Consider the partial sums {S t } of a real-valued functional F (Φ(t)) of a Markov chain {Φ(t)} with values in a general state space. Assuming only that the Markov chain is geometrically ergodic and that the functional F is bounded, the following conclusions are obtained:Spectral theory: Well-behaved solutionsf can be constructed for the "multiplicative Poisson equation" (e αF P )f = λf , where P is the transition kernel of the Markov chain, and α ∈ C is a constant. The functionf is an eigenfunction, with corresponding eigenvalue λ, for the kernel (e αF P ) = e αF (x) P (x, dy).A "multiplicative" mean ergodic theorem: For all complex α in a neighborhood of the origin, the normalized mean of exp(αS t ) (and not the logarithm of the mean) converges tof exponentially fast, wheref is a solution of the multiplicative Poisson equation.Edgeworth Expansions: Rates are obtained for the convergence of the distribution function of the normalized partial sums S t to the standard Gaussian distribution. The first term in this expansion is of order (1/ √ t), and it depends on the initial condition of the Markov chain through the solution F of the associated Poisson equation (and not the solutionf of the multiplicative Poisson equation).Large Deviations: The partial sums are shown to satisfy a large deviations principle in a neighborhood of the mean. This result, proved under geometric ergodicity alone, cannot in general be extended to the whole real line.Exact Large Deviations Asymptotics: Rates of convergence are obtained for the large deviations estimates above. The polynomial pre-exponent is of order (1/ √ t), and its coefficient depends on the initial condition of the Markov chain through the solutionf of the multiplicative Poisson equation.Extensions of these results to continuous-time Markov processes are also given.

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) = e F (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 Pf = 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 {S t } of the process with respect to any one of the functionals F (Φ(t)) considered above. The normalized mean E x [exp(S t )] (and not the logarithm of the mean) converges tof (x) exponentially fast, wheref 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 {S t }, 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 "τ W0 -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 {S t } are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.

We discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a Doeblin-type mixing condition. The estimators are Cesàro averages of longest match-lengths, and their consistency follows from a generalized ergodic theorem due to Maker. We provide examples of their performance on English text, and we generalize our results to countable alphabet processes and to random fields.

Suppose P is an arbitrary discrete distribution on a countable alphabet .Given an i.i.d. sample X 1 X n drawn from P, we consider the problem of estimating the entropy H P or some other functional F = F P of the unknown distribution P. We show that, for additive functionals satisfying mild conditions (including the cases of the mean, the entropy, and mutual information), the plug-in estimates of F are universally consistent. We also prove that, without further assumptions, no rate-of-convergence results can be obtained for any sequence of estimators. In the case of entropy estimation, under a variety of different assumptions, we get rate-of-convergence results for the plug-in estimate and for a nonparametric estimator based on match-lengths. The behavior of the variance and the expected error of the plug-in estimate is shown to be in sharp contrast to the finite-alphabet case. A number of other important examples of functionals are also treated in some detail.