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Probability Abstracts 106
This document contains abstracts 7442-7695
from September-1-2008 to October-31-2008.
They have been mailed on Nov 4, 2008.
Author(s): Milton Jara (CEREMADE) and Tomasz Komorowski (UMCS) and Stefano Olla (CEREMADE)
Abstract: Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability
measure $\pi$. Let $\Psi$ a function on the state space of the chain, with
$\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find sufficient
conditions on the probability transition to prove convergence in law of
$N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable law. "Martingale
approximation" approach and "coupling" approach give two different sets of
conditions. We extend these results to continuous time Markov jump processes
$X_t$, whose skeleton chain satisfies our assumptions. If waiting time between
jumps has finite expectation, we prove convergence of $N^{-1/\alpha}\int_0^{Nt}
V(X_s) ds$ to a stable process. In the case of waiting times with infinite
average, we prove convergence to a Mittag-Leffler process.
http://arxiv.org/abs/0809.0177
Author(s): Ali S\"uleyman \"Ust\"unel
Abstract: Let $(W,H,\mu)$ be the classical Wiener space, assume that $U=I_W+u$ is an
adapted perturbation of identity satisfying the Girsanov identity. Then, $U$ is
invertible if and only if the kinetic energy of $u$ is equal to the relative
entropy of the measure induced with the action of $U$ on the Wiener measure
$\mu$, in other words $U$ is invertible if and only if $$ \half
\int_W|u|_H^2d\mu=\int_W \frac{dU\mu}{d\mu}\log\frac{dU\mu}{d\mu}d\mu . $$
http://arxiv.org/abs/0809.0215
Author(s): Leonid Mytnik (Technion) and Edwin Perkins (The University of British Columbia)
Abstract: We prove pathwise uniqueness for solutions of parabolic stochastic pde's with
multiplicative white noise if the coefficient is H\"older continuous of index
$\gamma>3/4$. The method of proof is an infinite-dimensional version of the
Yamada-Watanabe argument for ordinary stochastic differential equations.
http://arxiv.org/abs/0809.0248
Author(s): Ryoki Fukushima
Abstract: We study two objects concerning the Wiener sausage among Poissonian
obstacles. The first is the asymptotics for the \textit{replica overlap}, which
is the intersection of two independent Wiener sausages. We show that it is
asymptotically equal to their union. This result confirms that the localizing
effect of the media is so strong as to completely determine the motional range
of particles. The second is an estimate on the \textit{covering time}. It is
known that the Wiener sausage avoiding Poissonian obstacles up to time $t$ is
confined in some `clearing' ball near the origin and almost fills it. We prove
here that the time needed to fill the confinement ball has the same order as
its volume.
http://arxiv.org/abs/0809.0262
Author(s): Antonio Di Crescenzo and Maria Longobardi
Abstract: We consider a competing risks model, in which system failures are due to one
out of two mutually exclusive causes, formulated within the framework of shock
models driven by bivariate Poisson process. We obtain the failure densities and
the survival functions as well as other related quantities under three
different schemes. Namely, system failures are assumed to occur at the first
instant in which a random constant threshold is reached by (a) the sum of
received shocks, (b) the minimum of shocks, (c) the maximum of shocks.
http://arxiv.org/abs/0809.0279
Author(s): Ernst Eberlein and Antonis Papapantoleon and Albert N. Shiryaev
Abstract: The duality principle in option pricing aims at simplifying valuation
problems that depend on several variables by associating them to the
corresponding dual option pricing problem. Here we analyze the duality
principle for options that depend on several assets. The asset price processes
are driven by general semimartingales, and the dual measures are constructed
via an Esscher transformation. As an application, we can relate swap and quanto
options to standard call and put options. Explicit calculations for jump models
are also provided.
http://arxiv.org/abs/0809.0301
Author(s): Nikolaos Fountoulakis and Ross J. Kang and Colin McDiarmid
Abstract: Given a graph G = (V,E), a vertex subset S is called t-stable (or
t-dependent) if the subgraph G[S] induced on S has maximum degree at most t.
The t-stability number of G is the maximum order of a t-stable set in G. We
investigate the typical values that this parameter takes on a random graph on n
vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed
non-negative integer t, we show that, with probability tending to 1 as n grows,
the t-stability number takes on at most two values which we identify as
functions of t, p and n. The main tool we use is an asymptotic expression for
the expected number of t-stable sets of order k. We derive this expression by
performing a precise count of the number of graphs on k vertices that have
maximum degree at most k. Using the above results, we also obtain asymptotic
bounds on the t-improper chromatic number of a random graph (this is the
generalisation of the chromatic number, where we partition of the vertex set of
the graph into t-stable sets).
http://arxiv.org/abs/0809.0141
Author(s): Louigi Addario-Berry and Nicolas Broutin and Gabor Lugosi
Abstract: We consider the minimum-weight path between any pair of nodes of the
$n$-vertex complete graph in which the weights of the edges are i.i.d.
exponentially distributed random variables. We show that the longest of these
minimum-weight paths has about $\alpha^\star \log n$ edges where
$\alpha^\star\approx 3.5911$ is the unique solution of the equation $\alpha
\log \alpha - \alpha =1$. This answers a question posed by Janson (1999).
http://arxiv.org/abs/0809.0275
Author(s): Mathew Joseph
Abstract: We consider an i.i.d. random environment with a strong form of transience on
the two dimensional integer lattice. Namely, the walk always moves forward in
the y-direction. We prove a functional CLT for the quenched expected position
of the random walk indexed by its level crossing times. We begin with a
variation of the Martingale Central Limit Theorem. The main part of the paper
checks the conditions of the theorem for our problem.
http://arxiv.org/abs/0809.0320
Author(s): Nicolas Bouleau (CIRED and Cermics)
Abstract: We propose a new method to apply the Lipschitz functional calculus of local
Dirichlet forms to Poisson random measures.
http://arxiv.org/abs/0809.0382
Author(s): Marie Amelie Morlais
Abstract: In this study, we consider the exponential utility maximization problem in
the context of a jump-diffusion model. To solve the problem, we rely on the
dynamic programming principle and we derive from it a quadratic BSDE with
jumps. Since this quadratic BSDE is driven both by a Wiener process and by a
Poisson random measure having a Levy measure with infinite mass, our main task
consists in establishing a new existence result for the specific BSDE
introduced.
http://arxiv.org/abs/0809.0423
Author(s): O.L.V. Costa and F. Dufour
Abstract: This paper deals with the long run average continuous control problem of
piecewise deterministic Markov processes (PDMP's) taking values in a general
Borel space and with compact action space depending on the state variable. The
control variable acts on the jump rate and transition measure of the PDMP, and
the running and boundary costs are assumed to be positive but not necessarily
bounded. Our first main result is to obtain an optimality equation for the long
run average cost in terms of a discrete-time optimality equation related to the
embedded Markov chain given by the post-jump location of the PDMP. Our second
main result guarantees the existence of a feedback measurable selector for the
discrete-time optimality equation by establishing a connection between this
equation and an integro-differential equation. Our final main result is to
obtain some sufficient conditions for the existence of a solution for a
discrete-time optimality inequality and an ordinary optimal feedback control
for the long run average cost using the so-called vanishing discount approach.
http://arxiv.org/abs/0809.0477
Author(s): Chih-Yuan Tseng and Ariel Caticha
Abstract: We develop a maximum relative entropy formalism to generate optimal
approximations to probability distributions. The central results consist in (a)
justifying the use of relative entropy as the uniquely natural criterion to
select a preferred approximation from within a family of trial parameterized
distributions, and (b) to obtain the optimal approximation by marginalizing
over parameters using the method of maximum entropy and information geometry.
As an illustration we apply our method to simple fluids. The "exact" canonical
distribution is approximated by that of a fluid of hard spheres. The proposed
method first determines the preferred value of the hard-sphere diameter, and
then obtains an optimal hard-sphere approximation by a suitably weighed average
over different hard-sphere diameters. This leads to a considerable improvement
in accounting for the soft-core nature of the interatomic potential. As a
numerical demonstration, the radial distribution function and the equation of
state for a Lennard-Jones fluid (argon) are compared with results from
molecular dynamics simulations.
http://arxiv.org/abs/0808.4160
Author(s): Robert Morris
Abstract: We study Glauber dynamics on Z^d, which is a dynamic version of the
celebrated Ising model of ferromagnetism. Spins are initially chosen according
to a Bernoulli distribution with density p, and then the states are
continuously (and randomly) updated according to the majority rule. This
corresponds to the sudden quenching of a ferromagnetic system at high
temperature with an external field, to one at zero temperature with no external
field. Define p_c(Z^d) to be the infimum over those p \ge 1/2 such that the
system fixates with probability 1. It is a folklore conjecture that p_c(Z^d) =
1/2 for every d \ge 2. We prove that p_c(Z^d) \to 1/2 as d \to \infty.
http://arxiv.org/abs/0809.0353
Author(s): Ali S\"uleyman \"Ust\"unel
Abstract: The existence and uniqueness of probabilistic solutions of variational
inequalities for the general American options are proved under the hypothesis
of hypoellipticity of the infinitesimal generator of the underlying diffusion
process which represents the risky assets of the stock market with which the
option is created. The main tool is an extension of the It\^o formula which is
valid for the tempered distributions on $\R^d$ and for nondegenerate It\^o
processes in the sense of the Malliavin calculus.
http://arxiv.org/abs/0809.0611
Author(s): Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
Abstract: The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particles
moving in $R^d$ according to a symmetric $\alpha$-stable L\'evy process
$(0<\alpha\leq 2)$, splitting with a critical $(1+\beta)$-branching law
$(0<\beta\leq 1)$, and starting from an inhomogeneous Poisson random measure
with intensity measure $\mu_\gamma(dx)=dx/(1+|x|^\gamma), \gamma\geq 0$. By
means of time rescaling $T$ and Poisson intensity measure $H_T\mu_\gamma$,
occupation time fluctuation limits for the system as $T\to\infty$ have been
obtained in two special cases: Lebesgue measure ($\gamma=0$, the homogeneous
case), and finite measures $(\gamma>d)$. In some cases $H_T\equiv 1$ and in
others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit
processes are quite different for Lebesgue and for finite measures. Therefore
the question arises of what kinds of limits can be obtained for Poisson
intensity measures that are intermediate between Lebesgue measure and finite
measures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$, are used
for investigating this question. Occupation time fluctuation limits are
obtained which interpolate in some way between the two previous extreme cases.
The limit processes depend on different arrangements of the parameters
$d,\alpha,\beta,\gamma$. Related results for the corresponding
$(d,\alpha,\beta,\gamma)$-superprocess are also given.
http://arxiv.org/abs/0809.0665
Author(s): Louis-Pierre Arguin
Abstract: In this note, we point out that infinite-volume Gibbs measures of spin glass
models on the hypercube can be identified as random probability measures on the
unit ball of a Hilbert space. This simple observation follows from a result of
Dovbysh and Sudakov on weakly exchangeable random matrices. Limiting Gibbs
measures can then be studied as single well-defined objects. This approach
naturally extends the space of Random Overlap Structures as defined by
Aizenman, Sims and Starr. We discuss the Ruelle Probability Cascades and the
stochastic stability within this framework. As an application, we use an idea
of Parisi and Talagrand to prove that if a sequence of finite-volume Gibbs
measures satisfies the Ghirlanda-Guerra identities, then the infinite-volume
measure must be singular as a measure on a Hilbert space.
http://arxiv.org/abs/0809.0683
Author(s): L. Saloff-Coste and J. Zuniga
Abstract: We give refined estimates for the discrete time and continuous time versions
of some basic random walks on the symmetric and alternating groups $S_n$ and
$A_n$. We consider the following models: random transposition, transpose top
with random, random insertion, and walks generated by the uniform measure on a
conjugacy class. In the case of random walks on $S_n$ and $A_n$ generated by
the uniform measure on a conjugacy class, we show that in continuous time the
$\ell^2$-cuttoff has a lower bound of $(n/2)\log n$. This result, along with
the results of M\"uller, Schlage-Puchta and Roichman, demonstrates that the
continuous time version of these walks may take much longer to reach
stationarity than its discrete time counterpart.
http://arxiv.org/abs/0809.0688
Author(s): Gordan Zitkovic
Abstract: We propose a mathematical framework for the study of a family of random
fields - called forward performances - which arise as numerical representation
of certain rational preference relations in mathematical finance. Their spatial
structure corresponds to that of utility functions, while the temporal one
reflects a Nisio-type semigroup property, referred to as self-generation. In
the setting of semimartingale financial markets, we provide a dual formulation
of self-generation in addition to the original one, and show equivalence
between the two, thus giving a dual characterization of forward performances.
Then, we focus on random fields with a log-affine structure and provide
necessary and sufficient conditions for self-generation in that case. Finally,
we illustrate our methods in financial markets driven by It\^o-processes, where
we obtain an explicit parametrization of all log-affine forward performances.
http://arxiv.org/abs/0809.0739
Author(s): D. Feyel and A. S. \"Ust\"unel
Abstract: We define, in the frame of an abstract Wiener space, the notions of convexity
and of concavity for the equivalence classes of random variables. As
application we show that some important inequalities of the finite dimensional
case have their natural counterparts in this setting.
http://arxiv.org/abs/0809.0812
Author(s): Anatoli Juditsky (LJK) and Arkadii S. Nemirovski (ISyE)
Abstract: We derive exponential bounds on probabilities of large deviations for "light
tail" martingales taking values in finite-dimensional normed spaces. Our
primary emphasis is on the case where the bounds are dimension-independent or
nearly so. We demonstrate that this is the case when the norm on the space can
be approximated, within an absolute constant factor, by a norm which is
differentiable on the unit sphere with a Lipschitz continuous gradient. We also
present various examples of spaces possessing the latter property.
http://arxiv.org/abs/0809.0813
Author(s): Jos\'e E. Figueroa-L\'opez and Christian Houdr\'e
Abstract: Let $X$ be a L\'evy process with absolutely continuous L\'evy measure $\nu$.
Small time expansions, polynomial in $t$, are obtained for the tails
$P(X_{t}>y)$ of the process. The conditions imposed on $X$ require for $X_{t}$
to have a $C^{\infty}$-transition density, whose derivatives remain uniformly
bounded away from the origin, as $t$ goes to 0. Such conditions are shown to be
satisfied for symmetric stable L\'evy processes as well as for other related
L\'evy processes of relevance in mathematical finance. Also, under very mild
conditions on the L\'evy density of the process and using a different
methodology, a second order power expansion is obtained by identifying
explicitly the limit of $(1/t){P(X_{t}>y)/t -\nu([y,\infty))}$ as $t$ goes to
0. The resulting limit seems to correct a result previously reported in the
literature and hints at the fact that our higher order expansions might be
valid under milder conditions.
http://arxiv.org/abs/0809.0849
Author(s): V.A. Vatutin V. Wachtel
Abstract: Let $T$ be the extinction moment of a critical branching process
$Z=(Z_{n},n\geq 0) $ in a random environment specified by iid probability
generating functions. We study the asymptotic behavior of the probability of
extinction of the process $Z$ at moment $n\to \infty$, and show that if the
logarithm of the (random) expectation of the offspring number belongs to the
domain of attraction of a non-gaussian stable law then the extinction occurs
owing to very unfavorable environment forcing the process, having at moment
$T-1$ exponentially large population, to die out. We also give an
interpretation of the obtained results in terms of random walks in random
environment.
http://arxiv.org/abs/0809.0986
Author(s): Daniil Ryabko (INRIA Lille - Nord Europe)
Abstract: Two series of binary observations $x_1,x_1,...$ and $y_1,y_2,...$ are
presented: at each time $n\in\N$ we are given $x_n$ and $y_n$. It is assumed
that the sequences are generated independently of each other by two
B-processes. We are interested in the question of whether the sequences
represent a typical realization of two different processes or of the same one.
We demonstrate that this is impossible to decide, in the sense that every
discrimination procedure is bound to err with non-negligible frequency when
presented with sequences from some B-processes. This contrasts earlier positive
results on B-processes, in particular those showing that there are consistent
$\bar d$-distance estimates for this class of processes.
http://arxiv.org/abs/0809.1053
Author(s): Steven Delvaux and Arno B.J.Kuijlaars
Abstract: We consider n non-intersecting Brownian motions with two fixed starting
positions and two fixed ending positions in the large n limit. We show that in
case of 'large separation' between the endpoints, the particles are
asymptotically distributed in two separate groups, with no interaction between
them, as one would intuitively expect. We give a rigorous proof using the
Riemann-Hilbert formalism. In the case of 'critical separation' between the
endpoints we are led to a model Riemann-Hilbert problem associated to the
Hastings-McLeod solution of the Painleve II equation. We show that the Painleve
II equation also appears in the large n asymptotics of the recurrence
coefficients of the multiple Hermite polynomials that are associated with the
Riemann-Hilbert problem.
http://arxiv.org/abs/0809.1000
Author(s): Joaquin Fontbona and Helene Guerin and Sylvie Meleard
Abstract: We consider the optimal mass transportation problem in $\RR^d$ with
measurably parameterized marginals, for general cost functions and under
conditions ensuring the existence of a unique optimal transport map. We prove a
joint measurability result for this map, with respect to the space variable and
to the parameter. The proof needs to establish the measurability of some
set-valued mappings, related to the support of the optimal transference plans,
which we use to perform a suitable discrete approximation procedure. A
motivation is the construction of a strong coupling between orthogonal
martingale measures. By this we mean that, given a martingale measure, we
construct in the same probability space a second one with specified covariance
measure. This is done by pushing forward one martingale measure through a
predictable version of the optimal transport map between the covariance
measures. This coupling allows us to obtain quantitative estimates in terms of
the Wasserstein distance between those covariance measures.
http://arxiv.org/abs/0809.1111
Author(s): Pierre Etor\'e (CMAP) and Gersende Fort (LTCI) and Benjamin Jourdain (CERMICS), Eric Moulines (LTCI)
Abstract: This paper investigates the use of stratified sampling as a variance
reduction technique for approximating integrals over large dimensional spaces.
The accuracy of this method critically depends on the choice of the space
partition, the strata, which should be ideally fitted to thesubsets where the
functions to integrate is nearly constant, and on the allocation of the number
of samples within each strata. When the dimension is large and the function to
integrate is complex, finding such partitions and allocating the sample is a
highly non-trivial problem. In this work, we investigate a novel method to
improve the efficiency of the estimator "on the fly", by jointly sampling and
adapting the strata and the allocation within the strata. The accuracy of
estimators when this method is used is examined in detail, in the so-called
asymptotic regime (i.e. when both the number of samples and the number of
strata are large). We illustrate the use of the method for the computation of
the price of path-dependent options in models with both constant and stochastic
volatility. The use of this adaptive technique yields variance reduction by
factors sometimes larger than 1000 compared to classical Monte Carlo
estimators.
http://arxiv.org/abs/0809.1135
Author(s): Bhupendra Gupta
Abstract: A random intersection graph is constructed by independently assigning a
subset of a given set of objects $W,$ to each vertex of the vertex set $V$ of a
simple graph $G.$ There is an edge between two vertices of $V,$ iff their
respective subsets(in $W$,) have at least one common element. The strong
threshold for the connectivity between any two arbitrary vertices of vertex set
$V,$ is derived. Also we determine the almost sure probability bounds for the
vertex degree of a typical vertex of graph $G.$
http://arxiv.org/abs/0809.1141
Author(s): Bhupendra Gupta
Abstract: In this article, we consider `$N$'spherical caps of area $4\pi p$ were
uniformly distributed over the surface of a unit sphere. We are giving the
strong threshold function for the size of random caps to cover the surface of a
unit sphere. We have shown that for large $N,$ if $\frac{Np}{\log\:N} > 1/2$
the surface of sphere is completely covered by the $N$ caps almost surely, and
if $\frac{Np}{\log\:N} \leq 1/2$ a partition of the surface of sphere is
remains uncovered by the $N$ caps almost surely.
http://arxiv.org/abs/0809.1142
Author(s): Bhupendra gupta
Abstract: In this article, we consider `$N$'spherical caps of area $4\pi p$ were
uniformly distributed over the surface of a unit sphere. We study the random
intersection graph $G_N$ constructed by these caps. We prove that for $p =
\frac{c}{N^{\al}},\:c >0$ and $\al >2,$ the number of edges in graph $G_N$
follow the Poisson distribution. Also we derive the strong law results for the
number of isolated vertices in $G_N$: for $p = \frac{c}{N^{\al}},\:c >0$ for
$\al < 1,$ there is no isolated vertex in $G_N$ almost surely i.e., there are
atleast $N/2$ edges in $G_N$ and for $\al >3,$ every vertex in $G_N$ is
isolated i.e., there is no edge in edge set $\cE_N.$
http://arxiv.org/abs/0809.1143
Author(s): Atilla Yilmaz
Abstract: In this work, we study the large deviation properties of random walk in a
random environment on $\mathbb{Z}^d$ with $d\geq1$.
We start with the quenched case, take the point of view of the particle, and
prove the large deviation principle (LDP) for the pair empirical measure of the
environment Markov chain. By an appropriate contraction, we deduce the quenched
LDP for the mean velocity of the particle and obtain a variational formula for
the corresponding rate function $I_q$. We propose an Ansatz for the minimizer
of this formula. This Ansatz is easily verified when $d=1$.
In his 2003 paper, Varadhan proves the averaged LDP for the mean velocity and
gives a variational formula for the corresponding rate function $I_a$. Under
the non-nestling assumption (resp. Kalikow's condition), we show that $I_a$ is
strictly convex and analytic on a non-empty open set $\mathcal{A}$, and that
the true velocity $\xi_o$ is an element (resp. in the closure) of
$\mathcal{A}$. We then identify the minimizer of Varadhan's variational formula
at any $\xi\in\mathcal{A}$.
For walks in high dimension, we believe that $I_a$ and $I_q$ agree on a set
with non-empty interior. We prove this for space-time walks when the dimension
is at least 3+1. In the latter case, we show that the cheapest way to condition
the asymptotic mean velocity of the particle to be equal to any $\xi$ close to
$\xi_o$ is to tilt the transition kernel of the environment Markov chain via a
Doob $h$-transform.
http://arxiv.org/abs/0809.1227
Author(s): Hui He
Abstract: A new result for the strong uniqueness for catalytic branching diffusions is
established, which improves the work of Dawson, D.A.; Fleischmann, K.; Xiong,
J.[Strong uniqueness for cyclically symbiotic branching diffusions. Statist.
Probab. Lett. 73, no. 3, 251--257 (2005)].
http://arxiv.org/abs/0809.1288
Author(s): Alexander Dukhovny and Jean-Luc Marichal
Abstract: A semicoherent system can be described by its structure function or,
equivalently, by a lattice polynomial function expressing the system lifetime
in terms of the component lifetimes. In this paper we point out the parallelism
between the two descriptions and use the natural connection of lattice
polynomial functions and relevant random events to collect exact formulas for
the system reliability. We also discuss the equivalence between calculating the
reliability of semicoherent systems and calculating the distribution function
of a lattice polynomial function of random variables.
http://arxiv.org/abs/0809.1332
Author(s): E. H. Essaky and M. Hassani
Abstract: In this paper, we are concerned with the problem of existence of solutions
for generalized reflected backward stochastic differential equations (GRBSDEs
for short) and generalized backward stochastic differential equations (GBSDEs
for short) when the generator $fds + gdA_s$ is continuous with general growth
with respect to the variable $y$ and stochastic quadratic growth with respect
to the variable $z$. We deal with the case of a bounded terminal condition
$\xi$ and a bounded barrier $L$ as well as the case of unbounded ones. This is
done by using the notion of generalized BSDEs with two reflecting barriers
studied in \cite{EH}. The work is suggested by the interest the results might
have in finance, control and game theory.
http://arxiv.org/abs/0809.1353
Author(s): I. Onur Filiz and Xin Guo and Jason Morton and Bernd Sturmfels
Abstract: A simple graphical model for correlated defaults is proposed, with explicit
formulas for the loss distribution. Algebraic geometry techniques are employed
to show that this model is well posed for default dependence: it represents any
given marginal distribution for single firms and pairwise correlation matrix.
These techniques also provide a calibration algorithm based on maximum
likelihood estimation. Finally, the model is compared with standard normal
copula model in terms of tails of the loss distribution and implied correlation
smile.
http://arxiv.org/abs/0809.1393
Author(s): Xinjia Chen
Abstract: In this paper, we have established a new framework of multistage parametric
estimation. Specially, we have developed sampling schemes for estimating
parameters of common important distributions. Without any information of the
unknown parameters, our sampling schemes rigorously guarantee prescribed levels
of precision and confidence, while achieving unprecedented efficiency in the
sense that the average sampling numbers are virtually the same as that are
computed as if the exact values of unknown parameters were available.
http://arxiv.org/abs/0809.1241
Author(s): Konstantin S. Turitsyn and Michael Chertkov and Marija Vucelja
Abstract: Equilibrium systems evolve according to Detailed Balance (DB). This principe
guided development of the Monte-Carlo sampling techniques, of which
Metropolis-Hastings (MH) algorithm is the famous representative. It is also
known that DB is sufficient but not necessary. We construct irreversible
deformation of a given reversible algorithm capable of dramatic improvement of
sampling from known distribution. Our transformation modifies transition rates
keeping the structure of transitions intact. To illustrate the general scheme
we design an Irreversible version of Metropolis-Hastings (IMH) and test it on
example of a spin cluster. Standard MH for the model suffers from the critical
slow down, while IMH is critical slow down free.
http://arxiv.org/abs/0809.0916
Author(s): Franco Flandoli and Massimiliano Gubinelli and Enrico Priola
Abstract: We consider the linear transport equation with a globally Holder continuous
and bounded vector field. While this deterministic PDE may not be well-posed,
we prove that a multiplicative stochastic perturbation of Brownian type is
enough to render the equation well-posed. This seems to be the first explicit
example of partial differential equation that become well-posed under the
influece of noise. The key tool is a differentiable stochastic flow constructed
and analysed by means of a special transformation of the drift of Tanaka type.
http://arxiv.org/abs/0809.1310
Author(s): Ben Hambly and Terry Lyons
Abstract: These notes cover background material on trees which are used in the paper
`On uniqueness of the signature of a path of variation and the reduced path
group'.
http://arxiv.org/abs/0809.1365
Author(s): Robert C. Griffiths and Dario Span\`o
Abstract: Multivariate versions of classical orthogonal polynomials such as Jacobi,
Hahn, Laguerre, Meixner are reviewed and their connection explored by adopting
a probabilistic approach. Hahn and Meixner polynomials are interpreted as
posterior mixtures of Jacobi and Laguerre polynomials, respectively. By using
known properties of Gamma point processes and related transformations, an
infinite-dimensional version of Jacobi polynomials is constructed with respect
to the size-biased version of the Poisson-Dirichlet weight measure and to the
law of the Gamma point process from which it is derived.
http://arxiv.org/abs/0809.1431
Author(s): Federico Bassetti and Lucia Ladelli and Daniel Matthes
Abstract: We introduce a class of Boltzmann equations on the real line, which
constitute extensions of the classical Kac caricature. The collisional gain
operators are defined by smoothing transformations with quite general
properties. By establishing a connection to the central limit problem, we are
able to prove long-time convergence of the equation's solutions towards a limit
distribution. If the initial condition for the Boltzmann equation belongs to
the domain of normal attraction of a certain stable law $g_\alpha$, then the
limit is non-trivial and is a statistical mixture of dilations of $g_\alpha$.
Under some additional assumptions, explicit exponential rates for the
equilibration in Wasserstein metrics are calculated, and strong convergence of
the probability densities is shown.
http://arxiv.org/abs/0809.1545
Author(s): Mark M. Meerschaert and Erkan Nane and Yimin Xiao
Abstract: Continuous time random walks impose a random waiting time before each
particle jump. Scaling limits of heavy tailed continuous time random walks are
governed by fractional evolution equations. Space-fractional derivatives
describe heavy tailed jumps, and the time-fractional version codes heavy tailed
waiting times. This paper develops scaling limits and governing equations in
the case of correlated jumps. For long-range dependent jumps, this leads to
fractional Brownian motion or linear fractional stable motion, with the time
parameter replaced by an inverse stable subordinator in the case of heavy
tailed waiting times. These scaling limits provide an interesting class of
non-Markovian, non-Gaussian self-similar processes.
http://arxiv.org/abs/0809.1612
Author(s): Federico Bonetto and Joel L. Lebowitz and Jani Lukkarinen and Stefano Olla (CEREMADE)
Abstract: We investigate a class of anharmonic crystals in $d$ dimensions, $d\ge 1$,
coupled to both external and internal heat baths of the Ornstein-Uhlenbeck
type. The external heat baths, applied at the boundaries in the 1-direction,
are at specified, unequal, temperatures $\tlb$ and $\trb$. The temperatures of
the internal baths are determined in a self-consistent way by the requirement
that there be no net energy exchange with the system in the non-equilibrium
stationary state (NESS). We prove the existence of such a stationary
self-consistent profile of temperatures for a finite system and show it
minimizes the entropy production to leading order in $(\tlb -\trb)$. In the
NESS the heat conductivity $\kappa$ is defined as the heat flux per unit area
divided by the length of the system and $(\tlb -\trb)$. In the limit when the
temperatures of the external reservoirs goes to the same temperature $T$,
$\kappa(T)$ is given by the Green-Kubo formula, evaluated in an equilibrium
system coupled to reservoirs all having the temperature $T$. This $\kappa(T)$
remains bounded as the size of the system goes to infinity. We also show that
the corresponding infinite system Green-Kubo formula yields a finite result.
Stronger results are obtained under the assumption that the self-consistent
profile remains bounded.
http://arxiv.org/abs/0809.0953
Author(s): Alexander Soshnikov
Abstract: We obtain the recursive identities for the joint moments of the traces of the
powers of the resolvent for Gaussian ensembles of random matrices at the soft
and hard edges of the spectrum. We also discuss the possible ways to extend
these results to the non-Gaussian case.
http://arxiv.org/abs/0809.1463
Author(s): Stefano Olla (CEREMADE)
Abstract: I review here some recent result on the thermal conductivity of chains of
oscillators whose hamiltonian dynamics is perturbed by a noise conserving
energy and momentum.
http://arxiv.org/abs/0809.1496
Author(s): Remco van der Hofstad and Akira Sakai
Abstract: We consider the critical spread-out contact process in Z^d with d\ge1, whose
infection range is denoted by L\ge1. In this paper, we investigate the r-point
function \tau_{\vec t}^{(r)}(\vec x) for r\ge3, which is the probability that,
for all i=1,...,r-1, the individual located at x_i\in Z^d is infected at time
t_i by the individual at the origin o\in Z^d at time 0. Together with the
results of the 2-point function in [van der Hofstad and Sakai, Electron. J.
Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially
rely, we prove that the r-point functions converge to the moment measures of
the canonical measure of super-Brownian motion above the upper-critical
dimension 4. We also prove partial results for d\le4 in a local mean-field
setting.
http://arxiv.org/abs/0809.1712
Author(s): Aleksandar Mijatovic
Abstract: A time-dependent double-barrier option is a derivative security that delivers
the terminal value $\phi(S_T)$ at expiry $T$ if neither of the continuous
time-dependent barriers $b_\pm:[0,T]\to \RR_+$ have been hit during the time
interval $[0,T]$. Using a probabilistic approach we obtain a decomposition of
the barrier option price into the corresponding European option price minus the
barrier premium for a wide class of payoff functions $\phi$, barrier functions
$b_\pm$ and linear diffusions $(S_t)_{t\in[0,T]}$. We show that the barrier
premium can be expressed as a sum of integrals along the barriers $b_\pm$ of
the option's deltas $\Delta_\pm:[0,T]\to\RR$ at the barriers and that the pair
of functions $(\Delta_+,\Delta_-)$ solves a system of Volterra integral
equations of the first kind. We find a semi-analytic solution for this system
in the case of constant double barriers and briefly discus a numerical
algorithm for the time-dependent case.
http://arxiv.org/abs/0809.1747
Author(s): Laure Coutin (MAP5) and Laurent Decreusefond (LTCI) and Jean-Stephane Dhersin (MAP5)
Abstract: We propose a Markov model for the spread of Hepatitis C virus (HCV) among
drug users who use injections. We then proceed to an asymptotic analysis (large
initial population) and show that the Markov process is close to the solution
of a non linear autonomous differential system. We prove both a law of large
numbers and functional central limit theorem to precise the speed of
convergence towards the limiting system. The deterministic system itself
converges, as time goes to infinity, to an equilibrium point. This corroborates
the empirical observations about the prevalence of HCV.
http://arxiv.org/abs/0809.1824
Author(s): Andrew Granville and Igor Wigman
Abstract: We study the asymptotic distribution of the number $Z_{N}$ of zeros of random
trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as
$N$ grows to infinity, the expected number of the zeros is asymptotic to
$\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted
by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that
$\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In
addition, we find that the analogous result is applicable for the number of
zeros in short intervals.
http://arxiv.org/abs/0809.1848
Author(s): Sara Brofferio and Dariusz Buraczewski and Ewa Damek
Abstract: We consider the autoregressive model on $\R^d$ defined by the following
stochastic recursion $X_n = A_n X_{n-1}+B_n$, where $\{(B_n,A_n)\}$ are i.i.d.
random variables valued in $\R^d\times \R^+$. The critical case, when
$\E\big[\log A_1\big]=0$, was studied by Babillot, Bougeorol and Elie, who
proved that there exists a unique invariant Radon measure $\nu$ for the Markov
chain $\{X_n \}$. In the present paper we prove that the weak limit of properly
dilated measure $\nu$ exists and defines a homogeneous measure on
$\R^d\setminus \{0\}$.
http://arxiv.org/abs/0809.1864
Author(s): N. Prasanth Anthapadmanabhan and Armand M. Makowski
Abstract: When studying networks using random graph models, one is sometimes faced with
situations where the notion of adjacency between nodes reflects multiple
constraints. Traditional random graph models are insufficient to handle such
situations.
A simple idea to account for multiple constraints consists in taking the
intersection of random graphs. In this paper we initiate the study of random
graphs so obtained through a simple example. We examine the intersection of an
Erdos-Renyi graph and of one-dimensional geometric random graphs. We
investigate the zero-one laws for the property that there are no isolated
nodes. When the geometric component is defined on the unit circle, a full
zero-one law is established and we determine its critical scaling. When the
geometric component lies in the unit interval, there is a gap in that the
obtained zero and one laws are found to express deviations from different
critical scalings. In particular, the first moment method requires a larger
critical scaling than in the unit circle case in order to obtain the one law.
This discrepancy is somewhat surprising given that the zero-one laws for the
absence of isolated nodes are identical in the geometric random graphs on both
the unit interval and unit circle.
http://arxiv.org/abs/0809.0918
Author(s): Shamgar Gurevich (UC Berkeley) and Ronny Hadani (University of Chicago)
Abstract: In this paper we formulate and prove a statistical version of the Candes-Tao
restricted isometry property (SRIP for short) which holds in general for any
incoherent dictionary which is a disjoint union of orthonormal bases. In
addition, we prove that, under appropriate normalization, the eigenvalues of
the associated Gram matrix fluctuate around \lambda = 1 according to the Wigner
semicircle distribution. The result is then applied to various dictionaries
that arise naturally in the setting of finite harmonic analysis, giving, in
particular, a better understanding on a remark of
Applebaum-Howard-Searle-Calderbank concerning RIP for the Heisenberg dictionary
of chirp like functions.
http://arxiv.org/abs/0809.1687
Author(s): Christa Cuchiero and Damir Filipovic and Josef Teichmann
Abstract: Affine term structure models have gained a lot of attention in the finance
literature, which is due to their analytic tractability and statistical
flexibility. The aim of this article is to present both, theoretical
foundations and empirical aspects. Starting from the first short rate models,
namely the Vasi\v{c}ek and the Cox-Ingersoll-Ross ones, we then give an
overview of some properties of affine processes and explain their relation to
affine term structure models. Pricing and estimation techniques are eventually
mentioned, showing how the analytic tractability of affine models can be
exploited for practical purposes.
http://arxiv.org/abs/0809.1985
Author(s): Aur\'elien Deya (IECN) and Samy Tindel (IECN)
Abstract: We define and solve Volterra equations driven by an irregular signal, by
means of a variant of the rough path theory called algebraic integration. In
the Young case, that is for a driving signal with H\"older exponent greater
than 1/2, we obtain a global solution, and are able to handle the case of a
singular Volterra coefficient. In case of a driving signal with H\"older
exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The
results are easily applied to the fractional Brownian motion with Hurst
coefficient H>1/3.
http://arxiv.org/abs/0809.2000
Author(s): Cl\'ement Dombry (LMA) and Christian Mazza
Abstract: Polygon spaces like $M_\ell=\{(u_1,...,u_n)\in S^1\times... S^1 ;\
\sum_{i=1}^n l_iu_i=0\}/SO(2)$ or they three dimensional analogues $N_\ell$
play an important r\^ole in geometry and topology, and are also of interest in
robotics where the $l_i$ model the lengths of robot arms. When $n$ is large,
one can assume that each $l_i$ is a positive real valued random variable,
leading to a random manifold. The complexity of such manifolds can be
approached by computing Betti numbers, the Euler characteristics, or the
related Poincar\'e polynomial. We study the average values of Betti numbers of
dimension $p_n$ when $p_n\to\infty$ as $n\to\infty$. We also focus on the
limiting mean Poincar\'e polynomial, in two and three dimensions. We show that
in two dimensions, the mean total Betti number behaves as the total Betti
number associated with the equilateral manifold where $l_i\equiv \bar l$. In
three dimensions, these two quantities are not any more asymptotically
equivalent. We also provide asymptotics for the Poincar\'e polynomials
http://arxiv.org/abs/0809.2082
Author(s): Pierre Andreoletti (MAPMO) and Roland Diel (MAPMO)
Abstract: We consider Brox's model: a one-dimensional diffusion in a Brownian
environment. We show the weak convergence of the normalized local time process
$(L(x+m_{\log t},t)/t,x\in I \subset \R)$, centered at the coordinate of the
bottom of the deepest valley $m_{\log t}$ reached by the process before time
$t$ to a functional of two independent 3-dimensional Bessel processes. We apply
that result to get the limit law of the supremum of the normalized local time.
These results are discussed and compared to the discrete time and space
analogous model whose same questions have been solved recently by N. Ganter, Y.
Peres and Z. Shi.
http://arxiv.org/abs/0809.2195
Author(s): Dapeng Zhan
Abstract: We use the whole-plane Loewner equation to define a family of continuous LERW
in finitely connected domains that are started from interior points. These
continuous LERW satisfy conformal invariance, preserve some continuous local
martingales, and are the scaling limits of the corresponding discrete LERW on
the discrete approximation of the domains.
http://arxiv.org/abs/0809.2230
Author(s): Michal Baran and Jacek Jakubowski and Jerzy Zabczyk
Abstract: The completeness of a bond market model with infinite number of sources of
randomness on a finite time interval in the Heath-Jarrow-Morton framework is
studied. It is proved that the market in the case of trading strategies is not
complete. An explicit construction of a bounded contingent claim, which can not
be replicated, is provided. Moreover, a new concept of generalized strategies
is introduced and sufficient conditions for the market completeness with such
strategies are given. An example of a complete model is provided.
http://arxiv.org/abs/0809.2270
Author(s): A. Berarducci and P. Majer and M. Novaga
Abstract: We prove some Ramsey-type theorems, where the colours are replaced by
measurable subsets of a probability space. Such results can be also interpreted
as percolation problems on a countable complete graph, without requiring any
specific assumption on the probability, such as independency. In particular, we
provide sharp estimates on the probability of finding an infinite clique or an
infinite monotone path in a random subgraph.
http://arxiv.org/abs/0809.2335
Author(s): Ravi Kannan
Abstract: A new probability inequality strengthening Azuma's is presented. Instead of
assuming an absolute bound on the Martingale differences as Azuma's does, we
use information on conditional moments of each variable. Information on moments
conditioned on "typical" as well as "worst-case" values are used. Also, a
weaker assumption than Martingale differences suffices here. The inequality is
strong enough to yield sub-Gaussian tail bounds in many situations. We present
several applications. For bin-packing optimal concentration intervals are
proved settling a question of Talagrand. For chromatic number of sparse random
graphs, the first sub-Gaussian tail bounds are proved and the first treatment
of the number of cliques of growing sizes in random graphs is given. For
Lonegest Sub-sequences, a result matching Talagrand's is given. Brief
comparisons to other recent inequalities are made.
http://arxiv.org/abs/0809.2477
Author(s): Fabrice Baudoin and Neil O'Connell
Abstract: We consider exponential functionals of a multi-dimensional Brownian motion
with drift, defined via a collection of linear functionals. We give a
characterisation of the Laplace transform of their joint law as the unique
bounded solution, up to a constant factor, to a Schrodinger-type partial
differential equation. We derive a similar equation for the probability
density. We then characterise all diffusions which can be interpreted as having
the law of the Brownian motion with drift conditioned on the law of its
exponential functionals. In the case where the family of linear functionals is
a set of simple roots, the Laplace transform of the joint law of the
corresponding exponential functionals can be expressed in terms of a class one
Whittaker function associated with the corresponding root system. In this
setting, we establish some basic properties of the corresponding diffusions,
which we call Whittaker processes.
http://arxiv.org/abs/0809.2506
Author(s): Ariel Yadin and Amir Yehudayoff
Abstract: Lawler, Schramm and Werner showed that the scaling limit of the loop-erased
random walk on the square lattice is SLE(2). We consider scaling limits of the
loop-erasure of random walks on other planar graphs (graphs embedded into the
complex plane so that edges do not cross one another). We show that if the
scaling limit of the random walk is planar Brownian motion, then the scaling
limit of its loop-erasure is SLE(2). Our main contribution is showing that for
such graphs, the discrete Poisson kernel can be approximated by the continuous
one.
One example is the infinite component of super-critical percolation on the
square lattice. Berger and Biskup showed that the scaling limit of the random
walk on this graph is planar Brownian motion. Our results imply that the
scaling limit of the loop-erased random walk on the super-critical percolation
cluster is SLE(2).
http://arxiv.org/abs/0809.2643
Author(s): Nils Berglund (MAPMO) and Barbara Gentz
Abstract: We consider a Ginzburg-Landau partial differential equation in a bounded
interval, perturbed by weak spatio-temporal noise. As the interval length
increases, a transition between activation regimes occurs, in which the
classical Kramers rate diverges [R.S. Maier and D.L. Stein, Phys. Rev. Lett.
87, 270601 (2001)]. We determine a corrected Kramers formula at the transition
point, yielding a finite, though noise-dependent prefactor, confirming a
conjecture by Maier and Stein [vol. 5114 of SPIE Proceeding (2003)]. For both
periodic and Neumann boundary conditions, we obtain explicit expressions of the
prefactor in terms of Bessel and error functions.
http://arxiv.org/abs/0809.2652
Author(s): Ivan Gentil (CEREMADE) and Cyril Imbert (CEREMADE)
Abstract: In this paper we study some applications of the L\'evy logarithmic Sobolev
inequality to the study of the regularity of the solution of the fractal heat
equation, i. e. the heat equation where the Laplacian is replaced with the
fractional Laplacian. It is also used to the study of the asymptotic behaviour
of the L\'evy-Ornstein-Uhlenbeck process.
http://arxiv.org/abs/0809.2654
Author(s): N.S. Walton
Abstract: A network of single server queues with routing is considered. These networks
have a product form stationary distribution. A new limit result proves a
sequence of such networks converges weakly to a stochastic flow level model.
This stochastic model is insensitive. A large deviation principle for the
stationary distribution of these queueing networks is found. Its rate function
has a dual formulation that coincides with proportional fairness. It is proven
that the stationary throughput of the queueing model converges to a
proportionally fair allocation.
The queueing models considered have no prescribed optimization structure.
Regardless of this, we find a proportionally fair allocation forms an entropy
minimizing state of these networks. Proportional fairness occurs as a
consequence of a collapse in the state space of the queueing model.
This work combines classical queueing networks with more recent work on
stochastic flow level models and proportional fairness. One could view these
seemingly different models as the same system described at different levels of
granularity: a microscopic, queueing level description; a macroscopic, flow
level description and a teleological, optimization description.
http://arxiv.org/abs/0809.2697
Author(s): Hans Crauel (Goethe-Universit\"at Frankfurt) and Georgi Dimitroff (ITWM Kaiserslautern), Michael Scheutzow (TU Berlin)
Abstract: The theory of random attractors has different notions of attraction, amongst
them pullback attraction and weak attraction. We investigate necessary and
sufficient conditions for the existence of pullback attractors as well as of
weak attractors.
http://arxiv.org/abs/0809.2719
Author(s): Michel Benaim and Pierre Tarres
Abstract: We generalize a result from Volkov (2001) and prove that, on an arbitrary
graph of bounded degree $(G,\sim)$ and for any symmetric reinforcement matrix
$a=(a_{i,j})_{i\sim j}$, the vertex-reinforced random walk (VRRW) eventually
localizes with positive probability on subsets which consist of a complete
$d$-partite subgraph plus its outer boundary.
We first show that, in general, any stable equilibrium of a linear symmetric
replicator dynamics with positive payoffs on a graph $G$ satisfies the property
that its support is a complete $d$-partite subgraph of $G$ for some $d\ge2$.
This result is used here for the study of VRRWs, but also applies to other
contexts such as evolutionary models in population genetics and game theory.
Next we generalize the result of Pemantle (1992) and Benaim (1997) relating
the asymptotic behaviour of the VRRW to replicator dynamics. This enables us to
conclude that, given any neighbourhood of a strictly stable equilibrium with
support $S$, the following event occurs with positive probability: the walk
localizes on $S\cup\partial S$, (where $\partial S$ is the outer boundary of
$S$) and the density of occupation of the VRRW converges, with polynomial rate,
to a strictly stable equilibrium in this neighbourhood.
http://arxiv.org/abs/0809.2739
Author(s): S.C. Harris and R. Knobloch and A.E. Kyprianou
Abstract: In the spirit of a classical results for Crump-Mode-Jagers processes, we
prove a strong law of large numbers for homogenous fragmentation processes.
Specifically, for self-similar fragmentation processes, including homogenous
processes, we prove the almost sure convergence of an empirical measure
associated with the stopping line corresponding to first fragments of size
strictly smaller than $\eta$ for $1\geq \eta >0$.
http://arxiv.org/abs/0809.2958
Author(s): Yuri Bakhtin
Abstract: We prove that under an easily verifiable set of conditions a sequence of
associated random fields converges under rescaling to the Poisson Point Process
and give a couple of examples.
http://arxiv.org/abs/0809.2971
Author(s): Yuri Bakhtin
Abstract: We consider Gibbs distributions on finite random plane trees with bounded
branching. We show that as the order of the tree grows to infinity, the
distribution of any finite neighborhood of the root of the tree converges to a
limit. We compute the limiting distribution explicitly and study its
properties. We introduce an infinite random tree consistent with these limiting
distributions and show that it satisfies a certain form of the Markov property.
We also study the growth of this tree and prove several limit theorems
including a diffusion approximation.
http://arxiv.org/abs/0809.2974
Author(s): Yehoram Gordon and Alexander Litvak and Carsten Sch\"utt and Elisabeth Werner
Abstract: We establish uniform estimates for order statistics of sequences of
independent identically distributed random variables with log-concave
distribution in terms of Orlicz norms associated with the distribution function
of the random variables.
http://arxiv.org/abs/0809.2989
Author(s): Anne Fey and Haiyan Liu and Ronald Meester
Abstract: We show that Zhang's sandpile model (N,[a,b]) on N sites and with uniform
additions on [a,b] has a unique stationary measure for all 0 <= a < b <= 1.
This generalizes earlier results where this was shown in some special cases.
We define the infinite volume Zhang's sandpile model in dimension d >= 1, in
which topplings occur according to a Markov toppling process, and we study the
stabilizability of initial configurations chosen according to some measure \mu.
We show that for a stationary ergodic measure \mu with density \rho, for all
\rho < 1/2, \mu is stabilizable; for all \rho >= 1, \mu is not stabilizable;
for 1/2 <= \rho < 1, when \rho is near to 1/2 or 1, both possibilities can
occur.
http://arxiv.org/abs/0809.2913
Author(s): Chris Preston
Abstract: These notes give an elementary approach to parts of the theory of standard
Borel and analytic spaces.
http://arxiv.org/abs/0809.3066
Author(s): Nicolas Privault
Abstract: These notes survey some aspects of discrete-time chaotic calculus and its
applications, based on the chaos representation property for i.i.d. sequences
of random variables. The topics covered include the Clark formula and
predictable representation, anticipating calculus, covariance identities and
functional inequalities (such as deviation and logarithmic Sobolev
inequalities), and an application to option hedging in discrete time.
http://arxiv.org/abs/0809.3168
Author(s): Xinjia Chen
Abstract: In this paper, we have established a new framework of multistage hypothesis
tests. Within the new framework, we have developed specific multistage tests
which guarantee prescribed level of power and are more efficient than previous
tests in terms of average sampling number and the number of sampling
operations. Without truncation, the maximum sampling numbers of our testing
plans are absolutely bounded.
http://arxiv.org/abs/0809.3170
Author(s): Tuomas Hyt\"onen
Abstract: The Lp operator norm of the generalized Beurling-Ahlfors transformation in n
variables is at most (n/2+1)(p-1) for p>2. This improves on earlier results in
all dimensions n>2. The proof is based on the heat extension and relies at the
bottom on Burkholder's sharp inequality for martingale transforms.
http://arxiv.org/abs/0809.3127
Author(s): Michael Roper
Abstract: We examine the small expiry behaviour of European call options in stock price
models of exponential L\'evy type. In most cases of interest, we are able to
identify the exact small expiry asymptotics. In "complete generality" we are
able to show that the time value of the call option has O(\tau) decay as \tau
(time to expiry) goes to zero. Using our results on the behaviour of call
options close to expiry we show that implied volatility explodes as
$\tau\to0^+$ in "most" exponential L\'evy models. Attention is restricted to
calls and implied volatilities that are not at-the-money.
http://arxiv.org/abs/0809.3305
Author(s): Ernst Eberlein and Kathrin Glau and Antonis Papapantoleon
Abstract: We discuss the valuation problem for a broad spectrum of plain vanilla and
path-dependent options in a general modeling framework, and specifically for
L\'evy driven models. Among the derivatives which we consider are digitals,
double digitals, asset-or-nothing options, self-quantos, lookback and one-touch
options. Extensions to the multivariate case, i.e. basket options and options
on the minimum or maximum of several assets are considered as well. As a link
to credit risk we derive a valuation formula for equity default swaps. The key
idea of this Fourier, resp. Laplace transform based approach is the clear
separation of two ingredients, namely the payoff function and the underlying
process. The latter can be the driving process itself, but also the supremum or
infimum or another process derived from it. For L\'evy processes the
analytically extended characteristic functions of the supremum and the infimum
process are derived.
http://arxiv.org/abs/0809.3405
Author(s): C. F. Coletti and E. S. Dias and L. R. G. Fontes
Abstract: We consider the two dimensional version of a drainage network model
introduced by Gangopadhyay, Roy and Sarkar, and show that the appropriately
rescaled family of its paths converges in distribution to the Brownian web. We
do so by verifying the convergence criteria proposed by Fontes, Isopi, Newman
and Ravishankar.
http://arxiv.org/abs/0809.3454
Author(s): L. R. G. Fontes and P. H. S. Lima
Abstract: We consider symmetric trap models in the d-dimensional hypercube whose
ordered mean waiting times, seen as weights of a measure in the natural
numbers, converge to a finite measure as d diverges, and show that the models
suitably represented converge to a K process as d diverges. We then apply this
result to get K processes as the scaling limits of the REM-like trap model and
the Random Hopping Times dynamics for the Random Energy Model in the hypercube
in time scales corresponding to the ergodic regime for these dynamics.
http://arxiv.org/abs/0809.3463
Author(s): Atilla Yilmaz
Abstract: In his 2003 paper, Varadhan proves the averaged large deviation principle
(LDP) for the mean velocity of a particle performing random walk in a random
environment (RWRE) on $\mathbb{Z}^d$ with $d\geq1$, and gives a variational
formula for the corresponding rate function $I_a$. Under the non-nestling
assumption (resp. Kalikow's condition), we show that $I_a$ is strictly convex
and analytic on a non-empty open set $\mathcal{A}$, and that the true velocity
$\xi_o$ of the particle is an element (resp. in the closure) of $\mathcal{A}$.
We then identify the minimizer of Varadhan's variational formula at any
$\xi\in\mathcal{A}$.
http://arxiv.org/abs/0809.3467
Author(s): A. Galves and N.L. Garcia and E. Loecherbach
Abstract: We consider a particle system on $Z^d$ with finite state space and
interactions of infinite range. Assuming that the rate of change is continuous
and decays sufficiently fast, we introduce a perfect simulation algorithm for
the stationary process. The algorithm follows from a representation of the
multicolor system as a finitary coding from a sequence of independent uniform
random variables. This implies that the process is exponentially ergodic. The
basic tool we use is a representation of the infinite range change rates as a
mixture of finite range change rates.
http://arxiv.org/abs/0809.3494
Author(s): Lutz Mattner and Uwe R\"osler
Abstract: The optimal function $f$ satisfying
$$
\mathbb{E} |\sum_{1}^n X_i |
\ge f(\mathrbb{E}|X_1|,...,\mathbb{E}|X_n|)
$$ for every martingale $(X_1,X_1+X_2, ...,\sum_{i=1}^n X_i)$ is shown to be
given by $$ f(a) = \max \Big\{a_k-\sum_{i=1}^{k-1} a_i\Big\}_{k=1}^n \cup
\Big\{\frac {a_k}2\Big\}_{k=3}^n $$ for $a\in{[0,\infty[}^n_{}$. A similar
result is obtained for submartingales $(0,X_1,X_1+X_2,..., \sum_{i=1}^n X_i)$.
The optimality proofs use a convex-analytic comparison lemma of independent
interest.
http://arxiv.org/abs/0809.3522
Author(s): Hanene Mohamed (INRIA Rocquencourt) and Philippe Robert
Abstract: In this paper, a general tree algorithm processing a random flow of arrivals
is analyzed. Capetanakis-Tsybakov-Mikhailov's protocol in the context of
communication networks with random access is an example of such an algorithm.
In computer science, this corresponds to a trie structure with a dynamic input.
Mathematically, it is related to a stopped branching process with exogeneous
arrivals (immigration). Under quite general assumptions on the distribution of
the number of arrivals and on the branching procedure, it is shown that there
exists a {\em positive} constant $\lambda_c$ so that if the arrival rate is
smaller than $\lambda_c$, then the algorithm is stable under the flow of
requests, i.e. that the total size of an associated tree is integrable. At the
same time a gap in the earlier proofs of stability of the literature is fixed.
When the arrivals are Poisson, an explicit characterization of $\lambda_c$ is
given. Under the stability condition, the asymptotic behavior of the average
size of a tree starting with a large number of individuals is analyzed. The
results are obtained with the help of a probabilistic rewriting of the
functional equations describing the dynamic of the system. The proofs use
extensively this stochastic background throughout the paper. In this analysis,
two basic limit theorems play a key role: the renewal theorem and the
convergence to equilibrium of an auto-regressive process with moving average.
http://arxiv.org/abs/0809.3577
Author(s): Alberto Lanconelli
Abstract: We establish the strong L2(P)-convergence of properly rescaled Wick powers as
the power index tends to infinity. The explicit representation of such limit
will also provide the convergence in distribution to normal and log-normal
random variables. The proofs rely on some estimates for the L2(P)-norm of Wick
products and on the properties of second quantization operators.
http://arxiv.org/abs/0809.3702
Author(s): M. Y. Mo
Abstract: We studied the universality of Wishart ensembles whose covariance matrix has
2 distinct eigenvalues. We studied the asymptotic limit when the number of both
eigenvalues goes to infinity and obtained universality results. In this case,
the limiting eigenvalue distribution can be supported on 1 or 2 disjoint
intervals. We obtained a necessary and sufficient condition on the parameters
such that the limiting distribution is supported on 2 disjoint intervals and
have computed the eigenvalue density in the limit. Furthermore, by using
Riemann-Hilbert analysis, we have shown that under proper rescaling of the
eigenvalues, the limiting correlation kernel is given by the sine kernel and
the Airy kernel in the bulk and the edge of the spectrum respectively. As a
consequence, the behavior of the largest eigenvalue in this model is described
by the Tracy-Widom distribution.
http://arxiv.org/abs/0809.3750
Author(s): Regis Ferriere and Viet Chi Tran (LPP)
Abstract: This work is a proceeding of the CANUM 2008 conference. Understanding how
stochastic and non-linear deterministic processes interact is a major challenge
in population dynamics theory. After a short review, we introduce a stochastic
individual-centered particle model to describe the evolution in continuous time
of a population with (continuous) age and trait structures. The individuals
reproduce asexually, age, interact and die. In a large population limit, the
random process converges to the solution of a Gurtin-McCamy type PDE. We show
that the random model has a long time behavior that differs from its
deterministic limit. However, the results on the limiting PDE and large
deviation techniques \textit{\`a la} Freidlin-Wentzell provide estimates of the
extinction time and a better understanding of the long time behavior of the
stochastic process. This has applications in the theory of Adaptive Dynamics.
In a last section, we present on simulations three biological issues dealing
with the consequences of size plasticity when taking growth into account, with
growth-reproduction trade-offs and with periodic behavior.
http://arxiv.org/abs/0809.3767
Author(s): Claude Auderset and Christian Mazza and Ernst Ruh
Abstract: This paper discusses the family of distributions on the Grassmannian of the
linear span of r central gaussian vectors parametrized by the covariance
matrix. Our main result is an existence and uniqueness criterion for the
maximum likelihood estimate of a sample.
http://arxiv.org/abs/0809.3697
Author(s): Michael Skeide
Abstract: For many Markov semigroups dilations in the sense of Hudson and
Parthasarathy, that is a dilation which is a cocycle perturbation of a noise,
have been constructed with the help of quantum stochastic calculi. In these
notes we show that every Markov semigroup on the algebra of all bounded
operators on a separable Hilbert space that is spatial in the sense of Arveson,
admits a Hudson-Parthasarathy dilation. In a sense, the opposite is also true.
The proof is based on general results on the the relation between spatial
E_0-semigroups and their product systems.
http://arxiv.org/abs/0809.3538
Author(s): Jocelyne Bion-Nadal
Abstract: In an incomplete financial market, the axiomatic of Time Consistent Pricing
Procedure (TCPP), recently introduced, is used to assign to any financial asset
a dynamic limit order book, taking into account both the dynamics of basic
assets and the limit order books for options.
Kreps-Yan fundamental theorem is extended to that context. A characterization
of TCPP calibrated on options is given in terms of their dual representation.
In case of perfectly liquid options, these options can be used as the basic
assets to hedge dynamically. A generic family of TCPP calibrated on option
prices is constructed, from cadlag BMO martingales.
http://arxiv.org/abs/0809.3824
Author(s): Alessandro De Gregorio and Stefano Maria Iacus
Abstract: In this paper a new dissimilarity measure to identify groups of assets
dynamics is proposed. The underlying generating process is assumed to be a
diffusion process solution of stochastic differential equations and observed at
discrete time. The mesh of observations is not required to shrink to zero. As
distance between two observed paths, the quadratic distance of the
corresponding estimated Markov operators is considered. Analysis of both
synthetic data and real financial data from NYSE/NASDAQ stocks, give evidence
that this distance seems capable to catch differences in both the drift and
diffusion coefficients contrary to other commonly used metrics.
http://arxiv.org/abs/0809.3902
Author(s): Milan \v{Z}ukovi\v{c} and Dionissios T. Hristopulos
Abstract: A problem of practical significance is the analysis of large, spatially
distributed data sets. The problem is more challenging for variables that
follow non-Gaussian distributions. We show that the spatial correlations
between variables can be captured by interactions between "spins". The spins
represent multilevel discretizations of the initial field with respect to a
number of pre-defined thresholds. The spatial dependence between the "spins" is
imposed by means of short-range interactions. We present two approaches,
inspired by the Ising and Potts models, that generate conditional simulations
from samples with missing data. The simulations of the "spin system" are forced
to respect locally the sample values and the system statistics globally. We
compare the two approaches in terms of their ability to reproduce the sample
statistical properties, to predict data at unsampled locations, as well as in
terms of their computational complexity. We discuss the impact of relevant
simulation parameters, such as the domain size, the number of discretization
levels, and the initial conditions.
http://arxiv.org/abs/0809.3918
Author(s): Yuri Kifer
Abstract: Various forms of the polynomial ergodic theorem (PET) which attracted
substantial attention in ergodic theory study the limits of expressions having
the form $1/N\sum_{n=1}^NT^{q_1(n)}f_1... T^{q_\ell (n)}f_\ell$ where $T$ is a
weakly mixing measure preserving transformation, $f_i$'s are bounded measurable
functions and $q_i$'s are polynomials taking on integer values on the integers.
Motivated partially by these results we obtain a central limit theorem for
expressions of the form $1/\sqrt{N}\sum_{n=1}^N (X_1(q_1(n))X_2(q_2(n))...
X_\ell(q_\ell(n))-a_1a_2... a_\ell)$ (sum-product limit theorem--SPLIT) where
$X_i$'s are fast $\alpha$-mixing bounded stationary processes, $a_j=EX_j(0)$
and $q_i$'s are positive functions taking on integer values on integers with
some growth conditions which are satisfied, for instance, when $q_i$'s are
polynomials of growing degrees. This result can be applied to the case when
$X_i(n)=T^nf_i$ where $T$ is a mixing subshift of finite type, a hyperbolic
diffeomorphism or an expanding transformation taken with a Gibbs invariant
measure, as well, as to the case when $X_i(n)=f_i(\xi_n)$ where $\xi_n$ is a
Markov chain satisfying the Doeblin condition considered as a stationary
process with respect to its invariant measure.
http://arxiv.org/abs/0809.4106
Author(s): Harry Kesten and Vladas Sidoravicius
Abstract: We consider the following problem in one-dimensional diffusion-limited
aggregation (DLA). At time $t$, we have an "aggregate" consisting of
$\Bbb{Z}\cap[0,R(t)]$ [with $R(t)$ a positive integer]. We also have $N(i,t)$
particles at $i$, $i>R(t)$. All these particles perform independent
continuous-time symmetric simple random walks until the first time $t'>t$ at
which some particle tries to jump from $R(t)+1$ to $R(t)$. The aggregate is
then increased to the integers in $[0,R(t')]=[0,R(t)+1]$ [so that
$R(t')=R(t)+1$] and all particles which were at $R(t)+1$ at time $t'{-}$ are
removed from the system. The problem is to determine how fast $R(t)$ grows as a
function of $t$ if we start at time 0 with $R(0)=0$ and the $N(i,0)$ i.i.d.
Poisson variables with mean $\mu>0$. It is shown that if $\mu<1$, then $R(t)$
is of order $\sqrt{t}$, in a sense which is made precise. It is conjectured
that $R(t)$ will grow linearly in $t$ if $\mu$ is large enough.
http://arxiv.org/abs/0809.4175
Author(s): J. van den Berg
Abstract: One of the most well-known classical results for site percolation on the
square lattice is the equation $p_c+p_c^*=1$. In words, this equation means
that for all values $\neq p_c$ of the parameter $p$, the following holds:
either a.s. there is an infinite open cluster or a.s. there is an infinite
closed "star" cluster. This result is closely related to the percolation
transition being sharp: below $p_c$, the size of the open cluster of a given
vertex is not only (a.s.) finite, but has a distribution with an exponential
tail. The analog of this result has been proven by Higuchi in 1993 for
two-dimensional Ising percolation (at fixed inverse temperature
$\beta<\beta_c$) with external field $h$, the parameter of the model. Using
sharp-threshold results (approximate zero-one laws) and a modification of an
RSW-like result by Bollob\'{a}s and Riordan, we show that these results hold
for a large class of percolation models where the vertex values can be "nicely"
represented (in a sense which will be defined precisely) by i.i.d. random
variables. We point out that the ordinary percolation model obviously belongs
to this class and we also show that the Ising model mentioned above belongs to
it.
http://arxiv.org/abs/0809.4184
Author(s): Bruno Schapira (LM-Orsay)
Abstract: We describe the set of bounded harmonic functions for the Heckman--Opdam
Laplacian, when the multiplicity function is larger than 1/2. We prove that
this set is a vector space of dimension the cardinality of the Weyl group. We
give some consequences in terms of the associated hypergeometric functions.
http://arxiv.org/abs/0809.4232
Author(s): John K. McSweeney and Boris G. Pittel
Abstract: We give an asymptotic expression for the expected coalescence time for a
non-uniform balls-into-boxes allocation model. Connections to coalescent
processes in population biology and computer science are discussed.
http://arxiv.org/abs/0809.4233
Author(s): Rishi Talreja and Ward Whitt
Abstract: We establish heavy-traffic stochastic-process limits for waiting times in
many-server queues with customer abandonment. If the system is asymptotically
critically loaded, as in the quality-and-efficiency-driven (QED) regime, then a
bounding argument shows that the abandonment does not affect waiting-time
processes so that the model behaves as if there were no abandonment
assumptions. If instead the system is overloaded, as in the efficiency-driven
(ED) regime, following Mandelbaum, Massey, Reiman and Stolyar, we treat
customer abandonment by studying the limiting behavior of the queueing models
with arrivals turned off at some time $t$. Then, the waiting time of an
infinitely patient customer arriving at time $t$ is the additional time it
takes for the queue to empty. To prove stochastic-process limits for virtual
waiting times, we establish a two-parameter version of Puhalskii's invariance
principle for first passage times. That in turn involves proving that
two-parameter versions of the composition and inverse mappings appropriately
preserve convergence.
http://arxiv.org/abs/0809.4275
Author(s): Birgit Rudloff and Ioannis Karatzas
Abstract: We study the problem of testing composite hypotheses versus composite
alternatives, using a convex duality approach. In contrast to classical results
obtained by Krafft & Witting (1967), where sufficient optimality conditions are
obtained via Lagrange duality, we obtain necessary and sufficient optimality
conditions via Fenchel duality under some compactness assumptions. This
approach also differs from the methodology developed in Cvitanic & Karatzas
(2001).
http://arxiv.org/abs/0809.4297
Author(s): Markus Heydenreich
Abstract: We consider a long-range version of self-avoiding walk in dimension $d >
2(\alpha \wedge 2)$, where $d$ denotes dimension and $\alpha$ the power-law
decay exponent of the coupling function. Under appropriate scaling we prove
convergence to Brownian motion for $\alpha \ge 2$, and to $\alpha$-stable
L\'evy motion for $\alpha < 2$. This complements results by Slade (1988), who
proves convergence to Brownian motion for nearest-neighbor self-avoiding walk
in high dimension.
http://arxiv.org/abs/0809.4333
Author(s): Dariusz Buraczewski and Ewa Damek and Yves Guivarc'h
Abstract: We consider a Markov chain $\{X_n\}_{n=0}^\8$ on $\R^d$ defined by the
stochastic recursion $X_{n}=M_n X_{n-1}+Q_n$, where $(Q_n,M_n)$ are i.i.d.
random variables taking values in the affine group $H=\R^d\rtimes {\rm
GL}(\R^d)$. Assume that $M_n$ takes values in the similarity group of $\R^d$,
and the Markov chain has a unique stationary measure $\nu$, which has unbounded
support. We denote by $|M_n|$ the expansion coefficient of $M_n$ and we assume
$\E |M|^\a=1$ for some positive $\a$. We show that the partial sums
$S_n=\sum_{k=0}^n X_k$, properly normalized, converge to a normal law ($\a\ge
2$) or to an infinitely divisible law, which is stable in a natural sense
($\a<2$). These laws are fully nondegenerate, if $\nu$ is not supported on an
affine hyperplane. Under a natural hypothesis, we prove also a local limit
theorem for the sums $S_n$. If $\a\le 2$, proofs are based on the homogeneity
at infinity of $\nu$ and on a detailed spectral analysis of a family of Fourier
operators $P_v$ considered as perturbations of the transition operator $P$ of
the chain $\{X_n \}$. The characteristic function of the limit law has a simple
expression in terms of moments of $\nu$ ($\a > 2$) or of the tails of $\nu$ and
of stationary measure for an associated Markov operator ($\a\le 2$). We extend
the results to the situation where $M_n$ is a random generalized similarity.
http://arxiv.org/abs/0809.4349
Author(s): Henrik Hult and Filip Lindskog
Abstract: In this paper we study the asymptotic decay of finite time ruin probabilities
for an insurance company that faces heavy-tailed claims, uses predictable
investment strategies and makes investments in risky assets whose prices evolve
according to quite general semimartingales. We show that the ruin problem
corresponds to determining hitting probabilities for the solution to a randomly
perturbed stochastic integral equation. We derive a large deviation result for
the hitting probabilities that holds uniformly over a family of semimartingales
and show that this result gives the asymptotic decay of finite time ruin
probabilities under arbitrary investment strategies, including optimal
investment strategies.
http://arxiv.org/abs/0809.4372
Author(s): H. Duminil-Copin
Abstract: We show that random walks on the infinite supercritical percolation clusters
in Z^d satisfy the usual Law of the Iterated Logarithm. The proof combines
Barlow's Gaussian heat kernel estimates and the ergodicity of the random walk
on the environment viewed from the random walker as derived by Berger and
Biskup.
http://arxiv.org/abs/0809.4380
Author(s): A.D. Barbour and A.V. Gnedin
Abstract: The paper is concerned with the classical occupancy scheme with infinitely
many boxes, in which $n$ balls are thrown independently into boxes $1,2,...$,
with probability $p_j$ of hitting the box $j$, where $p_1\geq p_2\geq...>0$ and
$\sum_{j=1}^\infty p_j=1$. We establish joint normal approximation as
$n\to\infty$ for the numbers of boxes containing $r_1,r_2,...,r_m$ balls,
standardized in the natural way, assuming only that the variances of these
counts all tend to infinity. The proof of this approximation is based on a
de-Poissonization lemma. We then review sufficient conditions for the variances
to tend to infinity. Typically, the normal approximation does not mean
convergence. We show that the convergence of the full vector of $r$-counts only
holds under a condition of regular variation, thus giving a complete
characterization of possible limit correlation structures.
http://arxiv.org/abs/0809.4387
Author(s): Paul Dupuis and Hitoshi Ishii
Abstract: Correction to The Annals of Probability 21 (1993) 554--580
[http://projecteuclid.org/euclid.aop/1176989415]
http://arxiv.org/abs/0809.4393
Author(s): Hui He
Abstract: Recently, it has been shown that stochastic spatial Lotka-Volterra models
when suitably rescaled can converge to a super-Brownian motion with drift. We
show that the limit process could be a super stable process if the kernel of
the underlying motion is in the domain of attraction of a stable law. As
applications of the convergence theorems, some new results on the voter model's
asymptotics are obtained.
http://arxiv.org/abs/0809.4520
Author(s): Achim Klenke and Leonid Mytnik
Abstract: Consider the mutually catalytic branching process with finite branching rate
$\gamma$. We show that as $\gamma\to\infty$ this process converges in finite
dimensional distributions (in time) to a certain discontinuous process. We give
descriptions of this process in terms of its semigroup, the infinitesimal
generator and as the solution of a martingale problem. We also give a strong
construction in terms of a planar Brownian motion from which we infer paths
properties of the process.
This is the first paper of a trilogy where we also construct an interacting
versions of this process and study its long-time behaviour.
http://arxiv.org/abs/0809.4554
Author(s): Enkelejd Hashorva
Abstract: Let B_0(s,t) be a Brownian pillow with continuous sample paths, and let
h,u:[0,1]^2\to R be two measurable functions. In this paper we derive upper and
lower bounds for the boundary non-crossing probability
\psi(u;h):=P{B_0(s,t)+h(s,t) \le u(s,t), \forall s,t\in [0,1]}. Further we
investigate the asymptotic behaviour of $\psi(u;\gamma h)$ with $\gamma$
tending to infinity, and solve a related minimisation problem.
http://arxiv.org/abs/0809.4560
Author(s): Holger Dette and Bettina Reuther
Abstract: In this paper we consider random block matrices, which generalize the general
beta ensembles, which were recently investigated by Dumitriu and Edelmann
(2002, 2005). We demonstrate that the eigenvalues of these random matrices can
be uniformly approximated by roots of matrix orthogonal polynomials which were
investigated independently from the random matrix literature. As a consequence
we derive the asymptotic spectral distribution of these matrices. The limit
distribution has a density, which can be represented as the trace of an
integral of densities of matrix measures corresponding to the Chebyshev matrix
polynomials of the first kind. Our results establish a new relation between the
theory of random block matrices and the field of matrix orthogonal polynomials,
which have not been explored so far in the literature.
http://arxiv.org/abs/0809.4601
Author(s): Ismael Bailleul
Abstract: This article investigates the question of sensitivity of the solutions of
Smoluchowski equation on R_+^* with respect to parameters \lambda in the
interaction kernel K^\lambda. It is proved that the solution is a C^1 function
of (t,\lambda) with values in a good space of measures under the hypotheses
K^{\lambda}(x,y)\leq \varphi(x)\varphi(y), for some sub-linear function
\varphi, a (4+epsilon)-moment assumption on the initial condition, and that the
derivative is a solution, in a suitable sense, of a linearized equation.
http://arxiv.org/abs/0809.4640
Author(s): Wilfrid S. Kendall
Abstract: Benjamini, Burdzy and Chen (2007) introduced the notion of a coupling: a
coupling of a Markov process such that, for suitable starting points, there is
a positive chance of the two component processes of the coupling staying a
positive distance away from each other for all time. Among other results, they
showed no shy couplings could exist for reflected Brownian motions in C^2
bounded convex planar domains whose boundaries contain no line segments. Here
we use potential-theoretic methods to extend this Benjamini et al. result (a)
to all bounded convex domains (whether planar and smooth or not) whose
boundaries contain no line segments, (b) to all bounded convex planar domains
regardless of further conditions on the boundary.
http://arxiv.org/abs/0809.4682
Author(s): Leonid Pastur and Anna Lytova
Abstract: We consider nxn real symmetric and hermitian Wigner random matrices n^{-1/2}W
with independent (modulo symmetry condition) entries and the (null) sample
covariance matrices n^{-1}X^*X with independent entries of mxn matrix X.
Assuming that 4th moments of entries of W and X satisfy a Lindeberg type
condition and the test functions of linear statistics of eigenvalues of the
above matrices are smooth enough (essentially of the class C^5), we prove that
linear statistics of eigenvalues satisfy the Central Limit Theorem as n, m tend
to infinity and m/n tends to finite c>0.
http://arxiv.org/abs/0809.4698
Author(s): Xinjia Chen
Abstract: In this paper, we develop a multistage approach for estimating the mean of a
bounded variable. We first focus on the multistage estimation of a binomial
parameter and then generalize the estimation methods to the case of general
bounded random variables. A fundamental connection between a binomial parameter
and the mean of a bounded variable is established. Our multistage estimation
methods rigorously guarantee prescribed levels of precision and confidence.
http://arxiv.org/abs/0809.4679
Author(s): Iva Kozakova and Mark Sapir
Abstract: We prove that with probability tending to 1, a 1-relator group with at least
3 generators and relator of length n is residually finite, virtually residually
(finite p)-group for all sufficiently large p, and coherent. The proof uses
both combinatorial group theory and non-trivial results about Brownian motions,
bridges and excursions in R^k.
http://arxiv.org/abs/0809.4693
Author(s): W. Bryc and D. Minda and S. Sethuraman
Abstract: Large deviation principles and related results are given for a class of
Markov chains associated to the "leaves" in random recursive trees and
preferential attachment random graphs, as well as the "cherries" in Yule trees.
In particular, the method of proof, combining analytic and Dupuis-Ellis type
path arguments, allows for an explicit computation of the large deviation
pressure.
http://arxiv.org/abs/0809.4741
Author(s): Erkan Nane
Abstract: We study solutions of a class of higher order partial differential equations
in bounded domains. These partial differential equations appeared first time in
the papers of Baeumer, Meerschaert and Nane \cite{bmn-07} and Meerschaert, Nane
and Vellaisamy \cite{MNV}, and Nane \cite{nane-h}. We express the solutions by
subordinating a killed Markov process by a hitting time of a stable
subordinator of index $0<\beta <1$, or by the absolute value of a symmetric
$\alpha$-stable process with $0<\alpha\leq 2$, independent of the Markov
process. In some special cases we represent the solutions by running
composition of $k$ independent Brownian motions, called $k$-iterated Brownian
motion for an integer $k\geq 2$.
http://arxiv.org/abs/0809.4824
Author(s): Malwina J. Luczak
Abstract: We consider Markovian models on graphs with local dynamics. We show that,
under suitable conditions, such Markov chains exhibit both rapid convergence to
equilibrium and strong concentration of measure in the stationary distribution.
We illustrate our results with applications to some known chains from computer
science and statistical mechanics.
http://arxiv.org/abs/0809.4856
Author(s): M. Birke and H. Dette
Abstract: We consider a uniform distribution on the set $\mathcal{M}_k$ of moments of
order $k \in \mathbb{N}$ corresponding to probability measures on the interval
$[0,1]$. To each (random) vector of moments in $\mathcal{M}_{2n-1}$ we consider
the corresponding uniquely determined monic (random) orthogonal polynomial of
degree $n$ and study the asymptotic properties of its roots if $n \to \infty$.
http://arxiv.org/abs/0809.4936
Author(s): Joaquin Fontbona
Abstract: We develop a McKean-Vlasov interpretation of Navier-Stokes equations with
external force field in the whole space, by associating with local mild
$L^p-$solutions of the 3d-vortex equation a generalized nonlinear diffusion
with random space-time birth, that probabilistically describes creation of
rotation in the fluid due to non-conservativeness of the force. We establish a
well-posedness result for this process and a stochastic representation formula
for the vorticity in terms of a vector-weighted version of its law after its
birth instant. Then, we introduce a stochastic system of 3d vortices with
mollified interaction and random space-time births, and prove the propagation
of chaos property, with the nonlinear process as limit, at an explicit pathwise
convergence rate. Convergence rates for stochastic approximation schemes of the
velocity and the vorticity fields are also obtained. We thus extend and refine
previous results concerning a probabilistic interpretation and a particle
approximation method for the non-forced equation, and generalize a recently
introduced random space-time-birth particle method for the 2d Navier-Stokes
equation with force.
http://arxiv.org/abs/0809.4947
Author(s): Bruce Driver and Maria Gordina
Abstract: We introduce a class of non-commutative, complex, infinite-dimensional
Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic
functions which are also square integrable with respect to a heat kernel
measure $\mu$ on these groups are studied. In particular, we establish a
unitary equivalence between the square integrable holomorphic functions and a
certain completion of the universal enveloping algebra of the "Lie algebra" of
this class of groups. Using quasi-invariance of the heat kernel measure, we
also construct a skeleton map which characterizes globally defined functions
from the $L^{2}(\nu)$-closure of holomorphic polynomials by their values on the
Cameron-Martin subgroup.
http://arxiv.org/abs/0809.4979
Author(s): Diana Dorobantu (SAF - EA2429)
Abstract: Our purpose is to study a particular class of optimal stopping problems for
Markov processes. We justify the value function convexity and we deduce that
there exists a boundary function such that the smallest optimal stopping time
is the first time when the Markov process passes over the boundary depending on
time. Moreover, we propose a method to find the optimal boundary function.
http://arxiv.org/abs/0809.4990
Author(s): Ross J. Kang and Colin McDiarmid
Abstract: We consider the $t$-improper chromatic number of the Erd{\H o}s-R{\'e}nyi
random graph $G(n,p)$. The t-improper chromatic number $\chi^t(G)$ of $G$ is
the smallest number of colours needed in a colouring of the vertices in which
each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$,
then this is the usual notion of proper colouring. When the edge probability
$p$ is constant, we provide a detailed description of the asymptotic behaviour
of $\chi^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.
http://arxiv.org/abs/0809.4726
Author(s): Michail Anthropelos and Nikolaos E. Frangos and Stylianos Z. Xanthopoulos and Athanasios N. Yannacopoulos
Abstract: In an incomplete market setting, we consider two financial agents, who wish
to price and trade a non-replicable contingent claim. Assuming that the agents
are utility maximizers, we propose a transaction price which is a result of the
minimization of a convex combination of their utility differences. We call this
price the risk sharing price, we prove its existence for a large family of
utility functions and we state some of its properties. As an example, we
analyze extensively the case where both agents report exponential utility.
http://arxiv.org/abs/0809.4781
Author(s): Qi Zhang and Huaizhong Zhao
Abstract: We prove a general theorem that the
$L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes
L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of an infinite
horizon backward doubly stochastic differential equation, if exists, gives the
stationary solution of the corresponding stochastic partial differential
equation. We prove the existence and uniqueness of the
$L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes
L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solutions for backward
doubly stochastic differential equations on finite and infinite horizon with
linear growth without assuming Lipschitz conditions, but under the monotonicity
condition. Therefore the solution of finite horizon problem gives the solution
of the initial value problem of the corresponding stochastic partial
differential equations, and the solution of the infinite horizon problem gives
the stationary solution of the SPDEs according to our general result.
http://arxiv.org/abs/0809.5089
Author(s): Samuel N. Cohen and Robert J. Elliott
Abstract: We consider backward stochastic differential equations (BSDEs) related to
finite state, continuous time Markov chains. We show that appropriate solutions
exist for arbitrary terminal conditions, and are unique up to sets of measure
zero. We do not require the generating functions to be monotonic, instead using
only an appropriate Lipschitz continuity condition.
http://arxiv.org/abs/0809.5102
Author(s): Emmanuel Abbe and Lizhong Zheng
Abstract: Over discrete memoryless channels (DMC), linear decoders (maximizing additive
metrics) afford several nice properties. In particular, if suitable encoders
are employed, the use of decoding algorithm with manageable complexities is
permitted. Maximum likelihood is an example of linear decoder. For a compound
DMC, decoders that perform well without the channel's knowledge are required in
order to achieve capacity. Several such decoders have been studied in the
literature. However, there is no such known decoder which is linear. Hence, the
problem of finding linear decoders achieving capacity for compound DMC is
addressed, and it is shown that under minor concessions, such decoders exist
and can be constructed. This paper also develops a "local geometric analysis",
which allows in particular, to solve the above problem. By considering very
noisy channels, the original problem is reduced, in the limit, to an inner
product space problem, for which insightful solutions can be found. The local
setting can then provide counterexamples to disproof claims, but also, it is
shown how in this problem, results proven locally can be "lifted" to results
proven globally.
http://arxiv.org/abs/0809.5217
Author(s): Steven R. Finch
Abstract: Simple analysis of the leftmost eigenvalue of Ince's equation (a boundary
value problem with periodicity) resolves an open issue surrounding a stochastic
Lyapunov exponent. Numerical verification is also provided.
http://arxiv.org/abs/0809.5274
Author(s): W. Liu and S. V. Lototsky
Abstract: A parameter estimation problem is considered for a one-dimensional stochastic
wave equation driven by additive space-time Gaussian white noise.
The estimator is of spectral type and utilizes a finite number of the spatial
Fourier coefficients of the solution. The asymptotic properties of the
estimator are studied as the number of the Fourier coefficients increases,
while the observation time and the noise intensity are fixed.
http://arxiv.org/abs/0810.0046
Author(s): Samuel N. Cohen and Robert J. Elliott
Abstract: Most previous contributions on BSDEs, and the related theories of nonlinear
expectation and dynamic risk measures, have been in the framework of continuous
time diffusions or jump diffusions. Using solutions of BSDEs on spaces related
to finite state, continuous time Markov Chains, we develop a theory of
nonlinear expectations in the spirit of Peng (2005). We prove basic properties
of these expectations, and show their applications to dynamic risk measures on
such spaces. In particular, we prove comparison theorems for scalar and vector
valued solutions to BSDEs, and discuss arbitrage and risk measures in the
scalar case.
http://arxiv.org/abs/0810.0055
Author(s): J.-R. Chazottes and F. Redig
Abstract: We obtain moment and Gaussian bounds for general Lipschitz functions
evaluated along the sample path of a Markov chain. We treat Markov chains on
general (possibly unbounded) state spaces via a coupling method. If the first
moment of the coupling time exists, then we obtain a variance inequality. If a
moment of order 1+epsilon of the coupling time exists, then depending on the
behavior of the stationary distribution, we obtain higher moment bounds. This
immediately implies polynomial concentration inequalities. In the case that a
moment of order 1+epsilon is finite uniformly in the starting point of the
coupling, we obtain a Gaussian bound. We illustrate the general results with
house of cards processes, in which both uniform and non-uniform behavior of
moments of the coupling time can occur.
http://arxiv.org/abs/0810.0097
Author(s): A. Faggionato
Abstract: We consider i.i.d. random variables {\omega (b):b \in E_d} parameterized by
the family of bonds in Z^d, d>1. The random variable \omega(b) is thought of as
the conductance of bond b and it ranges in a finite interval [0,c_0]. Assuming
the probability m of the event {\omega(b)>0} to be supercritical and denoting
by C(\omega) the unique infinite cluster associated to the bonds with positive
conductance, we study the zero range process on C(\omega) with
\omega(b)-proportional probability rate of jumps along bond b. For almost all
realizations of the environment we prove that the hydrodynamic behavior of the
zero range process is governed by the nonlinear heat equation $\partial_t \rho=
m \nabla \cdot (D \nabla\phi(\rho/m))$, where the matrix D and the function
\phi are \omega--independent. We do not require any ellipticity condition.
http://arxiv.org/abs/0810.0103
Author(s): James Norris and Amanda Turner
Abstract: We study a scaling limit associated to a model of planar aggregation. The
model is obtained by composing certain independent random conformal maps. The
evolution of harmonic measure on the boundary of the cluster is shown to
converge to the coalescing Brownian flow.
http://arxiv.org/abs/0810.0211
Author(s): Pierre Cardaliaguet (LM-Brest) and Catherine Rainer (LM)
Abstract: For zero-sum two-player continuous-time games with integral payoff and
incomplete information on one side, one shows that the optimal strategy of the
informed player can be computed through an auxiliary optimization problem over
some martingale measures. One also characterizes the optimal martingale
measures and compute it explicitely in several examples.
http://arxiv.org/abs/0810.0220
Author(s): Jonathon Peterson
Abstract: This thesis concerns the study of random walks in random environments (RWRE).
Since there are two levels of randomness for random walks in random
environments, there are two different distributions for the random walk that
can be studied. The quenched distribution is the law of the random walk
conditioned on a given environment. The annealed distribution is the quenched
law averaged over all environments. The main results of the thesis fall into
two categories: quenched limiting distributions for one-dimensional, transient
RWRE and annealed large deviations for multidimensional RWRE.
The analysis of the quenched distributions for transient, one-dimensional
RWRE falls into two separate cases. First, when an annealed central limit
theorem holds, we prove that a quenched central limit theorem also holds but
with a random (depending on the environment) centering. In contrast, when the
annealed limit distribution is not Gaussian, we prove that there is no quenched
limiting distribution for the RWRE. Moreover, we show that for almost every
environment, there exist two random (depending on the environment) sequences of
times, along which random walk has different quenched limiting distributions.
While an annealed large deviation principle for multidimensional RWRE was
known previously, very little qualitative information was available about the
annealed large deviation rate function. We prove that if the law on
environments is non-nestling, then the annealed large deviation rate function
is analytic in a neighborhood of its unique zero (which is the limiting
velocity of the RWRE).
http://arxiv.org/abs/0810.0257
Author(s): Carl Graham (CMAP) and Philippe Robert
Abstract: The mean-field limit of a Markovian model describing the interaction of
several classes of permanent connections in a network is analyzed. In the same
way as for the TCP algorithm, each of the connections has a self-adaptive
behavior in that its transmission rate along its route depends on the level of
congestion of the nodes of the route. Since several classes of connections
going through the nodes of the network are considered, an original mean-field
result in a multi-class context is established. It is shown that, as the number
of connections goes to infinity, the behavior of the different classes of
connections can be represented by the solution of an unusual non-linear
stochastic differential equation depending not only on the sample paths of the
process, but also on its distribution. Existence and uniqueness results for the
solutions of these equations are derived. Properties of their invariant
distributions are investigated and it is shown that, under some natural
assumptions, they are determined by the solutions of a fixed point equation in
a finite dimensional space.
http://arxiv.org/abs/0810.0347
Author(s): Iain M. MacPhee and Mikhail V. Menshikov and Stanislav Volkov and Andrew R. Wade
Abstract: We study the non-equilibrium dynamics of a one-dimensional interacting
particle system that is a mixture of the voter model and exclusion process.
With the process started from a finite perturbation of the ground-state
Heaviside configuration consisting of 1s to the left of the origin and 0s
elsewhere, we study the relaxation time $\tau$, that is, the first hitting time
of the ground-state configuration (up to translation). In particular, we give
conditions for $\tau$ to be finite and for certain moments of $\tau$ to be
finite or infinite, and prove a result that approaches a conjecture of Belitsky
{\em et al.} [{\em Bernoulli} {\bf 7} (2001) 119--144]. Ours are the first
non-existence of moments results for $\tau$ for the mixture model. Moreover, we
give almost-sure asymptotic results on the long-term evolution of the size of
the hybrid (disordered) region. Most of our results pertain to the
discrete-time setting, but several transfer to continuous-time. As well as the
mixture process, some of our results also cover the pure exclusion case. We
state several significant open problems that remain.
http://arxiv.org/abs/0810.0392
Author(s): Auguste Aman (LMAI) and Naoul Mrhardy
Abstract: This paper is intended to give a probabilistic representation for stochastic
viscosity solution of semi-linear reflected stochastic partial differential
equations with nonlinear Neumann boundary condition. We use it connection with
reflected generalized backward doubly stochastic differential equation.
http://arxiv.org/abs/0810.0436
Author(s): Xinjia Chen
Abstract: In this paper, we have developed a new class of sampling schemes for
estimating parameters of binomial and Poisson distributions. Without any
information of the unknown parameters, our sampling schemes rigorously
guarantee prescribed levels of precision and confidence.
http://arxiv.org/abs/0810.0430
Author(s): Brigitta Vermesi
Abstract: We prove existence of intersection exponents xi(k,lambda) for biased random
walks on d-dimensional half-infinite discrete cylinders, and show that, as
functions of lambda, these exponents are real analytic. As part of the
argument, we prove convergence to stationarity of a time-inhomogeneous Markov
chain on half-infinite random paths. Furthermore, we show this convergence
takes place at exponential rate, an estimate obtained via a coupling of
weighted half-infinite paths.
http://arxiv.org/abs/0810.0572
Author(s): Oliver Johnson
Abstract: We adapt arguments concerning information-theoretic convergence in the
Central Limit Theorem to the case of dependent random variables under
Rosenblatt mixing conditions. The key is to work with random variables
perturbed by the addition of a normal random variable, giving us good control
of the joint density and the mixing coefficient. We strengthen results of
Takano and of Carlen and Soffer to provide entropy-theoretic, not weak
convergence.
http://arxiv.org/abs/0810.0593
Author(s): A. D. Barbour and Torgny Lindvall
Abstract: The paper is concerned with approximating the distribution of a sum W of n
integer valued random variables Y_i, whose distributions depend on the state of
an underlying Markov chain X. The approximation is in terms of a translated
Poisson distribution, with mean and variance chosen to be close to those of W,
and the error is measured with respect to the total variation norm. Error
bounds comparable to those found for normal approximation with respect to the
weaker Kolmogorov distance are established, provided that the distribution of
the sum of the Y_i's between the successive visits of X to a reference state is
aperiodic. Without this assumption, approximation in total variation cannot be
expected to be good.
http://arxiv.org/abs/0810.0599
Author(s): Samuel Herrmann (IECN) and Pierre Vallois (IECN)
Abstract: We study a family of memory-based persistent random walks and we prove weak
convergences after space-time rescaling. The limit processes are not only
Brownian motions with drift. We have obtained a continuous but non-Markov
process $(Z_t)$ which can be easely expressed in terms of a counting process
$(N_t)$. In a particular case the counting process is a Poisson process, and
$(Z_t)$ permits to represent the solution of the telegraph equation. We study
in detail the Markov process $((Z_t,N_t); t\ge 0)$.
http://arxiv.org/abs/0810.0650
Author(s): M. Formentin and C. Kuelske
Abstract: We consider the free boundary condition Gibbs measure of the Potts model on a
random tree. We provide an explicit temperature interval below the
ferromagnetic transition temperature for which this measure is extremal,
improving older bounds of Mossel and Peres. In information theoretic language
extremality of the Gibbs measure corresponds to non-reconstructability for
symmetric q-ary channels. The bounds are optimal for the Ising model and appear
to be close to what we conjecture to be the true values up to a factor of
0.0150 in the case q = 3 and 0.0365 for q = 4. Our proof uses an iteration of
random boundary entropies from the outside of the tree to the inside, along
with a symmetrization argument.
http://arxiv.org/abs/0810.0677
Author(s): Robert W. Neel
Abstract: We prove that minimal graphs (other than planes) are parabolic in the sense
that any bounded harmonic function is determined by its boundary values. The
proof relies on using the coupling introduced in the author's earlier paper "A
martingale approach to minimal surfaces" to show that Brownian motion on such a
minimal graph almost surely strikes the boundary in finite time.
http://arxiv.org/abs/0810.0669
Author(s): Dmitry Panchenko
Abstract: We consider a symmetric positive definite weakly exchangeable infinite random
matrix whose elements take a finite number of values and we prove that if the
distribution of the matrix satisfies the Ghirlanda-Guerra identities then it is
ultrametric with probability one.
http://arxiv.org/abs/0810.0743
Author(s): Moawia Alghalith
Abstract: This paper synthesizes and analyzes some important current and recent
contributions to the theory of the firm under uncertainty. In so doing, it
examines the production and hedging decisions of the competitive firm under a
single source and multiple sources of uncertainty.
http://arxiv.org/abs/0810.0917
Author(s): Tom Alberts and Scott Sheffield
Abstract: We construct a natural measure mu supported on the intersection of a chordal
SLE(kappa) curve gamma with the real line R, in the range 4 < kappa < 8. The
measure is a function of the SLE path in question. Assuming that boundary
measures transform in a ``d-dimensional'' way (where d is the Hausdorff
dimension of gamma intersected with R), we show that the measure we construct
is (up to multiplicative constant) the unique measure-valued function of the
SLE path that satisfies the Domain Markov property.
http://arxiv.org/abs/0810.0940
Author(s): Federico Camia
Abstract: We present a review of the recent progress on percolation scaling limits in
two dimensions. In particular, we will consider the convergence of critical
crossing probabilities to Cardy's formula and of the critical exploration path
to chordal SLE(6), the full scaling limit of critical cluster boundaries, and
near-critical scaling limits.
http://arxiv.org/abs/0810.1002
Author(s): Manon Defosseux (PMA)
Abstract: A connection between representation of compact groups and some invariant
ensembles of Hermitian matrices is described. We focus on two types of
invariant ensembles which extend the Gaussian and the Laguerre Unitary
ensembles. We study them using projections and convolutions of invariant
probability measures on adjoint orbits of a compact Lie group. These measures
are described by semiclassical approximation involving tensor and restriction
mulltiplicities. We show that a large class of them are determinantal.
http://arxiv.org/abs/0810.1011
Author(s): Ju-Yi Yen and Marc Yor
Abstract: In this paper, we propose several "measurements" of the "non-stopping
timeness" of ends g of previsible sets, such that g avoids stopping times, in
an ambiant filtration. We then study several explicit examples, involving last
passage times of some remarkable martingales.
http://arxiv.org/abs/0810.1059
Author(s): Erkan Nane and Yimin Xiao and Aklilu Zeleke
Abstract: Let $p \in (0, \infty)$ be a constant and let $\{\xi_n\} \subset L^p(\Omega,
{\mathcal F}, \P)$ be a sequence of random variables. For any integers $m, n
\ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} \xi_k$. It is proved that, if
there exist a nondecreasing function $\varphi: \R_+\to \R_+$ (which satisfies a
mild regularity condition) and an appropriately chosen integer $a\ge 2$ such
that $$ \sum_{n=0}^\infty \sup_{k \ge 0} \E\bigg|\frac{S_{k, a^n}}
{\varphi(a^n)} \bigg|^p < \infty,$$ Then $$ \lim_{n \to \infty} \frac{S_{0, n}}
{\varphi(n)} = 0\qquad \hbox{a.s.} $$ This extends Theorem 1 in Levental,
Chobanyan and Salehi \cite{chobanyan-l-s} and can be applied conveniently to a
wide class of self-similar processes with stationary increments including
stable processes.
http://arxiv.org/abs/0810.1061
Author(s): Marcos Kiwi and Jos\'e Soto
Abstract: It is well known that, when normalized by n, the expected length of a longest
common subsequence of d sequences of length n over an alphabet of size sigma
converges to a constant gamma_{sigma,d}. We disprove a speculation by Steele
regarding a possible relation between gamma_{2,d} and gamma_{2,2}. In order to
do that we also obtain new lower bounds for gamma_{sigma,d}, when both sigma
and d are small integers.
http://arxiv.org/abs/0810.1066
Author(s): John A. Toth and Igor Wigman
Abstract: We compute the asymptotic expectation of the number of {\em open} nodal lines
for random waves on smooth planar domains. We find that for both the long
energy window $[0,\lambda]$ and the short one $[\lambda,\lambda+1]$ the
expected number of open nodal lines is proportional to $\lambda$,
asymptotically as $\lambda\to\infty$. Our results are consistent with the
predictions in the physics literature made by Blum, Gnutzmann and Smilansky
\cite{BGS}.
http://arxiv.org/abs/0810.1276
Author(s): Samy Tindel (IECN) and J\'er\'emie Unterberger (IECN)
Abstract: In this paper, we consider a complex-valued d-dimensional fractional Brownian
motion defined on the closure of the complex upper half-plane, called analytic
fractional Brownian motion. This process has been introduced by the second
author of the article, and both its real and imaginary parts, restricted on the
real axis, are usual fractional Brownian motions. The current note is devoted
to prove that a rough path based on the analytic fBm can be constructed for any
value of the Hurst parameter in (0,1/2). This allows in particular to solve
differential equations driven by this process in a neighborhood of 0 of the
complex upper half-plane, thanks to a variant of the usual rough path theory
due to Gubinelli.
http://arxiv.org/abs/0810.1408
Author(s): Michael Damron and C.L. Winter
Abstract: We introduce a new model for rill erosion. We start with a network similar to
that in the Discrete Web and instantiate a dynamics which makes the process
highly non-Markovian. The behavior of nodes in the streams is similar to the
behavior of Polya urns with time-dependent input. In this paper we use a
combination of rigorous arguments and simulation results.
http://arxiv.org/abs/0810.1483
Author(s): Cristian Giardina and Jorge Kurchan and Frank Redig and Kiamars Vafayi
Abstract: In the context of Markov processes, both in discrete and continuous setting,
we show a general relation between duality functions and symmetries of the
generator. If the generator can be written in the form of a Hamiltonian of a
quantum spin system, then the "hidden" symmetries are easily derived. We
illustrate our approach in processes of symmetric exclusion type, in which the
symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP)
model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an
instantaneous thermalization limit of the energy process associated to a large
family of models of interacting diffusions, which we call Brownian energy
process (BEP) and which all possess the SU(1,1) symmetry. We treat in details
the case where the system is in contact with reservoirs and the dual process
becomes absorbing.
http://arxiv.org/abs/0810.1202
Author(s): Richard C. Bradley and Alexander R. Pruss
Abstract: For an arbitrary integer N that is at least 2, this paper gives a
construction of a strictly stationary, N-tuplewise independent sequence of
(non-degenerate) bounded random variables such that the Central Limit Theorem
fails to hold. The sequence is in part an adaptation of a non-stationary
example with similar properties constructed by one of the authors (ARP) in a
paper published in 1998.
http://arxiv.org/abs/0810.1707
Author(s): Samy Tindel (IECN) and Aur\'elien Deya (IECN)
Abstract: We define and solve Volterra equations driven by an irregular signal, by
means of a variant of the rough path theory allowing to handle generalized
integrals weighted by an exponential coefficient. The results are applied to
the fractional Brownian motion with Hurst coefficient greater than 1/3
http://arxiv.org/abs/0810.1824
Author(s): Elvan Ceyhan
Abstract: For two or more classes of points in $\R^d$ with $d \ge 1$, the class cover
catch digraphs (CCCDs) can be constructed using the relative positions of the
points from one class with respect to the points from the other class. The
CCCDs were introduced by (Priebe, DeVinney, and Mar-chette, (2001). On the
distribution of the domination number of random class catch cover di-graphs.
Statistics and Probability Letters, 55:239-246) who investigated the case of
two classes, $\X$ and $\Y$. They calculated the exact (finite sample)
distribution of the domination number of the CCCDs based on $\X$ points
relative to $\Y$ points both of which were uniformly distri-buted on a bounded
interval. We investigate the distribution of the domination number of the CCCDs
based on data from non-uniform $\X$ points on an interval with end points from
$\Y$. Then we extend these calculations for multiple $\Y$ points on bounded
intervals.
http://arxiv.org/abs/0810.1893
Author(s): Juan Zhao
Abstract: We study a class of stochastic integral equations with jumps under
non-Lipschitz conditions. We use the method of Euler approximations to obtain
the existence of the solution and give some criteria for the strong solution.
http://arxiv.org/abs/0810.1908
Author(s): Serguei Foss and Dmitry Korshunov and Stan Zachary
Abstract: We study convolutions of long-tailed and subexponential distributions and
measures on the real line, proving some important new results through a simple
and coherent approach, and showing also that the standard properties of such
convolutions follow as easy consequences.
http://arxiv.org/abs/0810.1994
Author(s): Teemu Pennanen and Irina Penner
Abstract: We study superhedging of contingent claims with physical delivery in a
discrete-time market model with convex transaction costs. Our model extends
Kabanov's currency market model by allowing for nonlinear illiquidity effects.
We show that an appropriate generalization of Schachermayer's robust no
arbitrage condition implies that the set of claims hedgeable with zero cost is
closed in probability. Combined with classical techniques of convex analysis,
the closedness yields a dual characterization of premium processes that are
sufficient to superhedge a given claim process. We also extend the fundamental
theorem of asset pricing for general conical models.
http://arxiv.org/abs/0810.2016
Author(s): Kyle Siegrist
Abstract: We consider probability distributions with constant rate on partially ordered
sets, generalizing distributions in the usual reliability setting that have
constant failure rate. In spite of the minimal algebraic structure, there is a
surprisingly rich theory, including interesting moment results and an
associated stochastic process with "gamma" distributions. We concentrate mostly
on discrete posets and characterize constant rate distributions on trees. We
pose some questions on the existence of constant rate distributions for general
discrete posets.
http://arxiv.org/abs/0810.2105
Author(s): Tomoyuki Ichiba and Ioannis Karatzas
Abstract: We examine the behavior of n Brownian particles diffusing on the real line,
with bounded, measurable drift and bounded, piecewise continuous diffusion
coefficients that depend on the current configuration of particles. Sufficient
conditions are established for the absence of triple collisions, as well as for
the presence of (infinitely-many) triple collisions among the particles. As an
application to the Atlas model of equity markets, we study a special
construction of such systems of diffusing particles using Brownian motions with
reflection on polyhedral domains.
http://arxiv.org/abs/0810.2149
Author(s): Maxim Krikun and Anatoly Yambartsev
Abstract: Ising model without external field on an infinite Lorentzian triangulation
sampled from the uniform distribution is considered. We prove uniqueness of the
Gibbs measure in the high temperature region and coexistence of at least two
Gibbs measures at low temperature. The proofs are based on the disagreement
percolation method and on a variant of Peierls method. The critical temperature
is shown to be constant a.s.
http://arxiv.org/abs/0810.2182
Author(s): Kyle Siegrist
Abstract: We consider Markov chains on partially ordered sets that generalize the
success-runs and remaining life chains in reliability theory. We find
conditions for recurrence and transience and give simple expressions for the
invariant distributions. We study a number of special cases, including rooted
trees, uniform posets, and posets associated with positive semigroups.
http://arxiv.org/abs/0810.2289
Author(s): Dmitry Chelkak and Stanislav Smirnov
Abstract: We study discrete complex analysis and potential theory on a large family of
planar graphs, the so-called isoradial ones. Along with discrete analogues of
several classical results, we prove uniform convergence of discrete harmonic
measures, Green's functions and Poisson kernels to their continuous
counterparts. Among other applications, the results can be used to establish
universality of the critical Ising and other lattice models.
http://arxiv.org/abs/0810.2188
Author(s): Michael Blank
Abstract: We introduce and study a deterministic lattice model describing the motion of
an infinite system of oppositely charged particles under the action of a
constant electric field. As an application this model represents a traffic flow
of cars moving in opposite directions along a narrow road. Our main results
concern the Fundamental diagram of the system describing the dependence of
average particle velocities on their densities and the Phase diagram describing
the partition of the space of particle configurations into regions having
different qualitative properties, which we identify with free, jammed and
hysteresis phases.
http://arxiv.org/abs/0810.2205
Author(s): Gerhard Keller and Carlangelo Liverani
Abstract: We present a common framework to study decay and exchanges rates in a wide
class of dynamical systems. Several applications, ranging form the metric
theory of continuons fractions and the Shannon capacity of contrained systems
to the decay rate of metastable states, are given.
http://arxiv.org/abs/0810.2229
Author(s): K. D. Elworthy and Y. LeJan and Xue-Mei Li
Abstract: Geometry arising from two diffusion operators (smooth semi-elliptic, second
order differential operators) on different spaces but intertwined by a smooth
map is described. Particular cases arise from Riemannian submersions when the
operators are Laplace-Beltrami operators, from equivariant operators on the
total space of a principal bundle, and for the operators on the diffeomorphism
group arising from stochastic flows. Classical non-linear filtering problems
also lead to such conffigurations. A basic tool is the, possibly, non-linear
"semi-connection" induced by this set up, leading to a canonical decomposition
of the operator on the domain space. Topics discussed include: generalised
Wietzenbock curvatures arising in the equivariant case, skew -product
decompositions of diffusion processes, conditioned processes, classical
filtering, decomposition of stochastic flows, and connections determined by
stochastic differential equations.
http://arxiv.org/abs/0810.2253
Author(s): Youness Lamzouri
Abstract: We prove several results on the distribution of values of $L$-functions at
the edge of the critical strip, by constructing and studying a large class of
random Euler products. Among new applications, we provide a precise estimate
for the distribution of the values at $s=1$ of the family of $k$-th symmetric
power $L$-functions of primitive cusp forms of weight 2 in the level aspect
(assuming the automorphy of these $L$-functions), using results of Cogdell and
Michel on high complex moments of this family. Further we study families of
$L$-functions of the Selberg Class in the $t$ aspect at Re$(s)=1$, and of
$L$-functions of quadratic twists of cuspidal automorphic representations of
$GL(m)/{\Bbb Q}$ at $s=1$, and prove precise estimates for the corresponding
distribution functions assuming a uniform version of the Sato-Tate conjecture.
http://arxiv.org/abs/0810.2292
Author(s): Marco Lenci
Abstract: We prove the annealed Central Limit Theorem for random walks in bistochastic
random environments on $Z^d$ with zero local drift. The proof is based on a
"dynamicist's interpretation" of the system, and requires a much weaker
condition than the customary uniform ellipticity. Moreover, recurrence is
derived for $d \le 2$.
http://arxiv.org/abs/0810.2324
Author(s): Martin Barlow and Ben Hambly
Abstract: We consider the random walk on supercritical percolation clusters in the
d-dimensional Euclidean lattice. Previous papers have obtained Gaussian heat
kernel bounds, and a.s. invariance principles for this process. We show how
this information leads to a parabolic Harnack inequality, a local limit theorem
and estimates on the Green's function.
http://arxiv.org/abs/0810.2467
Author(s): A. V. Goltsev and S. N. Dorogovtsev and J. F. F. Mendes
Abstract: We reconsider the problem of percolation on an equilibrium random network
with degree-degree correlations between nearest-neighboring vertices focusing
on critical singularities at a percolation threshold. We obtain criteria for
degree-degree correlations to be irrelevant for critical singularities. We
present examples of networks in which assortative and disassortative mixing
leads to unusual percolation properties and new critical exponents.
http://arxiv.org/abs/0810.1742
Author(s): Olivier Durieu and Dalibor Volny
Abstract: We show by a constructive proof that in all aperiodic dynamical system, for
all sequences $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and
$\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a set $A\in\A$ having the
property that the sequence of the distributions of
$(\frac{1}{a_{n}}S_{n}(\ind_A-\mu(A)))_{n\in\N}$ is dense in the space of all
probability measures on $\R$.
http://arxiv.org/abs/0810.2452
Author(s): Emmanuel Breuillard and Peter Friz and Martin Huesmann
Abstract: Donsker's invariance principle is shown to hold for random walks in rough
path topology. As application, we obtain Donsker-type weak limit theorems for
stochastic integrals and differential equations.
http://arxiv.org/abs/0810.2681
Author(s): Martin Hairer and Jonathan C. Mattingly
Abstract: The aim of this note is to present an elementary proof of a variation of
Harris' ergodic theorem of Markov chains. This theorem, dating back to the
fifties essentially states that a Markov chain is uniquely ergodic if it admits
a ``small'' set which is visited infinitely often. This gives an extension of
the ideas of Doeblin to the unbounded state space setting. Often this is
established by finding a Lyapunov function with ``small'' level sets. This
topic has been studied by many authors (cf. Harris, Hasminskii, Nummelin, Meyn
and Tweedie). If the Lyapunov function is strong enough, one has a spectral gap
in a weighted supremum norm (cf. Meyn and Tweedie).
Traditional proofs of this result rely on the decomposition of the Markov
chain into excursions away from the small set and a careful analysis of the
exponential tail of the length of these excursions. There have been other
variations which have made use of Poisson equations or worked at getting
explicit constants. The present proof is very direct, and relies instead on
introducing a family of equivalent weighted norms indexed by a parameter
$\beta$ and to make an appropriate choice of this parameter that allows to
combine in a very elementary way the two ingredients (existence of a Lyapunov
function and irreducibility) that are crucial in obtaining a spectral gap.
The original motivation of this proof was the authors' work on spectral gaps
in Wasserstein metrics. The proof presented in this note is a version of our
reasoning in the total variation setting which we used to guide the
calculations in arXiv:math/0602479. While we initially produced it for that
purpose, we hope that it will be of interest in its own right.
http://arxiv.org/abs/0810.2777
Author(s): Matyas Barczy and Gyula Pap
Abstract: We study asymptotic behavior of maximum likelihood estimator for a time
inhomogeneous diffusion process given by a SDE $dX_t=\alpha b(t)X_t dt +
\sigma(t) dB_t$, $t\in[0,T)$, with a parameter $\alpha\in R$, where
$T\in(0,\infty]$ and $(B_t)_{t\in[0,T)}$ is a standard Wiener process. We
formulate sufficient conditions under which the MLE of $\alpha$ normalized by
Fisher information converges to the limit distribution of Dickey-Fuller
statistics. Next we study a SDE $dY_t=\alpha b(t)a(Y_t) dt + \sigma(t) dB_t$,
$t\in[0,T)$, with a perturbed drift satisfying $a(x)=x+O(1+|x|^\gamma)$ with
some $\gamma\in[0,1)$. We give again sufficient conditions under which the MLE
of $\alpha$ normalized by Fisher information converges to the limit
distribution of Dickey-Fuller statistics.
http://arxiv.org/abs/0810.2688
Author(s): Alexandru Nica
Abstract: Let k be a positive integer and let D_k denote the space of joint
distributions for k-tuples of selfadjoint elements in C*-probability space. The
paper studies the concept of "subordination distribution of \mu \boxplus \nu
with respect to \nu" for \mu, \nu \in D_k, where \boxplus is the operation of
free additive convolution on D_k. The main tools used in this study are
combinatorial properties of R-transforms for joint distributions and a related
operator model, with operators acting on the full Fock space
Multi-variable subordination turns out to have nice relations to a process of
evolution towards \boxplus-infinite divisibility on D_k that was recently found
by Belinschi and Nica (arXiv:0711.3787). Most notably, one gets better insight
into a connection which this process was known to have with free Brownian
motion.
http://arxiv.org/abs/0810.2571
Author(s): Daniel Remenik
Abstract: There is a widespread recent interest in using ideas from statistical physics
to model certain types of problems in economics and finance. The main idea is
to derive the macroscopic behavior of the market from the random local
interactions between agents. Our purpose is to present a general framework that
encompasses a broad range of models, by proving a law of large numbers and a
central limit theorem for certain interacting particle systems with very
general state spaces. To do this we draw inspiration from some work done in
mathematical ecology and mathematical physics. The first result is proved for
the system seen as a measure-valued process, while to prove the second one we
will need to introduce a chain of embeddings of some abstract Banach and
Hilbert spaces of test functions and prove that the fluctuations converge to
the solution of a certain generalized Gaussian stochastic differential equation
taking values in the dual of one of these spaces.
http://arxiv.org/abs/0810.2813
Author(s): Fabio Gagliardi Cozman
Abstract: This paper presents concentration inequalities and laws of large numbers
under weak assumptions of irrelevance, expressed through lower and upper
expectations. The results extend de Cooman and Miranda's recent results
concerning epistemic irrelevance. The proofs indicate connections between
concepts of irrelevance for lower/upper expectations and the standard theory of
martingales.
http://arxiv.org/abs/0810.2821
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X) and Gesine Reinert
Abstract: We compute explicit bounds in the Gaussian approximation of functionals of
infinite Rademacher sequences. Our tools involve Stein's method, as well as the
use of appropriate discrete Malliavin operators. Although our approach does not
require the classical use of exchangeable pairs, we employ a chaos expansion in
order to construct an explicit exchangeable pair vector for any random variable
which depends on a finite set of Rademacher variables. Among several examples,
which include random variables which depend on infinitely many Rademacher
variables, we provide three main applications: (i) to CLTs for multilinear
forms belonging to a fixed chaos, (ii) to the Gaussian approximation of
weighted infinite 2-runs, and (iii) to the computation of explicit bounds in
CLTs for multiple integrals over sparse sets. This last application provides an
alternate proof (and several refinements) of a recent result by Blei and
Janson.
http://arxiv.org/abs/0810.2890
Author(s): Matyas Barczy and Gyula Pap
Abstract: We consider a process $(X_t)_{t\in[0,T)}$ given by the SDE $dX_t = \alpha
b(t)X_t dt + \sigma(t) dB_t$, $t\in[0,T)$, with initial condition $X_0=0$,
where $T\in(0,\infty]$, $\alpha\in R$, $(B_t)_{t\in[0,T)}$ is a standard Wiener
process, $b:[0,T)\to R\setminus\{0\}$ and $\sigma:[0,T)\to(0,\infty)$ are
continuously differentiable functions. Assuming that $b$ and $\sigma$ satisfy a
certain differential equation we derive an explicit formula for the joint
Laplace transform of $\int_0^t\frac{b(s)^2}{\sigma(s)^2}(X_s)^2 ds$ and
$(X_t)^2$ for all $t\in[0,T)$. As an application, we study asymptotic behavior
of the maximum likelihood estimator of $\alpha$ for $\sign(\alpha-K)=\sign(K)$,
$K\ne0$, and for $\alpha=K$, $K\ne0$. As an example, we examine the so-called
$\alpha$-Wiener bridges given by SDE $dX_t = -\frac{\alpha}{T-t}X_t dt + dB_t$,
$t\in[0,T)$, with initial condition $X_0=0$.
http://arxiv.org/abs/0810.2930
Author(s): Christian Houdr\'e and Trevis J. Litherland
Abstract: Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain,
with state space an ordered finite alphabet of size $m$. Using combinatorial
constructions and weak invariance principles, we obtain the limiting shape of
the associated Young tableau as a multidimensional Brownian functional. Since
the length of the top row of the Young tableau is also the length of the
longest (weakly) increasing subsequence of $(X_k)_{1\le k \le n}$, the
corresponding limiting law follows. We relate our results to a conjecture of
Kuperberg by showing that, under a cyclic condition, a spectral
characterization of the Markov transition matrix delineates precisely when the
limiting shape is the spectrum of the traceless GUE. For $m=3$, all cyclic
Markov chains have such a limiting shape, a fact previously known for $m=2$.
However, this is no longer true for $m \ge 4$.
http://arxiv.org/abs/0810.2982
Author(s): Terence Tao and Van Vu
Abstract: The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$
matrix with iid complex entries of mean zero and unit variance, then the
empirical spectral distribution (ESD) of the normalized matrix
$\frac{1}{\sqrt{n}} M_n$ converges almost surely to the uniform distribution on
the unit disk $\{z \in \C: |z| \leq 1 \}$. After a long sequence of partial
results that verified this law under additional assumptions on the distribution
of the entries, the full circular law was recently established in
\cite{TVcir2}. In this survey we describe some of the key ingredients used in
the establishment of the circular law, in particular recent advances in
understanding the Littlewood-Offord problem and its inverse.
http://arxiv.org/abs/0810.2994
Author(s): Aijun Du and Jinqiao Duan and Hongjun Gao
Abstract: The stochastics two-layer quasi-geostrophic flow model is an intermediate
system between the single-layer two dimensional barotropic flow model and the
continuously stratified three dimensional baroclinic flow model. This model is
widely used to investigate basic mechanisms in geophysical flows, such as
baroclinic effects, the Gulf Stream and subtropical gyres.
A large deviation principle for the two-layer quasi-geostrophic flow model
under uncertainty is proved. The proof is based on the Laplace principle and a
variational approach. This approach does not require the exponential tightness
estimates which are needed in other methods for establishing large deviation
principles.
http://arxiv.org/abs/0810.2818
Author(s): Sylvie Corteel and Lauren Williams
Abstract: The partially asymmetric exclusion process (PASEP) is an important model from
statistical mechanics which describes a system of interacting particles hopping
left and right on a one-dimensional lattice of n sites. It has been cited as a
model for traffic flow and protein synthesis. In its most general form,
particles may enter and exit at the left with probabilities alpha and gamma,
and they may exit and enter at the right with probabilities beta and delta. In
the bulk, the probability of hopping left is q times the probability of hopping
right.
In previous work we used the matrix ansatz to give a combinatorial formula
for the steady state probability of each state of the PASEP, when
gamma=delta=0. The formula was the generating function for permutation tableaux
of a fixed shape, weighted according to three statistics. In this paper we give
a simple one-parameter generalization of the matrix ansatz, then use it to
generalize our results about the PASEP to the case of general alpha, beta,
gamma, delta (and q=1). We replace permutation tableaux by the slightly more
general bordered permutation tableaux, which we show have cardinality 4^n n!.
We also state our results in terms of alternative tableaux.
http://arxiv.org/abs/0810.2916
Author(s): Olivier Durieu and Dalibor Volny
Abstract: In this paper, we are interested in the limit theorem question for sums of
indicator functions. We show that in every aperiodic dynamical system, for
every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that
$a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exist a
measurable set $A$ such that the sequence of the partial sums
$\frac{1}{a_n}\sum_{i=0}^{n-1}(\ind_A-\mu(A))\circ T^i$ is dense in the set of
the probability measures on $\R$. Further, in the ergodic case, we prove that
there exists a dense $G_\delta$ of such sets.
http://arxiv.org/abs/0810.2917
Author(s): Matyas Barczy and Gyula Pap
Abstract: Let us consider the process $(X_t^{(\alpha)})_{t\in[0,T)}$ given by the SDE
$dX_t^{(\alpha)} = -\frac{\alpha}{T-t}X_t^{(\alpha)} dt+ dB_t$, $t\in[0,T)$,
where $\alpha\in R$, $T\in(0,\infty)$, and $(B_t)_{t\geq 0}$ is a standard
Wiener process. In case of $\alpha>0$ the process $X^{(\alpha)}$ is known as an
$\alpha$-Wiener bridge, in case of $\alpha=1$ as the usual Wiener bridge. We
prove that for all $\alpha,\beta\in R$, $\alpha\ne\beta$, the probability
measures induced by the processes $X^{(\alpha)}$ and $X^{(\beta)}$ are singular
on C[0,T). Further, we investigate regularity properties of $X_t^{(\alpha)}$ as
$t\uparrow T$.
http://arxiv.org/abs/0810.3070
Author(s): Philippe Robert and Florian Simatos
Abstract: An occupancy problem with an infinite number of bins and a random probability
vector for the locations of the balls is considered. The respective sizes of
bins are related to the split times of a Yule process. The asymptotic behavior
of the landscape of first empty bins, i.e., the set of corresponding indices
represented by point processes, is analyzed and convergences in distribution to
mixed Poisson processes are established. Additionally, the influence of the
random environment, the random probability vector, is analyzed. It is
represented by two main components: an i.i.d. sequence and a fixed random
variable. Each of these components has a specific impact on the qualitative
behavior of the stochastic model. It is shown in particular that for some
values of the parameters, some rare events, which are identified, play an
important role on average values of the number of empty bins in some regions.
http://arxiv.org/abs/0810.3079
Author(s): Katarzyna Pietruska-Pa{\l}uba
Abstract: We give a probabilistic characterisation of the Besov-Lipschitz spaces
$Lip(\alpha,p,q)(X)$ on domains which support a Markovian kernel with
appropriate exponential bounds. This extends former results of
\cite{Jon,KPP1,KPP2,GHL} which were valid for $\alpha=\frac{d_w}{2},p=2$,
$q=\infty,$ where $d_w$ is the walk dimension of the space $X.$
http://arxiv.org/abs/0810.3098
Author(s): Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
Abstract: We consider transient nearest neighbor random walks on the positive part of
the real line. We give criteria for the finiteness of the number of cutpoints
and strong cutpoints. Examples and open problems are presented.
http://arxiv.org/abs/0810.3123
Author(s): M.Draief and A. Ganesh
Abstract: We address the question of understanding the effect of the underlying network
topology on the spread of a virus and the dissemination of information when
users are mobile performing independent random walks on a graph. To this end we
propose a simple model of infection that enables to study the coincidence time
of two random walkers on an arbitrary graph. By studying the coincidence time
of a susceptible and an infected individual both moving in the graph we obtain
estimates of the infection probability. The main result of this paper is to
pinpoint the impact of the network topology on the infection probability. More
precisely, we prove that for homogeneous graph including regular graphs and the
classical Erdos-Renyi model, the coincidence time is inversely proportional to
the number of nodes in the graph. We then study the model on power-law graphs,
that exhibit heterogeneous connectivity patterns, and show the existence of a
phase transition for the coincidence time depending on the parameter of the
power-law of the degree distribution.
http://arxiv.org/abs/0810.3128
Author(s): M. Draief and A. Ganesh and L. Massoulie
Abstract: In this paper, we give an analytic solution for graphs with n nodes and E
edges for which the probability of obtaining a given graph G is specified in
terms of the degree sequence of G. We describe how this model naturally appears
in the context of load balancing in communication networks, namely Peer-to-Peer
overlays. We then analyse the degree distribution of such graphs and show that
the degrees are concentrated around their mean value. Finally, we derive
asymptotic results on the number of edges crossing a graph cut and use these
results $(i)$ to compute the graph expansion and conductance, and $(ii)$ to
analyse the graph resilience to random failures.
http://arxiv.org/abs/0810.3173
Author(s): Shanjian Tang and Wei Zhong
Abstract: In this paper, the optimal switching problem is proposed for one-dimensional
reflected backward stochastic differential equations (BSDEs, for short) where
the generators, the terminal values, and the barriers are all switched with
positive costs. The value process is characterized by a system of
multi-dimensional reflected BSDEs with oblique reflection, whose existence and
uniqueness is by no means trivial and is therefore carefully examined.
Existence is shown using both methods of the Picard iteration and penalization,
but under some different conditions. Uniqueness is proved by representation
either as the equilibrium value process to a stochastic mixed game of switching
and stopping, or as the value process to our optimal switching problem for
one-dimensional reflected BSDEs.
http://arxiv.org/abs/0810.3176
Author(s): Tomasz Schreiber
Abstract: In Piekniewski and Schreiber (2008) we have developed a simple mathematical
model for information flow structure in a class of recurrent neural networks
and shown that its asymptotic behaviour is scale-free and admits a description
in terms of the so-called winner-take-all dynamics. In the present paper we
establish a limit theorem for spectra of the spike-flow graphs induced by the
winner-take-all dynamics.
http://arxiv.org/abs/0810.3193
Author(s): Marco Lenci
Abstract: We prove the quenched Invariance Principle for random walks in random
environments on $Z^d$ with zero local drift, subject to a transitivity
hypothesis which is much weaker than the customary uniform ellipticity. We then
obtain recurrence for $d \le 2$. The proofs are based on a representation of
the system as an "uncountable union" of one-dimensional Markov maps, and on a
powerful theorem by Schmidt on cocycle recurrence.
http://arxiv.org/abs/0810.3197
Author(s): Valentin Konakov (CMI RAS) and Stephane Menozzi (PMA)
Abstract: Consider a multidimensional SDE of the form $X_t = x+\int_{0}^{t}
b(X_{s-})ds+\int{0}^{t} f(X_{s-})dZ_s$ where $(Z_s)_{s\ge 0}$ is a symmetric
stable process. Under suitable assumptions on the coefficients the unique
strong solution of the above equation admits a density w.r.t. the Lebesgue
measure and so does its Euler scheme. Using a parametrix approach, we derive an
error expansion at order 1 w.r.t. the time step for the difference of these
densities.
http://arxiv.org/abs/0810.3224
Author(s): Yuri Bakhtin
Abstract: We define analogues of Brownian motion on the triadic Cantor set by
introducing a few natural requirements on the Markov semigroup. We give a
detailed description of these symmetric self-similar processes and study their
properties such as mixing and moment asymptotics.
http://arxiv.org/abs/0810.3260
Author(s): Sourav Chatterjee and Nick Crawford
Abstract: In this paper we consider central limit theorems for various macroscopic
observables in the high temperature region of the Sherrington-Kirkpatrick spin
glass model. With a particular focus on obtaining a quenched central limit
theorem for the energy density of the system with non-zero external field, we
show how to combine the mean field cavity method with Stein's method in the
quenched regime. The result for the energy density extends the corresponding
result of Comets-Neveu.
http://arxiv.org/abs/0810.3279
Author(s): Ilia Binder and Mark Braverman
Abstract: In this paper we examine the rate of convergence of one of the standard
algorithms for emulating exit probabilities of Brownian motion, the Walk on
Spheres (WoS) algorithm. We obtain the complete characterization of the rate of
convergence of WoS in terms of the local geomnetry of a domain.
http://arxiv.org/abs/0810.3343
Author(s): Olivier Raimond (LM-Orsay and MODAL'X) and Bruno Schapira (LM-Orsay)
Abstract: We introduce and study a natural continuous time version of excited random
walks. In the case of nonnegative drift, we obtain a necessary and sufficient
condition for recurrence. This result is analogous to Zerner's result
\cite{Zer1} for excited (or cookie) random walks. We use similar arguments.
http://arxiv.org/abs/0810.3538
Author(s): Luigi Accardi and Andreas Boukas
Abstract: We study the non-trivial central extensions $CEHeis$ of the Heisenberg
algebra $Heis$ recently constructed in {AccBouCE}. We prove that a real form of
$CEHeis$ is one the fifteen classified real four--dimensional solvable Lie
algebras. We also show that $CEHeis$ can be realized (i) as a sub--Lie--algebra
of the Schroedinger algebra and (ii) in terms of two independent copies of the
canonical commutation relations (CCR). This gives a natural family of unitary
representations of $CEHeis$ and allows an explicit determination of the
associated group by exponentiation. In contrast with $Heis$, the group law for
$CEHeis$ is given by nonlinear (quadratic) functions of the coordinates.
http://arxiv.org/abs/0810.3365
Author(s): Nayantara Bhatnagar and Nick Crawford and Elchanan Mossel and Arnab Sen
Abstract: We study the structure of a uniformly randomly chosen partial order of width
2 on n elements. We analyze the local structure by studying the number of
elements incomparable to a random element in the poset. We show that under the
appropriate scaling, the number of incomparable elements converges to the
height of a one dimensional Brownian excursion at a uniformly chosen random
time in the interval [0,1], which follows the Rayleigh distribution.
http://arxiv.org/abs/0810.3670
Author(s): Grigori Olshanski
Abstract: We consider a family {P} of determinantal point processes arising in
representation theory and random matrix theory. The processes live on the
one-dimensional lattice and their correlation kernels correspond to projection
operators in the l^2 Hilbert space on the lattice. Moreover, these projections
are spectral projections associated to certain selfadjoint second order
difference operators on the lattice. The aim of the note is to demonstrate that
the difference operators in question can be efficiently employed in the study
of limit transitions inside the family {P}.
http://arxiv.org/abs/0810.3751
Author(s): Andrea Baronchelli and Michele Catanzaro and Romualdo Pastor-Satorras
Abstract: We study the properties of random walks on complex trees. We observe that the
absence of loops reflects in physical observables showing large differences
with respect to their looped counterparts. First, both the vertex discovery
rate and the mean topological displacement from the origin present a
considerable slowing down in the tree case. Second, the mean first passage time
(MFPT) displays a logarithmic degree dependence, in contrast to the inverse
degree shape exhibited in looped networks. This deviation can be ascribed to
the dominance of source-target topological distance in trees. To show this, we
study the distance dependence of a symmetrized MFPT and derive its logarithmic
profile, obtaining good agreement with simulation results. These unique
properties shed light on the recently reported anomalies observed in diffusive
dynamical systems on trees.
http://arxiv.org/abs/0801.1278
Author(s): Roland Schnaubelt and Mark Veraar
Abstract: In this paper we consider structurally damped plate and wave equations with
point and distributed random forces. In order to treat space dimensions more
than one, we work in the setting of $L^q$--spaces with (possibly small)
$q\in(1,2)$. We establish existence, uniqueness and regularity of mild and weak
solutions to the stochastic equations employing recent theory for stochastic
evolution equations in UMD Banach spaces.
http://arxiv.org/abs/0810.3898
Author(s): Anne Estrade (MAP5) and Jacques Istas (LJK)
Abstract: Throwing balls on Euclidean spaces have been considered since a while. With
suitable renormalization, it leads to fractional Brownian motion as limit
object. We investigate in this paper the throw of balls on spheres and
hyperbolic spaces. This leads to dramatically different behaviors. On spheres,
we obtain a Gaussian limit that is no more a fractinal Brownian motion, but is
locally self-similar. On hyperbolic spaces, we prove that there is no any
limit.
http://arxiv.org/abs/0810.4004
Author(s): St\'ephane Attal (ICJ) and Ion Nechita (ICJ)
Abstract: We prove that the free Fock space ${\F}(\R^+;\C)$, which is very commonly
used in Free Probability Theory, is the continuous free product of copies of
the space $\C^2$. We describe an explicit embedding and approximation of this
continuous free product structure by means of a discrete-time approximation:
the free toy Fock space, a countable free product of copies of $\C^2$. We show
that the basic creation, annihilation and gauge operators of the free Fock
space are also limits of elementary operators on the free toy Fock space. When
applying these constructions and results to the probabilistic interpretations
of these spaces, we recover some discrete approximations of the semi-circular
Brownian motion and of the free Poisson process. All these results are also
extended to the higher multiplicity case, that is, ${\F}(\R^+;\C^N)$ is the
continuous free product of copies of the space $\C^{N+1}$.
http://arxiv.org/abs/0810.4070
Author(s): Xinjia Chen
Abstract: In this paper, we have developed new multistage tests which guarantee
prescribed level of power and are more efficient than previous tests in terms
of average sampling number and the number of sampling operations. Without
truncation, the maximum sampling numbers of our testing plans are absolutely
bounded. Based on geometrical arguments, we have derived extremely tight bounds
for the operating characteristic function.
http://arxiv.org/abs/0810.3946
Author(s): Nobuo Yoshida
Abstract: We consider a discrete-time stochastic growth model on $d$-dimensional
lattice. The growth model describes various interesting examples such as
oriented site/bond percolation, directed polymers in random environment, time
discretizations of binary contact path process. We show the equivalence between
the slow population growth and a localization property in terms of "replica
overlap". This extends a result known for the directed polymers in random
environment to a large class of models. A new approach, based on the
multiplicative Doob's decomposition, is adopted to overcome the difficulty that
the total population may get extinct even at finite time.
http://arxiv.org/abs/0810.4218
Author(s): Sourav Chatterjee
Abstract: Disordered systems are an important class of models in statistical mechanics,
having the defining characteristic that the energy landscape is a fixed
realization of a random field. Examples include various models of glasses and
polymers. They also arise in other areas, like fitness models in evolutionary
biology. The ground state of a disordered system is the state with minimum
energy. The system is said to be chaotic if a small perturbation of the energy
landscape causes a drastic shift of the ground state. We present a rigorous
theory of chaos in disordered systems that confirms long-standing physics
intuition about connections between chaos, anomalous fluctuations of the ground
state energy, and the existence of multiple valleys in the energy landscape.
Combining these results with mathematical tools like hypercontractivity, we
establish the existence of the above phenomena in eigenvectors of GUE matrices,
the Kauffman-Levin model of evolutionary biology, directed polymers in random
environment, a subclass of the generalized Sherrington-Kirkpatrick model of
spin glasses, the discrete Gaussian free field, and continuous Gaussian fields
on Euclidean spaces. We also list several open questions.
http://arxiv.org/abs/0810.4221
Author(s): Nicoletta Cancrini and Fabio Martinelli and Roberto H. Schonmann and Cristina Toninelli
Abstract: We analyze the relaxation to equilibrium for kinetically constrained spin
models (KCSM) when the initial distribution $\nu$ is different from the
reversible one, $\mu$. This setting has been intensively studied in the physics
literature to analyze the slow dynamics which follows a sudden quench from the
liquid to the glass phase. We concentrate on two basic oriented KCSM: the East
model on $\bbZ$, for which the constraint requires that the East neighbor of
the to-be-update vertex is vacant and the model on the binary tree introduced
in \cite{Aldous:2002p1074}, for which the constraint requires the two children
to be vacant. While the former model is ergodic at any $p\neq 1$, the latter
displays an ergodicity breaking transition at $p_c=1/2$. For the East we prove
exponential convergence to equilibrium with rate depending on the spectral gap
if $\nu$ is concentrated on any configuration which does not contain a forever
blocked site or if $\nu$ is a Bernoulli($p'$) product measure for any $p'\neq
1$. For the model on the binary tree we prove similar results in the regime
$p,p'p_c$ and $p\leq p_c$ convergence to equilibrium cannot occur for all
local functions. Finally we present a very simple argument (different from the
one in \cite{Aldous:2002p1074}) based on a combination of combinatorial results
and ``energy barrier'' considerations, which yields the sharp upper bound for
the spectral gap of East when $p\uparrow 1$.
http://arxiv.org/abs/0810.4237
Author(s): Jos\'e Fajardo and Ernesto Mordecki
Abstract: In this paper we examine which Brownian Subordination with drift exhibits the
symmetry property introduced by Fajardo and Mordecki (2006). We obtain that
when the subordination results in a L\'evy process, a necessary and sufficient
condition for the symmetry to hold is that drift must be equal to -1/2.
http://arxiv.org/abs/0810.4271
Author(s): Christophe Garban and Steffen Rohde and Oded Schramm
Abstract: We prove that the $SLE_\kappa$ trace in any simply connected domain $G$ is
continuous (except possibly near its endpoints) if $\kappa<8$. We also prove an
SLE analog of Makarov's Theorem about the support of harmonic measure.
http://arxiv.org/abs/0810.4327
Author(s): S\'ergio Bezerra and Samy Tindel and Frederi Viens
Abstract: This paper provides information about the asymptotic behavior of a
one-dimensional Brownian polymer in random medium represented by a Gaussian
field $W$ on ${\mathbb{R}}_+\times{\mathbb{R}}$ which is white noise in time
and function-valued in space. According to the behavior of the spatial
covariance of $W$, we give a lower bound on the power growth (wandering
exponent) of the polymer when the time parameter goes to infinity: the polymer
is proved to be superdiffusive, with a wandering exponent exceeding any
$\alpha<3/5$.
http://arxiv.org/abs/0810.4378
Author(s): Konstancja Bobecka and Jacek Weso{\l}owski
Abstract: A new independence property of univariate beta distributions, related to the
results of Kshirsagar and Tan for beta matrices, is presented. Conversely, a
characterization of univariate beta laws through this independence property is
proved. A related characterization of a family of $2\times2$ random matrices
including beta matrices is also obtained. The main technical challenge was a
problem involving the solution of a related functional equation.
http://arxiv.org/abs/0810.4427
Author(s): Giovanni Peccati and Murad S. Taqqu
Abstract: Motivated by second order asymptotic results, we characterize the convergence
in law of double integrals, with respect to Poisson random measures, toward a
standard Gaussian distribution. Our conditions are expressed in terms of
contractions of the kernels. To prove our main results, we use the theory of
stable convergence of generalized stochastic integrals developed by Peccati and
Taqqu. One of the advantages of our approach is that the conditions are
expressed directly in terms of the kernel appearing in the multiple integral
and do not make any explicit use of asymptotic dependence properties such as
mixing. We illustrate our techniques by an application involving linear and
quadratic functionals of generalized Ornstein--Uhlenbeck processes, as well as
examples concerning random hazard rates.
http://arxiv.org/abs/0810.4432
Author(s): Mark Meerschaert and Dongsheng Wu and Yimin Xiao
Abstract: Denote by $H(t)=(H_1(t),...,H_N(t))$ a function in $t\in{\mathbb{R}}_+^N$
with values in $(0,1)^N$. Let
$\{B^{H(t)}(t)\}=\{B^{H(t)}(t),t\in{\mathbb{R}}^N_+\}$ be an
$(N,d)$-multifractional Brownian sheet (mfBs) with Hurst functional $H(t)$.
Under some regularity conditions on the function $H(t)$, we prove the
existence, joint continuity and the H\"{o}lder regularity of the local times of
$\{B^{H(t)}(t)\}$. We also determine the Hausdorff dimensions of the level sets
of $\{B^{H(t)}(t)\}$. Our results extend the corresponding results for
fractional Brownian sheets and multifractional Brownian motion to
multifractional Brownian sheets.
http://arxiv.org/abs/0810.4438
Author(s): Jos\'e Fajardo and Ernesto Mordecki
Abstract: We study the skewness premium (SK) introduced by Bates (1991) in a general
context using L\'evy Processes. Under a symmetry condition Fajardo and Mordecki
(2006) obtain that SK is given by the Bate's $x%$ rule. In this paper we study
SK under the absence of that symmetry condition. More exactly, we derive
sufficient conditions for SK to be positive, in terms of the characteristic
triplet of the L\'evy Process under the risk neutral measure.
http://arxiv.org/abs/0810.4485
Author(s): Bernard Bercu; Laure Coutin; Nicolas Savy
Abstract: We investigate the sharp large deviation properties of the energy and the
maximum likelihood estimator for the Ornstein-Uhlenbeck process driven by a
fractional Brownian motion with Hurst index greater than one half.
http://arxiv.org/abs/0810.4491
Author(s): Tomasz Downarowicz and Yves Lacroix and Didier L\'eandri
Abstract: In a stationary ergodic process, clustering is defined as the tendency of
events to appear in series of increased frequency separated by longer breaks.
Such behavior, contradicting the theoretical "unbiased behavior" with
exponential distribution of the gaps between appearances, is commonly observed
in experimental processes and often difficult to explain. In the last section
we relate one such empirical example of clustering, in the area of marine
technology. In the theoretical part of the paper we prove, using ergodic theory
and the notion of category, that clustering (even very strong) is in fact
typical for "rare events" defined as long cylinder sets in processes generated
by a finite partition of an arbitrary (infinite aperiodic) ergodic measure
preserving transformation.
http://arxiv.org/abs/0810.4509
Author(s): Giovanni Peccati and Murad S. Taqqu
Abstract: Motivated by second order asymptotic results, we characterize the convergence
in law of double integrals, with respect to Poisson random measures, toward a
standard Gaussian distribution. Our conditions are expressed in terms of
contractions of the kernels. To prove our main results, we use the theory of
stable convergence of generalized stochastic integrals developed by Peccati and
Taqqu. One of the advantages of our approach is that the conditions are
expressed directly in terms of the kernel appearing in the multiple integral
and do not make any explicit use of asymptotic dependence properties such as
mixing. We illustrate our techniques by an application involving linear and
quadratic functionals of generalized Ornstein--Uhlenbeck processes, as well as
examples concerning random hazard rates.
http://www.arxiv.org
Author(s): Tomasz Downarowicz and Yves Lacroix
Abstract: We consider an ergodic process on finitely many states, with positive
entropy. Our first main result asserts that the distribution function of the
normalized waiting time for the first visit to a small (i.e., over a long
block) cylinder set $B$ is, for majority of such cylinders and up to epsilon,
dominated by the exponential distribution function $1-e^{-t}$. That is, the
occurrences of so understood "rare event" $B$ along the time axis can appear
either with gap sizes of nearly exponential distribution (like in the
independent Bernoulli process), or they "attract" each-other. Our second main
result states that a {\it typical} ergodic process of positive entropy has the
following property: the distribution functions of the normalized hitting times
for the majority of cylinders $B$ of lengths $n'$ converge to zero along a \sq\
$n'$ whose upper density is 1. The occurrences of such a cylinder $B$ "strongly
attract", i.e., they appear in "series" of many frequent repetitions separated
by huge gaps of nearly complete absence.
These results, when properly and carefully interpreted, shed some new light,
in purely statistical terms, independently from physics, on a century old (and
so far rather avoided by serious science) common-sense phenomenon known as {\it
the law of series}, asserting that rare events in reality, once occurred, have
a mysterious tendency for untimely repetitions.
http://arxiv.org/abs/0810.4504
Author(s): T. Komorowski and S. Peszat and and T. Szarek
Abstract: We formulate a criterion for the existence, uniqueness of an invariant
measure for a Markov process taking values in a Polish phase space. In
addition, the weak star ergodicity, that is, the weak convergence of the
ergodic averages of the laws of the process starting with any initial
distribution, is established. The principal assumptions are the lower bound of
the ergodic averages of the transition probability function and the e- property
of the semigroup. The general result is applied to solutions of some stochastic
evolution equations in Hilbert spaces. As an example we consider an evolution
equation whose solution describes the Lagrangian observations of the velocity
field in the passive tracer model. We use the weak star mean ergodicity of the
respective invariant probability measure to derive the law of large numbers for
the tra jectory of the passive tracer.
http://arxiv.org/abs/0810.4609
Author(s): Andrey Pilipenko
Abstract: Some topological properties of stochastic flow $\varphi_t(x)$ generated by
stochastic differential equation in a ${\mathbb R}^d_+$ with normal reflection
at the boundary are investigated. Sobolev differentiability in initial
condition is received. The absolute continuity of the measure-valued process
$\mu\circ\varphi_t^{-1}$, where $\mu\ll\lambda^d,$ is studied.
http://arxiv.org/abs/0810.4644
Author(s): Xicheng Zhang
Abstract: By reversing the time variable we derive a stochastic representation for
backward incompressible Navier-Stokes equations in terms of stochastic
Lagrangian paths, which is similar to Constantin and Iyer's forward
formulations in {Co-Iy}. Using this representation, a self-contained proof of
local existence of solutions in Sobolev spaces are provided for the
incompressible Navier-Stokes equation in the whole space. In two dimensions, an
alternative proof to global existence is also given. Moreover, a large
deviation estimate for stochastic particle trajectories is presented when the
viscosity tends to zero.
http://arxiv.org/abs/0810.4664
Author(s): Albert Tarantola
Abstract: It is proposed that to the usual probability theory, three definitions and a
new theorem are added, the resulting theory allows one to displace the central
role usually given to the notion of conditional probability. When a mapping
$\varphi$ is defined between two measurable spaces, to each measure $\mu$
introduced on the first space, there corresponds an image $\varphi[\mu]$ on the
second space, and, reciprocally, to each measure $\nu$ defined on the second
space the corresponds a reciprocal image $\varphi^{-1}[\nu]$ on the first
space. As the intersection $\cap$ of two measures is easy to introduce, a
relation like $ \varphi[ \mu \cap \varphi^{-1} [\nu] ] = \varphi[\mu] \cap \nu
$ makes sense. It is, indeed, a theorem of the theory. This theorem gives
mathematical consistency to inferences drawn from physical measurements.
http://arxiv.org/abs/0810.4749
Author(s): Vytas Zacharovas and Hsien-Kuei Hwang
Abstract: A new approach to Poisson approximation is proposed. The basic idea is very
simple and based on properties of the Charlier polynomials and the Parseval
identity. Such an approach quickly leads to new effective bounds for several
Poisson approximation problems. A selected survey on diverse Poisson
approximation results is also given.
http://arxiv.org/abs/0810.4756
Author(s): Makoto Maejima and Ciprian Tudor (CES and SAMOS)
Abstract: Let $B^{H,K}=(B^{H,K}_{t}, t\geq 0)$ be a bifractional Brownian motion with
two parameters $H\in (0,1)$ and $K\in(0,1]$. The main result of this paper is
that the increment process generated by the bifractional Brownian motion
$(B^{H,K}_{h+t} -B^{H,K}_{h}, t\geq 0)$ converges when $h\to \infty$ to
$(2^{(1-K)/{2}}B^{HK}_{t}, t\geq 0)$, where $(B^{HK}_{t}, t\geq 0)$ is the
fractional Brownian motion with Hurst index $HK$. We also study the behavior of
the noise associated to the bifractional Brownian motion and limit theorems to
$B^{H,K}$.
http://arxiv.org/abs/0810.4764
Author(s): Omer Angel and Abraham D. Flaxman and and David B. Wilson
Abstract: In the complete graph on n vertices, when each edge has a weight which is an
exponential random variable, Frieze proved that the minimum spanning tree has
weight tending to zeta(3)=1/1^3+1/2^3+1/3^3+... as n goes to infinity. We
consider spanning trees constrained to have depth bounded by k from a specified
root. We prove that if k > log_2 log n+omega(1), where omega(1) is any function
going to infinity with n, then the minimum bounded-depth spanning tree still
has weight tending to zeta(3) as n -> infinity, and that if k < log_2 log n,
then the weight is doubly-exponentially large in log_2 log n - k. It is NP-hard
to find the minimum bounded-depth spanning tree, but when k < log_2 log n -
omega(1), a simple greedy algorithm is asymptotically optimal, and when k >
log_2 log n+omega(1), an algorithm which makes small changes to the minimum
(unbounded depth) spanning tree is asymptotically optimal. We prove similar
results for minimum bounded-depth Steiner trees, where the tree must connect a
specified set of m vertices, and may or may not include other vertices. In
particular, when m = const * n, if k > log_2 log n+omega(1), the minimum
bounded-depth Steiner tree on the complete graph has asymptotically the same
weight as the minimum Steiner tree, and if 1 <= k <= log_2 log n-omega(1), the
weight tends to (1-2^{-k}) sqrt{8m/n} [sqrt{2mn}/2^k]^{1/(2^k-1)} in both
expectation and probability. The same results hold for minimum bounded-diameter
Steiner trees when the diameter bound is 2k; when the diameter bound is
increased from 2k to 2k+1, the minimum Steiner tree weight is reduced by a
factor of 2^{1/(2^k-1)}.
http://arxiv.org/abs/0810.4908
Author(s): Xinjia Chen
Abstract: In this paper, we develop a computational approach for estimating the mean
value of a quantity in the presence of uncertainty. We demonstrate that, under
some mild assumptions, the upper and lower bounds of the mean value are
efficiently computable via a sample reuse technique, of which the computational
complexity is shown to posses a Poisson distribution.
http://arxiv.org/abs/0810.4727
Author(s): Bo'az Klartag
Abstract: We prove that any absolutely continuous probability measure on a
high-dimensional linear space has low-dimensional marginals that are
approximately spherically-symmetric.
http://arxiv.org/abs/0810.4700
Author(s): Yeong-Shyeong Tsai
Abstract: In order to estimate the average speed of mosquitoes, a simple experiment was
designed by Richard (Lu-Hsing Tsai), Tom (Po-Yu Tsai) and Robert (Hung-Ming
Tsai). The result of the experiment was posted in the science exhibitions
Taichung Taiwan 1993. The average speed of mosquitoes is inferred by the simple
relation is obtained In this paper, we will show how to get the data generated
by computer. Though the rigorous proof is not shown, a sketch proof will be
shown in this paper
http://arxiv.org/abs/0810.4902
Author(s): Samuel N. Cohen and Robert J. Elliott
Abstract: By analogy with the theory of Backward Stochastic Differential Equations, we
define Backward Stochastic Difference Equations on spaces related to discrete
time, finite state processes. This paper considers properties of these
processes as constructions in their own right, not as approximations to the
continuous case. We establish the existence and uniqueness of solutions under
weaker assumptions than are needed in the continuous time setting, and also
establish a comparison theorem for these solutions. The conditions of this
theorem are shown to approximate those required in the continuous time setting.
We also explore the relationship between the driver and the set of solutions;
in particular, we show under what conditions the driver can be uniquely
determined by the solution. Applications to the theory of nonlinear
expectations are explored, including a representation result in this context.
http://arxiv.org/abs/0810.4957
Author(s): Vincent Bansaye (PMA) and Julien Berestycki (PMA)
Abstract: A branching process in random environment $(Z_n, n \in \N)$ is a
generalization of Galton Watson processes where at each generation the
reproduction law is picked randomly. In this paper we give several results
which belong to the class of {\it large deviations}. By contrast to the
Galton-Watson case, here random environments and the branching process can
conspire to achieve atypical events such as $Z_n \le e^{cn}$ when $c$ is
smaller than the typical geometric growth rate $\bar L$ and $ Z_n \ge e^{cn}$
when $c > \bar L$. One way to obtain such an atypical rate of growth is to have
a typical realization of the branching process in an atypical sequence of
environments. This gives us a general lower bound for the rate of decrease of
their probability. When each individual leaves at least one offspring in the
next generation almost surely, we compute the exact rate function of these
events and we show that conditionally on the large deviation event, the
trajectory $t \mapsto \frac1n \log Z_{[nt]}, t\in [0,1]$ converges to a
deterministic function $f_c :[0,1] \mapsto \R_+$ in probability in the sense of
the uniform norm. The most interesting case is when $c < \bar L$ and we
authorize individuals to have only one offspring in the next generation. In
this situation, conditionally on $Z_n \le e^{cn}$, the population size stays
fixed at 1 until a time $ \sim n t_c$. After time $n t_c$ an atypical sequence
of environments let $Z_n$ grow with the appropriate rate ($\neq \bar L$) to
reach $c.$ The corresponding map $f_c(t)$ is piecewise linear and is 0 on
$[0,t_c]$ and $f_c(t) = c(t-t_c)/(1-t_c)$ on $[t_c,1].$
http://arxiv.org/abs/0810.4991
Author(s): Damir Filipovic and Stefan Tappe and Josef Teichmann
Abstract: By means of an original approach, called "method of the moving frame", we
establish existence, uniqueness and stability results for mild and weak
solutions of stochastic partial differential equations (SPDEs) with path
dependent coefficients driven by an infinite dimensional Wiener process and a
compensated Poisson random measure. Our approach is based on a time-dependent
coordinate transform, which reduces a wide class of SPDEs to a class of simpler
SDE problems. We try to present the most general results, which we can obtain
in our setting, within a self-contained framework to demonstrate our approach
in all details. Also an outlook towards a general theory of numerical
approaches to SPDEs is provided in the spirit of our setting.
http://arxiv.org/abs/0810.5023
Author(s): Zhen-Qing Chen
Abstract: In this paper, we address the equivalence of the analytic and probabilistic
notions of harmonicity in the context of general symmetric Hunt processes on
locally compact separable metric spaces. Extensions to general symmetric right
processes on Lusin spaces including infinite dimensional spaces are mentioned
at the end of this paper.
http://arxiv.org/abs/0810.5089
Author(s): Zhen-Qing Chen and Masatoshi Fukushima
Abstract: Let $D$ be an unbounded domain in $\RR^d$ with $d\geq 3$. We show that if $D$
contains an unbounded uniform domain, then the symmetric reflecting Brownian
motion (RBM) on $\overline D$ is transient. Next assume that RBM $X$ on
$\overline D$ is transient and let $Y$ be its time change by Revuz measure
${\bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function
$m$ on $\overline D$. We further show that if there is some $r>0$ so that
$D\setminus \overline {B(0, r)}$ is an unbounded uniform domain, then $Y$
admits one and only one symmetric diffusion that genuinely extends it and
admits no killings. In other words, in this case $X$ (or equivalently, $Y$) has
a unique Martin boundary point at infinity.
http://arxiv.org/abs/0810.5096
Author(s): O. Wittich
Abstract: We consider the heat equation with Dirichlet boundary conditions on the
tubular neighborhood of a closed Riemannian submanifold. We show that, as the
tube radius decreases, the semigroup of a suitably rescaled and renormalized
generator can be effectively described by a Hamiltonian on the submanifold with
a potential that depends on the geometry of the submanifold and of the
embedding.
http://arxiv.org/abs/0810.5047
Author(s): O. Wittich
Abstract: We consider the heat equation with Dirichlet boundary conditions on the
tubular neighborhood of a closed Riemannian submanifold. We show that, as the
tube diameter tends to zero, a suitably rescaled and renormalized semigroup
converges to a limit semigroup in Sobolev spaces of arbitrarily large Sobolev
index.
http://arxiv.org/abs/0810.5052
Author(s): Enrico Priola and Jerzy Zabczyk
Abstract: We consider a class of semilinear stochastic evolution equations driven by an
additive cylindrical stable noise.We investigate structural properties of the
solutions like Markov, irreducibility, stochastic continuity, Feller and strong
Feller properties, and study integrability of trajectories. The obtained
results can be applied to semilinear stochastic heat equations with Dirichlet
boundary conditions and bounded and Lipschitz nonlinearities.
http://arxiv.org/abs/0810.5063
Author(s): Michael Schmutz
Abstract: It turns out that a slightly generalised Margrabe formula exhibits symmetry
properties leading to semi-static hedges of rather general options in the
bivariate Black-Scholes economy. In order to increase the liquidity of the used
hedging instruments for currency options, the duality principle can be used to
set up the hedges in a foreign market by using only European vanilla options
sometimes along with a risk-less bond. Since the semi-static hedges in the
Black-Scholes economy are exact, a closed form valuation formula of a certain
weighted barrier swap-option can be easily derived.
http://arxiv.org/abs/0810.5146
Author(s): W. Jarczyk and J. Misiewicz
Abstract: In this paper we give a first attempt to define and study stable
distributions with respect to the weak generalized convolution, focusing our
attention on the symmetric weakly stable distribution. As in the case of the
classical convolution, characterization of distributions stable in the sense of
the weak generalized convolution depends on solving some functional equations
in the class of characteristic functions.
http://arxiv.org/abs/0810.5285
Author(s): Martin Hairer
Abstract: We study a model of two interacting Hamiltonian particles subject to a common
potential in contact with two Langevin heat reservoirs: one at finite and one
at infinite temperature. This is a toy model for 'extreme' non-equilibrium
statistical mechanics. We provide a full picture of the long-time behaviour of
such a system, including the existence / non-existence of a non-equilibrium
steady state, the precise tail behaviour of the energy in such a state, as well
as the speed of convergence toward the steady state.
Despite its apparent simplicity, this model exhibits a surprisingly rich
variety of long time behaviours, depending on the parameter regime: if the
surrounding potential is 'too stiff', then no stationary state can exist. In
the softer regimes, the tails of the energy in the stationary state can be
either algebraic, fractional exponential, or exponential. Correspondingly, the
speed of convergence to the stationary state can be either algebraic, stretched
exponential, or exponential. Regarding both types of claims, we obtain matching
upper and lower bounds.
http://arxiv.org/abs/0810.5431
Author(s): Patrick Cattiaux (LSProba) and Arnaud Guillin and Liming Wu
Abstract: We give by simple arguments sufficient conditions, so called Lyapunov
conditions, for Talagrand's transportation information inequality and for the
logarithmic Sobolev inequality. Those sufficient conditions work even in the
case where the Bakry-Emery curvature is not lower bounded. Several new examples
are provided.
http://arxiv.org/abs/0810.5435
Author(s): Xinjia Chen
Abstract: In this paper, we have established a new framework of truncated inverse
sampling for estimating mean values of non-negative random variables such as
binomial, Poisson, hyper-geometrical, and bounded variables. We have derived
explicit formulas and computational methods for designing sampling schemes to
ensure prescribed levels of precision and confidence for point estimators.
Moreover, we have developed interval estimation methods.
http://arxiv.org/abs/0810.5551
Author(s): Dang-Zheng Liu and Zheng-Dong Wang and Kui-Hua Yan
Abstract: We generally study the density of eigenvalues in unitary ensembles of random
matrices from the recurrence coefficients with regularly varying conditions for
the orthogonal polynomials. First we calculate directly the moments of the
density. Then, by studying some deformation of the moments, we get a family of
differential equations of first order which the densities satisfy (see Theorem
1.2), and give the densities by solving them. Further, we prove that the
density is invariant after the polynomial perturbation of the weight function
(see Theorem 1.5).
http://arxiv.org/abs/0810.5425
Author(s): Omar El-Dakkak (LSTA)
Abstract: In this paper, we obtain some uniform laws of large numbers and functional
central limit theorems for sequential empirical measure processes indexed by
classes of product functions satisfying appropriate Vapnik-Chervonenkis
properties.
http://arxiv.org/abs/0810.5565
Author(s): Jean B\'erard (ICJ)
Abstract: We consider a branching-selection particle system on $\Z$ with $N \geq 1$
particles. During a branching step, each particle is replaced by two new
particles, whose positions are shifted from that of the original particle by
independently performing two random walk steps according to the distribution $p
\delta_{1} + (1-p) \delta_{0}$, from the location of the original particle.
During the selection step that follows, only the N rightmost particles are kept
among the $2N$ particles obtained at the branching step, to form a new
population of $N$ particles. After a large number of iterated
branching-selection steps, the displacement of the whole population of $N$
particles is ballistic, with deterministic asymptotic speed $v_{N}(p)$. As $N$
goes to infinity, $v_{N}(p)$ converges to a finite limit $v_{\infty}(p)$. Our
main result is that, for every $0 < p < 1/2$, as $N$ goes to infinity, the
order of magnitude of the difference $v_{\infty}(p)- v_{N}(p)$ is
$\log(N)^{-2}$. This is called Brunet-Derrida behavior in reference to the
paper \cite{BruDer1} by E. Brunet and B. Derrida, where such a behavior is
established for a similar branching-selection particle system, using both
numerical simulations and heuristic arguments.
http://arxiv.org/abs/0810.5567
Author(s): Ismael Bailleul
Abstract: A new class of relativistic diffusions encompassing all the previously
studied examples has recently been introduced by C. Chevalier and F. Debbasch,
both in a heuristic and analytic way. A pathwise approach of these processes is
proposed here, in the general framework of Lorentzian geometry. In considering
the dynamics of the random motion in strongly causal spacetimes, we are able to
give a simple definition of the one-particle distribution function associated
with each process of the class and prove its fundamental property. This result
not only provides a dynamical justification of the analytical approach
developped up to now (enabling us to recover many of the results obtained so
far), but it provides a new general H-theorem. It also sheds some light on the
importance of the large scale structure of the manifold in the asymptotic
behaviour of the Franchi-Le Jan process. This pathwise approach is also the
source of many interesting questions that have no analytical counterparts.
http://arxiv.org/abs/0810.5662
Author(s): Thomas Duquesne
Abstract: We prove that the total range of Super-Brownian motion with quadratic
branching mechanism has an exact packing measure with respect to the gauge
function $g(r)=r^4 (\log \log1/r)^{-3}$ in super-critical dimensions $d\geq 5$.
More precisely, we prove that the total occupation measure of Super-Brownian
motion is equal to the $g$-packing measure restricted to its range, up to a
deterministic multiplicative constant that only depends on space dimension $d$.
http://arxiv.org/abs/0810.5673
Author(s): Said Hamadene and Jianfeng Zhang
Abstract: In this paper we study the nonzero-sum Dynkin game in continuous time which
is a two player non-cooperative game on stopping times. We show that it has a
Nash equilibrium point for general stochastic processes. As an application, we
consider the problem of pricing American game contingent claims by the utility
maximization approach.
http://arxiv.org/abs/0810.5698
Author(s): Enkelejd Hashorva
Abstract: In this paper we introduce the class of L_p random vectors which includes
beta-independent random vectors and L_p Dirichlet ones. We derive several
conditional limit results assuming that the associated random radius of the L_p
random vectors have distribution function in the max-domain of attraction of a
univariate extreme value distribution function. As an application we obtain the
joint asymptotic distribution of concomitants of order statics considering L_p
random samples.
http://arxiv.org/abs/0810.5706
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