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Probability Abstracts 99
This document contains abstracts 5757-5996 from
July-1-2007 to August-31-2007.
They have been mailed on September 14th, 2007.
Author(s): Michael Aizenman and Francois Germinet and Abel Klein and Simone Warzel
Abstract: As was noted already by A. N. Kolmogorov, any random variable has
a Bernoulli
component. This observation provides a tool for the extension of
results which
are known for Bernoulli random variables to arbitrary distributions. Two
applications are provided here: i. an anti-concentration bound for a
class of
functions of independent random variables, where probabilistic bounds
are
extracted from combinatorial results, and ii. a proof, based on the
Bernoulli
case, of spectral localization for random Schroedinger operators with
arbitrary
probability distributions for the single site coupling constants. For
a general
random variable, the Bernoulli component may be defined so that its
conditional
variance is uniformly positive. The natural maximization problem is
an optimal
transport question which is also addressed here.
http://arxiv.org/abs/0707.0095
Author(s): Kurt Johansson
Abstract: We show that the transition probability of the Markoc chain
$(G(j,1),...,G(j,n))_{j\ge 1}$, where the $G(i,j)'s$ are certain
directed
last-passage times, is given by a determinant of a special form. An
analogous
formula has recently been obtained by Warren in a Brownian motion model.
Furthermore we demonstrate that this formula leads to the Meixner
ensemble when
we compute the distribution function for $G(m,n)$. We also obtain the
Fredholm
determinant representation of this distribution, where the kernel has
a double
contour integral representation.
http://arxiv.org/abs/0707.0098
Author(s): Thomas Cass and Peter Friz and Nicolas Victoir
Abstract: Malliavin Calculus is about Sobolev-type regularity of functionals
on Wiener
space, the main example being the Ito map obtained by solving stochastic
differential equations. Rough path analysis is about strong
regularity of
solution to (possibly stochastic) differential equations. We combine
arguments
of both theories and discuss existence of a density for solutions to
stochastic
differential equations driven by a general class of non-degenerate
Gaussian
processes, including processes with sample path regularity worse than
Brownian
motion.
http://arxiv.org/abs/0707.0154
Author(s): Ilya Molchanov
Abstract: It is known that each symmetric stable distribution in $R^d$ is
related to a
norm on $R^d$ that makes $R^d$ embeddable in $L_p([0,1])$. In case of a
multivariate Cauchy distribution the unit ball in this norm
corresponds is the
polar set to a convex set in $R^d$ called a zonoid. This work
exploits recent
advances in convex geometry in order to come up with new
probabilistic results
for multivariate stable distributions. In particular, it provides
expressions
for moments of the Euclidean norm of a stable vector, mixed moments
and various
integrals of the density function. It is shown how to use geometric
inequalities in order to bound important parameters of stable laws.
It is shown
that each symmetric stable laws appears as the limit for the sum of
sub-Gaussian laws and an estimate for the probability distance to a
sub-Gaussian law is given. Operations with convex sets induce the
well-known
and new operations with stable vectors. Furthermore, covariation,
regression
and orthogonality concepts for stable laws acquire geometric
interpretations. A
similar collection of results is presented for one-sided stable laws.
http://arxiv.org/abs/0707.0221
Author(s): Arnaud Gloter (LAMA) and Emmanuel Gobet (LJK)
Abstract: In this paper we prove the Local Asymptotic Mixed Normality (LAMN)
property
for the statistical model given by the observation of local means of a
diffusion process $X$. Our data are given by $ \int_0^1 X_{\frac{s+i}
{n}} \dd
\mu (s)$ for $i=0,...,n-1$ and the unknown parameter appears in the
diffusion
coefficient of the process $X$ only. Although the data are nor Markovian
neither Gaussian we can write down, with help of Malliavin calculus, an
explicit expression for the log-likelihood of the model, and then
study the
asymptotic expansion. We actually find that the asymptotic
information of this
model is the same one as for a usual discrete sampling of $X$.
http://arxiv.org/abs/0707.0257
Author(s): Pavel Chigansky
Abstract: The paper studies large sample asymptotic properties of the Maximum
Likelihood Estimator (MLE) for the parameter of a continuous time
Markov chain,
observed in white noise. Using the method of weak convergence of
likelihoods
due to I.Ibragimov and R.Khasminskii, consistency, asymptotic
normality and
convergence of moments are established for MLE under certain strong
ergodicity
conditions of the chain.
http://arxiv.org/abs/0707.0271
Author(s): Xinjia Chen and Guoxiang Gu and Kemin Zhou
Abstract: In this paper, we propose a statistical theory on measurement and
estimation
of Rayleigh fading channels in wireless communications and provide
complete
solutions to the fundamental problems: What is the optimum estimator
for the
statistical parameters associated with the Rayleigh fading channel,
and how
many measurements are sufficient to estimate these parameters with the
prescribed margin of error and confidence level? Our proposed
statistical
theory suggests that two testing signals of different strength be
used. The
maximum likelihood (ML) estimator is obtained for estimation of the
statistical
parameters of the Rayleigh fading channel that is both sufficient and
complete
statistic. Moreover, the ML estimator is the minimum variance (MV)
estimator
that in fact achieves the Cramer-Rao lower bound.
http://arxiv.org/abs/0707.0284
Author(s): Itaru Mitoma and Seiki Nishikawa
Abstract: In an abstract Wiener space setting, we constract a rigorous
mathematical
model of the one-loop approximation of the perturbative Chern-Simons
integral,
and derive its explicit asymptotic expansion for stochastic Wilson
lines.
http://arxiv.org/abs/0707.0047
Author(s): Erhan Bayraktar and Masahiko Egami
Abstract: We characterize the optimal switching problem as coupled optimal
stoping
problems. We then use the optimal stopping theory to provide a
solution. As
opposed to the methods using quasi-variational inequalities and
verification
theorem we directly work with the value function.
http://arxiv.org/abs/0707.0100
Author(s): Peter Friz and Nicolas Victoir
Abstract: We consider multi-dimensional Gaussian processes and give a new
condition on
the covariance, simple and sharp, for the existence of stochastic area
(s).
Gaussian rough paths are constructed with a variety of weak and strong
approximation results. Together with a new RKHS embedding, we obtain
a powerful
- yet conceptually simple - framework in which to analysize differential
equations driven by Gaussian signals in the rough paths sense.
http://arxiv.org/abs/0707.0313
Author(s): Piotr Milos
Abstract: Limit theorems are presented for the rescaled occupation time
fluctuation
process of a critical finite variance branching particle system in
$\mathbb{R}^{d}$ with symmetric $\alpha$-stable motion starting off
from either
a standard Poisson random field or the equilibrium distribution for
critical
$d=2\alpha$ and large $d>2\alpha$ dimensions. The limit processes are
generalised Wiener processes. The obtained convergence is in space-time,
finite-dimensional distributions sense. With the addtional assumption
on the
branching law we obtain functional convergence.
http://arxiv.org/abs/0707.0316
Author(s): Jos\'e Trashorras
Abstract: In this paper we prove a Large Deviation Principle for the
sequence of
symmetrised empirical measures $\frac{1}{n} \sum_{i=1}^{n}
\delta_{(X^n_i,X^n_{\sigma_n(i)})}$ where $\sigma_n$ is a random
permutation
and $((X_i^n)_{1 \leq i \leq n})_{n \geq 1}$ is a triangular array of
random
variables with suitable properties. As an application we show how
this result
allows to improve the Large Deviation Principles for symmetrised
initial-terminal conditions bridge processes recently established by
Adams,
Dorlas and K\"{o}nig.
http://arxiv.org/abs/0707.0344
Author(s): Nizar Demni (PMA)
Abstract: We begin with the study of some properties of the radial Dunkl
process
associated to a reduced root system $R$. It is shown that this
diffusion is the
unique strong solution for all $t \geq 0$ of a SDE with singular
drift. Then,
we study $T_0$, the first hitting time of the positive Weyl chamber :
we prove,
via stochastic calculus, a result already obtained by Chybiryakov on the
finiteness of $T_0$. The second and new part deals with the law of
$T_0$ for
which we compute the tail distribution, as well as some insight via
stochastic
calculus on how root systems are connected with eigenvalues of standard
matrix-valued processes. This gives rise to the so-called $\beta$-
processes.
The ultraspherical $\beta$-Jacobi case still involves a reduced root
system
while the general case is closely connected to a non reduced one.
This process
lives in a convex bounded domain known as principal Weyl alcove and
the strong
uniqueness result remains valid. The last part deals with the first
hitting
time of the alcove's boundary and the semi group density which
enables us to
answer some open questions.
http://arxiv.org/abs/0707.0367
Author(s): Mark Adler and Jonathan Delepine and Pierre van Moerbeke
Abstract: Consider n non-intersecting Brownian particles on the real line
(Dyson
Brownian motions), all starting from the origin at time t=0, and
evolving up to
time t=1. Assume that, among those particles, r are forced to reach a
given
final target a >0 (outliers), while the (n-r) remaining ones return
to the
position x=0. Letting n tend to infinity, view this cloud of
particles from the
edge (i.e., near the largest particle), with the space and time
rescaling given
by the edge statistics of GUE. Also let the target point a go to
infinity with
n at the rate a=rho\sqrt{n/2} for rho between 0 and 1. Then a phase
transition
takes place at rho=1.
Indeed, for rho<1, the limit cloud is described by the Airy
process, which in
effect is rho-independent and also independent of the number r of
outlying
particles; it is as if rho were =0. For rho=1, the process depends on
the
number r of outliers, and leads to a new process: an Airy process with r
outliers (in short: r-Airy process), which is a kind of interpolation
between
the Airy and Pearcey processes. The log of the probability that at
time tau
(the new rescaled time) the cloud does not exceed x is given by the
Fredholm
determinant of a new kernel (extending the Airy kernel) and it
satisfies a
non-linear PDE in x and tau, from which the asymptotic behavior of
the process
can be deduced for tau tending to -infinity (remote past). This
kernel is
closely related to one found by Baik, Ben Arous and Peche in the
context of
multivariate statistics.
http://arxiv.org/abs/0707.0442
Author(s): Laurent Decreusefond and Nicolas Savy
Abstract: By a method inspired of the Stein's method, we derive an upper-
bound of the
Rubinstein distance between two absolutely continuous probability
measures on
configurations space. As an application, we show that the best way to
approximate a Modulated Poisson Process (see below for the
definition) by a
Poisson process is to equate their intensity.
http://arxiv.org/abs/0707.0445
Author(s): Guillaume Aubrun (ICJ) and Ion Nechita (ICJ)
Abstract: We study how iterated convolutions of probability measures compare
under
stochastic domination. We give necessary and sufficient conditions
for the
existence of an integer $n$ such that $\mu^{*n}$ is stochastically
dominated by
$\nu^{*n}$ for two given probability measures $\mu$ and $\nu$. As a
consequence
we obtain a similar theorem on the majorization order for vectors in $
\R^d$. In
particular we prove results about catalysis in quantum information
theory.
http://arxiv.org/abs/0707.0211
Author(s): Peter Elbau
Abstract: We show that in the large matrix limit, the eigenvalues of the
normal matrix
model for matrices with spectrum inside a compact domain with a
special class
of potentials homogeneously fill the interior of a polynomial curve
uniquely
defined by the area of its interior domain and its exterior harmonic
moments
which are all given as parameters of the potential.
Then we consider the orthogonal polynomials corresponding to this
matrix
model and show that, under certain assumptions, the density of the
zeros of the
highest relevant orthogonal polynomial in the large matrix limit is
(up to some
constant factor) given by the discontinuity of the Schwarz function
of this
polynomial curve.
http://arxiv.org/abs/0707.0425
Author(s): Mireille Chaleyat-Maurel (MAP5 and PMA) and Valentine Genon- Catalot (MAP5)
Abstract: We consider a Wright-Fisher diffusion (x(t)) whose current state
cannot be
observed directly. Instead, at times t1 < t2 < . . ., the
observations y(ti)
are such that, given the process (x(t)), the random variables (y(ti))
are
independent and the conditional distribution of y(ti) only depends on
x(ti).
When this conditional distribution has a specific form, we prove that
the model
((x(ti), y(ti)), i 1) is a computable filter in the sense that all
distributions involved in filtering, prediction and smoothing are
exactly
computable. These distributions are expressed as finite mixtures of
parametric
distributions. Thus, the number of statistics to compute at each
iteration is
finite, but this number may vary along iterations.
http://arxiv.org/abs/0707.0537
Author(s): Ken-iti Sato
Abstract: Let $X^{(\mu)}(ds)$ be an $\mathbb{R}^d$-valued homogeneous
independently
scattered random measure over $\mathbb{R}$ having $\mu$ as the
distribution of
$X^{(\mu)}((t,t+1])$. Let $f(s)$ be a nonrandom measurable function
on an open
interval $(a,b)$ where $-\infty\leqslant a
http://arxiv.org/abs/0707.0538
Author(s): Giuseppina Guatteri and Federica Masiero
Abstract: We study quadratic optimal stochastic control problems with
control dependent
noise state equation perturbed by an affine term and with stochastic
coefficients. Both infinite horizon case and ergodic case are
treated. To this
purpose we introduce a Backward Stochastic Riccati Equation and a
dual backward
stochastic equation, both considered in the whole time line. Besides
some
stabilizability conditions we prove existence of a solution for the two
previous equations defined as limit of suitable finite horizon
approximating
problems. This allows to perform the synthesis of the optimal control.
http://arxiv.org/abs/0707.0606
Author(s): Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
Abstract: We prove strong theorems for the local time at infinity of a
nearest neighbor
transient random walk. First, laws of the iterated logarithm are
given for the
large values of the local time. Then we investigate the length of
intervals
over which the walk runs through (always from left to right) without
ever
returning.
http://arxiv.org/abs/0707.0734
Author(s): Amine Asselah
Abstract: We obtain a large deviations principle for the self-intersection
local times
for a simple random walk in dimension d>4. As an application, we obtain
moderate deviations for random walk in random sceneries in some
region of
parameters.
http://arxiv.org/abs/0707.0813
Author(s): Francis Comets and Serguei Popov and Marina Vachkovskaia
Abstract: We study the asymptotic properties of the number of open paths of
length $n$
in an oriented $\rho$-percolation model. We show that this number is
$e^{n\alpha(\rho)(1+o(1))}$ as $n \to \infty$. The exponent $\alpha$ is
deterministic, it can be expressed in terms of the free energy of a
polymer
model, and it can be explicitely computed in some range of the
parameters.
Moreover, in a restricted range of the parameters, we even show that
the number
of such paths is $n^{-1/2} W e^{n\alpha(\rho)}(1+o(1))$ for some
nondegenerate
random variable $W$. We build on connections with the model of directed
polymers in random environment, and we use techniques and results
developed in
this context.
http://arxiv.org/abs/0707.0818
Author(s): Xinjia Chen
Abstract: In this article, we derive a new generalization of Chebyshev
inequality for
random vectors. We demonstrate that the new generalization is much less
conservative than the classical generalization.
http://arxiv.org/abs/0707.0805
Author(s): Xinjia Chen and Kemin Zhou and Jorge L. Aravena
Abstract: In this paper, we derive an explicit formula for constructing the
confidence
interval of binomial parameter with guaranteed coverage probability. The
formula overcomes the limitation of normal approximation which is
asymptotic in
nature and thus inevitably introduce unknown errors in applications.
Moreover,
the formula is very tight in comparison with classic Clopper-
Pearson's approach
from the perspective of interval width. Based on the rigorous
formula, we also
obtain approximate formulas with excellent performance of coverage
probability.
http://arxiv.org/abs/0707.0837
Author(s): Jean-Luc Marichal
Abstract: We give the cumulative distribution functions, the expected
values, and the
moments of weighted lattice polynomials when regarded as real
functions of
independent random variables. Since weighted lattice polynomial
functions
include ordinary lattice polynomial functions and, particularly, order
statistics, our results encompass the corresponding formulas for these
particular functions. We also provide an application to the reliability
analysis of coherent systems.
http://arxiv.org/abs/0707.0953
Author(s): Svante Janson and Niclas Petersson
Abstract: In this paper we study the integral of the supremum process of
standard
Brownian motion. We present an explicit formula for the moments of
the integral
(or area) A(T), covered by the process in the time interval [0,T].
The Laplace
transform of A(T) follows as a consequence. The main proof involves a
double
Laplace transform of A(T) and is based on excursion theory and local
time for
Brownian motion.
http://arxiv.org/abs/0707.0989
Author(s): Svante Janson and Guy Louchard
Abstract: Several Brownian areas are considered in this paper: the Brownian
excursion
area, the Brownian bridge area, the Brownian motion area, the
Brownian meander
area, the Brownian double meander area, the positive part of Brownian
bridge
area, the positive part of Brownian motion area. We are interested in
the
asymptotics of the right tail of their density function. Inverting a
double
Laplace transform, we can derive, in a mechanical way, all terms of an
asymptotic expansion. We illustrate our technique with the
computation of the
first four terms. We also obtain asymptotics for the right tail of the
distribution function and for the moments. Our main tool is the two-
dimensional
saddle point method.
http://arxiv.org/abs/0707.0991
Author(s): Ton\'ci Antunovi\'c and Ivan Veseli\'c
Abstract: We study homogeneous, independent percolation on general quasi-
transitive
graphs. We prove that in the disorder regime where all clusters are
finite
almost surely, in fact the expectation of the cluster size is finite.
This
extends a well-known theorem by Menshikov and Aizenman & Barsky to all
quasi-transitive graphs. Moreover we deduce that in this disorder
regime the
cluster size distribution decays exponentially, extending a result of
Aizenman
& Newman. Our results apply to both edge and site percolation, as
well as long
range (edge) percolation. The proof is based on a modification of the
Aizenman
& Barsky method.
http://arxiv.org/abs/0707.1089
Author(s): Giovanni Peccati (LSTA)
Abstract: Fix an integer k, and let I(l), l=1,2,..., be a sequence of k-
dimensional
vectors of multiple Wiener-It\^o integrals with respect to a general
Gaussian
process. We establish necessary and sufficient conditions to have
that, as l
diverges, the law of I(l) is asymptotically close (for example, in
the sense of
Prokhorov's distance) to the law of a k-dimensional Gaussian vector
having the
same covariance matrix as I(l). The main feature of our results is
that they
require minimal assumptions (basically, boundedness of variances) on the
asymptotic behaviour of the variances and covariances of the elements
of I(l).
In particular, we will not assume that the covariance matrix of I(l) is
convergent. This generalizes the results proved in Nualart and
Peccati (2005),
Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As
shown in
Marinucci and Peccati (2007b), the criteria established in this paper
are
crucial in the study of the high-frequency behaviour of stationary
fields
defined on homogeneous spaces.
http://arxiv.org/abs/0707.1220
Author(s): Julien Guyon (CERMICS)
Abstract: Given a smooth R^d-valued diffusion, we study how fast the Euler
scheme with
time step 1/n converges in law. To be precise, we look for which
class of test
functions f the approximate expectation E[f(X^{n,x}_1)] converges
with speed
1/n to E[f(X^x_1)]. If X is uniformly elliptic, we show that this class
contains all tempered distributions, and all measurable functions with
exponential growth. We give applications to option pricing and
hedging, proving
numerical convergence rates for prices, deltas and gammas.
http://arxiv.org/abs/0707.1243
Author(s): Jean-Baptiste Gouere (MAPMO) and Regine Marchand (IECN)
Abstract: We study a random growth model on $\R^d$ introduced by Deijfen.
This is a
continuous first-passage percolation model. The growth occurs by
means of
spherical outbursts with random radii in the infected region. We aim
at finding
conditions on the distribution of the random radii to determine
whether the
growth of the process is linear or not. To do so, we compare this
model with a
continuous analogue of the greedy lattice paths model and transpose
results in
the lattice setting to the continuous setting.
http://arxiv.org/abs/0707.1395
Author(s): Fabrice Gamboa (IMT) and Thierry Klein (IMT) and Cl\'ementine Prieur (IMT)
Abstract: We prove large and moderate deviation principles for the
distribution of an
empirical mean conditioned by the value of the sum of discrete i.i.d.
random
variables. Some applications for combinatoric problems are discussed.
http://arxiv.org/abs/0707.1461
Author(s): Volker Betz and Olaf Wittich
Abstract: We consider Gibbs measures relative to Brownian motion of Feynman-
Kac type,
with single site potential V. We show that for a large class of V,
including
the Coulomb potential, there exist infinitely many infinite volume Gibbs
measures.
http://arxiv.org/abs/0707.1462
Author(s): O. Khorunzhiy
Abstract: Regarding the adjacency matrices of n-vertex graphs and related graph
Laplacian, we introduce two families of discrete matrix models
constructed both
with the help of the Erdos-Renyi ensemble of random graphs.
Corresponding
matrix sums represent the characteristic functions of the average
number of
walks and closed walks over the random graph. These sums can be
considered as
discrete analogs of the matrix integrals of random matrix theory.
We study the diagram structure of the cumulant expansions of
logarithms of
these matrix sums and analyze the limiting expressions in the cases
of constant
and vanishing edge probabilities as n tends to infinity.
http://arxiv.org/abs/0707.0997
Author(s): Rui Dong
Abstract: Kingman derived the Ewens sampling formula for random partitions
from the
genealogy model defined by a Poisson process of mutations along lines of
descent governed by a simple coalescent process. M\"ohle described the
recursion which determines the generalization of the Ewens sampling
formula
when the lines of descent are governed by a coalescent with multiple
collisions. In a recent work by Dong, Gnedin and Pitman, authors
exploit an
analogy with the theory of regenerative composition and partition
structures,
and provide various characterizations of the associated exchangeable
random
partitions. This paper gives parallel results for the further
generalized model
with lines of descent following a coalescent with simultaneous multiple
collisions.
http://arxiv.org/abs/0707.1606
Author(s): Jean Bertoin (PMA and Dma)
Abstract: In the classical occupancy scheme, one considers a fixed discrete
probability
measure ${\bf p}=(p_i: {i\in{\cal I}})$ and throws balls
independently at
random in boxes labeled by ${\cal I}$, such that p_i is the
probability that a
given ball falls into the box i. In this work, we are interested in
asymptotic
regimes of this scheme in the situation induced by a refining
sequence $({\bf
p}(k) : k\in\N)$ of random probability measures which arise from some
multiplicative cascade. Our motivation comes from the study of the
asymptotic
behavior of certain fragmentation chains
http://arxiv.org/abs/0707.1640
Author(s): Yusuke Ide and Norio Konno and and Naoki Masuda
Abstract: The threshold network model is a type of finite random graphs. In
this paper,
we introduce a generalized threshold network model. A pair of
vertices with
random weights is connected by an edge when real-valued functions of
the pair
of weights belong to given Borel sets. We extend several known limit
theorems
for the number of prescribed subgraphs to show that the strong law of
large
numbers can be uniform convergence. We also prove two limit theorems
for the
local and global clustering coefficients.
http://arxiv.org/abs/0707.1744
Author(s): Mikhail Menshikov and Dimitri Petritis and Stanislav Volkov
Abstract: In this paper we study a regular rooted coloured tree with random
labels
assigned to its edges, where the distribution of the label assigned
to an edge
depends on the colours of its endpoints. We obtain some new results
relevant to
this model and also show how our model generalizes many other
probabilistic
models, including random walk in random environment on trees, recursive
distributional equations, and multi-type branching random walk on $
\mathbb{R}$.
http://arxiv.org/abs/0707.1746
Author(s): Abass Sagna (PMA)
Abstract: Let $ r, s>0 $. For a given probability measure $P$ on $\mathbb{R}
^d$, let
$(\alpha_n)_{n \geq 1}$ be a sequence of (asymptotically) $L^r(P)$-
optimal
quantizers. For all $\mu \in \mathbb{R}^d $ and for every $\theta >0
$, one
defines the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ by : $
\forall n \geq
1, \alpha_n^{\theta, \mu} = \mu + \theta(\alpha_n - \mu) = \{\mu +
\theta(a-
\mu), a \in \alpha_n \} $. In this paper, we are interested in the
asymptotics
of the $L^s$-quantization error induced by the sequence $(\alpha_n^
{\theta,
\mu})_{n \geq 1}$. We show that for a wide family of distributions, the
sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ is $L^s$-rate-optimal.
For the
Gaussian and the exponential distributions, one shows how to choose the
parameter $\theta$ such that $(\alpha_n^{\theta, \mu})_{n \geq 1}$
satisfies
the empirical measure theorem.
http://arxiv.org/abs/0707.1808
Author(s): Alex Gamburd and Shlomo Hoory and Mehrdad Shahshahani and Aner Shalev, Balint
Virag
Abstract: We prove that random d-regular Cayley graphs of the symmetric group
asymptotically almost surely have girth at least (log_{d-1}|G|)^{1/2}/
2 and
that random d-regular Cayley graphs of simple algebraic groups over F_q
asymptotically almost surely have girth at least log_{d-1}|G|/dim(G).
For the
symmetric p-groups the girth is between log log |G| and (log|G|)
^alpha with
alpha<1. Several conjectures and open questions are presented.
http://arxiv.org/abs/0707.1833
Author(s): A. B. Dieker and J. Warren
Abstract: We find the transition kernels for four Markovian interacting
particle
systems on the line, by proving that each of these kernels is
intertwined with
a Karlin-McGregor type kernel. The resulting kernels all inherit the
determinantal structure from the Karlin-McGregor formula, and have a
similar
form to Schutz's kernel for the totally asymmetric simple exclusion
process.
http://arxiv.org/abs/0707.1843
Author(s): Ming Yang
Abstract: We prove a theorem on additive Levy processes and give applications
http://arxiv.org/abs/0707.1845
Author(s): Ming Yang
Abstract: We prove a new theorem on additive Levy processes and show that
this theorem
implies several proved theorems and a hard conjectured theorem.
http://arxiv.org/abs/0707.1847
Author(s): Ming Yang
Abstract: We compute the Hausdorff dimension of the zero set of an additive
Levy
process.
http://arxiv.org/abs/0707.1849
Author(s): Svante Janson and Malwina Luczak
Abstract: We study the largest component of a random (multi)graph on n
vertices with a
given degree sequence. We let n tend to infinity. Then, under some
regularity
conditions on the degree sequences, we give conditions on the
asymptotic shape
of the degree sequence that imply that with high probability all the
components
are small, and other conditions that imply that with high probability
there is
a giant component and the sizes of its vertex and edge sets satisfy a
law of
large numbers; under suitable assumptions these are the only two
possibilities.
In particular, we recover the results by Molloy and Reed on the size
of the
largest component in a random graph with a given degree sequence.
We further obtain a new sharp result for the giant component just
above the
threshold, generalizing the case of G(n,p) with np=1+omega(n)n^
{-1/3}, where
omega(n) tends to infinity arbitrarily slowly.
Our method is based on the properties of empirical distributions of
independent random variables, and leads to simple proofs.
http://arxiv.org/abs/0707.1786
Author(s): Guang-hui Cai and Hang Wu
Abstract: Based on a law of the iterated logarithm for independent random
variables
sequences, an iterated logarithm theorem for NA sequences with non-
identical
distributions is obtained. The proof is based on a Kolmogrov-type
exponential
inequality.
http://arxiv.org/abs/0707.1968
Author(s): Yan Doumerc and John Moriarty
Abstract: We give the distribution of the first exit time of Brownian motion
from the
alcove of an affine Weyl group, in terms of the distributions of
first exit
times from simpler domains such as orthants. Applications are
explicitly given
in the different type cases. The results extend to any process for
which the
reflection arguments are valid. We also give the real eigenfunctions
of the
Laplacian for alcoves with Dirichlet and Neumann boundary conditions.
http://arxiv.org/abs/0707.2009
Author(s): Rick Durrett and Deena Schmidt and and Jason Schweinsberg
Abstract: We consider the population genetics problem: How long does it take
before
some member of the population has m specified mutations? The case m=2 is
relevant to onset of cancer due to the inactivation of both copies of
a tumor
suppressor gene. Models for larger m are needed for colon cancer and
other
diseases where a sequence of mutations leads to cells with
uncontrolled growth.
http://arxiv.org/abs/0707.2057
Author(s): Jacob van den Berg (CWI and VUA)
Abstract: One of the most well-known classical results for site percolation
on the
square lattice is the equation p_c + p_c^* = 1. In words, this
equation means
that for all values different from p_c of the parameter p the
following holds:
Either a.s. there is an infinite open cluster or a.s. there is an
infinite
closed `star' cluster. This result is closely related to the percolation
transition being sharp: Below p_c the size of the open cluster of a
given
vertex is not only (a.s.) finite, but has a distrubtion with an
exponential
tail. The analog of this result has been proved by Higuchi in 1993 for
two-dimensional Ising percolation, with fixed inverse temparature
beta
http://arxiv.org/abs/0707.2077
Author(s): V P Belavkin and L Gregory
Abstract: We study homogeneous quantum L\'{e}vy processes and fields with
independent
additive increments over a noncommutative *-monoid. These are
described by
infinitely divisible generating state functionals, invariant with
respect to an
endomorphic injective action of a symmetry semigroup. A strongly
covariant GNS
representation for the conditionally positive logarithmic functionals
of these
states is constructed in the complex Minkowski space in terms of
canonical
quadruples and isometric representations on the underlying pre-
Hilbert field
space. This is of much use in constructing quantum stochastic
representations
of homogeneous quantum L\'{e}vy fields on It\^{o} monoids, which is a
natural
algebraic way of defining dimension free, covariant quantum stochastic
integration over a space-time indexing set.
http://arxiv.org/abs/0707.2142
Author(s): Remi Leandre
Abstract: We translate in semigroup theory Bismut's way of the Malliavin
calculus.
http://arxiv.org/abs/0707.2143
Author(s): Un Cig Ji
Abstract: In this paper a quantum stochastic integral representation theorem is
obtained for unbounded regular martingales with respect to
multidimensional
quantum noise. This simultaneously extends results of Parthasarathy
and Sinha
to unbounded martingales and those of the author to multidimensions.
http://arxiv.org/abs/0707.2144
Author(s): Gerard Ben Arous and Alice Guionnet
Abstract: Let $X_N$ be an $N\ts N$ random symmetric matrix with independent
equidistributed entries. If the law $P$ of the entries has a finite
second
moment, it was shown by Wigner \cite{wigner} that the empirical
distribution of
the eigenvalues of $X_N$, once renormalized by $\sqrt{N}$, converges
almost
surely and in expectation to the so-called semicircular distribution
as $N$
goes to infinity. In this paper we study the same question when $P$
is in the
domain of attraction of an $\alpha$-stable law. We prove that if we
renormalize
the eigenvalues by a constant $a_N$ of order $N^{\frac{1}{\alpha}}$, the
corresponding spectral distribution converges in expectation towards
a law
$\mu_\alpha$ which only depends on $\alpha$. We characterize $\mu_
\alpha$ and
study some of its properties; it is a heavy-tailed probability
measure which is
absolutely continuous with respect to Lebesgue measure except
possibly on a
compact set of capacity zero.
http://arxiv.org/abs/0707.2159
Author(s): Matyas Barczy and Gyula Pap
Abstract: On the torus group, on the group of p-adic integers and on the p-adic
solenoid we give a construction of an arbitrary weakly infinitely
divisible
probability measure using real random variables. As a special case of
our
results, we have a new construction of the Haar measure on the p-adic
solenoid.
http://arxiv.org/abs/0707.2186
Author(s): Xing M. Wang
Abstract: In this paper, we continue to explore the consistence and
usability of
Probability Bracket Notation (PBN) proposed in our previous articles.
After a
brief review of PBN with dimensional analysis, we investigate
probability
spaces in terms of PBN by introducing probability spaces associated
with random
variables (R.V) or associated with stochastic processes (S.P). Next,
we express
several important properties of conditional expectation (CE) and some
their
proofs in PBN. Then, we introduce martingales based on sequence of
R.V or based
on filtration in PBN. In the process, we see PBN can be used to
investigate
some probability problems, which otherwise might need explicit usage
of Measure
theory. Whenever applicable, we use dimensional analysis to validate our
formulas and use graphs for visualization of concepts in PBN. We hope
this
study shows that PBN, stimulated by and adapted from Dirac notation
in Quantum
Mechanics (QM), may have the potential to be a useful tool in
probability
modeling, at least for those who are already familiar with Dirac
notation in
QM.
http://arxiv.org/abs/0707.2236
Author(s): Katrin Hofmann-Credner and Michael Stolz
Abstract: It is a classical result of Wigner that for an hermitian matrix with
independent entries on and above the diagonal, the mean empirical
eigenvalue
distribution converges weakly to the semicircle law as matrix size
tends to
infinity. In this paper, we prove analogs of Wigner's theorem for random
matrices taken from all infinitesimal versions of classical symmetric
spaces.
This is a class of models which contains those studied by Wigner and
Dyson,
along with seven others arising in condensed matter physics. Like
Wigner's, our
results are universal in that they only depend on certain assumptions
about the
moments of the matrix entries, but not on the specifics of their
distributions.
What is more, we allow for a certain amount of dependence among the
matrix
entries, in the spirit of a recent generalization of Wigner's
theorem, due to
Schenker and Schulz-Baldes. As a byproduct, we obtain a universality
result for
sample covariance matrices with dependent entries.
http://arxiv.org/abs/0707.2333
Author(s): Julius Borcea and Petter Br\"and\'en and Thomas M. Liggett
Abstract: We introduce the class of {\em strongly Rayleigh} probability
measures by
means of geometric properties of their generating polynomials that
amount to
the stability of the latter. This class contains e.g. product
measures, uniform
random spanning tree measures, and large classes of determinantal
probability
measures and distributions for symmetric exclusion processes. We show
that
strongly Rayleigh measures enjoy all virtues of negative dependence
and we also
prove a series of conjectures due to Liggett, Pemantle, and Wagner,
respectively. Moreover, we extend Lyons' recent results on determinantal
probability measures and we construct counterexamples to several
conjectures of
Pemantle and Wagner on negative dependence and ultra log-concave rank
sequences.
http://arxiv.org/abs/0707.2340
Author(s): Martin Hairer and Grigorios Pavliotis
Abstract: The long-time/large-scale, small-friction asymptotic for the one
dimensional
Langevin equation with a periodic potential is studied in this paper.
It is
shown that the Freidlin-Wentzell and central limit theorem
(homogenization)
limits commute. We prove that, in the combined small friction,
long-time/large-scale limit the particle position converges weakly to a
Brownian motion with a singular diffusion coefficient which we compute
explicitly. We show that the same result is valid for a whole one
parameter
family of space/time rescalings. The proofs of our main results are
based on
some novel estimates on the resolvent of a hypoelliptic operator.
http://arxiv.org/abs/0707.2352
Author(s): Xinjia Chen
Abstract: It is a common contention that it is an ``impossible mission'' to
exactly
determine the minimum sample size for the estimation of a binomial
parameter
with prescribed margin of error and confidence level. In this paper, we
investigate such a very old but also extremely important problem and
demonstrate that the difficulty for obtaining the exact solution is not
insurmountable. Unlike the classical approximate sample size method
based on
the central limit theorem, we develop a new approach for computing
the minimum
sample size that does not require any approximation. Moreover, our
approach
overcomes the conservatism of existing rigorous sample size methods
derived
from Bernoulli's theorem or Chernoff bounds.
Our computational machinery consists of two essential ingredients.
First, we
prove that the minimum of coverage probability with respect to a
binomial
parameter bounded in an interval is attained at a discrete set of
finite many
values of the binomial parameter. This allows for reducing infinite many
evaluations of coverage probability to finite many evaluations.
Second, a
recursive bounding technique is developed to further improve the
efficiency of
computation.
http://arxiv.org/abs/0707.2113
Author(s): Xinjia Chen
Abstract: In this paper, we develop an exact method for the determination of
the
minimum sample size for the estimation of the proportion of a finite
population
with prescribed margin of error and confidence level. By
characterizing the
behavior of the coverage probability with respect to the proportion,
we show
that the computational complexity can be significantly reduced and
bounded
regardless population size.
http://arxiv.org/abs/0707.2115
Author(s): Xinjia Chen
Abstract: In this paper, we develop an approach for the exact determination
of the
minimum sample size for the estimation of a Poisson parameter with
prescribed
margin of error and confidence level. The exact computation is made
possible by
reducing infinite many evaluations of coverage probability to finite
many
evaluations. Such reduction is based on our discovery that the
minimum of
coverage probability with respect to a Poisson parameter bounded in
an interval
is attained at a discrete set of finite many values.
http://arxiv.org/abs/0707.2116
Author(s): Benny Sudakov and Jan Vondrak
Abstract: In this paper we show how to find nearly optimal embeddings of
large trees in
several natural classes of graphs. The size of the tree T can be as
large as a
constant fraction of the size of the graph G, and the maximum degree
of T can
be close to the minimum degree of G. For example, we prove that any
graph of
minimum degree d without 4-cycles contains every tree of size
\epsilon d^2 and
maximum degree at most (1-2\epsilon)d - 2. As there exist d-regular
graphs
without 4-cycles of size O(d^2), this result is optimal up to
constant factors.
We prove similar nearly tight results for graphs of given girth,
graphs with no
complete bipartite subgraph K_{s,t}, random and certain pseudorandom
graphs.
These results are obtained using a simple and very natural randomized
embedding
algorithm, which can be viewed as a "self-avoiding tree-indexed
random walk".
http://arxiv.org/abs/0707.2079
Author(s): Rainer Buckdahn and Marc Quincampoix and Catherine Rainer and Josef Teichmann
Abstract: We provide a short and elementary proof for the recently proved
result by G.
da Prato and H. Frankowska that a closed set is stochastically
invariant if and
only if it is deterministically invariant.
http://arxiv.org/abs/0707.2353
Author(s): Litan Yan and Yu Sun and Yunsheng Lu
Abstract: In this paper, we introduce the linear fractional self-attracting
diffusion
driven by a fractional Brownian motion with Hurst index 1/2
http://arxiv.org/abs/0707.2627
Author(s): D. Marquez-Carreras and C. Rovira
Abstract: We prove a functional law of iterated logarithm for the following
kind of
anticipating stochastic differential equations
$$\xi^u_t=X_0^u+\frac{1}{\sqrt{\log\log u}}\sum_{j=1}^k \int_0^{t}
A_j^u(\xi^u_s)\circ dW_{s}^j+ \int_0^{t} A_0^u(\xi^u_s)ds,$$ where
$u>e$,
$W=\{(W_t^1,...,W_t^k), 0\le t\le 1\}$ is a standard $k$-dimensional
Wiener
process, $A_0^u,A_1^u,..., A_k^u:\mathbb{R}^d\longrightarrow \mathbb
{R}^d$ are
functions of class $\mathcal{C}^2$ with bounded partial derivatives
up to order
2, $X_0^u$ is a random vector not necessarily adapted and the first
integral is
a generalized Stratonovich integral .
http://arxiv.org/abs/0707.2650
Author(s): Boualem Djehiche and Said Hamadene and Alexandre Popier
Abstract: We consider the problem of optimal multiple switching in finite
horizon, when
the state of the system, including the switching costs, is a general
adapted
stochastic process. The problem is formulated as an extended impulse
control
problem and completely solved using probabilistic tools such as the
Snell
envelop of processes and reflected backward stochastic differential
equations.
Finally, when the state of the system is a Markov diffusion process,
we show
that the vector of value functions of the optimal problem is a viscosity
solution to a system of variational inequalities with inter-connected
obstacles.
http://arxiv.org/abs/0707.2663
Author(s): Benjamin Jourdain (CERMICS) and Sylvie M\'el\'eard (CMAP) and Wojbor Woyczynski
Abstract: In this paper we study general nonlinear stochastic differential
equations,
where the usual Brownian motion is replaced by a L\'evy process. We also
suppose that the coefficient multiplying the increments of this
process is
merely Lipschitz continuous and not necessarily linear in the time-
marginals of
the solution as is the case in the classical McKean-Vlasov model. We
first
study existence, uniqueness and particle approximations for these
stochastic
differential equations. When the driving process is a pure jump L
\'evy process
with a smooth but unbounded L\'evy measure, we develop a stochastic
calculus of
variations to prove that the time-marginals of the solutions are
absolutely
continuous with respect to the Lebesgue measure. In the case of a
symmetric
stable driving process, we deduce the existence of a function
solution to a
nonlinear integro-differential equation involving the fractional
Laplacian.
http://arxiv.org/abs/0707.2723
Author(s): Hiroki Sumi and Mariusz Urbanski
Abstract: We consider the dynamics of skew product maps associated with
finitely
generated semigroups of rational maps on the Riemann sphere. We show
that under
some conditions on the dynamics and the potential function \psi,
there exists a
unique equilibrium state for \psi and a unique $\exp(\P(\psi)-\psi)$-
conformal
measure, where P(\psi) denotes the topological pressure of \psi.
http://arxiv.org/abs/0707.2444
Author(s): Hiroki Sumi and Mariusz Urbanski
Abstract: We consider the dynamics of expanding semigroups generated by
finitely many
rational maps on the Riemann sphere. We show that for an analytic
family of
such semigroups, the Bowen parameter function is real-analytic and
plurisubharmonic. Combining this with a result obtained by the first
author, we
show that if for each semigroup of such an analytic family of expanding
semigroups satisfies the open set condition, then the function of the
Hausdorff
dimension of the Julia set is real-analytic and plurisubharmonic.
Moreover, we
provide an extensive collection of classes of examples of analytic
families of
semigroups satisfying all the above conditions and we analyze in
detail the
corresponding Bowen's parameters and Hausdorff dimension function.
http://arxiv.org/abs/0707.2447
Author(s): Pierre Collet and Antonio Galves and Florencia G. Leonardi
Abstract: We consider binary infinite order stochastic chains perturbed by a
random
noise. This means that at each time step, the value assumed by the
chain can be
randomly and independently flipped with a small fixed probability. We
show that
the transition probabilities of the perturbed chain are uniformly
close to the
corresponding transition probabilities of the original chain. As a
consequence,
in the case of stochastic chains with unbounded but otherwise finite
variable
length memory, we show that it is possible to recover the context
tree of the
original chain, using a suitable version of the algorithm Context,
provided
that the noise is small enough.
http://arxiv.org/abs/0707.2796
Author(s): Nathael Gozlan (LAMA)
Abstract: In this paper, we consider Poincar\'e inequalities for non
euclidean metrics
on $\mathbb{R}^d$. These inequalities enable us to derive precise
dimension
free concentration inequalities for product measures. This technique is
appropriate for a large scope of concentration rate: between
exponential and
gaussian and beyond. We give different equivalent functional forms of
these
Poincar\'e type inequalities in terms of transportation-cost
inequalities and
infimum convolution inequalities. Workable sufficient conditions are
given and
a comparison is made with generalized Beckner-Latala-Oleszkiewicz
inequalities.
http://arxiv.org/abs/0707.2834
Author(s): Asaf Nachmias and Yuval Peres
Abstract: We describe the component sizes in critical independent p-bond
percolation on
a random d-regular graph on n vertices, where d is fixed and n grows.
We prove
mean-field behavior around the critical probability p_c=1/(d-1).
In particular, we show that there is a scaling window of width n^
{-1/3}
around p_c in which the sizes of the largest components are roughly n^
{2/3} and
we describe their limiting joint distribution. We also show that for the
subcritical regime, i.e. p = (1-eps(n))p_c where eps(n)=o(1) but \eps
(n)n^{1/3}
tends to infinity, the sizes of the largest components are
concentrated around
an explicit function of n and eps(n) which is of order o(n^{2/3}). In
the
supercritical regime, i.e. p = (1+\eps(n))p_c where eps(n)=o(1) but
eps(n)n^{1/3} tends to infinity, the size of the largest component is
concentrated around the value (2d/(d-2))\eps(n)n and a duality
principle holds:
other component sizes are distributed as in the subcritical regime.
http://arxiv.org/abs/0707.2839
Author(s): David Coupier
Abstract: A $d$-dimensional ferromagnetic Ising model on a lattice torus is
considered.
As the size $n$ of the lattice tends to infinity, the magnetic field
$a=a(n)$
and the pair potential $b=b(n)$ depend on $n$. Precise bounds for the
probability for local configurations to occur in a large ball are
given. Under
some conditions bearing on potentials $a(n)$ and $b(n)$, the distance
between
copies of different local configurations is estimated according to their
weights. Finally, a sufficient condition ensuring that a given local
configuration occurs everywhere in the lattice is suggested.
http://arxiv.org/abs/0707.2889
Author(s): Sebastien Chambeu and Aline Kurtzmann
Abstract: This paper is concerned with some self-interacting diffusions $
(X_t,t\geq 0)$
living on $\mathbb{R}^d$. These diffusions are solutions to stochastic
differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t - g(t)\nabla V
(X_t -
\bar{\mu}_t) \mathrm{d}t,$$ where $\bar{\mu}_t$ is the mean of the
empirical
measure of the process $X$, $V$ is an asymptotically strictly convex
potential
and $g$ is a given function. We study the ergodic behavior of $X$ and
prove
that it is strongly related to $g$. Actually, we will show that $X$ and
$\bar{\mu}_t$ have the same asymptotic behavior and we will give
necessary and
sufficient conditions (on $g$ and $V$) for the almost sure
convergence of $X$.
http://arxiv.org/abs/0707.2908
Author(s): Sebastien Chambeu and Aline Kurtzmann
Abstract: The present paper is concerned with some self-interacting diffusions
$(X_t,t\geq 0)$ living on $\mathbb{R}^d$. These diffusions are
solutions to
stochastic differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t - g
(t)\nabla
V(X_t - \bar{\mu}_t) \mathrm{d}t$$ where $\bar{\mu}_t$ is the
empirical mean of
the process $X$, $V$ is an asymptotically strictly convex potential
and $g$ is
a given function. The authors have still studied the ergodic behavior
of $X$
and proved that it is strongly related to $g$. We go further and give
necessary
and sufficient conditions (for small $g$'s) in order that $X$
converges in
probability to $X_\infty$ (which is related to the global minima of $V
$).
http://arxiv.org/abs/0707.2910
Author(s): Xinjia Chen
Abstract: By our recently developed techniques, we have shown that the
minimum coverage
probability of an open binomial confidence interval with respect to the
corresponding binomial parameter is achieved at a discrete set of
finite many
values. Moreover, we have obtained similar results for the case of
Poisson
confidence interval and the case of confidence interval for the
proportion of
finite population.
http://arxiv.org/abs/0707.2814
Author(s): Xing M. Wang
Abstract: Dirac notation has been widely used for vectors in Hilbert spaces
of Quantum
Theories. It now has also been introduced to Information Retrieval.
In this
paper, we propose a new set of symbols, the Probability Bracket
Notation (PBN),
for representation of probability theories. The new are defined
similarly (but
not identically) as their counterparts in Dirac notation, which we
refer as
Vector Bracket Notation (VBN). By using PBN to represent fundamental
definitions and theorems for discrete and continuous random
variables, we show
that PBN could play a similar role in probability sample space as Dirac
notation in Hilbert space. We also find that there is a close
relation between
our probability state kets and probability vectors in Markov chains.
In the
end, we apply PBN to some important stochastic processes, and present
the time
evolution differential equations (TEDE) of time-continuous Markov
chains in
both Heisenberg and Schrodinger pictures. We summarize the
similarities and
differences between PBN and VBN in the two tables of Appendix A.
http://arxiv.org/abs/cs/0702021
Author(s): Xing M. Wang
Abstract: In this article, we continue to explore Probability Bracket
Notation (PBN),
proposed in our previous article. Using both Dirac vector bracket
notation
(VBN) and PBN, we define induced Hilbert space and induced sample
space, and
propose that there exists an equivalence relation between a Hilbert
space and a
probability sample space constructed from the same base observable
(s). Then we
investigate Markov transition matrices and their eigenvectors to make
diffusion
maps with two examples: a simple graph theory example, to serve as a
prototype
of bidirectional transition operator; a famous text document example
in IR
literature, to serve as a tutorial of diffusion map in text document
space. We
notice that, in both examples, the sample space of the Markov chain
and the
Hilbert space spanned by the eigenvectors of the transition matrix
are not
equivalent. At the end, we apply our PBN and equivalence proposal to
Thermophysics by associating phase space with Hilbert space or Fock
space of
many-particle systems.
http://arxiv.org/abs/cs/0702121
Author(s): Alexei Borodin and Eugene Kanzieper
Abstract: Two alternative, fairly compact proofs are presented of the Pfaffian
integration theorem that surfaced in the recent studies of spectral
properties
of Ginibre's Orthogonal Ensemble. The first proof is based on a
concept of the
Fredholm Pfaffian; the second proof is purely linear-algebraic.
http://arxiv.org/abs/0707.2784
Author(s): Omer Angel and Yuval Peres and David B. Wilson
Abstract: The ``overlapping-cycles shuffle'' mixes a deck of n cards by
moving either
the nth card or the (n-k)th card to the top of the deck, with
probability half
each. We determine the spectral gap for the location of a single
card, which,
as a function of k and n, has surprising behavior. For example,
suppose k is
the closest integer to alpha n for a fixed real alpha in (0,1). Then for
rational alpha the spectral gap is Theta(n^{-2}), while for poorly
approximable
irrational numbers alpha, such as the reciprocal of the golden ratio,
the
spectral gap is Theta(n^{-3/2}).
http://arxiv.org/abs/0707.2994
Author(s): S. Gerhold and R. Warnung
Abstract: We derive recursions for the probability distribution of random
sums by
computer algebra. Unlike the well-known Panjer-type recursions, they
are of
finite order and thus allow for computation in linear time. This
efficiency is
bought by the assumption that the probability generating function of
the claim
size be algebraic. The probability generating function of the claim
number is
supposed to be from the rather general class of D-finite functions.
http://arxiv.org/abs/0707.3028
Author(s): Frank Aurzada and Steffen Dereich
Abstract: We investigate the high resolution coding problem for general real-
valued
L\'evy processes under L^p[0,1]-norm distortion. Tight asymptotic
formulas are
found under mild regularity assumptions.
http://arxiv.org/abs/0707.3040
Author(s): S. V. Lototsky
Abstract: A change of variables is introduced to reduce certain nonlinear
stochastic
evolution equations with multiplicative noise to the corresponding
deterministic equation. The result is then used to investigate a
stochastic
porous medium equation.
http://arxiv.org/abs/0707.3155
Author(s): L. V. Bogachev
Abstract: Random walks provide a simple conventional model to describe various
transport processes, for example propagation of heat or diffusion of
matter
through a medium. However, in many practical cases the medium is highly
irregular due to defects, impurities, fluctuations etc., and it is
natural to
model this as random environment. In the random walks context, such
models are
referred to as Random Walks in Random Environments (RWRE). This is a
relatively
new chapter in applied probability and physics of disordered systems,
initiated
in the 1970s. Early interest was motivated by some problems in biology,
crystallography and metal physics, but later applications have spread
through
numerous areas. After 30 years of extensive work, RWRE remain a very
active
area of research, which has already led to many surprising
discoveries. The
goal of this article is to give a brief introduction to the beautiful
area of
RWRE. The principal model to be discussed is a random walk with
nearest-neighbor jumps in independent identically distributed
(i.i.d.) random
environment in one dimension, although we shall also comment on some
extensions
and generalizations. The focus is on rigorous results; however,
heuristics is
used freely to motivate the ideas and explain the approaches and
proofs. In a
few cases, sketches of the proofs have been included, which should
help the
reader to appreciate the flavor of results and methods.
http://arxiv.org/abs/0707.3160
Author(s): Tom Alberts (New York University) and Michael J. Kozdron (University of
Regina)
Abstract: We derive an estimate for the diameter of a chordal SLE path in
the upper
half plane H between two real boundary points 0 and x>0. In
particular, we
prove that if 0 < kappa < 8 and gamma:[0,1] to closure(H) is a
chordal SLE in H
from 0 to x, then P(gamma[0,1] cap C_R neq emptyset) asymp R^(1-4a)
where
a=2/kappa and C_R denotes the circle of radius Rx centred at 0 in the
upper
half plane. As an application of our result, we derive an estimate
that two
nearby points, one on the boundary and one in the interior, are
swallowed
together by a chordal SLE path, 4 < kappa <8.
http://arxiv.org/abs/0707.3163
Author(s): Ashkan Nikeghbali and Marc Yor
Abstract: We give a probabilistic interpretation for the Barnes G-function
which
appears in random matrix theory and in analytic number theory in the
important
moments conjecture due to Keating-Snaith for the Riemann zeta
function, via the
analogy with the characteristic polynomial of random unitary
matrices. We show
that the Mellin transform of the characteristic polynomial of random
unitary
matrices and the Barnes G-function are intimately related with
products and
sums of gamma, beta and log-gamma variables. In particular, we show
that the
law of the modulus of the characteristic polynomial of random unitary
matrices
can be expressed with the help of products of gamma or beta
variables, and that
the reciprocal of the Barnes G-function has a L\'{e}vy-Khintchin type
representation. These results lead us to introduce the so called
generalized
gamma convolution variables.
http://arxiv.org/abs/0707.3187
Author(s): Jan Palczewski and Lukasz Stettner
Abstract: This paper studies a portfolio optimization problem in a discrete-
time
Markovian model of a financial market, in which asset price dynamics
depend on
an external process of economic factors. There are transaction costs
with a
structure that covers, in particular, the case of fixed plus
proportional
costs. We prove that there exists a self-financing trading strategy
maximizing
the average growth rate of the portfolio wealth. We show that this
strategy has
a Markovian form. Our result is obtained by large deviations
estimates on
empirical measures of the price process and by a generalization of the
vanishing discount method to discontinuous transition operators.
http://arxiv.org/abs/0707.3198
Author(s): Elchanan Mossel and Allan Sly
Abstract: Gibbs sampling also known as Glauber dynamics is a popular
technique for
sampling high dimensional distributions defined on graphs. Of special
interest
is the behavior of Gibbs sampling on the Erd\H{o}s-R\'enyi random graph
G(n,d/n). While the average degree in G(n,d/n) is d(1-o(1)), it
contains many
nodes of degree of order $\log n / \log \log n$.
The existence of nodes of almost logarithmic degrees implies that
for many
natural distributions defined on G(n,p) such as uniform coloring or
the Ising
model, the mixing time of Gibbs sampling is at least $n^{1 + \Omega
(1 / \log
\log n)}$. High degree nodes pose a technical challenge in proving
polynomial
time mixing of the dynamics for many models including coloring.
In this work consider sampling q-colorings and show that for every
$d <
\infty$ there exists $q(d) < \infty$ such that for all $q \geq q(d)$
the mixing
time of Gibbs sampling on G(n,d/n) is polynomial in $n$ with high
probability.
Our results are the first polynomial time mixing results proven for the
coloring model on G(n,d/n) for d > 1 where the number of colors does
not depend
on n. They extend to much more general families of graphs which are
sparse in
some average sense and to much more general interactions. The results
also
generalize to the hard-core model at low fugacity and to general
models of soft
constraints at high temperatures.
http://arxiv.org/abs/0707.3241
Author(s): G. Giacomin (1) and F. L. Toninelli (2) ((1) Universite' de Paris 7, (2)
Laboratoire de Physique, ENS Lyon and CNRS)
Abstract: Recent results have lead to substantial progress in understanding
the role of
disorder in the (de)localization transition of polymer pinning
models. Notably,
there is an understanding of the crucial issue of disorder relevance and
irrelevance that, albeit still partial, is now rigorous. In this work we
exploit interpolation and replica coupling methods to get sharper
results on
the irrelevant disorder regime of pinning models. In particular, we
compute in
this regime the first order term in the expansion of the free energy
close to
criticality, which coincides with the first order of the formal
expansion
obtained by field theory methods. We also show that the quenched and the
quenched averaged correlation length exponents coincide, while in
general they
are expected to be different. Interpolation and replica coupling
methods in
this class of models naturally lead to studying the behavior of the
intersection of certain renewal sequences and one of the main tools
in this
work is precisely renewal theory and the study of these intersection
renewals.
http://arxiv.org/abs/0707.3340
Author(s): Annalisa Cerquetti
Abstract: We show that a Gibbs characterization of normalized generalized Gamma
processes, recently obtained in Lijoi, Pr\"unster and Walker (2007), can
alternatively be derived by exploiting a characterization of
exponentially
tilted Poisson-Kingman models stated in Pitman (2003). We also provide a
completion of this result investigating the existence of normalized
random
measures inducing exchangeable Gibbs partitions of type $\alpha \in (-
\infty,
0]$.
http://arxiv.org/abs/0707.3408
Author(s): Eben Kenah and Marc Lipsitch and James M. Robins
Abstract: The serial interval may be defined as the time between the onset
of symptoms
in an infectious person and the onset of symptoms in a person he or she
infects. Several methods of analyzing epidemic data, such as
estimates of
reproductive numbers, are based on a probability distribution for the
serial
interval. In this paper, we specify a general SIR epidemic model and
prove that
the mean serial interval must contract when susceptible persons are
at risk of
multiple infectious contacts. In an epidemic, the mean serial interval
contracts as the prevalence of infection increases. We illustrate two
mechanisms through which serial interval contraction can occur: In
global
competition among infectious contacts, risk of multiple infectious
contacts
results from a high global prevalence of infection. In local
competition among
infectious contacts, clustering of contacts places susceptible
persons at risk
of multiple infectious contacts even when the global prevalence of
infection is
low. We illustrate these patterns with simulations. We also find that
the
minimum mean serial interval in a compartmental SIR model becomes
arbitrarily
small with sufficiently high R_{0}. We conclude that the serial interval
distribution is not a stable characteristic of an infectious disease.
http://arxiv.org/abs/0706.2024
Author(s): Luigi Manca
Abstract: Given a real and separable Hilbert space H we consider the measure-
valued
equation \begin{equation*} \int_H\phi(x)\mu_t(dx)- \int_H\phi(x)\mu(dx)=
\int_0^t(\int_HK_0\phi(x)\mu_s(dx))ds, \end{equation*} where K_0
is the
Kolmogorov differential operator \[
K_0\phi(x)=\frac12\textrm{Trace}\big[BB^*D^2\phi(x)\big]+< x,A^*D
\phi(x)>+<
D\phi(x),F(x)>, \] $x\in H$, $\phi:H\to \Rset$ is a suitable smooth
function,
$A:D(A)\subset H\to H $ is linear, $F:H\to H$ is a globally Lipschitz
function
and $B:H\to H$ is linear and continuous. In order prove existence and
uniqueness of a solution for the above equation, we show that $K_0$
is a core,
in a suitable way, of the infinitesimal generator associated to the
solution of
a certain stochastic differential equation in H.
We also extend the above results to a reaction-diffusion operator
with
polinomial nonlinearities.
http://arxiv.org/abs/0707.3233
Author(s): Jiun-Chau Wang
Abstract: We study the distributional behavior for products, and for sums of
boolean
independent random variables in an infinitesimal triangular array. We
show that
the limit laws of boolean convolutions are determined by the limit
laws of free
convolutions, and vice versa. We further use these results to show
several
connections between the limiting distributional behavior of classical
convolutions and that of boolean convolutions. The proof of our
results is
based on the analytical apparatus developed for free convolutions.
http://arxiv.org/abs/0707.3401
Author(s): Peter Hegarty and Steven J. Miller
Abstract: We investigate the relationship between the sizes of the sum and
difference
sets attached to a subset of {0,1,...,N}, chosen randomly according to a
binomial model with parameter p(N), with N^{-1} = o(p(N)). We show
that the
random subset is almost surely difference dominated, as $N \to \infty
$, for any
choice of p(N) tending to zero, confirming a conjecture of Martin and
O'Bryant.
We exhibit a threshold phenomenon regarding the ratio of the size
of the
difference- to the sumset. If p(N) = o(N^{-1/2}) then almost all sums
and
differences in the random subset are almost surely distinct, and the
difference
set is almost surely about twice as large as the sumset. If N^{-1/2}
= o(p(N))
then both the sum and difference sets almost surely have size $(2N+1) -
(p(N)^{-2})$, and so the ratio in question is almost surely very
close to one.
If $p(N) = c \cdot N^{-1/2}$ then as c increases from zero to
infinity (i.e.:
as the threshold is crossed), the same ratio almost surely decreases
continuously from two to one according to an explicitly given
function of c.
We extend our results to the comparison of the generalized
difference sets
attached to an arbitrary pair of binary linear forms. For certain
pairs of
forms we show that there is a sharp threshold such that one form
almost surely
dominates below the threshold, and the other almost surely above it.
The heart of our approach involves proving strong concentration of
the sizes
of the sum and difference sets about their mean values.
http://arxiv.org/abs/0707.3417
Author(s): Ivan Nourdin (PMA) and David Nualart
Abstract: By means of Malliavin calculus, we prove the convergence in law
for certain
weighted quadratic variations of a fractional Brownian motion B with
Hurst
index H between 1/4 and 1/2.
http://arxiv.org/abs/0707.3448
Author(s): Nicolas Champagnat (INRIA Sophia Antipolis / INRIA Lorraine / IECN),
Sylvie Roelly
Abstract: A multitype Dawson-Watanabe process is conditioned, in subcritical
and
critical cases, on non-extinction in the remote future. On every
finite time
interval, its distribution is absolutely continuous with respect to
the law of
the unconditioned process. A martingale problem characterization is
also given.
The explicit form of the Laplace functional of the conditioned
process is used
to obtain several results on the long time behaviour of the mass of the
conditioned and unconditioned processes. The general case is
considered first,
where the mutation matrix which modelizes the interaction between the
types, is
irreducible. Several two-type models with decomposable mutation
matrices are
also analysed.
http://arxiv.org/abs/0707.3504
Author(s): Laurent Decreusefond (LTCI) and Eduardo Ferraz (LTCI) and Philippe Martins
(LTCI)
Abstract: For OFDMA systems, we find a rough but easily computed upper bound
for the
probability of loosing communications by insufficient number of sub-
channels on
downlink. We consider as random the positions of receiving users in
the system
as well as the number of sub-channels dedicated to each one. We use
recent
results of the theory of point processes which reduce our
calculations to the
first and second moments of the total required number of sub-carriers.
http://arxiv.org/abs/0707.3509
Author(s): F. Bassetti and M. Cosentino Lagomarsino and S. Mandr\'a
Abstract: We introduce and study a class of exchangeable random graph
ensembles. They
can be used as statistical null models for empirical networks, and as
a tool
for theoretical investigations. We provide general theorems that
carachterize
the degree distribution of the ensemble graphs, together with some
features
that are important for applications, such as subgraph distributions
and kernel
of the adjacency matrix. These results are used to compare to other
models of
simple and complex networks. A particular case of directed networks with
power-law out--degree is studied in more detail, as an example of the
flexibility of the model in applications.
http://arxiv.org/abs/0707.3545
Author(s): Akhtam Dzhalilov and Isabelle Liousse and Dieter Mayer
Abstract: Let $T_{f}$ be a circle homeomorphism with two break points $a_
{b},c_{b}$ and
irrational rotation number $\varrho_{f}$. Suppose that the derivative
$Df$ of
its lift $f$ is absolutely continuous on every connected interval of
the set
$S^{1}\backslash\{a_{b},c_{b}\}$, that $DlogDf \in L^{1}$ and the
product of
the jump ratios of $ Df $ at the break points is nontrivial, i.e.
$\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\frac{Df_{-}(c_{b})}{Df_{+}(c_
{b})}\neq1$.
We prove that the unique $T_{f}$- invariant probability measure $\mu_
{f}$ is
then singular with respect to Lebesgue measure $l$ on $S^{1}$.
http://arxiv.org/abs/0707.3528
Author(s): Joanna Jaroszewska and Michal Rams
Abstract: We consider measures which are invariant under a measurable
iterated function
system with positive, place-dependent probabilities in a separable
metric
space. We provide an upper bound of the Hausdorff dimension of such a
measure
if it is ergodic. We also prove that it is ergodic iff the related
skew product
is.
http://arxiv.org/abs/0707.3532
Author(s): Ingemar Kaj and Lasse Leskel\"a and Ilkka Norros and Volker Schmidt
Abstract: This paper studies the limits of a spatial random field generated by
uniformly scattered random sets, as the density $\lambda$ of the sets
grows to
infinity and the mean volume $\rho$ of the sets tends to zero.
Assuming that
the volume distribution has a regularly varying tail with infinite
variance, we
show that the centered and renormalized random field can have three
different
limits, depending on the relative speed at which $\lambda$ and $\rho$
are
scaled. If $\lambda$ grows much faster than $\rho$ shrinks, the limit is
Gaussian with long-range dependence, while in the opposite case, the
limit is
independently scattered with infinite second moments. In a special
intermediate
scaling regime, there exists a nontrivial limiting random field that
is not
stable.
http://arxiv.org/abs/0707.3729
Author(s): Jean-Fran\c{c}ois Renaud and Bruno R\'emillard
Abstract: In this paper, we construct a Malliavin derivative for functionals of
square-integrable L\'evy processes and derive a Clark-Ocone formula. The
Malliavin derivative is defined via chaos expansions involving
stochastic
integrals with respect to Brownian motion and Poisson random measure.
As an
illustration, we compute the explicit martingale representation for
the maximum
of a L\'evy process.
http://arxiv.org/abs/0707.3734
Author(s): Emmanuel Roy
Abstract: We show that a stationary IDp process (i.e., an infinitely divisible
stationary process without Gaussian part) can be written as the
independent sum
of four stationary IDp processes, each of them belonging to a
different class
characterized by its L\'{e}vy measure. The ergodic properties of each
class
are, respectively, nonergodicity, weak mixing, mixing of all order and
Bernoullicity. To obtain these results, we use the representation of
an IDp
process as an integral with respect to a Poisson measure, which, more
generally, has led us to study basic ergodic properties of these
objects.
http://arxiv.org/abs/0707.3746
Author(s): Bernard Bercu (IMB) and Abderrahmen Touati (IMB)
Abstract: We propose several exponential inequalities for self-normalized
martingales
similar to those established by De la Pena. The keystone is the
introduction of
a new notion of random variable heavy on left or right. Applications
associated
with linear regressions, autoregressive and branching processes are also
provided.
http://arxiv.org/abs/0707.3715
Author(s): Steven Lalley and Xinghua Zheng
Abstract: We consider a critical nearest neighbor branching random walk on the
$d-$dimensional integer lattice. Denote by $V_m$ the maximal number of
particles at a single site at time $m$, and by $G_{m}$ the event that
the
branching random walk survives to generation $m$. We show that if the
offspring
distribution has finite $n$-th moment, then in dimensions $d\geq 3$,
conditional on $G_{m}$, $V_m=O_p(m^{\frac{1}{n}})$; and if the offspring
distribution has exponentially decaying tail, then, conditional on $G_
{m}$, (a)
$V_m=O_p(\log m)$ in dimensions $d\geq 3$, and (b) $V_m=O_p((\log m)
^2)$ in
dimension $d=2$. On the other hand, we show that if the offspring
distribution
is non-degenerate then $P(V_m\geq \delta \log m | G_{m})\to 1$ for
some $\delta
> 0$. Therefore, in dimensions $d\geq 3$, if the offspring
> distribution has
exponentially decaying tail then conditional on $G_{m}$, the
distribution of
${V_m}/{\log m}$ must converge to a nontrivial limit as $m \to \infty$.
Furthermore, we show that, conditional on $G_{m}$, in dimensions $d
\geq 3$, the
number of multiplicity-$j$ sites, $j\geq 1$, and the number of
occupied sites,
normalized by $m$, converge jointly to multiples of an exponential
random
variable; in dimension $d=2$, however, the number of particles on a
`typical'
site is $O_p(\log m)$, and the number of occupied sites is $O_p(m/
\log m).$
http://arxiv.org/abs/0707.3829
Author(s): Michel Benaim and Itai Benjamini and Raphael Rossignol
Abstract: Take a big graph and make a random electrical network of it by
assigning
independent resistances on its edges. Now, ask for the behaviour of the
effective resistance between two vertices (two ``poles'') far apart.
We assume
in general that resistances are bounded away from 0 and infinity. In
this
paper, we study three cases of effective resistance in such random
electrical
networks: from one side to another in a box of $Z^d$, between two
points in
$Z^2$, and between two points on a cylinder graph $GxZ$. For all
these cases,
we obtain the right order of the fluctuations when the poles move
apart from
each other, and give corresponding subgaussian concentration
inequalities. For
the cylinder graphs, we prove two additional results: a central limit
theorem
and a result of uniform stability with respect to noise.
http://arxiv.org/abs/0707.3837
Author(s): Anna Amirdjanova and Matthew Linn
Abstract: The problem of nonlinear filtering of a random field observed in
the presence
of a noise, modeled by a persistent fractional Brownian sheet of
Hurst index
$(H_1,H_2)$ with $0.5
http://arxiv.org/abs/0707.3856
Author(s): Jean Brossard and Christophe Leuridan
Abstract: Nous \'{e}tudions les cha\^{{\i}}nes de Markov $(X_n)_{n\in\mathbf
{Z}}$
gouvern\'{e}es par une relation de r\'{e}currence de la forme
$X_{n+1}=f(X_n,V_{n+1})$, o\`{u} $(V_n)_{n\in\mathbf{Z}}$ est une
suite de
variables al\'{e}atoires ind\'{e}pendantes et de m\^{e}me loi telle
pour tout
$n\in \mathbf{Z}$, $V_{n+1}$ est ind\'{e}pendante de la suite
$((X_k,V_k))_{k\le n}$. L'objet de l'article est de donner une condition
n\'{e}cessaire et suffisante pour que les innovations $(V_n)_{n\in
\mathbf{Z}}$
d\'{e}terminent compl\`{e}tement la suite $(X_n)_{n\in \mathbf{Z}}$
et de
d\'{e}crire l'information manquante dans le cas contraire.
http://arxiv.org/abs/0707.3860
Author(s): F. Nazarov and M. Sodin and A. Volberg
Abstract: By random complex zeroes we mean the zero set of a random entire
function
whose Taylor coefficients are independent complex-valued Gaussian
variables,
and the variance of the k-th coefficient is 1/k!. This zero set is
distribution
invariant with respect to isometries of the complex plane. We study
large
fluctuations of random complex zeroes and show that they obey the
asymptotic
law that was discovered some time ago by Jancovici, Lebowitz and
Magnificat for
charge fluctuations of a Coulomb system of particles.
http://arxiv.org/abs/0707.3863
Author(s): A. Deniz Sezer
Abstract: We develop the mathematics of a filtration shrinkage model that
has recently
been considered in the credit risk modeling literature. Given a finite
collection of points $x_1<...
http://arxiv.org/abs/0707.3866
Author(s): Ming Yang
Abstract: Let $X_t$ be any additive process in $\mathbb{R}^d.$ There are
finite indices
$\delta_i, \beta_i, i=1,2$ and a function $u$, all of which are
defined in
terms of the characteristics of $X_t$, such that
\liminf_{t\to0}u(t)^{-1/\eta}X_t^*= \cases{0, \quad if $\eta>
\delta_1$,
\cr\infty, \quad if $\eta<\delta_2$,}
\limsup_{t\to0}u(t)^{-1/\eta}X_t^*= \cases{0, \quad if $\eta>
\beta_2$,
\cr\infty, \quad if $\eta<\beta_1$,}\qquad {a.s.},
where $X_t^*=\sup_{0\le s\le t}|X_s|.$ When $X_t$ is a L\'{e}vy
process with
$X_0=0$, $\delta_1=\delta_2$, $\beta_1=\beta_2$ and $u(t)=t.$ This is
a special
case obtained by Pruitt. When $X_t$ is not a L\'{e}vy process, its
characteristics are complicated functions of $t$. However, there are
interesting conditions under which $u$ becomes sharp to achieve
$\delta_1=\delta_2$, $\beta_1=\beta_2.$
http://arxiv.org/abs/0707.3886
Author(s): Itai Benjamini and Ariel Yadin and Ofer Zeitouni
Abstract: Let U(N) denote the maximal length of arithmetic progressions in a
random
uniform subset of {0,1}^N. By an application of the Chen-Stein
method, we show
that U(N)- 2 log(N)/log(2) converges in law to an extreme type
(asymmetric)
distribution. The same result holds for the maximal length W(N) of
arithmetic
progressions (mod N). When considered in the natural way on a common
probability space, we observe that U(N)/log(N) converges almost
surely to
2/log(2), while W(N)/log(N) does not converge almost surely (and in
particular,
limsup W(N)/log(N) is at least 3/log(2)).
http://arxiv.org/abs/0707.3888
Author(s): Mathew D. Penrose and Andrew R. Wade
Abstract: Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the
sum is
over points $x$ of a Poisson point process of intensity $\lambda$ on
a bounded
region in $d$-space, and $\xi_x$ is a functional determined by the
Poisson
points near to $x$, i.e. satisfying an exponential stabilization
condition,
along with a moments condition (examples include statistics for
proximity
graphs, germ-grain models and random sequential deposition models). A
known
general result says the $\mu_\lambda$-measures (suitably scaled and
centred) of
disjoint sets in $R^d$ are asymptotically independent normals as $
\lambda \to
\infty$; here we give an $O(\lambda^{-1/(2d + \epsilon)})$ bound on
the rate of
convergence. We illustrate our result with an explicit multivariate
central
limit theorem for the nearest-neighbour graph on Poisson points on a
finite
collection of disjoint intervals.
http://arxiv.org/abs/0707.3898
Author(s): Michael Marcus and Jay Rosen
Abstract: Let G be a mean zero Gaussian processes with stationary increments
and set
\si ^2(|x-y|)= E(G(x)-G(y))^2. Let f be a function with Ef^{2}(\eta)<
\ff, where
\eta=N(0,1). When \si^2 is convex and regularly varying at zero and
\lim_{h\to
0} \si(h)/h=\ff \quad {but} \quad ({d\over ds^2}\si^2(s))^{j_0} \mbox{is
locally integrable} for some integer j_0\ge 1, and satisfies some
additional
regularity conditions, then \int_a^bf(\frac{G(x+h)-G(x)}{\si (h)}) dx =
\sum_{j=0}^{j_0} (h/\si(h))^{j} {E(H_{j}(\eta) f(\eta))\over\sqrt {j!}}
:(G')^{j}:(I_{[a,b]}) +o({h\over\si (h)})^{j_0}\nn in L^2.
Here H_j is the j-th Hermite polynomial in the Hermite polynomial
expansion
of f. Also :(G')^{j}:(I_{[a,b]}) is a j-th order Wick power Gaussian
chaos
constructed from the Gaussian field G'(g)=\int g(x) dG(x) with
covariance
E(G'(g)G'(\wt g)) = \int \int \rho (x-y)g(x)\wt g(y) dx dy where
\rho(s)={1/2}{d^{2}\over ds^2}\si^2(s). Moreover, under the same
conditions
\lim_{h\downarrow0}\int_a^b :(\frac{G(x+h)-G(x)}{h})^{j_0}: dx =
:(G')^{j_0}:(I_{[a,b]}) \qquad {a.s.}
http://arxiv.org/abs/0707.3928
Author(s): Bruno Cernuschi-Frias (IRISA)
Abstract: New theoretical results are presented here on the recently
introduced model
called mixed states MRF. Such models were introduced in the context
of image
motion analysis and are useful to represent information which can
take both
discrete values accounting for symbolic states, and real values
corresponding
to continuous measurements. In particular, results are given when the
global
energy for the Gibbs formulation expressing the mixed states model,
can be
decomposed into one term accounting for the discrete part of the
model, and a
second term related to the continuous part. This decomposition
theorem permits
to define conditional mixed states models in a very simple way.
http://arxiv.org/abs/0707.3986
Author(s): Bojan Basrak and Johan Segers
Abstract: A multivariate, stationary time series is said to be jointly
regularly
varying if all its finite-dimensional distributions are multivariate
regularly
varying. This property is shown to be equivalent to weak convergence
of the
conditional distribution of the rescaled series given that, at a
fixed time
instant, its distance to the origin exceeds a threshold tending to
infinity.
The limit object, called the tail process, admits a decomposition in
independent radial and angular components. Under an appropriate mixing
condition, this tail process allows for a concise and explicit
description of
the limit of a sequence of point processes recording both the times
and the
positions of the time series when it is far away from the origin. The
theory is
applied to multivariate moving averages of finite order with random
coefficient
matrices.
http://arxiv.org/abs/0707.3989
Author(s): Nasir Ganikhodjaev and Fatimah Abdul Razak
Abstract: In this paper, correlation inequalities which have been considered
on Ising
model are extended to q-Potts model. It is considered on generalized
Potts
model with interaction of any number of spins. We replace the set of
spin
values $F=\{1,2,..., q\}$ by the centered set $F=\{-(q-1)/2,-(q-3)/2,...
,(q-3)/2,(q-1)/2\}$. Let $N$ be the subset of one-dimensional lattice
with $n$
vertices, $\g=(\s_1,\s_2,...,\s_n):N \to F^c$ be a configuration where
${(\s_i)}_\g$ is the number which appears as the ith spin (component)
in $\g$
and $\s_i$ be a random variable whose value at $\g$ is ${(\s_i)}_\g$.
Define
$\s^R=\prod_{i \in R}\s_i$ for any list $R$ where any $i \in R$
implies that $i
\in N$. We first prove that $<\s^R > \ge 0$ then we prove that for
any two
lists $R$ and $S$, we have $<\s^R \s^S >- < \s^R > < \s^S > \ge 0$.
http://arxiv.org/abs/0707.3848
Author(s): James Allen Fill
Abstract: A well-known theorem usually attributed to Keilson states that,
for an
irreducible continuous-time birth-and-death chain on the nonnegative
integers
and any d, the passage time from state 0 to state d is distributed as
a sum of
d independent exponential random variables. Until now, no
probabilistic proof
of the theorem has been known. In this paper we use the theory of strong
stationary duality to give a stochastic proof of a similar result for
discrete-time birth-and-death chains and geometric random variables,
and the
continuous-time result (which can also be given a direct stochastic
proof) then
follows immediately. In both cases we link the parameters of the
distributions
to eigenvalue information about the chain.
Intimately related to the passage-time theorem is a theorem of
Fill that any
fastest strong stationary time T for an ergodic birth-and-death chain
on {0,
> ..., d} in continuous time with generator G, started in state 0, is
distributed as a sum of d independent exponential random variables
whose rate
parameters are the nonzero eigenvalues of the negative of G. Our
approach
yields the first (sample-path) construction of such a T for which
individual
such exponentials summing to T can be explicitly identified.
http://arxiv.org/abs/0707.4042
Author(s): George P. Yanev and M. Ahsanullah and and M.I. Beg
Abstract: Bairamov et al. (Aust N Z J Stat 47:543-547, 2005) characterize the
exponential distribution in terms of the regression of a function of
a record
value with its adjacent record values as covariates. We extend these
results to
the case of non-adjacent covariates. We also consider a more general
setting
involving monotone transformations. As special cases, we present
characterizations involving weighted arithmetic, geometric, and
harmonic means.
http://arxiv.org/abs/0707.4121
Author(s): Alexei Borodin (1) and Patrik L. Ferrari (2) ((1) Caltech and (2) WIAS
Berlin)
Abstract: We consider a new interacting particle system on the one-
dimensional lattice
that interpolates between TASEP and Toom's model: A particle cannot
jump to the
right if the neighboring site is occupied, and when jumping to the
left it
simply pushes all the neighbors that block its way.
We prove that for flat and step initial conditions, the large time
fluctuations of the height function of the associated growth model
along any
space-like path are described by the Airy_1 and Airy_2 processes.
This includes
fluctuations of the height profile for a fixed time and fluctuations
of a
tagged particle's trajectory as special cases.
http://arxiv.org/abs/0707.2813
Author(s): Pablo A. Ferrari and James B. Martin
Abstract: In the Hammersley-Aldous-Diaconis process infinitely many
particles sit in R
and at most one particle is allowed at each position. A particle at x
$ whose
nearest neighbor to the right is at y, jumps at rate y-x to a position
uniformly distributed in the interval (x,y). The basic coupling between
trajectories with different initial configuration induces a process with
different classes of particles. We show that the invariant measures
for the
two-class process can be obtained as follows. First, a stationary M/M/
1 queue
is constructed as a function of two homogeneous Poisson processes,
the arrivals
with rate \lambda and the (attempted) services with rate \rho>
\lambda. Then put
the first class particles at the instants of departures (effective
services)
and second class particles at the instants of unused services. The
procedure is
generalized for the n-class case by using n-1 queues in tandem with n-1
priority-types of customers. A multi-line process is introduced; it
consists of
a coupling (different from Liggett's basic coupling), having as
invariant
measure the product of Poisson processes. The definition of the multi-
line
process involves the dual points of the space-time Poisson process
used in the
graphical construction of the system. The coupled process is a
transformation
of the multi-line process and its invariant measure the transformation
described above of the product measure.
http://arxiv.org/abs/0707.4202
Author(s): Marco Fuhrman (Dipartimento Di Matematica) and Ying Hu (IRMAR) and Gianmario
Tessitore (Dipartimento Di Matematica E Applicazioni)
Abstract: In this paper we introduce a new kind of Backward Stochastic
Differential
Equations, called ergodic BSDEs, which arise naturally in the study
of optimal
ergodic control. We study the existence, uniqueness and regularity of
solution
to ergodic BSDEs. Then we apply these results to the optimal ergodic
control of
a Banach valued stochastic state equation. We also establish the link
between
the ergodic BSDEs and the associated Hamilton-Jacobi-Bellman equation.
Applications are given to ergodic control of stochastic partial
differential
equations.
http://arxiv.org/abs/0707.4214
Author(s): Marton Balazs and Timo Seppalainen
Abstract: Take a random variable X with some finite exponential moments.
Define an
exponentially weighted expectation by E^t(f) = E(e^{tX}f)/E(e^{tX}) for
admissible values of the parameter t. Denote the weighted expectation
of X
itself by r(t) = E^t(X), with inverse function t(r). We prove that
for a convex
function f the expectation E^{t(r)}(f) is a convex function of the
parameter r.
Along the way we develop correlation inequalities for convex functions.
Motivation for this result comes from equilibrium investigations of some
stochastic interacting systems with stationary product distributions. In
particular, convexity of the hydrodynamic flux function follows in
some cases.
http://arxiv.org/abs/0707.4273
Author(s): R. Ba\~nuelos and T. Kulczycki and B. Siudeja
Abstract: We prove that the diagonal of the transition probabilities for the
d-dimensional Bessel processes on (0, 1], reflected at 1, which we
denote by
$p_R^N(t, r,r)$, is an increasing function of r for d>2 and that this
is false
for d=2.
http://arxiv.org/abs/0707.4299
Author(s): Xia Chen
Abstract: We study the upper tail behaviors of the local times of the
additive L\'{e}vy
processes and additive random walks. The limit forms we establish are
the
moderate deviations and the laws of the iterated logarithm for the
L_2-norms of
the local times and for the local times at a fixed site.
http://arxiv.org/abs/0707.4355
Author(s): Toshiro Watanabe
Abstract: A necessary and sufficient condition for the almost sure existence
of an
absolutely continuous (with respect to the branching measure) exact
Hausdorff
measure on the boundary of a Galton--Watson tree is obtained. In the
case where
the absolutely continuous exact Hausdorff measure does not exist
almost surely,
a criterion which classifies gauge functions $\phi$ according to whether
$\phi$-Hausdorff measure of the boundary minus a certain exceptional
set is
zero or infinity is given. Important examples are discussed in four
additional
theorems. In particular, Hawkes's conjecture in 1981 is solved.
Problems of
determining the exact local dimension of the branching measure at a
typical
point of the boundary are also solved.
http://arxiv.org/abs/0707.4358
Author(s): A. Popier
Abstract: In this paper we are concerned with one-dimensional backward
stochastic
differential equations (BSDE in short) of the following type: \[Y_t=\xi
-\int_{t\wedge \tau}^{\tau}Y_r|Y_r|^q dr-\int_{t\wedge \tau}^{\tau}Z_r
dB_r,\qquad t\geq 0,\] where $\tau$ is a stopping time, $q$ is a
positive
constant and $\xi$ is a $\mathcal{F}_{\tau}$-measurable random
variable such
that $\mathbf{P}(\xi =+\infty)>0$. We study the link between these
BSDE and the
Dirichlet problem on a domain $D\subset \mathbb{R}^d$ and with boundary
condition $g$, with $g=+\infty$ on a set of positive Lebesgue
measure. We also
extend our results for more general BSDE.
http://arxiv.org/abs/0707.4387
Author(s): N. Jain and N.V. Krylov
Abstract: Our aim is to unify and extend the large deviation upper and lower
bounds for
the empiricals of a Markov process with L_2 semigroups under minimal
conditions
on the state space and the process trajectories; for example, no
strong Markov
property is needed. The methods used here apply in both continuous
and discrete
time. We present the proofs for continuous time only because of the
inherent
technical difficulties in that situation; the proofs can be adapted for
discrete time in a straightforward manner.
http://arxiv.org/abs/0707.4469
Author(s): Rodrigo Banuelos and Tadeusz Kulczycki
Abstract: In this paper we study the behaviour in time of the trace (the
partition
function) of the heat semigroup associated with symmetric stable
processes in
domains of $\Rd$. In particular, we show that for domains with the so
called
{\it{$R$-smoothness}} property the second terms in the asymptotic as
$t\to 0$
involves the surface area of the domain, just as in the case of Brownian
motion.
http://arxiv.org/abs/0707.4313
Author(s): Milton D. Jara
Abstract: We prove an invariance principle for a tagged particle in a simple
exclusion
process out of equilibrium. The scaling limit is a time-inhomogeneous
process
of independent increments, related to the solution of a fractional heat
equation.
http://arxiv.org/abs/0707.4491
Author(s): Yoichi Nishiyama
Abstract: This note extends some results of Nishiyama [Ann. Probab. 28 (2000)
685--712]. A maximal inequality for stochastic integrals with respect to
integer-valued random measures which may have infinitely many jumps
on compact
time intervals is given. By using it, a tightness criterion is
obtained; if the
so-called quadratic modulus is bounded in probability and if a
certain entropy
condition on the parameter space is satisfied, then the tightness
follows. Our
approach is based on the entropy techniques developed in the modern
theory of
empirical processes.
http://arxiv.org/abs/0707.4536
Author(s): Laurent Massouli\'e
Abstract: In this article we provide a novel characterization of the
proportionally
fair bandwidth allocation of network capacities, in terms of the
Fenchel--Legendre transform of the network capacity region. We use this
characterization to prove stability (i.e., ergodicity) of network
dynamics
under proportionally fair sharing, by exhibiting a suitable Lyapunov
function.
Our stability result extends previously known results to a more
general model
including Markovian users routing. In particular, it implies that the
stability
condition previously known under exponential service time
distributions remains
valid under so-called phase-type service time distributions. We then
exhibit a
modification of proportional fairness, which coincides with it in some
asymptotic sense, is reversible (and thus insensitive), and has explicit
stationary distribution. Finally we show that the stationary
distributions
under modified proportional fairness and balanced fairness, a sharing
criterion
proposed because of its insensitivity properties, admit the same large
deviations characteristics. These results show that proportional
fairness is an
attractive bandwidth allocation criterion, combining the desirable
properties
of ease of implementation with performance and insensitivity.
http://arxiv.org/abs/0707.4542
Author(s): Laure Coutin and Peter Friz and Nicolas Victoir
Abstract: We consider anticipative Stratonovich stochastic differential
equations
driven by some stochastic process lifted to a rough path. Neither
adaptedness
of initial point and vector fields nor commuting conditions between
vector
field is assumed. Under a simple condition on the stochastic process,
we show
that the unique solution of the above SDE understood in the rough
path sense is
actually a Stratonovich solution. We then show that this condition is
satisfied
by the Brownian motion. As application, we obtain rather flexible
results such
as support theorems, large deviation principles and Wong--Zakai
approximations
for SDEs driven by Brownian motion along anticipating vectorfields. In
particular, this unifies many results on anticipative SDEs.
http://arxiv.org/abs/0707.4546
Author(s): Yu Miao
Abstract: In this paper, we give the central limit theorem and almost sure
central
limit theorem for products of some partial sums of independent
identically
distributed random variables.
http://arxiv.org/abs/0707.4549
Author(s): Feng Yu
Abstract: This paper deals with a model of sympatric speciation, that is,
speciation in
the absence of geographical separation, originally proposed by U.
Dieckmann and
M. Doebeli in 1999. We modify their original model to obtain a
Fleming--Viot
type model and study its stationary distribution. We show that
speciation may
occur, that is, the stationary distribution puts most of the mass on a
configuration that does not concentrate on the phenotype with maximum
carrying
capacity, if competition between phenotypes is intense enough.
Conversely, if
competition between phenotypes is not intense, then speciation will
not occur
and most of the population will have the phenotype with the highest
carrying
capacity. The length of time it takes speciation to occur also has a
delicate
dependence on the mutation parameter, and the exact shape of the
carrying
capacity function and the competition kernel.
http://arxiv.org/abs/0707.4553
Author(s): Konstantin Mischaikow and Thomas Wanner
Abstract: Homology has long been accepted as an important computable tool for
quantifying complex structures. In many applications, these
structures arise as
nodal domains of real-valued functions and are therefore amenable
only to a
numerical study based on suitable discretizations. Such an approach
immediately
raises the question of how accurate the resulting homology
computations are. In
this paper, we present a probabilistic approach to quantifying the
validity of
homology computations for nodal domains of random fields in one and
two space
dimensions, which furnishes explicit probabilistic a priori bounds
for the
suitability of certain discretization sizes. We illustrate our
results for the
special cases of random periodic fields and random trigonometric
polynomials.
http://arxiv.org/abs/0707.4588
Author(s): Zhiyi Chi
Abstract: Let (X_n,Y_n) be i.i.d. random vectors. Let W(x) be the partial
sum of Y_n
just before that of X_n exceeds x>0. Motivated by stochastic models
for neural
activity, uniform convergence of the form $\sup_{c\in I}|a(c,x)
\operatorname
{Pr}\{W(x)\gecx\}-1|=o(1)$, $x\to\infty$, is established for
probabilities of
large deviations, with a(c,x) a deterministic function and I an open
interval.
To obtain this uniform exact large deviations principle (LDP), we first
establish the exponentially fast uniform convergence of a family of
renewal
measures and then apply it to appropriately tilted distributions of
X_n and the
moment generating function of W(x). The uniform exact LDP is obtained
for cases
where X_n has a subcomponent with a smooth density and Y_n is not a
linear
transform of X_n. An extension is also made to the partial sum at the
first
exceedance time.
http://arxiv.org/abs/0707.4596
Author(s): H. Christian Gromoll and {\L}ukasz Kruk
Abstract: This paper considers a GI/GI/1 processor sharing queue in which
jobs have
soft deadlines. At each point in time, the collection of residual
service times
and deadlines is modeled using a random counting measure on the right
half-plane. The limit of this measure valued process is obtained under
diffusion scaling and heavy traffic conditions and is characterized as a
deterministic function of the limiting queue length process. As
special cases,
one obtains diffusion approximations for the lead time profile and
the profile
of times in queue. One also obtains a snapshot principle for sojourn
times.
http://arxiv.org/abs/0707.4600
Author(s): Alexei Borodin and Patrik L. Ferrari and Tomohiro Sasamoto
Abstract: We consider the polynuclear growth (PNG) model in 1+1 dimension
with flat
initial condition and no extra constraints. The joint distributions
of surface
height at finitely many points at a fixed time moment are given as
marginals of
a signed determinantal point process. The long time scaling limit of the
surface height is shown to coincide with the Airy_1 process. This
result holds
more generally for the observation points located along any space-
like path in
the space-time plane. We also obtain the corresponding results for
the discrete
time TASEP (totally asymmetric simple exclusion process) with
parallel update.
http://arxiv.org/abs/0707.4207
Author(s): A. Faggionato
Abstract: We show that the slopes between h-extrema of the drifted 1D
Brownian motion
form a stationary alternating marked point process, extending the
result of J.
Neveu and J. Pitman for the non drifted case. Our analysis covers the
results
on the statistics of h-extrema obtained by P. Le Doussal, C. Monthus
and D.
Fisher via a Renormalization Group analysis and gives a complete
description of
the slope between h-extrema covering the origin by means of the Palm--
Khinchin
theory. Moreover, we analyze the behavior of the Brownian motion near
its
h-extrema.
http://arxiv.org/abs/0708.0128
Author(s): Nathalie Krell (PMA)
Abstract: The main purpose of this work is to study self-similar branching
Markov
chains. First we will construct such a process. Then we will
establish certain
Limit Theorems using the theory of self-similar Markov processes.
http://arxiv.org/abs/0708.0138
Author(s): Mikhail Langovoy
Abstract: A general method for constructing tests of statistical hypotheses is
proposed. The method offers a generalization of the theory of score
tests. Our
tests are incorporated with model selection rules to choose
reasonable model
dimensions automatically by the data. A unified approach for proving
consistency of the tests is developed.
http://arxiv.org/abs/0708.0169
Author(s): Stanislav Smirnov
Abstract: Many 2D lattice models of physical phenomena are conjectured to have
conformally invariant scaling limits: percolation, Ising model, self-
avoiding
polymers, ... This has led to numerous exact (but non-rigorous)
predictions of
their scaling exponents and dimensions. We will discuss how to prove the
conformal invariance conjectures, especially in relation to Schramm-
Loewner
Evolution.
http://arxiv.org/abs/0708.0032
Author(s): Kei Takeuchi and Masayuki Kumon and Akimichi Takemura
Abstract: We introduce a new formulation of asset trading games in
continuous time in
the framework of the game-theoretic probability established by Shafer
and Vovk
(2001). In our formulation, the market moves continuously but an
investor
trades in discrete times, which can depend on the past path of the
market. We
prove that an investor can essentially force that the asset price
path behaves
with the variation exponent exactly equal to two. Our proof is based on
embedding high-frequency discrete time games into the continuous time
game and
the use of the Bayesian strategy of Kumon, Takemura and Takeuchi
(2007b) for
discrete time coin-tossing games. We also clarify that the main
growth part of
the investor's capital processes is lucidly described by the information
quantities, which are derived from the Kullback-Leibler information with
respect to the empirical fluctuation of the asset price.
http://arxiv.org/abs/0708.0275
Author(s): Vassili Kolokoltsov
Abstract: The general model of coagulation is considered.
For basic classes of unbounded coagulation kernels the central
limit theorem
(CLT) is obtained for the fluctuations around the dynamic law of
large numbers
(LLN). A rather precise rate of convergence is given both for LLN and
CLT.
http://arxiv.org/abs/0708.0329
Author(s): David F. Anderson and Jonathan C. Mattingly
Abstract: We consider biochemical reaction chains and investigate how random
external
fluctuations, as characterized by variance and coefficient of variation,
propagate down the chains. We perform such a study under the
assumption that
the number of molecules is high enough so that the behavior of the
concentrations of the system is well approximated by differential
equations. We
conclude that the variances and coefficients of variation of the
fluxes will
decrease as one moves down the chain and, through an example, show
that there
is no corresponding result for the variances of the chemical species.
We also
prove that the fluctuations of the fluxes as characterized by their time
averages decrease down reaction chains. The results presented give
insight into
how biochemical reaction systems are buffered against external
perturbations
solely by their underlying graphical structure and point out the
benefits of
studying the out-of-equilibrium dynamics of systems.
http://arxiv.org/abs/0708.0380
Author(s): Savas Dayanik and Harold Vincent Poor and Semih Onur Sezer
Abstract: Suppose that local characteristics of several independent compound
Poisson and Wiener processes change suddenly and simultaneously at
some
unobservable disorder time. The problem is to detect the disorder
time as
quickly as possible after it happens and minimize the rate of false
alarms at
the same time. These problems arise, for example, from managing
product quality
in manufacturing systems and preventing the spread of infectious
diseases. The
promptness and accuracy of detection rules improve greatly if multiple
independent information sources are available. Earlier work on
sequential
change detection in continuous time does not provide optimal rules for
situations in which several marked count data and continuously
changing signals
are simultaneously observable. In this paper, optimal Bayesian
sequential
detection rules are developed for such problems when the marked count
data is
in the form of independent compound Poisson processes, and the
continuously
changing signals form a multi-dimensional Wiener process. An
auxiliary optimal
stopping problem for a jump-diffusion process is solved by
transforming it
first into a sequence of optimal stopping problems for a pure
diffusion by
means of a jump operator. This method is new and can be very useful
in other
applications as well, because it allows the use of the powerful optimal
stopping theory for diffusions.
http://arxiv.org/abs/0708.0224
Author(s): Maria A. Avino-Diaz
Abstract: A probabilistic structure on sequential dynamical systems is
introduced here,
the new model will be called Probabilistic Sequential Network, PSN. The
morphisms of Probabilistic Sequential Networks are defined using two
algebraic
conditions, whose imply that the distribution of probabilities in the
systems
are close. It is proved here that two homomorphic Probabilistic
Sequential
Networks have the same equilibrium or steady state probabilities.
Additionally,
the proof of the set of PSN with its morphisms form the category PSN,
having
the category of sequential dynamical systems SDS, as a full
subcategory is
given. Several examples of morphisms, subsystems and simulations are
given.
http://arxiv.org/abs/0707.0026
Author(s): Saul Jacka and Abdelkarem Berkaoui
Abstract: We consider the problem of representing claims for coherent risk
measures.
For this purpose we introduce the concept of (weak and strong) time-
consistency
with respect to a portfolio of assets, generalizing the one defined
by Delbaen.
In a similar way we extend the notion of m-stability, by
introducing weak and
strong versions. We then prove that the two concepts of m-stability and
time-consistency are still equivalent, thus giving necessary and
sufficient
conditions for a coherent risk measure to be represented by a market
with
proportional transaction costs. We go on to deduce that, under a
separability
assumption, any coherent risk measure is strongly time-consistent
with respect
to a suitably chosen countable portfolio, and show the converse: that
any
market with proportional transaction costs is equivalent to a market
priced by
a coherent risk measure, essentially establishing the equivalence of
the two
concepts.
http://arxiv.org/abs/0708.0512
Author(s): David J. Aldous and Shankar Bhamidi
Abstract: Consider the complete n-vertex graph whose edge-lengths are
independent
exponentially distributed random variables. Simultaneously for each
pair of
vertices, put a constant flow between them along the shortest path.
Each edge
gets some random total flow. In the $n \to \infty$ limit we find
explicitly the
empirical distribution of these edge-flows, suitably normalized.
http://arxiv.org/abs/0708.0555
Author(s): Pawe{\l} J. Szab{\l}owski
Abstract: First we generalize to $q-$series case, a well known formula $(x+y)
^{n}%
=\sum_{i=0}^{n}\binom{n}{k}i^{k}H_{n-k}(x) H_{k}(-iy) ,$ where H_{k}
(x) denotes
k-th Hermite polynomial. Then we apply this generalization to Al-
Salam-Chihara
polynomials for specific values of parameters. We use this result to
prove the
existence of stationary random fields with linear regressions and
thus close an
open question posed by W. Bryc et al.. We prove this result by
describing a
discrete 1 dimensional conditional distribution. Its support consist
of zeros
of certain Al-Salam-Chihara polynomials.
http://arxiv.org/abs/0708.0563
Author(s): Michael Krivelevich and Po-Shen Loh and Benny Sudakov
Abstract: For a fixed integer r, consider the following random process. At
each round,
one is presented with r random edges from the edge set of the
complete graph on
n vertices, and is asked to choose one of them. The selected edges are
collected into a graph, which thus grows at the rate of one edge per
round.
This is a natural generalization of what is known in the literature
as an
Achlioptas process (the original version has r=2), which has been
studied by
many researchers, mainly in the context of delaying or accelerating the
appearance of the giant component.
In this paper, we investigate the small subgraph problem for
Achlioptas
processes. That is, given a fixed graph H, we study whether there is a
deterministic online algorithm that substantially delays or
accelerates a
typical appearance of H, compared to its threshold of appearance in
the random
graph G(n, M). It is easy to see that one cannot accelerate the
appearance of
any fixed graph by more than the constant factor r, so we concentrate
on the
task of avoiding H. We determine thresholds for the avoidance of all
cycles
C_t, cliques K_t, and complete bipartite graphs K_{t,t}, in every
Achlioptas
process with parameter r >= 2.
http://arxiv.org/abs/0708.0443
Author(s): Lancelot F. James
Abstract: An important line of research is the investigation of the laws of
random
variables known as Dirichlet means as discussed in Cifarelli and
Regazzini(1990). However there is not much information on inter-
relationships
between different Dirichlet means. Here we introduce two distributional
operations, which consist of multiplying a mean functional by an
independent
beta random variable and an operation involving an exponential change of
measure. These operations identify relationships between different
means and
their densities. This allows one to use the often considerable
analytic work to
obtain results for one Dirichlet mean to obtain results for an entire
family of
otherwise seemingly unrelated Dirichlet means. Additionally, it
allows one to
obtain explicit densities for the related class of random variables
that have
generalized gamma convolution distributions, and the finite-dimensional
distribution of their associated L\'evy processes. This has
implications in,
for instance, the explicit description of Bayesian nonparametric
prior and
posterior models, and more generally in a variety of applications in
probability and statistics involving Levy processes.
http://arxiv.org/abs/0708.0614
Author(s): Lancelot F. James
Abstract: This paper obtains density and cdf formula, and various
distributional
identities, for random variables defined as the ratio of two independent
positive random variables where one variable has an $\alpha$ stable
law, for
$0<\alpha<1,$ and the other variable has the law defined by power
tempering the
density of an $\alpha$ stable random variable by a factor $\theta>-
\alpha$.
When $\theta=0$, these variables equate with the ratio investigated
by Lamperti
which remarkably was shown to have a simple density. This variable
arises in a
variety of areas and gains importance from a close connection to the
stable
laws. This rationale motivates the investigations of its
generalizations which
we refer to as Lamperti type laws. Here specifically the results are
used to
obtain results for 3 interesting quantities, which appear in a
variety of
contexts. Explicit distributional formulae and identities are derived
for the
class of positive generalized Linnik random variables. Then the best
known
results for the density of the time spent positive of a Bessel bridge of
dimension $2-2\alpha$, and related quantities, are obtained.
Additionally,
integral representations and other identities for a class of generalized
Mittag-Leffler functions are obtained. We will also describe the
connections
between these results and show how they generalize previous results
in the
literature.
http://arxiv.org/abs/0708.0618
Author(s): Man-Wai Ho and Lancelot F. James and and John W. Lau
Abstract: This paper derives explicit results for the infinite Gibbs partitions
generated by the jumps of an $\alpha-$stable subordinator, derived in
Pitman
\cite{Pit02, Pit06}. We first show that for general $\alpha$ the
conditional
EPPF can be represented as ratios of Fox-$H$ functions, and in the
case of
rational $\alpha,$ Meijer-G functions. Furthermore the results show
that the
resulting unconditional EPPF's, can be expressed in terms of H and G
transforms
indexed by a function h. Hence when h is itself a H or G function the
EPPF is
also an H or G function. An implication, in the case of rational $
\alpha,$ is
that one can compute explicitly thousands of EPPF's derived from
possibly
exotic special functions. This would also apply to all $\alpha$
except that
computations for general Fox functions are not yet available.
However, moving
away from special functions, we demonstrate how results from
probability theory
may be used to obtain calculations. We show that a forward recursion
can be
applied that only requires calculation of the simplest components.
Additionally
we identify general classes of EPPF's where explicit calculations can be
carried out using distribution theory.
http://arxiv.org/abs/0708.0619
Author(s): Jonathon Peterson
Abstract: We consider a nearest-neighbor, one-dimensional random walk $\{X_n
\}_{n\geq
0}$ in a random i.i.d. environment, in the regime where the walk is
transient
with speed v_P > 0 and there exists an $s\in(1,2)$ such that the
annealed law
of $n^{-1/s} (X_n - n v_P)$ converges to a stable law of parameter s.
Under the
quenched law (i.e., conditioned on the environment), we show that no
limit laws
are possible. In particular we show that there exist sequences {t_k}
and {t_k'}
depending on the environment only, such that a quenched central limit
theorem
holds along the subsequence t_k, but the quenched limiting
distribution along
the subsequence t_k' is a centered reverse exponential distribution.
This
complements the results of a recent paper of Peterson and Zeitouni
(arXiv:0704.1778v1 [math.PR]) which handled the case when the parameter
$s\in(0,1)$.
http://arxiv.org/abs/0708.0649
Author(s): James A. Mingo (Queen's University) and Roland Speicher (Queen's University), Edward Tan (Queen's University)
Abstract: We derive a formula which expresses a second order cumulant whose
entries are
products as a sum of cumulants where the entries are single factors.
This
extends to the second order case the formula of Krawczyk and
Speicher. We apply
our result to the problem of calculating the second order cumulants of a
semi-circular and Haar unitary operator.
http://arxiv.org/abs/0708.0586
Author(s): Ivar Ekeland and Traian A. Pirvu
Abstract: In this paper, we investigate the Merton portfolio management
problem in the
context of non-exponential discounting. This gives rise to time-
inconsistency
of the decision-maker. If the decision-maker at time t=0 can commit
his/her
successors, he/she can choose the policy that is optimal from his/her
point of
view, and constrain the others to abide by it, although they do not
see it as
optimal for them. If there is no commitment mechanism, one must seek a
subgame-perfect equilibrium strategy between the successive decision-
makers. In
the line of the earlier work by Ekeland and Lazrak we give a precise
definition
of equilibrium strategies in the context of the portfolio management
problem,
with finite horizon, we characterize it by a system of partial
differential
equations, and we show existence in the case when the utility is CRRA
and the
terminal time T is small. We also investigate the infinite-horizon
case and we
give two different explicit solutions in the case when the utility is
CRRA (in
contrast with the case of exponential discount, where there is only
one). Some
of our results are proved under the assumption that the discount
function h(t)
is a linear combination of two exponentials, or is the product of an
exponential by a linear function.
http://arxiv.org/abs/0708.0588
Author(s): Souvik Ghosh and Gennady Samorodnitsky
Abstract: The large deviations of an infinite moving average process with
exponentially
light tails are very similar to those of an i.i.d. sequence as long
as the
coefficients decay fast enough. If they do not, the large deviations
change
dramatically. We study this phenomenon in the context of functional
large,
moderate and huge deviation principles.
http://arxiv.org/abs/0708.0865
Author(s): Pauline Barrieu (1) and Nicole El Karoui (2) ((1) and Statistics department,
London School of Economics, UK, (2) CMAP, Ecole Polytechnique,
France)
Abstract: The question of pricing and hedging a given contingent claim has a
unique
solution in a complete market framework. When some incompleteness is
introduced, the problem becomes however more difficult. Several
approaches have
been adopted in the literature to provide a satisfactory answer to this
problem, for a particular choice criterion. In this paper, in order
to price
and hedge a non-tradable contingent claim, we first start with a
(standard)
utility maximization problem and end up with an equivalent risk measure
minimization. This hedging problem can be seen as a particular case
of a more
general situation of risk transfer between different agents, one of them
consisting of the financial market. In order to provide constructive
answers to
this general optimal risk transfer problem, both static and dynamic
approaches
are considered. When considering a dynamic framework, our main
purpose is to
find a trade-off between static and very abstract risk measures as we
are more
interested in tractability issues and interpretations of the dynamic
risk
measures we obtain rather than the ultimate general results.
Therefore, after
introducing a general axiomatic approach to dynamic risk measures, we
relate
the dynamic version of convex risk measures to BSDEs.
http://arxiv.org/abs/0708.0948
Author(s): Haya Kaspi and Kavita Ramanan
Abstract: This work considers a many-server queueing system in which
customers with
i.i.d., generally distributed service times enter service in the
order of
arrival. The dynamics of the system is represented in terms of a
process that
describes the total number of customers in the system, as well as a
measure-valued process that keeps track of the ages of customers in
service.
Under mild assumptions on the service time distribution, as the
number of
servers goes to infinity, a law of large numbers (or fluid) limit is
established for this pair of processes. The limit is characterised as
the
unique solution to a coupled pair of integral equations, which admits
a fairly
explicit representation. As a corollary, the fluid limits of several
other
functionals of interest, such as the waiting time, are also obtained.
Furthermore, in the time-homogeneous setting, the fluid limit is
shown to
converge to its equilibrium. Along the way, some results of independent
interest are obtained, including a continuous mapping result and a
maximality
property of the fluid limit. A motivation for studying these systems
is that
they arise as models of computer data systems and call centers.
http://arxiv.org/abs/0708.0952
Author(s): Jan Obloj
Abstract: In this small note we use results derived in Berestycki et al. to
correct the
celebrated formulae of Hagan et al. We derive explicitly the correct
zero order
term in the expansion of the implied volatility in time to maturity.
The new
term is consistent as $\beta\to 1$. Furthermore, numerical
simulations show
that it reduces or eliminates known pathologies of the earlier formula.
http://arxiv.org/abs/0708.0998
Author(s): David Gamarnik and Dmitriy Katz
Abstract: One of the basic properties of a queueing network is stability.
Roughly
speaking it is the property that the total number of jobs in the network
remains bounded as a function of time. One of the key questions
related to the
stability issue is determining the exact conditions under which a given
queueing network operating under a given scheduling policy stable. While
initially there was a lot of progress in addressing this question,
most of the
obtained results were partial at best, and the complete
characterization of
stable queueing networks is lacking.
In this paper we resolve this important open problem, albeit in a
somewhat
unexpected way. We show that characterizing stable queueing networks
is an
algorithmically undecidable problem for the case of non-preemptive
static
buffer priority scheduling policies and deterministic interarrival
and service
times. Thus no constructive characterization of stable queueing networks
operating under this class of policies is possible. Our approach
builds on an
earlier related work and uses the so-called counter machine device as a
reduction tool.
http://arxiv.org/abs/0708.1034
Author(s): Enrico Priola and Jerzy Zabczyk
Abstract: We consider an Ornstein-Uhlenbeck process with values in R^n
driven by a
L\'evy process (Z_t) taking values in R^d with d possibly smaller
than n. The
L\'evy noise can have a degenerate or even vanishing Gaussian component.
Under a controllability condition and an assumption on the L\'evy
measure of
(Z_t), we prove that the law of the Ornstein-Uhlenbeck process at any
time t>0
has a density on R^n. Moreover, when the L\'evy process is of $\alpha
$-stable
type, $\alpha \in (0,2)$, we show that such density is a $C^{\infty}$-
function.
http://arxiv.org/abs/0708.1084
Author(s): Grace Lin and Yingdong Lu and David Yao
Abstract: The stochastic knapsack has been used as a model in wide ranging
applications
from dynamic resource allocation to admission control in
telecommunication. In
recent years, a variation of the model has become a basic tool in
studying
problems that arise in revenue management and dynamic/flexible
pricing; and it
is in this context that our study is undertaken. Based on a dynamic
programming
formulation and associated properties of the value function, we study
in this
paper a class of control that we call switch-over policies -- start from
accepting only orders of the highest price, and switch to including
lower
prices as time goes by, with the switch-over times optimally decided
via convex
programming. We establish the asymptotic optimality of the switch-
over policy,
and develop pricing models based on this policy to optimize the price
reductions over the decision horizon.
http://arxiv.org/abs/0708.1146
Author(s): Simone Scotti
Abstract: We present an application of error theory using Dirichlet Forms in
linear
partial differential equations (LPDE). We study the transmission of an
uncertainty on the terminal condition to the solution of the LPDE
thanks to the
decomposition of the solution on a wavelets basis. We analyze the basic
properties and a particular class of LPDE where the wavelets bases
show their
powerful, the combination of error theory and wavelets basis
justifies some
hypotheses, helpful to simplify the computation.
http://arxiv.org/abs/0708.1073
Author(s): Timothy C. Wallstrom
Abstract: It has recently been shown that the marginalization paradox (MP)
can be
resolved by interpreting improper inferences as probability limits.
The key to
the resolution is that probability limits need not satisfy the formal
Bayes'
law, which is used in the MP to deduce an inconsistency. In this
paper, I
explore the differences between probability limits and the more familiar
pointwise limits, which do imply the formal Bayes' law, and show how
these
differences underlie some key differences in the interpretation of
the MP.
http://arxiv.org/abs/0708.1350
Author(s): Nicola Cufaro Petroni
Abstract: We summarize the relations among three classes of laws: infinitely
divisible,
selfdecomposable and stable. First we look at them as the solutions
of the
Central Limit Problem; then their role is scrutinized in relation to
the Levy
and the additive processes with an emphasis on stationarity and
selfsimilarity.
Finally we analyze the Ornstein-Uhlenbeck processes driven by Levy
noises and
their selfdecomposable stationary distributions, and we end with a few
particular examples.
http://arxiv.org/abs/0708.1239
Author(s): Feng-Yu Wang
Abstract: By using coupling and Girsanov transformations, the dimension-free
Harnack
inequality and the strong Feller property are proved for transition
semigroups
of solutions to a class of stochastic generalized porous media
equations. As
applications, explicit upper bounds of the $L^p$-norm of the density
as well as
hypercontractivity, ultracontractivity and compactness of the
corresponding
semigroup are derived.
http://arxiv.org/abs/0708.1671
Author(s): Ron Doney and Ross Maller
Abstract: Let $R_n=\max_{0\leq j\leq n}S_j-S_n$ be a random walk $S_n$
reflected in its
maximum. Except in the trivial case when $P(X\ge0)=1$, $R_n$ will
pass over a
horizontal boundary of any height in a finite time, with probability
1. We
extend this by giving necessary and sufficient conditions for
finiteness of
passage times of $R_n$ above certain curved (power law) boundaries,
as well.
The intuition that a degree of heaviness of the negative tail of the
distribution of the increments of $S_n$ is necessary for passage of
$R_n$ above
a high level is correct in most, but not all, cases, as we show.
Conditions are
also given for the finiteness of the expected passage time of $R_n$
above
linear and square root boundaries.
http://arxiv.org/abs/0708.1676
Author(s): Hiroshi Kaneko and Anatoly N. Kochubei
Abstract: Study of stochastic differential equations on the field of p-adic
numbers was
initiated by the second author and has been developed by the first
author, who
proved several results for the p-adic case, similar to the theory of
ordinary
stochastic integral with respect to Levy processes on the Euclidean
spaces. In
this article, we present an improved definition of a stochastic
integral on the
field and prove the joint (time and space) continuity of the local
time for
p-adic stable processes. Then we use the method of random time change
to obtain
sufficient conditions for the existence of a weak solution of a
stochastic
differential equation on the field, driven by the p-adic stable
process, with a
Borel measurable coefficient.
http://arxiv.org/abs/0708.1706
Author(s): Ale\v{s} \v{C}ern\'y and Jan Kallsen
Abstract: We provide a new characterization of mean-variance hedging
strategies in a
general semimartingale market. The key point is the introduction of a
new
probability measure $P^{\star}$ which turns the dynamic asset allocation
problem into a myopic one. The minimal martingale measure relative to
$P^{\star}$ coincides with the variance-optimal martingale measure
relative to
the original probability measure $P$.
http://arxiv.org/abs/0708.1715
Author(s): Reuven Zeitak
Abstract: We introduce a new formalism for dealing with networks of queues. The
formalism is based on the Doi-Peliti second quantization method for
reaction
diffusion systems. As a demonstration of the method's utility we compute
perturbatively the different time busy-busy correlations between two
servers in
a Jackson network.
http://arxiv.org/abs/0708.1718
Author(s): Z. D. Bai and B. Q. Miao and G. M. Pan
Abstract: Let \{$X_{ij}$\}, $i,j=...,$ be a double array of i.i.d. complex
random
variables with $EX_{11}=0,E|X_{11}|^2=1$ and $E|X_{11}|^4<\infty$,
and let
$A_n=\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/
{2}}$ is
the square root of a nonnegative definite matrix $T_n$ and $X_n$ is the
$n\times N$ matrix of the upper-left corner of the double array. The
matrix
$A_n$ can be considered as a sample covariance matrix of an i.i.d.
sample from
a population with mean zero and covariance matrix $T_n$, or as a
multivariate
$F$ matrix if $T_n$ is the inverse of another sample covariance
matrix. To
investigate the limiting behavior of the eigenvectors of $A_n$, a new
form of
empirical spectral distribution is defined with weights defined by
eigenvectors
and it is then shown that this has the same limiting spectral
distribution as
the empirical spectral distribution defined by equal weights.
Moreover, if
\{$X_{ij}$\} and $T_n$ are either real or complex and some additional
moment
assumptions are made then linear spectral statistics defined by the
eigenvectors of $A_n$ are proved to have Gaussian limits, which
suggests that
the eigenvector matrix of $A_n$ is nearly Haar distributed when $T_n$
is a
multiple of the identity matrix, an easy consequence for a Wishart
matrix.
http://arxiv.org/abs/0708.1720
Author(s): Nina Gantert and Yuval Peres and Zhan Shi
Abstract: We consider a one-dimensional recurrent random walk in random
environment
(RWRE). We show that the - suitably centered - empirical
distributions of the
RWRE converge weakly to a certain limit law which describes the
stationary
distribution of a random walk in an infinite valley. The construction
of the
infinite valley goes back to Golosov. As a consequence, we show weak
convergence for both the maximal local time and the self-intersection
local
time of the RWRE and also determine the exact constant in the almost
sure upper
limit of the maximal local time.
http://arxiv.org/abs/0708.1739
Author(s): Aur\'elien Alfonsi (CERMICS) and Alexander Schied and Antje Schulz
Abstract: Following Obizhaeva and Wang (2005), we consider optimal execution
strategies
for block market orders placed in a limit order book (LOB). Our main
contribution is to allow for a general shape of the LOB defined via a
given
density function and thus to include the case of nonlinear price
impact of
market orders. In this setting, there are now two possibilities of
modeling the
resilience of the LOB after a large market order: the exponential
recovery of
the number of limit orders, i.e., of the volume of the LOB, or the
exponential
recovery of the bid-ask spread. We consider both situations and, in
each case,
derive explicit optimal execution strategies in discrete time.
Applying our
results to a block-shaped LOB, we obtain a new closed-form
representation for
the optimal strategy, which explicitly solves the recursive scheme
given in
Obizhaeva and Wang (2005). We also provide some evidence for the
robustness of
optimal strategies with respect to the choice of the shape function
and the
resilience-type.
http://arxiv.org/abs/0708.1756
Author(s): Steven N. Evans
Abstract: We consider the asymptotic behavior as $n \to \infty$ of the
spectra of
random matrices of the form \[\frac{1}{\sqrt{n-1}} \sum_{k=1}^{n-1} Z_
{nk}
\rho_n((k,k+1)),\] where for each $n$ the random variables $Z_{nk}$
are i.i.d.
standard Gaussian and the matrices $\rho_n((k,k+1))$ are obtained by
applying
an irreducible unitary representation $\rho_n$ of the symmetric group on
$\{1,2,...,n\}$ to the transposition $(k,k+1)$ that interchanges $k$
and $k+1$
(thus $\rho_n((k,k+1))$ is both unitary and self-adjoint, with all
eigenvalues
either +1 or -1). Irreducible representations of the symmetric group on
$\{1,2,...,n\}$ are indexed by partitions $\lambda_n$ of $n$. A
consequence of
the results we establish is that if $\lambda_{n,1} \ge \lambda_{n,2}
\ge ...
\ge 0$ is the partition of $n$ corresponding to $\rho_n$, $\mu_{n,1} \ge
\mu_{n,2} \ge ... \ge 0$ is the corresponding conjugate partition of
$n$ (that
is, the Young diagram of $\mu_n$ is the transpose of the Young
diagram of
$\lambda_n$), $\lim_{n \to \infty} \frac{\lambda_{n,i}}{n} = p_i$ for
each $i
\ge 1$, and $\lim_{n \to \infty} \frac{\mu_{n,j}}{n} = q_j$ for each
$j \ge 1$,
then the spectral measure of the resulting random matrix converges in
distribution to a random probability measure that is Gaussian with
mean $\theta
Z$ and variance $1 - \theta^2$, where $\theta$ is the constant $
\sum_i p_i^2 -
\sum_j q_j^2$ and $Z$ is a standard Gaussian random variable.
http://arxiv.org/abs/0708.1776
Author(s): Zhen-Qing Chen and Steffen Rohde
Abstract: We consider shape, size and regularity of the hulls of the chordal
Schramm-Loewner evolution driven by a symmetric alpha-stable process.
We obtain
derivative estimates, show that the complements of the hulls are Hoelder
domains, prove that the hulls have Hausdorff dimension 1, and show
that the
trace is right-continuous with left limits almost surely.
http://arxiv.org/abs/0708.1805
Author(s): Tom Lindstr{\o}m
Abstract: We present a random walk approximation to fractional Brownian
motion where
the increments of the fractional random walk are defined as a
weighted sum of
the past increments of a Bernoulli random walk.
http://arxiv.org/abs/0708.1905
Author(s): Matthieu Jonckheere (Centrum voor Wiskunde en Informatica) and Lasse
Leskel\"a (Eindhoven University of Technology)
Abstract: This paper studies multiclass loss systems with two layers of
servers, where
each server at the first layer is dedicated to a certain customer
class, while
the servers at the second layer can handle all customer classes. The
routing of
customers follows an overflow scheme, where arriving customers are
preferentially directed to the first layer. Stochastic comparison and
coupling
techniques are developed for studying how the system is affected by
packing of
customers, altered service rates, and altered server configurations.
This
analysis leads to easily computable upper and lower bounds for the
performance
of the system.
http://arxiv.org/abs/0708.1927
Author(s): Leonid Petrov
Abstract: The aim of the paper is to introduce a two-parameter family of
infinite-dimensional diffusion processes X(alpha,theta) related to
Pitman's
two-parameter Poisson-Dirichlet distributions PD(alpha,theta). The
diffusions
X(alpha,theta) are obtained in a scaling limit transition from
certain finite
Markov chains on partitions of natural numbers. The state space of
X(alpha,theta) is an infinite-dimensional simplex called the Kingman
simplex.
In the special case when parameter alpha vanishes, our finite Markov
chains are
similar to Moran-type model in population genetics, and our diffusion
processes
reduce to the infinitely-many-neutral-alleles diffusion model studied
by Ethier
and Kurtz (1981). Our main results extend those of Ethier and Kurtz
to the
two-parameter case and are as follows: The Poisson-Dirichlet
distribution
PD(alpha,theta) is a unique stationary distribution for the
corresponding
process X(alpha,theta); the process is ergodic and reversible; the
spectrum of
its generator is explicitly described. The general two-parameter case
seems to
fall outside the setting of models of population genetics, and our
approach
differs in some aspects from that of Ethier and Kurtz.
http://arxiv.org/abs/0708.1930
Author(s): B. Bollobas and O. Riordan
Abstract: Recently, Bollob\'as, Janson and Riordan introduced a very general
family of
random graph models, producing inhomogeneous random graphs with $
\Theta(n)$
edges. Roughly speaking, there is one model for each {\em kernel},
i.e., each
symmetric measurable function from $[0,1]^2$ to the non-negative reals,
although the details are much more complicated, to ensure the exact
inclusion
of many of the recent models for large-scale real-world networks.
A different connection between kernels and random graphs arises in
the recent
work of Borgs, Chayes, Lov\'asz, S\'os, Szegedy and Vesztergombi. They
introduced several natural metrics on dense graphs (graphs with $n$
vertices
and $\Theta(n^2)$ edges), showed that these metrics are equivalent,
and gave a
description of the completion of the space of all graphs with respect
to any of
these metrics in terms of {\em graphons}, which are essentially
kernels. One of
the most appealing aspects of this work is the message that sequences of
inhomogeneous quasi-random graphs are in a sense completely general: any
sequence of dense graphs contains such a subsequence. Alternatively,
their
results show that certain natural models of dense inhomogeneous
random graphs
(one for each kernel) cover the space of dense graphs: there is one
model for
each point of the completion, producing graphs that converge to this
point.
Our aim here is to investigate to what extent the results above
for dense
graphs can be generalized to graphs with $o(n^2)$ edges. Although
many of the
definitions extend in a simple way, the connections between the various
metrics, and between the metrics and random graph models, turn out to
be much
more complicated than in the dense case. We shall prove many partial
results,
and state even more conjectures and open problems.
http://arxiv.org/abs/0708.1919
Author(s): Enkelejd Hashorva
Abstract: Let (X,Y) be a bivariate elliptical random vector with associated
random
radius in the Gumbel max-domain of attraction. In this paper we
obtain a second
order asymptotic expansion of the joint survival probability P(X > x,
Y> y) for
x,y large. Further, based on the asymptotic bounds we discuss some
aspects of
the statistical modelling of joint survival probabilities and the
survival
conditional excess probability.
http://arxiv.org/abs/0708.1965
Author(s): A. Elices
Abstract: This paper presents a methodology to introduce time-dependent
parameters for
a wide family of models preserving their analytic tractability. This
family
includes hybrid models with stochastic volatility, stochastic
interest-rates,
jumps and their non-hybrid counterparts. The methodology is applied
to Heston's
model. A bootstrapping algorithm is presented for calibration. A case
study
works out the calibration of the time-dependent parameters to the
volatility
surface of the Eurostoxx 50 index. The methodology is also applied to
the
analytic valuation of forward start vanilla options driven by
Heston's model.
This result is used to explore the forward skew of the case study.
http://arxiv.org/abs/0708.2020
Author(s): Roberto Fern\'andez and Luiz Renato Fontes and E. Jord\~ao Neves
Abstract: We introduce jump processes in R^k, called density-profile
process, to model
biological signaling networks. They describe the macroscopic
evolution of
finite-size spin-flip models with k types of spins interacting through a
non-reversible Glauber dynamics. We focus on the the k-dimensional
empirical-magnetization vector in the thermodynamic limit, and prove
that,
within arbitrary finite time-intervals, its path converges almost
surely to a
deterministic trajectory determined by a first-order (non-linear)
differential
equation. As parameters of the spin-flip dynamics change, the associated
dynamical system may go through bifurcations, associated to phase
transitions
in the statistical mechanical setting. We present a simple example of
spin-flip
stochastic model leading to a dynamical system with Hopf and pitchfork
bifurcations; depending on the parameter values, the magnetization
random path
can either converge to a unique stable fixed point, converge to one
of a pair
of stable fixed points, or asymptotically evolve close to a
deterministic orbit
in R^k.
http://arxiv.org/abs/0708.2044
Author(s): Lyubov N. Positselskaya
Abstract: We study a nonzero-sum game of two players which is a
generalization of the
antagonistic noisy duel of discrete type. The game is considered from
the point
of view of various criterions of optimality. We prove existence of
epsilon-equilibrium situations and show that the epsilon-equilibrium
strategies
that we have found are epsilon-maxmin. Conditions under which the
equilibrium
plays are Pareto-optimal are given.
Keywords: noisy duel, payoff function, strategy, equilibrium
situation,
Pareto optimality, the value of a game.
http://arxiv.org/abs/0708.2023
Author(s): Jean Ren\'e Chazottes and Pierre Collet and Frank Redig
Abstract: We present a new approach to estimate the relaxation speed to
equilibrium of
interacting particle systems. It is based on concentration
inequalities and
coupling. We illustrate our approach in a variety of examples for
which we
obtain several new results with short and non technical proofs. These
examples
include the symmetric and asymmetric exclusion process and high-
temperature
spin-flip dynamics ("Glauber dynamics"). We also give a direct proof
of the
Poincar\'e inequality, based on coupling, in the context of one-
dimensional
Gibbs measures with possibly long-range potentials.
http://arxiv.org/abs/0708.2152
Author(s): Pierre Collet and Cristian Giardin\'a and Frank Redig
Abstract: We consider matching with shifts for Gibbsian sequences. We prove
that the
maximal overlap behaves as $c\log n$, where $c$ is explicitely
identified in
terms of the thermodynamic quantities (pressure) of the underlying
potential.
Our approach is based on the analysis of the first and second moment
of the
number of overlaps of a given size. We treat both the case of equal
sequences
(and non-zero shifts) and independent sequences.
http://arxiv.org/abs/0708.2165
Author(s): Nawaf Bou-Rabee and Houman Owhadi
Abstract: In this paper we introduce variational integrators for a class of
stochastic
mechanical systems driven by Wiener processes. The main result is to
derive
stochastic governing equations from a critical point of a stochastic
action.
With this result we derive Langevin-type equations for constrained
mechanical
systems, implement a stochastic analog of Lagrangian reduction, and
design
stochastic variational integrators.
http://arxiv.org/abs/0708.2187
Author(s): Hacene Belbachir (USTHB) and Sadek Bouroubi (USTHB) and Abdelkader Khelladi
(USTHB)
Abstract: Using an explicit computable expression of ordinary multinomials, we
establish three remarkable connections, with the q-generalized Fibonacci
sequence, the exponential partial Bell partition polynomials and the
density of
convolution powers of the discrete uniform distribution. Identities
and various
combinatorial relations are derived.
http://arxiv.org/abs/0708.2195
Author(s): Hacene Belbachir (USTHB)
Abstract: We establish the unimodality and the asymptotic strong unimodality
of the
ordinary multinomials and give their smallest mode leading to the
expression of
the maximal probability of convolution powers of the discrete uniform
distribution. We conclude giving the generating functions of the
sequence of
generalized ordinary multinomials and for an extension of the
sequence of
maximal probabilities for convolution power of discrete uniform
distribution.
http://arxiv.org/abs/0708.2341
Author(s): Vadim Gorin
Abstract: We compute the bulk limit of the correlation functions for the
uniform
measure on lozenge tilings of a hexagon. The limiting determinantal
process is
a translation invariant extension of the discrete sine process, which
also
describes the ergodic Gibbs measure of an appropriate slope.
http://arxiv.org/abs/0708.2349
Author(s): Loic Chaumont (LAREMA) and Andreas Kyprianou (UB) and Juan Carlos Pardo
Millan (PMA, UB)
Abstract: We consider some special classes of L\'evy processes with no gaussian
component whose L\'evy measure is of the type $\pi(dx)=e^{\gamma x}\nu
(e^x-1)
dx$, where $\nu$ is the density of the stable L\'evy measure and $
\gamma$ is a
positive parameter which depends on its characteristics. These
processes were
introduced in \cite{CC} as the underlying L\'evy processes in the
Lamperti
representation of conditioned stable L\'evy processes. In this paper, we
compute explicitly the law of these L\'evy processes at their first
exit time
from a finite or semi-finite interval, the law of their exponential
functional
and the first hitting time probability of a pair of points.
http://arxiv.org/abs/0708.2383
Author(s): Marcus Hutter and Andrej Muchnik
Abstract: Solomonoff's central result on induction is that the posterior of
a universal
semimeasure M converges rapidly and with probability 1 to the true
sequence
generating posterior mu, if the latter is computable. Hence, M is
eligible as a
universal sequence predictor in case of unknown mu. Despite some
nearby results
and proofs in the literature, the stronger result of convergence for all
(Martin-Loef) random sequences remained open. Such a convergence
result would
be particularly interesting and natural, since randomness can be
defined in
terms of M itself. We show that there are universal semimeasures M
which do not
converge for all random sequences, i.e. we give a partial negative
answer to
the open problem. We also provide a positive answer for some non-
universal
semimeasures. We define the incomputable measure D as a mixture over all
computable measures and the enumerable semimeasure W as a mixture
over all
enumerable nearly-measures. We show that W converges to D and D to mu
on all
random sequences. The Hellinger distance measuring closeness of two
distributions plays a central role.
http://arxiv.org/abs/0708.2319
Author(s): Sergio Albeverio and Olga Rozanova
Abstract: It is well known that the solutions to the non-viscous Burgers
equation
develop a gradient catastrophe at a critical time provided the
initial data
have a negative derivative in certain points. We consider this equation
assuming that the particle paths in the medium are governed by a
random process
with a variance which depends in a polynomial way on the velocity.
Given an
initial distribution of the particles which is uniform in space and
with the
initial velocity linearly depending on the position we show both
analytically
and numerically that there exists a threshold effect: if the power in
the above
variance is less or equal 1, then the noise does not influence the
solution
behavior, in the following sense: the conditional expectation of the
velocity
given the position goes to infinity outside the origin. If however
the power is
larger than 1, then this conditional expectation decays to zero as
the time
tends to a critical value.
http://arxiv.org/abs/0708.2320
Author(s): Dmitry Panchenko
Abstract: Talagrand's positivity principle states that one can slightly
perturb a
Hamiltonian in the Sherrington-Kirkpatrick model in such a way that
the overlap
of two configurations under the perturbed Gibbs' measure will become
typically
nonnegative. In this note we observe that abstracting from the
setting of the
SK model only improves the result and does not require any
modifications in
Talagrand's argument. In this version, for example, positivity principle
immediately applies to the setting of Aizenman-Sims-Starr
interpolation. Also,
abstracting from the SK model improves the conditions in the
Ghirlanda-Guerra
identities and as a consequence results in a perturbation of smaller
order
necessary to ensure positivity of the overlap.
http://arxiv.org/abs/0708.2453
Author(s): Nikolai Dokuchaev
Abstract: We study existence, uniqueness, and a priori estimates for
solutions for
backward parabolic Ito equations in domains with boundary. The proofs
are based
duality between forward and backward equations. This duality is used
also to
establish that backward parabolic equations have some causality (more
precisely, some anti-causality
http://arxiv.org/abs/0708.2497
Author(s): Vladimir Vovk
Abstract: This note makes the obvious observation that Hoeffding's original
proof of
his inequality remains valid in the game-theoretic framework. All
details are
spelled out for the convenience of future reference.
http://arxiv.org/abs/0708.2502
Author(s): Ori Gurel-Gurevich
Abstract: Pursuit-Evasion Games (in discrete time) are stochastic games with
nonnegative daily payoffs, with the final payoff being the cumulative
sum of
payoffs during the game. We show that such games admit a value even
in the
presence of incomplete information and that this value is uniform,
i.e. there
are epsilon-optimal strategies for both players that are epsilon-
optimal in any
long enough prefix of the game. We give an example to demonstrate that
nonnegativity is essential and expand the results to leavable games.
http://arxiv.org/abs/0708.2556
Author(s): Panki Kim and Renming Song and Zoran Vondracek
Abstract: We establish a boundary Harnack principle for a large class of
subordinate
Brownian motion, including mixtures of symmetric stable processes, in
bounded
$\kappa$-fat open set (disconnected analogue of John domains). As an
application of the boundary Harnack principle, we identify the Martin
boundary
and the minimal Martin boundary of bounded $\kappa$-fat open sets
with respect
to these processes with their Euclidean boundary.
http://arxiv.org/abs/0708.2583
Author(s): Timo Seppalainen
Abstract: This is a brief survey of laws of large numbers, fluctuation
results and
large deviation principles for asymmetric interacting particle
systems that
represent moving interfaces on the plane. We discuss the exclusion
process, the
Hammersley process and the related last-passage growth models.
http://arxiv.org/abs/0708.2721
Author(s): Lasse Leskel\"a
Abstract: Admission control can be employed to avoid congestion in queueing
networks
subject to overload. In distributed networks the admission decisions
are often
based on imperfect measurements on the network state. This paper
studies how
the lack of complete state information affects the system performance by
considering a simple network model for distributed admission control.
The
stability region of the network is characterized and it is shown how
feedback
signaling makes the system very sensitive to its parameters.
http://arxiv.org/abs/0708.2739
Author(s): Hock Peng Chan
Abstract: The extremal tail probabilities of moving sums in a marked Poisson
random
field is examined here. These sums are computed by adding up the
weighted
occurrences of events lying within a scanning set of fixed shape and
size.
Change of measure and analysis of local random fields are used to
provide tail
probabilities. The asymptotic constants are initially expressed in a
form that
seems hard to evaluate and do not seem to provide any additional
information on
the properties of the constants. A more sophisticated approach is then
undertaken giving rise to an expression that is not only neater but
also able
to provide computable bounds. The technique used to obtain this
constant can
also be modified to work on continuous processes.
http://arxiv.org/abs/0708.2764
Author(s): Dominic Schuhmacher and Aihua Xia
Abstract: Most metrics between finite point measures currently used in the
literature
have the flaw that they do not treat differing total masses in an
adequate
manner for applications. This paper introduces a new metric $\bar{d}_1
$ that
combines positional differences of points under a closest match with the
relative difference in total mass in a way that fixes this flaw. A
comprehensive collection of theoretical results about $\bar{d}_1$ and
its
induced Wasserstein metric $\bar{d}_2$ for point process
distributions are
given, including examples of useful $\bar{d}_1$-Lipschitz continuous
functions,
$\bar{d}_2$ upper bounds for Poisson process approximation, and $\bar
{d}_2$
upper and lower bounds between distributions of point processes of
i.i.d.
points. Furthermore, we present a statistical test for multiple point
pattern
data that demonstrates the potential of $\bar{d}_1$ in applications.
http://arxiv.org/abs/0708.2777
Author(s): Terence Tao and Van Vu
Abstract: Let $\a$ be a complex random variable with mean zero and bounded
variance
$\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with
entries being
i.i.d. copies of $\a$. Let $\lambda_{1}, ..., \lambda_{n}$ be the
eigenvalues
of $\frac{1}{\sigma \sqrt n}N_{n}$. Define the empirical spectral
distribution
$\mu_{n}$ of $N_{n}$ by the formula $$ \mu_n(s,t) := \frac{1}{n} # \
{k \leq n|
\Re(\lambda_k) \leq s; \Im(\lambda_k) \leq t \}.$$
The Circular law conjecture asserts that $\mu_{n}$ converges to
the uniform
distribution $\mu_\infty$ over the unit disk as $n$ tends to infinity.
We prove this conjecture under the slightly stronger assumption
that the
$(2+\eta)\th$-moment of $\a$ is bounded, for any $\eta >0$. Our
method builds
and improves upon earlier work of Girko, Bai, G\"otze-Tikhomirov, and
Pan-Zhou,
and also applies for sparse random matrices.
The new key ingredient in the paper is a general result about the
least
singular value of random matrices, which was obtained using tools and
ideas
from additive combinatorics.
http://arxiv.org/abs/0708.2895
Author(s): Akira Sakai
Abstract: We provide a complete proof of the diagrammatic bounds on the lace-
expansion
coefficients for oriented percolation, which are used in [arXiv:math/
0703455]
to investigate critical behavior for long-range oriented percolation
above
2\min{\alpha,2} spatial dimensions.
http://arxiv.org/abs/0708.2897
Author(s): Bernard Shiffman
Abstract: We show that the zeros of random sequences of Gaussian systems of
polynomials
of increasing degree almost surely converge to the expected limit
distribution
under very general hypotheses. In particular, the normalized
distribution of
zeros of systems of m polynomials of degree N, orthonormalized on a
regular
compact subset K of C^m, almost surely converge to the equilibrium
measure on K
as the degree N goes to infinity.
http://arxiv.org/abs/0708.2754
Author(s): Michael Farber
Abstract: Betti numbers of configuration spaces of mechanical linkages
(known also as
polygon spaces) depend on a large number of parameters -- the lengths
of the
bars of the linkage. Motivated by applications in topological robotics,
statistical shape theory and molecular biology, we view these lengths
as random
variables and study asymptotic values of the average Betti numbers as
the
number of links n tends to infinity. We establish a surprising fact
that for a
reasonably ample class of sequences of probability measures the
asymptotic
values of the average Betti numbers are independent of the choice of the
measure. The main results of the paper apply to planar linkages as
well as for
linkages in R^3. We also prove results about higher moments of Betti
numbers.
http://arxiv.org/abs/0708.2997
Author(s): Gabriela Gruninger and Wolfgang Konig
Abstract: We consider the parabolic Anderson model, the Cauchy problem for
the heat
equation with random potential in $Z^d$. We use i.i.d. potentials $
\xi: Z^d \to
\R$ in the third universality class, namely the class of almost bounded
potentials, in the classification of van der Hofstad, Konig and Morters
[HKM06]. This class consists of potentials whose logarithmic moment
generating
function is regularly varying with parameter $\gamma=1$, but do not
belong to
the class of so-called double-exponentially distributed potentials
studied by
Gartner and Molchanov (PTRF 1998).
In [HKM06] the asymptotics of the expected total mass was
identified in terms
of a variational problem that is closely connected to the well-known
logarithmic Sobolev inequality and whose solution, unique up to
spatial shifts,
is a perfect parabola. In the present paper we show that those
potentials whose
shape (after appropriate vertical shifting and spatial rescaling) is
away from
that parabola contribute only negligibly to the total mass. The
topology used
is the strong $L^1$-topology on compacts for the exponentials of the
potential.
In the course of the proof, we show that any sequence of approximate
minimisers
of the above variational formula approaches some spatial shift of the
minimiser, the parabola.
http://arxiv.org/abs/0708.3207
Author(s): Miklos Bona
Abstract: We study Stirling permutations defined by Gessel and Stanley in
\cite{stangess}. We prove that their generating function according to
the
number of descents has real roots only. We use that fact to prove
that the
distribution of these descents, and other, equidistributed statistics
on these
objects converge to a normal distribution.
http://arxiv.org/abs/0708.3223
Author(s): Ali Devin Sezer
Abstract: Importance sampling (IS) is a variance reduction method for
simulating rare
events. A recent paper by Dupuis, Wang and Sezer (Ann. App. Probab. 17
(4):1306-
1346, 2007) exploits connections between IS and stochastic games and
optimal
control problems to show how to design and analyze simple and
efficient IS
algorithms for various overflow events for tandem Jackson networks.
The present
paper uses the same approach to build asymptotically optimal IS
schemes for
stable open Jackson networks with a tree topology. Customers arrive
at the
single root of the tree. The rare overflow event we consider is the
following:
given that initially the network is empty, the system experiences a
buffer
overflow before returning to the empty state. Two types of buffer
structures
are considered: 1) A single system-wide buffer of size $n$ shared by
all nodes,
2) each node $i$ has its own buffer of size $\beta_i n$, $\beta_i \in
(0,1)$.
http://arxiv.org/abs/0708.3260
Author(s): M. Hairer
Abstract: We study a fairly general class of time-homogeneous stochastic
evolutions
driven by noises that are not white in time. As a consequence, the
resulting
processes do not have the Markov property. In this setting, we obtain
constructive criteria for the uniqueness of stationary solutions that
are very
close in spirit to the existing criteria for Markov processes.
In the case of discrete time, where the driving noise consists of a
stationary sequence of Gaussian random variables, we give optimal
conditions on
the spectral measure for our criteria to be applicable. In
particular, we show
that under a certain assumption on the spectral density, our
assumptions can be
checked in virtually the same way as one would check that the Markov
process
obtained by replacing the driving sequence by a sequence of independent
identically distributed Gaussian random variables is strong Feller and
topologically irreducible. The results of the present article are
based on
those obtained previously in the continuous time context of
diffusions driven
by fractional Brownian motion.
http://arxiv.org/abs/0708.3338
Author(s): Andras Balint and Federico Camia and Ronald Meester
Abstract: Consider subcritical Bernoulli bond percolation with fixed
parameter p
http://arxiv.org/abs/0708.3349
Author(s): Ramon van Handel
Abstract: This paper develops a connection between the asymptotic stability of
nonlinear filters and a notion of observability. We consider a
general class of
hidden Markov models in continuous time with compact signal state
space, and
call such a model observable if no two initial measures of the signal
process
give rise to the same law of the observation process. We demonstrate
that
observability implies stability of the filter, i.e., the filtered
estimates
become insensitive to the initial measure at large times. For the
special case
where the signal is a finite-state Markov process and the
observations are of
the white noise type, a complete (necessary and sufficient)
characterization of
filter stability is obtained in terms of a slightly weaker detectability
condition. In addition to observability, the role of controllability
in filter
stability is explored. Finally, the results are partially extended to
non-compact signal state spaces.
http://arxiv.org/abs/0708.3412
Author(s): Hassan Allouba
Abstract: Discretizing space and leaving time continuous, we view our recently
introduced fourth order space-time white noise driven BTP SPDE
through the eyes
of their associated SDDEs (stochastic differential-difference
equations) on
$d$-dimensional spatial lattices. BTP SPDEs are stochastic equations
in which
the fourth order PDE part is solved by running our recently introduced
Brownian-time processes (BTPs). To extend our SDDEs approach from
second order
SPDEs (see \cite{Adis,Asdde1,Asdde2}) to the fourth order BTP SPDEs, we
introduce Brownian-time random walk (BTRW). We then formulate the
associated
SDDEs in terms of the density function of a BTRW on the spatial
lattices. BTRWs
are random walks whose clock $t$ is replaced with the Brownian clock $
\lab
B_t\rab$, where $B$ is a Brownian motion. They are special cases of a
new class
of discrete-valued processes--which we introduce here as well--that
we call
Brownian-time chains (BTCs). BTCs are the discretized versions of
BTPs. In this
article, the density of BTRW plays the role of the Green function
(Brownian
motion density) for reaction diffusion (RD) and other second order
equations.
The system of BTRW SDDEs is a system of interacting diffusions
representing the
microscopic spatial behavior of the BTP SPDE. After defining the
notion of BTRW
SDDEs limits solutions to BTP SPDEs; we prove--among other things--the
existence of such a solution to these equations in spatial dimensions
$1\le
d\le3$, under less-than-Lipschitz conditions on the diffusion
coefficient. The
paths of such a solution are H\"older continuous with a dimension-
dependent
exponent $\gamma\in\lpa 0,\tf{4-d}{8}\rpa$, $d=1,2,3$. This is in sharp
contrast to RD SPDEs driven by space-time white noise, in which real-
valued
SDDEs weak limits solutions exist only in $d=1$.
http://arxiv.org/abs/0708.3419
Author(s): Charles Cuthbertson and Alison Etheridge and Feng Yu
Abstract: We consider the accumulation of beneficial and deleterious
mutations in large
asexual populations. The rate of adaptation is affected by the total
mutation
rate, proportion of beneficial mutations, and population size $N$. We
show that
regardless of mutation rates, as long as the proportion of beneficial
mutations
is strictly positive, the adaptation rate is at least $\Ocal(\log^{1-
\delta}
N)$, if the population size is sufficiently large. This shows that if
the
genome is modelled as continuous, there is no limit to natural
selection.
http://arxiv.org/abs/0708.3453
Author(s): A. N. Downes and K. Borovkov
Abstract: We consider the boundary crossing problem for time-homogeneous
diffusions and
general curvilinear boundaries. Bounds are derived for the
approximation error
of the one-sided (upper) boundary crossing probability when replacing
the
original boundary by a different one. In doing so we establish the
existence of
the first-passage time density and provide an upper bound for this
function. In
the case of processes with diffusion interval equal to whole real
line this is
extended to a lower bound, as well as bounds for the first crossing
time of a
lower boundary. An extension to some time-inhomogeneous diffusions is
given.
These results are illustrated by numerical examples.
http://arxiv.org/abs/0708.3562
Author(s): Akihiko Inoue and Vo Van Anh
Abstract: We introduce a class of Gaussian processes with stationary
increments which
exhibit long-range dependence. The class includes fractional Brownian
motion
with Hurst parameter $H>1/2$ as a typical example. We establish
infinite and
finite past prediction formulas for the processes in which the predictor
coefficients are given explicitly in terms of the MA$(\infty)$ and AR$
(\infty)$
coefficients. We apply the formulas to prove an analogue of Baxter's
inequality, which concerns the $L^{1}$-estimate of the difference
between the
finite and infinite past predictor coefficients.
http://arxiv.org/abs/0708.3631
Author(s): Dmitry Panchenko and Michel Talagrand
Abstract: New results about Poisson-Dirichlet point processes and Derrida-
Ruelle
cascades allow us to express Guerra's interpolation entirely in the
language of
Derrida-Ruelle cascades and to streamline Guerra's computations.
Moreover, our
approach clarifies the nature of the error terms along the
interpolation.
http://arxiv.org/abs/0708.3641
Author(s): Serguei Foss and Takis Konstantopoulos
Abstract: The standard functional central limit theorem for a renewal
process with
finite mean and variance, results in a Brownian motion limit. This
note shows
how to obtain a Brownian bridge process by a direct procedure that
does not
involve conditioning. Several examples are also considered.
http://arxiv.org/abs/0708.3667
Author(s): V.I. Bogachev and M.N. Lukintsova
Abstract: Two classes of topological spaces are introduced on which every
probability
Radon measure possesses a uniformly distributed sequence or a
uniformly tight
uniformly distributed sequence. It is shown that these classes are
stable under
multiplication by completely regular Souslin spaces
http://arxiv.org/abs/0708.3486
Author(s): Thomas Cass and Peter Friz
Abstract: We consider stochastic differential equations dY=V(Y)dX driven by a
multidimensional Gaussian process X in the rough path sense. Using
Malliavin
Calculus we show that Y(t) admits a density for t in (0,T] provided
(i) the
vector fields V=(V_1,...,V_d) satisfy Hoermander's condition and (ii)
the
Gaussian driving signal X satisfies certain conditions. Examples of
driving
signals include fractional Brownian motion with Hurst parameter
H>1/4, the
Brownian Bridge returning to zero after time T and the Ornstein-
Uhlenbeck
process.
http://arxiv.org/abs/0708.3730
Author(s): Lorenz Gilch
Abstract: Suppose we are given the free product $V$ of a finite family of
finite or
countable sets $(V_i)_{i\in\mathcal{I}}$ and probability measures on
each
$V_i$, which govern random walks on it. We consider a transient
random walk on
the free product arising naturally from the random walks on the $V_i
$. We prove
the existence of the rate of escape with respect to the block length,
that is,
the speed, at which the random walk escapes to infinity, and
furthermore we
compute formulas for it. For this purpose, we present three different
techniques providing three different, equivalent formulas.
http://arxiv.org/abs/0708.3763
Author(s): Lorenz Gilch
Abstract: Suppose we are given a homogeneous tree $\mathcal{T}_q$ of degree
$q\geq 3$,
where at each vertex sits a lamp, which can be switched on or off. This
structure can be described by the wreath product $(\mathbb{Z}/2)\wr
\Gamma$,
where $\Gamma=\ast_{i=1}^q \mathbb{Z}/2$ is the free product group of
$q$
factors $\mathbb{Z}/2$. We consider a transient random walk on a
Cayley graph
of $(\mathbb{Z}/2)\wr \Gamma$, for which we want to compute lower and
upper
bounds for the rate of escape, that is, the speed at which the random
walk
flees to infinity.
http://arxiv.org/abs/0708.3766
Author(s): Lorenz Gilch
Abstract: Suppose we are given an infinite, finitely generated group $G$ and a
transient random walk with bounded range on the wreath product $(\Z/ 2
\Z)\wr
G$, such that its projection on $G$ is transient. This random walk
can be
interpreted as a lamplighter random walk, where there is a lamp at
each element
of $G$, which can be switched on and off, and a lamplighter walks
along $G$ and
switches lamps randomly on and off. Our aim is to show that the
lamplighter
random walk escapes with respect to a suitable (pseudo-)metric on the
wreath
product faster to infinity than its projection onto $G$. For this
purpose, we
show that the asymptotic linear rate of burning lamps is non-zero,
providing an
acceleration of the lamplighter. If lamp switches are not charged by the
pseudo-metric and if $G\neq \Z$, we prove that the rate of escape
with respect
to the pseudo-metric, which becomes the length of a shortest
``travelling
salesman tour'', is strictly bigger than the rate of escape of the
lamplighter
random walk's projection on $G$. We prove the same for non-degenerate
cases if
$G=\Z$. Furthermore, we prove for $G$ having infinitely many ends the
acceleration with respect to a Markovian distance, which arises from
probabilities on $(\Z/ 2\Z)\wr G$ and the metric on $G$.
http://arxiv.org/abs/0708.3767
Author(s): Yukio Kasahara and Mohsen Pourahmadi and Akihiko Inoue
Abstract: Some of the most important results in prediction theory and time
series
analysis when finitely many values are removed from or added to its
infinite
past have been obtained using difficult and diverse techniques
ranging from
duality in Hilbert spaces of analytic functions (Nakazi, 1984) to linear
regression in statistics (Box and Tiao, 1975). We unify these results
via a
finite-dimensional duality lemma and elementary ideas from the linear
algebra.
The approach reveals the inherent finite-dimensional character of many
difficult prediction problems, the role of duality and
biorthogonality for a
finite set of random variables. The lemma is particularly useful when
the
number of missing values is small, like one or two, as in the case of
Kolmogorov and Nakazi prediction problems. The stationarity of the
underlying
process is not a requirement. It opens up the possibility of
extending such
results to nonstationary processes.
http://arxiv.org/abs/0708.3895
Author(s): Vincent Beffara (UMPA-Ensl)
Abstract: The aim of this paper is to explore possible ways of extending
Smirnov's
proof of Cardy's formula for critical site-percolation on the triangular
lattice to other cases (such as bond-percolation on the square
lattice); the
main question we address is that of the choice of the lattice
embedding into
the plane which gives rise to conformal invariance in the scaling
limit. Even
though we were not able to produce a complete proof, we believe that
the ideas
presented here go in the right direction.
http://arxiv.org/abs/0708.3908
Author(s): Lancelot F. James and Bernard Roynette and Marc Yor
Abstract: I. In Section 1, we present a number of classical results
concerning the
Generalized Gamma Convolution (: GGC) variables, their Wiener-Gamma
representations, and relation with the Dirichlet processes.
II. To a GGC variable, one may associate a unique Thorin measure.
Let $G$ a
positive r.v. and $\Gamma_{t} (G)$ \big(resp. $\Gamma_{t} (1/G)\big)$
the
Generalized Gamma Convolution with Thorin measure $t$-times the law
of $G$
(resp. the law of 1/G). In Section 2, we compare the laws of $\Gamma_
{t} (G)$
and $\Gamma_{t} (1/G)$.
III. In Section 3, we present some old and some new examples of GGC
variables, among which the lengths of excursions of Bessel processes
straddling
an independent exponential time.
http://arxiv.org/abs/0708.3932
Author(s): Tom Britton and Maria Deijfen and Andreas Nordvall Lager{\aa}s and Mathias
Lindholm
Abstract: In this paper, a branching process approximation for the spread of a
Reed-Frost epidemic on a network with tunable clustering is derived. The
approximation gives rise to expressions for the epidemic threshold
and the
probability of a large outbreak in the epidemic. It is investigated
how these
quantities varies with the clustering in the graph and it turns out for
instance that, as the clustering increases, the epidemic threshold
decreases.
The network is modelled by a random intersection graph, in which
individuals
are independently members of a number of groups and two individuals
are linked
to each other if and only if they share at least one group.
http://arxiv.org/abs/0708.3939
Author(s): Paul Balister and Bela Bollobas and Amites Sarkar and Mark Walters
Abstract: Let P be a Poisson process of intensity one in a square S_n of
area n. For a
fixed integer k, join every point of P to its k nearest neighbours,
creating an
undirected random geometric graph G_{n,k}. We prove that there exists a
critical constant c such that for c'c G_
{n,c'\log n} is
connected with probability tending to 1 as n tends to infinity. This
answers a
question previously posed by the authors.
http://arxiv.org/abs/0708.4007
Author(s): Yingdong Lu and Ana Radovanovi\'c
Abstract: The analysis of stochastic loss networks has long been of interest in
computer and communications networks and is becoming important in the
areas of
service and information systems. In traditional settings, computing
the well
known Erlang formula for blocking probability in these systems becomes
intractable for larger resource capacities. Using compound point
processes to
capture stochastic variability in the request process, we generalize
existing
models in this framework and derive simple asymptotic expressions for
blocking
probabilities. In addition, we extend our model to incorporate reserving
resources in advance. Although asymptotic, our experiments show an
excellent
match between derived formulas and simulation results even for
relatively small
resource capacities and relatively large values of blocking
probabilities.
http://arxiv.org/abs/0708.4059
Author(s): M. Mania and R. Tevzadze and T. Toronjadze
Abstract: We consider the mean-variance hedging problem under partial
information in
the case where the flow of observable events does not contain the full
information on the underlying asset price process. We introduce a
martingale
equation of a new type and characterize the optimal strategy in terms
of the
solution of this equation. We give relations between this equation
and backward
stochastic differential equations for the value process of the problem.
http://arxiv.org/abs/0708.4095
Author(s): Pierre Andreoletti (MAPMO)
Abstract: We consider an infinite system of particles in one dimension, each
particle
performs independant Sinai's random walk in random environment.
Considering an
instant $t$, large enough, we prove a result in probability showing
that the
particles are trapped in the neighborhood of well defined points of
the lattice
depending on the random environment the time $t$ and the starting
point of the
particles.
http://arxiv.org/abs/0708.4156
Author(s): Ou Zhao and Michael Woodroofe
Abstract: Consider additive functionals of a Markov chain $W_k$, with
stationary
(marginal) distribution and transition function denoted by $\pi$ and
$Q$, say
$S_n = g(W_1)+...+g(W_n)$, where $g$ is square integrable and has
mean 0 with
respect to $\pi$. If $S_n$ has the form $S_n = M_n+R_n$, where $M_n$
is a
square integrable martingale with stationary increments and $E(R_n^
{2}) =
o(n)$, then $g$ is said to admit a martingale approximation.
Necessary and
sufficient conditions for such an approximation are developed. Let
$Q^*$ denote
the adjoint operator to $Q$, regarded as a linear operator from $L^2
(\pi)$ into
itself, and consider co-isometries ($QQ^{*} = I$), an important
special case
that includes shift processes. In one main result a convenient
orthonormal
basis for $L_0^{2}(\pi)$ is identified along with a simple necessary and
sufficient condition for the existence of a martingale approximation
in terms
of the coefficients of the expansion of $g$ with respect to this
basis. Two
obvious necessary conditions for a martingale approximation are
$E[E(S_n|W_1)^2] = o(n)$ and $\lim_{n\to\infty} E(S_n^{2})/n < \infty$.
Assuming the first of these, let $\Vert g\Vert^2_{+} = \limsup_{n\to
\infty}
E(S_n^{2})/n$. Then $\Vert\cdot\Vert_{+}$ defines a pseudo norm on
the subspace
of $L^2(\pi)$ where it is finite. In another main result, a simple
necessary
and sufficient condition for a martingale approximation is developed
in terms
of $\Vert\cdot\Vert_{+}$.
http://arxiv.org/abs/0708.4183
Author(s): James Allen Fill
Abstract: An (upward) skip-free Markov chain with the set of nonnegative
integers as
state space is a chain for which upward jumps may be only of unit
size; there
is no restriction on downward jumps. In a 1987 paper, Brown and Shao
determined, for an irreducible continuous-time skip-free chain and
any d, the
passage time distribution from state 0 to state d. When the nonzero
eigenvalues
nu_j of the generator are all real, their result states that the
passage time
is distributed as the sum of d independent exponential random
variables with
rates nu_j. We give another proof of their theorem. In the case of
birth-and-death chains, our proof leads to an explicit representation
of the
passage time as a sum of independent exponential random variables.
Diaconis and
Miclo recently obtained the first such representation, but our
construction is
much simpler.
We obtain similar (and new) results for a fastest strong
stationary time T of
an ergodic continuous-time skip-free chain with stochastically monotone
time-reversal started in state 0, and we also obtain discrete-time
analogs of
all our results.
http://arxiv.org/abs/0708.4258
Author(s): Yuval Peres and Oded Schramm and Jeffrey E. Steif
Abstract: In dynamical percolation, the status of every bond is refreshed
according to
an independent Poisson clock. For graphs which do not percolate at
criticality,
the dynamical sensitivity of this property was analyzed extensively
in the last
decade. Here we focus on graphs which percolate at criticality, and
investigate
the dynamical sensitivity of the infinite cluster. We first give two
examples
of bounded degree graphs, one which percolates for all times at
criticality and
one which has exceptional times of nonpercolation. We then make a nearly
complete analysis of this question for spherically symmetric trees with
spherically symmetric edge probabilities bounded away from 0 and 1. One
interesting regime occurs when the expected number of vertices at the
nth level
that connect to the root at a fixed time is of order n(\log n)^
\alpha. R. Lyons
(1990) showed that at a fixed time, there is an infinite cluster a.s.
if and
only if \alpha >1. We prove that the probability that there is an
infinite
cluster at all times is 1 if \alpha > 2, while this probability is 0 if
1<\alpha \le 2. Within the regime where a.s. there is an infinite
cluster at
all times, there is yet another type of ``phase transition'' in the
behavior of
the process: if the expected number of vertices at the nth level
connecting to
the root at a fixed time is of order n^\theta with \theta > 2, then
the number
of connected components of the set of times in [0,1] at which the
root does not
percolate is finite a.s., while if 1<\theta < 2, then the number of such
components is infinite with positive probability.
http://arxiv.org/abs/0708.4287
Author(s): Lancelot F. James
Abstract: This paper explores large sample properties of the two parameter
$(\alpha,\theta)$ Poisson-Dirichlet Process in two contexts. In a
Bayesian
context of estimating an unknown probability measure, viewing this
process as a
natural extension of the Dirichlet process, we explore the
consistency and weak
convergence of the the two parameter Poisson Dirichlet posterior
process. We
also establish the weak convergence of properly centered two
parameter Poisson
Dirichlet processes for large $\theta+n\alpha.$ This latter result
complements
large $\theta$ results for the Dirichlet process and Poisson Dirichlet
sequences, and complements a recent result on large deviation
principles for
the two parameter Poisson Dirichlet process. A crucial component of
our results
is the use of distributional identities that may be useful in other
contexts.
http://arxiv.org/abs/0708.4294
Author(s): Jacob van den Berg and Antal A. J\'arai and B\'alint V\'agv\"olgyi
Abstract: We consider invasion percolation on the square lattice. It has
been proved by
van den Berg, Peres, Sidoravicius and Vares, that the probability
that the
radius of a so-called pond is larger than n, differs at most a factor
of order
log n from the probability that in critical Bernoulli percolation the
radius of
an open cluster is larger than n. We show that these two
probabilities are, in
fact, of the same order. Moreover, we prove an analogous result for
the volume
of a pond.
http://arxiv.org/abs/0708.4369
Author(s): Svante Janson
Abstract: It is shown that in a subcritical random graph with given vertex
degrees
satifying a power law degree distribution with exponent \gamma>3, the
largest
component is of order n^{1/(\gamma-1)}. More precisely, the order of the
largest component is approximatively given by a simple constant times
the
largest vertex degree. These results are extended to several other
random graph
models with power law degree distributions. This proves a conjecture by
Durrett.
http://arxiv.org/abs/0708.4404
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