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 (2003). 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 $\{\Phi(t)\}$ with transition kernel $P$ on a general state space $X$, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals $F$ on $X$, the kernel $\hat P(x,dy) = e^{F(x)} P(x,dy)$ has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a maximal, well-behaved solution to the multiplicative Poisson equation, defined as an eigenvalue problem for $\hat P$. Multiplicative Mean Ergodic Theorem: Consider the partial sums of this process with respect to any one of the functionals $F$ considered above. The normalized mean of their moment generating function (and not the logarithm of the mean) converges to the above maximal eigenfunction exponentially fast. 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 above partial sums. Large Deviations: The sequence of empirical measures of the process satisfies a large deviations principle in a topology finer that the usual tau-topology, generated by the above class of functionals. The rate function of this LDP is the convex dual of logarithm of the above maximal eigenvalue, 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 are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.
Highlights
Introduction and Main ResultsLet Φ = {Φ(t) : t ∈ T} be a Markov processes taking values in a Polish state space X, equipped with its associated Borel σ-field B
The distribution of Φ is determined by its initial state Ψ(0) = x ∈ X, and the transition semigroup {P t : t ∈ T}, where in discrete time all kernels P t are powers of the 1-step transition kernel P
In this paper we show that the foundations of the multiplicative ergodic theory and of the large deviations behavior of Φ can be developed in analogy to the linear theory, by shifting attention from the semigroup of linear operators {P t} to the family of nonlinear, convex operators {Wt}
Summary
Throughout the paper we assume that Φ is ψ-irreducible and aperiodic This means that there is a σ-finite measure ψ on (X, B) such that, for any A ∈ B satisfying ψ(A) > 0 and any initial condition x,. Let V : X → (0, ∞] be an extended-real valued function, with V (x0) < ∞ for at least one x0 ∈ X, and write A for the (extended) generator of the semigroup {P t : t ∈ T} This is equal to A = (P − I) in discrete time (where I = I(x, dy) denotes the identity kernel δx(dy)), and in continuous-time we think of. [Px] F ∈ LW ∞ , where in discrete time integrals with respect to time denote the corresponding sums, [0,T ) ≡ Tt=−01, and the measure Px denotes the conditional distribution of Φ given Φ(0) = x.
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