Large deviations asymptotics and the spectral theory of multiplicatively regular markov processes

Abstract

In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn [32]. We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process Φ = (Φ(t)) with transition kernel P on a general state space X, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals F : X → ℂ, the kernel P∧(x, dy) = eF(x)P(x, dy) has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a “maximal” solution (λ, f) to the multiplicative Poisson equation, defined as the eigenvalue problem P∧f = λf. The functional Λ(F) = log(Λ) is convex, smooth, and its convex dual Λ* is convex, with compact sublevel sets. Multiplicative Mean Ergodic Theorem: Consider the partial sums (St) of the process with respect to any one of the functionals F(Φ(t)) considered above. The normalized mean Ex[exp(St)] (and not the logarithm of the mean) converges to f(x) exponentially fast, where f is the above solution of the multiplicative Poisson equation. Multiplicative regularity: The Lyapunov drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for the partial sums (St), with respect to any functional F in the above class. Large Deviations: The sequence of empirical measures of (Φ(t)) satisfies a large deviations principle in the “τWo-topology,” a topology finer that the usual τ-topology, generated by the above class of functionals F on X which is strictly larger than L∞(X). The rate function of this LDP is Λ*, and it is shown to coincide with the Donsker-Varadhan rate function in terms of relative entropy. Exact Large Deviations Asymptotics: The above partial sums (St) are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables. © 2005 Applied Probability Trust.