From pas at www2.economia.unimi.it Mon Jan 1 15:25:59 2007 From: pas at www2.economia.unimi.it (pas@www2.economia.unimi.it) Date: Mon Jan 1 15:25:52 2007 Subject: [Pas] Probability Abstracts 95 Message-ID: Jan 1st, 2007 Letter 95 Probability Abstracts Service This document contains abstracts 4722-5092 from Oct-1-2006 to Dic-31-2006. They have been mailed on Jan 1st, 2007. This letter can be also found on line at http://www2.economia.unimi.it/PAS/Letters/letter_95.shtml --------------------------------------------------------------- 4722. CONDENSATION FOR A FIXED NUMBER OF INDEPENDENT RANDOM VARIABLES Pablo A. Ferrari and Claudio Landim and Valentin V. Sisko A family of m independent identically distributed random variables indexed by a chemical potential \phi\in[0,\gamma] represents piles of particles. As \phi increases to \gamma, the mean number of particles per site converges to a maximal density \rho_c<\infty. The distribution of particles conditioned on the total number of particles equal to n does not depend on \phi (canonical ensemble). For fixed m, as n goes to infinity the canonical ensemble measure behave as follows: removing the site with the maximal number of particles, the distribution of particles in the remaining sites converges to the grand canonical measure with density \rho_c; the remaining particles concentrate (condensate) on a single site. http://front.math.ucdavis.edu/math.PR/0612856 --------------------------------------------------------------- 4723. SURVIVAL PROBABILITY OF A DIFFUSING PARTICLE CONSTRAINED BY TWO MOVING, ABSORBING BOUNDARIES Alan J. Bray and Richard Smith We calculate the exact asymptotic survival probability, Q, of a one-dimensional Brownian particle, initially located located at the point x in (-L,L), in the presence of two moving absorbing boundaries located at \pm(L+ct). The result is Q(y,\lambda) = \sum_{n=-\infty}^\infty \cosh (ny) \exp(-n^2\lambda/4), where y=cx/D, \lambda = cL/D and D is the diffusion constant of the particle. The results may be extended to the case where the absorbing boundaries have different speeds. As an application, we compute the asymptotic survival probability for the trapping reaction A + B -> B, for evanescent traps with a long decay time. http://front.math.ucdavis.edu/cond-mat/0612563 --------------------------------------------------------------- 4724. HIGHLY ROBUST ERROR CORRECTION BY CONVEX PROGRAMMING Emmanuel J. Candes and Paige A. Randall This paper discusses a stylized communications problem where one wishes to transmit a real-valued signal x in R^n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably when a fraction of the transmitted codeword is corrupted by arbitrary gross errors, and when in addition, all the entries of the codeword are contaminated by smaller errors (e.g. quantization errors). We show that if one encodes the information as Ax where A is a suitable m by n coding matrix (m >= n), there are two decoding schemes that allow the recovery of the block of n pieces of information x with nearly the same accuracy as if no gross errors occur upon transmission (or equivalently as if one has an oracle supplying perfect information about the sites and amplitudes of the gross errors). Moreover, both decoding strategies are very concrete and only involve solving simple convex optimization programs, either a linear program or a second-order cone program. We complement our study with numerical simulations showing that the encoder/decoder pair performs remarkably well. http://front.math.ucdavis.edu/cs.IT/0612124 --------------------------------------------------------------- 4725. BILLIARDS IN A GENERAL DOMAIN WITH RANDOM REFLECTIONS Francis Comets and Serguei Popov and Gunter Sch\"utz and Marina Vachkovskaia We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain ${\mathcal D} \subset {\mathbb R}^d$ until it hits the boundary and bounces randomly inside according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord ``picked at random'' in ${\mathcal D}$, and we study the angle of intersection of the process with a $(d-1)$-dimensional manifold contained in ${\mathcal D}$. http://front.math.ucdavis.edu/math.PR/0612799 --------------------------------------------------------------- 4726. UNIQUENESS AND NON-UNIQUENESS IN PERCOLATION THEORY Olle H\"{a}ggstr\"{o}m and Johan Jonasson This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on ${\mathbb{Z}}^d $ and, more generally, on transitive graphs. For iid percolation on ${\mathbb {Z}}^d$, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed. http://front.math.ucdavis.edu/math.PR/0612812 --------------------------------------------------------------- 4727. ASYMPTOTIC NORMALITY OF THE K-CORE IN RANDOM GRAPHS Svante Janson and Malwina J. Luczak We study the $k$-core of a random (multi)graph on $n$ vertices with a given degree sequence. In our previous paper `A simple solution to the k-core problem' we used properties of empirical distributions of independent random variables to give a simple proof of the fact that the size of the giant $k$-core obeys a law of large numbers as $n$ tends to infinity. Here we develop the method further and show that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold. Further, we determine precisely the location of the phase transition window for the emergence of a giant $k$-core. Hence we deduce corresponding results for the $k$-core in $G(n,p)$ and $G(n,m)$. http://front.math.ucdavis.edu/math.PR/0612827 --------------------------------------------------------------- 4728. LARGE DEVIATIONS AND RANDOM ENERGY MODELS N. K. Jana and B. V. Rao A unified treatment for the existence of free energy in several random energy models is presented. If the sequence of distributions associated with the particle systems obeys a large deviation principle, then the free energy exists almost surely. This includes all the known cases as well as some heavy-tailed distributions. http://front.math.ucdavis.edu/math.PR/0612836 --------------------------------------------------------------- 4729. HIGH DIMENSIONAL PROBABILITY Evarist Gin\'{e} and Vladimir Koltchinskii and Wenbo Li and Joel Zinn About forty years ago it was realized by several researchers that the essential features of certain objects of Probability theory, notably Gaussian processes and limit theorems, may be better understood if they are considered in settings that do not impose structures extraneous to the problems at hand. For instance, in the case of sample continuity and boundedness of Gaussian processes, the essential feature is the metric or pseudometric structure induced on the index set by the covariance structure of the process, regardless of what the index set may be. This point of view ultimately led to the Fernique-Talagrand majorizing measure characterization of sample boundedness and continuity of Gaussian processes, thus solving an important problem posed by Kolmogorov. Similarly, separable Banach spaces provided a minimal setting for the law of large numbers, the central limit theorem and the law of the iterated logarithm, and this led to the elucidation of the minimal (necessary and/or sufficient) geometric properties of the space under which different forms of these theorems hold. However, in light of renewed interest in Empirical processes, a subject that has considerably influenced modern Statistics, one had to deal with a non-separable Banach space, namely $\mathcal{L}_{\infty}$. With separability discarded, the techniques developed for Gaussian processes and for limit theorems and inequalities in separable Banach spaces, together with combinatorial techniques, led to powerful inequalities and limit theorems for sums of independent bounded processes over general index sets, or, in other words, for general empirical processes. http://front.math.ucdavis.edu/math.PR/0612726 --------------------------------------------------------------- 4730. LAW OF THE ITERATED LOGARITHM FOR STATIONARY PROCESSES Ou Zhao and Michael Woodroofe There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes $... X_{-1}, X_0,X_1,...$ whose partial sums $S_n = X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $R_n$ is a remainder term for which $E(R_n^2) = o(n)$. Here we explore the Law of the Iterated Logarithm (LIL) for the same class of processes. Letting $\Vert\cdot\Vert$ denote the norm in $L^2(P)$, a sufficient condition for the partial sums of a stationary process to have the form $S_n = M_n+R_n$ is that $n^{-{3\over 2}}\Vert E(S_n|X_0,X_{-1},...)\Vert$ be summable. A sufficient condition for the LIL is only slightly stronger, requiring $n^{-{3\over 2}}\log^{3\over 2} (n)\Vert E(S_n|X_0,X_{-1},...)\Vert$ to be summable. As a by-product of our main result, we obtain an improved statement of the Conditional Central Limit Theorem. Invariance principles are obtained as well. http://front.math.ucdavis.edu/math.PR/0612747 --------------------------------------------------------------- 4731. CONVEX CHAINS IN Z^2 Nathanael Enriquez (PMA) A detailed combinatorial analysis of lattice convex polygonal lines of N^2 joining 0 to (n,n) is presented. We derive consequences on the line having the largest number of vertices as well as the cardinal and limit shape of lines having few vertices. The proof refines a statistical physical method used by Sinai to obtain the typical behavior of these lines, allied to some Fourier analysis. Limit shapes of convex lines joining 0 to (n,n) and having a given total length are also characterized. http://front.math.ucdavis.edu/math.PR/0612770 --------------------------------------------------------------- 4732. EMPIRICAL GRAPH LAPLACIAN APPROXIMATION OF LAPLACE--BELTRAMI OPERATORS: LARGE SAMPLE RESULTS Evarist Gin\'{e} and Vladimir Koltchinskii Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of dimension $\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the random operators $$ \Delta_{h_n,n}f(p):=\frac{1}{nh_n^{d+2}}\sum_{i=1}^n K(\frac{p-X_i}{h_n})(f(X_i)-f(p)), p\in M $$ where ${K(u):={\frac{1}{(4\pi)^{d/2}}}e^{-\|u\|^2/4}}$ is the Gaussian kernel and ${h_n\to 0}$ as ${n\to\infty.}$ Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of ${M,}$ ${\Delta_Mf}$ (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations ${\Delta_{h_n,n}f(p)-{\frac{1}{|\mu|}}\Delta_Mf(p)}$ for smooth functions ${f}$ both pointwise and uniformly in ${f}$ and ${p}$ (here ${|\mu|=\mu(M)} $ and ${\mu}$ is the Riemannian volume measure). In particular, we show that for any class ${{\cal F}}$ of three times differentiable functions on ${M}$ with uniformly bounded derivatives $$ \sup_{p\in M}\sup_{f\in F}\Big|\Delta_{h_n,p}f(p)-\frac{1}{|\mu|}\Delta_Mf(p)\Big|= O\Big(\sqrt{\frac{\log(1/h_n)}{nh_n^{d+2}}}\Big) a.s. $$ as soon as $$ nh_n^{d+2}/\log h_n^{-1}\to \infty and nh^{d+4}_n/\log h_n^{-1}\to 0, $$ and also prove asymptotic normality of ${\Delta_{h_n,p}f(p)-{\frac{1}{|\mu|}}\Delta_Mf(p)}$ (functional CLT) for a fixed ${p\in M}$ and uniformly in ${f}.$ http://front.math.ucdavis.edu/math.PR/0612777 --------------------------------------------------------------- 4733. ESTIMATION FOR THE DISCRETELY OBSERVED TELEGRAPH PROCESS stefano m. iacus and nakahiro yoshida The telegraph process $\{X(t), t>0\}$, is supposed to be observed at $n+1$ equidistant time points $t_i=i\Delta_n,i=0,1,..., n$. The unknown value of $\lambda$, the underlying rate of the Poisson process, is a parameter to be estimated. The asymptotic framework considered is the following: $\Delta_n \to 0$, $n\Delta_n = T \to \infty$ as $n \to \infty$. We show that previously proposed moment type estimators are consistent and asymptotically normal but not efficient. We study further an approximated moment type estimator which is still not efficient but comes in explicit form. For this estimator the additional assumption $n\Delta_n^3 \to 0$ is required in order to obtain asymptotic normality. Finally, we propose a new estimator which is consistent, asymptotically normal and asymptotically efficient under no additional hypotheses. http://front.math.ucdavis.edu/math.PR/0612784 --------------------------------------------------------------- 4734. A CLT FOR REGULARIZED SAMPLE COVARIANCE MATRICES Greg W Anderson and Ofer Zeitouni We consider the spectral properties of a class of {\em regularized estimators} of (large) empirical covariance matrices corresponding to stationary (but not necessarily Gaussian) sequences, obtained by {\em banding}. We prove a law of large numbers (similar to that proved in the Gaussian case by Bickel and Levina), which implies that the spectrum of a banded empirical covariance matrix is an efficient estimator. Our main result is a central limit theorem in the same regime, which to our knowledge is new, even in the Gaussian setup. http://front.math.ucdavis.edu/math.PR/0612791 --------------------------------------------------------------- 4735. LAW OF LARGE NUMBERS FOR SUPERDIFFUSIONS: THE NON-ERGODIC CASE Janos Englander In a previous paper of Winter and the author the Law of Large Numbers for the local mass of certain superdiffusions was proved under a spectral theoretical assumption, which is equivalent to the ergodicity (positive recurrence) of the motion component of an $H$-transformed (or weighted) superprocess. In fact the assumption is also equivalent to the property that the scaling for the expectation of the local mass is pure exponential. In this paper we go beyond ergodicity, that is we consider cases when the scaling is not purely exponential. Inter alia, we prove the analog of the Watanabe-Biggins Law of Large Numbers for super-Brownian motion (SBM). We will also prove another Law of Large Numbers for a bounded set moving with subcritical speed, provided the variance term decays sufficiently fast. http://front.math.ucdavis.edu/math.PR/0612797 --------------------------------------------------------------- 4736. STOCHASTIC INERTIAL MANIFOLDS FOR DAMPED WAVE EQUATIONS Zhenxin Liu In this paper, stochastic inertial manifold for damped wave equations subjected to additive white noise is constructed by the Lyapunov- Perron method. It is proved that when the intensity of noise tends to zero the stochastic inertial manifold converges to its deterministic counterpart almost surely. http://front.math.ucdavis.edu/math.DS/0612774 --------------------------------------------------------------- 4737. HOW TO CHOOSE A CHAMPION E. Ben-Naim and N.W. Hengartner League competition is investigated using random processes and scaling techniques. In our model, a weak team can upset a strong team with a fixed probability. Teams play an equal number of head-to-head matches and the team with the largest number of wins is declared to be the champion. The total number of games needed for the best team to win the championship with high certainty, T, grows as the cube of the number of teams, N, i.e., T ~ N^3. This number can be substantially reduced using preliminary rounds where teams play a small number of games and subsequently, only the top teams advance to the next round. When there are k rounds, the total number of games needed for the best team to emerge as champion, T_k, scales as follows, T_k ~N^(\gamma_k) with gamma_k=1/[1-(2/3)^(k+1)]. For example, gamma_k=9/5,27/19,81/65 for k=1,2,3. These results suggest an algorithm for how to infer the best team using a schedule that is linear in N. We conclude that league format is an ineffective method of determining the best team, and that sequential elimination from the bottom up is fair and efficient. http://front.math.ucdavis.edu/physics/0612217 --------------------------------------------------------------- 4738. ON THE EXCURSION THEORY FOR LINEAR DIFFUSIONS Paavo Salminen and Pierre Vallois and Marc Yor We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein's representations that, e.g., the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss Ornstein-Uhlenbeck processes. http://front.math.ucdavis.edu/math.PR/0612687 --------------------------------------------------------------- 4739. OPTION PRICING WITH LOG-STABLE L\'{E}VY PROCESSES Przemys{\l}aw Repetowicz and Peter Richmond We model the logarithm of the price (log-price) of a financial asset as a random variable obtained by projecting an operator stable random vector with a scaling index matrix $\underline{\underline{E}}$ onto a non-random vector. The scaling index $\underline{\underline{E}}$ models prices of the individual financial assets (stocks, mutual funds, etc.). We find the functional form of the characteristic function of real powers of the price returns and we compute the expectation value of these real powers and we speculate on the utility of these results for statistical inference. Finally we consider a portfolio composed of an asset and an option on that asset. We derive the characteristic function of the deviation of the portfolio, \mbox{${\mathfrak D}_t^ {({\mathfrak t})}$}, defined as a temporal change of the portfolio diminished by the the compound interest earned. We derive pseudo-differential equations for the option as a function of the log-stock-price and time and we find exact closed-form solutions to that equation. These results were not known before. Finally we discuss how our solutions correspond to other approximate results known from literature,in particular to the well known Black & Scholes equation. http://front.math.ucdavis.edu/math.PR/0612691 --------------------------------------------------------------- 4740. OSCILLATIONS OF EMPIRICAL DISTRIBUTION FUNCTIONS UNDER DEPENDENCE Wei Biao Wu We obtain an almost sure bound for oscillation rates of empirical distribution functions for stationary causal processes. For short-range dependent processes, the oscillation rate is shown to be optimal in the sense that it is as sharp as the one obtained under independence. The dependence conditions are expressed in terms of physical dependence measures which are directly related to the data-generating mechanism of the underlying processes and thus are easy to work with. http://front.math.ucdavis.edu/math.PR/0612692 --------------------------------------------------------------- 4741. KARHUNEN-LO\`{E}VE EXPANSIONS OF MEAN-CENTERED WIENER PROCESSES Paul Deheuvels For $\gamma>-{1/2}$, we provide the Karhunen-Lo\`{e}ve expansion of the weighted mean-centered Wiener process, defined by \[W _{\gamma}(t)=\frac{1}{\sqrt{1+2\gamma}}\Big\{W\big(t^{1+2\gamma}\big)- \int_0^1W\big(u^{1+2\gamma}\big)du\Big\},\] for $t\in(0,1]$. We show that the orthogonal functions in these expansions have simple expressions in term of Bessel functions. Moreover, we obtain that the $L^2[0,1]$ norm of $W_ {\gamma}$ is identical in distribution with the $L^2[0,1]$ norm of the weighted Brownian bridge $t^{\gamma}B(t)$. http://front.math.ucdavis.edu/math.PR/0612693 --------------------------------------------------------------- 4742. FRACTIONAL BROWNIAN FIELDS, DUALITY, AND MARTINGALES Vladimir Dobri\'{c} and Francisco M. Ojeda In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously. The field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. A mistake common to the existing literature regarding multifractional Brownian motions is pointed out and corrected. The Gaussian field, due to inherited ``duality'', reveals a new way of constructing martingales associated with the odd and even part of a fractional Brownian motion and therefore of the fractional Brownian motion. The existence of those martingales and their stochastic representations is the first step to the study of natural wavelet expansions associated to those processes in the spirit of our earlier work on a construction of natural wavelets associated to Gaussian-Markov processes. http://front.math.ucdavis.edu/math.PR/0612694 --------------------------------------------------------------- 4743. A GENERALIZED OCCUPATION TIME FORMULA FOR CONTINUOUS SEMIMARTINGALES Raouf Ghomrasni We show that for a wide class of functions $F$ that: $$ {\lim_{\epsilon \downarrow 0} {\frac{1}{\epsilon}} \int_0^t \Big\{F(s, X_s) - F(s, X_s - \epsilon)\Big\} d\big_s} = - \int_0^t\int_{\R} F(s, x) d L_s^x $$ where $X_t$ is a continuous semi-martingale, $(L_t^x, x \in \R, t \geq 0)$ its local time process and $(\big_t, t \geq 0)$ its quadratic variation process. http://front.math.ucdavis.edu/math.PR/0612699 --------------------------------------------------------------- 4744. FRACTAL PROPERTIES OF THE RANDOM STRING PROCESSES Dongsheng Wu and Yimin Xiao Let $\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}$ be a random string taking values in ${\mathbb{R}}^d$, specified by the following stochastic partial differential equation [Funaki (1983)]: \[\frac{\partial u_t(x)}{\partial t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},\] where $\dot{W} (x,t)$ is an ${\mathbb{R}}^d$-valued space-time white noise. Mueller and Tribe (2002) have proved necessary and sufficient conditions for the ${\mathbb{R}} ^d$-valued process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$ to hit points and to have double points. In this paper, we continue their research by determining the Hausdorff and packing dimensions of the level sets and the sets of double times of the random string process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$. We also consider the Hausdorff and packing dimensions of the range and graph of the string. http://front.math.ucdavis.edu/math.PR/0612700 --------------------------------------------------------------- 4745. MODIFIED EMPIRICAL CLT'S UNDER ONLY PRE-GAUSSIAN CONDITIONS Shahar Mendelson and Joel Zinn We show that a modified Empirical process converges to the limiting Gaussian process whenever the limit is continuous. The modification depends on the properties of the limit via Talagrand's characterization of the continuity of Gaussian processes. http://front.math.ucdavis.edu/math.PR/0612703 --------------------------------------------------------------- 4746. EMPIRICAL AND GAUSSIAN PROCESSES ON BESOV CLASSES Richard Nickl We give several conditions for pregaussianity of norm balls of Besov spaces defined over $\mathbb{R}^d$ by exploiting results in Haroske and Triebel (2005). Furthermore, complementing sufficient conditions in Nickl and P\"{o}tscher (2005), we give necessary conditions on the parameters of the Besov space to obtain the Donsker property of such balls. For certain parameter combinations Besov balls are shown to be pregaussian but not Donsker. http://front.math.ucdavis.edu/math.PR/0612706 --------------------------------------------------------------- 4747. INVARIANCE PRINCIPLE FOR STOCHASTIC PROCESSES WITH SHORT MEMORY Magda Peligrad and Sergey Utev In this paper we give simple sufficient conditions for linear type processes with short memory that imply the invariance principle. Various examples including projective criterion are considered as applications. In particular, we treat the weak invariance principle for partial sums of linear processes with short memory. We prove that whenever the partial sums of innovations satisfy the $L_p$--invariance principle, then so does the partial sums of its corresponding linear process. http://front.math.ucdavis.edu/math.PR/0612707 --------------------------------------------------------------- 4748. HOMOGENENOUS MULTITYPE FRAGMENTATIONS Jean Bertoin (PMA and DMA) A homogeneous mass-fragmentation, as it has been defined in \cite{RFC}, describes the evolution of the collection of masses of fragments of an object which breaks down into pieces as time passes. Here, we show that this model can be enriched by considering also the types of the fragments, where a type may represent, for instance, a geometrical shape, and can take finitely many values. In this setting, the dynamics of a randomly tagged fragment play a crucial role in the analysis of the fragmentation. They are determined by a Markov additive process whose distribution depends explicitly on the characteristics of the fragmentation. As applications, we make explicit the connexion with multitype branching random walks, and obtain multitype analogs of the pathwise central limit theorem and large deviation estimates for the empirical distribution of fragments. http://front.math.ucdavis.edu/math.PR/0612710 --------------------------------------------------------------- 4749. PATH INTEGRALS ON A COMPACT MANIFOLD WITH NON-NEGATIVE CURVATURE Adrian P.C. Lim A typical path integral on a manifold, $M$ is an informal expression of the form \frac{1}{Z}\int_{\sigma \in H(M)} f(\sigma) e^{-E(\sigma)}\mathcal{D}\sigma, \nonumber where $H(M)$ is a Hilbert manifold of paths with energy $E(\sigma) < \infty$, $f$ is a real valued function on $H(M)$, $\mathcal{D}\sigma$ is a \textquotedblleft Lebesgue measure \textquotedblright and $Z$ is a normalization constant. For a compact Riemannian manifold $M$, we wish to interpret $\mathcal{D}\sigma$ as a Riemannian \textquotedblleft volume form \textquotedblright over $H(M)$, equipped with its natural $G^{1}$ metric. Given an equally spaced partition, ${\mathcal{P}}$ of $[0,1],$ let $H_{{\mathcal{P}}%}(M)$ be the finite dimensional Riemannian submanifold of $H(M) $ consisting of piecewise geodesic paths adapted to $\mathcal{P.}$ Under certain curvature restrictions on $M,$ it is shown that \[ \frac{1}{Z_{{\mathcal{P}}}}e^{-{1/2}E(\sigma)}dVol_{H_{{\mathcal{P}}}% }(\sigma)\to\rho(\sigma)d\nu(\sigma)\text{as}\mathrm{mesh}% ({\mathcal{P}})\to0, \] where $Z_{{\mathcal{P}}}$ is a \textquotedblleft normalization\textquotedblright constant, $E:H(M) \to\lbrack0,\infty) $ is the energy functional, $Vol_{H_{{\mathcal{P}}%}}$ is the Riemannian volume measure on $H_{\mathcal{P}}(M) ,$ $\nu$ is Wiener measure on continuous paths in $M,$ and $\rho$ is a certain density determined by the curvature tensor of $M.$ http://front.math.ucdavis.edu/math.PR/0612711 --------------------------------------------------------------- 4750. RISK BOUNDS FOR THE NON-PARAMETRIC ESTIMATION OF L\'{E}VY PROCESSES Jos\'{e} E. Figueroa-L\'{o}pez and Christian Houdr\'{e} Estimation methods for the L\'{e}vy density of a L\'{e}vy process are developed under mild qualitative assumptions. A classical model selection approach made up of two steps is studied. The first step consists in the selection of a good estimator, from an approximating (finite- dimensional) linear model ${\mathcal{S}}$ for the true L\'{e}vy density. The second is a data-driven selection of a linear model ${\mathcal{S}}$, among a given collection $\{{\mathcal{S}}_m\}_{m\in {\mathcal{M}}}$, that approximately realizes the best trade-off between the error of estimation within ${\mathcal{S}}$ and the error incurred when approximating the true L \'{e}vy density by the linear model ${\mathcal{S}}$. Using recent concentration inequalities for functionals of Poisson integrals, a bound for the risk of estimation is obtained. As a byproduct, oracle inequalities and long-run asymptotics for spline estimators are derived. Even though the resulting underlying statistics are based on continuous time observations of the process, approximations based on high-frequency discrete-data can be easily devised. http://front.math.ucdavis.edu/math.ST/0612697 --------------------------------------------------------------- 4751. REVISITING TWO STRONG APPROXIMATION RESULTS OF DUDLEY AND PHILIPP Philippe Berthet and David M. Mason We demonstrate the strength of a coupling derived from a Gaussian approximation of Zaitsev (1987a) by revisiting two strong approximation results for the empirical process of Dudley and Philipp (1983), and using the coupling to derive extended and refined versions of them. http://front.math.ucdavis.edu/math.ST/0612701 --------------------------------------------------------------- 4752. ON THE BAHADUR SLOPE OF THE LILLIEFORS AND THE CRAM\'{E}R--VON MISES TESTS OF NORMALITY Miguel A. Arcones We find the Bahadur slope of the Lilliefors and Cram\'{e}r--von Mises tests of normality. http://front.math.ucdavis.edu/math.ST/0612708 --------------------------------------------------------------- 4753. IVY ON THE CEILING: FIRST-ORDER POLYMER DEPINNING TRANSITIONS WITH QUENCHED DISORDER Kenneth S. Alexander We consider a polymer, with monomer locations modeled by the trajectory of an underlying Markov chain, in the presence of a potential thatinteracts with the polymer when it visits a particular site 0. Disorder is introduced by having the interaction vary from one monomer to another, as a constant $u$ plus i.i.d. mean-0 randomness. There is a critical value of $u$ above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with high probability. When the excursions of the underlying chain have a finite mean but no finite exponential moment, it is known that the depinning transition (more precisely, the contact fraction) in the corresponding annealed system is discontinuous. One generally expects the presence of disorder to smooth transitions, and it was proved by Giacomin and Toninelli that when the excursion length distribution has power-law tails, the quenched system has a continuous transition even if the annealed system does not. We show here that when the underlying chain is transient but the finite part of the excursion length distribution has exponential tails, then the depinning transition is discontinuous even in the quenched system, and the quenched and annealed critical points are strictly different. By contrast, in the recurrent case, the depinning behavior depends on the subexponential prefactors on the exponential decay of the excursion length distribution, and when these prefactors decay with an appropriate power law, the quenched transition is continuous even though the annealed one is not. http://front.math.ucdavis.edu/math.PR/0612625 --------------------------------------------------------------- 4754. MERGING PERCOLATION ON $Z^D$ AND CLASSICAL RANDOM GRAPHS: PHASE TRANSITION Tatyana S. Turova and Thomas Vallier We study a random graph model which is a superposition of the bond percolation model on $Z^d$ with probability $p$ of an edge, and a classical random graph $G(n, c/n)$. We show that this model, being a {\it homogeneous} random graph, has a natural relation to the so-called "rank 1 case" of {\it inhomogeneous} random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters $c\geq 0$ and $0 \leq p0}is followed closely by the process {F_X(e^x): x>0}, with F_X(t) denoting the point with the most local time for the diffusion at time t. http://front.math.ucdavis.edu/math.PR/0612533 --------------------------------------------------------------- 4766. RANK DISTRIBUTIONS IN SEMIOTICS V. P. Maslov and T. V. Maslova The notions of real and user cardinality of a sign are introduced. Rank distributions can be extended to arbitrary sign objects, i.e., semiotic systems. The dynamics of the distribution of consumer durables, such as automobiles, is studied. http://front.math.ucdavis.edu/math.PR/0612540 --------------------------------------------------------------- 4767. ON EXPONENTIAL ERGODICITY OF MULTICLASS QUEUEING NETWORKS David Gamarnik and Sean Meyn One of the key performance measures in queueing systems is the exponential decay rate of the steady-state tail probabilities of the queue lengths. It is known that if a corresponding fluid model is stable and the stochastic primitives have finite moments, then the queue lengths also have finite moments, so that the tail probability \pr(\cdot >s) decays faster than s^{-n} for any n. It is natural to conjecture that the decay rate is in fact exponential. In this paper an example is constructed to demonstrate that this conjecture is false. For a specific stationary policy applied to a network with exponentially distributed interarrival and service times it is shown that the corresponding fluid limit model is stable, but the tail probability for the buffer length decays slower than s^{-\log s}. http://front.math.ucdavis.edu/math.PR/0612544 --------------------------------------------------------------- 4768. A CONTACT PROCESS WITH MUTATIONS ON A TREE Thomas M. Liggett and Rinaldo B. Schinazi and and Jason Schweinsberg Consider the following stochastic model for immune response. Each pathogen gives birth to a new pathogen at rate $\lambda$. When a new pathogen is born, it has the same type as its parent with probability $1 - r$. With probability $r$, a mutation occurs, and the new pathogen has a different type from all previously observed pathogens. When a new type appears in the population, it survives for an exponential amount of time with mean 1, independently of all the other types. All pathogens of that type are killed simultaneously. Schinazi and Schweinsberg (2006) have shown that this model on $\Z^d$ behaves rather differently from its non-spatial version. In this paper, we show that this model on a homogeneous tree captures features from both the non- spatial version and the $\Z^d$ version. We also obtain comparison results between this model and the basic contact process on general graphs. http://front.math.ucdavis.edu/math.PR/0612564 --------------------------------------------------------------- 4769. A NEW APPROACH FOR CAPACITY ANALYSIS OF LARGE DIMENSIONAL MULTI- ANTENNA CHANNELS Walid Hachem (LTCI) and Oleksiy Khorunzhiy and Philippe Loubaton (IGM-LabInfo), Jamal Najim (LTCI), Leonid Pastur This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to infinity at the same rate. Using a new and simple approach based on Poincar\'{e}-Nash inequality and on an integration by parts formula, it is rigorously established that the mutual information converges to a Gaussian random variable whose mean and variance are evaluated. These results confirm previous evaluations based on the powerful but non rigorous replica method. It is believed that the tools that are used in this paper are simple, robust, and of interest for the communications engineering community. http://front.math.ucdavis.edu/cs.IT/0612076 --------------------------------------------------------------- 4770. A NEW METHOD FOR QUEUING PERFORMANCE ESTIMATES USING MARKOV CHAINS Richard G. Clegg This paper gives an exact closed form solution for the expected queue length at equilibrium of a G/D/1 discrete time queuing system in which the arrival process is a specific Markov-modulated process. A system of equations is given which can calculate the probability that the queue has a given length. The results are tested in simulation. http://front.math.ucdavis.edu/math.PR/0612476 --------------------------------------------------------------- 4771. THE AREA OF EXPONENTIAL RANDOM WALK AND PARTIAL SUMS OF UNIFORM ORDER STATISTICS Vladislav Vysotsky Let S_i be a random walk with standard exponential increments. We call \sum_{i=1}^k S_i its k-step area. The random variable V = \inf_{k \ge 1} \frac{2}{k(k+1)} \sum_{i=1}^k S_i plays important role in the study of so-called one-dimensional sticky particles model. We find the distribution of V and prove that P(V > t) = \sqrt{1-t} exp(-t/2) for t in [0,1]. We also show that the variables \min_{1 \le k \le n} \frac{2n}{k(k+1)} \sum_{i=1} ^k U_{i, n} converge in distribution to V. Here U_{i, n} are the order statistics of n i.i.d. random variables uniformly distributed on [0,1]. http://front.math.ucdavis.edu/math.PR/0612490 --------------------------------------------------------------- 4772. MULTI-STEP RICHARDSON-ROMBERG EXTRAPOLATION: REMARKS ON VARIANCE CONTROL AND COMPLEXITY Gilles Pag\`{e}s (PMA) We propose a multi-step Richardson-Romberg extrapolation method for the computation of expectations $E f(X_{_T})$ of a diffusion $(X_t)_{t\in [0,T]}$ when the weak time discretization error induced by the Euler scheme admits an expansion at an order $R\ge 2$. The complexity of the estimator grows as $R^2$ (instead of $2^R$) and its variance is asymptotically controlled by considering some consistent Brownian increments in the underlying Euler schemes. Some Monte carlo simulations carried with path-dependent options (lookback, barriers) which support the conjecture that their weak time discretization error also admits an expansion (in a different scale). Then an appropriate Richardson-Romberg extrapolation seems to outperform the Euler scheme with Brownian bridge. http://front.math.ucdavis.edu/math.PR/0612523 --------------------------------------------------------------- 4773. CAPITAL ALLOCATION FOR CREDIT PORTFOLIOS WITH KERNEL ESTIMATORS Dirk Tasche Determining contributions by sub-portfolios or single exposures to portfolio-wide economic capital for credit risk is an important risk measurement task. Often economic capital is measured as Value-at-Risk (VaR) of the portfolio loss distribution. For many of the credit portfolio risk models used in practice, then the VaR contributions have to be estimated from Monte Carlo samples. In the context of a partly continuous loss distribution (i.e. continuous except for a positive point mass on zero), we investigate how to combine kernel estimation methods with importance sampling to achieve more efficient (i.e. less volatile) estimation of VaR contributions. http://front.math.ucdavis.edu/math.ST/0612470 --------------------------------------------------------------- 4774. ON THE HYPERPLANE CONJECTURE FOR RANDOM CONVEX SETS Bo'az Klartag and Gady Kozma Let N > n, and denote by K the convex hull of N independent standard gaussian random vectors in an n-dimensional Euclidean space. We prove that with high probability, the isotropic constant of K is bounded by a universal constant. Thus we verify the hyperplane conjecture for the class of gaussian random polytopes. http://front.math.ucdavis.edu/math.MG/0612517 --------------------------------------------------------------- 4775. A NEW REM CONJECTURE Gerard Ben Arous and Veronique Gayrard and Alexey Kuptsov We introduce here a new universality conjecture for levels of random Hamiltonians, in the same spirit as the local REM conjecture made by S. Mertens and H. Bauke. We establish our conjecture for a wide class of Gaussian and non-Gaussian Hamiltonians, which include the $p$-spin models, the Sherrington-Kirkpatrick model and the number partitioning problem. We prove that our universality result is optimal for the last two models by showing when this universality breaks down. http://front.math.ucdavis.edu/math.PR/0612373 --------------------------------------------------------------- 4776. TIGHTNESS FOR A FAMILY OF RECURSIVE EQUATIONS Maury Bramson and Ofer Zeitouni In this paper, we study the tightness of solutions for a family of recursive equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maximal displacement of branching random walk in one dimension, and the cover time of symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings. http://front.math.ucdavis.edu/math.PR/0612382 --------------------------------------------------------------- 4777. JOINT PROBABILITY FOR THE PEARCEY PROCESS Mark Adler & Pierre van Moerbeke This paper is a step in the direction of understanding the behavior of non-intersecting Brownian motions on the real line, when the number of particles becomes large. Consider 2k non-intersecting Brownian motions, all starting at the origin, such that the k left paths end up at -a and the k right paths end up at +a at time t=1. The Karlin-McGregor formula enables one to express the transition probability in terms of a matrix model, consisting of Gaussian Hermitian random matrices in a chain with external source. It is shown that the log of the probability for this model satisfies a fourth order PDE with a quartic non-linearity, obtained by means of the 3-component KP hierarchy and Virasoro constraints. When the number of particles grows very large, the particles will be concentrated on two intervals near t=0 and on one interval near t=1. The Pearcey process is the infinite-dimensional diffusion, near the critical transition from two to one interval. An appropriate scaling limit of the PDE for the finite model leads to a non-linear PDE for the multi-time transition probabilities of the Pearcey process. We conjecture that each of the Markov clouds (like the Pearcey process) arising near phase transitions is related to some integrable system. Moreover, there is an intimate connection between the integrable system and the associated Riemann-Hilbert problem. http://front.math.ucdavis.edu/math.PR/0612393 --------------------------------------------------------------- 4778. ON A DISTRIBUTION IN FREQUENCY PROBABILITY THEORY CORRESPONDING TO THE BOSE-EINSTEIN DISTRIBUTION V. P. Maslov The notion of density of a finite set is discussed. We proof a general theorem of set theory which refines Bose-Einstein distribution. http://front.math.ucdavis.edu/math.PR/0612394 --------------------------------------------------------------- 4779. INFLUENCES AND DECISION TREES Hamed Hatami A celebrated theorem of Friedgut says that every function $f:\{0,1\} ^n \to \{0,1\}$ can be approximated by a function $g:\{0,1\}^n \to \{0,1\}$ with $\|f-g\|_2^2 \le \epsilon$ which depends only on $e^{O(I_f/\epsilon)}$ variables where $I_f$ is the sum of the influences of the variables of $f$. Dinur and Friedgut later showed that this statement also holds if we replace the discrete domain $\{0,1\}^n$ with the continuous domain $[0,1]^n$, under the extra assumption that $f$ is monotone. They conjectured that the condition of monotonicity is unnecessary and can be removed. We show that certain constant-depth decision trees provide counter- examples to Dinur-Friedgut conjecture. This suggests a reformulation of the conjecture in which the function $g:[0,1]^n \to \{0,1\}$ instead of depending on a small number of variables has a decision tree of small depth. In fact we prove this reformulation by showing that the depth of the decision tree of $g$ can be bounded by $e^{O(I_f/\epsilon^2)}$. http://front.math.ucdavis.edu/math.PR/0612405 --------------------------------------------------------------- 4780. DYNAMICAL PROPERTIES AND CHARACTERIZATION OF GRADIENT DRIFT DIFFUSIONS S\'{e}bastien Darses (PMA) and Ivan Nourdin (PMA) We study the dynamical properties of the Brownian diffusions having $ \sigma {\rm Id}$ as diffusion coefficient matrix and $b=\nabla U$ as drift vector. We characterize this class through the equality $D^2_+=D^2_-$, where $D_ {+}$ (resp. $D_-$) denotes the forward (resp. backward) stochastic derivative of Nelson's type. Our proof is based on a remarkable identity for $D_+^2- D_-^2$ and on the use of the martingale problem. We also give a new formulation of a famous theorem of Kolmogorov concerning reversible diffusions. We finally relate our characterization to some questions about the complex stochastic embedding of the Newton equation which initially motivated of this work. http://front.math.ucdavis.edu/math.PR/0612413 --------------------------------------------------------------- 4781. AN L2 THEORY FOR DIFFERENTIAL FORMS ON PATH SPACES I K.D. Elworthy and Xue-Mei Li An L2 theory of differential forms is proposed for the Banach manifold of continuous paths on Riemannian manifolds M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H-tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H-forms, are perturbed by the curvature of M. A Hodge decomposition is given for L2 H-one- forms, and the structure of H-two -forms is described. The dual operator d* is analysed in terms of a natural connection on the H-tangent spaces. Malliavin calculus is a basic tool. http://front.math.ucdavis.edu/math.PR/0612416 --------------------------------------------------------------- 4782. HEAT KERNEL AND GREEN FUNCTION ESTIMATES ON AFFINE BUILDINGS OF TYPE $\TILDE{A}_R$ Jean-Philippe Anker (MAPMO) and Bruno Schapira (MAPMO and PMA) and Bartosz Trojan (MAPMO) We obtain a global estimate of the transition density $p^n(0,x)$ associated to a nearest neighbor random walk, called here "simple", on affine buildings of type $\widetilde{A}_r$. Then we deduce a global estimate of the Green function. This is the analogue of a result on Riemannian symmetric spaces of the noncompact type. http://front.math.ucdavis.edu/math.CA/0612385 --------------------------------------------------------------- 4783. SHARP THRESHOLDS FOR CONSTRAINT SATISFACTION PROBLEM AND GRAPH HOMOMORPHISMS Hamed Hatami and Michael Molloy We determine under which conditions certain natural models of random constraint satisfaction problems have sharp thresholds of satisfiability. These models include graph and hypergraph homomorphism, the $(d,k,t)$- model, and binary constraint satisfaction problems with domain size 3. http://front.math.ucdavis.edu/math.CO/0612391 --------------------------------------------------------------- 4784. ON THE MINIMIZATION OF OPERATIONAL RISKS V. P. Maslov We give a risk-minimizing formula for government investments taking into account the zero intelligence law for financial markets. http://front.math.ucdavis.edu/math.GM/0612395 --------------------------------------------------------------- 4785. CROSSING PROBABILITIES FOR DIFFUSION PROCESSES WITH PIECEWISE CONTINUOUS BOUNDARIES Liqun Wang and Klaus P\"otzelberger We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein-Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear boundaries are obtained, which are useful in evaluation and comparison of various omputational algorithms. Moreover, numerical computation can be easily done by Monte Carlo integration and the approximation errors for general boundaries are automatically calculated. Some numerical examples are presented. http://front.math.ucdavis.edu/math.PR/0612337 --------------------------------------------------------------- 4786. WHAT IS THE NATURAL SCALE FOR A L\'EVY PROCESS IN MODELLING TERM STRUCTURE OF INTEREST RATES? Jir\^o Akahori and Takahiro Tsuchiya This paper gives examples of explicit arbitrage-free term structure models with L\'evy jumps via state price density approach. By generalizing quadratic Gaussian models, it is found that the probability density function of a L\'evy process is a "natural" scale for the process to be the state variable of a market. http://front.math.ucdavis.edu/math.PR/0612341 --------------------------------------------------------------- 4787. ON THE LONGEST INCREASING SUBSEQUENCE FOR FINITE AND COUNTABLE ALPHABETS Christian houdr\'e and Trevis J. Litherland Let $X_1, X_2, ..., X_n, ... $ be a sequence of iid random variables with values in a finite alphabet $\{1,...,m\}$. Let $LI_n$ be the length of the longest increasing subsequence of $X_1, X_2, ..., X_n.$ We express the limiting distribution of $LI_n$ as functionals of $m$ and $(m-1)$-dimensional Brownian motions. These expressions are then related to similar functionals appearing in queueing theory, allowing us to further establish asymptotic behaviors as $m$ grows. The finite alphabet results are then used to treat the countable (infinite) alphabet. http://front.math.ucdavis.edu/math.PR/0612364 --------------------------------------------------------------- 4788. ANTICIPATING REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS Zongxia Liang and Tusheng Zhang In this paper, we establish the existence of the solutions $ (X, L)$ of reflected stochastic differential equations with possible anticipating initial random variables. The key is to obtain some substitution formula for Stratonovich integrals via a uniform convergence of the corresponding Riemann sums. http://front.math.ucdavis.edu/math.PR/0612294 --------------------------------------------------------------- 4789. ON RECURRENCE OF REFLECTED RANDOM WALK ON THE HALF-LINE. WITH AN APPENDIX ON RESULTS OF MARTIN BENDA Marc Peign\'e and Wolfgang Woess Let $(Y_n)$ be a sequence of i.i.d. real valued random variables. Reflected random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1} = |X_n - Y_{n+1}|$. In this note, we study recurrence of this process, extending a previous criterion. This is obtained by determining an invariant measure of the embedded process of reflections. http://front.math.ucdavis.edu/math.PR/0612306 --------------------------------------------------------------- 4790. ON A MODEL FOR THE STORAGE OF FILES ON A HARDWARE II : EVOLUTION OF A TYPICAL DATA BLOCK Vincent Bansaye (PMA) We consider a generalized version in continuous time of the parking problem of Knuth. Files arrive following a Poisson point process and are stored on a hardware identified with the real line, at the right of their arrival point. We study here the evolution of the extremities of the data block straddling 0, which is empty at time 0 and is equal to $\RRR$ at a deterministic time. http://front.math.ucdavis.edu/math.PR/0612312 --------------------------------------------------------------- 4791. FREE-KNOT SPLINE APPROXIMATION OF STOCHASTIC PROCESSES J. Creutzig and T. Mueller-Gronbach and K. Ritter We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the $s$-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average $L_p$-distance to the splines spaces, as the (expected) number $k$ of free knots tends to infinity. http://front.math.ucdavis.edu/math.PR/0612313 --------------------------------------------------------------- 4792. SCALING LIMITS OF BIPARTITE PLANAR MAPS ARE HOMEOMORPHIC TO THE 2-SPHERE J.F. Le Gall and F. Paulin We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic laminations of the disk. http://front.math.ucdavis.edu/math.PR/0612315 --------------------------------------------------------------- 4793. AUTOMORPHISMS OF THE TYPE II_1 ARVESON SYSTEM OF WARREN'S NOISE Boris Tsirelson Motions of the plane (shifts and rotations) correspond to automorphisms of the type I Arveson system of white noise. I prove that automorphisms corresponding to rotations cannot be extended to the type II Arveson system of Warren's noise. http://front.math.ucdavis.edu/math.OA/0612303 --------------------------------------------------------------- 4794. A LARGE CLOSED QUEUEING NETWORK IN MARKOV ENVIRONMENT AND ITS APPLICATION Vyacheslav M. Abramov A paper studies a closed queueing network containing a server station and $k$ client stations. The server station is an infinite server queueing system, and client stations are single server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. The total number of units in the network is $N$. The expected times between departures in client stations are $(N\mu_j)^{-1}$. After service completion in the server station a unit is transmitted to the $j$th client station with probability $p_{j}$ $(j=1,2,...,k)$, and being processed in the $j$th client station the unit returns to server station. The network is assumed to be in Markov environment. The Markov environment is defined by initial state, and phase space of dimension $d$. Then the routing matrix $p_{j}$ as well as transmission rates (which are expressed via parameters of the network) depend on the Markov state of the environment. The paper studies the queue- length processes in client stations of this network, and is aimed to analysis of performance measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex telecommunication networks. http://front.math.ucdavis.edu/math.PR/0612224 --------------------------------------------------------------- 4795. A FUNCTIONAL LIMIT THEOREM FOR THE POSITION OF A PARTICLE IN A LORENTZ TYPE MODEL Vladislav Vysotsky Consider a particle moving through a random medium, which consists of spherical obstacles, randomly distributed in R^d. The particle is accelerated by a constant external field; when colliding with an obstacle, the particle inelastically reflects. We study the asymptotics of X(t), which denotes the position of the particle at time t, as t tends to infinity. The result is a functional limit theorem for X(t). http://front.math.ucdavis.edu/math.PR/0612253 --------------------------------------------------------------- 4796. THE POISSON BOUNDARY OF TRIANGULAR MATRICES IN A NUMBER FIELD Bruno Schapira (MAPMO and PMA) The aim of this note is to describe the Poisson boundary of the group of invertible triangular matrices with coefficients in a number field. It generalizes to any dimension and to any number field a result of Brofferio \cite{Bro} concerning the Poisson boundary of random rational affinities. http://front.math.ucdavis.edu/math.PR/0612272 --------------------------------------------------------------- 4797. ERROR STRUCTURES AND PARAMETER ESTIMATION Nicolas Bouleau (CERMICS) and Christophe Chorro (CERMICS and CERMSEM) This article proposes a link between statistics and the theory of Dirichlet forms used to compute errors. The error calculus based on Dirichlet forms is an extension of classical Gauss' approach to error propagation. The aim of this paper is to derive error structures from measurements. The links with Fisher's information lay the foundations of a strong connection with experiment. We show that this connection behaves well towards changes of variables and is related to the theory of asymptotic statistics. http://front.math.ucdavis.edu/math.ST/0612258 --------------------------------------------------------------- 4798. ON MIXING AND ERGODICITY IN LOCALLY COMPACT MOTION GROUPS M. Anoussis and D. Gatzouras Let $G$ be a semi-direct product $G=A\times_\phi K$ with $A$ Abelian and $K$ compact. We characterize spread-out probability measures on $G$ that are mixing by convolutions by means of their Fourier transforms. A key tool is a spectral radius formula for the Fourier transform of a regular Borel measure on $G$ that we develop, and which is analogous to the well-known Beurling-- Gelfand spectral radius formula. For spread-out probability measures on $G$, we also characterize ergodicity by means of the Fourier transform of the measure. Finally, we show that spread-out probability measures on such groups are mixing if and only if they are weakly mixing. http://front.math.ucdavis.edu/math.FA/0612262 --------------------------------------------------------------- 4799. LIMIT THEOREMS FOR FREE MULTIPLICATIVE CONVOLUTIONS Hari Bercovici and Jiun-Chau Wang We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. In the case of positive variables, the main theorem extends a result proved earlier for arrays with identically distributed rows. The case of unitary variables is considered as well. http://front.math.ucdavis.edu/math.OA/0612278 --------------------------------------------------------------- 4800. THE SIMILARITY METRIC Ming Li (Univ. of Waterloo and BioInformatics Solutions Inc.) and Xin Chen (Univ. California, Santa Barbara), Xin Li (Univ. Western Ontario), Bin Ma (Univ. Western Ontario), Paul Vitanyi (CWI and Univ. of Amsterdam) A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new ``normalized information distance'', based on the noncomputable notion of Kolmogorov complexity, and show that it is in this class and it minorizes every computable distance in the class (that is, it is universal in that it discovers all computable similarities). We demonstrate that it is a metric and call it the {\em similarity metric}. This theory forms the foundation for a new practical tool. To evidence generality and robustness we give two distinctive applications in widely divergent areas using standard compression programs like gzip and GenCompress. First, we compare whole mitochondrial genomes and infer their evolutionary history. This results in a first completely automatic computed whole mitochondrial phylogeny tree. Secondly, we fully automatically compute the language tree of 52 different languages. http://front.math.ucdavis.edu/cs.CC/0111054 --------------------------------------------------------------- 4801. A NEW QUARTET TREE HEURISTIC FOR HIERARCHICAL CLUSTERING Rudi Cilibrasi and Paul M.B. Vitanyi We consider the problem of constructing an an optimal-weight tree from the 3*(n choose 4) weighted quartet topologies on n objects, where optimality means that the summed weight of the embedded quartet topologiesis optimal (so it can be the case that the optimal tree embeds all quartets as non-optimal topologies). We present a heuristic for reconstructing the optimal- weight tree, and a canonical manner to derive the quartet-topology weights from a given distance matrix. The method repeatedly transforms a bifurcating tree, with all objects involved as leaves, achieving a monotonic approximation to the exact single globally optimal tree. This contrasts to other heuristic search methods from biological phylogeny, like DNAML or quartet puzzling, which, repeatedly, incrementally construct a solution from a random order of objects, and subsequently add agreement values. http://front.math.ucdavis.edu/cs.DS/0606048 --------------------------------------------------------------- 4802. TWO-PLAYER KNOCK 'EM DOWN James Allen Fill and David B. Wilson We analyze the two-player game of Knock 'em Down, asymptotically as the number of tokens to be knocked down becomes large. Optimal play requires mixed strategies with deviations of order sqrt(n) from the naive law-of- large numbers allocation. Upon rescaling by sqrt(n) and sending n to infinity, we show that optimal play's random deviations always have bounded support and have marginal distributions that are absolutely continuous with respect to Lebesgue measure. http://front.math.ucdavis.edu/math.PR/0612205 --------------------------------------------------------------- 4803. SINAI'S WALK: A STATISTICAL ASPECT Pierre Andreoletti (MAPMO) We consider Sinai's random walk in random environment. We prove that the logarithm of the local time is a good estimator of the random potential associated to the random environment. We give a constructive method allowing us to built the random environment from a single trajectory of the random walk. http://front.math.ucdavis.edu/math.PR/0612209 --------------------------------------------------------------- 4804. A FILTERING APPROACH TO TRACKING VOLATILITY FROM PRICES OBSERVED AT RANDOM TIMES Jak\v{s}a Cvitani\'{c} and Robert Liptser and Boris Rozovskii This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process $S=(S_{t})_{t\geq0}$ is given by \[ dS_{t}=m(\theta_{t})S_{t} dt+v(\theta_{t})S_{t} dB_{t}, \] where $B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function and $\theta=(\theta_{t})_{t\geq0}$ is a c\'{a}dl\'{a}g strong Markov process. The random process $\theta$ is unobservable. We assume also that the asset price $S_{t}$ is observed only at random times $0<\tau_{1}<\tau_{2}<....$ This is an appropriate assumption when modeling high frequency financial data (e.g., tick-by-tick stock prices). In the above setting the problem of estimation of $\theta$ can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process $(\tau_{k},\log S_{\tau_{k}})$. While quite natural, this problem does not fit into the ``standard'' diffusion or simple point process filtering frameworks and requires more technical tools. We derive a closed form optimal recursive Bayesian filter for $\theta_{t}$, based on the observations of $(\tau_{k},\log S_{\tau_{k}})_{k\geq1}$. It turns out that the filter is given by a recursive system that involves only deterministic Kolmogorov-type equations, which should make the numerical implementation relatively easy. http://front.math.ucdavis.edu/math.PR/0612212 --------------------------------------------------------------- 4805. A SLOW TRANSIENT DIFFUSION IN A DRIFTED STABLE POTENTIAL Arvind Singh (PMA) We consider a diffusion process $X$ in a random potential $\V$ of the form $\V_x = \S_x -\delta x$ where $\delta$ is a positive drift and $\S$ is a strictly stable process of index $\alpha\in (1,2)$ with positive jumps. Then the diffusion is transient and $X_t / \log^\alpha t$ converges in law towards an exponential distribution. This behaviour contrasts with the case where $\V$ is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as "slow" as in the recurrent setting. http://front.math.ucdavis.edu/math.PR/0612220 --------------------------------------------------------------- 4806. DUALITY AND EXACT CORRELATIONS FOR A MODEL OF HEAT CONDUCTION C. Giardin\'a and J. Kurchan and F. Redig We study a model of heat conduction with stochastic diffusion of energy. We obtain a dual particle process which describes the evolution of all the correlation functions. An exact expression for the covariance of the energy exhibits long-range correlations in the presence of a current. We discuss the formal connection of this model with the simple symmetric exclusion process. http://front.math.ucdavis.edu/cond-mat/0612198 --------------------------------------------------------------- 4807. BESSEL POTENTIALS, HITTING DISTRIBUTIONS AND GREEN FUNCTIONS T. Byczkowski and M. Ryznar and J. Malecki The purpose of this paper is to find explicit formulas for basic objects pertaining the local potential theory of the operator $(I-\Delta)^ {\alpha/2}$, $0<\alpha<2$. The potential theory of this operator is based on Bessel potentials $J_{\alpha}=(I-\Delta)^{-\alpha/2}$. We compute the {\it harmonic measure} of the half-space and write a concise form of the corresponding {\it Green function} for the operator $(I-\Delta)^{\alpha/2}$. To achieve this we analyze the so-called {\it relativistic $\alpha$-stable process} on $ \R^d$ space, killed when exiting the half-space. In terms of this process we are dealing here with the 1-{\it potential theory} or, equivalently, potential theory of Schr{\"o}dinger operator based on the generator of the process with Kato's potential $q=-1$. http://front.math.ucdavis.edu/math.PR/0612176 --------------------------------------------------------------- 4808. THE LIMITING SPECTRA OF GIRKO'S BLOCK-MATRIX Tamer Oraby To analyze the limiting spectral distribution of some random block- matrices, Girko [Girko, 2000] uses a system of canonical equations from [Girko, 98]. In this paper, we use the method of moments to give an integral form for the almost sure limiting spectral distribution of such matrices. http://front.math.ucdavis.edu/math.PR/0612177 --------------------------------------------------------------- 4809. UTILITY MAXIMIZATION IN A JUMP MARKET MODEL Marie-Amelie Morlais In this paper, we consider the classical problem of utility maximization in a financial market allowing jumps. Assuming that the constraint set is a compact set, rather than a convex one, we use a dynamic method from which we derive a specific BSDE. We then aim at showing existence and uniqueness results for the introduced BSDE. This allows us to give an explicit expression of the value function and characterize optimal strategies for our problem. http://front.math.ucdavis.edu/math.PR/0612181 --------------------------------------------------------------- 4810. AN ARITHMETIC MODEL FOR THE TOTAL DISORDER PROCESS C. P. Hughes and A. Nikeghbali and M. Yor We prove a multidimensional extension of Selberg's central limit theorem for the logarithm of the Riemann zeta function on the critical line. The limit is a totally disordered process, whose coordinates are all independent and Gaussian. http://front.math.ucdavis.edu/math.PR/0612195 --------------------------------------------------------------- 4811. BROWNIAN SUPER-EXPONENTS Victor Goodman (Indiana University) We introduce a transform on the class of stochastic exponentials for d-dimensional Brownian motions. Each stochastic exponential generates another stochastic exponential under the transform. The new exponential process is often merely a supermartingale even in cases where the original process is a martingale. We determine a necessary and sufficient condition for the transform to be a martingale process. The condition links expected values of the transformed stochastic exponential to the distribution function of certain time-integrals. http://front.math.ucdavis.edu/math.PR/0612160 --------------------------------------------------------------- 4812. APPROCHE INTRINS\`{E}QUE DES FLUCTUATIONS QUANTIQUES EN M\'{E} CANIQUE STOCHASTIQUE (AN INTRINSIC APPROACH OF THE QUANTUM FLUCTUATIONS IN STOCHASTIC MECHANICS) Michel Fliess (INRIA Futurs) This note is answering an old questioning about the F\'{e}nyes-Nelson stochastic mechanics. The Brownian nature of the quantum fluctuations, which are associated to this mechanics, is deduced from Feynman's interpretation of the Heisenberg uncertainty principle via infinitesimal random walks stemming from nonstandard analysis. It is therefore no more necessary to combine those fluctuations with a background field, which has never been well understood. Most of the technical details are contained in an extended english abstract. http://front.math.ucdavis.edu/quant-ph/0612033 --------------------------------------------------------------- 4813. MARKOV LOOPS, DETERMINANTS AND GAUSSIAN FIELDS Yves Le Jan (LM-Orsay) The purpose of this note is to explore some simple relations between loop measures, determinants, and Gaussian Markov fields. http://front.math.ucdavis.edu/math.PR/0612112 --------------------------------------------------------------- 4814. SQUARE SUMMABILITY OF VARIATIONS AND CONVERGENCE OF THE TRANSFER OPERATOR Anders Johansson and Anders \"Oberg In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [13], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in [7]. We also prove uniqueness of so-called G-measures, introduced by Brown and Dooley [2], under square summability of variations. http://front.math.ucdavis.edu/math.DS/0612131 --------------------------------------------------------------- 4815. COMPUTABLE EXPONENTIAL BOUNDS FOR SCREENED ESTIMATION AND SIMULATION I. Kontoyiannis and S.P. Meyn Suppose the expectation E(F(X)) is to be estimated by the empirical averages of the values of F on independent and identically distributed samples {X_i}. A sampling rule called the ``screened'' estimator is introduced, and its performance is studied. When the mean E(U(X)) of a different function U is known, the estimates are ``screened,'' in that we only consider those which correspond to times when the empirical average of the {U(X_i)} is sufficiently close to its known mean. As long as U dominates F appropriately, the screened estimates admit exponential error bounds, even when F(X) is heavy- tailed. The main results are several nonasymptotic, explicit exponential bounds for the screened estimates. A geometric interpretation, in the spirit of Sanov's theorem, is given for the fact that the screened estimates always admit exponential error bounds, even if the standard estimates do not. And when they do, the screened estimates' error probability has a significantly better exponent. This implies that screening can be interpreted as a variance reduction technique. Our main mathematical tools come from large deviations techniques. The results are illustrated by a detailed simulation example. http://front.math.ucdavis.edu/math.PR/0612040 --------------------------------------------------------------- 4816. ON THE SUBMODULARITY OF INFLUENCE IN SOCIAL NETWORKS Elchanan Mossel and Sebastien Roch We prove and extend a conjecture of Kempe, Kleinberg, and Tardos (KKT) on the spread of influence in social networks. A social network can be represented by a directed graph where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or ``word-of-mouth'' effects on such a graph is to consider an increasing process of ``infected'' (or active) nodes: each node becomes infected once an activation function of the set of its infected neighbors crosses a certain threshold value. Such a model was introduced by KKT in \cite{KeKlTa:03,KeKlTa:05} where the authors also impose several natural assumptions: the threshold values are (uniformly) random; and the activation functions are monotone and submodular. For an initial set of active nodes $S$, let $\sigma(S)$ denote the expected number of active nodes at termination. Here we prove a conjecture of KKT: we show that the function $\sigma(S)$ is submodular under the assumptions above. We prove the same result for the expected value of any monotone, submodular function of the set of active nodes at termination. http://front.math.ucdavis.edu/math.PR/0612046 --------------------------------------------------------------- 4817. ATTRACTION TIME FOR STRONGLY REINFORCED WALKS C. Cotar and V. Limic We consider a class of strongly edge reinforced random walks, where the corresponding reinforcement weight function is non-decreasing. It is known by Limic and Tarr\`es (2006) that the attracting edge emerges with probability 1, whenever the underlying graph is locally bounded. We study the asymptotic behavior of the tail distribution of the (random) time of attraction. In particular, we obtain exact (up to multiplicative constant) asymptotics if the underlying graph has two edges. Next we show some extensions in the setting of finite and bounded degree infinite graphs. A nice corollary is that if the reinforcement weight has the form $W(k) = k^\rho$, $\rho>1$, then (universally over finite graphs) the expected time to attraction is infinite if and only if $\rho \leq 1+ \frac{1+\sqrt{5}}{2}$. http://front.math.ucdavis.edu/math.PR/0612048 --------------------------------------------------------------- 4818. CONVERGENCE OF SEQUENTIAL MARKOV CHAIN MONTE CARLO METHODS: I. NONLINEAR FLOW OF PROBABILITY MEASURES Andreas Eberle and Carlo Marinelli Sequential Monte Carlo Samplers are a class of stochastic algorithms for Monte Carlo integral estimation w.r.t. probability distributions, which combine elements of Markov chain Monte Carlo methods and importance sampling/ resampling schemes. We develop a stability analysis by functional inequalities for a nonlinear flow of probability measures describing the limit behavior of the algorithms as the number of particles tends to infinity. Stability results are derived both under global and local assumptions on the generator of the underlying Metropolis dynamics. This allows us to prove that the combined methods sometimes have good asymptotic stability properties in multimodal setups where traditional MCMC methods mix extremely slowly. For example, this holds for the mean field Ising model at all temperatures. http://front.math.ucdavis.edu/math.PR/0612074 --------------------------------------------------------------- 4819. OPTION PRICING WITHOUT PRICE DYNAMICS: A PROBABILISTIC APPROACH Dimitris Bertsimas and Natasha Bushueva Employing probabilistic techniques we compute best possible upper and lower bounds on the price of an option on one or two assets with continuous piecewise linear payoff function based on prices of simple call options of possibly distinct maturities and the no-arbitrage condition, but without any assumption on the price dynamics of underlying assets. We show that the problem reduces to solving linear optimization problems that we explicitly characterize. We report numerical results that illustrate the effectiveness of the algorithms we develop. http://front.math.ucdavis.edu/math.PR/0612075 --------------------------------------------------------------- 4820. A SINGULAR PERTURBATION APPROACH FOR CHOOSING PAGERANK DAMPING FACTOR Konstantin Avrachenkov and Nelly Litvak and Kim Son Pham The choice of the PageRank damping factor is not evident. The Google's choice for the value c=0.85 was a compromise between the true reflection of the Web structure and numerical efficiency. However, the Markov random walk on the original Web Graph does not reflect the importance of the pages because it absorbs in dead ends. Thus, the damping factor is needed not only for speeding up the computations but also for establishing a fair ranking of pages. In this paper, we propose new criteria for choosing the damping factor, based on the ergodic structure of the Web Graph and probability flows. Specifically, we require that the core component receives a fair share of the PageRank mass. Using singular perturbation approach we conclude that the value c=0.85 is too high and suggest that the damping factor should be chosen around 1/2. As a by-product, we describe the ergodic structure of the OUT component of the Web Graph in detail. Our analytical results are confirmed by experiments on two large samples of the Web Graph. http://front.math.ucdavis.edu/math.PR/0612079 --------------------------------------------------------------- 4821. HYDRODYNAMICS AND HYDROSTATICS FOR A CLASS OF ASYMMETRIC PARTICLE SYSTEMS WITH OPEN BOUNDARIES Christophe Bahadoran We consider asymmetric attractive particle systems with product invariant measures in any space dimension. We show that, in the presence of open boundaries, the hydrodynamic limit is a scalar conservation law with boundary conditions in the sense defined by Bardos, Leroux and N\'{e}d\'{e} lec. When the boundaries are parallel hyperplanes, we establish a large-time convergence result for the entropy solution and derive the stationary profile for the particle system. Models include current-density relations with arbitrarily many maxima and minima. http://front.math.ucdavis.edu/math.PR/0612094 --------------------------------------------------------------- 4822. THE PAVING PROPERTY FOR UNIFORMLY BOUNDED MATRICES: A NEW PROOF Joel A. Tropp This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison--Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systematic use of symmetrization and Khintchine inequalities to estimate the norm of some random matrices. http://front.math.ucdavis.edu/math.MG/0612070 --------------------------------------------------------------- 4823. RELATIVISTIC DIFFUSION IN G\"ODEL'S UNIVERSE Jacques Franchi K. G\"odel [G] discovered his celebrated solution to Einstein equations in 1949. Additional contributions were made by Kundt [K] and Hawking-Ellis ([H-E],5.7). On the other hand, a general Lorentz invariant operator, associated to the so-called "relativistic diffusion'', and making sense in any Lorentz manifold, was introduced by Franchi-Le Jan in [F-LJ]. Here is purposed a first study of the relativistic diffusion in the framework of G \"odel's universe, which contains matter. http://front.math.ucdavis.edu/math.PR/0612020 --------------------------------------------------------------- 4824. MODIFIED LOGARITHMIC SOBOLEV INEQUALITIES ON R Franck Barthe (LSProba) and Cyril Roberto (LAMA) We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and G\"{o}tze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo. http://front.math.ucdavis.edu/math.PR/0612026 --------------------------------------------------------------- 4825. EXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION Victor Goodman and Kyounghee Kim We find a simple expression for the probability density of $\int \exp (B_s - s/2) ds$ in terms of its distribution function and the distribution function for the time integral of $\exp (B_s + s/2)$. The relation is obtained with a change of measure argument where expectations over events determined by the time integral are replaced by expectations over the entire probability space. We develop precise information concerning the lower tail probabilities for these random variables as well as for time integrals of geometric Brownian motion with arbitrary constant drift. In particular, $E[ \exp\big (\theta / \int \exp (B_s)ds\big) ]$ is finite iff $\theta < 2$. We present a new formula for the price of an Asian call option. http://front.math.ucdavis.edu/math.PR/0612034 --------------------------------------------------------------- 4826. ONE-FACTOR TERM STRUCTURE WITHOUT FORWARD RATES Victor Goodman and Kyounghee Kim We construct a no-arbitrage model of bond prices where the long bond is used as a numeraire. We develop bond prices and their dynamics without developing any model for the spot rate or forward rates. The model is arbitrage free and all nominal interest rates remain positive in the model. We give examples where our model does not have a spot rate; other examples include both spot and forward rates. http://front.math.ucdavis.edu/math.PR/0612035 --------------------------------------------------------------- 4827. CONFORMAL BOUNDARY LOOP MODELS Jesper Lykke Jacobsen (LPTMS and SPhT) and Hubert Saleur (SPhT) We study a model of densely packed self-avoiding loops on the annulus, related to the Temperley Lieb algebra with an extra idempotent boundary generator. Four different weights are given to the loops, depending on their homotopy class and whether they touch the outer rim of the annulus. When the weight of a contractible bulk loop x = q + 1/q satisfies -2 < x <= 2, this model is conformally invariant for any real weight of the remaining three parameters. We classify the conformal boundary conditions and give exact expressions for the corresponding boundary scaling dimensions. The amplitudes with which the sectors with any prescribed number and types of non contractible loops appear in the full partition function Z are computed rigorously. Based on this, we write a number of identities involving Z which hold true for any finite size. When the weight of a contractible boundary loop y takes certain discrete values, y_r = [r+1]_q / [r]_q with r integer, other identities involving the standard characters K_{r,s} of the Virasoro algebra are established. The connection with Dirichlet and Neumann boundary conditions in the O(n) model is discussed in detail, and new scaling dimensions are derived. When q is a root of unity and y = y_r, exact connections with the A_m type RSOS model are made. These involve precise relations between the spectra of the loop and RSOS model transfer matrices, valid in finite size. Finally, the results where y=y_r are related to the theory of Temperley Lieb cabling. http://front.math.ucdavis.edu/math-ph/0611078 --------------------------------------------------------------- 4828. MARKOV CHAIN APPROXIMATIONS FOR SYMMETRIC JUMP PROCESSES R. Husseini and M. Kassmann Markov chain approximations of symmetric jump processes are investigated. Tightness results and a central limit theorem are established. Moreover, given the generator of a symmetric jump process with state space $\mathbbm {R}^d$ the approximating Markov chains are constructed explicitly. As a byproduct we obtain a definition of the Sobolev space $H^{\alpha/2}(\mathbbm{R}^d) $, $\alpha \in (0,2)$, that is equivalent to the standard one. http://front.math.ucdavis.edu/math.PR/0611934 --------------------------------------------------------------- 4829. DOES WASTE-RECYCLING REALLY IMPROVE METROPOLIS-HASTINGS MONTE CARLO ALGORITHM? Jean-Fran\c{c}ois Delmas (CERMICS) and Benjamin Jourdain (CERMICS) The waste-recycling Monte Carlo (WR) algorithm, introduced by Frenkel, is a modification of the Metropolis-Hastings algorithm, which makes use of all the proposals, whereas the standard Metropolis-Hastings algorithm only uses the accepted proposals. We prove the convergence of the WR algorithm and its asymptotic normality. We give an example which shows that in general the WR algorithm is not asymptotically better than the Metropolis-Hastings algorithm : the WR algorithm can have an asymptotic variance larger than the one of the Metropolis-Hastings algorithm. However, in the particular case of the Metropolis-Hastings algorithm called Boltzmann algorithm, we prove that the WR algorithm is asymptotically better than the Metropolis-Hastings algorithm. http://front.math.ucdavis.edu/math.PR/0611949 --------------------------------------------------------------- 4830. SPARSITY AND INCOHERENCE IN COMPRESSIVE SAMPLING Emmanuel Candes and Justin Romberg We consider the problem of reconstructing a sparse signal $x^0\in\R^n $ from a limited number of linear measurements. Given $m$ randomly selected samples of $U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimization recovers $x^0$ exactly when the number of measurements exceeds \[ m\geq \mathrm{Const}\cdot\mu^2(U)\cdot S\cdot\log n, \] where $S$ is the number of nonzero components in $x^0$, and $\mu$ is the largest entry in $U$ properly normalized: $\mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The smaller $\mu$, the fewer samples needed. The result holds for ``most'' sparse signals $x^0$ supported on a fixed (but arbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero entry on $T$ and the observed values of $Ux^0$ are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples. http://front.math.ucdavis.edu/math.ST/0611957 --------------------------------------------------------------- 4831. GRAVITATIONAL ALLOCATION TO POISSON POINTS Sourav Chatterjee and Ron Peled and Yuval Peres and Dan Romik For d>=3, we construct a non-randomized, fair and translation- equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R^d, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the "allocation diameter", defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail bound: P(X > R) < C exp[ -c R(log R)^(alpha_d) ], for all R>2, where: alpha_d = (d-2)/d for d>=4; alpha_3 can be taken as any number <-4/3; and C,c are positive constants that depend on d and alpha_d. This is the first construction of an allocation rule of Lebesgue measure to a Poisson point process with subpolynomial decay of the tail P(X>R). http://front.math.ucdavis.edu/math.PR/0611886 --------------------------------------------------------------- 4832. AN EXTENSION OF THE LEVY CHARACTERIZATION TO FRACTIONAL BROWNIAN MOTION Yulia Mishura and Esko Valkeila We extend the classical Levy characterization of Brownian motion to fractional Brownian motion. http://front.math.ucdavis.edu/math.PR/0611913 --------------------------------------------------------------- 4833. A FUNCTIONAL NON-CENTRAL LIMIT THEOREM FOR JUMP-DIFFUSIONS WITH PERIODIC COEFFICIENTS DRIVEN BY STABLE LEVY-NOISE Brice Franke We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by strictly stable Levy-processes with stability index bigger than one. The limit process turns out to be a strictly stable Levy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion, there is no drift related enhancement of diffusivity. http://front.math.ucdavis.edu/math.PR/0611852 --------------------------------------------------------------- 4834. HOW DO RANDOM FIBONACCI SEQUENCES GROW? Elise Janvresse (LMRS) and Beno\^{i}t Rittaud (IG) and Thierry De La Rue (LMRS) We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1 $ and for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0