lists.imstat.orglists.imstat.org/pipermail/pas/2009.txt · 2014-07-22From pas at lists.imstat.org...

Probability Abstracts 107This document contains abstracts 7696-7953from November-1-2008 to December-31-2008.They have been mailed on Jan 4, 2009.

This letter can be also found on line athttp://pas.imstat.org/Letters/letter_107.shtml

Wishing you all a great 2009!stefano

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7696. LARGE GAPS BETWEEN RANDOM EIGENVALUES

Benedek Valk\'o and B\'alint Vir\'ag

We show that in the point process limit of the bulk eigenvalues of$\beta$-ensembles of random matrices, the probability of having no eigenvaluein a fixed interval of size $\lambda$ is given by $$(\kappa_\beta+o(1))\lambda^{\gamma_\beta} \exp(-\frac{\beta}{64}\lambda^2+(\frac{\beta}{4}-\frac18)\lambda) $$ as$\lambda\to\infty$, where $$ \gamma_\beta={1/4}(\frac\beta{2}+\frac{2} {\beta}-3). $$ This is a slightly corrected version of a prediction by Dyson. Our proofuses the new Brownian carousel representation of the limit process, as well asthe Cameron-Martin-Girsanov transformation in stochastic calculus.

http://arxiv.org/abs/0811.0007

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7697. A CRITERION FOR THE VIABILITY OF STOCHASTIC SEMILINEAR CONTROL SYSTEMS VIA THE QUASI-TANGENCY CONDITION

Dan Goreac

In this paper we study a criterion for the viability of stochastic semilinearcontrol systems on a real, separable Hilbert space. The necessary andsufficient condition is given using the notion of stochastic quasi- tangency. Asa consequence, we prove that approximate viability and the viability propertycoincide for stochastic linear control systems. The paper generalizes recentresults from the deterministic framework.


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7698. COMPETITIVE OR WEAK COOPERATIVE STOCHASTIC LOTKA-VOLTERRA SYSTEMS CONDITIONED TO NON-EXTINCTION

Patrick Cattiaux (IMT) and Sylvie M\'el\'eard (CMAP)

We are interested in the long time behavior of a two-type density- dependentbiological population conditioned to non-extinction, in both cases ofcompetition or weak cooperation between the two species. This population isdescribed by a stochastic Lotka-Volterra system, obtained as limit ofrenormalized interacting birth and death processes. The weak cooperationassumption allows the system not to blow up. We study the existence anduniqueness of a quasi-stationary distribution, that is convergence toequilibrium conditioned to non extinction. To this aim we generalize intwo-dimensions spectral tools developed for one-dimensional generalized Fellerdiffusion processes. The existence proof of a quasi-stationary distribution isreduced to the one for a $d$-dimensional Kolmogorov diffusion process under asymmetry assumption. The symmetry we need is satisfied under a local balancecondition relying the ecological rates. A novelty is the outlined relationbetween the uniqueness of the quasi-stationary distribution and theultracontractivity of the killed semi-group. By a comparison between thekilling rates for the populations of each type and the one of the globalpopulation, we show that the quasi-stationary distribution can be eithersupported by individuals of one (the strongest one) type or supported byindividuals of the two types. We thus highlight two different long timebehaviors depending on the parameters of the model: either the model exhibitsan intermediary time scale for which only one type (the dominant trait) issurviving, or there is a positive probability to have coexistence of the twospecies.


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7699. ASYMPTOTICS FOR THE SURVIVAL PROBABILITY IN A SUPERCRITICAL BRANCHING RANDOM WALK

Nina Gantert and Yueyun Hu and Zhan Shi

Consider a discrete-time one-dimensional supercritical branching random walk.We study the probability that there exists an infinite ray in the branchingrandom walk that always lies above the line of slope $\gamma-\epsilon $, where$\gamma$ denotes the asymptotic speed of the right-most position in thebranching random walk. Under mild general assumptions upon the distribution ofthe branching random walk, we prove that when $\epsilon\to 0$, the probabilityin question decays like $\exp\{- {\beta + o(1)\over \epsilon^{1/2}}\} $, where$\beta$ is a positive constant depending on the distribution of the branchingrandom walk. In the special case of i.i.d. Bernoulli$(p)$ random variables(with $01$ should imply that $C_1-C_2$ contains aninterval. We prove that for 2-adic random Cantor sets generated by a vector ofprobabilities $(p_0,p_1)$ the interior of the region where the Palis conjecturedoes not hold is given by those $p_0,p_1$ which satisfy $p_0+p_1> \sqrt{2}$ and$p_0p_1(1+p_0^2+p_1^2)0$,and the first hitting time $\tau=\inf\{t\ge 0:B_t=0\}$, we find the probabilitydensity of $B_{u\tau}$ for a $u\in(0,1)$, i.e. of the Brownian motion on itsway to hitting zero.


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7730. ASYMPTOTIC INDEPENDENCE IN THE SPECTRUM OF THE GAUSSIAN UNITARY ENSEMBLE

P. Bianchi and M. Debbah and J. Najim

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE).Given a finite collection of bounded disjoint real Borel sets $ (\Delta_{i,n},\1\leq i\leq p)$, properly rescaled, and eventually included in anyneighbourhood of the support of Wigner's semi-circle law, we prove that therelated counting measures $({\mathcal N}_n(\Delta_{i,n}), 1\leq i\leq p)$,where ${\mathcal N}_n(\Delta)$ represents the number of eigenvalues within$\Delta$, are asymptotically independent as the size $n$ goes to infinity, $p$being fixed. As a consequence, we prove that the largest and smallest eigenvalues,properly centered and rescaled, are asymptotically independent; we finallydescribe the fluctuations of the condition number of a matrix from the GUE.


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7731. A STOCHASTIC EPIDEMIOLOGICAL MODEL AND A DETERMINISTIC LIMIT FOR BITTORRENT-LIKE PEER-TO-PEER FILE-SHARING NETWORKS

George Kesidis and Takis Konstantopoulos and Perla Sousi

In this paper, we propose a stochastic model for a file-sharing peer- to-peernetwork which resembles the popular BitTorrent system: large files are splitinto chunks and a peer can download or swap from another peer only one chunk ata time. We prove that the fluid limits of a scaled Markov model of this systemare of the coagulation form, special cases of which are well-knownepidemiological (SIR) models. In addition, Lyapunov stability and settling-timeresults are explored. We derive conditions under which the BitTorrentincentives under consideration result in shorter mean file-acquisition timesfor peers compared to client-server (single chunk) systems. Finally, adiffusion approximation is given and some open questions are discussed.


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7732. ON PERCOLATION AND THE BUNKBED CONJECTURE

Svante Linusson

We study a problem on percolation on product graphs G x K_2. Here G is anyfinite graph and K_2 consists of two vertices {0,1} connected by an edge. Inedge percolation every edge in G x K_2 is present with probability p. In [3]Olle H\"aggstr\"om stated a conjecture (which he claimed to be folklore) thatfor all G and p the probability that (u,0) is in the same component as (v,0) isgreater than the probability that (u,0) is in the same component as (v, 1) forevery pair of vertices u,v in G. We generalize this conjecture and formulate and prove similar statements forrandomly directed graphs. The methods lead to a proof of the originalconjecture for special classes of graphs $G$, in particular outerplanar graphs.


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7733. SPECTRUM OF LARGE RANDOM REVERSIBLE MARKOV CHAINS

Charles Bordenave (IMT) and Pietro Caputo and Djalil Chafai (IMT and UPTE)

In this work, we adopt a Random Matrix Theory point of view to study thespectrum of large reversible Markov chains in random environment. As the numberof states tends to infinity, we consider both the almost sure global behaviorof the spectrum, and the local behavior at the edge including the so calledspectral gap. We study presently two simple models. The first one is on thecomplete graph while the second is on the chain graph (birth-and-deathdynamics). These two models exhibit different scalings and limiting objects.The first model is related to the semi--circle law and Wigner's theorem. Itcontains as a special case a natural reversible Dirichlet Markov Ensemble. Thesecond model is related to homogenization and also to asymptotics for the rootsof random orthogonal polynomials. A special case gives rise to the arc--sinelaw as in a theorem by Erdos & Turan. This work raises several open problems.


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7734. ISOTROPIC ORNSTEIN-UHLENBECK FLOWS

Georgi Dimitroff and Holger van Bargen

Isotropic Brownian flows (IBFs) are a fairly natural class of stochasticflows which has been studied extensively by various authors. Their richstructure allows for explicit calculations in several situations and makes thema natural object to start with if one wants to study more general stochasticflows. Often the intuition gained by understanding the problem in the contextof IBFs transfers to more general situations. However, the obvious link betweenstochastic flows, random dynamical systems and ergodic theory cannot beexploited in its full strength as the IBF does not have an invariantprobability measure but rather an infinite one. Isotropic Ornstein- Uhlenbeckflows are in a sense localized IBFs and do have an invariant probabilitymeasure. The imposed linear drift destroys the translation invariance of theIBF, but many other important structure properties like the Markov property ofthe distance process remain valid and allow for explicit calculations incertain situations. The fact that isotropic Ornstein-Uhlenbeck flows haveinvariant probability measures allows one to apply techniques from randomdynamical systems theory. We demonstrate this by applying the results ofLedrappier and Young to calculate the Hausdorff dimension of the statisticalequilibrium of an isotropic Ornstein-Uhlenbeck flow.


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7735. RECONSTRUCTION OF SYMMETRIC POTTS MODELS

Allan Sly

The reconstruction problem on the tree has been studied in numerous contextsincluding statistical physics, information theory and computational biology.However, rigorous reconstruction thresholds have only been established in asmall number of models. We prove the first exact reconstruction threshold in anon-binary model establishing the Kesten-Stigum bound for the 3-state Pottsmodel on regular trees of large degree. We further establish that theKesten-Stigum bound is not tight for the $q$-state Potts model when $q \geq 5$.Moreover, we determine asymptotics for the reconstruction thresholds.


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7736. A SELF-REGULATING AND PATCH SUBDIVIDED POPULATION

Lamia Belhadji and Daniela Bertacchi and Fabio Zucca

We consider an interacting particle process on a graph which, from amacroscopic point of view, looks like $\Z^d$ and, at a microscopic level, is acomplete graph of degree $N$ (called a patch). There are two birth rates: aninter-patch one $\lambda$ and an intra-patch one $\phi$. Once a site isoccupied, there is no breeding from outside the patch and the probability$c(i)$ of success of an intra-patch breeding decreases with the size $i $ of thepopulation in the site. We prove the existence of a critical value$\lambda_{cr}(\phi, c, N)$ and a critical value $\phi_{cr}(\lambda, c, N)$. Weconsider a sequence of processes generated by the families of control functions$\{c_i\}_{i \in \N}$ and degrees $\{N_i\}_{i \in \N}$; we prove, under mildassumptions, the existence of a critical value $i_{cr}$. Roughly speaking weshow that, in the limit, these processes behave as the branching random walk on$\Z^d$ with external birth rate $\lambda$ and internal birth rate $\phi $. Someexamples of models that can be seen as particular cases are given.


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7737. MATRIX VALUED BROWNIAN MOTION AND A PAPER BY POLYA

Philippe Biane (IGM)

We give a geometric description of the motion of eigenvalues of a Brownianmotion with values in some matrix spaces. In the second part we consider apaper by Polya where he introduced a function close to the Riemann zetafunction, which satisfies Riemann hypothesis. We show that each of these twofunctions can be related to Brownian motion on a symmetric space.


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7738. ROBUST ADAPTIVE IMPORTANCE SAMPLING FOR NORMAL RANDOM VECTORS

Benjamin Jourdain and Jerome Lelong

Adaptive Monte Carlo methods are very efficient techniques designed to tunesimulation estimators on-line. In this work, we present an alternative tostochastic approximation to tune the optimal change of measure in the contextof importance sampling for normal random vectors. Unlike stochasticapproximation, which requires very fine tuning in practice, we propose to usesample average approximation and deterministic optimization techniques todevise a robust and fully automatic variance reduction methodology. The samesamples are used in the sample optimization of the importance samplingparameter and in the Monte Carlo computation of the expectation of interestwith the optimal measure computed in the previous step. We prove that thishighly non independent Monte Carlo estimator is convergent and satisfies acentral limit theorem with the optimal limiting variance. Numerical experimentsconfirm the performance of this estimator : in comparison with the crude MonteCarlo method, the computation time needed to achieve a given precision isdivided by a factor going from 2 to 10.


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7739. OPTIMAL SEQUENTIAL MULTIPLE HYPOTHESIS TESTS

Andrey Novikov

This work deals with a general problem of testing multiple hypotheses aboutthe distribution of a discrete-time stochastic process. Both the Bayesian andthe conditional settings are considered. The structure of optimal sequentialtests is characterized.


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7740. THE FUNDAMENTAL GROUP OF RANDOM 2-COMPLEXES

Eric Babson and Christopher Hoffman and Matthew Kahle

The random 2-complex Y=Y(n,p) is the probability space of all simplicialcomplexes on vertex set [n] and edge set [n] \choose 2, with each 2- dimensionalface included with probability p independently. Nathan Linial and Roy Meshulamshowed that if p >> 2\log{n}/n then the probability that H_{1}(Y,F_2) istrivial goes to 1 as n approaches infinity. This is an analogue of the phasetransition for connectivity of the Erd\H{o}s-R\'enyi random graph G(n,p). We show here that if p >> n^{-1/2}, then the probability that Y is simplyconnected goes to 1 as n approaches infinity, but if p 1) self- intersectionlocal time of a continuous time simple random walk on the d-dimensionallattice, in the critical dimension d=(2q)/(q-1). When q is integer, we obtainsimilar results for the intersection local times of q independent simple randomwalks.


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7858. POLYMORPHIC EVOLUTION SEQUENCE AND EVOLUTIONARY BRANCHING

Nicolas Champagnat (INRIA Sophia Antipolis / INRIA Lorraine / IECN) and Sylvie M\'el\'eard (CMAP)

We are interested in the study of models describing the evolution of apolymorphic population with mutation and selection in the specific scales ofthe biological framework of adaptive dynamics. The population size is assumedto be large and the mutation rate small. We prove that under a good combinationof these two scales, the population process is approximated in the long timescale of mutations by a Markov pure jump process describing the successivetrait equilibria of the population. This process, which generalizes theso-called trait substitution sequence, is called polymorphic evolutionsequence. Then we introduce a scaling of the size of mutations and we study thepolymorphic evolution sequence in the limit of small mutations. From this studyin the neighborhood of evolutionary singularities, we obtain a fullmathematical justification of a heuristic criterion for the phenomenon ofevolutionary branching. To this end we finely analyze the asymptotic behaviorof 3-dimensional competitive Lotka-Volterra systems.


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7859. TRANSPORT DIFFUSION COEFFICIENT FOR A KNUDSEN GAS IN A RANDOM TUBE

Francis Comets and Serguei Popov and Gunter M. Sch\"utz and Marina Vachkovskaia

We consider transport diffusion in a stochastic billiard in a random tubewhich is elongated in the direction of the first coordinate (the tube axis).Inside the random tube, which is stationary and ergodic, non-interactingparticles move straight with constant speed. Upon hitting the tube walls, theyare reflected randomly, according to the cosine law: the density of theoutgoing direction is proportional to the cosine of the angle between thisdirection and the normal vector. Steady state transport is studied byintroducing an open tube segment as follows: We cut out a large finite segmentof the tube with segment boundaries perpendicular to the tube axis. Particleswhich leave this piece through the segment boundaries disappear from thesystem. Through stationary injection of particles at one boundary of thesegment a steady state with non-vanishing stationary particle current ismaintained. We prove (i) that in the thermodynamic limit of an infinite openpiece the coarse-grained density profile inside the segment is linear, and (ii)that the transport diffusion coefficient obtained from the ratio of stationarycurrent and effective boundary density gradient equals the diffusioncoefficient of a tagged particle in an infinite tube. Thus we prove equality oftransport diffusion and self-diffusion coefficients for quite generic rough(random) tubes.


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7860. SUPERMARTINGALE DEOMPOSITION WITH GENERAL INDEX SET

Gianluca Casseses

We prove results on the existence of Dol\'{e}ans-Dade measures and of theDoob-Meyer decomposition for supermartingales indexed by a general index set


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7861. EXPONENTIAL INEQUALITIES FOR MARTINGALES AND ASYMPTOTIC PROPERTIES OF THE FREE ENERGY OF DIRECTED POLYMERS IN RANDOM ENVIRONMENT

Quansheng Liu (LMAM) and Fr\'ed\'erique Watbled (LMAM)

The objective of the present paper is to establish exponential largedeviation inequalities, and to use them to show exponential concentrationinequalities for the free energy of a polymer in general random environment,its rate of convergence, and an expression of its limit value in terms of thoseof some multiplicative cascades.


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7862. EXISTENCE AND UNIQUENESS OF SOLUTIONS OF STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

Max-K. von Renesse and Michael Scheutzow

We provide sufficient conditions on the coefficients of a stochasticfunctional differential equation with bounded memory driven by Brownian motionwhich guarantee existence and uniqueness of a maximal local and global strongsolution for each initial condition. Our results extend those of previousworks. For local existence and uniqueness, we only require the coefficients tobe continuous and to satisfy a one-sided local Lipschitz (or monotonicity)condition. In an appendix we formulate and prove four lemmas which may be ofindependent interest: three of them can be viewed as rather general stochasticversions of Gronwall's Lemma, the final one - which we call Dereich- Lemma -provides tail bounds for Hoelder norms of stochastic integrals.


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7863. PALINDROMIC RANDOM TRIGONOMETRIC POLYNOMIALS

J. Brian Conrey and David W. Farmer and and \"Ozlem Imamoglu

We show that if a real trigonometric polynomial has few real roots, then thetrigonometric polynomial obtained by writing the coefficients in reverse ordermust have many real roots. This is used to show that a class of randomtrigonometric polynomials has, on average, many real roots. In the case thatthe coefficients of a real trigonometric polynomial are independently andidentically distributed, but with no other assumptions on the distribution, theexpected fraction of real zeros is at least one-half. This result is bestpossible.


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7864. OCCUPATION TIMES VIA BESSEL FUNCTIONS

Yevgeniy Kovchegov and Nick Meredith and Eyal Nir

This study of occupation time densities for continuous-time Markov processeswas inspired by the work of E.Nir et al (2006) in the field of Single MoleculeFRET spectroscopy. There, a single molecule fluctuates between two or morestates, and the experimental observable depends on the state's occupation timedistribution. To mathematically describe the observable there was a need tocalculate a single state occupation time distribution. In this paper, we consider a Markov process with countably many states. Inorder to find a one-stete occupation time density, we use a combination ofFourier and Laplace transforms in the way that allows for inversion of theFourier transform. We derive an explicit expression for an occupation timedensity in the case of a simple continuous time random walk on Z. Also weexamine the spectral measures in Karlin-McGregor diagonalization in an attemptto represent occupation time densities via modified Bessel functions.


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7865. ORTHOGONALITY AND PROBABILITY: BEYOND NEAREST NEIGHBOR TRANSITIONS

Yevgeniy Kovchegov

In this article, we will explore why Karlin-McGregor method of usingorthogonal polynomials in the study of Markov processes was so successful forone dimensional nearest neighbor processes, but failed beyond nearest neighbortransitions. We will proceed by suggesting and testing possible fixtures.


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7866. COMPLETENESS OF BOND MARKET DRIVEN BY L\'EVY PROCESS

Michal Baran and Jerzy Zabczyk

The completeness problem of the bond market model with noise given by theindependent Wiener process and Poisson random measure is studied. Hedgingportfolios are assumed to have maturities in a countable, dense subset of afinite time interval. It is shown that under some assumptions the market is notcomplete unless the support of the Levy measure consists of a finite number ofpoints. Explicit constructions of contingent claims which can not be replicatedare provided.


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7867. BROWNIAN MOTION ON THE SIERPINSKI CARPET

Martin T. Barlow and Richard F. Bass and Takashi Kumagai and and Alexander Teplyaev

We prove that, up to scalar multiples, there exists only one local regularDirichlet form on a generalized Sierpinski carpet that is invariant withrespectto the local symmetries of the carpet. Consequently for each suchfractalthe law of Brownian motion is uniquely determined and theLaplacian is welldefined.


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7868. ALTERNATING I-DIVERGENCE MINIMIZATION IN FACTOR ANALYSIS

Lorenzo Finesso and Peter Spreij

In this paper we attempt at understanding how to build an optimal normalfactor analysis model. The criterion we have chosen to evaluate the distancebetween different models is the I-divergence between the corresponding normallaws. The algorithm that we propose for the construction of the bestapproximation is of an the alternating minimization kind.


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7869. STOCHASTIC HEAT EQUATION WITH MULTIPLICATIVE FRACTIONAL-COLORED NOISE

Raluca Balan and Ciprian Tudor (CES and SAMOS)

We consider the stochastic heat equation with multiplicative noise$u_t={1/2}\Delta u+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where$\diamond$ denotes the Wick product, and the solution is interpreted in themild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq1/2$), and colored in space (with spatial covariance kernel $f$). We prove thatif $f$ is the Riesz kernel of order $\alpha$, or the Bessel kernel of order$\alpha1/2$), respectively $d a} and the expected value E(logk(f)).


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7911. EVOLUTION BY MEAN CURVATURE IN SUB-RIEMANNIAN GEOMETRIES: A STOCHASTIC APPROACH

Nicolas Dirr and Federica Dragoni and Max von Renesse

We study the phenomenon of evolution by horizontal mean curvature flow insub-Riemannian geometries. We use a stochastic approach to prove the existenceof a generalized evolution in these spaces. In particular we show that thevalue function of suitable family of stochastic control problems solves in theviscosity sense the level set equation for the evolution by horizontal meancurvature flow.


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7912. EPIDEMIC MODELLING: ASPECTS WHERE STOCHASTICITY MATTERS

Tom Britton and David Lindenstrand

Epidemic models are always simplifications of real world epidemics. Whichreal world features to include, and which simplifications to make, depend bothon the disease of interest and on the purpose of the modelling. In the presentpaper we discuss some such purposes for which a \emph{stochastic} model ispreferable to a \emph{deterministic} counterpart. The two main examplesillustrate the importance of allowing the infectious and latent periods to berandom when focus lies on the \emph{probability} of a large epidemic outbreakand/or on the initial \emph{speed}, or growth rate, of the epidemic. Aconsequence of the latter is that estimation of the basic reproduction number$R_0$ is sensitive to assumptions about the distributions of the infectious andlatent periods when using the data from the early stages of an outbreak, whichwe illustrate with data from the SARS outbreak. Some further examples are alsodiscussed as are some practical consequences related to these stochasticaspects.


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7913. ON THE ALMOST SURE CENTRAL LIMIT THEOREM FOR VECTOR MARTINGALES: CONVERGENCE OF MOMENTS AND STATISTICAL APPLICATIONS

Bernard Bercu (IMB and INRIA Bordeaux - Sud-Ouest) and Peggy C\'enac (IMB) and Guy Fayolle (INRIA Rocquencourt)

We investigate the almost sure asymptotic properties of vector martingaletransforms. Assuming some appropriate regularity conditions both on theincreasing process and on the moments of the martingale, we prove thatnormalized moments of any even order converge in the almost sure cental limittheorem for martingales. A conjecture about almost sure upper bounds underwider hypotheses is formulated. The theoretical results are supported byexamples borrowed from statistical applications, including linearautoregressive models and branching processes with immigration, for which newasymptotic properties are established on estimation and prediction errors.


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7914. ESTIMATION OF THE INSTANTANEOUS VOLATILITY AND DETECTION OF VOLATILITY JUMPS

A. Alvarez and F. Panloup and M. Pontier and N. Savy

Concerning price processes, the fact that the volatility is not constant hasbeen observed for a long time. So we deal with models as$dX_t=\mu_tdt+\sigma_tdW_t$ where $\sigma$ is a stochastic process. Recentworks on volatility modeling suggest that we should incorporate jumps in thevolatility process. Empirical observations suggest that simultaneous jumps onthe price \underline{and} the volatility \cite{BarShep1,ConTan} exist. Thehypothesis that jumps occur simultaneously makes the problem of volatility jumpdetection reduced to the prices jump detection. But in case of this hypothesisfailure, we try to work in this direction. Among others, we use Jacod and Ait-Sahalia' recent work \cite{jac1} givingestimators of cumulated volatility $\int_0^t|\sigma_s|^pds$ for any $p \geq 2.$ This tool allows us to deliver an estimator of instantaneous volatility.Moreover we prove a central limit theorem for it. Obviously, such a theoremprovides a confidence interval for the instantaneous volatility and leads us toa test of the jump existence hypothesis. For instance, we consider a simplestmodel having volatility jumps, when volatility is piecewise constant:$\sigma_t=\sum_{i=0}^{N_t-1} \sigma_i \1_{[\tau_i,\tau_{i+1}[}(t).$ The jumptimes are $\tau_i, i\geq 1,$ and $\sigma_i$ is a $\F_{\tau_{i}}$- measurablerandom variable. Another example is studied: $\sigma_t=|Y_t|$ where $ (Y_t)$ isa solution to a L\'evy driven SDE, with suitable coefficients.


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7915. ON THE ANNEALED LARGE DEVIATION RATE FUNCTION FOR A MULTI- DIMENSIONAL RANDOM WALK IN RANDOM ENVIRONMENT

Jonathon Peterson and Ofer Zeitouni

We derive properties of the rate function in Varadhan's (annealed) largedeviation principle for multidimensional, ballistic random walk in randomenvironment, in a certain neighborhood of the zero set of the rate function.Our approach relates the LDP to that of regeneration times and distances. Theanalysis of the latter is possible due to the i.i.d. structure ofregenerations.


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7916. OPTIMAL DETECTION OF HOMOGENEOUS SEGMENT OF OBSERVATIONS IN STOCHASTIC SEQUENCE

Wojciech Sarnowski and Krzysztof Szajowski

We register a Markov process. At random moment $\theta$ the distribution ofobserved sequence changes. Using probability maximizing approach the optimalstopping rule is identified. For the particular case of disorder the explicitsolution is obtained.


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7917. TIME MANAGEMENT IN A POISSON FISHING MODEL

Anna Karpowicz and Krzysztof Szajowski

The aim of the paper is to extend the model of "fishing problem". The simpleformulation is following. The angler goes to fishing. He buys fishing ticketfor a fixed time. There are two places for fishing at the lake. The fishes arecaught according to renewal processes which are different at both places. Thefishes' weights and the inter-arrival times are given by the sequences ofi.i.d. random variables with known distribution functions. These distributionsare different for the first and second fishing place. The angler's satisfactionmeasure is given by difference between the utility function dependent on sizeof the caught fishes and the cost function connected with time. On each placethe angler has another utility functions and another cost functions. In thisway, the angler's relative opinion about these two places is modeled. Forexample, on the one place better sort of fish can be caught with biggerprobability or one of the places is more comfortable. Obviously our anglerwants to have as much satisfaction as possible and additionally he have toleave the lake before the fixed moment. Therefore his goal is to find twooptimal stopping times in order to maximize his satisfaction. The first timecorresponds to the moment, when he eventually should change the place and thesecond time, when he should stop fishing. These stopping times have to be lessthan the fixed time of fishing. The value of the problem and the optimalstopping times are derived.


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7918. MAXIMUM EMPIRICAL LIKELIHOOD ESTIMATION OF THE SPECTRAL MEASURE OF AN EXTREME VALUE DISTRIBUTION

John H. J. Einmahl and Johan Segers

Consider a random sample from a bivariate distribution function $F$ in themax-domain of attraction of an extreme value distribution function $G $. This$G$ is characterized by two extreme value indices and a spectral measure, thelatter determining the tail dependence structure of $F$. A major issue inmultivariate extreme value theory is the estimation of the spectral measure$\Phi_p$ with respect to the $L_p$ norm. For every $p \in [1, \infty] $, anonparametric maximum empirical likelihood estimator is proposed for $ \Phi_p$.The main novelty is that these estimators are guaranteed to satisfy the momentconstraints by which spectral measures are characterized. Asymptotic normalityof the estimators is proved under conditions that allow for tail independence.Moreover, the conditions are easily verifiable as we demonstrate through anumber of theoretical examples. A simulation study shows substantially improvedperformance of the new estimators. Two case studies illustrate how to implementthe methods in practice.


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7919. A RANK-BASED SELECTION WITH CARDINAL PAYOFFS AND A COST OF CHOICE

Krzysztof Szajowski

A version of the secretary problem is considered. The ranks of items, whosevalues are independent, identically distributed random variables$X_1,X_2,...,X_n$ from a uniform distribution on $[0; 1]$, are observedsequentially by the grader. He has to select exactly one item, when it appears,and receives a payoff which is a function of the unobserved realization ofrandom variable assigned to the item diminished by some cost. The methods ofanalysis are based on the existence of an embedded Markov chain and use thetechnique of backward induction. The result is a generalization of theselection model considered by Bearden(2006). The asymptotic behaviour of thesolution is also investigated.


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7920. ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR ORTHOGONAL POLYNOMIALS

Nizar Demni

We characterize probability distributions of all order finite moments gavingultraspherical type generating functions for orthogonal polynomials.


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7921. SIMULTANEOUS ASYMPTOTICS FOR THE SHAPE OF RANDOM YOUNG TABLEAUX WITH GROWINGLY RESHUFFLED ALPHABETS

Jean-Christophe Breton and Christian Houdr\'e

Given a random word of size n whose letters are drawn independently from anordered alphabet of size m, the fluctuations of the shape of the associatedrandom Young tableaux are investigated, when both n and m converge together toinfinity. If m does not grow too fast and if the draws are uniform, thelimiting shape is the same as the limiting spectrum of the GUE. In thenon-uniform case, a control of both highest probabilities will ensure theconvergence of the first row of the tableau towards the Tracy-Widomdistribution.


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7922. THE GAUSSIAN APPROXIMATION FOR MULTI-COLOR GENERALIZED FRIEDMAN'S URN MODEL

Li-Xin Zhang and Feifang Hu

The Friedman's urn model is a popular urn model which is widely used in manydisciplines. In particular, it is extensively used in treatment allocationschemes in clinical trials. In this paper, we prove that both the urncomposition process and the allocation proportion process can be approximatedby a multi-dimensional Gaussian process almost surely for a multi-colorgeneralized Friedman's urn model with non-homogeneous generating matrices. TheGaussian process is a solution of a stochastic differential equation. ThisGaussian approximation together with the properties of the Gaussian process isimportant for the understanding of the behavior of the urn process and is alsouseful for statistical inferences. As an application, we obtain the asymptoticproperties including the asymptotic normality and the law of the iteratedlogarithm for a multi-color generalized Friedman's urn model as well as therandomized-play-the-winner rule as a special case.


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7923. THE DURATION PROBLEM WITH MULTIPLE EXCHANGES

Charles E.M. Pearce and Krzysztof Szajowski and Mitsushi Tamaki

We treat a version of the multiple-choice secretary problem called themultiple-choice duration problem, in which the objective is to maximize thetime of possession of relatively best objects. It is shown that, for the$m$--choice duration problem, there exists a sequence (s1,s2,...,sm) ofcritical numbers such that, whenever there remain k choices yet to be made,then the optimal strategy immediately selects a relatively best object if itappears at or after time $s_k$ ($1\leq k\leq m$). We also exhibit anequivalence between the duration problem and the classical best-choicesecretary problem. A simple recursive formula is given for calculating thecritical numbers when the number of objects tends to infinity. Extensions aremade to models involving an acquisition or replacement cost.


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7924. THE CRITICAL Z-INVARIANT ISING MODEL VIA DIMERS: THE PERIODIC CASE

C\'edric Boutillier and B\'eatrice de Tili\`ere

We study a large class of critical two-dimensional Ising models namelycritical Z-invariant Ising models on periodic graphs, example of which are theclassical square, triangular and honeycomb lattice at the critical temperature.Fisher introduced a correspondence between the Ising model and the dimer modelon a decorated graph, thus setting dimer techniques as a powerful tool forunderstanding the Ising model. In this paper, we give a full description of thedimer model corresponding to the critical Z-invariant Ising model. We provethat the dimer characteristic polynomial is equal (up to a constant) to thecritical Laplacian characteristic polynomial, and defines a Harnack curve ofgenus 0. We prove an explicit expression for the free energy, and for the Gibbsmeasure obtained as weak limit of Boltzmann measures.


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7925. IMMIGRATED URN MODELS - ASYMPTOTIC PROPERTIES AND APPLICATIONS

Li-Xin Zhang and Feifang Hu and Siu Hung Cheung and Wei Sum Chan

Urn models have been widely studied and applied in both scientific and socialdisciplines. In clinical studies, the adoption of urn models in treatmentallocation schemes has been proved to be beneficial to both researchers, byproviding more efficient clinical trials, and patients, by increasing theprobability of receiving the better treatment. In this paper, we endeavor toderive a very general class of immigrated urn models that incorporates theimmigration mechanism into the urn process. Important asymptotic properties aredeveloped and illustrative examples are provided to demonstrate theapplicability of our proposed class of urn models. In general, the immigratedurn model has smaller variability than the corresponding urn model. Therefore,it is more powerful when used in clinical trials.


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7926. ARTIFICIAL INTELLIGENCE FOR BIDDING HEX

Sam Payne and Elina Robeva

We present a Monte Carlo algorithm for efficiently finding near optimal movesand bids in the game of Bidding Hex. The algorithm is based on the recentsolution of Random-Turn Hex by Peres, Schramm, Sheffield, and Wilson togetherwith Richman's work connecting random-turn games to bidding games.


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7927. PREDICTABILITY IN SPATIALLY EXTENDED SYSTEMS WITH MODEL UNCERTAINTY

Jinqiao Duan

Macroscopic models for spatially extended systems under random influences areoften described by stochastic partial differential equations (SPDEs). Some techniques for understanding solutions of such equations, such asestimating correlations, Liapunov exponents and impact of noises, arediscussed. They are relevant for understanding predictability in spatiallyextended systems with model uncertainty, for example, in physics, geophysicsand biological sciences. The presentation is for a wide audience.


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7928. A NEW FAMILY OF COVARIATE-ADJUSTED RESPONSE ADAPTIVE DESIGNS AND THEIR ASYMPTOTIC PROPERTIES

Li-Xin Zhang and Feifang Hu

It is often important to incorporating covariate information in the design ofclinical trials. In literature, there are many designs of using stratificationand covariate-adaptive randomization to balance on certain known covariate.Recently Zhang, Hu, Cheung and Chan (2007) have proposed a family ofcovariate-adjusted response-adaptive (CARA) designs and studied theirasymptotic properties. However, these CARA designs often have highvariabilities. In this paper, we propose a new family of covariate- adjustedresponse-adaptive (CARA) designs. We show that the new designs have smallervariabilities and therefore more efficient.


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7929. MULTI-COLOR RANDOMLY REINFORCED URN FOR ADAPTIVE DESIGNS

Li-Xin Zhang and Feifang Hu and Siu Hung Cheung and Wei Sum Chan

The response-adaptive design driven by randomly reinforced urn model isoptimal in the sense that it allocate patients to the best treatment withprobability converging to one. This paper illustrates asymptotic properties formulti-color reinforced urn models. Results on the rate of convergence of thenumber of patients assigned to each treatment are obtained under minimumrequirement of conditions and the distributions of the limits are found.Asymptotic distributions of the Wald test statistic for testing meandifferences are obtained both under the null hypothesis and alternatehypothesis. The asymptotic behavior for the non-homogenous is also studied.


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7930. A PROBABLISTIC ORIGIN FOR A NEW CLASS OF BIVARIATE POLYNOMIALS

Michael R. Hoare and Mizan Rahman

We present here a probabilistic approach to the generation of new polynomialsin two discrete variables. This extends our earlier work on the 'classical'orthogonal polynomials in a previously unexplored direction, resulting in thediscovery of an exactly soluble eigenvalue problem corresponding to a bivariateMarkov chain with a transition kernel formed by a convolution of simplebinomial and trinomial distributions. The solution of the relevanteigenfunction problem, giving the spectral resolution of the kernel, leads towhat we believe to be a new class of orthogonal polynomials in two discretevariables. Possibilities for the extension of this approach are discussed.


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7931. SCALING LIMITS FOR SYMMETRIC ITO-LEVY PROCESSES IN RANDOM MEDIUM

Remi Rhodes; Vincent Vargas

We are concerned with scaling limits of the solutions to stochasticdifferential equations with stationary coefficients driven by Poisson randommeasures and Brownian motions. We state an annealed convergence theorem, inwhich the limit exhibits a diffusive or superdiffusive behavior, depending onthe integrability properties of the Poisson random measure


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7932. OPTIMAL STOPPING OF A RISK PROCESS WHEN CLAIMS ARE COVERED IMMEDIATELY

Bogdan K. Muciek and Krzysztof J. Szajowski

The optimal stopping problem for the risk process with interests rates andwhen claims are covered immediately is considered. An insurance companyreceives premiums and pays out claims which have occured according to a renewalprocess and which have been recognized by them. The capital of the company isinvested at interest rate $\alpha\in\Re^{+}$, the size of claims increase atrate $\beta\in\Re^{+}$ according to inflation process. The immediate payment ofclaims decreases the company investment by rate $\alpha_1$. The aim is to findthe stopping time which maximizes the capital of the company. The improvementto the known models by taking into account different scheme of claims paymentand the possibility of rejection of the request by the insurance company ismade. It leads to essentially new risk process and the solution of optimalstopping problem is different.


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7933. REFLECTED BACKWARD SDES WITH GENERAL JUMPS

S.Hamadene and Y.Ouknine

In the first part of this paper we give a solution for the one- dimensionalreflected backward stochastic differential equation (BSDE for short) when thenoise is driven by a Brownian motion and an independent Poisson point process.The reflecting process is right continuous with left limits (rcll for short)whose jumps are arbitrary. We first prove existence and uniqueness of thesolution for a specific coefficient in using a method based on a combination ofpenalization and the Snell envelope theory. To show the result in the generalframework we use a fixed point argument in an appropriate space. The secondpart of the paper is related to BSDEs with two reflecting barriers. Once morewe prove the existence and uniqueness of the solution of the BSDE.


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7934. RADIAL DUNKL PROCESSES ASSOCIATED WITH DIHEDRAL SYSTEMS

Nizar Demni

We stduy radial Dunkl processes associated with dihedral systems: we derivethe semi group, the generalized Bessel function, the Dunkl-Hermite polynomials.Then we give a skew product decomposition by means of independent Besselprocesses and we compute the tail distribution of the first hitting time of theboundary of Weyl chamber.


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7935. ISING (CONFORMAL) FIELDS AND CLUSTER AREA MEASURES

Federico Camia and Charles M. Newman

We provide a representation for the scaling limit of the d=2 critical Isingmagnetization field as a (conformal) random field using SLE (Schramm- LoewnerEvolution) clusters and associated renormalized area measures. The renormalizedareas are from the scaling limit of the critical FK (Fortuin-Kasteleyn)clusters and the random field is a convergent sum of the area measures withrandom signs. Extensions to off-critical scaling limits, to d=3 and to Pottsmodels are also considered.


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7936. ON THE SUPREMUM OF CERTAIN FAMILIES OF STOCHASTIC PROCESSES

Wenbo V. Li and Natesh S. Pillai and Robert L. Wolpert

We consider a family of stochastic processes $\{X_t^\epsilon, t \in T\} $ on ametric space $T$, with a parameter $\epsilon \downarrow 0$. We study theconditions under which \lim_{\e \to 0} \P \Big(\sup_{t \in T} |X_t^\e| < \delta\Big) =1 when one has the \textit{a priori} estimate on the modulus ofcontinuity and the value at one point. We compare our problem to the celebratedKolmogorov continuity criteria for stochastic processes, and finally give anapplication of our main result for stochastic intergrals with respect tocompound Poisson random measures with infinite intensity measures.


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7937. DEFAULT TIMES, NON ARBITRAGE CONDITIONS AND CHANGE OF PROBABILITY MEASURES

Delia Coculescu and Monique Jeanblanc and Ashkan Nikeghbali

In this paper we give a financial justification, based on non arbitrageconditions, of the $(H)$ hypothesis in default time modelling. We also show howthe $(H)$ hypothesis is affected by an equivalent change of probabilitymeasure. The main technique used here is the theory of progressive enlargementsof filtrations.


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7938. FLUCTUATIONS OF THE EMPIRICAL QUANTILES OF INDEPENDENT BROWNIAN MOTIONS

Jason Swanson

We consider $n$ independent, identically distributed one-dimensional Brownianmotions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth densityfunction $f$. The empirical quantiles, or pointwise order statistics, aredenoted by $B_{j:n}(t)$, and we are interested in a sequence of quantiles$Q_n(t) = B_{j(n):n}(t)$, where $j(n)/n \to \alpha \in (0,1)$. This sequenceconverges in probability in $C[0,\infty)$ to $q(t)$, the $\alpha$- quantile ofthe law of $B_j(t)$. Our main result establishes the convergence in law in$C[0,\infty)$ of the fluctuation processes $F_n = n^{1/2}(Q_n - q)$. The limitprocess $F$ is a centered Gaussian process and we derive an explicit formulafor its covariance function. We also show that $F$ has many of the same localproperties as $B^{1/4}$, the fractional Brownian motion with Hurst parameter $H= 1/4$. For example, it is a quartic variation process, it has H\"oldercontinuous paths with any exponent $\gamma < 1/4$, and (at least locally) ithas increments whose correlation is negative and of the same order of magnitudeas those of $B^{1/4}$.


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7939. ASYMPTOTICS OF THE NORM OF ELLIPTICAL RANDOM VECTORS

Enkelejd Hashorva

In this paper we consider elliptical random vectors X in R^d,d>1 withstochastic representation A R U where R is a positive random radius independentof the random vector U which is uniformly distributed on the unit sphere of R^dand A is a given matrix. The main result of this paper is an asymptoticexpansion of the tail probability of the norm of X derived under the assumptionthat R has distribution function is in the Gumbel or the Weibull max- domain ofattraction.


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7940. NON-EQUILIBRIUM DYNAMICS OF DYSON'S MODEL WITH INFINITE PARTICLES

Makoto Katori and Hideki Tanemura

Dyson's model is a one-dimensional system of Brownian motions with long-rangerepulsive forces acting between any pair of particles with strengthproportional to the inverse of distances. We give sufficient conditions forinitial configurations so that Dyson's model with infinite number of particlesis well defined in the sense that any multitime correlation function is givenby a determinant with a locally integrable kernel. The class ofinfinite-dimensional configurations satisfying our conditions is large enoughto study non-equilibrium dynamics. For example, a relaxation process startingfrom a configuration, in which each lattice point of $\Z$ is occupied by oneparticle, to the stationary state, which is the determinantal point processwith the sine kernel $\mu_{\sin}$, is determined. The invariant measure$\mu_{\sin}$ also satisfies our conditions and Dyson's model starting from$\mu_{\sin}$, which is a reversible process, is identified with the infiniteparticle system, which is determinantal with the extended sine kernel studiedin the random matrix theory. We also show that this infinite-dimensionalreversible process is Markovian.


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7941. THRESHOLD BEHAVIOUR AND FINAL OUTCOME OF AN EPIDEMIC ON A RANDOM NETWORK WITH HOUSEHOLD STRUCTURE

Frank Ball and David Sirl and Pieter Trapman

This paper considers a stochastic SIR (susceptible$\to$infective$\to $removed)epidemic model in which individuals may make infectious contacts in two ways,both within `households' (which for ease of exposition are assumed to haveequal size) and along the edges of a random graph describing additional socialcontacts. Heuristically-motivated branching process approximations aredescribed, which lead to a threshold parameter for the model and methods forcalculating the probability of a major outbreak, given few initial infectives,and the expected proportion of the population who are ultimately infected bysuch a major outbreak. These approximate results are shown to be exact as thenumber of households tends to infinity by proving associated limit theorems.Moreover, simulation studies indicate that these asymptotic results providegood approximations for modestly sized finite populations. The extension tounequal sized households is discussed briefly.


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7942. A USEFUL RELATIONSHIP BETWEEN EPIDEMIOLOGY AND QUEUEING THEORY

Pieter Trapman and Martin Bootsma

In this paper we establish a relation between the spread of infectiousdiseases and the dynamics of so called M/G/1 queues with processor sharing. Thein epidemiology well known relation between the spread of epidemics andbranching processes and the in queueing theory well known relation betweenM/G/1 queues and birth death processes will be combined to provide a frameworkin which results from queueing theory can be used in epidemiology and viceversa. In particular, we consider the number of infectious individuals in a standardSIR epidemic model at the moment of the first detection of the epidemic, whereinfectious individuals are detected at a constant per capita rate. We use aresult from the literature on queueing processes to show that this number ofinfectious individuals is geometrically distributed.


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7943. NOTE ON RADIAL DUNKL PROCESSES

Nizar Demni

This note encloses relatively short proofs of the following known results:the radial Dunkl process associated with a reduced system and a strictlypositive multiplicity function is the unique strong solution for all time t ofa stochastic differential equation of a singular drift (see [11] for theoriginal proof and [4] for a proof under additional restrictions), the firsthitting time of the Weyl chamber by a radial Dunkl process is finite almostsurely for small values of the multiplicity function. Our proof of the secondmentioned result gives more information than the original one.


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7944. STOCHASTICALLY STABLE GLOBALLY COUPLED MAPS WITH BISTABLE THERMODYNAMIC LIMIT

Jean-Baptiste Bardet (IRMAR and LMRS) and Gerhard Keller and Roland Zweim\"uller

We study systems of globally coupled interval maps, where the identicalindividual maps have two expanding, fractional linear, onto branches, and wherethe coupling is introduced via a parameter - common to all individual maps -that depends in an analytic way on the mean field of the system. We show: 1)For the range of coupling parameters we consider, finite-size coupled systemsalways have a unique invariant probability density which is strictly positiveand analytic, and all finite-size systems exhibit exponential decay ofcorrelations. 2) For the same range of parameters, the self-consistentPerron-Frobenius operator which captures essential aspects of the correspondinginfinite-size system (arising as the limit of the above when the system sizetends to infinity), undergoes a supercritical pitchfork bifurcation from aunique stable equilibrium to the coexistence of two stable and one unstableequilibrium.


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7945. BOUNDING BASIC CHARACTERISTICS OF SPATIAL EPIDEMICS WITH A NEW PERCOLATION MODEL

Ronald Meester and Pieter Trapman

We introduce a new percolation model to describe and analyze the spread of anepidemic on a general directed and locally finite graph. We assign atwo-dimensional random weight vector to each vertex of the graph in such a waythat the weights of different vertices are i.i.d., but the two entries of thevector assigned to a vertex need not be independent. The probability for anedge to be open depends on the weights of its end vertices, but conditionallyon the weights, the states of the edges are independent of each other. In anepidemiological setting, the vertices of a graph represent the individuals in a(social) network and the edges represent the connections in the network. Theweights assigned to an individual denote its (random) infectivity andsusceptibility, respectively. We show that one can bound the percolationprobability and the expected size of the cluster of vertices that can bereached by an open path starting at a given vertex from above and below by thecorresponding quantities for respectively independent bond and site percolationwith certain densities; this generalizes a result of Kuulasmaa. Many models inthe literature are special cases of our general model.


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7946. MARTINGALE-COBOUNDARY REPRESENTATION FOR A CLASS OF RANDOM FIELDS

Mikhail Gordin

A stationary random sequence admits under some assumptions a representationas the sum of two others: one of them is a martingale difference sequence, andanother is a so-called coboundary. Such a representation can be used forproving some limit theorems by means of the martingale approximation. Amultivariate version of such a decomposition is presented in the paper for aclass of random fields generated by several commuting non-invertibleprobability preserving transformations. In this representation summands ofmixed type appear which behave with respect to some groupof directions of theparameter space as reversed multiparameter martingale differences (in the senseof one of several known definitions) while they look as coboundaries relativeto the other directions. Applications to limit theorems will be publishedelsewhere.


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7947. CONSTRUCTION OF SIGNED MULTIPLICATIVE CASCADES

Julien Barral and Xiong Jin and Benoit Mandelbrot

The theory of positive $T$-martingales was developed in order to set up ageneral framework including the positive measure-valued martingales initiallyconsidered for intermittent turbulence modelling. We consider the naturalextension consisting in allowing the martingale to take complex values. Wefocus on martingales constructed on the line: $T$ is the interval $[0,1]$.Then, random measures are replaced by random functions. We specify a largeclass of such martingales, which contains the complex extension of $b$- adiccanonical cascades, compound Poisson cascades, and more generally infinitelydivisible cascades. For the elements of this class, we find a sufficientcondition for their almost sure uniform convergence to a non-trivial limit.Such limit provide new examples of multifractal processes.


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7948. CONVERGENCE OF SIGNED MULTIPLICATIVE CASCADES

Julien Barral and Xiong Jin and Benoit Mandelbrot

This paper extends the familiar sequences of random measures obtained on$[0,1]$ via $b$-adic independent cascades by allowing the random weightsinvoked in the cascades to take real, or complex values. This yields sequencesof random functions. The asymptotic behavior of these sequences isinvestigated. We obtain a sufficient condition for the almost sure convergenceof these signed cascades to non-trivial statistically self-similar limit. Undersuitable assumptions, the limit function can be represented almost surely as amonofractal function in multifractal time. When the sufficient condition forconvergence does not hold, in most of the cases we show that either the limitis 0 or the sequence diverges almost surely. In the later case, under somecondition we prove a functional central limit theorem, which claims that thereis a natural normalization making the sequence convergent in law to a standardBrownian motion in multifractal time.


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7949. POLYNOMIAL PROCESSES AND THEIR APPLICATIONS TO MATHEMATICAL FINANCE

Christa Cuchiero and Martin Keller-Ressel and Josef Teichmann

We introduce a class of Markov stochastic processes called $m$- polynomial,for which the calculation of (mixed) moments up to order $m$ only requires thecomputation of matrix exponentials. This class contains affine processes,Feller processes with quadratic squared diffusion coefficient, as well asL\'evy-driven SDEs with affine vector fields. Thus, many popular models such asthe classical Black-Scholes, exponential L\'evy or affine models are covered bythis setting. The applications range from statistical GMM estimation to optionpricing. For instance, the efficient and easy computation of moments cansuccessfully be used for variance reduction techniques in Monte Carlosimulations.


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7950. POLYNOMIAL BIRTH-DEATH DISTRIBUTION APPROXIMATION IN WASSERSTEIN DISTANCE

Aihua Xia and Fuxi Zhang

The polynomial birth-death distribution (abbr. as PBD) on $\ci=\{0,1,2, >...\}$ or $\ci=\{0,1,2, ..., m\}$ for some finite $m$ introduced in Brown &Xia (2001) is the equilibrium distribution of the birth-death process withbirth rates $\{\alpha_i\}$ and death rates $\{\beta_i\}$, where $\a_i \ge0$ and$\b_i\ge0$ are polynomial functions of $i\in\ci$. The family includes Poisson,negative binomial, binomial and hypergeometric distributions. In this paper, wegive probabilistic proofs of various Stein's factors for the PBD approximationwith $\a_i=a$ and $\b_i=i+bi(i-1)$ in terms of the Wasserstein distance. Thepaper complements the work of Brown & Xia (2001) and generalizes the work ofBarbour & Xia (2006) where Poisson approximation ($b=0$) in the Wassersteindistance is investigated. As an application, we establish an upper bound forthe Wasserstein distance between the PBD and Poisson binomial distribution andshow that the PBD approximation to the Poisson binomial distribution is muchmore precise than the approximation by the Poisson or shifted Poissondistributions.


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7951. ON THE BOSE-EINSTEIN DISTRIBUTION AND BOSE CONDENSATION

V. P. Maslov (1 and 2) and V. E. Nazaikinskii (2) ((1) Moscow State University, (2) Institute for Problems in Mechanics, RAS, Moscow)

For a system of identical Bose particles sitting on integer energy levels, wegive sharp estimates for the convergence of the sequence of occupation numbersto the Bose-Einstein distribution and for the Bose condensation effect.


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7952. A NEW APPROACH OF POINT ESTIMATION FROM TRUNCATED OR GROUPED AND CENSORED DATA

Ahmed Guellil (USTHB) and Tewfik Kernane (USTHB)

We propose a new approach for estimating the parameters of a probabilitydistribution. It consists on combining two new methods of estimation. The firstis based on the definition of a new distance measuring the difference betweenvariations of two distributions on a finite number of points from their supportand on using this measure for estimation purposes by the method of minimumdistance. For the second method, given an empirical discrete distribution, webuild up an auxiliary discrete theoretical distribution having the same supportof the first and depending on the same parameters of the parent distribution ofthe data from which the empirical distribution emanated. We estimate then theparameters from the empirical distribution by the usual statistical methods. Inpractice, we propose to compute the two estimations, the second based onmaximum likelihood principle of known theoretical properties, and the firstbeing as a control of the effectiveness of the obtained estimation, and forwhich we prove the convergence in probability, so we have also a criterion onthe quality of the information contained in the observations. We apply theapproach to truncated or grouped and censored data situations to give theflavour on the effectiveness of the approach. We give also some interestingperspectives of the approach including model selection from truncated data,estimation of the initial trial value in the celebrate EM algorithm in the caseof truncation and merged normal populations, a test of goodness of fit based onthe new distance, quality of estimations and data.


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7953. RANDOM COMPLEX DYNAMICS AND SEMIGROUPS OF HOLOMORPHIC MAPS

Hiroki Sumi

We investigate the random dynamics of rational maps and the dynamics ofsemigroups of rational maps on the Riemann sphere. We see that the both fieldsare related to each other very deeply. We investigate spectral properties oftransition operators and the dynamics of associated semigroups of rationalmaps. We define several kinds of Julia sets of the associated Markov processesand we study the properties and the dimension of them. Moreover, we investigate"singular functions on the complex plane". In particular, we consider thefunctions $T$ which represent the probability of tending to infinity withrespect to the random dynamics of polynomials. Under certain conditions thesefunctions $T$ are complex analogues of the devil's staircase and Lebesgue'ssingular functions. More precisely, we show that these functions $T$ arecontinuous on the Riemann sphere and vary only on the Julia sets of associatedsemigroups. Furthermore, by using ergodic theory and potential theory, weinvestigate the non-differentiability and regularity of these functions. Wefind many phenomena which can hold in the random complex dynamics and thedynamics of semigroups of rational maps, but cannot hold in the usual iterationdynamics of a single holomorphic map. We carry out a systematic study of thesephenomena and their mechanisms.


-----------------------------------Stefano M. IacusDepartment of Economics,Business and StatisticsUniversity of MilanVia Conservatorio, 7I-20123 Milan - ItalyPh.: +39 02 50321 461Fax: +39 02 50321 505http://www.economia.unimi.it/iacus------------------------------------------------------------------------------------Please don't send me Word or PowerPoint attachments if notabsolutely necessary. See:http://www.gnu.org/philosophy/no-word-attachments.html

Probability Abstracts 108This document contains abstracts 7954-8212from Jan-1-2009 to February-28-2009.They have been mailed on Mar 3, 2009.

This letter can be also found on line athttp://pas.imstat.org/Letters/letter_108.shtml

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7954. Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy white noiseAuthor(s): Zdzis{\l}aw Brze{\'z}niak and Jerzy Zabczyk

Abstract: The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by L\'evy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.


7955. Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert EquationAuthor(s): Z. Brzezniak and B. Goldys

Abstract: The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative space-dependent noise is considered for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere $\mathbb S^2$. The regularity of weak solutions is also discussed. Some of the regularity results are new even for the deterministic Landau-Lifshitz-Gilbert equation.


7956. Conditions for certain ruin for the generalised Ornstein- Uhlenbeck process and the structure of the upper and lower boundsAuthor(s): Damien Bankovsky

Abstract: For a bivariate \Levy process $(\xi_t,\eta_t)_{t\geq 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as \ [V_t:=e^{\xi_t}(z+\int_0^t e^{-\xi_{s-}}\ud \eta_s), t\ge0,\]where $z \in\mathbb{R}.$ We present conditions on the characteristic triplet of $(\xi,\eta)$ which ensure certain ruin for the GOU. We present a detailed analysis on the structure of the upper and lower bounds and the sets of values on which the GOU is almost surely increasing, or decreasing. This paper is the sequel to \cite{BankovskySly08}, which stated conditions for zero probability of ruin, and completes a significant aspect of the study of the GOU.


7957. Current and density fluctuations for interacting particle systems with anomalous diffusive behaviorAuthor(s): M. Jara

Abstract: We prove density and current fluctuations for two examples of symmetric, interacting particle systems with anomalous diffusive behavior: the zero-range process with long jumps and the zero-range process with degenerated bond disorder. As an application, we obtain subdiffusive behavior of a tagged particle in a simple exclusion process with variable diffusion coefficient.


7958. Order-invariant Measures on Causal SetsAuthor(s): Graham Brightwell and Malwina Luczak

Abstract: A causal set is a partially ordered set on a countably infinite ground-set such that each element is above finitely many others. A natural extension of a causal set is an enumeration of its elements which respects the order. We bring together two different classes of random processes. In one class, we are given a fixed causal set, and we consider random natural extensions of this causal set: we think of the random enumeration as being generated one point at a time. In the other class of processes, we generate a random causal set, again working from the bottom up, adding one new maximal element at each stage. Processes of both types can exhibit a property called order-invariance: if we stop the process after some fixed number of steps, then, conditioned on the structure of the causal set, every possible order of generation of its elements is equally likely. We develop a framework for the study of order-invariance which includes both types of example: order-invariance is then a property of probability measures on a certain space. Our main result is a description of the extremal order-invariant measures.


7959. Spatial Epidemics and Local Times for Critical Branching Random Walks in Dimensions 2 and 3Author(s): Steven P. Lalley and Xinghua Zheng

Abstract: The behavior at criticality of spatial SIR (susceptible/ infected/recovered) epidemic models in dimensions two and three is investigated. In these models, finite populations of size N are situated at the vertices of the integer lattice, and infectious contacts are limited to individuals at the same or at neighboring sites. Susceptible individuals, once infected, remain contagious for one unit of time and then recover, after which they are immune to further infection. It is shown that the measure-valued processes associated with these epidemics, suitably scaled, converge, in the large-N limit, either to a standard Dawson-Watanabe process (super- Brownian motion) or to a Dawson-Watanabe process with location- dependent killing, depending on the size of the the initially infected set. A key element of the argument is a proof of Adler's 1993 conjecture that the local time processes associated with branching random walks converge to the local time density process associated with the limiting super-Brownian motion.


7960. Representation of gaussian small ball probabilities in $l_2$Author(s): Andr\'e Mas (I3M)

Abstract: Let $z=\sum_{i=1}^{+\infty}x_{i}^{2}/a_{i}^{2}$ where the $x_{i}$'s are i.d.d centered with unit variance gaussian random variables and $(a_{i}) _{i\in\mathbb{N}}$ an increasing sequence such that $\sum _{i=1}^{+\infty}a_{i}^{-2} S be the left shift acting on S, a one- sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of $\sigma$-invariant Borel probabilities that maximize the integral of a given locally H\"older continuous potential A:S -> R. Under certain conditions, we are able to show not only that A-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions).


8079. Ergodicity of multiplicative statisticsAuthor(s): Yuri Yakubovich

Abstract: For a subfamily of multiplicative measures on integer partitions we give conditions for properly rescaled associated Young diagrams to converge in probability to a certain deterministic curve named the limit shape of partitions. We provide explicit formulas for the scaling function and the limit shape covering some known and some new examples.


8080. Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensembleAuthor(s): Joseph Najnudel and Ashkan Nikeghbali and Felix Rubin

Abstract: In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble $GCy$, whose eigenvalues PDF is given by $$\textrm{const}\cdot\prod_{1\leq j-1/2$ and where $N$ is the size of the matrix ensemble. Using results by Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove that for this ensemble, the largest eigenvalue divided by $N$ converges in law to some probability distribution for all $s$ such that $ \Re(s)>-1/2$. Using results by Forrester and Witte \cite{Forrester- Witte2} on the distribution of the largest eigenvalue for fixed $N$, we also express the limiting probability distribution in terms of some non-linear second order differential equation. Eventually, we show that the convergence of the probability distribution function of the re-scaled largest eigenvalue to the limiting one is at least of order $ (1/N)$.


8081. Wick Calculus For Nonlinear Gaussian FunctionalsAuthor(s): Yaozhong Hu and Jia-an Yan

Abstract: This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormalization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormalization are integrable random variables. Relevant results on multiple Wiener integrals, second quantization operator, Malliavin calculus and their relations with the Wick product and Wick renormalization are also briefly presented. A useful tool for Wick product is the $S$-transform which is also described without the introduction of generalized random variables.


8082. Parameter estimation for fractional Ornstein-Uhlenbeck processesAuthor(s): Yaozhong Hu and David Nualart

Abstract: We study a least squares estimator $\hat {\theta}_T$ for the Ornstein-Uhlenbeck process, $dX_t=\theta X_t dt+\sigma dB^H_t$, driven by fractional Brownian motion $B^H$ with Hurst parameter $H\ge \frac12$. We prove the strong consistence of $\hat {\theta}_T$ (the almost surely convergence of $\hat {\theta}_T$ to the true parameter $ {% \theta}$). We also obtain the rate of this convergence when $1/2\le H4$ corresponds to finite {\it third} moment of the degrees. When, the proportion of vertices with degree at least $k$ is asymptotically equal to $ck^{-\tau+1}$ for some $\tau\in (3,4),$ the largest critical connected component is of order $n^{(\tau-2)/(\tau-1)},$ instead. Our results show that, for inhomogeneous random graphs with a power-law degree sequence, the critical behavior admits a transition when the third moment of the degrees turns from finite to infinite. Similar phase transitions have been shown to occur for typical distances in such random graphs when the variance of the degrees turns from finite to infinite. We present further results related to the size of the critical or scaling window, and state conjectures for this and related random graph models.


8091. Shelf Life of Candidates in the Generalized Secretary ProblemAuthor(s): Krzysztof Szajowski and Mitsushi Tamaki

Abstract: A version of the secretary problem called the duration problem, in which the objective is to maximize the time of possession of relatively best objects or the second best, is treated. It is shown that in this duration problem there are threshold numbers $(k_1^ \star,k_2^\star)$ such that the optimal strategy immediately selects a relatively best object if it appears after time $k_1^\star$ and a relatively second best object if it appears after moment $k_2^\star$. When number of objects tends to infinity the thresholds values are $ \lfloor 0.417188N\rfloor$ and $\rfloor 0.120381N\rfloor$, respectively. The asymptotic mean time of shelf life of the object is $0.403827N$.


8092. On Stein's method for multivariate normal approximationAuthor(s): Elizabeth S. Meckes

Abstract: The purpose of this paper is to synthesize the approaches taken by Chatterjee-Meckes and Reinert-R\"ollin in adapting Stein's method of exchangeable pairs for multivariate normal approximation. The more general linear regression condition of Reinert-R\"ollin allows for wider applicability of the method, while the method of bounding the solution of the Stein equation due to Chatterjee-Meckes allows for improved convergence rates. Two abstract normal approximation theorems are proved, one for use when the underlying symmetries of the random variables are discrete, and one for use in contexts in which continuous symmetry groups are present. The application to runs on the line from Reinert-R\"ollin is reworked to demonstrate the improvement in convergence rates, and a new application to joint value distributions of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold is presented.


8093. Fermionic construction of tau functions and random processesAuthor(s): John Harnad and Alexander Yu. Orlov

Abstract: Tau functions expressed as fermionic expectation values are shown to provide a natural and straightforward description of a number of random processes and statistical models involving hard core configurations of identical particles on the integer lattice, like a discrete version simple exclusion processes (ASEP), nonintersecting random walkers, lattice Coulomb gas models and others, as well as providing a powerful tool for combinatorial calculations involving paths between pairs of partitions. We study the decay of the initial step function within the discrete ASEP (d-ASEP) model as an example.


8094. Clustering Bounds on N-Point Correlations for Unbounded Spin SystemsAuthor(s): Abdelmalek Abdesselam and Aldo Procacci and Benedetto Scoppola

Abstract: We prove clustering estimates for the truncated correlations, i.e., cumulants of an unbounded spin system on the lattice. We provide a unified treatment, based on cluster expansion techniques, of four different regimes: large mass, small interaction between sites, large self-interaction, as well as the more delicate small self-interaction or `low temperature' regime. A clustering estimate in the latter regime is needed for the Bosonic case of the recent result obtained by Lukkarinen and Spohn on the rigorous control on kinetic scales of quantum fluids.


8095. Very large graphsAuthor(s): Laszlo Lovasz

Abstract: In the last decade it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks: separable elements, with connections (or interactions) between certain pairs of them. These huge networks pose exciting challenges for the mathematician. Graph Theory (the mathematical theory of networks) faces novel, unconventional problems: these very large networks (like the Internet) are never completely known, in most cases they are not even well defined. Data about them can be collected only by indirect means like random local sampling. Dense networks (in which a node is adjacent to a positive percent of others nodes) and sparse networks (in which a node has a bounded number of neighbors) show very different behavior. From a practical point of view, sparse networks are more important, but at present we have more complete theoretical results for dense networks. The paper surveys relations with probability, algebra, extrema graph theory, and analysis.


8096. Carries, shuffling, and symmetric functionsAuthor(s): Persi Diaconis and Jason Fulman

Abstract: The "carries" when n random numbers are added base b form a Markov chain with an "amazing" transition matrix determined by Holte. This same Markov chain occurs in following the number of descents or rising sequences when n cards are repeatedly riffle shuffled. We give generating and symmetric function proofs and determine the rate of convergence of this Markov chain to stationarity. Similar results are given for type B shuffles. We also develop connections with Gaussian autoregressive processes and the Veronese mapping of commutative algebra.


8097. Poset limits and exchangeable random posetsAuthor(s): Svante Janson

Abstract: We develop a theory of limits of finite posets in close analogy to the recent theory of graph limits. In particular, we study representations of the limits by functions of two variables on a probability space, and connections to exchangeable random infinite posets.


8098. Random symmetrizations of measurable setsAuthor(s): Aljosa Volcic

Abstract: In this paper we prove almost sure convergence to the ball, in the Nikodym metric, of sequences of random Steiner symmetrizations of bounded Caccioppoli and bounded measurable sets, paralleling a result due to Mani-Levitska concerning convex bodies.


8099. A L\'{e}vy input model with additional state-dependent servicesAuthor(s): Zbigniew Palmowski and Maria Vlasiou

Abstract: We consider a queuing model with the workload evolving between consecutive i.i.d. exponential timers $\{e_q^{(i)} \}_{i=1,2,...}$ according to a spectrally positive L\'{e}vy process $Y(t)$ which is reflected at 0. When the exponential clock $e_q^{(i)}$ ends, the additional state-dependent service requirement modifies the workload so that the latter is equal to $F_i(Y(e_q^{(i)}))$ at epoch $e^{(1)}_q+...+e^{(i)}_q$ for some random nonnegative i.i.d. functionals $F_i$. In particular, we focus on the case when $F_i(y)=(B_i-y)^+$, where $\{B_i\}_{i=1,2,...}$ are i.i.d. nonnegative random variables. We analyse the steady-state workload distribution for this model.


8100. Discretizing the fractional Levy areaAuthor(s): Andreas Neuenkirch and Samy Tindel (IECN) and J\'er\'emie Unterberger (IECN)

Abstract: In this article, we give sharp bounds for the Euler- and trapezoidal discretization of the Levy area associated to a d- dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean-square convergence rate of the Euler scheme. For H3/4 the exact rate is n^{-1}. Moreover, the trapezoidal scheme has exact convergence rate n^{-2H+1/2} for H>1/2. Finally, we also derive the asymptotic error distribution of the Euler scheme. For H lesser than 3/4 one obtains a Gaussian limit, while for H>3/4 the limit distribution is of Rosenblatt type.


8101. Convergence of multi-class systems of fixed possibly infinite sizesAuthor(s): Carl Graham (CMAP)

Abstract: Multi-class systems having possibly both finite and infinite classes are investigated under a natural partial exchangeability assumption. It is proved that the conditional law of such a system, given the vector of the empirical measures of its finite classes and directing measures of its infinite ones (given by the de Finetti Theorem), corresponds to sampling independently from each class, without replacement from the finite classes and i.i.d. from the directing measure for the infinite ones. The equivalence between the convergence of multi-exchangeable systems with fixed class sizes and the convergence of the corresponding vectors of measures is then established.


8102. A Bernstein type inequality and moderate deviations for weakly dependent sequencesAuthor(s): Florence Merlev\`ede (LAMA) and Magda Peligrad and Emmanuel Rio (LM-Versailles, INRIA Bordeaux - Sud-Ouest)

Abstract: In this paper we present a tail inequality for the maximum of partial sums of a weakly dependent sequence of random variables that are not necessarily bounded. The class considered includes geometrically and subgeometrically strongly mixing sequences. The result is then used to derive asymptotic moderate deviations results. Applications include classes of Markov chains, functions of linear processes with absolutely regular innovations and ARCH m

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