General Limit Theorems with $o$-Rates and Markov Processes under Pseudo-Moment Conditions

Abstract

Making use of the Dvoretzky extension of the Trotter-operator method, a general convergence theorem with $o$-rates for dependent, real-valued random variables satisfying a pseudo-moment condition is established. Applications are to general convergence theorems, a central limit theorem as well as a weak law of large numbers for Markov processes with discrete time parameter. Further, pseudo-moment conditions are discussed in detail and the results are compared with the theory of probability-metrics.