Maximal inequalities and ergodic theorems for Cesàro-α or weighted averages

Abstract

This chapter discusses the problems involving the relationship between ergodic theorems and summability methods. It presents a resume of the strong law of large numbers and the ergodic theorem for Cesaro-α average. It also presents a general form of the maximal ergodic inequality for weighted averages and some of its corollaries. Such a maximal inequality is essential in the proof of the ergodic theorem for Cesaro-α average. The chapter also presents an extension of the precise form of the maximal inequality to the Cesaro-α average. For Classical Tauberian theory, it is known that for bounded sequences, all (C, α) convergences and Abel convergence are equivalent; however, they are not equivalent to Riesz harmonic convergence.