General limit theorems with \({\mathfrak o}\)-rates and Markov processes under pseudo-moment conditions (Q1113518)

The authors establish a general convergence theorem with little-\({\mathfrak o}\)-rates for real-valued, dependent random variables satisfying a generalized pseudo-Lindeberg condition as well as a pseudomoment condition. The proof is a modification of the Dvoretzky extension (telescoping argument) of the Lindeberg-Trotter operator method. The dependency structure, characterized solely by the pseudomoment condition, covers in any case martingale difference sequences and Markov-processes with discrete time parameter. Applications are to a central limit theorem as well as to a weak law of large numbers with \({\mathfrak o}\)-rates for such Markov-processes. The pseudomoments in question, which are special probability metrics, are treated in detail. The results are also compared with other convergence assertions [e.g. \textit{V. M. Zolotarev}, Teor. Veroyatn. Primen. 23, 284- 294 (1978; Zbl 0421.60006)].