Explicit Limit Results for Minimal Sufficient Statistics and Maximum Likelihood Estimators in Some Markov Processes: Exponential Families Approach

Abstract

Finite-state Markov chains with either a discrete or continuous time parameter, Markov renewal processes and Markov-additive processes are considered. We prove that their likelihood functions, in the nonsequential as well as in various sequential cases, belong to special $(n + k, n)$-curved exponential families in general, for which limit results are easily established. Subsequently, asymptotic normality of the corresponding nonsequential and sequential maximum likelihood estimators is established. Also in the case of Markov renewal and Markov-additive processes, stopping times are determined which reduce the corresponding curved exponential families in general to noncurved ones. The latter, together with results of Stefanov, are combined with results of Serfozo to imply explicit solutions in functional limit theorems for the considered processes. In particular, we derive explicit solutions for the important variance parameter in the functional central limit theorems and functional laws of iterated logarithm for those processes. Indeed, our explicit solutions cover more general cases than the known ones, even in the case of finite-state Markov chains. Moreover, we supply explicit solutions, not previously available, in functional limit theorems for Markov renewal processes and Markov-additive processes.