion stochastic processes when
alpha=beta in
(0,1). The latters have marginal probability density function governed
by
time-fractional diffusion equations of order beta. The ggBm is defined
through
the explicit construction of the underline probability space. However,
in this
paper we show that it is possible to define it in an unspecified
probability
space. For this purpose, we write down explicitly all the finite
dimensional
probability density functions. Moreover, we provide different ggBm
characterizations. The role of the M-Wright function, which is related
to the
fundamental solution of the time-fractional diffusion equation,
emerges as a
natural generalization of the Gaussian distribution. Furthermore, we
show that
ggBm can be represented in terms of the product of a random variable,
which is
related to the M-Wright function, and an independent fractional Brownian
motion. This representation highlights the $H$-{\bf sssi} nature of
the ggBm
and provides a way to study and simulate the trajectories. For this
purpose, we
developed a random walk model based on a finite difference
approximation of a
partial integro-differenital equation of fractional type.
http://arxiv.org/abs/0801.4879
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7247. ON A SURPRISING RELATION BETWEEN RECTANGULAR AND SQUARE FREE
CONVOLUTIONS
Florent Benaych-Georges (PMA)
Debbah and Ryan have recently proved a result about the limit empirical
singular distribution of the sum of two rectangular random matrices
whose
dimensions tend to infinity. In this paper, we reformulate it in terms
of the
rectangular free convolution introduced in a previous paper and then
we give a
new, shorter, proof of this result under weaker hypothesis: we do not
suppose
the \pro measure in question in this result to be compactly supported
anymore.
At last, we discuss the inclusion of this result in the family of
relations
between rectangular and square random matrices.
http://arxiv.org/abs/0807.0505
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7248. ASYMPTOTIC ANALYSIS FOR BIFURCATING AUTOREGRESSIVE PROCESSES VIA
A MARTINGALE APPROACH
Bernard Bercu and Benoite de Saporta and Anne Gegout-Petit
We study the asymptotic behavior of the least squares estimators of the
unknown parameters of bifurcating autoregressive processes. Under very
weak
assumptions on the driven noise of the process, namely conditional
pair-wise
independence and suitable moment conditions, we establish the almost
sure
convergence of our estimators together with the quadratic strong law
and the
central limit theorem. All our analysis relies on non-standard
asymptotic
results for martingales.
http://arxiv.org/abs/0807.0528
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7249. A NEW FAMILY OF MARKOV BRANCHING TREES: THE ALPHA-GAMMA MODEL
Bo Chen and Daniel Ford and Matthias Winkel
We introduce a simple tree growth process that gives rise to a new
two-parameter family of discrete fragmentation trees that extends
Ford's alpha
model to multifurcating trees and includes the trees obtained by uniform
sampling from Duquesne and Le Gall's stable continuum random tree. We
call
these new trees the alpha-gamma trees. In this paper, we obtain their
splitting
rules, dislocation measures both in ranked order and in sized-biased
order, and
we study their limiting behaviour.
http://arxiv.org/abs/0807.0554
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7250. REPRESENTATIONS OF SO(3) AND ANGULAR PLYSPECTRA
Domenico Marinucci and Giovanni Peccati (LSTA)
We characterize the angular polyspectra, of arbitrary order,
associated with
isotropic fields defined on the sphere S^2. Our techniques rely
heavily on
group representation theory, and specifically on the properties of
Wigner
matrices and Clebsch-Gordan coefficients. The findings of the present
paper
constitute a basis upon which one can build formal procedures for the
statistical analysis and the probabilistic modelization of the Cosmic
Microwave
Background radiation, which is currently a crucial topic of
investigation in
cosmology. We also outline an application to random data compression and
"simulation" of Clebsch-Gordan coefficients.
http://arxiv.org/abs/0807.0687
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7251. DETECTION OF CELLULAR AGING IN A GALTON-WATSON PROCESS
Jean-Fran\c{c}ois Delmas (CERMICS) and Laurence Marsalle (LPP)
We consider the bifurcating Markov chain model introduced by Guyon to
detect
cellular aging from cell lineage. To take into account the possibility
for a
cell to die, we use an underlying Galton-Watson process to describe the
evolution of the cell lineage. We give in this more general framework
a weak
law of large number, an invariance principle and thus fluctuation
results for
the average over one generation or up to one generation. We also prove
the
fluctuations over each generation are independent. Then we present the
natural
modifications of the tests given by Guyon in cellular aging detection
within
the particular case of the auto-regressive model.
http://arxiv.org/abs/0807.0749
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7252. UNCONSTRAINED RECURSIVE IMPORTANCE SAMPLING
Vincent Lemaire (PMA) and Gilles Pag\`es (PMA)
We propose an unconstrained stochastic approximation method of finding
the
optimal measure change (in an a priori parametric family) for Monte
Carlo
simulations. We consider different parametric families based on the
Girsanov
theorem and the Esscher transform (or exponential-tilting). In a
multidimensional Gaussian framework, Arouna uses a projected Robbins-
Monro
procedure to select the parameter minimizing the variance. In our
approach, the
parameter (scalar or process) is selected by a classical Robbins-Monro
procedure without projection or truncation. To obtain this unconstrained
algorithm we intensively use the regularity of the density of the law
without
assume smoothness of the payoff. We prove the convergence for a large
class of
multidimensional distributions and diffusion processes. We illustrate
the
effectiveness of our algorithm via pricing a Basket payoff under a
multidimensional NIG distribution, and pricing a barrier options in
different
markets.
http://arxiv.org/abs/0807.0762
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7253. LOCALISABLE MOVING AVERAGE STABLE AND MULTISTABLE PROCESSES
Kenneth Falconer and Ronan Le Gu\'evel (LMJL) and Jacques L\'evy-V
\'ehel (INRIA Saclay - Ile de France)
We study a particular class of moving average processes which possess a
property called localisability. This means that, at any given point,
they admit
a ``tangent process'', in a suitable sense. We give general conditions
on the
kernel g defining the moving average which ensures that the process is
localisable and we characterize the nature of the associated tangent
processes.
Examples include the reverse Ornstein-Uhlenbeck process and the
multistable
reverse Ornstein-Uhlenbeck process. In the latter case, the tangent
process is,
at each time t, a L\'evy stable motion with stability index possibly
varying
with t. We also consider the problem of path synthesis, for which we
give both
theoretical results and numerical simulations.
http://arxiv.org/abs/0807.0764
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7254. FROM BLACK-SCHOLES AND DUPIRE FORMULAE TO LAST PASSAGE TIMES OF
LOCAL MARTINGALES. PART B : THE FINITE TIME HORIZON
Amel Bentata (PMA) and Marc Yor (PMA and Iuf)
These notes are the second half of the contents of the course given by
the
second author at the Bachelier Seminar (8-15-22 February 2008) at IHP.
They
also correspond to topics studied by the first author for her
Ph.D.thesis.
http://arxiv.org/abs/0807.0788
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7255. ON A LOWER BOUND FOR THE TIME CONSTANT OF FIRST-PASSAGE
PERCOLATION
Xian-Yuan Wu and Ping Feng
We consider the Bernoulli first-passage percolation on $\mathbb Z^d (d
\ge
2)$. That is, the edge passage time is taken independently to be 1 with
probability $1-p$ and 0 otherwise. Let ${\mu(p)}$ be the time
constant. We
prove in this paper that \[ \mu(p_1)-\mu({p_2})\ge
\frac{\mu(p_2)}{1-p_2}(p_2-p_1)\] for all $ 0\leq p_1<p_2< 1$ by using
Russo's
formula.
http://arxiv.org/abs/0807.0839
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7256. MATRIX RANDOM PRODUCTS WITH SINGULAR HARMONIC MEASURE
Vadim A. Kaimanovich and Vincent Le Prince
Any Zariski dense countable subgroup of $SL(d,R)$ is shown to carry a
non-degenerate finitely supported symmetric random walk such that its
harmonic
measure on the flag space is singular. The main ingredients of the
proof are:
(1) a new upper estimate for the Hausdorff dimension of the
projections of the
harmonic measure onto Grassmannians in $R^d$ in terms of the associated
differential entropies and differences between the Lyapunov exponents;
(2) an
explicit construction of random walks with uniformly bounded entropy and
Lyapunov exponents going to infinity.
http://arxiv.org/abs/0807.1015
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7257. GAUSSIAN MULTIPLICATIVE CHAOS REVISITED
Raoul Robert (IF) and Vincent Vargas (CEREMADE)
In this article, we extend the theory of multiplicative chaos for
positive
definite functions in Rd of the form f(x) = 2 ln+ T|x|+ g(x) where g
is a
continuous and bounded function. The construction is simpler and more
general
than the one defined by Kahane in 1985. As main application, we give a
rigorous
mathematical meaning to the Kolmogorov-Obukhov model of energy
dissipation in a
turbulent flow.
http://arxiv.org/abs/0807.1030
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7258. KPZ FORMULA FOR LOG-INFINITELY DIVISIBLE MULTIFRACTAL RANDOM
MEASURES
R\'emi Rhodes (CEREMADE) and Vincent Vargas (CEREMADE)
We consider the continuous model of log-infinitely divisible
multifractal
random mea- sures (MRM) introduced in [1]. If M is a non degenerate
multifractal measure with associated metric rho(x, y) = M ([x, y]) and
structure function zeta, we show that we have the following relation
between
the (Euclidian) Hausdorff dimension dimH of a measurable set K and the
Hausdorff dimension dim_H with respect to rho of the same set: $
\zeta({\rm
dim}_H^{\rho}(K))={\r m dim}_H(K)$.
http://arxiv.org/abs/0807.1036
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7259. HYDRODYNAMIC TURBULENCE AND INTERMITTENT RANDOM FIELDS
Raoul Robert (IF) and Vincent Vargas (CEREMADE)
In this article, we construct two families of nonsymmetrical
multifractal
fields. One of these families is used for the modelization of the
velocity
field of turbulent flows.
http://arxiv.org/abs/0807.1042
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7260. DISCRETE TIME NONLINEAR FILTERS WITH REGULAR OBSERVATIONS ARE
ALWAYS STABLE
Ramon van Handel
The nonlinear filter associated with the discrete time signal-
observation
model $(X_k,Y_k)$ is known to forget its initial condition as $k\to
\infty$
regardless of the observation structure when the signal possesses
sufficiently
strong ergodic properties. Conversely, it stands to reason that if the
observations are sufficiently informative, then the nonlinear filter
should
forget its initial condition regardless of the properties of the
signal. We
show that the latter is indeed the case for the additive noise model
$Y_k=h(X_k)+\xi_k$ with invertible observation function $h$ under mild
regularity assumptions on $h$ and on the distribution of the noise
variables
$\xi_k$.
http://arxiv.org/abs/0807.1072
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7261. EXACT DISTRIBUTION OF THE MAXIMAL HEIGHT OF WATERMELONS
Gregory Schehr and Satya N. Majumdar and Alain Comtet and Julien
Randon-Furling
We study p non intersecting one-dimensional Brownian walks, either
excursions
(p-watermelons with a wall) or bridges (p-watermelons without wall).
We focus
on the maximal height H_p of these p-watermelons configurations on the
unit
time interval. Using path integral techniques associated to
corresponding
models of free Fermions, we compute exactly the distribution of H_p
for generic
integer p. For large p, one obtains < H_p > \sim \sqrt{2p} for p-
watermelons
with a wall whereas < H_p > \sim \sqrt{p} for p-watermelons without
wall. We
point out and solve a discrepancy between these exact asymptotic
behaviors and
numerical experiments, which recently attracted much attention, and we
show
that only the pre-asymptotic behaviors of these averages were actually
measured. In addition, our method, using tools of many-body physics,
provides a
simpler physical derivation of the connection between vicious walkers
and
random matrix theory.
http://arxiv.org/abs/0807.0522
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7262. A NONCOMMUTATIVE DE FINETTI THEOREM: INVARIANCE UNDER QUANTUM
PERMUTATIONS IS EQUIVALENT TO FREENESS WITH AMALGAMATION
Claus K\"ostler and Roland Speicher
We show that the classical de Finetti theorem has a canonical
noncommutative
counterpart if we strengthen `exchangeability' (i.e., invariance of
the joint
distribution of the random variables under the action of the
permutation group)
to invariance under the action of the quantum permutation group. More
precisely, for an infinite sequence of noncommutative random
variables, we
prove that invariance of their joint distribution under quantum
permutations is
equivalent to the fact that the random variables are identically
distributed
and free with respect to the conditional expectation onto their tail
algebra.
http://arxiv.org/abs/0807.0677
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7263. COMPONENTWISE CONDITION NUMBERS OF RANDOM SPARSE MATRICES
Dennis Cheung and Felipe Cucker
We prove an O(log n) bound for the expected value of the logarithm of
the
componentwise (and, a fortiori, the mixed) condition number of a
random sparse
n x n matrix. As a consequence, small bounds on the average loss of
accuracy
for triangular linear systems follow.
http://arxiv.org/abs/0807.0956
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7264. SPATIAL HOMOGENIZATION IN A STOCHASTIC NETWORK WITH MOBILITY
Florian Simatos and Danielle Tibi (PMA and INRIA Rocquencourt)
A stochastic model for a mobile network is studied. Users enter the
network,
and then perform independent Markovian routes between nodes, where
they receive
service according to the Processor-Sharing policy. Once their service
requirement is satisfied, they leave the system. The stability region is
identified via a fluid limit approach, and strongly relies on a
``spatial
homogenization'' property: At the fluid level, customers are
instantaneously
distributed across the network according to the stationary
distribution of
their Markovian dynamics and stay distributed as such as long as the
network is
not empty. In the unstable regime, spatial homogenization almost
surely holds
asymptotically as time goes to infinity (on the normal scale), telling
how the
system fills up. One of the technical achievements of the paper is the
construction of a family of martingales associated to the
multidimensional
process of interest, which makes it possible to get crucial estimates
for
certain exit times.
http://arxiv.org/abs/0807.1205
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7265. SELF-SIMILARITY PARAMETER ESTIMATION AND REPRODUCTION PROPERTY
FOR NON-GAUSSIAN HERMITE PROCESSES
Alexandra Chronopoulou and Ciprian Tudor (CES and SAMOS) and
Frederi Viens
We consider the class of all the Hermite processes $(Z_{t}^{(q,H)})_{t
\in
\lbrack 0,1]}$ of order $q\in \mathbf{N}^{\ast}$ and with Hurst
parameter $%
H\in ({1/2},1)$. The process $Z^{(q,H)}$ is $H$-selfsimilar, it has
stationary
increments and it exhibits long-range dependence identical to that of
fractional Brownian motion (fBm). For $q=1$, $Z^{(1,H)}$ is fBm, which
is
Gaussian; for $q=2$, $Z^{(2,H)}$ is the Rosenblatt process, which
lives in the
second Wiener chaos; for any $q>2$, $Z^{(q,H)}$ is a process in the $q
$th
Wiener chaos. We study the variations of $Z^{(q,H)}$ for any $q$, by
using
multiple Wiener -It\^{o} stochastic integrals and Malliavin calculus.
We prove
a reproduction property for this class of processes in the sense that
the terms
appearing in the chaotic decomposition of their variations give rise
to other
Hermite processes of different orders and with different Hurst
parameters. We
apply our results to construct a strongly consistent estimator for the
self-similarity parameter $H$ from discrete observations of $Z^{(q,H)}
$; the
asymptotics of this estimator, after appropriate normalization, are
proved to
be distributed like a Rosenblatt random variable (value at time 1 of a
Rosenblatt process).with self-similarity parameter $1+2(H-1)/q$.
http://arxiv.org/abs/0807.1208
---------------------------------------------------------------
7266. NEGATIVE VOLATILITY FOR A 2-DIMENSIONAL SQUARE ROOT SDE
Peter Spreij (University of Amsterdam) and Enno Veerman (University
of Amsterdam)
In affine term structure models the short rate is modelled as an affine
transformation of a multi-dimensional square root process. Sufficient
conditions to avoid negative volatility factors are the multivariate
Feller
conditions. We will prove their necessity for a 2-dimensional square
root SDE
with one volatility factor by presenting a methodology based on measure
transformations and solving linear systems of ordinary differential
equations.
http://arxiv.org/abs/0807.1224
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7267. ON THE ESSCHER TRANSFORMS AND OTHER EQUIVALENT MARTINGALE
MEASURES FOR BARNDORFF-NIELSEN AND SHEPHARD STOCHASTIC VOLATILITY
MODELS WITH JUMPS
Friedrich Hubalek (Technical University of Vienna) Carlo Sgarra
(Technical University of Milan)
We compute and discuss the Esscher martingale transform for exponential
processes, the Esscher martingale transform for linear processes, the
minimal
martingale measure, the class of structure preserving martingale
measures, and
the minimum entropy martingale measure for stochastic volatility
models of
Ornstein-Uhlenbeck type as introduced by Barndorff-Nielsen and
Shephard. We
show, that in the model with leverage, with jumps both in the
volatility and in
the returns, all those measures are different, whereas in the model
without
leverage, with jumps in the volatility only and a continuous return
process,
several measures coincide, some simplifications can be made and the
results are
more explicit. We illustrate our results with parametric examples used
in the
literature.
http://arxiv.org/abs/0807.1227
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7268. INFORMED TRADERS
Dorje C. Brody and Mark H. A. Davis and Robyn L. Friedman and Lane
P. Hughston
A model is introduced in which there is a small agent who is more
susceptible
to the flow of information in the market than the general market
participant,
and who tries to implement strategies based on the additional
information. In
this model market participants have access to a stream of noisy
information
concerning the future return of an asset, whereas the informed trader
has
access to a further information source which is obscured by an
additional noise
that may be correlated with the market noise. The informed trader uses
the
extraneous information source to seek statistical arbitrage
opportunities,
while at the same time accommodating the additional risk. The amount of
information available to the general market participant concerning the
asset
return is measured by the mutual information of the asset price and the
associated cash flow. The worth of the additional information source
is then
measured in terms of the difference of mutual information between the
general
market participant and the informed trader. This difference is shown
to be
nonnegative when the signal-to-noise ratio of the information flow is
known in
advance. Explicit trading strategies leading to statistical arbitrage
opportunities, taking advantage of the additional information, are
constructed,
illustrating how excess information can be translated into profit.
http://arxiv.org/abs/0807.1253
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7269. QUANTITATIVE COMPARISONS BETWEEN FINITARY POSTERIOR
DISTRIBUTIONS AND BAYESIAN POSTERIOR DISTRIBUTIONS
Federico Bassetti
The main object of Bayesian statistical inference is the determination
of
posterior distributions. Sometimes these laws are given for quantities
devoid
of empirical value. This serious drawback vanishes when one confines
oneself to
considering a finite horizon framework. However, assuming infinite
exchangeability gives rise to fairly tractable {\it a posteriori}
quantities,
which is very attractive in applications. Hence, with a view to a
reconciliation between these two aspects of the Bayesian way of
reasoning, in
this paper we provide quantitative comparisons between posterior
distributions
of finitary parameters and posterior distributions of allied parameters
appearing in usual statistical models.
http://arxiv.org/abs/0807.1201
---------------------------------------------------------------
7270. MONTE CARLO GREEKS FOR FINANCIAL PRODUCTS VIA APPROXIMATIVE
TRANSITION DENSITIES
Joerg Kampen and Anastasia Kolodko and and John Schoenmakers
In this paper we introduce efficient Monte Carlo estimators for the
valuation
of high-dimensional derivatives and their sensitivities (''Greeks'').
These
estimators are based on an analytical, usually approximative
representation of
the underlying density. We study approximative densities obtained by
the WKB
method. The results are applied in the context of a Libor market model.
http://arxiv.org/abs/0807.1213
---------------------------------------------------------------
7271. ALMOST SURE STABILIZATION FOR ADAPTIVE CONTROLS OF REGIME-
SWITCHING LQ SYSTEMS WITH A HIDDEN MARKOV CHAIN
Bernard Bercu and Francois Dufour and G. George Yin
This work is devoted to the almost sure stabilization of adaptive
control
systems that involve an unknown Markov chain. The control system
displays
continuous dynamics represented by differential equations and discrete
events
given by a hidden Markov chain. Different from previous work on
stabilization
of adaptive controlled systems with a hidden Markov chain, where average
criteria were considered, this work focuses on the almost sure
stabilization or
sample path stabilization of the underlying processes. Under simple
conditions,
it is shown that as long as the feedback controls have linear growth
in the
continuous component, the resulting process is regular. Moreover, by
appropriate choice of the Lyapunov functions, it is shown that the
adaptive
system is stabilizable almost surely. As a by-product, it is also
established
that the controlled process is positive recurrent.
http://arxiv.org/abs/0807.1413
---------------------------------------------------------------
7272. ERGODIC BSDES AND RELATED PDES WITH NEUMANN BOUNDARY CONDITIONS
Adrien Richou (IRMAR)
We study a new class of ergodic backward stochastic differential
equations
(EBSDEs for short) which is linked with semi-linear Neumann type
boundary value
problems related to ergodic phenomenas. The particularity of these
problems is
that the ergodic constant appears in Neumann boundary conditions. We
study the
existence and uniqueness of solutions to EBSDEs and the link with
partial
differential equations. Then we apply these results to optimal ergodic
control
problems.
http://arxiv.org/abs/0807.1521
---------------------------------------------------------------
7273. RANDOM REGULAR GRAPHS OF NON-CONSTANT DEGREE: DISTRIBUTION OF
EDGES AND APPLICATIONS
Sonny Ben-Shimon and Michael Krivelevich
In this work we analyze the distribution of the number of edges
spanned by a
single set and between two disjoint sets of vertices in the random
regular
graph model $\mathcal{G}_{n,d}$ in the range $d=o(\sqrt n)$. We show
it to be
very similar to that of the binomial random graph model, $\mathcal{G}
(n,p)$
with $p=\frac{d}{n}$. Some graph properties are known to exist solely
based on
the edge distribution of the graph. We demonstrate how our analysis
can be used
to prove the high probability of existence of such graph properties in
the
$\mathcal{G}_{n,d}$ probability space, specifically spectral gap and
Hamiltonicity. Next, using our results on the distribution of edges
spanned by
a single set of vertices, we show that for every fixed $\epsilon>0$ and
$d=o(n^{1/5})$, the chromatic number of $\mathcal{G}_{n,d}$ is
concentrated in
two consecutive values with probability at least $1-\epsilon$ for
large enough
values of $n$.
http://arxiv.org/abs/math/0511343
---------------------------------------------------------------
7274. SEQUENTIAL CAVITY METHOD FOR COMPUTING FREE ENERGY AND SURFACE
PRESSURE
David Gamarnik and Dmitriy Katz
We propose a new method for the problems of computing free energy and
surface
pressure for various statistical mechanics models on a lattice $\Z^d$.
Our
method is based on representing the free energy and surface pressure
in terms
of certain marginal probabilities in a suitably modified sublattice of
$\Z^d$.
Then recent deterministic algorithms for computing marginal
probabilities are
used to obtain numerical estimates of the quantities of interest. The
method
works under the assumption of Strong Spatial Mixing (SSP), which is a
form of a
correlation decay.
We illustrate our method for the hard-core and monomer-dimer
models, and
improve several earlier estimates. For example we show that the
exponent of the
monomer-dimer coverings of $\Z^3$ belongs to the interval
$[0.78595,0.78599]$,
improving best previously known estimate of (approximately)
$[0.7850,0.7862]$
obtained in \cite{FriedlandPeled},\cite{FriedlandKropLundowMarkstrom}.
Moreover, we show that given a target additive error $\epsilon>0$, the
computational effort of our method for these two models is
$(1/\epsilon)^{O(1)}$ \emph{both} for free energy and surface
pressure. In
contrast, prior methods, such as transfer matrix method, require
$\exp\big((1/\epsilon)^{O(1)}\big)$ computation effort.
http://arxiv.org/abs/0807.1551
---------------------------------------------------------------
7275. PERCOLATION OF ARBITRARY WORDS IN ONE DIMENSION
Geoffrey R. Grimmett and Thomas M. Liggett and Thomas Richthammer
We consider a type of long-range percolation problem on the positive
integers, motivated by earlier work of others on the appearance of
(in)finite
words within a site percolation model. The main issue is whether a given
infinite binary word appears within an iid Bernoulli sequence at
locations that
satisfy certain constraints. We settle the issue in some cases, and
provide
partial results in others.
http://arxiv.org/abs/0807.1676
---------------------------------------------------------------
7276. THE EYRING-KRAMERS LAW FOR POTENTIALS WITH NONQUADRATIC SADDLES
Nils Berglund (MAPMO) and Barbara Gentz
The Eyring-Kramers law describes the mean transition time of an
overdamped
Brownian particle between local minima in a potential landscape. In the
weak-noise limit, the transition time is to leading order exponential
in the
potential difference to overcome. This exponential is corrected by a
prefactor
which depends on the principal curvatures of the potential at the
starting
minimum and at the highest saddle crossed by an optimal transition
path. The
Eyring-Kramers law, however, does not hold whenever one of these
principal
curvatures vanishes, since it would predict a vanishing or infinite
transition
time. We derive the correct prefactor up to multiplicative errors that
tend to
one in the zero-noise limit. As an illustration, we discuss the case
of a
symmetric pitchfork bifurcation, in which the prefactor can be
expressed in
terms of modified Bessel functions. The results extend work by Bovier,
Eckhoff,
Gayrard and Klein, who rigorously analysed the case of quadratic
saddles, using
methods from potential theory.
http://arxiv.org/abs/0807.1681
---------------------------------------------------------------
7277. CONVERGENCE OF THE LAW OF THE ENVIRONMENT SEEN BY THE PARTICLE
FOR DIRECTED POLYMERS IN RANDOM MEDIA IN THE $L^2$ REGION
Gregorio Moreno Flores (PMA)
We consider the model of Directed Polymers in an i.i.d. gaussian or
bounded
Environment in the $L^2$ region. We prove the convergence of the law
of the
environment seen by the particle. As a main technical step, we
establish a
lower tail concentration inequality for the partition function for
bounded
environments. Our proof is based on arguments developed by Talagrand
in the
context of the Hopfield Model. This improves in some sense a
concentration
inequality obtained by Carmona and Hu for gaussian environments. We
use this
and a Local Limit Theorem to prove the $L^1$ convergence of the
density of the
law of the environment seen by the particle with respect to the product
measure.
http://arxiv.org/abs/0807.1685
---------------------------------------------------------------
7278. ASYMPTOTICS FOR AN INDIVIDUAL PARTICLE IN ASEP
Craig A. Tracy and Harold Widom
In previous work the authors considered the asymmetric simple exclusion
process on the integer lattice in the case of step initial condition,
particles
beginning at the positive integers. There it was shown that the
probability
distribution for the position of an individual particle is given by an
integral
whose integrand involves a Fredholm determinant. Here we use this
formula to
obtain two asymptotic results for the positions of these particles.
http://arxiv.org/abs/0807.1713
---------------------------------------------------------------
7279. STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS: WELL POSEDNESS AND
LARGE DEVIATIONS
Igor Chueshov and Annie Millet (PMA and Matisse and Samos)
We deal with a class of abstract nonlinear stochastic models, which
covers
many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D
MHD
models and 2D magnetic B\'enard problem and also some shell models of
turbulence. We first prove the existence and uniqueness theorem for
the class
considered. Our main result is a Wentzell-Freidlin type large deviation
principle for small multiplicative noise which we prove by weak
convergence
method.
http://arxiv.org/abs/0807.1810
---------------------------------------------------------------
7280. LP-SOLUTIONS FOR REECTED BACKWARD STOCHASTIC DIFFERENTIAL
EQUATIONS
Said Hamadene and Alexandre Popier
This paper deals with the problem of existence and uniqueness of a
solution
for a backward stochastic differential equation (BSDE for short) with
one
reflecting barrier in the case when the terminal value, the generator
and the
obstacle process are Lp-integrable with p in ]1,2[. To construct the
solution
we use two methods: penalization and Snell envelope. As an application
we
broaden the class of functions for which the related obstacle partial
differential equation problem has a unique viscosity solution.
http://arxiv.org/abs/0807.1846
---------------------------------------------------------------
7281. ON THE GAUSSIAN Q-DISTRIBUTION
Rafael Diaz and Eddy Pariguan
We present study of the Gaussian q-measure introduced by Diaz and
Teruel from
a probabilistic and a categorical viewpoint. We show that the Gaussian
q-measure interpolates between that uniform measure on the interval
[-1,1] and
the Gaussian measure on the real line.
http://arxiv.org/abs/0807.1918
---------------------------------------------------------------
7282. THE BEST CONSTANT IN A FRACTIONAL HARDY INEQUALITY
Krzysztof Bogdan and Bart{\l}omiej Dyda
We prove an optimal Hardy inequality for the fractional Laplacian on the
half-space.
http://arxiv.org/abs/0807.1825
---------------------------------------------------------------
7283. ENERGY IMAGE DENSITY PROPERTY AND LOCAL GRADIENT FOR POISSON
RANDOM MEASURES
Nicolas Bouleau (CIRED and Cermics) and Laurent Denis (DP)
We introduce a new approach to absolute continuity of laws of Poisson
functionals. It is based on the {\it energy image density} property for
Dirichlet forms and on what we call {\it the lent particle method} which
consists in adding a particle and taking it back after some calculation.
http://arxiv.org/abs/0807.1963
---------------------------------------------------------------
7284. SPARSE RANDOM GRAPHS WITH CLUSTERING
Bela Bollobas and Svante Janson and Oliver Riordan
In 2007 we introduced a general model of sparse random graphs with
independence between the edges. The aim of this paper is to present an
extension of this model in which the edges are far from independent,
and to
prove several results about this extension. The basic idea is to
construct the
random graph by adding not only edges but also other small graphs. In
other
words, we first construct an inhomogeneous random hypergraph with
independent
hyperedges, and then replace each hyperedge by a (perhaps complete)
graph.
Although flexible enough to produce graphs with significant dependence
between
edges, this model is nonetheless mathematically tractable. Indeed, we
find the
critical point where a giant component emerges in full generality, in
terms of
the norm of a certain integral operator, and relate the size of the
giant
component to the survival probability of a certain (non-Poisson) multi-
type
branching process. While our main focus is the phase transition, we
also study
the degree distribution and the numbers of small subgraphs. We
illustrate the
model with a simple special case that produces graphs with power-law
degree
sequences with a wide range of degree exponents and clustering
coefficients.
http://arxiv.org/abs/0807.2040
---------------------------------------------------------------
7285. REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH
CONTINUOUS COEFFICIENT AND L^2 BARRIERS
Shaolin Ji and Zhen Wu and Li Zhou
In this paper we study reflected backward stochastic differential
equations
with a continuous, linear growth coefficient and two barriers which
belong to
L^2. We prove that there exists at least by penalization method.
http://arxiv.org/abs/0807.2075
---------------------------------------------------------------
7286. REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY L
\'{E}VY PROCESS
Yong Ren and Xiliang Fan
In this paper, we deal with a class of reflected backward stochastic
differential equations associated to the subdifferential operator of a
lower
semi-continuous convex function driven by Teugels martingales
associated with
L\'{e}vy process. We obtain the existence and uniqueness of solutions
to these
equations by means of the penalization method. As its application, we
give a
probabilistic interpretation for the solutions of a class of partial
differential-integral inclusions.
http://arxiv.org/abs/0807.2076
---------------------------------------------------------------
7287. GENERALIZED FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES
Kotaro Endo and Muneya Matsui
We introduce an extended version of the fractional Ornstein-Uhlenbeck
(FOU)
process where the integrand is replaced by the exponential of an
independent
L\'evy process. We call the process the generalized fractional
Ornstein-Uhlenbeck (GFOU) process. Alternatively, the process can be
constructed from a generalized Ornstein-Uhlenbeck (GOU) process using an
independent fractional Brownian motion (FBM) as integrator. We show
that the
GFOU process is well-defined by checking the existence of the integral
included
in the process, and investigate its properties. It is proved that the
process
has a stationary version and exhibits long memory. We also find that the
process satisfies a certain stochastic differential equation. Our
underlying
intention is to introduce long memory into the GOU process which has
short
memory without losing the possibility of jumps. Note that both FOU and
GOU
processes have found application in a variety of fields as useful
alternatives
to the Ornstein-Uhlenbeck (OU) process.
http://arxiv.org/abs/0807.2110
---------------------------------------------------------------
7288. AN INFORMATION-BASED FRAMEWORK FOR ASSET PRICING: X-FACTOR
THEORY AND ITS APPLICATIONS
Andrea Macrina
A new framework for asset pricing based on modelling the information
available to market participants is presented. Each asset is
characterised by
the cash flows it generates. Each cash flow is expressed as a function
of one
or more independent random variables called market factors or "X-
factors". Each
X-factor is associated with a "market information process", the values
of which
become available to market participants. In addition to true
information about
the X-factor, the information process contains an independent "noise"
term
modelled here by a Brownian bridge. The information process thus gives
partial
information about the X-factor, and the value of the market factor is
only
revealed at the termination of the process. The market filtration is
assumed to
be generated by the information processes associated with the X-
factors. The
price of an asset is given by the risk-neutral expectation of the sum
of the
discounted cash flows, conditional on the information available from the
filtration. The theory is developed in some detail, with a variety of
applications to credit risk management, share prices, interest rates,
and
inflation. A number of new exactly solvable models are obtained for
the price
processes of various types of assets and derivative securities; and a
novel
mechanism is proposed to account for the dynamics of stochastic
volatility and
dynamic correlation. A discrete-time version of the information-based
framework
is also developed, and is used to construct a new class of models for
the real
and nominal interest rate term structures, and the dynamics of the
associated
price index.
http://arxiv.org/abs/0807.2124
---------------------------------------------------------------
7289. WINDINGS OF PLANAR RANDOM WALKS AND AVERAGED DEHN FUNCTION
Bruno Schapira (LM-Orsay) and Robert Young (IHES)
We prove a sharp estimate on the expected value of the integral of the
index
of a simple random walk on the square or triangular lattice. This
gives new
lower bounds on the averaged Dehn function, which measures the
expected area
needed to fill a random curve with a disc.
http://arxiv.org/abs/0807.2192
---------------------------------------------------------------
7290. MARKOV STOCHASTIC OPERATORS OF HEREDITY
Nasir Ganikhodjaev
Applying non-ergodic quadratic stochastic operator the continual
family of
weak ergodic non-homogeneous Markov chains is constructed.
http://arxiv.org/abs/0807.2012
---------------------------------------------------------------
7291. RATE OF ESCAPE OF RANDOM WALKS ON REGULAR LANGUAGES AND FREE
PRODUCTS BY AMALGAMATION OF FINITE GROUPS
Lorenz A. Gilch
We consider random walks on the set of all words over a finite
alphabet such
that in each step only the last two letters of the current word may be
modified
and only one letter may be adjoined or deleted. We assume that the
transition
probabilities depend only on the last two letters of the current word.
Furthermore, we consider also the special case of random walks on free
products
by amalgamation of finite groups which arise in a natural way from
random walks
on the single factors. The aim of this paper is to compute several
equivalent
formulas for the rate of escape with respect to natural length
functions for
these random walks using different techniques.
http://arxiv.org/abs/0807.2264
---------------------------------------------------------------
7292. LARGE DEVIATIONS OF THE FRONT IN A ONE DIMENSIONAL MODEL OF $X+Y
\TO 2X$
Jean B\'erard (ICJ) and Alejandro Ram\'irez
We investigate the probabilities of large deviations for the position
of the
front in a stochastic model of the reaction $X+Y \to 2X$ on the
integer lattice
in which $Y$ particles do not move while $X$ particles move as
independent
simple continuous time random walks of total jump rate $2$. For a wide
class of
initial conditions, we prove that a large deviations principle holds
and we
show that the zero set of the rate function is the interval $[0,v]$,
where $v$
is the velocity of the front given by the law of large numbers. We
also give
more precise estimates for the rate of decay of the slowdown
probabilities. Our
results indicate a gapless property of the generator of the process as
seen
from the front, as it happens in the context of nonlinear diffusion
equations
describing the propagation of a pulled front into an unstable state.
http://arxiv.org/abs/0807.2349
---------------------------------------------------------------
7293. THE HEIGHT OF RANDOM BINARY UNLABELLED TREES
Nicolas Broutin and Philippe Flajolet
This extended abstract is dedicated to the analysis of the height of
non-plane unlabelled rooted binary trees. The height of such a tree
chosen
uniformly among those of size $n$ is proved to have a limiting theta
distribution, both in a central and local sense. Moderate as well as
large
deviations estimates are also derived. The proofs rely on the analysis
(in the
complex plane) of generating functions associated with trees of
bounded height.
http://arxiv.org/abs/0807.2365
---------------------------------------------------------------
7294. BROWNIAN SURVIVAL AND LIFSHITZ TAIL IN PERTURBED LATTICE DISORDER
Ryoki Fukushima
We consider the annealed asymptotics for the survival probability of
Brownian
motion among randomly distributed traps. The configuration of traps is
given by
independent displacements of the lattice points. We determined the
asymptotics
for the logarithm of the survival probability up to multiplicative
constant. As
applications, we show the Lifshitz tail effect of the density of
states of
associated random Schr\"{o}dinger operator and intermittency for the
parabolic
Anderson problem.
http://arxiv.org/abs/0807.2486
---------------------------------------------------------------
7295. ARBITRAGE AND DEFLATORS IN ILLIQUID MARKETS
Teemu Pennanen
This paper presents a stochastic model for discrete-time trading in
financial
markets where trading costs are given by convex cost functions and
portfolios
are constrained by convex sets. The model does not assume the
existence of a
cash account/numeraire. In addition to classical frictionless markets
and
markets with transaction costs or bid-ask spreads, our framework
covers markets
with nonlinear illiquidity effects for large instantaneous trades. In
the
presence of nonlinearities, the classical notion of arbitrage turns
out to have
two equally meaningful generalizations, a marginal and a scalable one.
We study
their relations to state price deflators by analyzing two auxiliary
market
models describing the local and global behavior of the cost functions
and
constraints.
http://arxiv.org/abs/0807.2526
---------------------------------------------------------------
7296. QUENCHED LDP FOR WORDS IN A LETTER SEQUENCE
Matthias Birkner and Andreas Greven and Frank den Hollander
When we cut an i.i.d. sequence of letters into words according to an
independent renewal process, we obtain an i.i.d. sequence of words. In
the
annealed large deviation principle (LDP) for the empirical process of
words,
the rate function is the specific relative entropy of the observed law
of words
w.r.t. the reference law of words. In the present paper we consider the
quenched LDP, i.e., we condition on a typical letter sequence. We
focus on the
case where the renewal process has an algebraic tail. The rate
function turns
out to be a sum of two terms, one being the annealed rate function,
the other
being proportional to the specific relative entropy of the observed
law of
letters w.r.t. the reference law of letters, with the former being
obtained by
concatenating the words and randomising the location of the origin. The
proportionality constant equals the tail exponent of the renewal
process.
Earlier work by Birkner considered the case where the renewal process
has an
exponential tail, in which case the rate function turns out to be the
first
term on the set where the second term vanishes and to be infinite
elsewhere.
We apply our LDP to prove that the radius of convergence of the
moment
generating function of the collision local time of two strongly
transient
random walks on \Z^d, d \geq 1, strictly increases when we condition
on one of
the random walks, both in discrete time and in continuous time. The
presence of
these gaps implies the existence of an intermediate phase for the long-
time
behaviour of a class of coupled branching processes, interacting
diffusions,
respectively, directed polymers in random environments.
http://arxiv.org/abs/0807.2611
---------------------------------------------------------------
7297. DECAY OF COVARIANCES, UNIQUENESS OF ERGODIC COMPONENT AND
SCALING LIMIT FOR A CLASS OF \NABLA\PHI SYSTEMS WITH NON-CONVEX
POTENTIAL
Codina Cotar and Jean-Dominique Deuschel
We consider a gradient interface model on the lattice with interaction
potential which is a nonconvex perturbation of a convex potential.
Using a
technique which decouples the neighboring vertices sites into even and
odd
vertices, we show for a class of non-convex potentials: the uniqueness
of
ergodic component for \nabla\phi-Gibbs measures, the decay of
covariances, the
scaling limit and the strict convexity of the surface tension.
http://arxiv.org/abs/0807.2621
---------------------------------------------------------------
7298. A DECOMPOSITION RESULT FOR THE HAAR DISTRIBUTION ON THE
ORTHOGONAL GROUP
Morris L. Eaton and Robb J. Muirhead
Let H be a Haar distributed random matrix on the group of pxp real
orthogonal
matrices. Partition H into four blocks: (1) the (1,1) element, (2)the
rest of
the first row, (3) the rest of the first column, and (4)the remaining
(p-1)x(p-1) matrix. The marginal distribution of (1) is well known. In
this
paper, we give the conditional distribution of (2) and (3) given (1),
and the
conditional distribution of (4) given (1), (2), (3). This conditional
specification uniquely determines the Haar distribution. The two
conditional
distributions involve well known probability distributions namely, the
uniform
distribution on the unit sphere in p-1 dimensional space and the Haar
distribution on (p-2)x(p-2) orthogonal matrices. Our results show how to
construct the Haar distribution on pxp orthogonal matrices from the Haar
distribution on (p-2)x(p-2) orthogonal matrices coupled with the uniform
distribution on the unit sphere in p-1 dimensions.
http://arxiv.org/abs/0807.2598
---------------------------------------------------------------
7299. ON ASYMPTOTICS OF EXCHANGEABLE COALESCENTS WITH MULTIPLE
COLLISIONS
Alexander Gnedin and Alex Iksanov and Martin M\"ohle
We study the number of collisions $X_n$ of an exchangeable coalescent
with
multiple collisions ($\Lambda$-coalescent) which starts with $n$
particles and
is driven by rates determined by a finite characteristic measure $
\nu({\rm
d}x)=x^{-2}\Lambda({\rm d}x)$. Via a coupling technique we derive
limiting laws
of $X_n$, using previous results on regenerative compositions derived
from
stick-breaking partitions of the unit interval. The possible limiting
laws of
$X_n$ include normal, stable with index $1\le\alpha<2$ and Mittag-
Leffler
distributions. The results apply, in particular, to the case when $\nu
$ is a
beta$(a-2,b)$ distribution with parameters $a>2$ and $b>0$. The
approach taken
allows to derive asymptotics of three other functionals of the
coalescent, the
absorption time, the length of an external branch chosen at random
from the $n$
external branches, and the number of collision events that occur
before the
randomly selected external branch coalesces with one of its neighbours.
http://arxiv.org/abs/0807.2742
---------------------------------------------------------------
7300. ANNEALED VS QUENCHED CRITICAL POINTS FOR A RANDOM WALK PINNING
MODEL
Matthias Birkner and Rongfeng Sun
We study a random walk pinning model, where conditioned on a simple
random
walk Y on Z^d acting as a random medium, the path measure of a second
independent simple random walk X up to time t is Gibbs transformed with
Hamiltonian -L_t(X,Y), where L_t(X,Y) is the collision local time
between X and
Y up to time t. This model arises naturally in various contexts,
including the
study of the parabolic Anderson model with moving catalysts, the
parabolic
Anderson model with Brownian noise, and the directed polymer model. It
falls in
the same framework as the pinning and copolymer models, and exhibits a
localization-delocalization transition as the inverse temperature
\beta varies.
We show that in dimensions d=1,2, the annealed and quenched critical
values of
\beta are both 0, while in dimensions d\geq 4, the quenched critical
value of
\beta is strictly larger than the annealed critical value (which is
positive).
This implies the existence of certain intermediate regimes for the
parabolic
Anderson model with Brownian noise and the directed polymer model. For
d\geq 5,
the same result has recently been established by Birkner, Greven and den
Hollander via a quenched large deviation principle. Our proof is based
on a
fractional moment method used recently by Derrida, Giacomin, Lacoin and
Toninelli to establish the non-coincidence of annealed and quenched
critical
points for the pinning model in the disorder-relevant regime. The
critical case
d=3 remains open.
http://arxiv.org/abs/0807.2752
---------------------------------------------------------------
7301. PHASE TRANSITION ON THE DEGREE SEQUENCE OF A MIXED RANDOM GRAPH
PROCESS
Xian-Yuan Wu and Zhao Dong and Ke Liu and Kai-Yuan Cai
This paper focuses on the problem of the degree sequence for a mixed
random
graph process which continuously combines the {\it classical} model
and the BA
model. Note that the number of step added edges for the mixed model
are random
and unbounded. By developing a comparing argument, phase transition on
the
degree distributions of the mixed model is revealed: while the {\it pure
classical model possesses a {\it exponential} degree sequence, the
{\it pure}
BA model and the mixed model possess {\it power law} degree sequences.
We point
out that the intermediate mixed model can be looked as a BA model with
{\it
sublinear preferential attachment}.
http://arxiv.org/abs/0807.2811
---------------------------------------------------------------
7302. THE DEGREE SEQUENCE OF A SCALE-FREE RANDOM GRAPH PROCESS WITH
HARD COPYING
Gao-Rong Ning and Xian-Yuan Wu and Kai-Yuan Cai
We consider a simple random graph process with {\it hard} copying as
following: At any Time-Step $t$, with probability $0<\alpha\leq 1$ a
new vertex
$v_t$ is added and then $m$ edges incident with $v_t$ are added in the
manner
of {\it preferential attachment}; or with probability $1-\alpha$ a
existing
vertex is copied uniformly at random. We prove in the paper that, when
$\alpha$
large enough, the model possesses a mean degree sequence as $ d_{k}
\thicksim
Ck^{-(1+2\alpha)}$, where $d_k$ be the limit mean proportion of
vertices of
degree $k$. Note that in the present model, while a vertex with large
degree is
copied, the number of added edges is just its degree, so the number of
added
edges is not upper bounded.
http://arxiv.org/abs/0807.2819
---------------------------------------------------------------
7303. METHOD OF MOMENTS ESTIMATION OF ORNSTEIN-UHLENBECK PROCESSES
DRIVEN BY GENERAL L\'{E}VY PROCESS
Konstantinos Spiliopoulos
Ornstein-Uhlenbeck processes driven by general L\'{e}vy process are
considered in this paper. We derive strongly consistent estimators for
the
moments of the underlying L\'{e}vy process and for the mean reverting
parameter
of the Ornstein-Uhlenbeck process. Moreover, we prove that the
estimators are
asymptotically normal. Finally, we test the empirical performance of our
estimators in a simulation study and we fit the model to real data.
http://arxiv.org/abs/0807.2832
---------------------------------------------------------------
7304. LEARNING FROM EXPERTS: A SURVEY
Irene Valsecchi
The survey is concerned with the issue of information transmission from
experts to non-experts. Two main approaches to the use of experts can be
traced. According to the game-theoretic approach expertise is a case of
asymmetric information between the expert, who is the better informed
agent,
and the non-expert, who is either a decision-maker or an evaluator of
the
expert's performance. According to the Bayesian decision-theoretic
approach the
expert is the agent who announces his probabilistic opinion, and the
non-expert
has to incorporate that opinion into his beliefs in a consistent way,
despite
his poor understanding of the expert's substantive knowledge. The two
approaches ground the relationships between experts and non-experts on
so
different premises that their results are very poorly connected.
http://arxiv.org/abs/0807.2931
---------------------------------------------------------------
7305. LIMIT THEOREMS FOR SOME ADAPTIVE MCMC ALGORITHMS WITH
SUBGEOMETRIC KERNELS
Yves Atchad\'e and Gersende Fort (LTCI)
This paper deals with the ergodicity and the existence of a strong law
of
large numbers for adaptive Markov Chain Monte Carlo. We show that a
diminishing
adaptation assumption together with a drift condition for positive
recurrence
is enough to imply ergodicity. Strengthening the drift condition to a
polynomial drift condition yields a strong law of large numbers for
possibly
unbounded functions. These results broaden considerably the class of
adaptive
MCMC algorithms for which rigorous analysis is now possible. As an
example, we
give a detailed analysis of the Adaptive Metropolis Algorithm of
Haario et al.
(2001) when the target distribution is sub-exponential in the tails.
http://arxiv.org/abs/0807.2952
---------------------------------------------------------------
7306. COEXISTENCE IN THREE TYPE LAST PASSAGE PERCOLATION MODEL
D. Coupier and P. Heinrich
A three types competition model governed by directed last passage
percolation
on $\mathbb{N}^{2}$ is considered. We prove that coexistence of the
three
types, i.e. the sets of vertices of the three types are simultaneously
unbounded, occurs with positive probability. Moreover, the asymptotic
angles
formed by the two competition interfaces with the horizontal axis are
determined and their probability of being different is positive. As a
key step,
a stochastic domination between subtrees of the last passage
percolation tree
is obtained.
http://arxiv.org/abs/0807.2987
---------------------------------------------------------------
7307. SUPERHEDGING IN ILLIQUID MARKETS
Teemu Pennanen
We study contingent claims in a discrete-time market model where trading
costs are given by convex functions and portfolios are constrained by
convex
sets. In addition to classical frictionless markets and markets with
transaction costs or bid-ask spreads, our framework covers markets with
nonlinear illiquidity effects for large instantaneous trades. We
derive dual
characterizations of superhedging conditions for contingent claim
processes in
a market without a cash account. The characterizations are given in
terms of
stochastic discount factors that correspond to martingale densities in
a market
with a cash account. The dual representations are valid under a
topological
condition and a weak consistency condition reminiscent of the ``law of
one
price'', both of which are implied by the no arbitrage condition in
the case of
classical perfectly liquid market models. We give alternative sufficient
conditions that apply to market models with nonlinear cost functions and
portfolio constraints.
http://arxiv.org/abs/0807.2962
---------------------------------------------------------------
7308. STOCHASTIC MAXIMUM PRINCIPLE FOR A PDES WITH NOISE AND CONTROL
ON THE BOUNDARY
Giuseppina Guatteri
In this paper we prove necessary conditions for optimality of a
stochastic
control problem for a class of stochastic partial differential
equations that
is controlled through the boundary. This kind of problems can be
interpreted as
a stochastic control problem for an evolution system in an Hilbert
space. The
regularity of the solution of the adjoint equation, that is a backward
stochastic equation in infinite dimension, plays a crucial role in the
formulation of the maximum principle.
http://arxiv.org/abs/0807.3096
---------------------------------------------------------------
7309. FUNCTIONAL INEQUALITIES FOR HEAVY TAILS DISTRIBUTIONS AND
APPLICATION TO ISOPERIMETRY
Patrick Cattiaux (LSProba) and Nathael Gozlan (LAMA) and Arnaud
Guillin (LATP), Cyril Roberto (LAMA)
This paper is devoted to the study of probability measures with heavy
tails.
Using the Lyapunov function approach we prove that such measures satisfy
different kind of functional inequalities such as weak Poincar\'e and
weak
Cheeger, weighted Poincar\'e and weighted Cheeger inequalities and
their dual
forms. Proofs are short and we cover very large situations. For product
measures on $\R^n$ we obtain the optimal dimension dependence using
the mass
transportation method. Then we derive (optimal) isoperimetric
inequalities.
Finally we deal with spherically symmetric measures. We recover and
improve
many previous results.
http://arxiv.org/abs/0807.3112
---------------------------------------------------------------
7310. THE PATTERN OF GENETIC HITCHHIKING UNDER RECURRENT MUTATION
Joachim Hermisson and Peter Pfaffelhuber
Genetic hitchhiking describes evolution at a neutral locus that is
linked to
a selected locus. If a beneficial allele rises to fixation at the
selected
locus, a characteristic polymorphism pattern (so-called selective sweep)
emerges at the neutral locus. The classical model assumes that
fixation of the
beneficial allele occurs from a single copy of this allele that arises
by
mutation. However, recent theory (Pennings and Hermisson, 2006a;
Pennings and
Hermisson, 2006b) has shown that recurrent beneficial mutation at
biologically
realistic rates can lead to markedly different polymorphism patterns,
so called
soft selective sweeps. We extend an approach that has recently been
developed
for the classical hitchhiking model (Schweinsbergand Durrett, 2005;
Etheridge,
Pfaffelhuber, Wakolbinger, 2006) to study the recurrent mutation
scenario. We
show that the genealogy at the neutral locus can be approximated (to
leading
orders in the selection strength) by a marked Yule process with
immigration.
Using this formalism, we derive an improved analytical approximation
for the
expected heterozygosity at the neutral locus at the time of fixation
of the
beneficial allele.
http://arxiv.org/abs/0807.3167
---------------------------------------------------------------
7311. ON THE LOCALIZED PHASE OF A COPOLYMER IN AN EMULSION:
SUBCRITICAL PERCOLATION REGIME
Frank den Hollander and Nicolas Petrelis
The present paper is a continuation of \cite{dHP07b}. The object of
interest
is a two-dimensional model of a directed copolymer, consisting of a
random
concatenation of hydrophobic and hydrophilic monomers, immersed in an
emulsion,
consisting of large blocks of oil and water arranged in a percolation-
type
fashion. The copolymer interacts with the emulsion through an
interaction
Hamiltonian that favors matches and disfavors mismatches between the
monomers
and the solvents, in such a way that the interaction with the oil is
stronger
than with the water.
The model has two regimes, supercritical and subcritical, depending
on
whether the oil blocks percolate or not. In \cite{dHP07b} we focussed
on the
supercritical regime and obtained a complete description of the phase
diagram,
which consists of two phases separated by a single critical curve. In
the
present paper we focus on the subcritical regime and show that the phase
diagram consists of four phases separated by three critical curves
meeting in
two tricritical points.
http://arxiv.org/abs/0807.3190
---------------------------------------------------------------
7312. REVERSIBILITY OF SOME CHORDAL SLE$(\KAPPA;\RHO)$ TRACES
Dapeng Zhan
We prove that, for $\kappa\in(0,4)$ and $\rho\ge (\kappa-4)/2$, the
chordal
SLE$(\kappa;\rho)$ trace started from $(0;0^+)$ or $(0;0^-)$ satisfies
the
reversibility property. And we obtain the equation for the reversal of
the
chordal SLE$(\kappa;\rho)$ trace started from $(0;b_0)$, where $b_0>0$.
http://arxiv.org/abs/0807.3265
---------------------------------------------------------------
7313. CONVERGENCE OF SYMMETRIC MARKOV CHAINS ON $\Z^D$
R.F. Bass and T. Kumagai and and T. Uemura
For each $n$ let $Y^n_t$ be a continuous time symmetric Markov chain
with
state space $n^{-1} \Z^d$. A condition in terms of the conductances is
given
for the convergence of the $Y^n_t$ to a symmetric Markov process $Y_t$
on
$\R^d$. We have weak convergence of $\{Y^n_t: t\leq t_0\}$ for every
$t_0$ and
every starting point. The limit process $Y$ has a continuous part and
may also
have jumps.
http://arxiv.org/abs/0807.3268
---------------------------------------------------------------
7314. VISIBILITY TO INFINITY IN THE HYPERBOLIC PLANE, DESPITE OBSTACLES
Itai Benjamini and Johan Jonasson and Oded Schramm and Johan Tykesson
Suppose that $Z$ is a random closed subset of the hyperbolic plane $
\H^2$,
whose law is invariant under isometries of $\H^2$. We prove that if the
probability that $Z$ contains a fixed ball of radius 1 is larger than
some
universal constant $p<1$, then there is positive probability that $Z$
contains
(bi-infinite) lines.
We then consider a family of random sets in $\H^2$ that satisfy some
additional natural assumptions. An example of such a set is the
covered region
in the Poisson Boolean model. Let $f(r)$ be the probability that a
line segment
of length $r$ is contained in such a set $Z$. We show that if $f(r)$
decays
fast enough, then there are almost surely no lines in $Z$. We also
show that if
the decay of $f(r)$ is not too fast, then there are almost surely
lines in $Z$.
In the case of the Poisson Boolean model with balls of fixed radius $R
$ we
characterize the critical intensity for the almost sure existence of
lines in
the covered region by an integral equation.
We also determine when there are lines in the complement of a
Poisson process
on the Grassmannian of lines in $\H^2$.
http://arxiv.org/abs/0807.3308
---------------------------------------------------------------
7315. SPACE-TIME CURRENT PROCESS FOR INDEPENDENT RANDOM WALKS IN ONE
DIMENSION
Rohini Kumar
In a system made up of independent random walks, fluctuations of order
$n^{1/4}$ from the hydrodynamic limit come from particle current across
characteristics. We show that a two-parameter space-time particle
current
process converges to a two-parameter Gaussian process. These Gaussian
processes
also appear as the limit for the one-dimensional random average
process. The
final section of this paper looks at large deviations of the current
process.
http://arxiv.org/abs/0807.3313
---------------------------------------------------------------
7316. CUTTING CAKES CORRECTLY
Theodore P. Hill
Without additional hypotheses, Proposition 7.1 in Brams and Taylor's
book
"Fair Division" (Cambridge University Press, 1996) is false, as are
several
related Pareto-optimality theorems of Brams, Jones and Klamler in
their 2006
cake-cutting paper.
http://arxiv.org/abs/0807.3117
---------------------------------------------------------------
7317. ON THE SPECTRAL GAP OF THE KAC WALK AND OTHER BINARY COLLISION
PROCESSES
Pietro Caputo
We give a new and elementary computation of the spectral gap of the
Kac walk
on the N-sphere. The result is obtained as a by-product of a more
general
observation which allows to reduce the analysis of the spectral gap of
an
N-component system to that of the same system for N=3. The method
applies to a
number of random 'binary collision' processes with complete-graph
structure,
including non-homogeneous examples such as exclusion and colored
exclusion
processes with site disorder.
http://arxiv.org/abs/0807.3415
---------------------------------------------------------------
7318. LIMIT THEOREM FOR RANDOM WALK IN WEAKLY DEPENDENT RANDOM SCENERY
Nadine Guillotin-Plantard (ICJ) and Cl\'ementine Prieur (LSProba)
Let $S=(S_k)_{k\geq 0}$ be a random walk on $\mathbb{Z}$ and
$\xi=(\xi_{i})_{i\in\mathbb{Z}}$ a stationary random sequence of
centered
random variables, independent of $S$. We consider a random walk in
random
scenery that is the sequence of random variables $(\Sigma_n)_{n\geq
0}$ where
$$\Sigma_n=\sum_{k=0}^n \xi_{S_k}, n\in\mathbb{N}.$$ Under a weak
dependence
assumption on the scenery $\xi$ we prove a functional limit theorem
generalizing Kesten and Spitzer's theorem (1979).
http://arxiv.org/abs/0807.3441
---------------------------------------------------------------
7319. A STOCHASTIC SIR MODEL WITH CONTACT-TRACING: LARGE POPULATION
LIMITS AND STATISTICAL INFERENCE
St\'ephan Cl\'emen\c{c}on (LTCI and METaRISK) and Viet Chi Tran
(LPP) and Hector De Arazoza (MATCOM)
A stochastic epidemic model accounting for the effect of contact-
tracing on
the spread of an infectious disease is studied. Precisely, individuals
identified as infected may contribute to detecting other infectious
individuals
by providing information related to persons with whom they have had
possibly
infectious contacts. The population evolves through demographic,
infection and
detection processes, in a way that its temporal evolution is described
by a
stochastic Markov process, of which the component accounting for the
contact-tracing feature is assumed to be valued in a space of point
measures.
For adequate scalings of the demographic, infection and detection
rates, it is
shown to converge to the weak deterministic solution of a PDE system,
as a
parameter n, interpreted as the population size roughly speaking,
becomes
large. From the perspective of the analysis of infectious disease
data, this
approximation result may serve as a key tool for exploring the
asymptotic
properties of standard inference methods such as maximum likelihood
estimation.
We state preliminary statistical results in this context. Eventually,
relation
of the model to the available data of the HIV epidemic in Cuba, in which
country a contact-tracing detection system has been set up since 1986,
is
investigated and numerical applications are carried out.
http://arxiv.org/abs/0807.3462
---------------------------------------------------------------
7320. JOINT ANALYSIS AND ESTIMATION OF STOCK PRICES AND TRADING VOLUME
IN BARNDORFF-NIELSEN AND SHEPHARD STOCHASTIC VOLATILITY MODELS
Friedrich Hubalek and Petra Posedel
We introduce a variant of the Barndorff-Nielsen and Shephard stochastic
volatility model where the non Gaussian Ornstein-Uhlenbeck process
describes
some measure of trading intensity like trading volume or number of
trades
instead of unobservable instantaneous variance. We develop an explicit
estimator based on martingale estimating functions in a bivariate
model that is
not a diffusion, but admits jumps. It is assumed that both the
quantities are
observed on a discrete grid of fixed width, and the observation
horizon tends
to infinity. We show that the estimator is consistent and
asymptotically normal
and give explicit expressions of the asymptotic covariance matrix. Our
method
is illustrated by a finite sample experiment and a statistical
analysis on the
International Business Machines Corporation (IBM) stock from the New
York Stock
Exchange (NYSE) and the Microsoft Corporation (MSFT) stock from Nasdaq
during a
history of five years.
http://arxiv.org/abs/0807.3464
---------------------------------------------------------------
7321. ASYMPTOTIC ANALYSIS FOR A SIMPLE EXPLICIT ESTIMATOR IN BARNDORFF-
NIELSEN AND SHEPHARD STOCHASTIC VOLATILITY MODELS
Friedrich Hubalek and Petra Posedel
We provide a simple explicit estimator for discretely observed
Barndorff-Nielsen and Shephard models, prove rigorously consistency and
asymptotic normality based on the single assumption that all moments
of the
stationary distribution of the variance process are finite, and give
explicit
expressions for the asymptotic covariance matrix.
We develop in detail the martingale estimating function approach
for a
bivariate model, that is not a diffusion, but admits jumps. We do not
use
ergodicity arguments.
We assume that both, logarithmic returns and instantaneous variance
are
observed on a discrete grid of fixed width, and the observation
horizon tends
to infinity. As the instantaneous variance is not observable in
practice, our
results cannot be applied immediately. Our purpose is to provide a
theoretical
analysis as a starting point and benchmark for further developments
concerning
optimal martingale estimating functions, and for theoretical and
empirical
investigations, that replace the variance process with a substitute,
such as
number or volume of trades or implied variance from option data.
http://arxiv.org/abs/0807.3479
---------------------------------------------------------------
7322. ON THE ADJUSTMENT COEFFICIENT, DRAWDOWNS AND LUNDBERG-TYPE
BOUNDS FOR RANDOM WALK
Isaac Meilijson
Consider a random walk whose (light-tailed) increments have positive
mean.
Lower and upper bounds are provided for the expected maximal value of
the
random walk until it experiences a given drawdown d. These bounds,
related to
the Calmar ratio in Finance, are of the form (exp{alpha d}-1)/alpha
and (K
exp{alpha d}-1)/alpha for some K>1, in terms of the adjustment
coefficient
alpha (E[exp{-alpha X}]=1) of the insurance risk literature. Its inverse
1/alpha has been recently derived by Aumann and Serrano as an index of
riskiness of the random variable X. This article also complements the
Lundberg
exponential stochastic upper bound and the Cramer-Lundberg
approximation for
the expected minimum of the random walk, with an exponential
stochastic lower
bound. The tail probability bounds are of the form C exp{-alpha x} and
exp{-alpha x} respectively, for some 1/K < C < 1. Our treatment of the
problem
involves Skorokhod embeddings of random walks in Martingales,
especially via
the Azema-Yor and Dubins stopping times, adapted from standard
Brownian Motion
to exponential Martingales.
http://arxiv.org/abs/0807.3506
---------------------------------------------------------------
7323. WILL THE ANNOUNCED INFLUENZA PANDEMIC REALLY HAPPEN?
Rinaldo B. Schinazi
We propose two simple probability models to compute the probability of
an
influenza pandemic. Under a random walk model the probability that all
pandemics between times 0 and 300 occur by time 150 is 1/2. Under a
Poisson
model with mean inter arrival time of 30 years the probability that no
pandemic
occurs during at least 60 years is 14%. These probabilities are much
higher
than generally perceived. So yes the next influenza pandemic will
happen but
maybe much later than generally thought.
http://arxiv.org/abs/0807.3524
---------------------------------------------------------------
7324. QUANTUM HOMODYNE TOMOGRAPHY AS AN INFORMATIONALLY COMPLETE
POSITIVE OPERATOR VALUED MEASURE
P. Albini and E. De Vito and A. Toigo
We define a positive operator valued measure $E$ on $[0,2\pi]\times R$
describing the measurement of randomly sampled quadratures in quantum
homodyne
tomography, and we study its probabilistic properties. Moreover, we
give a
mathematical analysis of the relation between the description of a
state in
terms of $E$ and the description provided by its Wigner transform.
http://arxiv.org/abs/0807.3437
---------------------------------------------------------------
7325. ON THE EXPECTED DIAMETER OF AN L2-BOUNDED MARTINGALE
Lester E. Dubins and David Gilat and Isaac Meilijson
It is shown that the ratio between the expected diameter of an L2-
bounded
martingale and the standard deviation of its last term cannot exceed
sqrt(3).
Moreover, a one-parameter family of stopping times on standard
Brownian Motion
is exhibited, for which the sqrt(3) upper bound is attained. These
stopping
times, one for each cost-rate c, are optimal when the payoff for
stopping at
time t is the diameter D(t) obtained up to time t minus the hitherto
accumulated cost c t. A quantity related to diameter, maximal drawdown
(or
rise), is introduced and its expectation is shown to be bounded by
sqrt(2)
times the standard deviation of the last term of the martingale. These
results
complement the Dubins and Schwarz respective bounds 1 and sqrt(2) for
the
ratios between the expected maximum and maximal absolute value of the
martingale and the standard deviation of its last term. Dynamic
programming
(gambling theory) methods are used for the proof of optimality.
http://arxiv.org/abs/0807.3571
---------------------------------------------------------------
7326. EIGENVECTORS OF RANDOM GRAPHS: NODAL DOMAINS
Yael Dekel and James R. Lee and Nathan Linial
We initiate a systematic study of eigenvectors of random graphs.
Whereas much
is known about eigenvalues of graphs and how they reflect properties
of the
underlying graph, relatively little is known about the corresponding
eigenvectors. Our main focus in this paper is on the nodal domains
associated
with the different eigenfunctions. In the analogous realm of
Laplacians of
Riemannian manifolds, nodal domains have been the subject of intensive
research
for well over a hundred years. Graphical nodal domains turn out to have
interesting and unexpected properties. Our main theorem asserts that
there is a
constant c such that for almost every graph G, each eigenfunction of G
has at
most two large nodal domains, and in addition at most c exceptional
vertices
outside these primary domains. We also discuss variations of these
questions
and briefly report on some numerical experiments which, in particular,
suggest
that almost surely there are just two nodal domains and no exceptional
vertices.
http://arxiv.org/abs/0807.3675
---------------------------------------------------------------
7327. UNIVERSALITY OF THE LIMIT SHAPE OF RANDOM CONVEX LATTICE POLYGONS
Leonid V. Bogachev and Sakhavat M. Zarbaliev
Let $\Pi_n$ be the set of planar convex lattice polygons $\Gamma$
(i.e., with
vertices on $\mathbb{Z}_+^2$ and non-negative inclination of all
edges) with
fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the
limit
shape of a typical polygon $\Gamma\in\Pi_n$ as $n\to\infty$ with
respect to a
certain parametric family of probability measures $\{P_n^r\}$ ($0<r<
\infty$) on
the space $\Pi_n$, including the uniform distribution ($r=1$). We show
that if
$0<C_1\le n_2/n_1\le C_2<\infty$ then, under the scaling $(1/n_1,1/
n_2)$, the
limit shape is universal in the class $\{P_n^r\}$ and thus coincides
with that
for the uniform distribution $P^1_n$ (found independently by Vershik,
B\'ar\'any, and Sinai). Our result gives a partial affirmative answer to
Vershik--Prokhorov's universality conjecture. The measure $P^r_n$ is
constructed, using Sinai's approach, as a conditional distribution
induced by a
suitable product measure $Q^r$ defined on the space $\Pi=\cup_n\Pi_n$ of
polygons with a free right end. The proof involves subtle analytical
tools
including the M\"obius inversion formula and properties of zeroes of the
Riemann zeta function.
http://arxiv.org/abs/0807.3682
---------------------------------------------------------------
7328. COUNTING THE FACES OF RANDOMLY-PROJECTED HYPERCUBES AND
ORTHANTS, WITH APPLICATIONS
David L. Donoho and Jared Tanner
Let $A$ be an $n$ by $N$ real valued random matrix, and $\h$ denote the
$N$-dimensional hypercube. For numerous random matrix ensembles, the
expected
number of $k$-dimensional faces of the random $n$-dimensional zonotope
$A\h$
obeys the formula $E f_k(A\h) /f_k(\h) = 1-P_{N-n,N-k}$, where $P_{N-
n,N-k}$ is
a fair-coin-tossing probability. The formula applies, for example,
where the
columns of $A$ are drawn i.i.d. from an absolutely continuous symmetric
distribution. The formula exploits Wendel's Theorem\cite{We62}.
Let $\po$ denote the positive orthant; the expected number of $k$-
faces of
the random cone$A \po$ obeys $ {\cal E} f_k(A\po) /f_k(\po) = 1 - P_{N-
n,N-k}$.
The formula applies to numerous matrix ensembles, including those with
iid
random columns from an absolutely continuous, centrally symmetric
distribution.
There is an asymptotically sharp threshold in the behavior of face
counts of
the projected hypercube; thresholds known for projecting the simplex
and the
cross-polytope, occur at very different locations. We briefly consider
face
counts of the projected orthant when $A$ does not have mean zero;
these do
behave similarly to those for the projected simplex. We consider non-
random
projectors of the orthant; the 'best possible' $A$ is the one
associated with
the first $n$ rows of the Fourier matrix.
These geometric face-counting results have implications for signal
processing, information theory, inverse problems, and optimization.
Most of
these flow in some way from the fact that face counting is related to
conditions for uniqueness of solutions of underdetermined systems of
linear
equations.
http://arxiv.org/abs/0807.3590
---------------------------------------------------------------
7329. REFLECTION PRINCIPLE AND OCONE MARTINGALES
Lo\"ic Chaumont (LAREMA) and L. Vostrikova (LAREMA)
Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic
process. We
prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real
numbers which
converges to 0 and such that $M$ satisfies the reflection property at
all
levels $a_n$ and $2a_n$ with $n\geq 1$, then $M$ is an Ocone local
martingale
with respect to its natural filtration. We state the subsequent open
question:
is this result still true when the property only holds at levels $a_n
$? Then we
prove that the later question is equivalent to the fact that for
Brownian
motion, the $\sigma$-field of the invariant events by all reflections
at levels
$a_n$, $n\ge1$ is trivial. We establish similar results for skip free
$\mathbb{Z}$-valued processes and use them for the proof in continuous
time,
via a discretisation in space.
http://arxiv.org/abs/0807.3816
---------------------------------------------------------------
7330. ON THE STRONG APPROXIMATION OF NON-OVERLAPPING M-SPACINGS
PROCESSES
Salim Bouzebda (LSTA) and Nabil Nessigha (LSTA)
In this paper we establish strong approximations of the uniform
non-overlapping m-spacings process extending the preceding results.
Our methods
rely on the Mason and Van Zwet's invariance principle.
http://arxiv.org/abs/0807.3868
---------------------------------------------------------------
7331. A SHORT PROOF AND A GENERALIZATION OF THE BKR-INEQUALITY
Martin Hutzenthaler
The BKR-inequality (a.k.a. BK-conjecture) asserts that the probability
of two
events occurring on disjoint subsets of a finite family of independent
variables is dominated by the product of the probabilities of the two
events.
This correlation inequality has been conjectured and partially
established by
van den Berg and Kesten (1985) and has been proved by Reimer (2000).
We give a
short proof of the BKR-inequality and generalize it to arbitrary
families of
independent variables.
http://arxiv.org/abs/0807.4038
---------------------------------------------------------------
7332. LONG-TIME BEHAVIOR OF STOCHASTICALLY PERTURBED NEURONAL NETWORKS
Stefano Bonaccorsi and Delio Mugnolo
Our investigation is specially motivated by the stochastic version of a
common model of potential spread in a dendritic tree. We do not assume
the
noise in the junction points to be Markovian. In fact, we allow for
long-range
dependence in time of the stochastic perturbation. This leads to an
abstract
formulation in terms of a stochastic diffusion with dynamic boundary
conditions, featuring fractional Brownian motion. We prove results on
existence, uniqueness and asymptotics of weak and strong solutions to
such a
stochastic differential equation.
http://arxiv.org/abs/0807.4057
---------------------------------------------------------------
7333. AGGREGATION OF RISKS AND ASYMPTOTIC INDEPENDENCE
Abhimanyu Mitra and Sidney I. Resnick
We study the tail behavior of the distribution of the sum of
asymptotically
independent risks whose marginal distributions belong to the maximal
domain of
attraction of the Gumbel distribution. We impose conditions on the
distribution
of the risks $(X,Y)$ such that $P(X + Y > x) \sim (const)P (X > x)$.
With the
further assumption of non-negativity of the risks, the result is
extended to
more than two risks. We note a sufficient condition for a distribution
to
belong to both the maximal domain of attraction of the Gumbel
distribution and
the subexponential class. We provide examples of distributions which
satisfy
our assumptions. The examples include cases where the marginal
distributions of
$X$ and $Y$ are subexponential and also cases where they are not. In
addition,
the asymptotic behavior of linear combinations of such risks with
positive
coefficients is explored leading to an approximate solution of an
optimization
problem which is applied to portfolio design.
http://arxiv.org/abs/0807.4200
---------------------------------------------------------------
7334. LARGE SCALE BEHAVIOR OF SEMIFLEXIBLE HETEROPOLYMERS
Francesco Caravenna and Giambattista Giacomin and Massimiliano
Gubinelli
We consider a general discrete model for heterogeneous semiflexible
polymer
chains. Both the thermal noise and the inhomogeneous character of the
chain
(the disorder) are modeled in terms of random rotations. We focus on the
quenched regime, i.e., the analysis is performed for a given
realization of the
disorder. Semiflexible models differ substantially from random walks
on short
scales, but on large scales a Brownian behavior emerges. By exploiting
techniques from tensor analysis and non-commutative Fourier analysis, we
establish the Brownian character of the model on large scale and we
obtain an
expression for the diffusion constant. We moreover give conditions
yielding
quantitative mixing properties.
http://arxiv.org/abs/0807.4232
---------------------------------------------------------------
7335. THE $\LAMBDA$-COALESCENT SPEED OF COMING DOWN FROM INFINITY
Julien Berestycki and Nathanael Berestycki and and Vlada Limic
Consider a $\Lambda$-coalescent that comes down from infinity (meaning
that
it starts from a configuration containing infinitely many blocks at
time 0, yet
it has a finite number $N_t$ of blocks at any positive time $t>0$). We
exhibit
a deterministic function $v:(0,\infty)\to (0,\infty)$, such that $N_t/
v(t)\to
1$, almost surely and in $L^p$ for any $p\geq 1$, as $t\to 0$. Our
approach
relies on a novel martingale technique.
http://arxiv.org/abs/0807.4278
---------------------------------------------------------------
7336. PENALISING SYMMETRIC STABLE L\'EVY PATHS
Kouji Yano and Yuko Yano and Marc Yor
Limit theorems for the normalized laws with respect to two kinds of
weight
functionals are studied for any symmetric stable L\'evy process of
index $ 1 <
\alpha \le 2 $. The first kind is a function of the local time at the
origin,
and the second kind is the exponential of an occupation time integral.
Special
emphasis is put on the role played by a stable L\'evy counterpart of the
universal $ \sigma $-finite measure, found in [9] and [10], which
unifies the
corresponding limit theorems in the Brownian setup for which $ \alpha
=2 $.
http://arxiv.org/abs/0807.4336
---------------------------------------------------------------
7337. NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS FOR RELAXED AND
STRICT CONTROL PROBLEMS OF BACKWARD SYSTEMS
Seid Bahlali
We consider a stochastic control problem where the set of strict
(classical)
controls is not necessarily convex, and the system is governed by a
nonlinear
backward stochastic differential equation. By introducing a new
approach, we
establish necessary as well as sufficient conditions of optimality for
two
models. The first concerns the relaxed controls, who are measure-valued
processes. The second is a particular case of the first and relates to
strict
control problems.
http://arxiv.org/abs/0807.4297
---------------------------------------------------------------
7338. ON THE RANDOM SATISFIABLE PROCESS
Michael Krivelevich and Benny Sudakov and Dan Vilenchik
In this work we suggest a new model for generating random satisfiable
k-CNF
formulas. To generate such formulas -- randomly permute all 2^k
\binom{n}{k}
possible clauses over the variables x_1, ..., x_n, and starting from
the empty
formula, go over the clauses one by one, including each new clause as
you go
along if after its addition the formula remains satisfiable. We study
the
evolution of this process, namely the distribution over formulas
obtained after
scanning through the first m clauses (in the random permutation's
order).
Random processes with conditioning on a certain property being
respected are
widely studied in the context of graph properties. This study was
pioneered by
Ruci\'nski and Wormald in 1992 for graphs with a fixed degree
sequence, and
also by Erd\H{o}s, Suen, and Winkler in 1995 for triangle-free and
bipartite
graphs. Since then many other graph properties were studied such as
planarity
and H-freeness. Thus our model is a natural extension of this approach
to the
satisfiability setting.
Our main contribution is as follows. For m \geq cn, c=c(k) a
sufficiently
large constant, we are able to characterize the structure of the
solution space
of a typical formula in this distribution. Specifically, we show that
typically
all satisfying assignments are essentially clustered in one cluster,
and all
but e^{-\Omega(m/n)} n of the variables take the same value in all
satisfying
assignments. We also describe a polynomial time algorithm that finds
with high
probability a satisfying assignment for such formulas.
http://arxiv.org/abs/0807.4326
---------------------------------------------------------------
7339. BOUNDS FOR THE TRANSITION DENSITY OF TIME-HOMOGENEOUS DIFFUSION
PROCESSES
Andrew N. Downes
The paper presents new simple sharp bounds for transition density
functions
for time-homogeneous diffusions processes. The bounds are obtained
under mild
conditions on the drift and diffusion coefficients, extending and
substantially
improving previous results in the literature which were limited to
drifts
satisfying a linear growth condition. They lead to an asymptotic
expression for
the transition density as the transition time approaches zero. While
the focus
is on the one-dimensional case, an extension to multiple dimensions is
discussed. Results are illustrated by numerical examples.
http://arxiv.org/abs/0807.4586
---------------------------------------------------------------
7340. THE $M_T/M_T/K_T+M_T$ QUEUE IN HEAVY TRAFFIC
Anatolii Puhalskii
We prove a heavy traffic limit theorem for a non-time homogeneous
Markovian
many server queue with customer abandonment.
http://arxiv.org/abs/0807.4621
---------------------------------------------------------------
7341. A STRONG LAW OF LARGE NUMBERS FOR STRONGLY MIXING PROCESSES
Aryeh Kontorovich and Anthony Brockwell
We prove a strong law of large numbers for a class of strongly mixing
processes. Our result rests on recent advances in understanding of
concentration of measure. It is simple to apply and gives finite-
sample (as
opposed to asymptotic) bounds, with readily computable rate constants.
In
particular, this makes it suitable for analysis of inhomogeneous Markov
processes. We demonstrate how it can be applied to establish an almost-
sure
convergence result for a class of models that includes as a special
case a
class of adaptive Markov chain Monte Carlo algorithms.
http://arxiv.org/abs/0807.4665
---------------------------------------------------------------
7342. RESCALED WEIGHTED RANDOM BALLS MODELS AND STABLE SELF-SIMILAR
RANDOM FIELDS
Jean-Christophe Breton (MIA) and Cl\'ement Dombry (LMA)
We consider weighted random balls in $\real^d$ distributed according
to a
random Poisson measure with heavy-tailed intensity and study the
asymptotic
behaviour of the total weight of some configurations in $\real^d$. This
procedure amounts to be very rich and several regimes appear in the
limit,
depending on the intensity of the balls, the zooming factor, the tail
parameters of the radii and of the weights. Statistical properties of
the limit
fields are also evidenced, such as isotropy, self-similarity or
dependence. One
regime is of particular interest and yields $\alpha$-stable stationary
isotropic self-similar generalized random fields which recovers Takenaka
fields, Telecom process or fractional Brownian motion.
http://arxiv.org/abs/0807.4700
---------------------------------------------------------------
7343. ON A CONJECTURE OF LAUGESEN AND MORPURGO
Mihai N. Pascu and Maria E. Gageonea
A well known conjecture of R. Laugesen and C. Morpurgo asserts that the
diagonal element of the Neumann heat kernel of the unit ball in
$\mathbb{R}^{n}$ ($n\geq 1$) is a radially increasing function. In
this paper,
we use probabilistic arguments to settle this conjecture, and, as an
application, we derive a new proof of the Hot Spots conjecture of J.
Rauch in
the case of the unit disk.
http://arxiv.org/abs/0807.4726
---------------------------------------------------------------
7344. HIERARCHICAL PINNING MODEL WITH SITE DISORDER: DISORDER IS
MARGINALLY IRRELEVANT
Hubert Lacoin
We study a hierarchical disordered pinning model with site disorder for
which, like in the bond disordered case [5, 8], there exists a value
of a
parameter b (enters in the definition of the hierarchical lattice) that
separates an irrelevant disorder regime and a relevant disorder
regime. We show
that for such a value of b the critical point of the disordered system
is
different from the critical point of the annealed version of the
model. The
proof goes beyond the technique used in [8] and it takes explicitly
advantage
of the inhomogeneous character of the Green function of the model.
http://arxiv.org/abs/0807.4864
---------------------------------------------------------------
7345. RANDOM MATRICES: UNIVERSALITY OF ESDS AND THE CIRCULAR LAW
Terence Tao and Van Vu
Given an $n \times n$ complex matrix $A$, let
$$\mu_{A}(x,y):= \frac{1}{n} |\{1\le i \le n, \Re \lambda_i \le x,
\Im
\lambda_i \le y\}|$$ be the empirical spectral distribution (ESD) of its
eigenvalues
$\lambda_i \in \BBC, i=1, ... n$.
We consider the limiting distribution (both in probability and in
the almost
sure convergence sense) of the normalized ESD $\mu_{\frac{1}{\sqrt{n}}
A_n}$ of
a random matrix $A_n = (a_{ij})_{1 \leq i,j \leq n}$ where the random
variables
$a_{ij} - \E(a_{ij})$ are iid copies of a fixed random variable $x$
with unit
variance. We prove a \emph{universality principle} for such ensembles,
namely
that the limit distribution in question is {\it independent} of the
actual
choice of $x$. In particular, in order to compute this distribution,
one can
assume that $x$ is real of complex gaussian. As a related result, we
show how
laws for this ESD follow from laws for the \emph{singular} value
distribution
of $\frac{1}{\sqrt{n}} A_n - zI$ for complex $z$. As a corollary we
establish
the Circular Law conjecture (in both strong and weak forms), that
asserts that
$\mu_{\frac{1}{\sqrt{n}} A_n}$ converges to the uniform measure on the
unit
disk when the $a_{ij}$ have zero mean.
http://arxiv.org/abs/0807.4898
---------------------------------------------------------------
7346. RANDOM NETWORKS WITH SUBLINEAR PREFERENTIAL ATTACHMENT: DEGREE
EVOLUTIONS
Steffen Dereich and Peter Morters
We define a dynamic model of random networks, where new vertices are
connected to old ones with a probability proportional to a sublinear
function
of their degree. We first give a strong limit law for the empirical
degree
distribution, and then have a closer look at the temporal evolution of
the
degrees of individual vertices, which we describe in terms of large and
moderate deviation principles. Using these results, we expose an
interesting
phase transition: in cases of strong preference of large degrees,
eventually a
single vertex emerges forever as vertex of maximal degree, whereas in
cases of
weak preference, the vertex of maximal degree is changing infinitely
often.
Loosely speaking, the transition between the two phases occurs in the
case when
a new edge is attached to an existing vertex with a probability
proportional to
the root of its current degree.
http://arxiv.org/abs/0807.4904
---------------------------------------------------------------
7347. FDR CONTROL FOR MULTIPLE HYPOTHESIS TESTING ON COMPOSITE NULLS
Zhiyi Chi
Multiple hypothesis testing often involves composite nulls, i.e.,
nulls that
are associated with two or more distributions. In many cases, it is
reasonable
to assume that there is a prior distribution on the distributions
despite it is
unknown. When the number of distributions under true nulls is finite,
we show
that under the above assumption, the false discover rate (FDR) can be
controlled using $p$-values computed under constraints imposed by the
empirical
distribution of the observations. Comparing to FDR control using $p$-
values
defined as maximum significance level over all null distributions, the
proposed
FDR control can have substantially more power.
http://arxiv.org/abs/0807.4879
---------------------------------------------------------------
7348. MOD-GAUSSIAN CONVERGENCE: NEW LIMIT THEOREMS IN PROBABILITY AND
NUMBER THEORY
Jean Jacod and Emmanuel Kowalski and Ashkan Nikeghbali
We introduce a new type of convergence in probability theory, which we
call
``mod-Gaussian convergence''. It is directly inspired by theorems and
conjectures, in random matrix theory and number theory, concerning
moments of
values of characteristic polynomials or zeta functions. We study this
type of
convergence in detail in the framework of infinitely divisible
distributions,
and exhibit some unconditional occurrences in number theory, in
particular for
families of $L$-functions over function fields in the Katz-Sarnak
framework. A
similar phenomenon of ``mod-Poisson convergence'' turns out to also
appear in
the classical Erd\H{o}s-K\'ac Theorem.
http://arxiv.org/abs/0807.4739
---------------------------------------------------------------
7349. HAZARD PROCESSES AND MARTINGALE HAZARD PROCESSES
Delia Coculescu and Ashkan Nikeghbali
In this paper, we provide a solution to two problems which have been
open in
default time modeling in credit risk. We first show that if $\tau$ is an
arbitrary random (default) time such that its Az\'ema's supermartingale
$Z_t^\tau=\P(\tau>t|\F_t)$ is continuous, then $\tau$ avoids stopping
times. We
then disprove a conjecture about the equality between the hazard
process and
the martingale hazard process, which first appeared in
\cite{jenbrutk1}, and we
show how it should be modified to become a theorem. The pseudo-
stopping times,
introduced in \cite{AshkanYor}, appear as the most general class of
random
times for which these two processes are equal. We also show that these
two
processes always differ when $\tau$ is an honest time.
http://arxiv.org/abs/0807.4958
---------------------------------------------------------------
7350. DECOMPOSITION OF ORDER STATISTICS OF SEMIMARTINGALES USING LOCAL
TIMES
Raouf Ghomrasni and Olivier Menoukeu Pamen
In a recent work \cite{BG}, given a collection of continuous
semimartingales,
authors derive a semimartingale decomposition from the corresponding
ranked
processes in the case that the ranked processes can meet more than two
original
processes at the same time. This has led to a more general
decomposition of
ranked processes. In this paper, we derive a more general result for
semimartingales (not necessarily continuous) using a simpler approach.
Furthermore, we also give a generalization of Ouknine \cite{O1, O2}
and Yan's
\cite{Y1} formula for local times of ranked processes
http://arxiv.org/abs/0807.5001
---------------------------------------------------------------
7351. STEIN'S METHOD AND NORMAL APPROXIMATION OF POISSON FUNCTIONALS
Giovanni Peccati (LSTA) and Josep Llu\'is Sol\'e (Universitat Aut
\'Onoma De Barcelona), Murad S. Taqqu (Boston University), Frederic
Utzet
(Universitat Aut\'Onoma De Barcelona)
We combine Stein's method with a version of Malliavin calculus on the
Poisson
space. As a result, we obtain explicit Berry-Ess\'een bounds in
Central Limit
Theorems (CLTs) involving multiple Wiener-It\^o integrals with respect
to a
general Poisson measure. We provide several applications to CLTs
related to
Ornstein-Uhlenbeck L\'evy processes.
http://arxiv.org/abs/0807.5035
---------------------------------------------------------------
7352. COEXISTENCE OF ORDERED AND DISORDERED PHASES IN POTTS MODELS IN
THE CONTINUUM
Anna De Masi and Immacolata Merola and Errico Presutti and Yvon
Vignaud
This is the second of two papers on a continuum version of the Potts
model,
where particles are points in $\mathbb R^d$, $d\ge 2$, with a spin
which may
take $S\ge 3$ possible values. Particles with different spins repel
each other
via a Kac pair potential of range $\ga^{-1}$, $\ga>0$. In this paper
we prove
phase transition, namely we prove that if the scaling parameter of the
Kac
potential is suitably small, given any temperature there is a value of
the
chemical potential such that at the given temperature and chemical
potential
there exist $S+1$ mutually distinct DLR measures.
http://arxiv.org/abs/0807.5080
---------------------------------------------------------------
7353. AN ANTICIPATING IT\^O FORMULA FOR L\'EVY PROCESSES
Elisa Al\`os and Jorge A. Le\'on and Josep Vives
In this paper, we use the Malliavin calculus techniques to obtain an
anticipative version of the change of variable formula for L\'evy
processes.
Here the coefficients are in the domain of the anihilation (gradient)
operator
in the "future sense", which includes the family of all adapted and
square-integrable processes. This domain was introduced on the Wiener
space by
Al\`os and Nualart.
http://arxiv.org/abs/0808.0035
---------------------------------------------------------------
7354. NON-STANDARD APPROXIMATIONS OF THE ITO-MAP
Peter Friz and Harald Oberhauser
The Wong-Zakai theorem asserts that ODEs driven by "reasonable" (e.g.
piecewise linear) approximations of Brownian motion converge to the
corresponding Stratonovich stochastic differential equation. With the
aid of
rough path analysis, we study "non-reasonable" approximations and go
beyond a
well-known criterion of [Ikeda--Watanabe, North Holland 1989] in the
sense that
our result applies to perturbations on all levels, exhibiting
additional drift
terms involving any iterated Lie brackets of the driving vector
fields. In
particular, this applies to the approximations by McShane ('72) and
Sussmann
('91). Our approach is not restricted to Brownian driving signals. At
last,
these ideas can be used to prove optimality of certain rough path
estimates.
http://arxiv.org/abs/0808.0337
---------------------------------------------------------------
7355. A MODIFIED LOOKDOWN CONSTRUCTION FOR THE XI-FLEMING-VIOT PROCESS
WITH MUTATION AND POPULATIONS WITH RECURRENT BOTTLENECKS
Matthias Birkner and Jochen Blath and Martin Moehle and Matthias
Steinruecken and Johanna Tams
Let $\Lambda$ be a finite measure on the unit interval. A
$\Lambda$-Fleming-Viot process is a probability measure valued Markov
process
which is dual to a coalescent with multiple collisions ($\Lambda$-
coalescent)
in analogy to the duality known for the classical Fleming Viot process
and
Kingman's coalescent, where $\Lambda$ is the Dirac measure in 0.
We explicitly construct a dual process of the coalescent with
simultaneous
multiple collisions ($\Xi$-coalescent) with mutation, the $\Xi$-
Fleming-Viot
process with mutation, and provide a representation based on the
empirical
measure of an exchangeable particle system along the lines of Donnelly
and
Kurtz (1999). We establish pathwise convergence of the approximating
systems to
the limiting $\Xi$-Fleming-Viot process with mutation. An alternative
construction of the semigroup based on the Hille-Yosida theorem is
provided and
various types of duality of the processes are discussed.
In the last part of the paper a population is considered which
undergoes
recurrent bottlenecks. In this scenario, non-trivial $\Xi$-Fleming-Viot
processes naturally arise as limiting models.
http://arxiv.org/abs/0808.0412
---------------------------------------------------------------
7356. THE HAUSDORFF DIMENSION OF THE DOUBLE POINTS ON THE BROWNIAN
FRONTIER
Richard Kiefer and Peter Morters
The frontier of a planar Brownian motion is the boundary of the
unbounded
component of the complement of its range. In this paper we find the
Hausdorff
dimension of the set of double points on the frontier.
http://arxiv.org/abs/0808.0425
---------------------------------------------------------------
7357. STRONG LAWS FOR BALANCED TRIANGULAR URNS
Arup Bose and Amites Dasgupta and Krishanu Maulik
Consider an urn model whose replacement matrix is triangular, has all
entries
nonnegative and the row sums are all equal to one. We obtain the
strong laws
for the counts of balls corresponding to each color. The scalings for
these
laws depend on the diagonal elements of a rearranged replacement
matrix. We use
the strong laws obtained to study further behavior of certain three
color urn
models.
http://arxiv.org/abs/0808.0426
---------------------------------------------------------------
7358. SERIES JACKSON NETWORKS AND NON-CROSSING PROBABILITIES
A.B. Dieker and J. Warren
We show that Markov transition probabilities for the queue length
process in
series Jackson networks can be written as a finite sum of non-crossing
probabilities. Using this, we prove a conjecture of Blanc that the
relaxation
time (i.e., the reciprocal of the `spectral gap') of a positive
recurrent
system equals the relaxation time of an M/M/1 queue with the same
arrival and
service rates as the network's bottleneck station.
http://arxiv.org/abs/0808.0513
---------------------------------------------------------------
7359. GLOBAL GEOMETRY UNDER ISOTROPIC BROWNIAN FLOWS
Sreekar Vadlamani and Robert J. Adler
We consider global geometric properties of a codimension one manifold
embedded in Euclidean space, as it evolves under an isotropic and volume
preserving Brownian flow of diffeomorphisms. In particular, we obtain
expressions describing the expected rate of growth of the Lipschitz-
Killing
curvatures, or intrinsic volumes, of the manifold under the flow.
These results shed new light on some of the intriguing growth
properties of
flows from a global perspective, rather than the local perspective, on
which
there is a much larger literature.
http://arxiv.org/abs/0808.0720
---------------------------------------------------------------
7360. BACKWARD STOCHASTIC VARIATIONAL INEQUALITIES UNDER WEAK
ASSUMPTIONS ON THE DATA
Lucian Maticiuc and Aurel Rascanu and Adrian Zalinescu
The aim of this paper is to study the existence and uniqueness of the
solution of the backward stochastic differential equations involving the
subdifferential operator $\partial\varphi$ (also called backward
stochastic
variational inequalities): \[ {\begin{array} [c]{l}%
-dY_{t}+\partial\varphi(Y_{t}) dt\ni F(t,Y_{t}%, Z_{t}) dt-
Z_{t}dB_{t}, 0\leq
t<T\medskip Y_{T}=\eta. \end{array} . \] Our results generalize those
of E.
Pardoux and A. R\u{a}\c{s}canu (Stochastic Processes and their
Applications 76,
1998) to the case in which the function $F$ satisfies a local boundeness
condition (instead of sublinear growth condition with respect to $y$).
http://arxiv.org/abs/0808.0801
---------------------------------------------------------------
7361. DIVERGENCES TEST STATISTICS FOR DISCRETELY OBSERVED DIFFUSION
PROCESSES
de gregorio alessandro and stefano iacus
In this paper we propose the use of $\phi$-divergences as test
statistics to
verify simple hypotheses about a one-dimensional parametric diffusion
process
$\de X_t = b(X_t, \theta)\de t + \sigma(X_t, \theta)\de W_t$, from
discrete
observations $\{X_{t_i}, i=0, ..., n\}$ with $t_i = i\Delta_n$, $i=0,
1, >...,
n$, under the asymptotic scheme $\Delta_n\to0$, $n\Delta_n\to\infty$ and
$n\Delta_n^2\to 0$. The class of $\phi$-divergences is wide and includes
several special members like Kullback-Leibler, R\'enyi, power and
$\alpha$-divergences. We derive the asymptotic distribution of the test
statistics based on $\phi$-divergences. The limiting law takes
different forms
depending on the regularity of $\phi$. These convergence differ from the
classical results for independent and identically distributed random
variables.
Numerical analysis is used to show the small sample properties of the
test
statistics in terms of estimated level and power of the test.
http://arxiv.org/abs/0808.0853
---------------------------------------------------------------
7362. THERMAL CONDUCTIVITY FOR A NOISY DISORDERED HARMONIC CHAIN
Cedric Bernardin (UMPA-Ensl)
We consider a $d$-dimensional disordered harmonic chain (DHC)
perturbed by an
energy conservative noise. We obtain uniform in the volume upper and
lower
bounds for the thermal conductivity defined through the Green-Kubo
formula.
These bounds indicate a positive finite conductivity. We prove also
that the
infinite volume homogenized Green-Kubo formula converges.
http://arxiv.org/abs/0808.0660
---------------------------------------------------------------
7363. SUPERDIFFUSIVITY OF ASYMMETRIC ENERGY MODEL IN DIMENSION ONE AND
TWO
Cedric Bernardin (UMPA-Ensl)
We discuss an asymmetric energy model (AEM) introduced by Giardina et
al. in
\cite{7}. This model is expected to belong to the KPZ class. We obtain
lower
bounds for the diffusion coefficient. In particular, the diffusion
coefficient
is diverging in dimension one and two as it is expected in the KPZ
picture.
http://arxiv.org/abs/0808.0661
---------------------------------------------------------------
7364. STATIONARY NON-EQUILIBRIUM PROPERTIES FOR A HEAT CONDUCTION MODEL
Cedric Bernardin (UMPA-Ensl)
We consider a stochastic heat conduction model for solids composed by N
interacting atoms. The system is in contact with two heat baths at
different
temperature $T_\ell$ and $T_r$. The bulk dynamics conserve two
quantities: the
energy and the deformation between atoms. If $T_\ell \neq T_r$, a heat
flux
takes place in the system. For large $N$, the system adopts a linear
temperature profile between $T_\ell$ and $T_r$. We establish the
hydrodynamic
limit for the two conserved quantities. We introduce the fluctuations
field of
the energy and of the deformation in the non-equilibrium steady state.
As $N$
goes to infinity, we show that this field converges to a Gaussian
field and we
compute the limiting covariance matrix. The main contribution of the
paper is
the study of large deviations for the temperature profile in the
non-equilibrium stationary state. A variational formula for the rate
function
is derived following the recent macroscopic fluctuation theory of
Bertini et
al.
http://arxiv.org/abs/0808.0662
---------------------------------------------------------------
7365. AN INTERMEDIATE REGIME FOR EXIT PHENOMENA DRIVEN BY NON-GAUSSIAN
LEVY NOISES
Zhihui Yang and Jinqiao Duan
A dynamical system driven by non-Gaussian L\'evy noises of small
intensity is
considered. The first exit time of solution orbits from a bounded
neighborhood
of an attracting equilibrium state is estimated. For a class of non-
Gaussian
L\'evy noises, it is shown that the mean exit time is asymptotically
faster
than exponential (the well-known Gaussian Brownian noise case) but
slower than
polynomial (the stable L\'evy noise case), in terms of the reciprocal
of the
small noise intensity.
http://arxiv.org/abs/0808.1085
---------------------------------------------------------------
7366. RELATIONS AMONG CONDITIONAL PROBABILITIES
Jason Morton
We describe a Groebner basis of relations among conditional
probabilities in
a discrete probability space, with any set of conditioned-upon events.
They may
be specialized to the partially-observed random variable case, the
purely
conditional case, and other special cases. We also investigate the
connection
to generalized permutohedra and describe a conditional probability
simplex.
http://arxiv.org/abs/0808.1149
---------------------------------------------------------------
7367. MICROSCOPIC CONCAVITY AND FLUCTUATION BOUNDS IN A CLASS OF
DEPOSITION PROCESSES
M. Bal\'azs and J. Komj\'athy and T. Sepp\"al\"ainen
This paper develops a general approach to proving order of magnitude
t^{1/3}
for fluctuations in characteristic directions for asymmetric
deposition models,
or asymmetric conservative particle systems, in the case of a strictly
concave
flux function. For hypothesis the argument requires control of second
class
particles in a manner that deserves to be called microscopic concavity
by
analogy with the effects of concavity of the flux. This hypothesis has
been
verified for the asymmetric simple exclusion process and is verified
in the
present paper for totally asymmetric zero range processes with jump
rates that
increase with exponentially decaying slope.
http://arxiv.org/abs/0808.1177
---------------------------------------------------------------
7368. GENERAL MATRIX-VALUED INHOMOGENEOUS LINEAR STOCHASTIC
DIFFERENTIAL EQUATIONS AND APPLICATIONS
Jinqiao Duan and Jia-an Yan
The expressions of solutions for general $n\times m$ matrix-valued
inhomogeneous linear stochastic differential equations are derived. This
generalizes a result of Jaschke (2003) for scalar inhomogeneous linear
stochastic differential equations. As an application, some $\R^n$
vector-valued
inhomogeneous nonlinear stochastic differential equations are reduced
to random
differential equations, facilitating pathwise study of the solutions.
http://arxiv.org/abs/0808.1112
---------------------------------------------------------------
7369. LINEAR SYSTEMS AND DETERMINANTAL RANDOM POINT FIELDS
Gordon Blower
Tracy and Widom showed that fundamentally important kernels in random
matrix
theory arise from differential equations with rational coefficients.
More
generally, this paper considers symmetric Hamiltonian systems abd
determines
the properties of kernels that arise from them. The inverse spectral
problem
for self-adjoint Hankel operators gives a sufficient condition for a
self-adjoint operator to be the Hankel operator on $L^2(0, \infty)$
from a
linear system in continuous time; thus this paper expresses certain
kernels as
squares of Hankel operators. For a suitable linear system $(-A,B,C)$
with one
dimensional input and output spaces, there exists a Hankel operator $
\Gamma$
with kernel $\phi_{(x)}(s+t)=Ce^{-(2x+s+t)A}B$ such that $\det
(I+(z-1)\Gamma\Gamma^\dagger)$ is the generating function of a
determinantal
random point field.
http://arxiv.org/abs/0808.1276
---------------------------------------------------------------
7370. OPTIMAL CONTROL OF A STOCHASTIC NETWORK DRIVEN BY A FRACTIONAL
BROWNIAN MOTION INPUT
Arka P. Ghosh and Alexander Roitershtein and Ananda Weerasinghe
We consider a stochastic control model driven by a fractional Brownian
motion. This model is a formal approximation to a queueing network
with an
on-off input process. We study stochastic control problems associated
with the
long-run average cost, the infinite horizon discounted cost, and the
finite
horizon cost. In addition, we find a solution to a constrained
minimization
problem as an application of our solution to the long-run average cost
problem.
We also establish Abelian limit relationships among the value
functions of the
above control problems.
http://arxiv.org/abs/0808.1299
---------------------------------------------------------------
7371. A THEORY OF HYPOELLIPTICITY AND UNIQUE ERGODICITY FOR
SEMILINEAR STOCHASTIC PDES
Martin Hairer and Jonathan C. Mattingly
We present a theory of hypoellipticity and unique ergodicity for
semilinear
parabolic stochastic PDEs with ``polynomial'' nonlinearities and
additive
noise, considered as abstract evolution equations in some Hilbert
space. It is
shown that if Hormander's bracket condition holds at every point of this
Hilbert space, then a lower bound on the Malliavin covariance operator
$M_t$
can be obtained. Informally, this bound can be read as ``Fix any
finite-dimensional projection $\Pi$ on a subspace of sufficiently
regular
functions. Then the eigenfunctions of $M_t$ with small eigenvalues
have only a
very small component in the image of $\Pi$.''
We also show how to use a priori bounds on the solutions to the
equation to
obtain good control on the dependency of the bounds on the Malliavin
matrix on
the initial condition. These bounds are sufficient in many cases to
obtain the
asymptotic strong Feller property introduced in [HairerMattingly06].
One of the main novel technical tools is an almost sure bound from
below on
the size of ``Wiener polynomials,'' where the coefficients are possibly
non-adapted stochastic processes satisfying a Lipschitz condition. By
exploiting the polynomial structure of the equations, this result can
be used
to replace Norris' lemma, which is unavailable in the present context.
We conclude by showing that the two-dimensional stochastic Navier-
Stokes
equations and a large class of reaction-diffusion equations fit the
framework
of our theory.
http://arxiv.org/abs/0808.1361
---------------------------------------------------------------
7372. A STOCHASTIC DIFFERENTIAL GAME FOR THE INHOMOGENEOUS $\INFTY$-
LAPLACE EQUATION
Rami Atar and Amarjit Budhiraja
Given a bounded $\calC^2$ domain $G\subset\R^m$ and functions $g\in
\calC(\pl
G,\R)$ and $h\in\calC(\bar G,\R\setminus\{0\})$, let $u$ denote the
unique
viscosity solution to the equation $-2\Del_\infty u=h$ in $G$ with
boundary
data $g$. We provide a representation for $u$ as the value of a two-
player
zero-sum stochastic differential game.
http://arxiv.org/abs/0808.1457
---------------------------------------------------------------
7373. CIRCULAR LAW THEOREM FOR RANDOM MARKOV MATRICES
Djalil Chafai (IMT and Upte)
Let $(X_{i,j})$ be an infinite array of i.i.d. non negative real random
variables with unit mean, finite positive variance $\sigma^2$, and
finite
fourth moment. Let $M$ be the $n\times n$ random Markov matrix with
i.i.d. rows
defined by $M_{i,j}=X_{i,j}/(X_{i,1}+...+X_{i,n})$. It belongs to the
Dirichlet
Markov Ensemble when $X_{1,1}$ follows an exponential law. We show
that with
probability one, the empirical spectral distribution
$\frac{1}{n}(\delta_{\lambda_1}+...+\delta_{\lambda_n})$ of $\sqrt{n}M$
converges weakly as $n\to\infty$ to the uniform law on the disc
$\{z\in\mathbb{C};|z|\leq \sigma\}$ and moreover, the spectral gap of
$M$ is of
order $n^{-1/2}$ for large enough $n$. There is for now a gap in the
proof of
Theorem 1.3 (convergence to the circular law).
http://arxiv.org/abs/0808.1502
---------------------------------------------------------------
7374. REGULAR DEPENDENCE ON INITIAL DATA FOR STOCHASTIC EVOLUTION
EQUATIONS WITH MULTIPLICATIVE POISSON NOISE
Carlo Marinelli and Claudia Pr\'ev\^ot and Michael R\"ockner
We prove existence, uniqueness and Lipschitz dependence on the initial
datum
for mild solutions of stochastic partial differential equations with
Lipschitz
coefficients driven by Wiener and Poisson noise. Under additional
assumptions,
we prove Gateaux and Frechet differentiability of solutions with
respect to the
initial datum. As an application, we obtain gradient estimates for the
resolvent associated to the mild solution. Finally, we prove the
strong Feller
property of the associated semigroup.
http://arxiv.org/abs/0808.1509
---------------------------------------------------------------
7375. LIOUVILLE QUANTUM GRAVITY AND KPZ
Bertrand Duplantier and Scott Sheffield
Consider a bounded planar domain D, an instance h of the Gaussian free
field
on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0
< gamma <
2. The Liouville quantum gravity measure on D is the weak limit as
\epsilon
tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_
\epsilon(z)}dz,
where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean
value of h
on the circle of radius \epsilon centered at z. Given a random (or
deterministic) subset X of D one can define the scaling dimension of X
using
either Lebesgue measure or this random measure. We derive a general
quadratic
relation between these two dimensions, which we view as a probabilistic
formulation of the KPZ relation from conformal field theory. We also
present a
boundary analog of KPZ (for subsets of the boundary of D). We
illustrate (via
heuristics and announced results) the connection between discrete and
continuum
quantum gravity and provide a framework for understanding Euclidean
scaling
exponents via quantum gravity.
http://arxiv.org/abs/0808.1560
---------------------------------------------------------------
7376. HEAT CONDUCTIVITY FROM MOLECULAR CHAOS HYPOTHESIS IN LOCALLY
CONFINED BILLIARD SYSTEMS
Thomas Gilbert and Raphael Lefevere
We discuss the transport properties of a class of Hamiltonian dynamics
with
local confinement, in which interactions between neighboring particles
occur
through hard core elastic collisions. Such dynamics may be described as
high-dimensional billiards. We consider the case where the collisions
are rare
and, for large systems, derive a Boltzmann-like equation for the
evolution of
the probability densities. We solve this equation in the linear regime
and
compute the heat conductivity in the approximate stationary state and
with the
help of the Green-Kubo formula. We demonstrate the validity of the
molecular
chaos hypothesis by comparing our theoretical predictions to the
results of
numerical simulations performed on a new class of models, which are
defocusing
chaotic billiards, likened to higher-dimensional stadia.
http://arxiv.org/abs/0808.1179
---------------------------------------------------------------
7377. EXTREMAL DISTRIBUTIONS FOR TAIL PROBABILITIES OF SUMS OF IID
RANDOM VARIABLES ON [0,1]
Ludolf E. Meester
Two old conjectures from problem sections, one of which from SIAM
Review,
concern the question of finding distributions that maximize P(Sn <=
t), where
Sn is the sum of i.i.d. random variables X1, ..., Xn on the interval
[0,1],
satisfying E[X1]=m. In this paper a Lagrange multiplier technique is
applied to
this problem, yielding necessary conditions for distributions to be
extremal,
for arbitrary n. For n=2, a complete solution is derived from them:
extremal
distributions are discrete and have one of the following supports,
depending on
m and t: {0,t}, {t-1,1}, {t/2,1}, or {0,t,1}. These results suffice to
refute
both conjectures. However, acquired insight naturally leads to a revised
conjecture: that extremal distributions always have at most three
support
points and belong to a (for each n, specified) finite collection of
two and
three point distributions.
http://arxiv.org/abs/0808.1669
---------------------------------------------------------------
7378. FINDING CORES OF RANDOM 2-SAT FORMULAE VIA POISSON CLONING
Jeong Han Kim
For the random 2-SAT formula $F(n,p)$, let $F_C (n,p)$ be the formula
left
after the pure literal algorithm applied to $F(n,p)$ stops. Using the
recently
developed Poisson cloning model together with the cut-off line algorithm
(COLA), we completely analyze the structure of $F_{C} (n,p)$. In
particular, it
is shown that, for $\gl:= p(2n-1) = 1+\gs $ with $\gs\gg n^{-1/3}$,
the core of
$F(n,p)$ has $\thl^2 n +O((\thl n)^{1/2})$ variables and $\thl^2 \gl n
+O((\thl
n))^{1/2}$ clauses, with high probability, where $\thl$ is the larger
solution
of the equation $\th- (1-e^{-\thl \gl})=0$. We also estimate the
probability of
$F(n,p)$ being satisfiable to obtain $$ \pr[ F_2(n, \sfrac{\gl}{2n-1})
is
satisfiable ] = \caseth{1-\frac{1+o(1)}{16\gs^3 n}}{if $\gl= 1-\gs$ with
$\gs\gg n^{-1/3}$}{}{}{e^{-\Theta(\gs^3n)}}{if $\gl=1+\gs$ with $\gs\gg
n^{-1/3}$,} $$ where $o(1)$ goes to 0 as $\gs$ goes to 0. This
improves the
bounds of Bollob\'as et al. \cite{BBCKW}.
http://arxiv.org/abs/0808.1599
---------------------------------------------------------------
7379. ON IDEMPOTENT STATES ON QUANTUM GROUPS
Uwe Franz and Adam Skalski
Idempotent states on a compact quantum group are shown to yield group-
like
projections in the multiplier algebra of the dual discrete quantum
group. This
allows to deduce that every idempotent state on a finite quantum group
arises
as the Haar state on a finite quantum hypergroup. A natural order
structure on
the set of idempotent states is also studied and some examples
discussed.
http://arxiv.org/abs/0808.1683
---------------------------------------------------------------
7380. ON THE NUMBER OF ALLELIC TYPES FOR SAMPLES TAKEN FROM
EXCHANGEABLE COALESCENTS WITH MUTATION
Fabian Freund and Martin M\"ohle
Let $K_n$ denote the number of types of a sample of size $n$ taken
from an
exchangeable coalescent process ($\Xi$-coalescent) with mutation. A
distributional recursion for the sequence $(K_n)_{n\in{\mathbb N}}$ is
derived.
If the coalescent does not have proper frequencies, i.e., if the
characterizing
measure $\Xi$ on the infinite simplex $\Delta$ does not have mass at
zero and
satisfies $\int_\Delta |x|\Xi(dx)/(x,x)<\infty$, where $|x|:=
\sum_{i=1}^\infty
x_i$ and $(x,x):=\sum_{i=1}^\infty x_i^2$ for $x=(x_1,x_2,...)\in\Delta
$, then
$K_n/n$ converges weakly as $n\to\infty$ to a limiting variable $K$
which is
characterized by an exponential integral of the subordinator
associated with
the coalescent process. For so-called simple measures $\Xi$ satisfying
$\int_\Delta\Xi(dx)/(x,x)<\infty$ we characterize the distribution of
$K$ via a
fixed-point equation.
http://arxiv.org/abs/0808.1792
---------------------------------------------------------------
7381. CONFLATIONS OF PROBABILITY DISTRIBUTIONS: AN OPTIMAL METHOD FOR
CONSOLIDATING DATA FROM DIFFERENT EXPERIMENTS
Theodore P. Hill
The conflation of a finite number of probability distributions
P_1,...,P_n is
the probability distribution that minimizes the loss of Shannon
Information in
consolidating the combined information from P_1,...,P_n into a single
distribution Q, and is also the minimax likelihood ratio consolidation
of the
distributions. Intuitively, the conflation is the conditional
distribution of
independent random variables, given that they are all equal, so in large
classes of distributions the conflation is the distribution determined
by the
normalized product of the probability density or probability mass
functions.
When P_1,...,P_n are Gaussian, Q is Gaussian with mean the classical
weighted-mean-squares reciprocal of variances. For unbiased
estimators, the
mean of the conflation thus yields a BLUE (best linear unbiased
estimator) and
if the underlying data is Gaussian, the conflation mean is also an MLE
(maximum
likelihood estimator). A version of the classical convolution theorem
holds for
conflations of a large class of a.c. measures.
http://arxiv.org/abs/0808.1808
---------------------------------------------------------------
7382. FRONT PROPAGATION WITH REJUVENATION IN FLIPPING PROCESSES
T. Antal and D. ben-Avraham and E. Ben-Naim and P.L. Krapivsky
We study a directed flipping process that underlies the performance of
the
random edge simplex algorithm. In this stochastic process, which takes
place on
a one-dimensional lattice whose sites may be either occupied or vacant,
occupied sites become vacant at a constant rate and simultaneously
cause all
sites to the right to change their state. This random process exhibits
rich
phenomenology. First, there is a front, defined by the position of the
left-most occupied site, that propagates at a nontrivial velocity.
Second, the
front involves a depletion zone with an excess of vacant sites. The
total
excess D_k increases logarithmically, D_k ~ ln k, with the distance k
from the
front. Third, the front exhibits rejuvenation -- young fronts are
vigorous but
old fronts are sluggish. We investigate these phenomena using a quasi-
static
approximation, direct solutions of small systems, and numerical
simulations.
http://arxiv.org/abs/0808.0159
---------------------------------------------------------------
7383. THE FINITE HARMONIC OSCILLATOR AND ITS ASSOCIATED SEQUENCES
Shamgar Gurevich and Ronny Hadani and Nir Sochen
A system of functions (signals) on the finite line, called the
oscillator
system, is described and studied. Applications of this system for
discrete
radar and digital communication theory are explained.
Keywords: Weil representation, commutative subgroups,
eigenfunctions, random
behavior, deterministic construction
http://arxiv.org/abs/0808.1417
---------------------------------------------------------------
7384. AVERAGING AND LARGE DEVIATION PRINCIPLES FOR FULLY-COUPLED
PIECEWISE DETERMINISTIC MARKOV PROCESSES AND APPLICATIONS TO
MOLECULAR MOTORS
A. Faggionato and D. Gabrielli and M. Ribezzi Crivellari
We consider Piecewise Deterministic Markov Processes (PDMPs) with a
finite
set of discrete states. In the regime of fast jumps between discrete
states, we
prove a law of large number and a large deviation principle. In the
regime of
fast and slow jumps, we analyze a coarse-grained process associated to
the
original one and prove its convergence to a new PDMP with effective
force
fields and jump rates. In all the above cases, the continuous
variables evolve
slowly according to ODEs.
Finally, we discuss some applications related to the
mechanochemical cycle of
macromolecules, including strained--dependent power--stroke molecular
motors.
Our analysis covers the case of fully--coupled slow and fast motions.
http://arxiv.org/abs/0808.1910
---------------------------------------------------------------
7385. THE NUMBER OF 2X2 INTEGER MATRICES HAVING A PRESCRIBED INTEGER
EIGENVALUE
Greg Martin and Erick B. Wong
Random matrices arise in many mathematical contexts, and it is natural
to ask
about the properties that such matrices satisfy. If we choose a matrix
with
integer entries at random, for example, what is the probability that
it will
have a particular integer as an eigenvalue, or an integer eigenvalue
at all? If
we choose a matrix with real entries at random, what is the
probability that it
will have a real eigenvalue in a particular interval? The purpose of
this paper
is to resolve these questions, once they are made suitably precise, in
the
setting of 2x2 matrices.
http://arxiv.org/abs/0808.1922
---------------------------------------------------------------
7386. NEW COINS FROM OLD, SMOOTHLY
Olga Holtz and Fedor Nazarov and Yuval Peres
Given a (known) function $f:[0,1] \to (0,1)$, we consider the problem of
simulating a coin with probability of heads $f(p)$ by tossing $N$
times a coin
with unknown heads probability $p$, where $N$ may be random. Keane and
O'Brien
(1994) showed that such a simulation scheme with $\P_p(N<\infty)=1$
exists iff
$f$ is continuous. Nacu and Peres (2005) proved that $f$ is real
analytic iff
such a simulation scheme exists with $\P_p(N>n)$ decaying
exponentially. We
prove that for $\alpha>0$ noninteger, $f$ is in the space $C^\alpha
[0,1]$ if
and only if a simulation scheme as above exists with $\P_p(N>n) \le
C\Delta_n(p)$,where $\Delta_n(x)\eqbd \max \{\sqrt{x(1-x)/n},1/n \}$.
The key
to the proof is a new result in approximation theory:
Let $H_n^+$ be the space of homogenous polynomials of degree $n$ with
non-negative coefficients in two variables. We show that a function $f:
[0,1]
\to (0,1)$ is in $C^\alpha [0,1]$ if and only if $f$ has a series
representation $\sum_{n=1}^\infty F_n(x,1-x)$ with $F_n(x,y) \in H_n^+
$ and
$\sum_{k>n} F_k(x,1-x) \le C\Delta_n(x)$ for all $ x \in [0,1]$ and $n
\ge 1$.
We also provide a counterexample to a theorem stated without proof by
Lorentz
(1963), who claimed that if some $\phi_n(x,y) \in H_n^+$ satisfy
$|f(x)-\phi_n(x,1-x)| \le C \Delta_n(x)$ for all $ x \in [0,1]$ and $n
\ge 1$,
then $f \in C^\alpha [0,1]$.
http://arxiv.org/abs/0808.1936
---------------------------------------------------------------
7387. REGULAR INDUCED SUBGRAPHS OF A RANDOM GRAPH
Michael Krivelevich and Benny Sudakov and Nicholas Wormald
An old problem of Erd\H{o}s, Fajtlowicz and Staton asks for the order
of a
largest induced regular subgraph that can be found in every graph on n
vertices. Motivated by this problem, we consider the order of such a
subgraph
in a typical graph on n vertices, i.e., in a binomial random graph G(n,
1/2). We
prove that with high probability a largest induced regular subgraph of
G(n,1/2)
has about n^{2/3} vertices.
http://arxiv.org/abs/0808.2023
---------------------------------------------------------------
7388. DENSITY ESTIMATES AND CONCENTRATION INEQUALITIES WITH MALLIAVIN
CALCULUS
Ivan Nourdin (University of Paris 6) and Frederi G. Viens (Purdue
University)
We show how to use the Malliavin calculus to obtain density estimates
of the
law of general centered random variables. In particular, under a non-
degeneracy
condition, we prove and use a new formula for the density of a random
variable
which is measurable and differentiable with respect to a given isonormal
Gaussian process. Among other results, we apply our techniques to
bound the
density of the maximum of a general Gaussian process from above and
below;
several new results ensue, including improvements on the so-called
Borell-Sudakov inequality. We then explain what can be done when one
is only
interested in or capable of deriving concentration inequalities, i.e.
tail
bounds from above or below but not necessarily both simultaneously.
http://arxiv.org/abs/0808.2088
---------------------------------------------------------------
7389. ERDOS-RENYI RANDOM GRAPHS + FOREST FIRES = SELF-ORGANIZED
CRITICALITY
Balazs Rath (TU Budapest and Institute of Mathematics) and Balint
Toth (TU Budapest, Institute of Mathematics)
We modify the usual Erdos-Renyi random graph evolution by letting
connected
clusters 'burn down' (i.e. fall apart to disconnected single sites)
due to a
Poisson flow of lightnings. In a range of the intensity of rate of
lightnings
the system sticks to a permanent critical state.
http://arxiv.org/abs/0808.2116
---------------------------------------------------------------
7390. ON THE GEOMETRY OF A CLASS OF INVARIANT MEASURES AND A PROBLEM
OF ALDOUS
Tim Austin (UCLA)
In his 1985 survey of notions of exchangeability, Aldous introduced a
form of
exchangeability corresponding to the symmetries of the infinite
discrete cube,
and asked whether these exchangeable probability measures enjoy a
representation theorem similar to those for exchangeable sequences,
arrays and
set-indexed families. In this note we to prove that, whereas the known
representation theorems for different classes of partially exchangeable
probability measure imply that the compact convex set of such measures
is a
Bauer simplex (that is, its subset of extreme points is closed), in
the case of
cube-exchangeability it is a copy of the Poulsen simplex (in which the
extreme
points are dense). This follows from the arguments used by Glasner and
Weiss'
for their characterization of property (T) in terms of the geometry of
the
simplex of invariant measures for associated generalized Bernoulli
actions.
The emergence of this Poulsen simplex suggests that, if a
representation
theorem for these processes is available at all, it must take a very
different
form from the case of set-indexed exchangeable families.
http://arxiv.org/abs/0808.2268
---------------------------------------------------------------
7391. MARKOV PATHS, LOOPS AND FIELDS
Yves Le Jan (LM-Orsay)
This is a preliminary version of the notes of a series of lectures
given in
St Flour. It includes a discussion of relations between the occupation
field of
Markov loops with the corresponding free field and new developments on
reflection positivity.
http://arxiv.org/abs/0808.2303
---------------------------------------------------------------
7392. FITTING MARTINGALES TO GIVEN MARGINALS
George Lowther
We consider the problem of finding a real valued martingale fitting
specified
marginal distributions. For this to be possible, the marginals must be
increasing in the convex order and have constant mean. We show that,
under the
extra condition that they are weakly continuous, the marginals can
always be
fitted in a unique way by a martingale which lies in a particular
class of
strong Markov processes.
It is also shown that the map that this gives from the sets of
marginal
distributions to the martingale measures is continuous. Furthermore,
we prove
that it is the unique continuous method of fitting martingale measures
to the
marginal distributions.
http://arxiv.org/abs/0808.2319
---------------------------------------------------------------
7393. THE AGE INCIDENCE OF ANY CANCER CAN BE EXPLAINED BY A ONE-
MUTATION MODEL
Rinaldo B. Schinazi
We propose a one mutation model for cancer with a mutation rate that
increases with time. Under rather general hypotheses the number of
mutations is
necessarily a (non homogeneous) Poisson process with the prescribed
mutation
rate. We show that the cumulative probability of cancer up to time $t$
is, up
to a multiplicative constant, an antiderivative of the mutation rate.
http://arxiv.org/abs/0808.2424
---------------------------------------------------------------
7394. DEDUCING THE MULTIDIMENSIONAL SZEMEREDI THEOREM FROM THE
INFINITARY HYPERGRAPH REMOVAL LEMMA
Tim Austin (UCLA)
We offer a new proof of the Furstenberg-Katznelson multiple recurrence
theorem for several commuting probability-preserving transformations
$T_1,T_2,...,T_d:\bbZ\curvearrowright (X,\S,\mu)$, and so, via the
Furstenberg
correspondence principle, a new proof of the multi-dimensional Szemer
\'edi
Theorem. We bypass the detailed analysis of certain towers of factors
of a
probability-preserving systems that underlies the Furstenberg-Katznelson
analysis, instead modifying an approach recently developed for the
analysis of
nonconventional ergodic averages to pass to a large extension of our
original
system in which this analysis greatly simplifies. In particular, we
find that
this simplifies our setting quite quickly to data that can be analyzed
using
the infinitary version of the hypergraph removal lemma studied by Tao,
and we
complete the proof by a simple application of that lemma. This
addresses the
difficulty, highlighted by Tao, of establishing a direct connection
between his
infinitary, probabilistic approach to the hypergraph removal lemma and
the
infinitary, ergodic-theoretic approach to Szemer\'edi's Theorem set in
motion
by Furstenberg.
http://arxiv.org/abs/0808.2267
---------------------------------------------------------------
7395. AN OBSERVATION ABOUT SUBMATRICES
Sourav Chatterjee and Michel Ledoux
Let M be an arbitrary Hermitian matrix of order n, and k be a positive
integer less than or equal to n. We show that if k is large, the
distribution
of eigenvalues on the real line is almost the same for almost all
principal
submatrices of M of order k. The proof uses results about random walks
on
symmetric groups and concentration of measure. In a similar way, we
also show
that almost all k x n submatrices of M have almost the same
distribution of
singular values.
http://arxiv.org/abs/0808.2521
---------------------------------------------------------------
7396. UNIVERSAL MALLIAVIN CALCULUS IN FOCK AND L\'{E}VY-IT\^{O} SPACES
David Applebaum
We review and extend Lindsay's work on abstract gradient and divergence
operators in Fock space over a general complex Hilbert space. Precise
expressions for the domains are given, the $L^2$-equivalence of norms
is proved
and an abstract version of the It\^{o}-Skorohod isometry is
established. We
then outline a new proof of It\^{o}'s chaos expansion of complex
L\'{e}vy-It\^{o} space in terms of multiple Wiener-L\'{e}vy integrals
based on
Brownian motion and a compensated Poisson random measure. The duality
transform
now identifies L\'{e}vy-It\^{o} space as a Fock space. We can then
easily
obtain key properties of the gradient and divergence of a general L
\'{e}vy
process. In particular we establish maximal domains of these operators
and
obtain the It\^{o}-Skorohod isometry on its maximal domain.
http://arxiv.org/abs/0808.2593
---------------------------------------------------------------
7397. EDGE SCALING LIMITS FOR A FAMILY OF NON-HERMITIAN RANDOM MATRIX
ENSEMBLES
Martin Bender
A family of random matrix ensembles interpolating between the GUE and
the
Ginibre ensemble of $n\times n$ matrices with iid centered complex
Gaussian
entries is considered. The asymptotic spectral distribution in these
models is
uniform in an ellipse in the complex plane, which collapses to an
interval of
the real line as the degree of non-Hermiticity diminishes. Scaling limit
theorems are proven for the eigenvalue point process at the rightmost
edge of
the spectrum, and it is shown that a non-trivial transition occurs
between
Poisson and Airy point process statistics when the ratio of the axes
of the
supporting ellipse is of order $n^{-1/3}$. In this regime, the family of
limiting probability distributions of the maximum of the real parts of
the
eigenvalues interpolates between the Gumbel and Tracy-Widom
distributions.
http://arxiv.org/abs/0808.2608
---------------------------------------------------------------
7398. A CONNECTION BETWEEN THE STOCHASTIC HEAT EQUATION AND
FRACTIONAL BROWNIAN MOTION, AND A SIMPLE PROOF OF A RESULT OF TALAGRAND
Carl Mueller and Zhixin Wu
We give a new representation of fractional Brownian motion with Hurst
parameter H<=1/2 using stochastic partial differential equations. This
representation allows us to use the Markov property and time reversal,
tools
which are not usually available for fractional Brownian motion. We
then give
simple proofs that fractional Brownian motion does not hit points in the
critical dimension, and that it does not have double points in the
critical
dimension. These facts were already known, but our proofs are quite
simple and
use some ideas of Levy.
http://arxiv.org/abs/0808.2634
---------------------------------------------------------------
7399. EFFICIENT RARE-EVENT SIMULATION FOR THE MAXIMUM OF HEAVY-TAILED
RANDOM WALKS
Jose Blanchet and Peter Glynn
Let $(X_n:n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean.
Set
$S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling
algorithm to estimate the tail of $M=\max \{S_n:n\geq 0\}$ that is
strongly
efficient for both light and heavy-tailed increment distributions.
Moreover, in
the case of heavy-tailed increments and under additional technical
assumptions,
our estimator can be shown to have asymptotically vanishing relative
variance
in the sense that its coefficient of variation vanishes as the tail
parameter
increases. A key feature of our algorithm is that it is state-
dependent. In the
presence of light tails, our procedure leads to Siegmund's (1979)
algorithm.
The rigorous analysis of efficiency requires new Lyapunov-type
inequalities
that can be useful in the study of more general importance sampling
algorithms.
http://arxiv.org/abs/0808.2731
---------------------------------------------------------------
7400. UNIFYING PRACTICAL UNCERTAINTY REPRESENTATIONS: I. GENERALIZED P-
BOXES
Sebastien Destercke and Didier Dubois and Eric Chojnacki
There exist several simple representations of uncertainty that are
easier to
handle than more general ones. Among them are random sets, possibility
distributions, probability intervals, and more recently Ferson's p-
boxes and
Neumaier's clouds. Both for theoretical and practical considerations,
it is
very useful to know whether one representation is equivalent to or can
be
approximated by other ones. In this paper, we define a generalized
form of
usual p-boxes. These generalized p-boxes have interesting connections
with
other previously known representations. In particular, we show that
they are
equivalent to pairs of possibility distributions, and that they are
special
kinds of random sets. They are also the missing link between p-boxes and
clouds, which are the topic of the second part of this study.
http://arxiv.org/abs/0808.2747
---------------------------------------------------------------
7401. UNIFYING PRACTICAL UNCERTAINTY REPRESENTATIONS: II. CLOUDS
Sebastien Destercke and Didier Dubois and Eric Chojnacki
There exist many simple tools for jointly capturing variability and
incomplete information by means of uncertainty representations. Among
them are
random sets, possibility distributions, probability intervals, and the
more
recent Ferson's p-boxes and Neumaier's clouds, both defined by pairs of
possibility distributions. In the companion paper, we have extensively
studied
a generalized form of p-box and situated it with respect to other
models . This
paper focuses on the links between clouds and other representations.
Generalized p-boxes are shown to be clouds with comonotonic
distributions. In
general, clouds cannot always be represented by random sets, in fact
not even
by 2-monotone (convex) capacities.
http://arxiv.org/abs/0808.2779
---------------------------------------------------------------
7402. BAYESIAN NONPARAMETRIC ESTIMATORS DERIVED FROM CONDITIONAL
GIBBS STRUCTURES
Antonio Lijoi and Igor Pr\"unster and Stephen G. Walker
We consider discrete nonparametric priors which induce Gibbs-type
exchangeable random partitions and investigate their posterior
behavior in
detail. In particular, we deduce conditional distributions and the
corresponding Bayesian nonparametric estimators, which can be readily
exploited
for predicting various features of additional samples. The results
provide
useful tools for genomic applications where prediction of future
outcomes is
required.
http://arxiv.org/abs/0808.2863
---------------------------------------------------------------
7403. CENTRAL LIMIT THEOREM FOR A MANY-SERVER QUEUE WITH RANDOM
SERVICE RATES
Rami Atar
Given a random variable $N$ with values in ${\mathbb{N}}$, and $N$
i.i.d.
positive random variables $\{\mu_k\}$, we consider a queue with renewal
arrivals and $N$ exponential servers, where server $k$ serves at rate $
\mu_k$,
under two work conserving routing schemes. In the first, the service
rates
$\{\mu_k\}$ need not be known to the router, and each customer to
arrive at a
time when some servers are idle is routed to the server that has been
idle for
the longest time (or otherwise it is queued). In the second, the
service rates
are known to the router, and a customer that arrives to find idle
servers is
routed to the one whose service rate is greatest. In the many-server
heavy
traffic regime of Halfin and Whitt, the process that represents the
number of
customers in the system is shown to converge to a one-dimensional
diffusion
with a random drift coefficient, where the law of the drift depends on
the
routing scheme. A related result is also provided for nonrandom
environments.
http://arxiv.org/abs/0808.2865
---------------------------------------------------------------
7404. STEIN'S METHOD FOR DISCRETE GIBBS MEASURES
Peter Eichelsbacher and Gesine Reinert
Stein's method provides a way of bounding the distance of a probability
distribution to a target distribution $\mu$. Here we develop Stein's
method for
the class of discrete Gibbs measures with a density $e^V$, where $V$
is the
energy function. Using size bias couplings, we treat an example of Gibbs
convergence for strongly correlated random variables due to Chayes and
Klein
[Helv. Phys. Acta 67 (1994) 30--42]. We obtain estimates of the
approximation
to a grand-canonical Gibbs ensemble. As side results, we slightly
improve on
the Barbour, Holst and Janson [Poisson Approximation (1992)] bounds
for Poisson
approximation to the sum of independent indicators, and in the case of
the
geometric distribution we derive better nonuniform Stein bounds than
Brown and
Xia [Ann. Probab. 29 (2001) 1373--1403].
http://arxiv.org/abs/0808.2877
---------------------------------------------------------------
7405. ON HONEST TIMES IN FINANCIAL MODELING
Ashkan Nikeghbali and Eckhard Platen
This paper demonstrates the usefulness and importance of the concept of
honest times to financial modeling. It studies a financial market with
asset
prices that follow jump-diffusions with negative jumps. The central
building
block of the market model is its growth optimal portfolio (GOP), which
maximizes the growth rate of strictly positive portfolios. Primary
security
account prices, when expressed in units of the GOP, turn out to be
nonnegative
local martingales. In the proposed framework an equivalent risk neutral
probability measure need not exist. Derivative prices are obtained as
conditional expectations of corresponding future payoffs, with the GOP
as
numeraire and the real world probability as pricing measure. The time
when the
global maximum of a portfolio with no positive jumps, when expressed
in units
of the GOP, is reached, is shown to be a generic representation of an
honest
time. We provide a general formula for the law of such honest times
and compute
the conditional distributions of the global maximum of a portfolio in
this
framework. Moreover, we provide a stochastic integral representation for
uniformly integrable martingales whose terminal values are functions
of the
global maximum of a portfolio. These formulae are model independent and
universal. We also specialize our results to some examples where we
hedge a
payoff that arrives at an honest time.
http://arxiv.org/abs/0808.2892
---------------------------------------------------------------
7406. ON THE LARGEST COMPONENT OF A RANDOM GRAPH WITH A SUBPOWER-LAW
DEGREE SEQUENCE IN A SUBCRITICAL PHASE
B. G. Pittel
A uniformly random graph on $n$ vertices with a fixed degree sequence,
obeying a $\gamma$ subpower law, is studied. It is shown that, for $
\gamma>3$,
in a subcritical phase with high probability the largest component
size does
not exceed $n^{1/\gamma+\varepsilon_n}$, $\varepsilon_n=O(\ln\ln n/\ln
n)$,
$1/\gamma$ being the best power for this random graph. This is similar
to the
best possible $n^{1/(\gamma-1)}$ bound for a different model of the
random
graph, one with independent vertex degrees, conjectured by Durrett,
and proved
recently by Janson.
http://arxiv.org/abs/0808.2907
---------------------------------------------------------------
7407. ON THE CONCENTRATION AND THE CONVERGENCE RATE WITH A MOMENT
CONDITION IN FIRST PASSAGE PERCOLATION
Yu Zhang
We consider the first passage percolation model on the ${\bf Z}^d$
lattice.
In this model, we assign independently to each edge $e$ a non-negative
passage
time $t(e)$ with a common distribution $F$. Let $a_{0,n}$ be the
passage time
from the origin to $(n,0,..., 0)$. Under the exponential tail
assumption,
Kesten (1993) and Talagrand (1995) investigated the concentration of
$a_{0,n}$ from its mean using different methods. With this
concentration and
the exponential tail assumption, Alexander gave an estimate for the
convergence
rate for ${\bf E} a_{0,n}$. In this paper, focusing on a moment
condition, we
reinvestigate the concentration and the convergence rate for $a_{0,n}$
using a
special martingale structure.
http://arxiv.org/abs/0808.3021
---------------------------------------------------------------
7408. SKEW-PRODUCT REPRESENTATIONS OF MULTIDIMENSIONAL DUNKL MARKOV
PROCESSES
Oleksandr Chybiryakov
In this paper we obtain skew-product representations of the
multidimensional
Dunkl processes which generalize the skew-product decomposition in
dimension 1
obtained in L. Gallardo and M. Yor. Some remarkable properties of the
Dunkl
martingales. S\'{e}minaire de Probabilit\'{e}s XXXIX, 2006. We also
study the
radial part of the Dunkl process, i.e. the projection of the Dunkl
process on a
Weyl chamber.
http://arxiv.org/abs/0808.3033
---------------------------------------------------------------
7409. LIMIT LAWS FOR THE ENERGY OF A CHARGED POLYMER
Xia Chen
In this paper we obtain the central limit theorems, moderate
deviations and
the laws of the iterated logarithm for the energy \[H_n=\sum_{1\le j<k
\le
n}\omega_j\omega_k1_{\{S_j=S_k\}}\] of the polymer $\{S_1,...,S_n\}$
equipped
with random electrical charges $\{\omega_1,...,\omega_n\}$. Our
approach is
based on comparison of the moments between $H_n$ and the self-
intersection
local time \[Q_n=\sum_{1\le j<k\le n}1_{\{S_j=S_k\}}\] run by the
$d$-dimensional random walk $\{S_k\}$. As partially needed for our main
objective and partially motivated by their independent interest, the
central
limit theorems and exponential integrability for $Q_n$ are also
investigated in
the case $d\ge3$.
http://arxiv.org/abs/0808.3037
---------------------------------------------------------------
7410. ON MEAN CENTRAL LIMIT THEOREMS FOR STATIONARY SEQUENCES
J\'er\^ome Dedecker and Emmanuel Rio
In this paper, we give estimates of the minimal ${\mathbb{L}}^1$
distance
between the distribution of the normalized partial sum and the limiting
Gaussian distribution for stationary sequences satisfying projective
criteria
in the style of Gordin or weak dependence conditions.
http://arxiv.org/abs/0808.3048
---------------------------------------------------------------
7411. JOINT CONTINUITY OF THE LOCAL TIMES OF FRACTIONAL BROWNIAN SHEETS
Antoine Ayache and Dongsheng Wu and Yimin Xiao
Let $B^H=\{B^H(t),t\in{{\mathbb{R}}_+^N}\}$ be an $(N,d)$-fractional
Brownian
sheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by
$B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where
$B^H_1,...,B^H_d$ are independent copies of a real-valued fractional
Brownian
sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, then
the
local times of $B^H$ are jointly continuous. This verifies a
conjecture of Xiao
and Zhang (Probab. Theory Related Fields 124 (2002)). We also
establish sharp
local and global H\"{o}lder conditions for the local times of $B^H$.
These
results are applied to study analytic and geometric properties of the
sample
paths of $B^H$.
http://arxiv.org/abs/0808.3054
---------------------------------------------------------------
7412. UNIFORM DETERMINISTIC EQUIVALENT OF ADDITIVE FUNCTIONALS AND
NON-PARAMETRIC DRIFT ESTIMATION FOR ONE-DIMENSIONAL RECURRENT DIFFUSIONS
D. Loukianova and O. Loukianov
Usually the problem of drift estimation for a diffusion process is
considered
under the hypothesis of ergodicity. It is less often considered under
the
hypothesis of null-recurrence, simply because there are fewer limit
theorems
and existing ones do not apply to the whole null-recurrent class. The
aim of
this paper is to provide some limit theorems for additive functionals
and
martingales of a general (ergodic or null) recurrent diffusion which
would
allow us to have a somewhat unified approach to the problem of non-
parametric
kernel drift estimation in the one-dimensional recurrent case. As a
particular
example we obtain the rate of convergence of the Nadaraya--Watson
estimator in
the case of a locally H\"{o}lder-continuous drift.
http://arxiv.org/abs/0808.3069
---------------------------------------------------------------
7413. SOME PROPERTIES OF MULTIVARIATE MEASURES OF CONCORDANCE
M. D. Taylor
We explore the consequences of a set of axioms which extend Scarsini's
axioms
for bivariate measures of concordance to the multivariate case and
exhibit the
following results: (1) A method of extending measures of concordance
from the
bivariate case to arbitrarily high dimensions. (2) A formula
expressing the
measure of concordance of the random vectors $(\pm X_1,...,\pm X_n)$
in terms
of the measures of concordance of the "marginal" random vectors
$(X_{i_1},...,X_{i_k})$. (3) A method of expressing the measure of
concordance
of an odd-dimensional copula in terms of the measures of concordance
of its
even-dimensional marginals. (4) A family of relations which exist
between the
measures of concordance of the marginals of a given copula.
http://arxiv.org/abs/0808.3105
---------------------------------------------------------------
7414. EXPLICIT AND ALMOST EXPLICIT SPECTRAL CALCULATIONS FOR
DIFFUSION OPERATORS
Ross G. Pinsky
The diffusion operator $$ H_D=-\frac12\frac d{dx}a\frac d{dx}-b\frac
d{dx}=-\frac12\exp(-2B)\frac d{dx}a\exp(2B)\frac d{dx}, $$ where
$B(x)=\int_0^x\frac ba(y)dy$, defined either on $R^+=(0,\infty)$ with
the
Dirichlet boundary condition at $x=0$, or on $R$, can be realized as a
self-adjoint operator with respect to the density $\exp(2Q(x))dx$. The
operator
is unitarily equivalent to the Schr\"odinger-type operator $H_S=-
\frac12\frac
d{dx}a\frac d{dx}+V_{b,a}$, where $V_{b,a}=\frac12(\frac{b^2}a+b')$.
We obtain
an explicit criterion for the existence of a compact resolvent and
explicit
formulas up to the multiplicative constant 4 for the infimum of the
spectrum
and for the infimum of the essential spectrum for these operators. We
give some
applications which show in particular how $\inf\sigma(H_D)$ scales
when $a=\nu
a_0$ and $b=\gamma b_0$, where $\nu$ and $\gamma$ are parameters, and
$a_0$ and
$b_0$ are chosen from certain classes of functions. We also give
applications
to self-adjoint, multi-dimensional diffusion operators.
http://arxiv.org/abs/0808.3044
---------------------------------------------------------------
7415. ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH
GEOMETRIC DISTRIBUTION
Eugen J. Ionascu and Alin A. Stancu
We are interested in constructing concrete independent events in purely
atomic probability spaces with geometric distribution. Among other
facts we
prove that there are uncountable many sequences of independent events.
http://arxiv.org/abs/0808.3155
---------------------------------------------------------------
7416. ENTROPY AND CHAOS IN THE KAC MODEL
E.A. Carlen and M.C. Carvalho and J. Le Roux and M. Loss and C.
Villani
We investigate the behavior in $N$ of the $N$--particle entropy
functional
for Kac's stochastic model of Boltzmann dynamics, and its relation to
the
entropy function for solutions of Kac's one dimensional nonlinear model
Boltzmann equation. We prove a number of results that bring together
the notion
of propagation of chaos, which Kac introduced in the context of this
model,
with the problem of estimating the rate of equilibration in the model in
entropic terms, and obtain a bound showing that the entropic rate of
convergence can be arbitrarily slow. Results proved here show that one
can in
fact use entropy production bounds in Kac's stochastic model to obtain
entropic
convergence bounds for his non linear model Boltzmann equation, though
the
problem of obtaining optimal lower bounds of this sort for the
original Kac
model remains open, and the upper bounds obtained here show that this
problem
is somewhat subtle.
http://arxiv.org/abs/0808.3192
---------------------------------------------------------------
7417. ON HOMOGENIZATION OF SPACE-TIME DEPENDENT AND DEGENERATE RANDOM
FLOWS II
R\'emi Rhodes
We study the long time behavior (homogenization) of a diffusion in
random
medium with time and space dependent coefficients. The diffusion
coefficient
may degenerate. In Stochastic Process. Appl. (2007) (to appear), an
invariance
principle is proved for the critical rescaling of the diffusion. Here,
we
generalize this approach to diffusions whose space-time scaling
differs from
the critical one.
http://arxiv.org/abs/0808.3301
---------------------------------------------------------------
7418. PERCOLATION FOR THE VACANT SET OF RANDOM INTERLACEMENTS
Vladas Sidoravicius and Alain-Sol Sznitman
We investigate random interlacements on Z^d, d bigger or equal to 3.
This
model recently introduced in arXiv:0704.2560 corresponds to a Poisson
cloud on
the space of doubly infinite trajectories modulo time-shift tending to
infinity
at positive and negative infinite times. A non-negative parameter u
measures
how many trajectories enter the picture. Our main interest lies in the
percolative properties of the vacant set left by random interlacements
at level
u. We show that for all d bigger or equal to 3 the vacant set at level u
percolates when u is small. This solves an open problem of arXiv:
0704.2560,
where this fact has only been established when d is bigger or equal to
7. It
also completes the proof of the non-degeneracy in all dimensions d
bigger or
equal to 3 of the critical parameter introduced in arXiv:0704.2560.
http://arxiv.org/abs/0808.3344
---------------------------------------------------------------
7419. MAXIMAL INEQUALITIES IN BILATERAL GRAND LEBESQUE SPACES OVER
UNBOUNDED MEASURE
E. Ostrovsky and E. Rogover
In this paper non-asymptotic exact rearrangement invariant norm
estimates are
derived for the maximum distribution of the family elements of some
rearrangement invariant (r.i.) space over unbounded measure in the
entropy
terms and in the terms of generic chaining.
We consider some applications in the martingale theory and in the
theory of
Fourier series.
http://arxiv.org/abs/0808.3247
---------------------------------------------------------------
7420. SUPPORT OF BORELIAN MEASURES IN SEPARABLE BANACH SPACES
E. Ostrovsky
We prove in this article that every Borelian measure, for example, the
distribution of a random variable, in separable Banach space has a
support
which is compact embedded Banach subspace; and prove that if the norm
of the
random variable belongs to some exponential Orlicz space, then the new
subspace
can be choose such that the norm of this variable in the new space
also belongs
to other exponential Orlicz space.
http://arxiv.org/abs/0808.3248
---------------------------------------------------------------
7421. CALL OPTION PRICES BASED ON BESSEL PROCESSES
Ju-Yi Yen and Marc Yor
As a complement to some recent work by Pal and Protter, "Strict local
martingales, bubbles, and no early exercise", we show that the call
option
prices associated with the Bessel strict local martingales are
integrable over
time, and we discuss the probability densities obtained thus.
http://arxiv.org/abs/0808.3402
---------------------------------------------------------------
7422. A CENTRAL LIMIT THEOREM FOR THE RESCALED L\'EVY AREA OF TWO-
DIMENSIONAL FRACTIONAL BROWNIAN MOTION WITH HURST INDEX $H<1/4$
Jeremie Unterberger
Let $B=(B^{(1)},B^{(2)})$ be a two-dimensional fractional Brownian
motion
with Hurst index $\alpha\in (0,1/4)$. Using an analytic approximation
$B(\eta)$
of $B$ introduced in \cite{Unt08}, we prove that the rescaled L\'evy
area
process $(s,t)\to \eta^{\half(1-4\alpha)}\int_s^t dB_{t_1}^{(1)}(\eta)
\int_s^{t_1} dB_{t_2}^{(2)}(\eta)$ converges in law to $W_t-W_s$ where
$W$ is a
Brownian motion independent from $B$. The method relies on a very
general
scheme of analysis of singularities of analytic functions, applied to
the
moments of finite-dimensional distributions of the L\'evy area.
http://arxiv.org/abs/0808.3458
---------------------------------------------------------------
7423. K-PROCESSES, SCALING LIMIT AND AGING FOR THE TRAP MODEL IN THE
COMPLETE GRAPH
L. R. G. Fontes and P. Mathieu
We study K-processes, which are Markov processes in a denumerable state
space, all of whose elements are stable, with the exception of a
single state,
starting from which the process enters finite sets of stable states with
uniform distribution. We show how these processes arise, in a particular
instance, as scaling limits of the trap model in the complete graph, and
subsequently derive aging results for those models in this context.
http://arxiv.org/abs/0808.3494
---------------------------------------------------------------
7424. LARGE DEVIATIONS FOR A RANDOM SIGN LINDLEY RECURSION
Maria Vlasiou and Zbigniew Palmowski
We investigate the tail behaviour of the steady state distribution of a
stochastic recursion that generalises Lindley's recursion. This
recursion
arises in queuing systems with dependent interarrival and service
times, and
includes alternating service systems and carousel storage systems as
special
cases. We obtain precise tail asymptotics in three qualitatively
different
cases, and compare these with existing results for Lindley's recursion
and for
alternating service systems.
http://arxiv.org/abs/0808.3495
---------------------------------------------------------------
7425. EDGE PERCOLATION ON A RANDOM REGULAR GRAPH OF LOW DEGREE
Boris Pittel
Consider a uniformly random regular graph of a fixed degree $d\ge3$,
with $n$
vertices. Suppose that each edge is open (closed), with probability
$p(q=1-p)$,
respectively. In 2004 Alon, Benjamini and Stacey proved that
$p^*=(d-1)^{-1}$
is the threshold probability for emergence of a giant component in the
subgraph
formed by the open edges. In this paper we show that the transition
window
around $p^*$ has width roughly of order $n^{-1/3}$. More precisely,
suppose
that $p=p(n)$ is such that $\omega:=n^{1/3}|p-p^*|\to\infty$. If $p<p^*
$, then
with high probability (whp) the largest component has $O((p-
p^*)^{-2}\log n)$
vertices. If $p>p^*$, and $\log\omega\gg\log\log n$, then whp the
largest
component has about $n(1-(p\pi+q)^d)\asymp n(p-p^*)$ vertices, and the
second
largest component is of size $(p-p^*)^{-2}(\log n)^{1+o(1)}$, at most,
where
$\pi=(p\pi+q)^{d-1},\pi\in(0,1)$. If $\omega$ is merely
polylogarithmic in $n$,
then whp the largest component contains $n^{2/3+o(1)}$ vertices.
http://arxiv.org/abs/0808.3516
---------------------------------------------------------------
7426. EXCEPTIONAL TIMES FOR THE DYNAMICAL DISCRETE WEB
L. R. G. Fontes and C. M. Newman and K. Ravishankar and E. Schertzer
The dynamical discrete web (DyDW),introduced in recent work of Howitt
and
Warren, is a system of coalescing simple symmetric one-dimensional
random walks
which evolve in an extra continuous dynamical time parameter \tau. The
evolution is by independent updating of the underlying Bernoulli
variables
indexed by discrete space-time that define the discrete web at any
fixed \tau.
In this paper, we study the existence of exceptional (random) values
of \tau
where the paths of the web do not behave like usual random walks and the
Hausdorff dimension of the set of exceptional such \tau. Our results are
motivated by those about exceptional times for dynamical percolation
in high
dimension by H\"{a}ggstrom, Peres and Steif, and in dimension two by
Schramm
and Steif. The exceptional behavior of the walks in the DyDW is rather
different from the situation for the dynamical random walks of
Benjamini,
H\"{a}ggstrom, Peres and Steif. For example, we prove that the walk
from the
origin S^\tau_0 violates the law of the iterated logarithm (LIL) on a
set of
\tau of Hausdorff dimension one. We also discuss how these and other
results
extend to the dynamical Brownian web, the natural scaling limit of the
DyDW.
http://arxiv.org/abs/0808.3599
---------------------------------------------------------------
7427. LARGE DEVIATIONS FOR INFINITE DIMENSIONAL STOCHASTIC DYNAMICAL
SYSTEMS
Amarjit Budhiraja and Paul Dupuis and Vasileios Maroulas
The large deviations analysis of solutions to stochastic differential
equations and related processes is often based on approximation. The
construction and justification of the approximations can be onerous,
especially
in the case where the process state is infinite dimensional. In this
paper we
show how such approximations can be avoided for a variety of infinite
dimensional models driven by some form of Brownian noise. The approach
is based
on a variational representation for functionals of Brownian motion.
Proofs of
large deviations properties are reduced to demonstrating basic
qualitative
properties (existence, uniqueness and tightness) of certain
perturbations of
the original process.
http://arxiv.org/abs/0808.3631
---------------------------------------------------------------
7428. REVERSIBILITY OF CHORDAL SLE
Dapeng Zhan
We prove that the chordal SLE$_{\kappa}$ trace is reversible for
$\kappa\in(0,4]$.
http://arxiv.org/abs/0808.3649
---------------------------------------------------------------
7429. MARTINGALE APPROACH TO STOCHASTIC DIFFERENTIAL GAMES OF CONTROL
AND STOPPING
Ioannis Karatzas and Ingrid-Mona Zamfirescu
We develop a martingale approach for studying continuous-time stochastic
differential games of control and stopping, in a non-Markovian
framework and
with the control affecting only the drift term of the state-process.
Under
appropriate conditions, we show that the game has a value and
construct a
saddle pair of optimal control and stopping strategies. Crucial in this
construction is a characterization of saddle pairs in terms of
pathwise and
martingale properties of suitable quantities.
http://arxiv.org/abs/0808.3656
---------------------------------------------------------------
7430. ASYMPTOTICS OF RANDOMLY STOPPED SUMS IN THE PRESENCE OF HEAVY
TAILS
Denis Denisov and Sergey Foss and Dmitry Korshunov
We study conditions under which $P(S_\tau>x)\sim P(M_\tau>x)\sim E\tau
P(\xi_1>x)$ as $x\to\infty$, where $S_\tau$ is a sum $\xi_1+...+\xi_
\tau$ of
random size $\tau$ and $M_\tau$ is a maximum of partial sums
$M_\tau=\max_{n\le\tau}S_n$. Here $\xi_n$, $n=1$, 2, ..., are
independent
identically distributed random variables whose common distribution is
assumed
to be subexponential. We consider mostly the case where $\tau$ is
independent
of the summands; also, in a particular situation, we deal with a
stopping time.
Also we consider the case where $E\xi>0$ and where the tail of $\tau
$ is
comparable with or heavier than that of $\xi$, and obtain the
asymptotics
$P(S_\tau>x) \sim E\tau P(\xi_1>x)+P(\tau>x/E\xi)$ as $x\to\infty$.
This case
is of a primary interest in the branching processes.
In addition, we obtain new uniform (in all $x$ and $n$) upper
bounds for the
ratio $P(S_n>x)/P(\xi_1>x)$ which substantially improve Kesten's bound
in the
subclass ${\mathcal S}^*$ of subexponential distributions.
http://arxiv.org/abs/0808.3697
---------------------------------------------------------------
7431. SURVIVAL OF CONTACT PROCESSES ON THE HIERARCHICAL GROUP
Siva R. Athreya and Jan M. Swart
We consider contact processes on the hierarchical group $\Omega_N$ with
freedom $N$, where sites infect other sites at hierarchical distance $k
$ with
rate $\alpha_kN^{-k}$, and sites become healthy with recovery rate $
\delta$. We
show that the critical recovery rate $\delta_{\rm c}$ is zero (i.e., the
process dies out for any $\delta>0$) if
$\liminf_{k\to\infty}N^{-k}\log(\beta_k)=-\infty$, where
$\beta_k:=\sum_{n=k}^\infty\alpha_n$. On the other hand, in the
special case
that $N$ is a power of two, we show that $\delta_{\rm c}>0$ provided
that
$\sum_kN^{-k}\log(\alpha_k)>-\infty$. The proof of this latter fact is
based on
a coupling argument that compares contact processes on $\Omega_2$ with
contact
processes on a renormalized lattice.
http://arxiv.org/abs/0808.3732
---------------------------------------------------------------
7432. NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF THE Q-
OPTIMAL MEASURE
Sotirios Sabanis
This paper presents the general form and essential properties of the
q-optimal measure following the approach of Delbaen and Schachermayer
(1996)
and proves its existence under mild conditions. Most importantly, it
states a
necessary and sufficient condition for a candidate measure to be the q-
optimal
measure in the case even of signed measures. Finally, an updated
characterization of the q-optimal measure for continuous asset price
processes
is presented in the light of the counterexample appearing in Cerny and
Kallsen
(2006) concerning Hobson's (2004) approach.
http://arxiv.org/abs/0808.3751
---------------------------------------------------------------
7433. OCCUPATION TIMES OF SUBCRITICAL BRANCHING IMMIGRATION SYSTEM
WITH MARKOV MOTIONS
Piotr Milos
We consider a branching system consisting of particles moving
according to a
Markov family in $\Rd$ and undergoing subcritical branching with a
constant
rate $V>0$. New particles immigrate to the system according to
homogeneous
space-time Poisson random field. The process of the fluctuations of the
rescaled occupation time is studied with very mild assumptions on the
Markov
family. In this general setting a functional central limit theorem is
proved.
The subcriticality of the branching law is crucial for the limit
behaviour and
in a sense overwhelms the properties of the particles' motion. It is
for this
reason that the limit is the same for all dimensions and can be
obtained for a
wide class of Markov processes. Another consequence is the form of the
limit -
$\SP$-valued Wiener process with a simple temporal structure and a
complicated
spatial one. This behaviour contrasts sharply with the case of critical
branching systems.
http://arxiv.org/abs/0808.3755
---------------------------------------------------------------
7434. THE STABILITY OF GROWING NETWORKS
Zhenting Hou; Xiangxing Kong; Qinggui Zhao
In this paper we abstract a kind of stochastic processes from evolving
processes of growing networks, which are called as growing network
Markov
chains, threrefore the existence of the steady degree distribution and
the
formulas of the degree distribution are transformed to the corresponding
problems of growing network Markov chains. We divide growing network
markov
chains into two classes: non-multiple and multiple, and then, obtain the
condition in which the steady degree distribution exists and the exact
formulas
respectively, and then applied it to the various growing networks. So
we have
rigorous, exact and united solution of the steady degree distribution
of the
growing networks.
http://arxiv.org/abs/0808.3661
---------------------------------------------------------------
7435. CRITIQUE DU RAPPORT SIGNAL \`A BRUIT EN COMMUNICATIONS NUM
\'ERIQUES -- QUESTIONING THE SIGNAL TO NOISE RATIO IN DIGITAL
COMMUNICATIONS
Michel Fliess (LIX and INRIA Saclay - Ile de France)
The signal to noise ratio, which plays such an important r\^ole in
information theory, is shown to become pointless for digital
communications
where the demodulation is achieved via new fast estimation techniques.
Operational calculus, differential algebra, noncommutative algebra and
nonstandard analysis are the main mathematical tools.
http://arxiv.org/abs/0808.3712
---------------------------------------------------------------
7436. DIRECTED POLYMER IN RANDOM ENVIRONMENT AND LAST PASSAGE
PERCOLATION
Philippe Carmona (LMJL)
The sequence of random probability measures $\nu_n$ that gives a path of
length $n$, $\unsur{n}$ times the sum of the random weights collected
along the
paths, is shown to satisfy a large deviations principle with good rate
function
the Legendre transform of the free energy of the associated directed
polymer in
a random environment. Consequences on the asymptotics of the typical
number of
paths whose collected weight is above a fixed proportion are then drawn.
http://arxiv.org/abs/0808.3842
---------------------------------------------------------------
7437. ON A SURPRISING RELATION BETWEEN THE MARCHENKO-PASTUR LAW,
RECTANGULAR AND SQUARE FREE CONVOLUTIONS
Florent Benaych-Georges (PMA)
In this paper, we prove a result linking the square and the rectangular
R-transforms, which consequence is a surprising relation between the
square and
rectangular free convolutions, involving the Marchenko-Pastur law.
Consequences
on infinite divisibility and on the arithmetics of Voiculescu's free
additive
and multiplicative convolutions are given.
http://arxiv.org/abs/0808.3938
---------------------------------------------------------------
7438. A PRIORI HOLDER ESTIMATE, PARABOLIC HARNACK PRINCIPLE AND HEAT
KERNEL ESTIMATES FOR DIFFUSIONS WITH JUMPS
Zhen-Qing Chen and Takashi Kumagai
In this paper, we consider the following type of non-local
(pseudo-differential) operators $\LL $ on $\R^d$:
$$
\LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i}
(a_{ij}(x)
\frac{\partial}{\partial x_j}) +
\lim_{\eps \downarrow 0} \int_{\{y\in \R^d: |y-x|>\eps\}}
(u(y)-u(x)) J(x, y) dy,
$$ where $A(x)=(a_{ij}(x))_{1\leq i, j\leq d}$ is a measurable $d
\times d$
matrix-valued function on $\R^d$ that is uniform elliptic and bounded
and $J$
is a symmetric measurable non-trivial non-negative kernel on $\R^d
\times \R^d$
satisfying certain conditions. Corresponding to $\LL$ is a symmetric
strong
Markov process $X$ on $\R^d$ that has both the diffusion component and
pure
jump component. We establish a priori H\"older estimate for bounded
parabolic
functions of $\LL$ and parabolic Harnack principle for positive
parabolic
functions of $\LL$. Moreover, two-sided sharp heat kernel estimates
are derived
for such operator $\LL$ and jump-diffusion $X$. In particular, our
results
apply to the mixture of symmetric diffusion of uniformly elliptic
divergence
form operator and mixed stable-like processes on $\R^d$. To establish
these
results, we employ methods from both probability theory and analysis.
http://arxiv.org/abs/0808.4010
---------------------------------------------------------------
7439. ROBUST HEDGING OF DOUBLE TOUCH BARRIER OPTIONS
Alexander M. G. Cox and Jan K. Ob{\l}\'oj
We consider model-free pricing of digital options, which pay out if the
underlying asset has crossed both upper and lower barriers. We make
only weak
assumptions about the underlying process (typically continuity), but
assume
that the initial prices of call options with the same maturity and all
strikes
are known. Under such circumstances, we are able to give upper and
lower bounds
on the arbitrage-free prices of the relevant options, and further, using
techniques from the theory of Skorokhod embeddings, to show that these
bounds
are tight. Additionally, martingale inequalities are derived, which
provide the
trading strategies with which we are able to realise any potential
arbitrages.
We show that, depending of the risk aversion of the investor, the
resulting
hedging strategies can outperform significantly the standard delta/
vega-hedging
in presence of market frictions and/or model misspecification.
http://arxiv.org/abs/0808.4012
---------------------------------------------------------------
7440. THE CENTER OF MASS FOR SPATIAL BRANCHING PROCESSES AND AN
APPLICATION FOR SELF-INTERACTION
Janos Englander
In this paper we prove that the center of mass of a supercritical
branching-Brownian motion, or that of a supercritical super-Brownian
motion
tends to a limiting position almost surely, which, in a sense
complements a
result of Tribe on the final behavior of a critical super-Brownian
motion. This
is shown to be true also for a model where branching Brownian motion is
modified by attraction/repulsion between particles.
We then put this observation together with the description of the
interacting
system as viewed from its center of mass, and get the following
asymptotic
behavior: the system asymptotically becomes a branching Ornstein
Uhlenbeck
process (inward for attraction and outward for repulsion), but the
origin is
shifted to a random point which has normal distribution, and the
Ornstein
Uhlenbeck particles are not independent but constitute a system with a
degree
of freedom which is less by their number by precisely one.
http://arxiv.org/abs/0808.4024
---------------------------------------------------------------
7441. THE DIAMETER OF SPARSE RANDOM GRAPHS
Oliver Riordan and Nicholas Wormald
In this paper we study the diameter of the random graph $G(n,p)$,
i.e., the
largest distance between two vertices in the same component, for a
wide range
of functions $p=p(n)$. For $p=\la/n$ with $\la>1$ constant, we give a
simple
proof of an essentially best possible result, with an $\Op(1)$ additive
correction term. For $p=(1+\eps)/n$ with $\eps\to 0$, we obtain a
corresponding
result that applies almost all the way down to the scaling window of
the phase
transition, with an $\Op(1/\eps)$ additive correction term whose
(appropriately
scaled) limiting distribution we describe. Throughout we use branching
process
methods, rather than the more common approach of separate analysis of
the
2-core.
http://arxiv.org/abs/0808.4067
-----------------------------
Stefano Iacus
IMS Groups Editor
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