Statistical Inference for Markov Chains with Known Structure

For Markov chains arising in many applications, the structure of the stochastic recursion defining the chain is known except for the distribution of the iid "driving noise" variables (the source of randomness in the chain). Given a dataset of such driving noise samples, and for a nonparametric model of the driving noise distribution, we study statistically efficient estimation of equilibrium and non-equilibrium expectation performance measures for the (general state space) Markov chain. Such expectations can be viewed as statistical functionals of the driving noise distribution, and we establish a framework based on semiparametric efficiency to characterize asymptotically minimum variance estimation for such functionals. In particular, we provide Foster-Lyapunov drift conditions to ensure suitable directional differentiability of the functionals and the existence of influence functions. Surprisingly, the nonparametric plug-in estimator, in which the empirical driving noise distribution is "plugged in" to the functional, can be inconsistent when the state space is infinite. To resolve the inconsistency issue, we introduce uniform versions of the classical Foster-Lyapunov drift conditions, which ensures that the conditions hold with the population kernels replaced by their empirical counterparts. Under such conditions, we establish asymptotic normality and efficiency of plug-in estimators of the performance measures of interest.