Large deviations for empirical measures of Markov chains

In this paper we obtain large-deviation upper and lower bounds for the empirical measure of a Markov chain with general state space, as well as for the associated multivariate empirical measure and empirical process. In each of these instances we improve in various ways the results in the literature.     

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.     

Large deviations for empirical measures of switching diffusion processes with small parameters

We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small parameters.     

Large deviations for invariant measures of SPDEs with two reflecting walls

In this article, we establish a large deviation principle for invariant measures of solutions of stochastic partial differential equations with two reflecting walls driven by a space–time white noise.     

Large deviations for the empirical measure of a diffusion via weak convergence methods

We consider the large deviation principle for the empirical measure of a diffusion in Euclidean space, which was first established by Donsker and Varadhan. We employ a weak convergence approach and obtain a characterization for the rate function that is dual to the one obtained by Donsker and Varadhan, and which allows us to evaluate the variational form of the rate function for both reversible and nonreversible diffusions.     

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.     

Large deviations for the empirical mean of an M/M/1 queue

Let $(Q(k):k\ge 0)$ be an $M/M/1$ queue with traffic intensity $\rho \in (0,1).$ Consider the quantity $$\begin{aligned} S_{n}(p)=\frac{1}{n}\sum _{j=1}^{n}Q\left( j\right) ^{p} \end{aligned}$$ for any $p>0.$ The ergodic theorem yields that $S_{n}(p) \rightarrow \mu (p) :=E[Q(\infty )^{p}]$ , where $Q(\infty )$ is geometrically distributed with mean $\rho /(1-\rho ).$ It is known that one can explicitly characterize $I(\varepsilon )>0$ such that $$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n}\log P\big (S_{n}(p)<\mu \left( p\right) -\varepsilon \big ) =-I\left( \varepsilon \right) ,\quad \varepsilon >0. \end{aligned}$$ In this paper, we show that the approximation of the right tail asymptotics requires a different logarithm scaling, giving $$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n^{1/(1+p)}}\log P\big (S_{n} (p)>\mu \big (p\big )+\varepsilon \big )=-C\big (p\big ) \varepsilon ^{1/(1+p)}, \end{aligned}$$ where $C(p)>0$ is obtained as the solution of a variational problem. We discuss why this phenomenon—Weibullian right tail asymptotics rather than exponential asymptotics—can be expected to occur in more general queueing systems.     

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 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 in total variation of occupation measures of one-dimensional diffusions

For one-dimensional diffusion processes, we find an explicit necessary and sufficient condition for the large deviation principle of the occupation measures in the total variation and of local times in L¹$L^1$.