Random pinning model with finite range correlations: Disorder relevant regime

The purpose of this paper is to show how one can extend some results on disorder relevance obtained for the random pinning model with i.i.d disorder to the model with finite range correlated disorder. In a previous work, the annealed critical curve of the latter model was computed, and equality of quenched and annealed critical points, as well as exponents, was proved under some conditions on the return exponent of the interarrival times. Here we complete this work by looking at the disorder relevant regime, where annealed and quenched critical points differ. All these results show that the Harris criterion, which was proved to be correct in the i.i.d case, remains valid in our setup. We strongly use Markov renewal constructions that were introduced in the solving of the annealed model.     

随机矩阵乘积的尾概率

在一定条件下,本文给出了一列正随机矩阵乘积的尾概率估计,它以指数的速度消失;然后,在一维的情形,基于更新过程的残差等待时间的拉普拉斯变换,建立了极限常数的两种不同形式表达式之间的联系.     

Sanov results for Glauber spin-glass dynamics

Summary. In this paper we prove a Sanov result, i.e. a Large Deviation Principle (LDP) for the distribution of the empirical measure, for the annealed Glauber dynamics of the Sherrington-Kirkpatrick spin-glass. Without restrictions on time or temperature we prove a full LDP for the asymmetric dynamics and the crucial upper large deviations bound for the symmetric dynamics. In the symmetric model a new order-parameter arises corresponding to the response function in [SoZi83]. In the asymmetric case we show that the corresponding rate function has a unique minimum, given as the solution of a self-consistent equation. The key argument used in the proofs is a general result for mixing of what is known as Large Deviation Systems (LDS) with measures obeying an independent LDP. Received: 18 May 1995 / In revised form: 14 March 1996     

The limit theorems for random walk with state space ℝ in a space-time random environment

We consider a discrete time random environment. We state that when the random walk on real number space in a environment is i.i.d., under the law, the law of large numbers, iterated law and CLT of the process are correct space-time random marginal annealed Using a martingale approach, we also state an a.s. invariance principle for random walks in general random environment whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.     

The relation between quenched and annealed Lyapunov exponents in random potential on trees

Our subject of interest is a simple symmetric random walk on the integers which faces a random risk to be killed. This risk is described by random potentials, which are defined by a sequence of independent and identically distributed non-negative random variables. To determine the risk of taking a walk in these potentials we consider the decay of the Green function. There are two possible tools to describe this decay: The quenched Lyapunov exponent and the annealed Lyapunov exponent. It turns out that on the integers and on regular trees we can state a precise relation between these two.     

Quenched point-to-point free energy for random walks in random potentials

We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and steps of the walk. The potential is subject to a moment assumption whose strictness is tied to the mixing of the environment, the best case being the i.i.d. environment. We prove that the infinite volume quenched point-to-point free energy exists and has a variational formula in terms of entropy. We establish regularity properties of the point-to-point free energy, and link it to the infinite volume point-to-line free energy and quenched large deviations of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function.     

Conditional large deviations for a sequence of words

Cut an i.i.d. sequence (_Xi)$\left(X_i\right)$ of ‘letters’ into ‘words’ according to an independent renewal process. Then one obtains an i.i.d. sequence of words, and thus the level 3 large deviation behaviour of this sequence of words is governed by the specific relative entropy. We consider the corresponding problem for the conditional   empirical process of words, where one conditions on a typical underlying (_Xi)$\left(X_i\right)$. We find that if the tails of the word lengths decay exponentially, the large deviations under the conditional distribution are almost surely again governed by the specific relative entropy, but the set of attainable limits is restricted.     

On relative stability and weighted laws of large numbers

Weighted laws of large numbers are established for components which are independent copies of a positive relatively stable law and the weights comprise a regularly varying sequence. The index of regular variation of the weights must be at least ?1 for a weak law and be exactly ?1 for a strong law. Consideration is given to the special case where the truncated moment function is proportional to the logarithm, a case arising from quotients of independent random variables and quotients of successive order statistics. Closure of the class of positive relatively stable laws under independent multiplication (i.e, Mellin convolution) is reprised and tail equivalences are extended.     

Tail bounds for the distribution of the deficit in the renewal risk model

We obtain upper and lower bounds for the tail of the deficit at ruin in the renewal risk model, which are (i) applicable generally; and (ii) based on reliability classifications. We also derive two-side bounds, in the general case where a function satisfies a defective renewal equation, and we apply them to the renewal model, using the function Λ_u introduced by [Psarrakos, G., Politis, K., 2007. A generalisation of the Lundberg condition in the Sparre Andersen model and some applications (submitted for publication)]. Finally, we construct an upper bound for the integrated function and an asymptotic result when the adjustment coefficient exists.     

Asymptotics of Subexponential Max Plus Networks: the Stochastic Event Graph Case

We calculate the exact tail asymptotics of stationary response times for open stochastic event graphs, in the irreducible and reducible cases. These networks admit a representation as (max,?plus)-linear systems in a random medium. We study the case of renewal input and i.i.d. service times with subexponential distributions. We show that the stationary response times have tail asymptotics of the same order as the integrated tail of service times. The mutiplicative constants only involve the intensity of the arrival process and the (max,?plus)-Lyapunov exponents of the sequence of (max,?plus)-matrices.     

Large deviations for random dynamical systems and applications to hidden Markov models

In this paper, we prove the large deviation principle (LDP) for the occupation measures of not necessarily irreducible random dynamical systems driven by Markov processes. The LDP for not necessarily irreducible dynamical systems driven by i.i.d. sequence is derived. As a further application we establish the LDP for extended hidden Markov models, filling a gap in the literature, and obtain large deviation estimations for the log-likelihood process and maximum likelihood estimator of hidden Markov models.     

Large deviations for Langevin spin glass dynamics

Summary We study the asymptotic behaviour of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime. We study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureQ which is not markovian. We deduce that the quenched law of the empirical measure converges to _Q . Extending then the preceeding results to replicated dynamics, we investigate the quenched behavior of a single spin. We get quenched convergence toQ in the case of a symmetric initial law and even potential for the free spin.     

General study of the distribution of N-tuples of letters or words based on the distributions of the single letters or words

This paper establishes the general relation between the distribution of N-tuples of letters (e.g., N-truncations, N-grams) or words (e.g., N-word phrases) and the distributions of the single letters or words. Here the very general case is treated: the case where there is dependence on the place i in the N-tuple (i = 1,…, N) in the sense that, for each i = 1,…, N, a different distribution of the letters or words is supposed.Concrete calculations are performed in the important case of Zipfian distributions (i.e., power laws) for the single letters or words. In this case, we prove that the distribution of the N-tuples (N-fixed) is the sum of power laws.     

Tsallis statistics and turbulence

In order to reveal the underlying statistics describing properly the fully developed turbulence, the probability density function of the local dissipation is derived by taking extremal of a generalized entropy (Tsallis entropy) under the two constraints, i.e., one is the normalization of probability and the other is to fix the intermittency exponent being constant. The generalized entropy includes the Boltzmann–Gibbs entropy as a special case where the Tsallis index q is equal to 1. The multifractal spectrum f(α) corresponding to the probability density function is determined self-consistently in the sense that all quantities can be determined by the observed value of the intermittency exponent. It is shown that the scaling exponents ζ_m of velocity structure function derived by making use of f(α) explains experimental data very well. It is also revealed that the asymptotic expression of ζ_m for m≫1 has a log term. The Tsallis index q turns out to be 0.380 which manifests itself that the system of fully developed turbulence has a nonextensive character.     

The statistics of words on rings

We analyze sequences of letters on a ring. Our objective is to determine the statistics of the occurrences of a set of r‐letter words when the sequence is chosen as a periodic Markov chain of order ≤ r ? 1. We first obtain a generating function for the associated probability distribution and then display its Poisson limit. For an i.i.d. letter sequence, correction terms to the Poisson limit are given. Finally, we indicate how a hidden Markov chain fits into this scheme. © 2005 Wiley Periodicals, Inc.     

Precise large deviation estimates for a one-dimensional random walk in a random environment

ω_x } (taking values in the interval [1/2, 1)), which serve as an environment. This environment defines a random walk {X _k } (called a RWRE) which, when at x, moves one step to the right with probability ω _x , and one step to the left with probability 1 −ωx. Solomon (1975) determined the almost-sure asymptotic speed (= rate of escape) of a RWRE, in a more general set-up. Dembo, Peres and Zeitouni (1996), following earlier work by Greven and den Hollander (1994) on the quenched case, have computed rough tail asymptotics for the empirical mean of the annealed RWRE. They conjectured the form of the rate function in a full LDP. We prove in this paper their conjecture. The proof is based on a “coarse graining scheme” together with comparison techniques. Received: 22 July 1997/Revised version: 15 June 1998     

Cover times for words in symmetric and nonsymmetric cases: A comparison

Considering an infinite string of i.i.d. random letters drawn from a finite alphabet X, we are interested in the number of letters needed until one or more given words are observed as a substring. Given a large subset of words of length n (typically the whole Xⁿ), one can define the corresponding cover time as the maximum of the waiting times associated with each of the words considered, that is, the number of random letters needed until each word appears at least once. The symmetric case of uniformly distributed random letters has been studied throughly, but it is less well known how the results change if the distribution of random letters is no longer uniform. In the paper, a comparison is made between results in the symmetric and nonsymmetric cases, both known and new ones. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Martingale methods for pricing inventory penalties under continuous replenishment and compound renewal demands

This paper addresses the problem of inventory penalty pricing under the risk-neutral valuation principle. The underlying production-inventory system has a constant replenishment rate and a compound renewal demand stream (i.e., iid demand interarrival times are independent of iid demand sizes), and is subject to underage and overage penalties. Our pricing approach treats the penalties as a series of perpetual American options, and constructs auxiliary martingale processes in term of the inventory process. We provide a necessary and sufficient martingale condition for general compound renewal demands. Explicit expressions of penalty functions for underage and overage are obtained for the case where demand arrivals follow a Poisson process.