|
|
|
|
 |
|
|
|
|
|
|
|
Probability Abstracts 114
This document contains abstracts 9660-9932
from Jan-1-2010 to Feb-28-2010.
They have been mailed on Mar 1st, 2010.
Author(s): Thomas Simon (LPP)
Abstract: It is known that real Non-Gaussian stable distributions are unimodal, not
additive strongly unimodal, and multiplicative strongly unimodal in the
symmetric case. By a theorem of Cuculescu-Theodorescu, the only remaining
relevant situation for the multiplicative strong unimodality of stable laws is
the one-sided. In this paper, we show that positive $\alpha-$stable laws are
multiplicative strongly unimodal iff $\alpha\le 1/2.$
http://arxiv.org/abs/1002.4977
Author(s): Jean-Christophe Breton and Clement Dombry
Abstract: We consider a generalization of the weighted random ball model. The model is
driven by a random Poisson measure with a product heavy tailed intensity
measure. Such a model typically represents the transmission of a network of
stations with a fading effect. In a previous article, the authors proved the
convergence of the finite-dimensional distributions of related generalized
random fields under various scalings and in the particular case when the fading
function is the indicator function of the unit ball. In this paper, tightness
and functional convergence are investigated. Using suitable moment estimates,
we prove functional convergences for some parametric classes of configurations
under the so-called large ball scaling and intermediate ball scaling.
Convergence in the space of distributions is also discussed.
http://arxiv.org/abs/1002.4985
Author(s): Augusto Teixeira
Abstract: In this paper we establish some properties of percolation for the vacant set
of random interlacements, for d at least 5 and small intensity u. The model of
random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560.
It is known that, for small u, almost surely there is a unique infinite
connected component in the vacant set left by the random interlacements at
level u, see arXiv:0808.3344 and arXiv:0805.4106. We estimate here the
distribution of the diameter and the volume of the vacant component at level u
containing the origin, given that it is finite. This comes as a by-product of
our main theorem, which proves a stretched exponential bound on the probability
that the interlacement set separates two macroscopic connected sets in a large
cube. As another application, we show that with high probability, the unique
infinite connected component of the vacant set is `ubiquitous' in large
neighborhoods of the origin.
http://arxiv.org/abs/1002.4995
Author(s): Mindaugas Bloznelis
Abstract: Given b>0, integers n, m=bn and a probability measure Q on {0, 1,..., m},
consider the random intersection graph on the vertex set [n]={1, ..., n}, where
i and j are declared adjacent whenever S(i) and S(j) intersect. Here S(1), ...,
S(n) denote iid random subsets of [m] such that P(|S(i)|=k)=Q(k). For sparse
random intersection graphs we establish a first order asymptotic for the order
of the largest connected component N=n(1-Q(0))g+o(n) in probability. Here g is
an average of nonextinction probabilities of a related multi-type Poisson
branching process.
http://arxiv.org/abs/1002.4649
Author(s): Hubert Lacoin
Abstract: This paper presents a very simple and self-contained proof of disorder
irrelevance for inhomogeneous pinning models with return exponent \alpha\in
(0,1/2). We also give a new upper bound for the contact fraction of the
disordered model at criticality.
http://arxiv.org/abs/1002.4753
Author(s): Vassili Kolokoltsov
Abstract: Stochastic monotonicity and the related duality are well studied for
one-dimensional diffusions and discrete Markov chains. In this note we extend
the theory to arbitrary one-dimensional Markov Feller processes. This seems to
be relevant in connection with the recent increase of interest to the analysis
of general processes containing jumps, in particular in financial mathematics.
http://arxiv.org/abs/1002.4773
Author(s): Mikko S. Pakkanen
Abstract: In this note, we study the infinite-dimensional conditional laws of Brownian
semistationary processes. Motivated by the fact that these processes are
typically not semimartingales, we present sufficient conditions ensuring that a
Brownian semistationary process has conditional full support, a property
introduced by Guasoni, R\'asonyi, and Schachermayer [Ann. Appl. Probab., 18
(2008) pp. 491--520]. By the results of Guasoni, R\'asonyi, and Schachermayer,
this property has two important implications. It ensures, firstly, that the
process admits no free lunches under proportional transaction costs, and
secondly, that it can be approximated pathwise (in the sup norm) by
semimartingales that admit equivalent martingale measures.
http://arxiv.org/abs/1002.4774
Author(s): Frank Redig and Feijia Wang
Abstract: We study single-site stochastic and deterministic transforma- tions of
one-dimensional Gibbs measures in the uniqueness regime with infinite-range
interactions. We prove conservation of Gibbsianness and give quantitative
estimates on the decay of the transformed potential. As examples, we consider
exponentially decaying potentials, and potentials decaying as a power-law.
http://arxiv.org/abs/1002.4796
Author(s): Enza Orlandi and Eva Loecherbach
Abstract: We consider Markov Random Fields defined by finite-region conditional
probabilities depending on a neighborhood of the region which changes with the
boundary conditions. The formal definition of these models requires partitions
of the set of configurations according to their projections on finite
neighborhoods of each lattice site. Each of these projections is called a
context for the site.
This framework is a natural extension, to d-dimensional fields, of the notion
of variable-length Markov chains introduced by Rissanen (1983) in his classical
paper. We define an algorithm to estimate the radius of the smallest ball
containing the context based on a realization of the field. We prove the
consistency of this estimator when the Dobrushin uniqueness condition for the
one point conditional probabilities holds. Our proofs are constructive and
yield explicit upper bounds for the probability of wrong estimation of the
radius of the context.
http://arxiv.org/abs/1002.4850
Author(s): Ehud Hrushovski and Anand Pillay and Pierre Simon
Abstract: We study stable like behaviour in first order theories without the
independence property. We introduce generically stable measures, give
characterizatiions, and show their ubiquity. We also introduce generic compact
domination. We also prove the approximate definability of arbitrary Borel
probability measures on definable sets in the real and p-adic fields.
http://arxiv.org/abs/1002.4763
Author(s): E. Di Nardo and P. Petrullo and D. Senato
Abstract: We provide an unifying polynomial expression giving moments in terms of
cumulants, and viceversa, holding in the classical, boolean and free setting.
This is done by using a symbolic treatment of Abel polynomials. As a
by-product, we show that in the free cumulant theory the volume polynomial of
Pitman and Stanley plays the role of the complete Bell exponential polynomial
in the classical theory. Moreover via generalized Abel polynomials we construct
a new class of cumulants, including the classical, boolean and free ones, and
the convolutions linearized by them. Finally, via an umbral Fourier transform,
we state a explicit connection between boolean and free convolution.
http://arxiv.org/abs/1002.4803
Author(s): Nikita Alexeev (Saint-Petersburg State University and Russia) and Friedrich G\"otze (University of Bielefeld, Germany), and Alexander Tikhomirov
(Syktyvkar State University, Russia)
Abstract: Let $x$ be a complex random variable such that ${\E {x}=0}$, ${\E |x|^2=1}$,
${\E |x|^{4} < \infty}$. Let $x_{ij}$, $i,j \in \{1,2,...\}$ be independet
copies of $x$. Let ${\Xb=(N^{-1/2}x_{ij})}$, $1\leq i,j \leq N$ be a random
matrix. Writing $\Xb^*$ for the adjoint matrix of $\Xb$, consider the product
$\Xb^m{\Xb^*}^m$ with some $m \in \{1,2,...\}$. The matrix $\Xb^m{\Xb^*}^m$ is
Hermitian positive semi-definite. Let $\lambda_1,\lambda_2,...,\lambda_N$ be
eigenvalues of $\Xb^m{\Xb^*}^m$ (or squared singular values of the matrix
$\Xb^m$). In this paper we find the asymptotic distribution function \[
G^{(m)}(x)=\lim_{N\to\infty}\E{F_N^{(m)}(x)} \] of the empirical distribution
function \[ {F_N^{(m)}(x)} = N^{-1} \sum_{k=1}^N {\mathbb{I}{\{\lambda_k \leq
x\}}}, \] where $\mathbb{I} \{A\}$ stands for the indicator function of event
$A$. The moments of $G^{(m)}$ satisfy \[ M^{(m)}_p=\int_{\mathbb{R}}{x^p
dG^{(m)}(x)}=\frac{1}{mp+1}\binom{mp+p}{p}. \] In Free Probability Theory
$M^{(m)}_p$ are known as Fuss--Catalan numbers. With $m=1$ our result turns to
a well known result of Marchenko--Pastur 1967.
http://arxiv.org/abs/1002.4442
Author(s): Fran\c{c}ois Bolley (CEREMADE) and Ivan Gentil (CEREMADE)
Abstract: We present new $\Phi$-entropy inequalities for diffusion semigroups under the
curvature-dimension criterion. They include the isoperimetric function of the
Gaussian measure. Applications to the long time behaviour of solutions to
Fokker-Planck equations are given.
http://arxiv.org/abs/1002.4478
Author(s): Shige Peng
Abstract: In this book, we introduce a new approach of sublinear expectation to deal
with the problem of probability and distribution model uncertainty. We a new
type of (robust) normal distributions and the related central limit theorem
under sublinear expectation. We also present a new type of Brownian motion
under sublinear expectations and the related stochastic calculus of Ito's type.
The results provide robust tools for the problem of probability model
uncertainty arising from financial risk management, statistics and stochastic
controls.
http://arxiv.org/abs/1002.4546
Author(s): I. M. MacPhee and M. V. Menshikov and M. Vachkovskaia
Abstract: We consider the long term behaviour of a Markov chain \xi(t) on \Z^N based on
the N station supermarket model. Different routing policies for the supermarket
model give different Markov chains. We show that for a general class of local
routing policies, "join the least weighted queue" (JLW), the N one-dimensional
components \xi_i(t) can be partitioned into disjoint clusters C_k. Within each
cluster C_k the "speed" of each component \xi_j converges to a constant V_k and
under certain conditions \xi is recurrent in shape on each cluster. To
establish these results we have assembled methods from two distinct areas of
mathematics, semi-martingale techniques used for showing stability of Markov
chains together with the theory of optimal flows in networks. As corollaries to
our main result we obtain the stability classification of the supermarket model
under any JLW policy and can explicitly compute the C_k and V_k for any
instance of the model and specific JLW policy.
http://arxiv.org/abs/1002.4570
Author(s): F. P. Kelly and R. J. Williams
Abstract: Unlimited access to a motorway network can, in overloaded conditions, cause a
loss of capacity. Ramp metering (signals on slip roads to control access to the
motorway) can help avoid this loss of capacity. The design of ramp metering
strategies has several features in common with the design of access control
mechanisms in communication networks.
Inspired by models and rate control mechanisms developed for Internet
congestion control, we propose a Brownian network model as an approximate model
for a controlled motorway and consider it operating under a proportionally fair
ramp metering policy.We present an analysis of the performance of this model.
http://arxiv.org/abs/1002.4591
Author(s): Joe Suzuki
Abstract: We proposed a learning algorithm for nonparametric estimation and on-line
prediction for general stationary ergodic sources. We prepare histograms each
of which estimates the probability as a finite distribution, and mixture them
with weights to construct an estimator. The whole analysis is based on measure
theory. The estimator works whether the source is discrete or continuous. If it
is stationary ergodic, then the measure theoretically given Kullback-Leibler
information divided by the sequence length $n$ converges to zero as $n$ goes to
infinity. In particular, for continuous sources, the method does not require
existence of a probability density function.
http://arxiv.org/abs/1002.4453
Author(s): Paolo Da Pelo and Alberto Lanconelli
Abstract: We obtain a new probabilistic representation for the solution of the heat
equation in terms of a product for smooth random variables which is introduced
and studied in this paper. This multiplication, expressed in terms of the
Hida-Malliavin derivatives of the random variables involved, exhibits many
useful properties which are to some extents opposite to some peculiar features
of the Wick product.
http://arxiv.org/abs/1002.4269
Author(s): Dmitry Ioffe and Yvan Velenik
Abstract: We consider a model of a polymer in Z^{d+1}, constrained to join 0 and a
hyperplane at distance N. The polymer is subject to a quenched non-negative
random environment. Alternatively, the model describes crossing random walks in
a random potential (see Chapter 5 of [Sznitman] for the original Brownian
motion formulation). It was recently shown, by Flury and by Zygouras, that, in
such a setting, the quenched and annealed free energies coincide in the limit N
to infinity, when d is at least 3 and the temperature is sufficiently high. We
first strengthen this result by proving that, under somewhat weaker assumptions
on the distribution of disorder which, in particular, enable a small
probability of traps, the ratio of quenched and annealed partition functions
actually converges. We then conclude that, in this case, the polymer obeys a
diffusive scaling, with the same diffusivity constant as the annealed model.
http://arxiv.org/abs/1002.4289
Author(s): Xicheng Zhang
Abstract: In this article we prove the existence and uniqueness for degenerate
stochastic differential equations with Sobolev (possibly singular) drift and
diffusion coefficients in a generalized sense. In particular, our result covers
the classical DiPerna-Lions flows and, we also obtain the well-posedness for
degenerate Fokker-Planck equations with irregular coefficients. Moreover, a
large deviation principle of Freidlin-Wenzell type for this type of SDEs is
established.
http://arxiv.org/abs/1002.4297
Author(s): Wilfrid S. Kendall
Abstract: We exhibit some explicit co-adapted couplings for n-dimensional Brownian
motion and all its Levy stochastic areas. In the two-dimensional case we show
how to derive exact asymptotics for the coupling time under various mixed
coupling strategies, using Dufresne's formula for the distribution of
exponential functionals of Brownian motion. This yields quantitative
asymptotics for the distributions of random times required for certain
simultaneous couplings of stochastic area and Brownian motion. The approach
also applies to higher dimensions, but will then lead to upper and lower bounds
rather than exact asymptotics.
http://arxiv.org/abs/1002.4348
Author(s): Shankar Bhamidi and Remco van der Hofstad
Abstract: In the recent past, there has been a concerted effort to develop mathematical
models for real-world networks and analyze various dynamics on these models.
One particular problem of significant importance is to understand the effect of
random edge lengths or costs on the geometry and flow transporting properties
of the network. Two different regimes are of great interest, the weak disorder
regime where optimality of a path is determined by the sum of edge weights on
the path and the strong disorder regime where optimality of a path is
determined by the maximal edge weight on the path. In the context of the
stochastic mean-field model of distance, we provide the first mathematically
tractable model of weak disorder and show that no transition occurs at finite
temperature. Indeed we show that for all fixed finite temperatures, the number
of edges on the minimal weight path (i.e the hopcount) is always
$\Theta(\log{n})$ and satisfies a central limit theorem with asymptotic means
and variances of order $\Theta(\log{n})$, with limiting constants expressible
in terms of the Malthusian rate of growth and the mean of the stable-age
distribution of the associated continuous-time branching process. More
precisely, we take independent and identically distributed edge weights with
distribution $E^s$ for some parameter $s>0$, where $E$ is an exponential random
variable with mean 1. Then, the asymptotic mean and variance of the central
limit theorem for the hopcount are $s\log{n}$ and $s^2 \log{n}$ respectively.
We also find limiting distributional asymptotics for the value of the minimal
weight path in terms of extreme value distributions, Cox processes and
martingale limits of branching processes.
http://arxiv.org/abs/1002.4362
Author(s): L\'aszl\'o Lov\'asz and Bal\'azs Szegedy
Abstract: We highlight a topological aspect of the graph limit theory. Graphons are
limit objects for convergent sequences of dense graphs. We introduce the
representation of a graphon on a unique metric space and we relate the
dimension of this metric space to the size of regularity partitions. We prove
that if a graphon has an excluded induced sub-bigraph then the underlying
metric space is compact and has finite packing dimension. It implies in
particular that such graphons have regularity partitions of polynomial size.
http://arxiv.org/abs/1002.4377
Author(s): Stephen Curran
Abstract: We construct spaces of quantum increasing sequences, which give quantum
families of maps in the sense of Soltan. We then introduce a notion of quantum
spreadability for a sequence of noncommutative random variables, by requiring
their joint distribution to be invariant under taking quantum subsequences. Our
main result is a free analogue of a theorem of Ryll-Nardzewski: for an infinite
sequence of noncommutative random variables, quantum spreadability is
equivalent to free independence and identical distribution with respect to a
conditional expectation.
http://arxiv.org/abs/1002.4390
Author(s): Hua-Huai Chern and Hsien-Kuei Hwang and Conrado Mart\'inez
Abstract: An unusual and surprising expansion of the form \[ p_n = \rho^{-n-1}(6n
+\tfrac{18}5+ \tfrac{336}{3125} n^{-5}+\tfrac{1008}{3125} n^{-6} +\text{smaller
order terms}), \] as $n\to\infty$, is derived for the probability $p_n$ that
two randomly chosen binary search trees are identical (in shape and in labels
of all corresponding nodes). A quantity arising in the analysis of phylogenetic
trees is also proved to have a similar asymptotic expansion. Our method of
proof is new in the literature of discrete probability and analysis of
algorithms, and based on the psi-series expansions for nonlinear differential
equations. Such an approach is very general and applicable to many other
problems involving nonlinear differential equations; many examples are
discussed and several attractive phenomena are discovered.
http://arxiv.org/abs/1002.3859
Author(s): Janko Gravner and Alexander E. Holroyd and Robert Morris
Abstract: Two-dimensional bootstrap percolation is a cellular automaton in which sites
become 'infected' by contact with two or more already infected nearest
neighbors. We consider these dynamics, which can be interpreted as a monotone
version of the Ising model, on an n x n square, with sites initially infected
independently with probability p. The critical probability p_c is the smallest
p for which the probability that the entire square is eventually infected
exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim
\pi^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the
second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining
it up to a poly(log log n)-factor.
http://arxiv.org/abs/1002.3881
Author(s): Matthew I. Roberts
Abstract: We consider a (random permutation model) binary search tree with n nodes and
give asymptotics on the loglog scale for the height H_n and saturation level
h_n of the tree as n\to\infty, both almost surely and in probability. We then
consider the number F_n of particles at level H_n at time n, and show that F_n
is unbounded almost surely.
http://arxiv.org/abs/1002.3896
Author(s): Igor Cialenco
Abstract: We study parameter estimation problem for diagonalizable parabolic stochastic
partial differential equations driven by a multiplicative fractional noise with
any Hurst parameter $H\in(0,1)$. Two classes of estimates are investigated:
traditional maximum likelihood type estimates, and a new class called
closed-form exact estimates. Finally several examples are discussed, including
statistical inference for stochastic heat equation driven by a fractional
Brownian motion.
http://arxiv.org/abs/1002.3911
Author(s): Alexander Stolyar
Abstract: We consider a system with N unit-service-rate queues in tandem, with
exogenous arrivals of rate lambda at queue 1, under a back-pressure (MaxWeight)
algorithm: service at queue n is blocked unless its queue length is greater
than that of next queue n+1. The question addressed is how steady-state queues
scale as N goes to infinity. We show that the answer depends on whether lambda
is below or above the critical value 1/4: in the former case queues remain
uniformly stochastically bounded, while otherwise they grow to infinity.
The problem is essentially reduced to the behavior of the system with
infinite number of queues in tandem, which is studied using tools from
interacting particle systems theory. In particular, the criticality of load 1/4
is closely related to the fact that this is the maximum possible flux (flow
rate) of a stationary totally asymmetric simple exclusion process.
http://arxiv.org/abs/1002.3940
Author(s): Bo'az Klartag and Sasha Sodin
Abstract: We analyze the quality of the gaussian approximation to linear combinations
of n independent, identically-distributed random variables with finite fourth
moments. It turns out that there exist universal, simple linear combinations
that perform better than the sum of the variables. We also investigate the case
in which the random variables are independent, yet they are not necessarily
identically distributed.
http://arxiv.org/abs/1002.3970
Author(s): Svante Janson
Abstract: We study positive random variables whose moments can be expressed by products
and quotients of Gamma functions; this includes many standard distributions.
General results are given on existence, series expansion and asymptotics of
density functions. It is shown that the integral of the supremum process of
Brownian motion has moments of this type, as well as a related random variable
occuring in the study of hashing with linear displacement, and the general
results are applied to these variables.
http://arxiv.org/abs/1002.4135
Author(s): Alexander Vandenberg-Rodes
Abstract: Using the recently discovered strong negative dependence properties of the
symmetric exclusion process, we derive general conditions for when the
normalized current of particles between regions converges to the Gaussian
distribution. The main novelty is that the results do not assume any
translation invariance, and hold for most initial configurations.
http://arxiv.org/abs/1002.4148
Author(s): Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
Abstract: Occupation time fluctuation limits of particle systems in R^d with
independent motions (symmetric stable Levy process, with or without critical
branching) have been studied assuming initial distributions given by Poisson
random measures (homogeneous and some inhomogeneous cases). In this paper, with
d=1 for simplicity, we extend previous results to a wide class of initial
measures obeying a quasi-homogeneity property, which includes as special cases
homogeneous Poisson measures and many deterministic measures (simple example:
one atom at each point of Z), by means of a new unified approach. In previous
papers, in the homogeneous Poisson case, for the branching system in "low"
dimensions, the limit was characterized by a long-range dependent Gaussian
process called sub-fractional Brownian motion (sub-fBm), and this effect was
attributed to the branching because it had appeared only in that case. An
unexpected finding in this paper is that sub-fBm is more prevalent than
previously thought. Namely, it is a natural ingredient of the limit process in
the non-branching case (for "low" dimension), as well. On the other hand,
fractional Brownian motion is not only related to systems in equilibrium (e.g.,
non-branching system with initial homogeneous Poisson measure), but it also
appears here for a wider class of initial measures of quasi-homogeneous type.
http://arxiv.org/abs/1002.4152
Author(s): Xavier Bardina and Khalifa Es-Sebaiy (SAMM)
Abstract: In this paper we introduce and study a self-similar Gaussian process that is
the bifractional Brownian motion $B^{H,K}$ with parameters $H\in(0,1)$ and
$K\in(1,2)$ such that $HK\in(0,1)$. A remarkable difference between the case
$K\in(0,1)$ and our situation is that this process is a semimartingale when
$2HK=1$.
http://arxiv.org/abs/1002.3680
Author(s): J\'er\'emie Bettinelli (LM-Orsay)
Abstract: We discuss scaling limits of large bipartite quadrangulations of positive
genus. For a given $g$, we consider, for every $n \ge 1$, a random
quadrangulation $\q_n$ uniformly distributed over the set of all rooted
bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric
space by endowing its set of vertices with the graph distance. We show that, as
$n$ tends to infinity, this metric space, with distances rescaled by the factor
$n^{-1/4}$, converges in distribution, at least along some subsequence, toward
a limiting random metric space. This convergence holds in the sense of the
Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of
the choice of the subsequence, the Hausdorff dimension of the limiting space is
almost surely equal to 4. Our main tool is a bijection introduced by Chapuy,
Marcus, and Schaeffer between the quadrangulations we consider and objects they
call well-labeled $g$-trees. An important part of our study consists in
determining the scaling limits of the latter.
http://arxiv.org/abs/1002.3682
Author(s): Henrik Renlund
Abstract: We consider first-passage percolation on a ladder, i.e. the graph
{0,1,...}*{0,1} where nodes at distance 1 are joined by an edge, and the times
are exponentially i.i.d. with mean 1. We find an appropriate Markov chain to
calculate an explicit expression for the time constant whose numerical value is
approximately 0.6827. This time constant is the long-term average inverse speed
of the process. We also calculate the average residual time.
http://arxiv.org/abs/1002.3709
Author(s): Henrik Renlund
Abstract: We collect, survey and develop methods of (one-dimensional) stochastic
approximation in a framework that seems suitable to handle fairly broad
generalizations of Polya urns. To show the applicability of the results we
determine the limiting fraction of balls in an urn with balls of two colors. We
consider two models generalizing the Polya urn, in the first one ball is drawn
and replaced with balls of (possibly) both colors according to which color was
drawn. In the second, two balls are drawn simultaneously and replaced along
with balls of (possibly) both colors according to what combination of colors
were drawn.
http://arxiv.org/abs/1002.3716
Author(s): Martin Hairer
Abstract: We consider a class of singular perturbations to the stochastic heat equation
or semilinear variations thereof. The interesting feature of these
perturbations is that, as the small parameter epsilon tends to zero, their
solutions converge to the 'wrong' limit, i.e. they do not converge to the
solution obtained by simply setting epsilon = 0. A similar effect is also
observed for some (formally) small stochastic perturbations of a deterministic
semilinear parabolic PDE.
Our proofs are based on a detailed analysis of the spatially rough component
of the equations, combined with a judicious use of Gaussian concentration
inequalities.
http://arxiv.org/abs/1002.3722
Author(s): Paavo Salminen
Abstract: In this paper we study the optimal stopping problem for L\'evy processes
studied by Novikov and Shiryayev, Stochastics, 2007 In particular, we are
interested in finding the representing measure of the value function. It is
seen that that this can be expressed in terms of the Appell polynomials. An
important tool in our approach and computations is the Wiener-Hopf
factorization.
http://arxiv.org/abs/1002.3746
Author(s): Moritz Deger and Moritz Helias and Stefano Cardanobile and Fatihcan M. Atay and Stefan Rotter
Abstract: The Poisson process with dead time (PPD) is a widely used model for time
series of events. Here we analyze non-equilibrium properties of an ensemble of
PPDs. We derive a delay differential equation that describes the dynamics of
the state of the ensemble. Analytical solutions are obtained for the
time-dependent ensemble output rate in response to a step input. We also derive
the mapping of periodic input to steady-state output, which we solve
specifically for sinusoidal inputs. We are able to generalize the dynamics of
the PPD to the case of random dead times, by which the method becomes
applicable to a much larger class of stochastic point processes. Transient
properties of the PPD are a recurring theme in many quantitative sciences,
since a dead time after event detection is a feature of most technical counting
devices. Our results are also relevant for the neurosciences because
refractoriness is a characteristic of trains of action potentials emitted by
nerve cells.
http://arxiv.org/abs/1002.3798
Author(s): Thomas Simon (LPP)
Abstract: We observe that the function $F_\alpha (x) = (1+ \alpha
x^\alpha)e^{-x^\alpha}$ is completely monotone iff $\alpha \le \alpha_0$ for
some $\alpha_0 \in ]2/3, 3/4[.$ This property is equivalent to the unimodality
of the inverse positive $\alpha$-stable law. The random variable associated
with $F_\alpha$ appears then in two different factorizations of the positive
$\alpha$-stable distribution. Furthermore, it is infinitely divisible iff
$\alpha \le \alpha_1$ for some $\alpha_1 \in ]2/3, \alpha_0[$ and
self-decomposable iff $\alpha \le \alpha_2$ for some $\alpha_2 \in ]2/3,
\alpha_1[.$
http://arxiv.org/abs/1002.3813
Author(s): Beatrice Acciaio and Hans Foellmer and Irina Penner
Abstract: We study the risk assessment of uncertain cash flows in terms of dynamic
convex risk measures for processes as introduced in Cheridito, Delbaen, and
Kupper (2006). These risk measures take into account not only the amounts but
also the timing of a cash flow. We discuss their robust representation in terms
of suitably penalized probability measures on the optional sigma-field. This
yields an explicit analysis both of model and discounting ambiguity. We focus
on supermartingale criteria for different notions of time consistency. In
particular we show how bubbles may appear in the dynamic penalization, and how
they cause a breakdown of asymptotic safety of the risk assessment procedure.
http://arxiv.org/abs/1002.3627
Author(s): Turdebek N. Bekjan and Zeqian Chen
Abstract: This paper is devoted to the study of $\Phi$-moment inequalities for
noncommutative martingales. In particular, we prove the noncommutative
$\Phi$-moment analogues of martingale transformations, Stein's inequalities,
Khintchine's inequalities for Rademacher's random variables, and
Burkholder-Gundy's inequalities. The key ingredient is a noncommutative version
of Marcinkiewicz type interpolation theorem for Orlicz spaces which we
establish in this paper.
http://arxiv.org/abs/1002.3670
Author(s): Behamar Chouaf and Serguei Pergamenchtchikov (LMRS)
Abstract: We consider the optimal investment problem for Black-Scholes type financial
market with bounded VaR measure on the whole investment interval $[0,T]$. The
explicit form for the optimal strategies is found.
http://arxiv.org/abs/1002.3681
Author(s): Beatrice Acciaio and Irina Penner
Abstract: This paper gives an overview of the theory of dynamic convex risk measures
for random variables in discrete time setting. We summarize robust
representation results of conditional convex risk measures, and we characterize
various time consistency properties of dynamic risk measures in terms of
acceptance sets, penalty functions, and by supermartingale properties of risk
processes and penalty functions.
http://arxiv.org/abs/1002.3794
Author(s): Joaquin Fontbona and Nathalie Krell (IRMAR) and Servet Martinez
Abstract: We present a ?rst study on the energy required to reduce a unit mass fragment
by consecutively using several devices, as it happens in the mining industry.
Two devices are considered, which we represent as different stochastic
fragmentation processes. Following the self-similar energy model introduced by
Bertoin and Martinez, we compute the average energy required to attain a size x
with this two-device procedure. We then asymptotically compare, as x goes to 0
or 1, its energy requirement with that of individual fragmentation processes.
In particular, we show that for certain range of parameters of the
fragmentation processes and of their energy cost-functions, the consecutive use
of two devices can be asymptotically more efficient than using each of them
separately, or conversely.
http://arxiv.org/abs/1002.3460
Author(s): Raphael Cerf and Pierre Petit
Abstract: We expose here a short proof of Cramer's theorem in R based on convex
duality.
http://arxiv.org/abs/1002.3496
Author(s): Kristi Kuljus and J\"uri Lember
Abstract: We consider the maximum likelihood (Viterbi) alignment of a hidden Markov
model (HMM). In an HMM, the underlying Markov chain is usually hidden and the
Viterbi alignment is often used as the estimate of it. This approach will be
referred to as the Viterbi segmentation. The goodness of the Viterbi
segmentation can be measured by several risks. In this paper, we prove the
existence of asymptotic risks. Being independent of data, the asymptotic risks
can be considered as the characteristics of the model that illustrate the
long-run behavior of the Viterbi segmentation.
http://arxiv.org/abs/1002.3509
Author(s): James Martin and Philipp Schmidt
Abstract: The TASEP (totally asymmetric simple exclusion process) is a basic model for
an one-dimensional interacting particle system with non-reversible dynamics.
Despite the simplicity of the model it shows a very rich and interesting
behaviour. In this paper we study some aspects of the TASEP in discrete time
and compare the results to the recently obtained results for the TASEP in
continuous time. In particular we focus on stationary distributions for
multi-type models, speeds of second-class particles, collision probabilities
and the "speed process". In discrete time, jump attempts may occur at different
sites simultaneously, and the order in which these attempts are processed is
important; we consider various natural update rules.
http://arxiv.org/abs/1002.3539
Author(s): Lutz Duembgen and Richard Samworth and Dominic Schuhmacher
Abstract: We study the approximation of arbitrary distributions P on d-dimensional
space by distributions with log-concave density. Approximation means minimizing
a Kullback-Leibler type functional. We show that such an approximation exists
if, and only if, P has finite first moments and is not concentrated on some
hyperplane. Furthermore we show that this approximation depends continuously on
P with respect to Mallows' distance D_1. This result implies consistency of the
maximum likelihood estimator of a log-concave density under fairly general
conditions. It also allows us to prove existence and consistency of estimators
in regression models with a response Y = m(X) + E, where X and E are
independent, m(.) belongs to a certain class of regression functions while E is
a random error with log-concave density.
http://arxiv.org/abs/1002.3448
Author(s): Olivier Marchal and Mattia Cafasso
Abstract: In this article, we show that the double scaling limit correlation functions
of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when
$y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the
determinental formulae defined by conformal $(2m,1)$ models. Our approach
follows the one developped by Berg\`{e}re and Eynard in \cite{BergereEynard}
and uses a Lax pair representation of the conformal $(2m,1)$ models (giving
Painleve II integrable hierarchy) as suggested by Bleher and Eynard in
\cite{BleherEynard}. In particular we define Baker-Akhiezer functions
associated to the Lax pair to construct a kernel which is then used to compute
determinental formulae giving the correlation functions of the double scaling
limit of a matrix model near the merging of two cuts.
http://arxiv.org/abs/1002.3347
Author(s): Takahiro Hasebe
Abstract: This article is focused on properties of monotone convolutions. A criterion
for infinite divisibility and time evolution of convolution semigroups are
mainly studied. In particular, we clarify that many analogues of the classical
results of L\'{e}vy processes hold such as characterizations of subordinators
and strictly stable distributions.
http://arxiv.org/abs/1002.3430
Author(s): Nikolaos Fountoulakis and Konstantinos Panagiotou
Abstract: Broadcasting algorithms are important building blocks of distributed systems.
In this work we investigate the typical performance of the classical and
well-studied push model. Assume that initially one node in a given network
holds some piece of information. In each round, every one of the informed nodes
chooses independently a neighbor uniformly at random and transmits the message
to it.
In this paper we consider random networks where each vertex has degree d,
which is at least 3, i.e., the underlying graph is drawn uniformly at random
from the set of all d-regular graphs with n vertices. We show that with
probability 1 - o(1) the push model broadcasts the message to all nodes within
(1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In
particular, we can characterize precisely the effect of the node degree to the
typical broadcast time of the push model. Moreover, we consider pseudo-random
regular networks, where we assume that the degree of each node is very large.
There we show that the broadcast time is (1+o(1))C ln n with probability 1 -
o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.
http://arxiv.org/abs/1002.3518
Author(s): Vasileios Maroulas
Abstract: Uniform large deviation principles for positive functionals of all equivalent
types of infinite dimensional Brownian motions acting together with a Poisson
random measure are established. The core of our approach is a variational
representation formula which for an infinite sequence of i.i.d real Brownian
motions and a Poisson random measure was shown in [5].
http://arxiv.org/abs/1002.3290
Author(s): Anastasia Papavasiliou
Abstract: We study the problem of estimating parameters of the limiting equation of a
multiscale diffusion in the case of averaging and homogenization, given data
from the corresponding multiscale system. First, we review some recent results
that make use of the maximum likelihood of the limiting equation. In
particular, it has been shown that in the averaging case, the MLE will be
asymptotically consistent in the limit while in the homogenization case, the
MLE will be asymptotically consistent only if we subsample the data. Then, we
focus on the problem of estimating the diffusion coefficient. We suggest a
novel approach that makes use of the total $p$-variation, as defined in the
theory of rough paths and avoids the subsampling step. The method is applied to
a multiscale OU process.
http://arxiv.org/abs/1002.3241
Author(s): B. Cessac
Abstract: We provide rigorous and exact results characterizing the statistics of spike
trains in a network of leaky integrate and fire neurons, where time is discrete
and where neurons are submitted to noise, without restriction on the synaptic
weights. We show the existence and uniqueness of an invariant measure of Gibbs
type and discuss its properties. We also discuss Markovian approximations and
relate them to the approaches currently used in computational neuroscience to
analyse experimental spike trains statistics.
http://arxiv.org/abs/1002.3275
Author(s): Elvan Ceyhan
Abstract: The data-random graphs called proximity catch digraphs (PCDs) have been
introduced recently and have applications in pattern recognition and spatial
pattern analysis. A PCD is a random directed graph (i.e., digraph) which is
constructed from data using the relative positions of the points from various
classes. Different PCDs result from different definitions of the proximity
region associated with each data point. We consider the underlying graphs based
on a family of PCDs which is determined by a family of parameterized proximity
maps called proportional-edge proximity map. The graph invariant we investigate
is the relative edge density of the underlying graphs. We demonstrate that,
properly scaled, relative edge density of the underlying graphs is a
U-statistic, and hence obtain the asymptotic normality of the relative edge
density for data from any distribution that satisfies mild regulatory
conditions. By detailed probabilistic and geometric calculations, we compute
the explicit form of the asymptotic normal distribution for uniform data on a
bounded region. We also compare the relative edge densities of the two types of
the underlying graphs and the relative arc density of the PCDs. The approach
presented here is also valid for data in higher dimensions.
http://arxiv.org/abs/1002.2957
Author(s): Jianhai Bao and Xuerong Mao and Chenggui Yuan
Abstract: In this paper we study the well-known Khasminskii-Type Theorem, i.e. the
existence and uniqueness of solutions of stochastic evolution delay equations,
under local Lipschitz condition, but without linear growth condition. We then
establish one stochastic LaSalle-type theorem for asymptotic stability analysis
of strong solutions. Moreover, several examples are established to illustrate
the power of our theories.
http://arxiv.org/abs/1002.3116
Author(s): Brendan D McKay
Abstract: Let d = (d1, d2, ..., dn) be a vector of non-negative integers with even sum.
We prove some basic facts about the structure of a random graph with degree
sequence d, including the probability of a given subgraph or induced subgraph.
Although there are many results of this kind, they are restricted to the sparse
case with only a few exceptions. Our focus is instead on the case where the
average degree is approximately a constant fraction of n. Our approach is the
multidimensional saddle-point method. This extends the enumerative work of
McKay and Wormald (1990) and is analogous to the theory developed for bipartite
graphs by Greenhill and McKay (arXiv:math/0701600, 2009).
http://arxiv.org/abs/1002.3018
Author(s): Enrico Priola and Feng-Yu Wang
Abstract: We introduce a new condition on elliptic operators $L= {1/2}\triangle + b
\cdot \nabla $ which ensures the validity of the Liouville property for bounded
solutions to $Lu=0$ on $\R^d$. Such condition is sharp when $d=1$. We extend
our Liouville theorem to more general second order operators in non-divergence
form assuming a Cordes type condition.
http://arxiv.org/abs/1002.3055
Author(s): Leonid Petrov
Abstract: In this note we present new examples of determinantal point processes with
infinitely many particles. The particles live on the half-lattice {1,2,...} or
on the open half-line (0,+\infty). The main result is the computation of the
correlation kernels. They have integrable form and are expressed through the
Euler gamma function (the lattice case) and the classical Whittaker functions
(the continuous case). Our processes are obtained via a limit transition from a
model of random strict partitions introduced by Borodin (1997) in connection
with the problem of harmonic analysis for projective characters of the infinite
symmetric group.
http://arxiv.org/abs/1002.2714
Author(s): Souvik Ghosh and Gennady Samorodnitsky
Abstract: We obtain the rate of growth of long strange segments and the rate of decay
of infinite horizon ruin probabilities for a class of infinite moving average
processes with exponentially light tails. The rates are computed explicitly. We
show that the rates are very similar to those of an i.i.d. process as long as
moving average coefficients decay fast enough. If they do not, then the rates
are significantly different. This demonstrates the change in the length of
memory in a moving average process associated with certain changes in the rate
of decay of the coefficients.
http://arxiv.org/abs/1002.2751
Author(s): Itai Benjamini and Sebastian M\"uller
Abstract: We study branching random walk on Cayley graphs. A first result is that the
trace of a transient branching random walk on a Cayley graph is a.s. transient
for simple random walk. In addition, it has a.s. critical percolation
probability less than one and exponential volume growth. The proofs rely on the
fact that the trace induces an invariant percolation on the family tree of the
branching random walk. Furthermore, we prove that the trace is a.s. strongly
recurrent for any branching random walk. This follows from the observation that
the trace, after appropriate biasing of the root, defines a unimodular measure.
All the results hold more generally for branching random walk on unimodular
random graphs.
http://arxiv.org/abs/1002.2781
Author(s): Mykhaylo Shkolnikov
Abstract: We consider finite and infinite systems of particles on the real line and
half-line evolving in continuous time. Hereby, the particles are driven by
i.i.d. Levy processes endowed with rank-dependent drift and diffusion
coefficients. In the finite systems we show that the processes of gaps in the
respective particle configurations possess unique invariant distributions and
prove the convergence of the gap processes to the latter in the total variation
distance, assuming a bound on the jumps of the Levy processes. In the infinite
case we show that the gap process of the particle system on the half-line is
tight for appropriate initial conditions and same drift and diffusion
coefficients for all particles. Applications of such processes include the
modelling of capital distributions among the ranked participants in a financial
market, the stability of certain stochastic queueing and storage networks and
the study of the Sherrington-Kirkpatrick model of spin glasses.
http://arxiv.org/abs/1002.2811
Author(s): Feng-Yu Wang
Abstract: By using Hsu's multiplicative functional for the Neumann heat equation, a
natural damped gradient operator is defined for the reflecting Brownian motion
on compact manifolds with boundary. This operator is linked to quasi-invariant
flows in terms of a integration by parts formula, which leads to the standard
log-Sobolev inequality for the associated Dirichlet form on the path space.
http://arxiv.org/abs/1002.2887
Author(s): Feng-Yu Wang
Abstract: Coupling and strong Feller property are investigated for the linear SDE on
$\R^d$: $$\d X_t= A X_t\d t+ \d L_t,$$ where $A$ is a $d\times d$ real matrix
and $L_t$ is a L\'evy process with L\'evy measure $\nu$ on $\R^d$. Assume that
$\nu(\d z)\ge \rr_0(z)\d z$ for some $\rr_0\ge 0$. If $A \le 0$ and
$\int_{B(x_0,\vv)} \rr_0(z)^{-1}\d z<\infty$ holds for some $x_0\in \R^d$ and
some $\vv>0$, then the associated Markov transition probability $P_t(x,\d y)$
satisfies $$\|P_t (x, \cdot)- P_t (y, \cdot)\|_{var} \le \ff{C(1+|x-y|)}{\ss
t}, x,y\in \R^d, t>0$$ for some constant $C>0$, which is sharp for large $t$
and implies that the process has successful couplings. If $\rr_0\in
C(\R^d\setminus \{0\})$ with $\int_{\R^d}\rr_0(z)\d z=\infty$, then the process
is strong Feller.
http://arxiv.org/abs/1002.2890
Author(s): J. Bouttier and E. Guitter
Abstract: We present a detailed calculation of the distance-dependent two-point
function for quadrangulations with no multiple edges. Various discrete
observables measuring this two-point function are computed and analyzed in the
limit of large maps. For large distances and in the scaling regime, we recover
the same universal scaling function as for general quadrangulations. We then
explore the geometry of "minimal neck baby universes" (minbus), which are the
outgrowths to be removed from a general quadrangulation to transform it into a
quadrangulation with no multiple edges, the "mother universe". We give a number
of distance-dependent characterizations of minbus, such as the two-point
function inside a minbu or the law for the distance from a random point to the
mother universe.
http://arxiv.org/abs/1002.2552
Author(s): Joerg Kampen
Abstract: The characteristic functions of multivariate Feller processes with generator
of affine type, and with smooth symbol functions have an explicit
representation in terms of power series with rational number coefficients and
with monmoms consisting of powers of the the symbol functions and formal
derivatives of the symbol functions. The power series repesentations are
convergent globally in time and on bounded domains of arbitrary size.
Generalized symbol functions can be derived leading to power series expansions
which are convergent on arbitrary domains in special cases. The rational number
coefficients can be efficiently computed by an integer recursion. As a
numerical consequence characteristic functions of multivariate affine processes
can be efficiently computed from the symbol function avoiding computation of
the generalized Riccati equations (an observation first made recently in a more
general context).
http://arxiv.org/abs/1002.2764
Author(s): Claudia Kluppelberg and Serguei Pergamenchtchikov (LMRS)
Abstract: We investigate optimal consumption and investment problems for a
Black-Scholes market under uniform restrictions on Value-at-Risk and Expected
Shortfall. We formulate various utility maximization problems, which can be
solved explicitly. We compare the optimal solutions in form of optimal value,
optimal control and optimal wealth to analogous problems under additional
uniform risk bounds. Our proofs are partly based on solutions to
Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification
theorem. This work was supported by the European Science Foundation through the
AMaMeF programme.
http://arxiv.org/abs/1002.2487
Author(s): Rama Cont and David-Antoine Fournie
Abstract: We develop a non-anticipative calculus for functionals of a continuous
semimartingale, using a notion of pathwise functional derivative. A functional
extension of the Ito formula is derived and used to obtain a constructive
martingale representation theorem for a class of continuous martingales
verifying a regularity property. By contrast with the Clark-Haussmann-Ocone
formula, this representation involves non-anticipative quantities which can be
computed pathwise.
These results are used to construct a weak derivative acting on
square-integrable martingales, which is shown to be the inverse of the Ito
integral, and derive an integration by parts formula for Ito stochastic
integrals. We show that this weak derivative may be viewed as a
non-anticipative "lifting" of the Malliavin derivative.
Regular functionals of an Ito martingale which have the local martingale
property are characterized as solutions of a functional differential equation,
for which a uniqueness result is given.
http://arxiv.org/abs/1002.2446
Author(s): Geoffrey R. Grimmett and Alexander E. Holroyd
Abstract: The high-density plaquette percolation model in d dimensions contains a
surface that is homeomorphic to the (d-1)-sphere and encloses the origin. This
is proved by a path-counting argument in a dual model. When d=3, this permits
an improved lower bound on the critical point p_e of entanglement percolation,
namely p_e >= \mu^-2 where \mu is the connective constant for self-avoiding
walks on Z^3. Furthermore, when the edge density p is below this bound, the
radius of the entanglement cluster containing the origin has an exponentially
decaying tail.
http://arxiv.org/abs/1002.2623
Author(s): Claudia Kluppelberg and Serguei Pergamenchtchikov (LMRS)
Abstract: We investigate optimal consumption problems for a Black-Scholes market under
uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic
utility functions. We find the solutions in terms of a dynamic strategy in
explicit form, which can be compared and interpreted. This paper continues our
previous work, where we solved similar problems for power utility functions.
http://arxiv.org/abs/1002.2486
Author(s): Claudia Kluppelberg and Serguei Pergamenchtchikov (LMRS)
Abstract: We investigate optimal consumption and investment problems for a
Black-Scholes market under uniform restrictions on Value-at-Risk and Expected
Shortfall. We formulate various utility maximization problems, which can be
solved explicitly. We compare the optimal solutions in form of optimal value,
optimal control and optimal wealth to analogous problems under additional
uniform risk bounds. Our proofs are partly based on solutions to
Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification
theorem. This work was supported by the European Science Foundation through the
AMaMeF programme.
http://arxiv.org/abs/1002.2487
Author(s): Leonid Galtchouk (IRMA) and Serguei Pergamenchtchikov (LMRS)
Abstract: In this paper we find nonasymptotic exponential upper bounds for the
deviation in the ergodic theorem for families of homogeneous Markov processes.
We find some sufficient conditions for geometric ergodicity uniformly over a
parametric family. We apply this property to the nonasymptotic nonparametric
estimation problem for ergodic diffusion processes.
http://arxiv.org/abs/1002.2341
Author(s): Ronald Getoor
Abstract: This is the first part of a possible monograph on the duality of Markov
processes. It contains a proof of Fitzsimmons' existence theorem of a moderate
Markov dual process relative to an excessive measure, m, together with the
necessary preliminary material. Then this is applied to prove the
correspondence between optional copredictable homogenous random measures and
sigma finite measures not charging m-exceptional sets again following
Fitzsimmons. The second part which may never be written would deal with duality
proper including results from, but not limited to, my joint paper with P. J.
Fitzsimmons"Potential Theory of Moderate Markov Dual Processes" which appeared
in Potential Anal.(2009) 31:275-310. Complete proofs of all results not
appearing in standard books are given with the one exception of Dellacherie's
result characterizing semipolar sets.
http://arxiv.org/abs/1002.2399
Author(s): D. Goreac
Abstract: We aim at characterizing viability, invariance and some reachability
properties of controlled piecewise deterministic Markov processes (PDMPs).
Using analytical methods from the theory of viscosity solutions, we establish
criteria for viability and invariance in terms of the first order normal cone.
We also investigate reachability of arbitrary open sets. The method is based on
viscosity techniques and duality for some associated linearized problem. The
theoretical results are applied to general On/Off systems, Cook's model for
haploinssuficiency, and a stochastic model for bacteriophage lambda.
http://arxiv.org/abs/1002.2242
Author(s): Constantinos Kardaras and Gordan Zitkovic
Abstract: For a sequence in $\mathbb{L}^0_+$, we provide simple necessary and
sufficient conditions to ensure that each sequence of its forward convex
combinations converges to the same limit. These conditions correspond to a
measure-free version of the notion of uniform integrability and are related to
the numeraire problem of mathematical finance.
http://arxiv.org/abs/1002.1889
Author(s): Rim Amami
Abstract: We consider an impulse control problem in infinite horizon applied with
switching technology. We suppose that the firm decides at certain moments
(impulse moments) to switch technology, leading to a jump of the firm value. We
show that the value function for such problems satisfies a dynamic programming
principle version. Our objective is to look for an optimal strategy which
maximizes the value function associated with a switching problem.
http://arxiv.org/abs/1002.2086
Author(s): Charles M. Goldie (University of Sussex) and Rosie Cornish (University of Bristol), Carol L. Robinson (Loughborough University)
Abstract: Computer-based tests with randomly generated questions allow a large number
of different tests to be generated. Given a fixed number of alternatives for
each question, the number of tests that need to be generated before all
possible questions have appeared is surprisingly low.
http://arxiv.org/abs/1002.2114
Author(s): Maria Joao Oliveira and Habib Ouerdiane and Jose Luis da Silva and R. Vilela Mendes
Abstract: The Mittag-Leffler function $E_{\alpha}$ being a natural generalization of
the exponential function, an infinite-dimensional version of the fractional
Poisson measure would have a characteristic functional \[ C_{\alpha}(\varphi)
:=E_{\alpha}(\int (e^{i\varphi(x)}-1)d\mu (x)) \] which we prove to fulfill all
requirements of the Bochner-Minlos theorem.
The identity of the support of this new measure with the support of the
infinite-dimensional Poisson measure ($\alpha =1$) allows the development of a
fractional infinite-dimensional analysis modeled on Poisson analysis through
the combinatorial harmonic analysis on configuration spaces. This setting
provides, in particular, explicit formulas for annihilation, creation, and
second quantization operators. In spite of the identity of the supports, the
fractional Poisson measure displays some noticeable differences in relation to
the Poisson measure, which may be physically quite significant.
http://arxiv.org/abs/1002.2124
Author(s): Fuqing Gao and Arnaud Guillin and Liming Wu
Abstract: Using the method of transportation-information inequality introduced in
\cite{GLWY}, we establish Bernstein type's concentration inequalities for
empirical means $\frac 1t \int_0^t g(X_s)ds$ where $g$ is a unbounded
observable of the symmetric Markov process $(X_t)$. Three approaches are
proposed : functional inequalities approach ; Lyapunov function method ; and an
approach through the Lipschitzian norm of the solution to the Poisson equation.
Several applications and examples are studied.
http://arxiv.org/abs/1002.2163
Author(s): Dmitry Panchenko
Abstract: We show that, under the conditions known to imply the validity of the Parisi
formula, if the generic Sherrington-Kirkpatrick Hamiltonian contains a $p$-spin
term then the Ghirlanda-Guerra identities for the $p$th power of the overlap
hold in a strong sense without averaging. This implies strong version of the
extended Ghirlanda-Guerra identities for mixed $p$-spin models than contain
terms for all even $p\geq 2$ and $p=1.$
http://arxiv.org/abs/1002.2190
Author(s): Xinpeng Li
Abstract: In this paper, we prove two forms of strict comparison theorem for $X,Y\in
L_G^1(\Omega)$. Furthermore, if $X,Y\in Lip(\Omega)$ and $\x>0$, we give a
necessary and sufficient condition under which the strict comparison theorem
holds.
http://arxiv.org/abs/1002.1765
Author(s): Alexander V. Kolesnikov and Roman I. Zhdanov
Abstract: We study the isoperimetric problem for the radially symmetric measures.
Applying the spherical symmetrization procedure and variational arguments we
reduce this problem to a one-dimensional ODE of the second order. Solving
numerically this ODE we get an empirical description of isoperimetric regions
of the planar radially symmetric exponential power laws. We also prove some
isoperimetric inequalities for the log-convex measures. We show, in particular,
that the symmetric balls of large size are isoperimetric sets for strictly
log-convex and radially symmetric measures. We also establish a comparison
theorem for the products of the one-dimensional log-convex measures.
http://arxiv.org/abs/1002.1829
Author(s): Fabienne Castell (LATP) and Nadine Guillotin-Plantard (UCB and ICJ) and Fran\c{c}oise P\`ene (LM), Bruno Schapira (LM-Orsay)
Abstract: Random walks in random scenery are processes defined by
$Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and
$(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random
variables. We assume here that their distributions belong to the normal domain
of attraction of stable laws with index $\alpha\in (0,2]$ and $\beta\in (0,2]$
respectively. These processes were first studied by H. Kesten and F. Spitzer,
who proved the convergence in distribution when $\alpha\neq 1$ and as $n\to
\infty$, of $n^{-\delta}Z_n$, for some suitable $\delta>0$ depending on
$\alpha$ and $\beta$. Here we are interested in the convergence, as $n\to
\infty$, of $n^\delta{\mathbb P}(Z_n=\lfloor n^{\delta} x\rfloor)$, when $x\in
\RR$ is fixed. We also consider the case of random walks on randomly oriented
lattices for which we obtain similar results.
http://arxiv.org/abs/1002.1878
Author(s): Maria Deijfen and Olle Haggstrom and and Alexander E. Holroyd
Abstract: Let each point of a homogeneous Poisson process in R^d independently be
equipped with a random number of stubs (half-edges) according to a given
probability distribution mu on the positive integers. We consider
translation-invariant schemes for perfectly matching the stubs to obtain a
simple graph with degree distribution mu. Leaving aside degenerate cases, we
prove that for any mu there exist schemes that give only finite components as
well as schemes that give infinite components. For a particular matching scheme
that is a natural extension of Gale-Shapley stable marriage, we give sufficient
conditions on mu for the absence and presence of infinite components.
http://arxiv.org/abs/1002.1943
Author(s): Bartek Blaszczyszyn (INRIA Rocquencourt) and Paul Muhlethaler (INRIA Rocquencourt)
Abstract: In this paper we propose two analytically tractable stochastic models of
non-slotted Aloha for Mobile Ad-hoc NETworks (MANETs): one model assumes a
static pattern of nodes while the other assumes that the pattern of nodes
varies over time. Both models feature transmitters randomly located in the
Euclidean plane, according to a Poisson point process with the receivers
randomly located at a fixed distance from the emitters. We concentrate on the
so-called outage scenario, where a successful transmission requires a
Signal-to-Interference-and-Noise Ratio (SINR) larger than a given threshold.
With Rayleigh fading and the SINR averaged over the duration of the packet
transmission, both models lead to closed form expressions for the probability
of successful transmission. We show an excellent matching of these results with
simulations. Using our models we compare the performances of non-slotted Aloha
to previously studied slotted Aloha. We observe that when the path loss is not
very strong both models, when appropriately optimized, exhibit similar
performance. For stronger path loss non-slotted Aloha performs worse than
slotted Aloha, however when the path loss exponent is equal to 4 its density of
successfully received packets is still 75% of that in the slotted scheme. This
is still much more than the 50% predicted by the well-known analysis where
simultaneous transmissions are never successful. Moreover, in any path loss
scenario, both schemes exhibit the same energy efficiency.
http://arxiv.org/abs/1002.1629
Author(s): N. Alon and B. Sudakov
Abstract: What is the minimum number of edges that have to be added to the random graph
$G=G_{n,0.5}$ in order to increase its chromatic number $\chi=\chi(G)$ by one
percent ? One possibility is to add all missing edges on a set of $1.01 \chi$
vertices, thus creating a clique of chromatic number $1.01 \chi$. This
requires, with high probability, the addition of $\Omega(n^2/\log^2 n)$ edges.
We show that this is tight up to a constant factor, consider the question for
more general random graphs $G_{n,p}$ with $p=p(n)$, and study a local version
of the question as well.
The question is motivated by the study of the resilience of graph properties,
initiated by the second author and Vu, and improves one of their results.
http://arxiv.org/abs/1002.1748
Author(s): Vittorio Addona and Stan Wagon and and Herb Wilf
Abstract: Suppose Alice has a coin with heads probability $q$ and Bob has one with
heads probability $p>q$.
Now each of them will toss their coin $n$ times, and Alice will win iff she
gets more heads than Bob does. Evidently the game favors Bob, but for the given
$p,q$, what is the choice of $n$ that maximizes Alice's chances of winning? The
problem of determining the optimal $N$ first appeared in \cite{wa}. We show
that there is an essentially unique value $N(q,p)$ of $n$ that maximizes the
probability $f(n)$ that the weak coin will win, and it satisfies
$\frac{1}{2(p-q)}-\frac12\le N(q,p)\le \frac{\max{(1-p,q)}}{p-q}$. The analysis
uses the multivariate form of Zeilberger's algorithm to find an indicator
function $J_n(q,p)$ such that $J>0$ iff $n
http://arxiv.org/abs/1002.1763
Author(s): Constantinos Kardaras and Gordan Zitkovic
Abstract: For a sequence in $\mathbb{L}^0_+$, we provide simple necessary and
sufficient conditions to ensure that each sequence of its forward convex
combinations converges to the same limit. These conditions correspond to a
measure-free version of the notion of uniform integrability and are related to
the numeraire problem of mathematical finance.
http://arxiv.org/abs/1002.1889
Author(s): Atsushi Takeuchi
Abstract: Consider jump-type stochastic differential equations with the drift,
diffusion and jump terms. Logarithmic derivatives of densities for the solution
process are studied, and the Bismut-Elworthy-Li type formulae can be obtained
under the uniformly elliptic condition on the coefficients of the diffusion and
jump terms. Our approach is based upon the Kolmogorov backward equation by
making full use of the Markovian property of the process.
http://arxiv.org/abs/1002.1384
Author(s): Nobuo Yoshida and Yutaka Terasawa
Abstract: We consider a SPDE (stochastic partial differential equation) which describes
the velocity field of a viscous, incompressible non-Newtonian fluid subject to
a random perturbation. Here, the extra stress tensor of the fluid is given by a
polynomial of degree $p-1$ of the deformation rate tensor, while the colored
noise is considered as the random perturbation. We investigate the existence
and the uniqueness of the weak solution to this SPDE.
http://arxiv.org/abs/1002.1431
Author(s): Imen Boutouria and Abdelhamid Hassairi and Helene Massam
Abstract: The Wishart distribution on an homogeneous cone is a generalization of the
Riesz distribution on a symmetric cone which corresponds to a given graph. The
paper extends to this distribution, the famous Olkin and Rubin characterization
of the ordinary Wishart distribution on symmetric matrices.
http://arxiv.org/abs/1002.1451
Author(s): Thomas Simon (LPP)
Abstract: A multiplicative identity in law connecting the hitting times of completely
asymmetric $\alpha-$stable L\'evy processes in duality is established. In the
spectrally positive case, this identity allows with an elementary argument to
compute fractional moments and to get series representations for the density.
We also prove that the hitting times are unimodal as soon as $\alpha\le 3/2.$
Analogous results are obtained, in a much simplified manner, for the first
passage time across a positive level.
http://arxiv.org/abs/1002.1540
Author(s): Franck Barthe and Fabrice Gamboa and Li-Vang Lozada-Chang and Alain Rouault
Abstract: The geometry of unit $N$-dimensional $\ell_{p}$ balls has been intensively
investigated in the past decades. A particular topic of interest has been the
study of the asymptotics of their projections. Apart from their intrinsic
interest, such questions have applications in several probabilistic and
geometric contexts (Barthe et al. 2005). In this paper, our aim is to revisit
some known results of this flavour with a new point of view. Roughly speaking,
we will endow the ball with some kind of Dirichlet distribution that
generalizes the uniform one and will follow the method developed in Skibinsky
(1967), Chang et al. (1993) in the context of the randomized moment space. The
main idea is to build a suitable coordinate change involving independent random
variables. Moreover, we will shed light on a nice connection between the
randomized balls and the randomized moment space.
http://arxiv.org/abs/1002.1544
Author(s): Francois M. Dunlop
Abstract: The serial harness introduced by Hammersley is equivalent, in the Gaussian
case, to the Gaussian Solid-On-Solid interface model with parallel heat bath
dynamics. Here we consider sub-lattice parallel dynamics, and give exact
results about relaxation dynamics, based on the equivalence to the infinite
time limit of a time periodic random field. We also give a numerical comparison
to the harness process in continuous time studied by Hsiao and by Ferrari,
Niederhauser and Pechersky.
http://arxiv.org/abs/1002.1604
Author(s): A. De Gregorio and S.M. Iacus
Abstract: The LASSO is a widely used statistical methodology for simultaneous
estimation and variable selection. In the last years, many authors analyzed
this technique from a theoretical and applied point of view. We introduce and
study the adaptive LASSO problem for discretely observed ergodic diffusion
processes. We prove oracle properties also deriving the asymptotic distribution
of the LASSO estimator. Our theoretical framework is based on the random field
approach and it applied to more general families of regular statistical
experiments in the sense of Ibragimov-Hasminskii (1981). Furthermore, we
perform a simulation and real data analysis to provide some evidence on the
applicability of this method.
http://arxiv.org/abs/1002.1312
Author(s): R. M. Abrarov and R. M. Abrarov
Abstract: We obtained the probabilities for the values of the M\"obius function for
arbitrary numbers and found that the asymptotic densities of the squarefree
integers among the odd and even numbers are $8/\pi^2$ and $4/\pi^2$,
respectively. It is determined that statistics of successive outcomes of the
M\"obius function for very large squarefree odd and even numbers behaves
similar to statistics of heads and tails of two flipping coins. These
preliminary results are giving arguments supporting the Riemann Hypothesis. Its
plausibility is based on statistical phenomena for integers.
http://arxiv.org/abs/1002.1682
Author(s): Zhen-Qing Chen and Panki Kim and Renming Song
Abstract: We consider a family of pseudo differential operators $\{\Delta+ a^\alpha
\Delta^{\alpha/2}; a\in (0, 1]\}$ on $\bR^d$ for every $d\geq 1$ that evolves
continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $\alpha \in
(0, 2)$. It gives rise to a family of L\'evy processes $\{X^a, a\in (0, 1]\}$
in $\bR^d$, where $X^a$ is the sum of a Brownian motion and an independent
symmetric $\alpha$-stable process with weight $a$. We establish sharp two-sided
estimates for the heat kernel of $\Delta + a^{\alpha} \Delta^{\alpha/2}$ with
zero exterior condition in a family of open subsets, including bounded $C^{1,
1}$ (possibly disconnected) open sets. This heat kernel is also the transition
density of the sum of a Brownian motion and an independent symmetric
$\alpha$-stable process with weight $a$ in such open sets.
Our result is the first sharp two-sided estimates for the transition density
of a Markov process with both diffusion and jump components in open sets.
Moreover, our result is uniform in $a$ in the sense that the constants in the
estimates are independent of $a\in (0, 1]$ so that it recovers the Dirichlet
heat kernel estimates for Brownian motion by taking $a\to 0$. Integrating the
heat kernel estimates in time $t$, we recover the two-sided sharp uniform Green
function estimates of $X^a$ in bounded $C^{1,1}$ open sets in $\bR^d$, which
were recently established in \cite{CKSV2} by using a completely different
approach.
http://arxiv.org/abs/1002.1121
Author(s): Lucas Gerin (MODAL'x)
Abstract: We discuss a Monte Carlo Markov Chain (MCMC) procedure for the random
sampling of some one-dimensional lattice paths with constraints, for various
constraints. We show that an approach inspired by optimal transport allows us
to bound efficiently the mixing time of the associated Markov chain. The
algorithm is robust and easy to implement, and samples an "almost" uniform path
of length $n$ in $n^{3+\eps}$ steps. This bound makes use of a certain
contraction property of the Markov chain, and is also used to derive a bound
for the running time of Propp-Wilson's CFTP algorithm.
http://arxiv.org/abs/1002.1183
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati
Abstract: In the paper [25], written in collaboration with Gesine Reinert, we proved a
universality principle for the Gaussian Wiener chaos. In the present work, we
aim at providing an original example of application of this principle in the
framework of random matrix theory. More specifically, by combining the result
in [25] with some combinatorial estimates, we are able to prove
multi-dimensional central limit theorems for the spectral moments (of arbitrary
degrees) associated with random matrices with real-valued i.i.d. entries,
satisfying some appropriate moment conditions. Our approach has the advantage
of yielding, without extra effort, bounds over classes of smooth (i.e., thrice
differentiable) functions, and it allows to deal directly with discrete
distributions. As a further application of our estimates, we provide a new
"almost sure central limit theorem", involving logarithmic means of functions
of vectors of traces.
http://arxiv.org/abs/1002.1212
Author(s): Fran\c{c}ois Baccelli (INRIA Rocquencourt) and Bartek Blaszczyszyn (INRIA Rocquencourt)
Abstract: We consider Mobile Ad-hoc Network (MANET) with transmitters located according
to a Poisson point in the Euclidean plane, slotted Aloha Medium Access (MAC)
protocol and the so-called outage scenario, where a successful transmission
requires a Signal-to-Interference-and-Noise (SINR) larger than some threshold.
We analyze the local delays in such a network, namely the number of times slots
required for nodes to transmit a packet to their prescribed next-hop receivers.
The analysis depends very much on the receiver scenario and on the variability
of the fading. In most cases, each node has finite-mean geometric random delay
and thus a positive next hop throughput. However, the spatial (or large
population) averaging of these individual finite mean-delays leads to infinite
values in several practical cases, including the Rayleigh fading and positive
thermal noise case. In some cases it exhibits an interesting phase transition
phenomenon where the spatial average is finite when certain model parameters
are below a threshold and infinite above. We call this phenomenon, contention
phase transition. We argue that the spatial average of the mean local delays is
infinite primarily because of the outage logic, where one transmits full
packets at time slots when the receiver is covered at the required SINR and
where one wastes all the other time slots. This results in the "RESTART"
mechanism, which in turn explains why we have infinite spatial average.
Adaptive coding offers a nice way of breaking the outage/RESTART logic. We show
examples where the average delays are finite in the adaptive coding case,
whereas they are infinite in the outage case.
http://arxiv.org/abs/1002.0855
Author(s): Lung-Chi Chen and Akira Sakai
Abstract: We consider random walk and self-avoiding walk whose 1-step distribution is
given by D, and oriented percolation whose bond-occupation probability is
proportional to D. Suppose that D(x) decays as |x|^{-d-a} with a>0. For random
walk in any dimension and for self-avoiding walk and critical/subcritical
oriented percolation above the common upper-critical dimension 2min{a,2}, we
prove large-t asymptotics of the gyration radius, which is the average
end-to-end distance of random walk/self-avoiding walk of length t or the
average spatial size of an oriented-percolation cluster at time t. This proves
the conjecture for long-range self-avoiding walk by Heydenreich and for
long-range oriented percolation in Chen and Sakai (2009).
http://arxiv.org/abs/1002.0875
Author(s): Xuepeng Bai and Yiqing Lin
Abstract: In this paper, we study the existence and uniqueness of solutions to
stochastic differential equations driven by G-Brownian motion with an
integral-Lipschitz condition for the coefficients.
http://arxiv.org/abs/1002.1046
Author(s): Krzysztof Burdzy and Soumik Pal and Jason Swanson
Abstract: We study two models consisting of reflecting one-dimensional Brownian
"particles" of positive radius. We show that the stationary empirical
distributions for the particle systems do not converge to the harmonic function
for the generator of the individual particle process, unlike in the case when
the particles are infinitely small.
http://arxiv.org/abs/1002.1057
Author(s): Miquel Montero and Javier Villarroel
Abstract: By appealing to renewal theory we determine the equations that the mean exit
time of a continuous-time random walk with drift satisfies both when the
present coincides with a jump instant or when it does not. Particular attention
is paid to the corrections ensuing from the non-Markovian nature of the
process. We show that when drift and jumps have the same sign the relevant
integral equations can be solved in closed form. The case when holding times
have the classical Erlang distribution is considered in detail.
http://arxiv.org/abs/1002.0571
Author(s): Pablo A. Ferrari and Eugene A. Pechersky and Valentin V. Sisko and Anatoly A. Yambartsev
Abstract: Consider a discrete locally finite subset $\Gamma$ of $R^d$ and the complete
graph $(\Gamma,E)$, with vertices $\Gamma$ and edges $E$. We consider Gibbs
measures on the set of sub-graphs with vertices $\Gamma$ and edges $E'\subset
E$. The Gibbs interaction acts between open edges having a vertex in common. We
study percolation properties of the Gibbs distribution of the graph ensemble.
The main results concern percolation properties of the open edges in two cases:
(a) when the $\Gamma$ is a sample from homogeneous Poisson process and (b) for
a fixed $\Gamma$ with exponential decay of connectivity.
http://arxiv.org/abs/1002.0610
Author(s): Alexey Kuznetsov
Abstract: We derive explicit asymptotic expansions of the density of the supremum of a
strictly stable process when the index $\alpha$ is not rational. In the case
when parameters $\alpha$ and $\rho=\p(X_1>0)$ satisfy $\rho+k=l/\alpha$ for
some integers $k,l \ge 1$ we prove that these asymptotic expansions are in fact
convergent series representations of the density of supremum.
http://arxiv.org/abs/1002.0614
Author(s): Frank Aurzada and Leif Doering and and Mladen Savov
Abstract: We prove Chung-type laws of the iterated logarithm for general L\'evy
processes at zero. In particular, we provide a tool to translate small
deviation estimates directly into laws of the iterated logarithm without any
loss of constants nor any extra conditions. This reveals new laws of the
iterated logarithm for L\'evy processes at small times in many concrete
examples. In some cases, exotic norming functions are derived.
http://arxiv.org/abs/1002.0675
Author(s): Boubaker Smii
Abstract: We consider the generalized Ornstein- Uhlenbeck equation $\partial_t X=-m
X_t+\eta$. In this paper We construct the L\'evy noise $\eta$. The generalized
Ornstein- Uhlenbeck process $X_t$ will be represented by a special types of
graphs called rooted trees with two types of leaves.
http://arxiv.org/abs/1002.0744
Author(s): Heikki Tikanm\"aki
Abstract: We define fractional L\'evy processes by two different integral
transformations by taking integral representation of fractional Brownian motion
and replacing the driving Brownian motion by more general square integrable
L\'evy process. The definition using infinitely supported transformation kernel
is well known in the literature but the definition by compact interval
representation is new to the best of my knowledge in this setup. We prove that
the processes defined by different transformations do not have the same finite
dimensional distributions in general, even though it is the case in fractional
Brownian motion setup. Hovever, we prove a connection between the two concepts.
We consider different properties of fractional L\'evy processes by compact
interval transformation and compare them to the properties of fractional L\'evy
processes by infinite interval transformation. We also consider financial
applications and represent a no-arbitrage theorem for a model including
fractional L\'evy processes by any of the two integral transformations.
http://arxiv.org/abs/1002.0780
Author(s): D. Tibi
Abstract: Two models of loss networks, introduced by Gibbens et al. and by Antunes et
al., are known to exhibit a mean field limiting regime with several stable
equilibria. This paper first provides an interpretation of the Lyapunov
function given by Antunes et al., in terms of entropy dissipation. The argument
extends to another, similar but closed model. The two main models are next
reexamined in the light of Freidlin and Wentzell's large deviation approach of
randomly perturbed dynamical systems. Assuming that some of their results still
hold under slightly relaxed conditions, the metastability property is derived
for both systems. The Lyapunov function of the second model is then identified
with the quasipotential associated with a slightly modified, asymptotically
reversible, Markovian perturbation of the same dynamical system.
http://arxiv.org/abs/1002.0796
Author(s): Miquel Montero and Javier Villarroel
Abstract: By appealing to renewal theory we determine the equations that the mean exit
time of a continuous-time random walk with drift satisfies both when the
present coincides with a jump instant or when it does not. Particular attention
is paid to the corrections ensuing from the non-Markovian nature of the
process. We show that when drift and jumps have the same sign the relevant
integral equations can be solved in closed form. The case when holding times
have the classical Erlang distribution is considered in detail.
http://www.arxiv.org
Author(s): Jason Miller
Abstract: The object of our study is the massless field on $D_n = D \cap \tfrac{1}{n}
\Z^2$, where $D \subseteq \R^2$ is a bounded domain with smooth boundary, with
Hamiltonian $\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))$ and $h(x) = f(x)$ if $x
\in \partial D_n$ for a given continuous function $f \colon \R^2 \to \R$. The
interaction $\CV$ is assumed to be symmetric, strictly convex, and have bounded
second derivatives. This is a general model for a $(2+1)$-dimensional effective
interface where $h$ represents the height. We prove that linear functionals of
$h$ converge in the limit to a Gaussian free field on $D$, the standard
Gaussian with respect to the Dirichlet inner product $(f,g)_\nabla = \int_D
\nabla f \cdot \nabla g$. The main step in the proof is to establish a general
estimate that serves to quantify the degree to which the presence of boundary
conditions affect the behavior of the model. In particular, our estimate
implies that $\E h(x)$ converges to the harmonic extension of $f$ from
$\partial D$ to $D$. In a subsequent article, we will employ the tools
developed here to resolve a conjecture made by Sheffield that the zero contour
lines of $h$ are asymptotically described by a family of conformally invariant
random curves which are variants of SLE(4).
http://arxiv.org/abs/1002.0381
Author(s): Svante Janson and Guy Louchard and Anders Martin-L\"of
Abstract: We study the maximum of a Brownian motion with a parabolic drift; this is a
random variable that often occurs as a limit of the maximum of discrete
processes whose expectations have a maximum at an interior point. We give
series expansions and integral formulas for the distribution and the first two
moments, together with numerical values to high precision.
http://arxiv.org/abs/1002.0497
Author(s): Mats Brod\'en and Magnus Wiktorsson
Abstract: The problem of approximating/tracking the value of a Wiener process is
considered. The discretization points are placed at times when the value of the
process differs from the approximation by some amount, here denoted by eta. It
is found that the limiting difference, as eta goes to 0, between the
approximation and the value of the process normalized with eta converges in
distribution to a triangularly distributed random variable.
http://arxiv.org/abs/1002.0528
Author(s): Annalisa Cerquetti
Abstract: We present an alternative approach to the Bayesian nonparametric analysis of
conditional species richness under two-parameter Poisson Dirichlet priors. We
rely on a known characterization by deletion of classes property and on results
for Beta-Binomial distributions. Besides leading to simplified and much more
direct proofs, our proposal provides a new scale mixture representation of the
conditional asymptotic law.
http://arxiv.org/abs/1002.0535
Author(s): Pedro Catuogno and Christian Olivera
Abstract: We introduced a new algebra of stochastic generalized functions which
contains to the space of stochastic distributions G, [25]. As an application,
we prove existence and uniqueness of the solution of a stochastic Cauchy
problem involving singularities.
http://arxiv.org/abs/1002.0454
Author(s): Soumik Pal
Abstract: We study a family of multidimensional diffusions taking values in the unit
simplex of vectors with non-negative coordinates that add up to one. The family
of processes satisfy stochastic differential equations which are similar to the
ones for the classical Wright-Fisher model, except that the "mutation rates"
are now nonpositive. This model, suggested by Aldous, appears in the study of a
conjectured diffusion limit for a Markov chain on Cladograms. The striking
feature of these models is that the boundary is not reflecting, and we kill the
process once it hits the boundary. We derive the explicit exit distribution
from the simplex, and probabilistic bounds on the exit time. We also prove that
these processes can be viewed as a "stochastic time-reversal" of a
Wright-Fisher process of increasing dimensions and conditioned at a random
time. A key idea in our proofs is a skew-product construction using certain
one-dimensional diffusions called Bessel-square processes of negative
dimensions which have been recently introduced by Going-Jaeschke and Yor.
http://arxiv.org/abs/1002.0159
Author(s): Patrick Cheridito and Mitja Stadje
Abstract: We provide existence results and comparison principles for solutions of
backward stochastic difference equations (BS$\Delta$Es) and then prove
convergence of these to solutions of backward stochastic differential equations
(BSDEs) when the mesh size of the time-discretizaton goes to zero. The
BS$\Delta$Es and BSDEs are governed by drivers $f^N(t,\omega,y,z)$ and
$f(t,\omega,y,z),$ respectively. The new feature of this paper is that they may
be non-Lipschitz in $z$. For the convergence results it is assumed that the
BS$\Delta$Es are based on $d$-dimensional random walks $W^N$ approximating the
$d$-dimensional Brownian motion $W$ underlying the BSDE and that $f^N$
converges to $f$. Conditions are given under which for any terminal condition
$\xi$, there exist terminal conditions $\xi^N$ for the sequence of
BS$\Delta$Es, converging to $\xi$ in $L^2$, such that for the solutions $Y^N$
and $Y$ of the corresponding BS$\Delta$Es and the limiting BSDE one has
$\sup_{0\le t\le T} |Y^N_t - Y_t| \to 0$ in $L^2$. An important special case is
when $f^N(t,\omega,y,z)$ and $f(t,\omega,y,z)$ are convex in $z.$ We show that
in this situation, $\sup_{0\le t\le T} |Y^N_t - Y_t| \to 0$ in $L^2$ for every
sequence of discrete terminal conditions $\xi^N$ converging to $\xi$ in $L^2$.
As a consequence, one obtains that the BSDE is robust in the sense that if
$(W^N,\xi^N)$ is close to $(W,\xi)$ in distribution, then $Y^N$ is close to $Y$
in distribution too.
http://arxiv.org/abs/1002.0175
Author(s): P. Del Moral and F. Patras and S. Rubenthaler
Abstract: The convergence of U-statistics has been intensively studied for estimators
based on families of i.i.d. random variables and variants of them. In most
cases, the independence assumption is crucial [Lee90, de99]. When dealing with
Feynman-Kac and other interacting particle systems of Monte Carlo type, one
faces a new type of problem. Namely, in a sample of N particles obtained
through the corresponding algorithms, the distributions of the particles are
correlated -although any finite number of them is asymptotically independent
with respect to the total number N of particles. In the present article,
exploiting the fine asymptotics of particle systems, we prove convergence
theorems for U-statistics in this framework.
http://arxiv.org/abs/1002.0224
Author(s): J. Inglis and M. Neklyudov and B. Zegarlinski
Abstract: We analyse certain degenerate infinite dimensional sub-elliptic generators
and obtain estimates on the long-time behaviour of the corresponding Markov
semigroups.
http://arxiv.org/abs/1002.0282
Author(s): Vladimir Y.Chernyak and Michael Chertkov and David A. Goldberg and Konstantin Turitsyn
Abstract: A stable open queuing network is considered as a steady non-equilibrium
system of interacting particles. The network is completely specified by its
underlying graphical structure, type of interaction at each node, and the
Poisson transition rates between nodes. For such systems we identify two
regimes in which the system may operate depending on the value of currents
accumulated on the graph edges over time, large compared to the system
correlation time scale. In the first regime of moderate currents, the
large-deviation distribution of currents is universal (independent of the
interaction details), and the system behaves in an "uncongested" mode. In the
second regime of larger currents, the large-deviation current distribution is
sensitive to interaction details, and the system is in a "congested" mode. The
transition between the two regimes can be described as a dynamical second order
phase transition. We illustrate these ideas using a simple, yet non-trivial,
example of a single node with feedback.
http://arxiv.org/abs/1001.5454
Author(s): Yaiza Canzani and Dmitry Jakobson and Igor Wigman
Abstract: We study Gauss curvature for random Riemannian metrics on a compact surface,
lying in a fixed conformal class; our questions are motivated by comparison
geometry. Next, analogous questions are considered for the scalar curvature in
dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.
http://arxiv.org/abs/1002.0030
Author(s): Christian Borgs and Jennifer Chayes and Jeff Kahn and L\'aszl\'o Lov\'asz
Abstract: The theory of convergent graph sequences has been worked out in two extreme
cases, dense graphs and bounded degree graphs. One can define convergence in
terms of counting homomorphisms from fixed graphs into members of the sequence
(left-convergence), or counting homomorphisms into fixed graphs
(right-convergence). Under appropriate conditions, these two ways of defining
convergence was proved to be equivalent in the dense case by Borgs, Chayes,
Lov\'asz, S\'os and Vesztergombi. In this paper a similar equivalence is
established in the bounded degree case.
In terms of statistical physics, the implication that left convergence
implies right convergence means that for a left-convergent sequence, partition
functions of a large class of statistical physics models converge. The proof
relies on techniques from statistical physics, like cluster expansion and
Dobrushin Uniqueness.
http://arxiv.org/abs/1002.0115
Author(s): N.V. Krylov
Abstract: We consider divergence form uniformly parabolic SPDEs with VMO bounded
leading coefficients, bounded coefficients in the stochastic part, and possibly
growing lower-order coefficients in the deterministic part. We look for
solutions which are summable to the $p$th power, $p\geq2$, with respect to the
usual Lebesgue measure along with their first-order derivatives with respect to
the spatial variable.
Our methods allow us to include Zakai's equation for the Kalman-Bucy filter
into the general filtering theory.
http://arxiv.org/abs/1002.0306
Author(s): Thomas Duquesne (PMA)
Abstract: In this paper we discuss Hausdorff and packing measures of random continuous
trees called stable trees. Stable trees form a specific class of L\'evy trees
(introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum
random tree (1991) which corresponds to the Brownian case. We provide results
for the whole stable trees and for their level sets that are the sets of points
situated at a given distance from the root. We first show that there is no
exact packing measure for levels sets. We also prove that non-Brownian stable
trees and their level sets have no exact Hausdorff measure with regularly
varying gauge function, which continues previous results from a joint work with
J-F Le Gall (2006).
http://arxiv.org/abs/1001.5329
Author(s): I. Corwin and P.L. Ferrari and S. Peche
Abstract: We demonstrate that, under minimal hypothesis, a wide class of growth models
diplays a phenomenon known as slow decorrelation, where along certain
characteristic directions the range of correlation for fluctuations of the
growth surface height is much longer than other directions. We apply this
result to certain models known to be in the Kardar-Parisi-Zhang (KPZ)
universality class in 1+1 dimension for which the necessary hypothesis holds.
These models are the totally asymmetric simple exclusion process (TASEP), last
passage percolation (LPP), and the polynuclear growth (PNG) model. Utilizing
the slow decorrelation of fluctuations in these models we are able to extend
known fluctuation limit process results away from the fixed curves on which
they were proved, to general space-time curves.
Using the monotonicity of the basic coupling we additionally prove that the
partially asymmetric simple exclusion process (PASEP) displays slow
decorrelation.
http://arxiv.org/abs/1001.5345
Author(s): A. D. Barbour and G. Reinert
Abstract: Using an associated branching process as the basis of our approximation, we
show that typical inter-point distances in a multitype random intersection
graph have a defective distribution, which is well described by a mixture of
translated and scaled Gumbel distributions, the missing mass corresponding to
the event that the vertices are not in the same component of the graph.
http://arxiv.org/abs/1001.5357
Author(s): O.Hryniv and M.Menshikov
Abstract: We study a continuous time stochastic process on strings made of two types of
particles, whose dynamics mimics the behaviour of microtubules in a living
cell; namely, the strings evolve via a competition between (local)
growth/shrinking as well as (global) hydrolysis processes. We give a complete
characterization of the phase diagram of the model, and derive several criteria
of the transient and recurrent regimes for the underlying stochastic process.
http://arxiv.org/abs/1001.5469
Author(s): Carlo Marinelli and Michael R\"ockner
Abstract: In the semigroup approach to stochastic evolution equations, the fundamental
issue of uniqueness of mild solutions is often "reduced" to the much easier
problem of proving uniqueness for strong solutions. This reduction is usually
carried out in a formal way, without really justifying why and how one can do
that. We provide sufficient conditions for uniqueness of mild solutions to a
broad class of semilinear stochastic evolution equations with coefficients
satisfying a monotonicity assumption.
http://arxiv.org/abs/1001.5413
Author(s): Arnaud Debussche (IRMAR) and Julien Vovelle (ICJ)
Abstract: We show that the Cauchy Problem for a randomly forced, periodic
multi-dimensional scalar first-order conservation law with additive or
multiplicative noise is well-posed: it admits a unique solution, characterized
by a kinetic formulation of the problem, which is the limit of the solution of
the stochastic parabolic approximation.
http://arxiv.org/abs/1001.5415
Author(s): Abhimanyu Mitra and Sidney I. Resnick
Abstract: Hidden regular variation defines a subfamily of distributions satisfying
multivariate regular variation on $\mathbb{E} = [0, \infty]^d \backslash
\{(0,0, ..., 0) \} $ and models another regular variation on the sub-cone
$\mathbb{E}^{(2)} = \mathbb{E} \backslash \cup_{i=1}^d \mathbb{L}_i$, where
$\mathbb{L}_i$ is the $i$-th axis. We extend the concept of hidden regular
variation to sub-cones of $\mathbb{E}^{(2)}$ as well. We suggest a procedure of
detecting the presence of hidden regular variation, and if it exists, propose a
method of estimating the limit measure exploiting its semi-parametric
structure. We exhibit examples where hidden regular variation yields better
estimates of probabilities of risk sets.
http://arxiv.org/abs/1001.5058
Author(s): Piotr Milos
Abstract: In the paper the rescaled occupation time fluctuation process of a certain
empirical system is investigated. The system consists of particles evolving
independently according to \alpha-stable motion in R^d, \alpha0. We study how the limit behaviour of the fluctuations of the occupation
time depends on the \emph{initial particle configuration}. We obtain a
functional central limit theorem for a vast class of infinitely divisible
distributions. Our findings extend and put in a unified setting results which
previously seemed to be disconnected. The limit processes form a one
dimensional family of long-range dependance centred Gaussian processes.
http://arxiv.org/abs/1001.5142
Author(s): A. Faggionato and D. Gabrielli
Abstract: Inspired by some recent results on fluctuation theory for piecewise
deterministic Markov processes, we consider a generic diffusion on the 1D torus
and give a simple representation formula for the large deviation rate
functional of its invariant probability measure, in the limit of vanishing
noise. Previously, this rate functional had been characterized by M.I. Freidlin
and A.D. Wentzell as solution of a rather complex optimization problem. We
discuss this last problem in full generality and show that it leads to our
formula. Finally, we discuss some geometric and regularity properties of the
rate functional. In particular, we prove a universality result showing that the
rate functional is a viscosity solution of the stationary Hamilton--Jacobi
equation associated to any Hamiltonian H satisfying weak suitable conditions.
http://arxiv.org/abs/1001.5160
Author(s): V\'ictor Ortiz-L\'opez and Marta Sanz-Sol\'e
Abstract: We consider a stochastic wave equation in spatial dimension three, driven by
a Gaussian noise, white in time and with a stationary spatial covariance. The
free terms are nonlinear with Lipschitz continuous coefficients. Under suitable
conditions on the covariance measure, Dalang and Sanz-Sol\'e [Memoirs of the
AMS, Vol 199, 2009] have proved the existence of a random field solution with
H\"older continuous sample paths, jointly in both arguments, time and space. By
perturbing the driving noise with a multiplicative parameter
$\varepsilon\in]0,1]$, a family of probability laws corresponding to the
respective solutions to the equation is obtained. Using the weak convergence
approach to large deviations developed in [P. Dupuis, R. S. Ellis, 1997], we
prove that this family satisfies a Laplace principle in the H\"older norm.
http://arxiv.org/abs/1001.5228
Author(s): Wei Wang and A. J. Roberts and Jinqiao Duan
Abstract: A large deviation principle is derived for stochastic partial differential
equations with slow-fast components. The result shows that the rate function is
exactly that of the averaged equation plus the fluctuating deviation which is a
stochastic partial differential equation with small Gaussian perturbation. This
also confirms the effectiveness of the approximation of the averaged equation
plus the fluctuating deviation to the slow-fast stochastic partial differential
equations.
http://arxiv.org/abs/1001.4826
Author(s): Michel Bena\"im (UNINE) and Mathieu Faure (UNINE)
Abstract: This paper considers a stochastic approximation algorithm, with decreasing
step size and martingale difference noise. Under very mild assumptions, we
prove the non convergence of this process toward a certain class of repulsive
sets for the associated ordinary differential equation (ODE). We then use this
result to derive the convergence of the process when the ODE is cooperative in
the sense of [Hirsch, 1985]. In particular, this allows us to extend
significantly the main result of [Hofbauer and Sandholm, 2002] on the
convergence of stochastic fictitious play in supermodular games.
http://arxiv.org/abs/1001.4871
Author(s): R. A. Doney and M. S. Savov
Abstract: If $X$ is a stable process of index $\alpha\in(0,2)$ whose L\'{e}vy measure
has density $cx^{-\alpha-1}$ on $(0,\infty)$, and $S_1=\sup_{0x)\backsim A\alpha ^{-1}x^{-\alpha}$ as $x\to\infty$ and
$P(S_1\leq x)\backsim B\alpha^{-1}\rho^{-1}x^{\alpha\rho}$ as $x\downarrow0$.
[Here $\rho =P(X_1>0)$ and $A$ and $B$ are known constants.] It is also known
that $S_1$ has a continuous density, $m$ say. The main point of this note is to
show that $m(x)\backsim Ax^{-(\alpha+1)}$ as $x\to\infty$ and $m(x)\backsim
Bx^{\alpha\rho-1}$ as $x\downarrow0$. Similar results are obtained for related
densities.
http://arxiv.org/abs/1001.4872
Author(s): Yongsheng Song
Abstract: In this article, we consider the properties of hitting times for
$G$-martingale and the stopped processes. We prove that the stopped processes
for $G$-martingales are still $G$-martingales and that the hitting times for a
class of $G$-martingales including $G$-Brownian motion are quasi-continuous.
http://arxiv.org/abs/1001.4907
Author(s): Alexander Gnedin and Alexander Iksanov and and Alexander Marynych
Abstract: The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy
scheme, in which random frequencies of infinitely many boxes are produced by a
multiplicative renewal process, also known as the residual allocation model or
stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one
of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is
derived from the properties of associated perturbed random walks. Refining the
approach based on the standard renewal theory we remove a moment constraint to
cover the cases left open in previous studies.
http://arxiv.org/abs/1001.4920
Author(s): Jean-Pierre Conze (IRMAR) and St\'ephane Le Borgne (IRMAR)
Abstract: An example due to Erdos and Fortet shows that, for a lacunary sequence of
integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of
normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a
generalization of their example interpreted as the limiting behavior of some
modified ergodic sums in the framework of dynamical systems.
http://arxiv.org/abs/1001.4862
Author(s): Antonio Avil\'es and Grzegorz Plebanek and Jos\'e Rodr\'iguez
Abstract: Di Piazza and Preiss asked whether every Pettis integrable function defined
on [0,1] and taking values in a weakly compactly generated Banach space is
McShane integrable. In this paper we answer this question in the negative.
http://arxiv.org/abs/1001.4896
Author(s): Bela Bollobas and Oliver Riordan
Abstract: Recently, Scullard and Ziff noticed that a broad class of planar percolation
models are self-dual under a simple condition that, in a parametrized version
of such a model, reduces to a single equation. They state that the solution of
the resulting equation gives the critical point. However, just as in the
classical case of bond percolation on the square lattice, self-duality is
simply the starting point: the mathematical difficulty is precisely showing
that self-duality implies criticality. Here we do so for a generalization of
the models considered by Scullard and Ziff. In these models, the states of the
bonds need not be independent; furthermore, increasing events need not be
positively correlated, so new techniques are needed in the analysis. The main
new ingredients are a generalization of Harris's Lemma to products of partially
ordered sets, and a new proof of a type of Russo-Seymour-Welsh Lemma with
minimal symmetry assumptions.
http://arxiv.org/abs/1001.4674
Author(s): Enkelejd Hashorva
Abstract: Let R be a positive random variable independent of S which is beta
distributed. In this paper we are interested on the relation between the
distribution function of $R$ and that of RS. For this model we derive first
some distributional properties, and then investigate the lower tail asymptotics
of RS when R is regularly varying at 0, and vice-versa. The applications we
present in this paper concern a) the simplicity of Dirichlet distributions, b)
asymptotics of the sample minima of elliptical distributions, and c) the effect
of the scaling on the asymptotics of aggregated risks.
http://arxiv.org/abs/1001.4684
Author(s): M. T. Barlow and J.-D. Deuschel
Abstract: We study a continuous time random walk $X$ in an environment of i.i.d. random
conductances $\mu_e\in[1,\infty)$. We obtain heat kernel bounds and prove a
quenched invariance principle for $X$. This holds even when
${\mathbb{E}}\mu_e=\infty$.
http://arxiv.org/abs/1001.4702
Author(s): E.A. Cator and L.P.R. Pimentel
Abstract: In this paper we will prove a shape theorem for the last passage percolation
model on a two dimensional $F$-compound Poisson process, called the Hammersley
model with random weights. We will also provide diffusive upper bounds for
shape fluctuations. Finally we will indicate how these results can be used to
prove existence and coalescence of semi-infinite geodesics in some fixed
direction $\alpha$, following an approach developed by Newman and co-authors,
and applied to the classical Hammersley process by W\"uthrich. These results
will be crucial in the development of an upcoming paper on the relation between
Busemann functions and equilibrium measures in last passage percolation models.
http://arxiv.org/abs/1001.4706
Author(s): Daniel Conus and Davar Khoshnevisan
Abstract: We study a family of non-linear stochastic heat equations in (1+1)
dimensions, driven by the generator of a L\'evy process and space-time white
noise. We assume that the underlying L\'evy process has finite exponential
moments in a neighborhood of the origin and that the initial condition has
exponential decay at infinity. Then we prove that under natural conditions on
the non-linearity: (i) The absolute moments of the solution to our stochastic
heat equation grow exponentially with time; and (ii) The distances to the
origin of the farthest high peaks of those moments grow exactly linearly with
time. Very little else seems to be known about the location of the high peaks
of the solution to the non-linear stochastic heat equation. Finally, we show
that these results extend to the stochastic wave equation driven by Laplacian.
http://arxiv.org/abs/1001.4759
Author(s): Craig A. Tracy and Harold Widom
Abstract: For the asymmetric simple exclusion process on the integer lattice with
two-sided Bernoulli initial condition, we derive exact formulas for the
following quantities: (1) the probability that site x is occupied at time t;
(2) a correlation function, the probability that site 0 is occupied at time 0
and site x is occupied at time t; (3) the distribution function for the total
flux across 0 at time t and its exponential generating function.
http://arxiv.org/abs/1001.4766
Author(s): Valentina Cammarota and Aime Lachal
Abstract: Consider the high-order heat-type equation $\partial u/\partial t=\pm
\partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related
Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study the sojourn
time $T(t)$ in the interval $[0,+\infty)$ up to a fixed time $t$ for this
pseudo-process. We provide explicit expressions for the joint distribution of
the couple $(T(t),X(t))$.
http://arxiv.org/abs/1001.4201
Author(s): Alexandros Beskos and Natesh S. Pillai and Gareth O. Roberts and Jesus M. Sanz-Serna, Andrew M. Stuart
Abstract: We investigate the properties of the Hybrid Monte-Carlo algorithm (HMC) in
high dimensions. HMC develops a Markov chain reversible w.r.t. a given target
distribution $\Pi$ by using separable Hamiltonian dynamics with potential
$-\log\Pi$. The additional momentum variables are chosen at random from the
Boltzmann distribution and the continuous-time Hamiltonian dynamics are then
discretised using the leapfrog scheme. The induced bias is removed via a
Metropolis-Hastings accept/reject rule. In the simplified scenario of
independent, identically distributed components, we prove that, to obtain an
$\mathcal{O}(1)$ acceptance probability as the dimension $d$ of the state space
tends to $\infty$, the leapfrog step-size $h$ should be scaled as $h= l \times
d^{-1/4}$. Therefore, in high dimensions, HMC requires $\mathcal{O}(d^{1/4})$
steps to traverse the state space. We also identify analytically the
asymptotically optimal acceptance probability, which turns out to be 0.651 (to
three decimal places). This is the choice which optimally balances the cost of
generating a proposal, which {\em decreases} as $l$ increases, against the cost
related to the average number of proposals required to obtain acceptance, which
{\em increases} as $l$ increases.
http://arxiv.org/abs/1001.4460
Author(s): Yutaka Shikano and Kota Chisaki and Etsuo Segawa and Norio Konno
Abstract: Quantum walks are powerful tools not only to construct the quantum speedup
algorithms but also to describe specific models in physical processes.
Furthermore, the discrete time quantum walk has been experimentally realized in
various setups. We apply the concept of the quantum walk to the problems in
quantum foundations. We show that randomness and the arrow of time in the
quantum walk gradually emerge by periodic projective measurements from the
mathematically obtained limit distribution under the time scale transformation.
http://arxiv.org/abs/1001.3989
Author(s): Zemer Kosloff
Abstract: We extend the notion of zero-type to the whole class of non-singular
transformations and then prove that every non-singular Bernoulli shift is
either zero-type or there is an equivalent invariant probability.
http://arxiv.org/abs/1001.4261
Author(s): Alexander I. Bufetov
Abstract: Vershik and Kerov conjectured in 1985 that suitably normalized dimensions of
irreducible representations of finite symmetric groups converge to a constant
with respect to the Plancherel family of measures on the space of Young
diagrams. The main result of this paper is the proof of the Vershik-Kerov
conjecture. The argument is based on the methods of Borodin, Okounkov and
Olshanski.
http://arxiv.org/abs/1001.4275
Author(s): Guenter Last and Mathew D. Penrose
Abstract: We consider a Poisson process $\eta$ on a measurable space
$(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict
almost everwhwere with respect to the intensity measure $\lambda$ of $\eta$. We
give a Clark-Ocone type formula providing an explicit representation of square
integrable martingales (defined with respect to the natural filtration
associated with $\eta$), which was previously known only in the special case,
when $\lambda$ is the product of Lebesgue measure on $\R_+$ and a
$\sigma$-finite measure on another space $\BX$. Our proof is new and based on
only a few basic properties of Poisson processes and stochastic integrals. We
also consider the more general case of an independent random measure in the
sense of It\^o of pure jump type and show that the Clark-Ocone type
representation leads to an explicit version of the Kunita-Watanabe
decomposition of square integrable martingales. We also find the explicit
minimal variance hedge in a quite general financial market driven by an
independent random measure.
http://arxiv.org/abs/1001.3972
Author(s): Raphael Cerf and Francesco Manzo
Abstract: We analyze the relaxation time of a ferromagnetic d dimensional growth model
on the lattice. The model is characterized by d param- eters which represent
the activation energies of a site, depending on the number of occupied nearest
neighbours. This model is a natural generalisation of the model studied by
Dehghanpour and Schonmann [DS97a], where the activation energy of a site with
more than two occupied neighbours is zero.
http://arxiv.org/abs/1001.3990
Author(s): Zdzislaw Brzezniak and Jan van Neerven and Donna Salopek
Abstract: Let H be a Hilbert space and E a Banach space. We set up a theory of
stochastic integration of L(H,E)-valued functions with respect to H-cylindrical
Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in
the interval (0,1). For Hurst parameters in (0,1/2) we show that a function
F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical
Liouville fBm if and only if it is stochastically integrable with respect to an
H-cylindrical fBm with the same Hurst parameter. As an application we show that
second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by
space-time noise which is white in space and Liouville fractional in time with
Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous
both and space.
http://arxiv.org/abs/1001.4013
Author(s): Richard Kenyon
Abstract: The classical matrix-tree theorem relates the determinant of the
combinatorial laplacian on a graph to the number of spanning trees. We
generalize this result to laplacians on one- and two-dimensional vector
bundles, giving a combinatorial interpretation of their determinants in terms
of so-called cycle rooted spanning forests. We construct natural measures on
CRSFs for which the edges form a determinantal process.
This theory gives a natural generalization of the spanning tree process
adapted to graphs embedded on surfaces. We give a number of other applications,
for example we compute the probability that a loop-erased random walk on a
planar graph between two vertices on the outer boundary passes left of two
given faces. This probability can not be computed using the standard Laplacian
alone.
http://arxiv.org/abs/1001.4028
Author(s): Mathias Beiglboeck and Peter Friz and Stefan Sturm
Abstract: We discuss the possibility of obtaining model-free bounds on volatility
derivatives, given present market data in the form of a calibrated local
volatility model. A counter-example to a wide-spread conjecture is given.
http://arxiv.org/abs/1001.4031
Author(s): A.V. Plyukhin
Abstract: In a simple model of a continuous random walk a particle moves in one
dimension with the velocity fluctuating between V and -V. If V is associated
with the thermal velocity of a Brownian particle and allowed to be position
dependent, the model accounts readily for the particle's drift along the
temperature gradient and recovers basic results of the conventional
thermophoresis theory.
http://arxiv.org/abs/0903.3584
Author(s): A.V. Plyukhin
Abstract: Stochastic processes are proposed whose master equations coincide with
classical wave, telegraph, and Klein-Gordon equations. Similar to predecessors
based on the Goldstein-Kac telegraph process, the model describes the motion of
particles with constant speed and transitions between discreet allowed velocity
directions. A new ingredient is that transitions into a given velocity state
depend on spatial derivatives of other states populations, rather than on
populations themselves. This feature requires the sacrifice of the
single-particle character of the model, but allows to imitate the Huygens'
principle and to recover wave equations in arbitrary dimensions.
http://arxiv.org/abs/1001.3821
Author(s): Jean Bertoin (PMA and Dma)
Abstract: We point out that aging occurs for the following simple model of
fragmentation-coagulation inspired by Pitman's coalescent random forests. For
every $n\in \N$, we consider a uniform random tree with $n$ vertices, and at
each step, depending on the outcome of an independent fair coin tossing, either
we remove one edge chosen uniformly at random amongst the remaining edges, or
we replace one edge chosen uniformly at random amongst the edges which have
been removed previously. The process that records the sizes of the
tree-components evolves by fragmentation and coagulation. It exhibits aging in
the sense that when it is observed after $k$ steps in the regime $k\sim
tn+s\sqrt n$ with $t>0$ fixed, it seems to reach a statistical equilibrium as
$n\to\infty$; but different values of $t$ yield distinct pseudo-stationary
distributions. The approach owes much to the construction by Aldous and Pitman
of the standard additive coalescent via Poissonian cuts on the skeleton of a
Continuum Random Tree.
http://arxiv.org/abs/1001.3721
Author(s): K. Borovkov and G. Decrouez and J. Hinz
Abstract: Mandatory emission trading schemes are being established around the world.
Participants of such market schemes are always exposed to risks. This leads to
the creation of an accompanying market for emission-linked derivatives. To
evaluate the fair prices of such financial products, one needs appropriate
models for the evolution of the underlying assets, emission allowance
certificates. In this paper, we discuss continuous time diffusion and
jump-diffusion models, the latter enabling one to model information shocks that
cause jumps in allowance prices. We show that the resulting martingale dynamics
can be described in terms of non-linear partial differential and
integro-differential equations and use a finite difference method to
investigate numerical properties of their discretizations. The results are
illustrated by a small numerical study.
http://arxiv.org/abs/1001.3728
Author(s): H.M. Soner; N. Touzi; J. Zhang
Abstract: This paper considers the nonlinear theory of G-martingales as introduced by
Peng. A martingale representation theorem for this theory is proved by using
the techniques and the results established in an accompanying paper for the
second order stochastic target problems and the second order backward
stochastic differential equations. In particular, this representation provides
a hedging strategy in a market with an uncertain volatility.
http://arxiv.org/abs/1001.3802
Author(s): Elena Permyakova
Abstract: In this paper is proved the limit theorem for randomly indexed sequence of
random processes in the case where sequences of random index and random
processes are independent, also the estimation of convergence rate is obtained.
http://arxiv.org/abs/1001.3844
Author(s): Jonathan Touboul and Bard Ermentrout and Olivier Faugeras and Bruno Cessac
Abstract: We review a recent approach to the mean-field limits in neural networks that
takes into account the stochastic nature of input current and the uncertainty
in synaptic coupling. This approach was proved to be a rigorous limit of the
network equations in a general setting, and we express here the results in a
more customary and simpler framework. We propose a heuristic argument to derive
these equations providing a more intuitive understanding of their origin. These
equations are characterized by a strong coupling between the different moments
of the solutions. We analyse the equations, present an algorithm to simulate
the solutions of these mean-field equations, and investigate numerically the
equations. In particular, we build a bridge between these equations and
Sompolinsky and collaborators approach (1988, 1990), and show how the coupling
between the mean and the covariance function deviates from customary
approaches.
http://arxiv.org/abs/1001.3872
Author(s): Wei Dou and David Pollard and and Harrison H. Zhou
Abstract: The paper derives a minimax lower bound for rates of convergence for an
infinite-dimensional parameter in an exponential family model. An estimator
that achieves the optimal rate is constructed by maximum likelihood on
finite-dimensional approximations with parameter dimension that grows with
sample size.
http://arxiv.org/abs/1001.3742
Author(s): Pedro C. Pinto and Joao Barros and Moe Z. Win
Abstract: Information-theoretic security -- widely accepted as the strictest notion of
security -- relies on channel coding techniques that exploit the inherent
randomness of the propagation channels to significantly strengthen the security
of digital communications systems. Motivated by recent developments in the
field, this paper aims at a characterization of the fundamental secrecy limits
of wireless networks. Based on a general model in which legitimate nodes and
potential eavesdroppers are randomly scattered in space, the intrinsically
secure communications graph (iS-graph) is defined from the point of view of
information-theoretic security. Conclusive results are provided for the local
connectivity of the Poisson iS-graph, in terms of node degrees and isolation
probabilities. It is shown how the secure connectivity of the network varies
with the wireless propagation effects, the secrecy rate threshold of each link,
and the noise powers of legitimate nodes and eavesdroppers. Sectorized
transmission and eavesdropper neutralization are explored as viable strategies
for improving the secure connectivity. Lastly, the maximum secrecy rate between
a node and each of its neighbours is characterized, and the case of colluding
eavesdroppers is studied. The results help clarify how the spatial density of
eavesdroppers can compromise the intrinsic security of wireless networks.
http://arxiv.org/abs/1001.3697
Author(s): Michael Ludkovski
Abstract: We study optimal behavior of energy producers under a CO_2 emission abatement
program. We focus on a two-player discrete-time model where each producer is
sequentially optimizing her emission and production schedules. The
game-theoretic aspect is captured through a reduced-form price-impact model for
the CO_2 allowance price. Such duopolistic competition results in a new type of
a non-zero-sum stochastic switching game on finite horizon. Existence of game
Nash equilibria is established through generalization to randomized switching
strategies. No uniqueness is possible and we therefore consider a variety of
correlated equilibrium mechanisms. We prove existence of correlated equilibrium
points in switching games and give a recursive description of equilibrium game
values. A simulation-based algorithm to solve for the game values is
constructed and a numerical example is presented.
http://arxiv.org/abs/1001.3455
Author(s): Elchanan Mossel and Sebastien Roch and Allan Sly
Abstract: Recent work has highlighted deep connections between sequence-length
requirements for high-probability phylogeny reconstruction and the related
problem of the estimation of ancestral sequences. In [Daskalakis et al.'09],
building on the work of [Mossel'04], a tight sequence-length requirement was
obtained for the CFN model. In particular the required sequence length for
high-probability reconstruction was shown to undergo a sharp transition (from
$O(\log n)$ to $\hbox{poly}(n)$, where $n$ is the number of leaves) at the
"critical" branch length $\critmlq$ (if it exists) of the ancestral
reconstruction problem.
Here we consider the GTR model. For this model, recent results of [Roch'09]
show that the tree can be accurately reconstructed with sequences of length
$O(\log(n))$ when the branch lengths are below $\critksq$, known as the
Kesten-Stigum (KS) bound. Although for the CFN model $\critmlq = \critksq$, it
is known that for the more general GTR models one has $\critmlq \geq \critksq$
with a strict inequality in many cases. Here, we show that this phenomenon also
holds for phylogenetic reconstruction by exhibiting a family of symmetric
models $Q$ and a phylogenetic reconstruction algorithm which recovers the tree
from $O(\log n)$-length sequences for some branch lengths in the range
$(\critksq,\critmlq)$. Second we prove that phylogenetic reconstruction under
GTR models requires a polynomial sequence-length for branch lengths above
$\critmlq$.
http://arxiv.org/abs/1001.3480
Author(s): Bernard Lapeyre (CERMICS) and J\'er\^ome Lelong (LJK)
Abstract: Adaptive Monte Carlo methods are powerful variance reduction techniques. In
this work, we propose a mathematical setting which greatly relaxes the
assumptions needed by for the adaptive importance sampling techniques presented
by Arouna in 2003. We establish the convergence and asymptotic normality of the
adaptive Monte Carlo estimator under local assumptions which are easily
verifiable in practice. We present one way of approximating the optimal
importance sampling parameter using a randomly truncated stochastic algorithm.
Finally, we apply this technique to the valuation of financial derivatives and
our numerical experiments show that the computational time needed to achieve a
given accuracy is divided by a factor up to 5.
http://arxiv.org/abs/1001.3551
Author(s): Tianxiao Wang and Yufeng Shi
Abstract: In this paper we study the unique solvability of backward stochastic Volterra
integral equations (BSVIEs in short), in terms of both the M-solutions
introduced in [17] and the adapted solutions in [6], [12] or [14]. A general
existence and uniqueness of M-solutions is proved under non-Lipschitz
conditions by virtue of a briefer argument than the one in [17], which also
extends the results in [17]. For the adapted solutions, the unique solvability
of BSVIEs, under more general stochastic non-Lipschitz conditions, is obtained
which generalizes the results in [6], [12] and [14].
http://arxiv.org/abs/1001.3557
Author(s): Tianxiao Wang
Abstract: This paper is concerned with existence and uniqueness of M-solutions of
backward stochastic Volterra integral equations (BSVIEs for short), which
Lipschitz coefficients are allowed to be random, which generalize the results
in [15]. Then a class of continuous time dynamic dynamic coherent risk measures
is derived, allowing the riskless interest rate to be random, which is
different from the case in [15].
http://arxiv.org/abs/1001.3558
Author(s): Richard F. Bass
Abstract: Under very general conditions the hitting time of a set by a stochastic
process is a stopping time. We give a new simple proof of this fact. The
section theorems foroptional and predictable sets are easy corollaries of the
proof.
http://arxiv.org/abs/1001.3619
Author(s): Alexei Borodin
Abstract: We construct discrete time Markov chains that preserve the class of Schur
processes on partitions and signatures.
One application is a simple exact sampling algorithm for
q^{volume}-distributed skew plane partitions with an arbitrary back wall.
Another application is a construction of Markov chains on infinite
Gelfand-Tsetlin schemes that represent deterministic flows on the space of
extreme characters of the infinite-dimensional unitary group.
http://arxiv.org/abs/1001.3442
Author(s): Michael Ludkovski
Abstract: We study optimal behavior of energy producers under a CO_2 emission abatement
program. We focus on a two-player discrete-time model where each producer is
sequentially optimizing her emission and production schedules. The
game-theoretic aspect is captured through a reduced-form price-impact model for
the CO_2 allowance price. Such duopolistic competition results in a new type of
a non-zero-sum stochastic switching game on finite horizon. Existence of game
Nash equilibria is established through generalization to randomized switching
strategies. No uniqueness is possible and we therefore consider a variety of
correlated equilibrium mechanisms. We prove existence of correlated equilibrium
points in switching games and give a recursive description of equilibrium game
values. A simulation-based algorithm to solve for the game values is
constructed and a numerical example is presented.
http://arxiv.org/abs/1001.3455
Author(s): Zdzislaw Brzezniak and Rafael Serrano
Abstract: We study an optimal relaxed control problem for a class of semilinear
stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven
by a cylindrical Wiener process. The state equation is controlled through the
nonlinear part of the drift coefficient which satisfies a dissipative-type
condition with respect to the state variable. The main tools of our study are
the factorization method for stochastic convolutions in UMD type-2 Banach
spaces and certain compactness properties of the factorization operator and of
the class of Young measures on Suslin metrisable control sets.
http://arxiv.org/abs/1001.3165
Author(s): Dapeng Zhan
Abstract: We use the coupling technique to prove that there exists a loop-erasure of a
plane Brownian motion stopped on exiting a simply connected domain, and the
loop-erased curve is the reversal of a radial SLE$_2$ curve.
http://arxiv.org/abs/1001.3189
Author(s): Arijit Chakrabarty and Gennady Samorodnitsky
Abstract: We address the important question of the extent to which random variables and
vectors with truncated power tails retain the characteristic features of random
variables and vectors with power tails. We define two truncation regimes, soft
truncation regime and hard truncation regime, and show that, in the soft
truncation regime, truncated power tails behave, in important respects, as if
no truncation took place. On the other hand, in the hard truncation regime much
of "heavy tailedness" is lost. We show how to estimate consistently the tail
exponent when the tails are truncated, and suggest statistical tests to decide
on whether the truncation is soft or hard. Finally, we apply our methods to two
recent data sets arising from computer networks.
http://arxiv.org/abs/1001.3218
Author(s): Myriam Fradon and Sylvie Roelly
Abstract: We consider an infinite system of non overlapping globules undergoing
Brownian motions in R^3. The term globules means that the objects we are
dealing with are spherical, but with a radius which is random and
time-dependent. The dynamics is modelized by an infinite-dimensional Stochastic
Differential Equation with local time. Existence and uniqueness of a strong
solution is proven for such an equation with fixed deterministic initial
condition. We also find a class of reversible measures.
http://arxiv.org/abs/1001.3252
Author(s): Thomas Meinguet and Johan Segers
Abstract: When a spatial process is recorded over time and the observation at a given
time instant is viewed as a point in a function space, the result is a time
series taking values in a Banach space. To study the spatio-temporal extremal
dynamics of such a time series, the latter is assumed to be jointly regularly
varying. This assumption is shown to be equivalent to convergence in
distribution of the rescaled time series conditionally on the event that at a
given moment in time it is far away from the origin. The limit is called the
tail process or the spectral process depending on the way of rescaling. These
processes provide convenient starting points to study, for instance, joint
survival functions, tail dependence coefficients, extremograms, extremal
indices, and point processes of extremes. The theory applies to linear
processes composed of infinite sums of linearly transformed independent random
elements whose common distribution is regularly varying.
http://arxiv.org/abs/1001.3262
Author(s): Boualem Djehiche (KTH Stockolm) and Said Hamad\`ene (LMM) and Marie Am\'elie Morlais (LMM)
Abstract: We consider a finite horizon optimal stopping problem related to trade-off
strategies between expected profit and cost cash-flows of an investment under
uncertainty. The optimal problem is first formulated in terms of a system of
Snell envelopes for the profit and cost yields which act as obstacles to each
other. We then construct both a minimal and a maximal solutions using an
approximation scheme of the associated system of reflected backward SDEs. When
the dependence of the cash-flows on the sources of uncertainty, such as
fluctuation market prices, assumed to evolve according to a diffusion process,
is made explicit, we also obtain a connection between these solutions and
viscosity solutions of a system of variational inequalities (VI) with
interconnected obstacles. We also provide two counter-examples showing that
uniqueness of solutions of (VI) does not hold in general.
http://arxiv.org/abs/1001.3289
Author(s): Aur\'elien Deya (IECN) and Andreas Neuenkirch and Samy Tindel (IECN)
Abstract: In this article, we study the numerical approximation of stochastic
differential equations driven by a multidimensional fractional Brownian motion
(fBm) with Hurst parameter greater than 1/3. We introduce an implementable
scheme for these equations, which is based on a second order Taylor expansion,
where the usual Levy area terms are replaced by products of increments of the
driving fBm. The convergence of our scheme is shown by means of a combination
of rough paths techniques and error bounds for the discretisation of the Levy
area terms.
http://arxiv.org/abs/1001.3344
Author(s): Ana Bela Cruzeiro and Evelina Shamarova
Abstract: We describe a probabilistic construction of $H^s$-regular solutions for the
spatially periodic Burgers equation by using a characterization of this
solution through a forward-backward stochastic system.
http://arxiv.org/abs/1001.3367
Author(s): Anne Fey and Lionel Levine and and David B. Wilson
Abstract: A popular theory of self-organized criticality relates the critical behavior
of driven dissipative systems to that of systems with conservation. In
particular, this theory predicts that the stationary density of the abelian
sandpile model should be equal to the threshold density of the corresponding
fixed-energy sandpile. This "density conjecture" has been proved for the
underlying graph Z. We show (by simulation or by proof) that the density
conjecture is false when the underlying graph is any of Z^2, the complete graph
K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower
graph. Driven dissipative sandpiles continue to evolve even after a constant
fraction of the sand has been lost at the sink. These results cast doubt on the
validity of using fixed-energy sandpiles to explore the critical behavior of
the abelian sandpile model at stationarity.
http://arxiv.org/abs/1001.3401
Author(s): Aernout van Enter and Evgeny Verbitskiy
Abstract: Recently Verdu and Weissman introduced erasure entropies, which are meant to
measure the information carried by one or more symbols given all of the
remaining symbols in the realization of the random process or field. A natural
relation to Gibbs measures has also been observed. In his short note we study
this relation further, review a few earlier contributions from statistical
mechanics, and provide the formula for the erasure entropy of a Gibbs measure
in terms of the corresponding potentia. For some
2-dimensonal Ising models, for which Verdu and Weissman suggested a numerical
procedure, we show how to obtain an exact formula for the erasure entropy. l
http://arxiv.org/abs/1001.3122
Author(s): Michael Borokhovich and Chen Avin and Zvi Lotker
Abstract: We study the stopping times of gossip algorithms for network coding. We
analyze algebraic gossip (i.e., random linear coding) and consider three gossip
algorithms for information spreading Pull, Push, and Exchange. The stopping
time of algebraic gossip is known to be linear for the complete graph, but the
question of determining a tight upper bound or lower bounds for general graphs
is still open. We take a major step in solving this question, and prove that
algebraic gossip on any graph of size n is O(D*n) where D is the maximum degree
of the graph. This leads to a tight bound of Theta(n) for bounded degree graphs
and an upper bound of O(n^2) for general graphs. We show that the latter bound
is tight by providing an example of a graph with a stopping time of Omega(n^2).
Our proofs use a novel method that relies on Jackson's queuing theorem to
analyze the stopping time of network coding; this technique is likely to become
useful for future research.
http://arxiv.org/abs/1001.3265
Author(s): Evelina Shamarova
Abstract: The classical Chernoff's theorem is a statement about discrete-time
approximations of semigroups, where the approximations are consturcted as
products of time-dependent contraction operators strongly differentiable at
zero. We generalize the version of Chernoff's theorem for semigroups proved in
a paper by Smolyanov et al., and obtain a theorem about descrete-time
approximations of backward propagators.
http://arxiv.org/abs/1001.3373
Author(s): Jorge Littin and Servet Martinez
Abstract: We revisit the $R-$positivity of nearest neighbors matrices on ${\ZZ_+}$ and
the Gibbs measures on the set of nearest neighbors trajectories on ${\ZZ_+}$
whose Hamiltonians award either visits to sites a or visits to edges. We give
conditions that guarantee the $R-$positivity or equivalently the existence of
the infinite volume Gibbs measure, and we show geometrical recurrence of the
associated Markov chain. In this work we generalize and sharpen results
obtained in [3] and [6].
http://arxiv.org/abs/1001.2782
Author(s): Yongsheng Song
Abstract: In this article, a sublinear expectation induced by $G$-expectation is
introduced, which is called $G$-evaluation for convenience. As an application,
we prove that any $\xi\in L^\beta_G(\Omega_T)$ with some $\beta>1$ the
decomposition theorem holds and any $\beta>1$ integrable symmetric
$G$-martingale can be represented as an It$\hat{o}'s$ integral w.r.t
$G$-Brownian motion. As a byproduct, we prove a regular property for
$G$-martingale: Any $G$-martingale $\{M_t\}$ has a quasi-continuous version
http://arxiv.org/abs/1001.2802
Author(s): Xuehong Zhu
Abstract: This paper is concerned with the Dynamic Programming Principle (DPP in short)
with SDEs on Riemannian manifolds. Moreover, through the DPP, we conclude that
the cost function is the unique viscosity solution to the related PDEs on
manifolds.
http://arxiv.org/abs/1001.2820
Author(s): Ricardo Castro Santis and Alberto Barchielli
Abstract: A natural formulation of the theory of quantum measurements in continuous
time is based on quantum stochastic differential equations
(Hudson-Parthasarathy equations). However, such a theory was developed only in
the case of Hudson-Parthasarathy equations with bounded coefficients. By using
some results on Hudson-Parthasarathy equations with unbounded coefficients, we
are able to extend the theory of quantum continuous measurements to cases in
which unbounded operators on the system space are involved. A significant
example of a quantum optical system (the degenerate parametric oscillator) is
shown to fulfill the hypotheses introduced in the general theory.
http://arxiv.org/abs/1001.2826
Author(s): Iosif Pinelis
Abstract: Exact lower bounds on the exponential moments of min(y,X) and XI{X
http://arxiv.org/abs/1001.2901
Author(s): Serik Sagitov and Peter Jagers and Vladimir Vatutin
Abstract: We study the genealogy of a geographically - or otherwise - structured
version of the Wright-Fisher population model with fast migration. The new
feature is that migration probabilities may change in a random fashion.
Applying Takahashi's results on Markov chains with random transition matrices,
we establish convergence to the Kingman coalescent, as the population size goes
to infinity.
This brings a novel formula for the coalescent effective population size
(EPS). We call it a quenched EPS to emphasize the key feature of our model -
random environment. The quenched EPS is compared with an annealed (mean-field)
EPS which describes the case of constant migration probabilities obtained by
averaging the random migration probabilities over possible environments.
http://arxiv.org/abs/1001.2907
Author(s): P. Friz and S. Gerhold and A. Gulisashvili and S. Sturm
Abstract: It is known that Heston's stochastic volatility model exhibits moment
explosion, and that the critical moment $s^{*}$ can be obtained by solving
(numerically) a simple equation. This yields a leading order expansion for the
implied volatility at large strikes: $\sigma_{BS}(k,T)^{2}T\sim \Psi (s^*-1)
\times k$ (Roger Lee's moment formula). Motivated by recent "tail-wing"
refinements of this moment formula, we first derive a novel tail expansion for
the Heston density, sharpening previous work of Dr{\u{a}}gulescu and Yakovenko
[Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined
expansion of the type $% \sigma_{BS}(k,T) ^{2}T=(\beta
_{1}k^{1/2}+\beta_{2}+...)^{2}$, where all constants are explicitly known as
functions of $s^*$, the Heston model parameters, spot vol and maturity $T$. In
the case of the "zero-correlation" Heston model such an expansion was derived
by Gulisashvili and Stein [Appl. Math. Opt., DOI: 10.1007/s002450099085]. Our
methods and results may prove useful beyond the Heston model: the entire
quantitative analysis is based on affine principles; at no point do we need
knowledge of the (explicit, but cumbersome) closed form expression of the
Fourier transform of $\log S_{T}$ (equivalently: Mellin transform of $S_{T}$%
). Secondly, our analysis reveals a new parameter ("\textit{critical slope}"%
), defined in a model free manner, which drives the second and higher order
terms in tail- and implied volatility expansions.
http://arxiv.org/abs/1001.3003
Author(s): Shizan Fang and Dejun Luo and Anto Thalmaier
Abstract: We consider It\^o SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$
on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the
Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear
growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is
continuous whose distributional divergence $\delta(A_0)$ w.r.t. the Gaussian
measure $\gamma_d$ exists, (ii) $A_0$ has the Sobolev regularity
$W_\text{loc}^{1,p'}$ for some $p'>1$. Assume $\int_{\R^d}
\exp\big[\lambda_0\bigl(|\delta(A_0)| + \sum_{j=1}^m (|\delta(A_j)|^2 +|\nabla
A_j|^2)\bigr)\big] \d\gamma_d<+\infty$ for some $\lambda_0>0$, in the case (i),
if the pathwise uniqueness of solutions holds, then the push-forward $(X_t)_#
\gamma_d$ admits a density with respect to $\gamma_d$. In particular, if the
coefficients are bounded Lipschitz continuous, then $X_t$ leaves the Lebesgue
measure $\Leb_d$ quasi-invariant. In the case (ii), we develop a method used by
G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to
establish the existence and uniqueness of stochastic flow of maps.
http://arxiv.org/abs/1001.3007
Author(s): Lifen An and Shaolin Ji
Abstract: In this paper, we firstly establish the discrete time and finite state
reflected backward stochastic difference equations(DF-RBSDE for short); then we
explore the corresponding basic properties and theorems including the Existence
and Uniqueness Theorem as well as the Comparison Theorem in our framework by
"one step" method; afterwards, we show the connections between DF-RBSDE and
optimal stopping time problems. For applications, we study the connection
between DF-RBSDE and the general theory of g-martingales and multiple prior
martingale including Doob-Mayer Decomposition Theorem and Optional Sampling
Theorem in our framework; and then we apply the theory of DF-RBSDEs to multiple
prior martingale and optimal stopping problems under Knightian uncertainty;
finally, applying the above theories, we consider the pricing models of
American Option in complete and incomplete markets.
http://arxiv.org/abs/1001.3054
Author(s): Ronan Le Gu\'evel (LMJL) and Jacques L\'evy-V\'ehel (INRIA Saclay - Ile de France)
Abstract: Multistable processes, that is, processes which are, at each ``time'',
tangent to a stable process, but where the index of stability varies along the
path, have been recently introduced as models for phenomena where the intensity
of jumps is non constant. In this work, we give further results on
(multifractional) multistable processes related to their local structure. We
show that, under certain conditions, the incremental moments display a scaling
behaviour, and that the pointwise exponent is, as expected, equal to the
localisability index.
http://arxiv.org/abs/1001.3130
Author(s): Markus Rei\ss
Abstract: The basic model for high-frequency data in finance is considered, where an
efficient price process is observed under microstructure noise. It is shown
that this nonparametric model is in Le Cam's sense asymptotically equivalent to
a Gaussian shift experiment in terms of the square root of the volatility
function $\sigma$. As an application, simple rate-optimal estimators of the
volatility and efficient estimators of the integrated volatility are
constructed.
http://arxiv.org/abs/1001.3006
Author(s): Miaomiao Fu and Zhenxin Liu
Abstract: The concept of square-mean almost automorphy for stochastic processes is
introduced. The existence and uniqueness of square-mean almost automorphic
solutions to some linear and non-linear stochastic differential equations are
established provided the coefficients satisfy some conditions. The asymptotic
stability of the unique square-mean almost automorphic solution in square-mean
sense is discussed.
http://arxiv.org/abs/1001.3049
Author(s): Frederic Bernicot (LPP) and Juliette Venel (LAMAV)
Abstract: Here we present well-posedness results for first order stochastic
differential inclusions, more precisely for sweeping process with a stochastic
perturbation. These results are provided in combining both deterministic
sweeping process theory and methods concerning the reflection of a Brownian
motion. In addition, we prove convergence results for a Euler scheme,
discretizing theses stochastic differential inclusions.
http://arxiv.org/abs/1001.3128
Author(s): Tadeusz Banek
Abstract: A simple nonlinear integral equation for Ito's map is obtained. Although, it
does not include stochastic integrals, it does give causal construction of
diffusion processes which can be easily implemented by iteration systems.
Applications in financial modelling and extension to fBm are discussed.
http://arxiv.org/abs/1001.2715
Author(s): Arnulf Jentzen and Michael Roeckner
Abstract: A new numerical method for stochastic partial differential equations (SPDEs)
of evolutionary type, which is in some sense the infinite dimensional analog of
Milstein's scheme for finite dimensional stochastic ordinary differential
equations (SODEs), is introduced and analyzed in this article. The Milstein
scheme is known to be impressively efficient for scalar one-dimensional SODEs
but only for some special multidimensional SODEs due to difficult simulations
of iterated stochastic integrals in the general multidimensional SODE case. It
is a key observation of this article that, in contrast to what one may expect,
its infinite dimensional counterpart introduced here is very easy to simulate
and this, therefore, leads to a break of the complexity (number of
computational operations and random variables needed to compute the scheme) in
comparison to previously considered algorithms for simulating nonlinear SPDEs
with multiplicative trace class noise.
http://arxiv.org/abs/1001.2751
Author(s): Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg
Abstract: We consider a system of particles which perform branching Brownian motion
with negative drift and are killed upon reaching zero, in the near-critical
regime where the total population stays roughly constant with approximately N
particles. We show that the characteristic time scale for the evolution of this
population is of order (log N)^3, in the sense that when time is measured in
these units, the scaled number of particles converges to a variant of Neveu's
continuous-state branching process. Furthermore, the genealogy of the particles
is then governed by a coalescent process known as the Bolthausen-Sznitman
coalescent. This validates the non-rigorous predictions by Brunet, Derrida,
Muller, and Munier for a closely related model.
http://arxiv.org/abs/1001.2337
Author(s): Guodong Pang and Ward Whitt
Abstract: We establish continuity of the integral representation
$y(t)=x(t)+\int_0^th(y(s)) ds$, $t\ge0$, mapping a function $x$ into a function
$y$ when the underlying function space $D$ is endowed with the Skorohod $M_1$
topology. We apply this integral representation with the continuous mapping
theorem to establish heavy-traffic stochastic-process limits for many-server
queueing models when the limit process has jumps unmatched in the converging
processes as can occur with bursty arrival processes or service interruptions.
The proof of $M_1$-continuity is based on a new characterization of the $M_1$
convergence, in which the time portions of the parametric representations are
absolutely continuous with respect to Lebesgue measure, and the derivatives are
uniformly bounded and converge in $L_1$.
http://arxiv.org/abs/1001.2381
Author(s): C. Boeinghoff and E.E. Dyakonova and G. Kersting and V.A. Vatutin
Abstract: Let ${Z_{n},n\geq 0} $ be a critical branching process in random environment
and let $T$ be its moment of extinction. Under the annealed approach we prove,
as $n\to \infty ,$ a limit theorem for the number of particles in the process
at moment $n$ given $T=n+1$ and a functional limit theorem for the properly
scaled process ${Z_{nt},\delta \leq t\leq 1-\delta} $ given $T=n+1$ and $\delta
\in (0,1/2)$.
http://arxiv.org/abs/1001.2413
Author(s): Jean-Christophe Mourrat
Abstract: Attributing a positive value \tau_x to each x in Z^d, we investigate a
nearest-neighbour random walk which is reversible for the measure with weights
(\tau_x), often known as "Bouchaud's trap model". We assume that these weights
are independent, identically distributed and non-integrable random variables
(with polynomial tail), and that d > 4. We obtain the quenched subdiffusive
scaling limit of the model, the limit being the fractional kinetics process. We
begin our proof by expressing the random walk as a time change of a random walk
among random conductances. We then focus on proving that the time change
converges, under the annealed measure, to a stable subordinator. This is
achieved using previous results concerning the mixing properties of the
environment viewed by the time-changed random walk.
http://arxiv.org/abs/1001.2459
Author(s): Simon C. Harris and Matthew I. Roberts
Abstract: For a set $A\subset C[0,\infty)$, we give new results on the growth of the
number of particles in a dyadic branching Brownian motion whose paths fall
within A. We show that it is possible to work without rescaling the paths. We
give large deviations probabilities as well as a more sophisticated proof of a
result on growth in the number of particles along certain sets of paths. Our
results reveal that the number of particles can oscillate dramatically. As a
byproduct of our methods we also obtain new results on the number of particles
near the frontier of the model. The methods used are entirely probabilistic.
http://arxiv.org/abs/1001.2471
Author(s): K. Pakdaman and M. Thieullen and G. Wainrib
Abstract: This paper establishes limit theorems for a class of stochastic hybrid
systems (continuous deterministic dynamic coupled with jump Markov processes)
in the fluid limit (small jumps at high frequency), thus extending known
results for jump Markov processes. We prove a functional law of large numbers
with exponential convergence speed, derive a diffusion approximation and
establish a functional central limit theorem. We apply these results to neuron
models with stochastic ion channels, as the number of channels goes to
infinity, estimating the convergence to the deterministic model. In terms of
neural coding, we apply our central limit theorems to estimate numerically
impact of channel noise both on frequency and spike timing coding.
http://arxiv.org/abs/1001.2474
Author(s): Christophe Gallesco and Sebastian Muller and Serguei Popov and Marina Vachkovskaia
Abstract: A spider consists of several, say $N$, particles. Particles can jump
independently according to a random walk if the movement does not violate some
given restriction rules. If the movement violates a rule it is not carried out.
We consider random walk in random environment (RWRE) on $\Z$ as underlying
random walk. We suppose the environment $\omega=(\omega_x)_{x \in \Z}$ to be
elliptic, with positive drift and nestling, so that there exists a unique
positive constant $\kappa$ such that $\E[((1-\omega_0)/\omega_0)^{\kappa}]=1$.
The restriction rules are kept very general; we only assume transitivity and
irreducibility of the spider. The main result is that the speed of a spider is
positive if $\kappa/N>1$ and null if $\kappa/N<1$. In particular, if $\kappa/N
<1$ a spider has null speed but the speed of a (single) RWRE is positive.
http://arxiv.org/abs/1001.2533
Author(s): Sriram Vishwanath
Abstract: This paper studies the low-rank matrix completion problem from an information
theoretic perspective. The completion problem is rephrased as a communication
problem of an (uncoded) low-rank matrix source over an erasure channel. The
paper then uses achievability and converse arguments to present order-wise
optimal bounds for the completion problem.
http://arxiv.org/abs/1001.2331
Author(s): T. N. Narasimhan
Abstract: Although the same mathematical expression is used to describe physical
diffusion and stochastic diffusion, there are intrinsic similarities and
differences in their nature. A comparative study shows that characteristic
terms of physical and stochastic diffusion cannot be placed exactly in
one-to-one correspondence. Therefore, judgment needs to be exercised in
transferring ideas between physical and stochastic diffusion.
http://arxiv.org/abs/1001.2357
Author(s): Hiroaki Hata and Hideo Nagai and Shuenn-Jyi Sheu
Abstract: We consider a long-term optimal investment problem where an investor tries to
minimize the probability of falling below a target growth rate. From a
mathematical viewpoint, this is a large deviation control problem. This problem
will be shown to relate to a risk-sensitive stochastic control problem for a
sufficiently large time horizon. Indeed, in our theorem we state a duality in
the relation between the above two problems. Furthermore, under a
multidimensional linear Gaussian model we obtain explicit solutions for the
primal problem.
http://arxiv.org/abs/1001.2131
Author(s): Roland Schnaubelt and Mark Veraar
Abstract: We study the wellposedness and pathwise regularity of semilinear
non-autonomous parabolic evolution equations with boundary and interior noise
in an $L^p$ setting. We obtain existence and uniqueness of mild and weak
solutions. The boundary noise term is reformulated as a perturbation of a
stochastic evolution equation with values in extrapolation spaces.
http://arxiv.org/abs/1001.2137
Author(s): Lech Jankowski and Zbigniew J. Jurek
Abstract: The Nevanlinna transform K(z), of a measure and a real constant, plays an
important role in the complex analysis and more recently in the free
probability theory (boolean convolution). It is shown that its restriction
k(it) (the restricted Nevanlinna transform) to the imaginary axis can be
expressed as the Laplace transform of the Fourier transform (characteristic
function) of the corresponding measure. Finally, a relation between the
Voiculescu and the boolean convolution is indicated.
http://arxiv.org/abs/1001.2154
Author(s): Anatolii A. Puhalskii and Josh E. Reed
Abstract: We establish a heavy-traffic limit theorem on convergence in distribution for
the number of customers in a many-server queue when the number of servers tends
to infinity. No critical loading condition is assumed. Generally, the limit
process does not have trajectories in the Skorohod space. We give conditions
for the convergence to hold in the topology of compact convergence. Some new
results for an infinite server are also provided.
http://arxiv.org/abs/1001.2163
Author(s): Manuel S. Santos
Abstract: This paper considers a simulation-based estimator for a general class of
Markovian processes and explores some strong consistency properties of the
estimator. The estimation problem is defined over a continuum of invariant
distributions indexed by a vector of parameters. A key step in the method of
proof is to show the uniform convergence (a.s.) of a family of sample
distributions over the domain of parameters. This uniform convergence holds
under mild continuity and monotonicity conditions on the dynamic process. The
estimator is applied to an asset pricing model with technology adoption. A
challenge for this model is to generate the observed high volatility of stock
markets along with the much lower volatility of other real economic aggregates.
http://arxiv.org/abs/1001.2173
Author(s): Assane Diop (PMA)
Abstract: In this paper, we study the asymptotic behavior of sums of functions of the
increments of a given semimartingale, taken along a regular grid whose mesh
goes to 0. The function of the $i$th increment may depend on the current time,
and also on the past of the semimartingale before this time. We study the
convergence in probability of two types of such sums, and we also give
associated central limit theorems. This extends known results when the summands
are a function depending only on the increments, and this is motivated mainly
by statistical applications.
http://arxiv.org/abs/1001.2182
Author(s): Valentin Feray (LaBRI) and Pierre-Lo\"ic M\'eliot (IGM-LabInfo)
Abstract: In this paper, we are interested in the asymptotic size of rows and columns
of a random Young diagram under a natural deformation of the Plancherel measure
coming from Hecke algebras. The first lines of such diagrams are typically of
order $n$, so it does not fit in the context studied by P. Biane and P.
\'Sniady. Using the theory of polynomial functions on Young diagrams of Kerov
and Olshanski, we are able to compute explicitly the first- and second-order
asymptotics of the length of the first rows. Our method works also for other
measures, for instance those coming from Schur-Weyl representations.
http://arxiv.org/abs/1001.2180
Author(s): Mikhail Menshikov and Serguei Popov and Alejandro Ramirez and Marina Vachkovskaia
Abstract: The generalized excited random walk is a generalization of the excited random
walk, introduced in 2003 by Benjamini and Wilson, which is a discrete-time
stochastic process $(X_n,n=0,1,2,...)$ taking values on $\Z^d$, $d\geq 2$,
described as follows: when the particle visits a site for the first time, it
has a uniformly positive drift in a given direction $\ell$; when the particle
is at a site which was already visited before, it has zero drift. Assuming
uniform ellipticity and that the jumps of the process are uniformly bounded, we
prove that the process is ballistic in the direction $\ell$ so that
$\liminf_{n\to\infty}\frac{X_n\cdot \ell}{n}>0$. A key ingredient in the proof
of this result is an estimate on the probability that the process visits less
than $n^{{1/2}+\alpha}$ distinct sites by time $n$, where $\alpha$ is some
positive number depending on the parameters of the model. This approach
completely avoids the use of tan points and coupling methods specific to the
excited random walk. Furthermore, we apply this technique to prove that the
excited random walk in an i.i.d. random environment satisfies a ballistic law
of large numbers and a central limit theorem.
http://arxiv.org/abs/1001.1741
Author(s): Patrick Cattiaux (IMT) and Arnaud Guillin
Abstract: We review here some recent results by the authors, and various coauthors, on
(weak,super) Poincar\'e inequalities, transportation-information inequalities
or logarithmic Sobolev inequality via a quite simple and efficient technique:
Lyapunov conditions.
http://arxiv.org/abs/1001.1822
Author(s): Ansgar Steland
Abstract: Aiming at monitoring a time series to detect stationarity as soon as
possible, we introduce monitoring procedures based on kernel-weighted
sequential Dickey-Fuller (DF) processes, and related stopping times, which may
be called weighted Dickey-Fuller control charts. Under rather weak assumptions,
(functional) central limit theorems are established under the unit root null
hypothesis and local-to-unity alternatives. For gen- eral dependent and
heterogeneous innovation sequences the limit processes depend on a nuisance
parameter. In this case of practical interest, one can use estimated control
limits obtained from the estimated asymptotic law. Another easy-to-use approach
is to transform the DF processes to obtain limit laws which are invariant with
respect to the nuisance pa- rameter. We provide asymptotic theory for both
approaches and compare their statistical behavior in finite samples by
simulation.
http://arxiv.org/abs/1001.1833
Author(s): Ansgar Steland
Abstract: The question whether a time series behaves as a random walk or as a station-
ary process is an important and delicate problem, particularly arising in
financial statistics, econometrics, and engineering. This paper studies the
problem to detect sequentially that the error terms in a polynomial regression
model no longer behave as a random walk but as a stationary process. We provide
the asymptotic distribution theory for a monitoring procedure given by a
control chart, i.e., a stopping time, which is related to a well known unit
root test statistic calculated from sequentially updated residuals. We provide
a functional central limit theorem for the corresponding stochastic process
which implies a central limit theorem for the control chart. The finite sample
properties are investigated by a simulation study.
http://arxiv.org/abs/1001.1845
Author(s): Nathanael Berestycki and Oded Schramm and Ofer Zeitouni
Abstract: Let S_n be the permutation group on n elements, and consider a random walk on
S_n whose step distribution is uniform on k-cycles. We prove a well-known
conjecture that the mixing time of this process is (1/k) n \log n, with
threshold of width linear in n. Our proofs are elementary and purely
probabilistic, and do not appeal to the representation theory of S_n.
http://arxiv.org/abs/1001.1894
Author(s): Elizabeth S. Meckes and Mark W. Meckes
Abstract: Let $T$ be a self-adjoint operator on a finite dimensional Hilbert space. It
is shown that the distribution of the eigenvalues of a compression of $T$ to a
subspace of a given dimension is almost the same for almost all subspaces. This
is a coordinate-free analogue of a recent result of Chatterjee and Ledoux on
principal submatrices. The proof is based on measure concentration and entropy
techniques, and the result improves on some aspects of the result of Chatterjee
and Ledoux.
http://arxiv.org/abs/1001.1954
Author(s): Andrzej {\L}uczak
Abstract: The paper deals with various centering problems for probability measures on
finite dimensional vector spaces. We show that for every such measure there
exists a vector $h$ satisfying $\mu*\delta(h)=S(\mu*\delta (h))$ for each
symmetry $S$ of $\mu$, generalizing thus Jurek's result obtained for full
measures. An explicit form of the $h$ is given for infinitely divisible $\mu$.
The main result of the paper consists in the analysis of quasi-decomposable
(operator-semistable and operator-stable) measures and finding conditions for
the existence of a `universal centering' of such a measure to a strictly
quasi-decomposable one.
http://arxiv.org/abs/1001.1963
Author(s): Kathrin Bringmann and Karl Mahlburg
Abstract: We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and
Romik on metastability thresholds for generalized two-dimensional bootstrap
percolation models, and answer several of their open problems and conjectures.
Specifically, we prove slow convergence and localization bounds for Holroyd,
Liggett, and Romik's k-percolation models, and in the process provide a unified
and improved treatment of existing results for bootstrap, modified bootstrap,
and Frobose percolation. Furthermore, we prove improved asymptotic bounds for
the generating functions of partitions without k-gaps, which are also related
to certain infinite probability processes relevant to these percolation models.
One of our key technical probability results is also of independent interest.
We prove new upper and lower bounds for the probability that a sequence of
independent events with monotonically increasing probabilities contains no
``k-gap'' patterns, which interpolates the general Markov chain solution that
arises in the case that all of the probabilities are equal.
http://arxiv.org/abs/1001.1977
Author(s): Ansgar Steland
Abstract: In this paper sequential monitoring schemes to detect nonparametric drifts
are studied for the random walk case. The procedure is based on a kernel
smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson
estimator and its as- sociated sequential partial sum process under
non-standard sampling. The asymptotic behavior differs substantially from the
stationary situation, if there is a unit root (random walk component). To
obtain meaningful asymptotic results we consider local nonpara- metric
alternatives for the drift component. It turns out that the rate of convergence
at which the drift vanishes determines whether the asymptotic properties of the
monitoring procedure are determined by a deterministic or random function.
Further, we provide a theoretical result about the optimal kernel for a given
alternative.
http://arxiv.org/abs/1001.1828
Author(s): Keisuke Hara and Masanori Hino
Abstract: We prove the neo-classical inequality with the optimal constant, which was
conjectured by T. J. Lyons [Rev. Mat. Iberoamericana 14 (1998) 215-310]. For
the proof, we introduce the fractional order Taylor's series with residual
terms. Their application to a particular function provides an identity that
deduces the optimal neo-classical inequality.
http://arxiv.org/abs/1001.1775
Author(s): Behrouz Touri and Angelia Nedi'c
Abstract: We consider the ergodicity and consensus problem for a discrete-time linear
dynamic model driven by random matrices, which is equivalent to studying these
concepts for the product of random matrices. Our focus is on the model where
the matrices are "stochastic". We introduce a new phenomena, the infinite flow,
and we study its fundamental properties and relations with the ergodicity and
consensus. We establish several new and important results. The central result
of this work is the infinite flow theorem establishing the role of infinite
flow in the ergodicity of a general independent random model. Through the use
of infinite flow, we show that the ergodicity of the model is equivalent to the
ergodicity of the expected model when the matrices in the model have a common
steady state in expectation and a feedback property. This result demonstrates
that for such models, the expected infinite flow is both necessary and
sufficient for the ergodicity. The result is providing us with a powerful
deterministic characterization of the ergodicity, which renders a new elegant
tool that can be used for studying the consensus and average consensus over
random graphs, as well as random consensus algorithms.
http://arxiv.org/abs/1001.1890
Author(s): Andrzej {\L}uczak
Abstract: We construct quantum stochastic integrals for the integrator being a
martingale in a von Neumann algebra, and the integrand -- a suitable process
with values in the same algebra, as densely defined operators affiliated with
the algebra. In the case of a finite algebra we allow the integrator to be an
$L^2$--martingale in which case the integrals are $L^2$--martingales too.
http://arxiv.org/abs/1001.1959
Author(s): Subhankar Ghosh
Abstract: Let $(\mathbf{W,W'})$ be an exchangeable pair of vectors in $\mathbb{R}^k$.
Suppose this pair satisfies \beas
E(\mathbf{W}'|\mathbf{W})=(I_k-\Lambda)\mathbf{W}+\mathbf{R(W)}. \enas If
$||\mathbf{W-W'}||_2\le K$ and $\mathbf{R(W)}=0$, then concentration of measure
results of following form is proved for all $\mathbf{w}\succeq 0$ when the
moment generating function of $\mathbf{W}$ is finite. \beas
P(\mathbf{W}\succeq\mathbf{w}),P(\mathbf{W}\preceq -\mathbf{w})\le
\exp(-\frac{||\mathbf{w}||_2^2}{2K^2\nu_1}), \enas for an explicit constant
$\nu_1$, where $\succeq$ stands for coordinate wise $\ge$ ordering.
This result is applied to examples like complete non degenerate U-statistics.
Also, we deal with the example of doubly indexed permutation statistics where
$\mathbf{R(W)}\neq 0$ and obtain similar concentration of measure inequalities.
Practical examples from doubly indexed permutation statistics include
Mann-Whitney-Wilcoxon statistic and random intersection of two graphs. Both
these two examples are used in nonparametric statistical testing. We conclude
the paper with a multivariate generalization of a recent concentration result
due to Ghosh and Goldstein \cite{cnm} involving bounded size bias couplings.
http://arxiv.org/abs/1001.1396
Author(s): Alan Hammond
Abstract: This is the first in a series of three papers that addresses the behaviour of
the droplet that results, in the percolating phase, from conditioning the
Fortuin-Kasteleyn random cluster model on the presence of an open dual circuit
Gamma_0 encircling the origin and enclosing an area of at least (or exactly)
n^2. (By the Fortuin-Kasteleyn representation, the model is a close relative of
the droplet formed by conditioning the Potts model on an excess of spins of a
given type.) We consider local deviation of the droplet boundary, measured in a
radial sense by the maximum local roughness, MLR(Gamma_0), this being the
maximum distance from a point in the circuit Gamma_0 to the boundary of the
circuit's convex hull; and in a longitudinal sense by what we term maximum
facet length, MFL(Gamma_0), namely, the length of the longest line segment of
which the boundary of the convex hull is formed. The principal conclusion of
the series of papers is the following uniform control on local deviation: that
there are positive constants c and C such that the conditional probability that
the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) lies in
the interval [c,C] tends to 1 in the high n-limit; and that the same statement
holds for n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_0). In this way, we
confirm the anticipated n^{1/3} scaling of maximum local roughness, and provide
a sharp logarithmic power-law correction. This local deviation behaviour occurs
by means of locally Gaussian effects constrained globally by curvature, and we
believe that it arises in a range of radially defined stochastic interface
models, including several in the Kardar-Parisi-Zhang universality class. This
paper is devoted to proving the upper bounds in these assertions, and includes
a heuristic overview of the surgical technique used in the three papers.
http://arxiv.org/abs/1001.1527
Author(s): Alan Hammond
Abstract: We study the droplet that results from conditioning the subcritical
Fortuin-Kasteleyn random cluster model on the presence of an open circuit
Gamma_0 encircling the origin and enclosing an area of at least (or exactly)
n^2. We consider local deviation of the droplet boundary, measured in a radial
sense by the maximum local roughness, MLR(Gamma_0), this being the maximum
distance from a point in the circuit Gamma_0 to the boundary of the circuit's
convex hull; and in a longitudinal sense by what we term maximum facet length,
MFL(Gamma_0), namely, the length of the longest line segment of which the
boundary of the convex hull is formed. We prove that that there exists a
constant c > 0 such that the conditional probability that the normalised
quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) exceeds c tends to 1 in
the high n-limit; and that the same statement holds for n^{-2/3}\big(\log n
\big)^{-1/3} MFL(Gamma_0). To obtain these bounds, we exhibit the random
cluster measure conditional on the presence of an open circuit trapping high
area as the invariant measure of a Markov chain that resamples sections of the
circuit boundary. We analyse the chain at equilibrium to prove the local
roughness lower bounds. Alongside complementary upper bounds provided in
arXiv:1001.1527, the fluctuations MLR(Gamma_0) and MFL(Gamma_0) are determined
up to a constant factor.
http://arxiv.org/abs/1001.1528
Author(s): Alan Hammond
Abstract: We study the droplet that results from conditioning the subcritical
Fortuin-Kasteleyn random cluster model on the presence of an open circuit
Gamma_0 encircling the origin and enclosing an area of at least (or exactly)
n^2. In this paper, we prove that the resulting circuit is highly regular: we
define a notion of a regeneration site in such a way that, for any such element
v of Gamma_0, the circuit Gamma_0 cuts through the radial line segment through
v only at v. We show that, provided that the conditioned circuit is centred at
the origin in a natural sense, the set of regeneration sites reaches into all
parts of the circuit, with maximal distance from one such site to the next
being at most logarithmic in n with high probability. The result provides a
flexible control on the conditioned circuit that permits the use of surgical
techniques to bound its fluctuations, and, as such, it plays a crucial role in
the derivation of bounds on the local fluctuation of the circuit carried out in
arXiv:1001.1527 and arXiv:1001.1528.
http://arxiv.org/abs/1001.1529
Author(s): Sem Borst and Matthieu Jonckheere and Lasse Leskel\"a
Abstract: This paper considers a parallel system of queues fed by independent arrival
streams, where the service rate of each queue depends on the number of
customers in all of the queues. Necessary and sufficient conditions for the
stability of the system are derived, based on stochastic monotonicity and
marginal drift properties of multiclass birth and death processes. These
conditions yield a sharp characterization of stability for systems, where the
service rate of each queue is decreasing in the number of customers in other
queues, and has uniform limits as the queue lengths tend to infinity. The
results are illustrated with applications where the stability region may be
nonconvex.
http://arxiv.org/abs/1001.1560
Author(s): Eyal Lubetzky and Allan Sly
Abstract: The Ising model is widely regarded as the most studied model of spin-systems
in statistical physics. The focus of this paper is its dynamic (stochastic)
version, the Glauber dynamics, introduced in 1963 and by now the most popular
means of sampling the Ising measure. Intensive study throughout the last three
decades has yielded a rigorous understanding of the spectral-gap of the
dynamics on $\Z^2$ everywhere except at criticality. While the critical
behavior of the Ising model has long been the focus for physicists,
mathematicians have only recently developed an understanding of its critical
geometry with the advent of SLE, CLE and new tools to study conformally
invariant systems.
A rich interplay exists between the static and dynamic models. At the static
phase-transition for Ising, the dynamics is conjectured to undergo a critical
slowdown: At high temperature the inverse-gap is O(1), at the critical
$\beta_c$ it is polynomial in the side-length and at low temperature it is
exponential in it. A seminal series of papers verified this on $\Z^2$ except at
$\beta=\beta_c$ where the behavior remained a challenging open problem.
Here we establish the first rigorous polynomial upper bound for the critical
mixing, thus confirming the critical slowdown for the Ising model in $\Z^2$.
Namely, we show that on a finite box with arbitrary (e.g. fixed, free,
periodic) boundary conditions, the inverse-gap at $\beta=\beta_c$ is polynomial
in the side-length. The proof harnesses recent understanding of the scaling
limit of critical Fortuin-Kasteleyn representation of the Ising model together
with classical tools from the analysis of Markov chains.
http://arxiv.org/abs/1001.1613
Author(s): Marianna Bolla and Tamas Koi and Andras Kramli
Abstract: Testable weighted graph parameters and equivalent notions of testability are
investigated based on papers of Laszlo Lovasz and coauthors. We prove that
certain balanced minimum multiway cut densities are testable. Using this fact,
quadratic programming techniques are applied to approximate some of these
quantities. The problem is related to cluster analysis and statistical physics.
Convergence of special noisy graph sequences is also discussed.
http://arxiv.org/abs/1001.1623
Author(s): V.I. Afanasyev and C. Boeinghoff and G. Kersting and V.A. Vatutin
Abstract: For a branching process in random environment it is assumed that the
offspring distribution of the individuals varies in a random fashion,
independently from one generation to the other. Interestingly there is the
possibility that the process may at the same time be subcritical and,
conditioned on nonextinction, 'supercritical'. This so-called weakly
subcritical case is considered in this paper. We study the asymptotic survival
probability and the size of the population conditioned on non-extinction. Also
a functional limit theorem is proven, which makes the conditional
supercriticality manifest. A main tool is a new type of functional limit
theorems for conditional random walks.
http://arxiv.org/abs/1001.1672
Author(s): E B Davies
Abstract: We give an account of some results, both old and new, about any $n\times n$
Markov matrix that is embeddable in a one-parameter Markov semigroup. These
include the fact that its eigenvalues must lie in a certain region in the unit
ball. We prove that a well-known procedure for approximating a non-embeddable
Markov matrix by an embeddable one is optimal in a certain sense.
http://arxiv.org/abs/1001.1693
Author(s): Jon. Aaronson
Abstract: Relative complexity measures the complexity of a probability preserving
transformation relative to a factor being a sequence of random variables whose
exponential growth rate is the relative entropy of the extension. We prove
distributional limit theorems for the relative complexity of certain zero
entropy extensions: RWRSs whose associated random walks satisfy the
alpha-stable CLT (alpha>1). The results give invariants for relative
isomorphism of these.
http://arxiv.org/abs/1001.1433
Author(s): Piotr Graczyk and Tomasz Jakubowski
Abstract: We give a series representation of the logarithm of the bivariate Laplace
exponent $\kappa$ of $\alpha$-stable processes for almost all $\alpha \in
(0,2]$.
http://arxiv.org/abs/1001.1230
Author(s): Heinrich Matzinger and Felipe Torres
Abstract: The problem of the fluctuation of the Longest Common Subsequence (LCS) of two
i.i.d. sequences of length $n>0$ has been open for decades. There exist
contradicting conjectures on the topic. Chvatal and Sankoff conjectured in 1975
that asymptotically the order should be $n^{2/3}$, while Waterman conjectured
in 1994 that asymptotically the order should be $n$. A contiguous substring
consisting only of one type of symbol is called a block. In the present work,
we determine the order of the fluctuation of the LCS for a special model of
sequences consisting of i.i.d. blocks whose lengths are uniformly distributed
on the set $\{l-1,l,l+1\}$, with $l$ a given positive integer. We showed that
the fluctuation in this model is asymptotically of order $n$, which confirm
Waterman's conjecture. For achieving this goal, we developed a new method which
allows us to reformulate the problem of the order of the variance as a
(relatively) low dimensional optimization problem.
http://arxiv.org/abs/1001.1273
Author(s): Bojan Basrak and Danijel Krizmani\'c and and Johan Segers
Abstract: Under an appropriate regular variation condition, the affinely normalized
partial sums of a sequence of independent and identically distributed random
variables converges weakly to a non-Gaussian stable random variable. A
functional version of this is known to be true as well, the limit process being
a stable L\'evy process. The main result in the paper is that for a stationary,
regularly varying sequence for which clusters of high-threshold excesses can be
broken down into asymptotically independent blocks, the properly centered
partial sum process still converges to a stable L\'evy process. Due to
clustering, the L\'evy triple of the limit process can be different from the
one in the independent case. The convergence takes place in the space of
c\`adl\`ag functions endowed with Skorohod's $M_1$ topology, the more usual
$J_1$ topology being inappropriate as the partial sum processes may exhibit
rapid successions of jumps within temporal clusters of large values, collapsing
in the limit to a single jump. The result rests on a new limit theorem for
point processes which is of independent interest. The theory is applied to
moving average processes, squared GARCH(1,1) processes, and stochastic
volatility models.
http://arxiv.org/abs/1001.1345
Author(s): Vincent Lemaire (PMA) and Stephane Menozzi (PMA)
Abstract: We obtain non asymptotic bounds for the Monte Carlo algorithm associated to
the Euler discretization of some diffusion processes. The key tool is the
Gaussian concentration satisfied by the density of the discretization scheme.
This Gaussian concentration is derived from a Gaussian upper bound of the
density of the scheme and a modification of the so-called ``Herbst argument''
used to prove Logarithmic Sobolev inequalities. We eventually establish a
Gaussian lower bound for the density of the scheme that emphasizes the
concentration is sharp.
http://arxiv.org/abs/1001.1347
Author(s): Nathan Keller
Abstract: We show that certain statements related to the Fourier-Walsh expansion of
functions with respect to a biased measure on the discrete cube can be deduced
from the respective results for the uniform measure by a simple reduction. In
particular, we present simple generalizations to the biased measure $\mu_p$ of
the Bonami-Beckner hypercontractive inequality, and of Talagrand's lower bound
on the size of the boundary of subsets of the discrete cube. Our
generalizations are tight up to constant factors.
http://arxiv.org/abs/1001.1167
Author(s): Tomasz Schreiber and Christoph Thaele
Abstract: Since the seminal work of Mecke, Nagel and Weiss, the iteration stable (STIT)
tessellations have attracted considerable interest in stochastic geometry as a
natural and flexible yet analytically tractable model for hierarchical spatial
cell-splitting and crack-formation processes. The purpose of this paper is to
describe large scale asymptotic geometry of STIT tessellations in
$\mathbb{R}^d$ and more generally that of non-stationary iteration infinitely
divisible tessellations. We study several aspects of the typical first-order
geometry of such tessellations resorting to martingale techniques as providing
a direct link between the typical characteristics of STIT tessellations and
those of suitable mixtures of Poisson hyperplane tessellations. Further, we
also consider second-order properties of STIT and iteration infinitely
divisible tessellations, such as the variance of the total surface area of cell
boundaries inside a convex observation window. Our techniques, relying on
martingale theory and tools from integral geometry, allow us to give explicit
and asymptotic formulae. Based on these results, we establish a functional
central limit theorem for the length/surface increment processes induced by
STIT tessellations. We conclude a central limit theorem for total edge
length/facet surface, with normal limit distribution in the planar case and
non-normal ones in all higher dimensions.
http://arxiv.org/abs/1001.0990
Author(s): Alexey Kuznetsov
Abstract: We study Wiener-Hopf factorization and distribution of extrema for general
stable processes. By connecting Wiener-Hopf factors with a certain
elliptic-like function we are able to obtain many explicit and general results,
such as expressions for Wiener-Hopf factors and Mellin transform of supremum in
terms of double gamma functions, quasi-periodicity and functional identities
for these functions, finite product representations in some special cases and
identities in distribution satisfied by the supremum functional.
http://arxiv.org/abs/1001.0991
Author(s): David Sivakoff
Abstract: The d-dimensional Hamming torus is the graph whose vertices are all of the
integer points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constants
a_1, ..., a_d > 0), and whose edges connect all vertices within Hamming
distance one. We study the size of the largest connected component of the
subgraph generated by independently removing each vertex of the Hamming torus
with probability 1-p. We show that if p=\lambda / n, then there exists
\lambda_c > 0, which is the positive root of a degree d polynomial whose
coefficients depend on a_1, ..., a_d, such that for \lambda < \lambda_c the
largest component has O(log n) vertices (a.a.s. as n \to \infty), and for
\lambda > \lambda_c the largest component has (1-q) \lambda (\prod_i a_i)
n^{d-1} + o(n^{d-1}) vertices and the second largest component has O(log n)
vertices (a.a.s.). An implicit formula for q < 1 is also given. Additionally,
we show that if p = c log n / n, then when c < (d-1) / (\sum a_i) the site
subgraph of the Hamming torus is not connected, and when c > (d-1) / (\sum a_i)
the subgraph is connected (a.a.s.). We also show that the subgraph is connected
precisely when it contains no isolated vertices.
http://arxiv.org/abs/1001.1007
Author(s): Antonio Auffinger and Oren Louidor
Abstract: We study the model of Directed Polymers in Random Environment in 1+1
dimensions, where the distribution at a site has a tail which decays regularly
polynomially with power \alpha, where \alpha \in (0,2). After proper scaling of
temperature \beta^{-1}, we show strong localization of the polymer to a
favorable region in the environment where energy and entropy are best balanced.
We prove that this region has a weak limit under linear scaling and identify
the limiting distribution as an (\alpha, \beta)-indexed family of measures on
Lipschitz curves lying inside the 45-degrees-rotated square with unit diagonal.
In particular, this shows order n transversal fluctuations of the polymer. If,
and only if, \alpha is small enough, we find that there exists a random
critical temperature below which, but not above, the effect of the environment
is macroscopic. The results carry over to d+1 dimensions for d>1 with minor
modifications.
http://arxiv.org/abs/1001.1028
Author(s): Shai Covo (Bar Ilan University)
Abstract: Let {X_{t_1,t_2}: t_1,t_2 >= 0} be a two-parameter L\'evy process on R^d. We
study basic properties of the one-parameter process {X_{x(t),y(t)}: t \in T}
where x and y are, respectively, nondecreasing and nonincreasing nonnegative
continuous functions on the interval T. We focus on and characterize the case
where the process has stationary increments.
http://arxiv.org/abs/1001.1134
Author(s): Seung Ki Baek and Sebastian Bernhardsson
Abstract: We address the equilibrium concept of a reverse auction game so that no one
can enhance the individual payoff by a unilateral change when all the others
follow a certain strategy. In this approach the combinatorial possibilities to
consider become very much involved even for a small number of players, which
has hindered a precise analysis in previous works. We here present a systematic
way to reach the solution for a general number of players, and show that this
game is an example of conflict between the group and the individual interests.
http://arxiv.org/abs/1001.1065
Author(s): Robert C. Dalang and Lluis Quer-Sardanyons
Abstract: We present the Walsh theory of stochastic integrals with respect to
martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic
integrals with respect to Hilbert-space-valued Wiener processes and some other
approaches to stochastic integration, and we explore the links between these
theories. We then show how each theory can be used to study stochastic partial
differential equations, with an emphasis on the stochastic heat and wave
equations driven by spatially homogeneous Gaussian noise that is white in time.
We compare the solutions produced by the different theories.
http://arxiv.org/abs/1001.0856
Author(s): M.J. Luczak and J.R. Norris
Abstract: We set out a general procedure which allows the approximation of certain
Markov chains by the solutions of differential equations. The chains considered
have some components which oscillate rapidly and randomly, while others are
close to deterministic. The limiting dynamics are obtained by averaging the
drift of the latter with respect to a local equilibrium distribution of the
former. Some general estimates are proved under a uniform mixing condition on
the fast variable which give explicit error probabilities for the fluid
approximation.
Mitzenmacher, Prabhakar and Shah \cite{MPS} introduced a variant with memory
of the `join the shortest queue' or `supermarket' model, and obtained a limit
picture for the case of a stable system in which the number of queues and the
total arrival rate are large. In this limit, the empirical distribution of
queue sizes satisfies a differential equation, while the memory of the system
oscillates rapidly and randomly. We illustrate our general fluid limit estimate
in giving a proof of this limit picture.
http://arxiv.org/abs/1001.0895
Author(s): Martin T. Barlow and Jian Ding and Asaf Nachmias and Yuval Peres
Abstract: The cover time of a graph is a celebrated example of a parameter that is easy
to approximate using a randomized algorithm, but for which no constant factor
deterministic polynomial time approximation is known. A breakthrough due to
Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation.
We refine this upper bound, and show that the resulting bound is sharp and
explicitly computable in random graphs. Cooper and Frieze showed that the cover
time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the
supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where
f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover
time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows
how the cover time evolves from the critical window to the supercritical phase.
Our general estimate also yields the order of the cover time for a variety of
other concrete graphs, including critical percolation clusters on the Hamming
hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large
d. For the graphs we consider, our results show that the blanket time,
introduced by Winkler and Zuckerman, is within a constant factor of the cover
time. Finally, we prove that for any connected graph, adding an edge can
increase the cover time by at most a factor of 4.
http://arxiv.org/abs/1001.0609
Author(s): Jeff Kahn and Michael Neiman
Abstract: Competing urns refers to the random experiment where m balls are dropped,
randomly and independently, into urns 1,...,n. Formally, we have a random map
$\sigma$ from {1,...,m} to {1,...,n} with the $\sigma(i)$'s i.i.d. With $x_j$
the indicator of the event that at least $t_j$ balls land in urn j (for some
threshold $t_j$), we prove conditional negative association for the random
variables $x_1,...,x_n$. We mostly deal with the more general situation in
which the $\sigma(i)$'s need not be identically distributed, proving results
which imply conditional negative association in the i.i.d. case. Some of the
results--particularly Lemma 8 on graph orientations--are thought to be of
independent interest. We also give a counterexample to a negative correlation
conjecture of D. Welsh, a strong version of a (still open) conjecture of G.
Farr.
http://arxiv.org/abs/1001.0610
Author(s): Larry Goldstein and Haimeng Zhang
Abstract: In the so called lightbulb process, on days $r=1,...,n$, out of $n$
lightbulbs, all initially off, exactly $r$ bulbs, selected uniformly and
independent of the past, have their status changed from off to on, or vice
versa. With $X$ the number of bulbs on at the terminal time $n$, an even
integer, and $\mu=n/2, \sigma^2={Var}(X)$, we have $$ \sup_{z \in \mathbb{R}}
|P(\frac{X-\mu}{\sigma})-P(Z \le z)| \le \frac{n}{2\sigma^2}\Delta_0 + 1.64
\frac{n}{\sigma^3}+ \frac{2}{\sigma} $$ where $$ \Delta_0 \le
\frac{1}{2\sqrt{n}} + \frac{1}{2n} + e^{-n/2} \qmq {for $n \ge 4$,} $$ yielding
a bound of order $O(n^{-1/2})$ as $n \to \infty$. A similar, though slightly
larger bound holds for $n$ odd. The results are shown using a version of
Stein's method for bounded, monotone size bias couplings. The argument for even
$n$ depends on the construction of a variable $X^s$ on the same space as $X$
which has the $X$ size bias distribution, that is, that satisfies E X g(X)=\mu
Eg(X^s) \quad for all bounded continuous $g$, and for which there exists a $B
\ge 0$, in this case B=2, such that $X \le X^s \le X+B$ almost surely. The
argument for $n$ odd is similar to that for $n$ even, but one first couples $X$
closely to $V$, a symmetrized version of $X$, for which a size bias coupling of
$V$ to $V^s$ can proceed as in the even case.
http://arxiv.org/abs/1001.0612
Author(s): Konstantin Borovkov
Abstract: We present functional versions of recent results on the univariate
distributions of the process $V_{x,u} = x + W_{u\tau(x)},$ $0\le u\le 1$, where
$W_\bullet$ is the standard Brownian motion process, $x>0$ and $\tau (x)
=\inf\{t>0 : W_{t}=-x\}$.
http://arxiv.org/abs/1001.0628
Author(s): Emmanuel Roy (LAGA)
Abstract: We give a second look at stable processes (especially stationary) by
interpreting the self-similar property at the level of the L\'evy measure as
characteristic of a Maharam system. This allows us to derive structural results
and their ergodic consequences. As a byproduct, we obtain a ?stable processes?
proof of Banach-Lamperti Theorem for \alpha<2.
http://arxiv.org/abs/1001.0638
Author(s): Romuald Lenczewski
Abstract: We study the asymptotics of sums of matricially free random variables called
random pseudomatrices, and we compare it with that of random matrices with
block-identical variances. For objects of both types we find the limit joint
distributions of blocks and give their Hilbert space realizations, using
operators called `matricially free Gaussian operators'. In particular, if the
variance matrices are symmetric, the asymptotics of symmetric blocks of random
pseudomatrices agrees with that of symmetric random blocks. We also show that
blocks of random pseudomatrices are `asymptotically matricially free' whereas
the corresponding symmetric random blocks are `asymptotically symmetrically
matricially free', where symmetric matricial freeness is obtained from
matricial freeness by an operation of symmetrization. Finally, we show that row
blocks of square, lower-block-triangular and block-diagonal pseudomatrices are
asymptotically free, monotone independent and boolean independent,
respectively.
http://arxiv.org/abs/1001.0667
Author(s): G. D'Agostini
Abstract: A quite old problem has been recently revitalized by Leonard Mlodinow's book
The Drunkard's Walk, where it is presented in a way that has definitely
confused several people, that wonder why the prevalence of the name of one
daughter among the population should change the probability that the other
child is a girl too. I try here to discuss the problem from scratch, showing
that the rarity of the name plays no role, unless the strange assumption of two
identical names in the same family is taken into account. But also the name
itself does not matter. What is really important is `identification', meant in
an acceptation broader than usual, in the sense that a child is characterized
by a set of attributes that make him/her uniquely identifiable (`that one')
inside a family. The important point of how the information is acquired is also
commented, suggesting an explanation of why several people tend to consider the
informations "at least one boy" and "a well defined boy" (elder/youngest or of
a given name) equivalent.
http://arxiv.org/abs/1001.0708
Author(s): A.D. Barbour and M.J. Luczak
Abstract: When modelling metapopulation dynamics, the influence of a single patch on
the metapopulation depends on the number of individuals in the patch. Since the
population size has no natural upper limit, this leads to systems in which
there are countably infinitely many possible types of individual. Analogous
considerations apply in the transmission of parasitic diseases. In this paper,
we prove a law of large numbers for rather general systems of this kind,
together with a rather sharp bound on the rate of convergence in an
appropriately chosen weighted $\ell_1$ norm.
http://arxiv.org/abs/1001.0044
Author(s): Xicheng Zhang
Abstract: In this article we prove the existence of a stochastic optimal transference
plan for a stochastic Monge-Kantorovich problem by measurable selection
theorem. A stochastic version of Kantorovich duality and the characterization
of stochastic optimal transference plan are also established. Moreover,
Wasserstein distance between two probability kernels are discussed too.
http://arxiv.org/abs/1001.0094
Author(s): Li Ma and Wei Sun
Abstract: Suppose $X$ is a right process which is associated with a non-symmetric
Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on $L^{2}(E;m)$. For $u\in
D(\mathcal{E})_{e}$, we have Fukushima's decomposition:
$\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$. In this paper, we
investigate the strong continuity of the generalized Feynman-Kac semigroup
defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$. Let
$Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in
D(\mathcal{E})_{b}$. Denote by $J_1$ the dissymmetric part of the jumping
measure $J$ of $(\mathcal{E},D(\mathcal{E}))$. Under the assumption that $J_1$
is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if
and only if there exists a constant $\alpha_0\ge 0$ such that
$\|P^{u}_{t}\|_2\leq e^{\alpha_0 t}$ for every $t>0$. If one of these
conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on
$L^{2}(E;m)$. If $X$ is equipped with a differential structure, then this
result also holds without assuming that $J_1$ is finite.
http://arxiv.org/abs/1001.0203
Author(s): Huyen Pham (PMA and Crest)
Abstract: We formulate and investigate a general stochastic control problem under a
progressive enlargement of filtration. The global information is enlarged from
a reference filtration and the knowledge of multiple random times together with
associated marks when they occur. By working under a density hypothesis on the
conditional joint distribution of the random times and marks, we prove a
decomposition of the original stochastic control problem under the global
filtration into classical stochastic control problems under the reference
filtration, which are determined in a finite backward induction. Our method
revisits and extends in particular stochastic control of diffusion processes
with finite number of jumps. This study is motivated by optimization problems
arising in default risk management, and we provide applications of our
decomposition result for the indifference pricing of defaultable claims, and
the optimal investment under bilateral counterparty risk. The solutions are
expressed in terms of BSDEs involving only Brownian filtration, and remarkably
without jump terms coming from the default times and marks in the global
filtration.
http://arxiv.org/abs/1001.0206
Author(s): Annalisa Cerquetti
Abstract: We derive explicit Bayesian nonparametric analysis for a species sampling
model with finitely many types of Gibbs form of type $\alpha= -1$ recently
introduced in Gnedin (2009). Our results complement existing analysis under
Gibbs priors of type $\alpha \in [0, 1)$ proposed in Lijoi et al. (2008).
Calculations rely on a groups sequential construction of Gibbs partitions
introduced in Cerquetti (2008).
http://arxiv.org/abs/1001.0245
Author(s): N. Modarresi and S. Rezakhah
Abstract: In this paper we introduce a new class of non-stationary processes called,
Periodically correlated-locally stationary (PC-LS) processes. It is concerned
with spectral analysis of the harmonizable representation of the processes. Let
$X(t)=X^s(t)+X^p(t)$ represents a stochastic process, where $X^s(t)$ is a
continuous time stationary process and $X^p(t)$ is a discrete time periodically
correlated (PC) process, then $X(t)$ is PC-LS. We also show that $X(t)$ is
linearly correlated, which is include of periodically correlated and locally
stationary (LS) processes.
http://arxiv.org/abs/1001.0296
Author(s): David Windisch
Abstract: We study the entropy of the distribution of the set R_n of vertices visited
by a simple random walk on a graph with bounded degrees in its first n steps.
It is shown that this quantity grows linearly in the expected size of R_n if
the graph is uniformly transient, and sublinearly in the expected size if the
graph is uniformly recurrent with subexponential volume growth. This in
particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff
(arXiv:0903.3179v1). In the recurrent setting, our proof shows that R_n can be
compressed into a string of 0-1-bits of length sublinear in its expected size
with low probability of error.
http://arxiv.org/abs/1001.0355
Author(s): Adrien Richou (IRMAR)
Abstract: We consider Markovian backward stochastic differential equations (BSDEs) with
drivers of quadratic growth and bounded terminal conditions. We first show some
bound estimations on the process $Z$. Then we give a new time discretization
scheme for such BSDEs and we obtain an explicit convergence rate for this
scheme.
http://arxiv.org/abs/1001.0401
Author(s): Samuel N. Cohen and Robert J. Elliott
Abstract: We present a theory of Backward Stochastic Differential Equations in
continuous time with an arbitrary filtered probability space. No assumptions
are made regarding the continuity of the filtration, or of the predictable
quadratic variations of martingales in this space. We present conditions for
existence and uniqueness of square-integrable solutions, using Lipschitz
continuity of the driver. These conditions unite the requirements for existence
in continuous and discrete time, and allow discrete processes to be embedded
with continuous ones. We also present conditions for a comparison theorem, and
hence construct time consistent nonlinear expectations in these general spaces.
http://arxiv.org/abs/1001.0439
Author(s): Maria Joao Oliveira and Jose Luis da Silva and and Ludwig Streit
Abstract: In this work we present expansions of intersection local times of fractional
Brownian motions in $\R^d$, for any dimension $d\geq 1$, with arbitrary Hurst
coefficients in $(0,1)^d$. The expansions are in terms of Wick powers of white
noises (corresponding to multiple Wiener integrals), being well-defined in the
sense of generalized white noise functionals. As an application of our
approach, a sufficient condition on $d$ for the existence of intersection local
times in $L^2$ is derived, extending the results of D. Nualart and S.
Ortiz-Latorre in "Intersection Local Time for Two Independent Fractional
Brownian Motions" (J. Theoret. Probab.,20(4)(2007), 759-767) to different and
more general Hurst coefficients.
http://arxiv.org/abs/1001.0513
Author(s): Alexandre Belloni and Victor Chernozhukov
Abstract: In this paper we study the post-penalized estimator which applies ordinary,
unpenalized linear regression to the model selected by the first step penalized
estimators, typically the LASSO. We show that post-LASSO can perform as well or
nearly as well as the LASSO in terms of the rate of convergence. We show that
this performance occurs even if the LASSO-based model selection "fails", in the
sense of missing some components of the "true" regression model. Furthermore,
post-LASSO can perform strictly better than LASSO, in the sense of a strictly
faster rate of convergence, if the LASSO-based model selection correctly
includes all components of the "true" model as a subset and enough sparsity is
obtained. Of course, in the extreme case, when LASSO perfectly selects the true
model, the past-LASSO estimator becomes the oracle estimator. We show that the
results hold in both parametric and non-parametric models; and by the "true"
model we mean the best $s$-dimensional approximation to the true regression
model, where the dimension $s$ is can be chosen to maximize the rate of
convergence of LASSO or post-LASSO estimators. Moreover, our analysis is not
limited to the LASSO estimator in the first step, and also applies to other
estimators, for example, the trimmed LASSO or Dantzig selector estimator. Our
analysis also highlights the importance of sparsity induced by the first
estimators. That motivated us to also study the impact of trimming small
components of the initial estimator to achieve a sparser support for the
post-LASSO. Our analysis covers both traditional trimming, as well as a new
practical completely data-driven trimming scheme that induces maximal sparsity
subject to maintaining a certain goodness-of-fit.
http://arxiv.org/abs/1001.0188
Author(s): Hsien-Kuei Hwang and Michael Fuchs and Vytas Zacharovas
Abstract: Asymptotics of the variances of many cost measures in random digital search
trees are often notoriously messy and involved to obtain. A new approach is
proposed to facilitate such an analysis for several shape parameters on random
symmetric digital search trees. Our approach starts from a more careful
normalization at the level of Poisson generating functions, which then provides
an asymptotically equivalent approximation to the variance in question. Several
new ingredients are also introduced such as a combined use of Laplace and
Mellin transforms and a simple, mechanical technique for justifying the
analytic de-Poissonization procedures involved. The methodology we develop can
be easily adapted to many other problems with an underlying binomial
distribution. In particular, the less expected and somewhat surprising $n(\log
n)^2$-variance for certain notions of total path-length is also clarified.
http://arxiv.org/abs/1001.0095
|
|
|
|
|
|
