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