Correction added in proof

Summary We extend Sanov's theorem on i.i.d. large deviations to independent but not identically distributed random variables, and study the generalization of relative entropy that appears as the rate function.     

Large deviations for stochastic partial differential equations driven by a Poisson random measure

Stochastic partial differential equations driven by Poisson random measures (PRMs) have been proposed as models for many different physical systems, where they are viewed as a refinement of a corresponding noiseless partial differential equation (PDE). A systematic framework for the study of probabilities of deviations of the stochastic PDE from the deterministic PDE is through the theory of large deviations. The goal of this work is to develop the large deviation theory for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. Although the analogous questions for finite dimensional systems have been well studied, there are currently no general results in the infinite dimensional setting. This is in part due to the fact that in this setting solutions may have little spatial regularity, and thus classical approximation methods for large deviation analysis become intractable. The approach taken here, which is based on a variational representation for nonnegative functionals of general PRMs, reduces the proof of the large deviation principle to establishing basic qualitative properties for controlled analogues of the underlying stochastic system. As an illustration of the general theory, we consider a particular system that models the spread of a pollutant in a waterway.     

Large deviations for statistics of the Jacobi process

This paper aims to derive large deviations for statistics of the Jacobi process already conjectured by M. Zani in her thesis. To proceed, we write in a simpler way the Jacobi semi-group density. Being given by a bilinear sum involving Jacobi polynomials, it differs from Hermite and Laguerre cases by the quadratic form of its eigenvalues. Our attempt relies on subordinating the process using a suitable random time change. This gives a Mehler-type formula whence we recover the desired semi-group density. Once we do, an adaptation of Zani’s result [M. Zani, Large deviations for squared radial Ornstein–Uhlenbeck processes, Stochastic. Process. Appl. 102 (1) (2002) 25–42] to the non-steep case will provide the required large deviations principle.     

Conditional distribution of heavy tailed random variables on large deviations of their sum

It is known that large deviations of sums of subexponential random variables are most likely realised by deviations of a single random variable. In this article we give a detailed picture of how subexponential random variables are distributed when a large deviation of the sum is observed.     

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.     

Effect of truncation on large deviations for heavy-tailed random vectors

This paper studies the effect of truncation on the large deviations behavior of the partial sum of a triangular array coming from a truncated power law model. Each row of the triangular array consists of i.i.d. random vectors, whose distribution matches a power law on a ball of radius going to infinity, and outside that it has a light-tailed modification. The random vectors are assumed to be R^d-valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the same is lost.     

Lower deviation probabilities for supercritical Galton-Watson processes

There is a well-known sequence of constants cn describing the growth of supercritical Galton-Watson processes Zn. By lower deviation probabilities we refer to P(Zn=kn) with kn=o(cn) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Zn+1/Zn. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramér method for proving large deviations of sums of independent variables to our needs.     

Large deviations for empirical measures with degenerate limiting distribution

Summary This paper studies the large deviations of the empirical measure associated withn independent random variables with a degenerate limiting distribution asn. A large deviations principle — quite unlike the classical Sanov type results — is established for such empirical measures in a general Polish space setting. This result is applied to the large deviations for the empirical process of a system of interacting particles, in which the diffusion coefficient vanishes as the number of particles tends to infinity. A second way in which the present example differs from previous work on similar weakly interacting systems is that there is a singularity in the mean-field type interaction.     

Action functional for diffusions in discontinuous media

Summary The action functional, i.e. the rate function governing the large deviations is obtained for a family of stochastic processes with discontinuous drift and small diffusion. A well-known method of continuous mapping is developed which proves to be efficient in a so called stable case.     

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 for the radial processes of the Brownian motions on hyperbolic spaces

Using the heat kernel estimates by Davies (1989) and Anker et al. (1996), we show large deviations for the radial processes of the Brownian motions on hyperbolic spaces.     

Limit theorems for some polynomial statistics of the poisson process

The central limit theorem and the theorem on large deviations for the functionals of the Poisson random process are proved. The formulas for cumulants of multiple stochastic integrals (m.s.i.) with respect to the Poisson process are obtained. The m.s.i. may be considered as anU-statistics arising in queueing theory as well as a generalization of the well-known Poisson shot-noise process, having wide applications.     

Long strange segments,ruin probabilities and the effect of memory on moving average processes

We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as the moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.     

Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion processes

We prove the hydrostatics of boundary driven gradient exclusion processes, Fick’s law and we present a simple proof of the dynamical large deviations principle which holds in any dimension.     

Large deviations and related problems for absorbing Markov chains

In this paper, large deviations and their connections with several other fundamental topics are investigated for absorbing Markov chains. A variational representation for the Dirichlet principal eigenvalues is given by the large deviation approach. Kingman’s decay parameters and mean ratio quasi-stationary distributions of the chains are also characterized by the large deviation rate function. As an application of these results, we interpret the “stationarity” of mean ratio quasi-stationary distributions via a concrete example. An application to quasi-ergodicity is also discussed.     

Estimation de la densité de la solution de l'équation de Zakaï robuste

We study the behaviour in small time of the density of the robust Zakaï equation under the weak Hörmander's hypothesis. We avoid large deviations phenomena by supposing that the drift at the departure is equal to zero. We get an asymptotic expansion over the diagonal of the density where the observation is involved. In the case where all the terms of that asymptotic expansion are equal to zero, a pathology which is related to the Bismut's condition, we give an estimation of the decay by using a probabilistic analogous of the Gevrey methods.

     

On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations

Summary We consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous Boltzmann type equation, as the number of particles tends to infinity. These laws of large numbers were proved for the Maxwellian molecules by H. Tanaka [Tal] and for the hard spheres by A.S. Sznitman [Szl]. In the present paper we investigate the corresponding large deviations: the large deviation upper bound is obtained and, using convex analysis, a non-variational formulation of the rate function is given. Our results hold for Maxwellian molecules with a cutoff potential and for hard spheres.     

Refinements of the Gibbs conditioning principle

Summary Refinements of Sanov's large deviations theorem lead via Csiszár's information theoretic identity to refinements of the Gibbs conditioning principle which are valid for blocks whose length increase with the length of the conditioning sequence. Sharp bounds on the growth of the block length with the length of the conditioning sequence are derived.Partially supported by NSF DMS92-09712 grant and by a US-Israel BSF grantPartially supported by a US-Israel BSF grant and by the fund for promotion of research at the Technion     

Sample path large deviations in finer topologies

In this paper we present sufficient conditions for sample path large deviation principles to be extended to finer topologies. We consider extensions of the uniform topology by Orlicz functional and we consider Lipschitz spaces: the former are concerned with cumulative path behavior while the latter are more sensitive to extremes in local variation. We also consider sample paths indexed by the half line, where the usual projective limit topologies are not strong enough for many applications. We introduce and apply a new technique extending large deviation principles to finer topologies. We show how to apply the results to obtain large deviations for weighted statistics, to improve Schilder's theorem as well as to obtain large deviations in queueing theory     

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