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.     

Large deviations for Glauber dynamics of continuous gas

This paper is devoted to the large deviation principles of the Glauber-type dynamics of finite or infinite volume continuous particle systems.We prove that the level-2 empirical process satisfies the large deviation principles in the weak convergence topology,while it does not satisfy the large deviation principles in the T-topology.     

Large deviations and occupation times for spin particle systems with long range interactions

The large deviation principle for spin particle systems with long range interactions has been studied. It is shown that most of the results in Chen J. W. and Dai Pra P. ’s previous papers can be extended to the present situation. A particularly interesting result is the variational principle which characterizes the stationary Markov measures of such systems as the zeros of the governing LD rate functions. Uniqueness of such measure is studied from this as well as other point of view. We then apply the results to the occupation times of the systems. New large deviation and convergence results are obtained.     

Large deviations for M-estimators

We study the large deviation principle for M-estimators (and maximum likelihood estimators in particular). We obtain the rate function of the large deviation principle for M-estimators. For exponential families, this rate function agrees with the Kullback–Leibler information number. However, for location or scale families this rate function is smaller than the Kullback–Leibler information number. We apply our results to obtain confidence regions of minimum size whose coverage probability converges to one exponentially. In the case of full exponential families, the constructed confidence regions agree with the ones obtained by inverting the likelihood ratio test with a simple null hypothesis.     

Averaged and quenched propagation of chaos for spin glass dynamics

Summary. We study the asymptotic behaviour for both asymmetric and symmetric spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove, without any condition on time and temperature, averaged propagation of chaos results. Extending this result to replicated systems, we conclude that the law of a single spin converges to a non Markovian probability measure, in law with respect to the random interaction. Received: 3 April 1995/In revised form: 2 April 1996     

Large deviations for sample quantities

One obtains certain theorems for the probabilities of large deviations of uniform order statistics.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 144–150, 1980.     

Large deviations for renewal processes

We investigate large deviations for the empirical measure of the forward and backward recurrence time processes associated with a classical renewal process with arbitrary waiting-time distribution. The Donsker-Varadhan theory cannot be applied in this case, and indeed it turns out that the large deviations rate functional differs from the one suggested by such a theory. In particular, a non-strictly convex and non-analytic rate functional is obtained.     

Large deviations forU-statistics

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol.32, No. 1, pp. 7–19, January–March,1992.     

Large deviations for independent random walks

Summary We consider a system of independent random walks on . Let _n (x) be the number of particles atx at timen, and letL _n (x)=₀(x)+ ... + _n (x) be the total occupation time ofx by timen. In this paper we study the large deviations ofL _n (0)–L _n (1). The behavior we find is much different from that ofL _n (0). We investigate the limiting behavior when the initial configurations has asymptotic density 1 and when ₀(x) are i.i.d Poisson mean 1, finding that the asymptotics are different in these two cases.This work was done while the first author was on sabbatical at Cornell University. Both authors were partially supported by the National Science Foundation and the Army Research Office through the Mathematical Sciences Institute at Cornell     

Large deviations for two scaled diffusions

Summary. We formulate large deviations principle (LDP) for diffusion pair (X ^ɛ ,ξ ^ɛ )=(X _t ^ɛ ,ξ _t ^ɛ ), where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for (X ^ɛ ,ν ^ɛ ) with ν ^ɛ (dt, dz) being an occupation type measure corresponding to ξ _t ^ɛ . In some sense we obtain a combination of Freidlin–Wentzell’s and Donsker–Varadhan’s results. Our approach relies on the concept of the exponential tightness and Puhalskii’s theorem. Received: 29 June 1995/In revised form: 14 February 1996     

Large deviations for almost markovian processes

Summary Large deviation theorems for the empirical distribution of almost markovian processes are proven. Relative entropy identifies the rate function and its definition depends only on the process.Work partially supported by grant NSF-DMR81-14726     

Large deviations for conditional Volterra processes

In this article we investigate a problem of large deviations for continuous Gaussian Volterra processes, conditioned to follow a fixed trajectory up to a fixed time T > 0, in order to establish the behavior of the process in the near future after T and to give an asymptotic estimate of the exit probability of its bridge. Some examples are considered.     

Large deviations for martingales and derivatives

Fix a sequence of positive integers (mn) and a sequence of positive real numbers (wn). Two closely related sequences of linear operators (Tn) are considered. One sequence has given by the Lebesgue derivatives . The other sequence has given by the dyadic martingale when (l−1)/n²?x<l/n² for l=1,…,n². We prove both positive and negative results concerning the convergence of .     

Large deviations for Gibbs random fields

A large deviation principle for Gibbs random fields on Z^d is proven and a corresponding large deviations proof of the Gibbs variational formula is given. A generalization of the Lanford theory of large deviations is also obtained.This work was partially supported by NSF-DMR81-14726     

次线性期望下独立随机变量列的大偏差和中偏差

本文研究在次线性期望下的独立随机变量列的大偏差和中偏差原理. 利用次可加方法, 我们得 到次线性期望下的大偏差原理. 与次线性期望下的中心极限定理相应的中偏差原理也被建立.     

Large deviations and moderate deviations for kernel density estimators of directional data

Let fn be the non-parametric kernel density estimator of directional data based on a kernel function K and a sequence of independent and identically distributed random variables taking values in d-dimensional unit sphere Sd-1. It is proved that if the kernel function is a function with bounded variation and the density function f of the random variables is continuous, then large deviation principle and moderate deviation principle for {sup x∈sd-1 ｜fn（x） - E（fn（x））｜, n ≥ 1} hold.