A general approach to the strong law of large numbers. (English) Zbl 0991.60021

Fazekas, I.; Klesov, O.

Theory Probab. Appl. 45, No. 3, 436-449 (2000) and Teor. Veroyatn. Primen. 45, No. 3, 568-583 (2000).

A basic approach to proving strong laws of large numbers is to use directly a Hájek-Rényi maximal inequality for normed sums. The aim of this paper is to show that such an inequality is, in fact, a consequence of an appropriate maximal inequality for cumulative sums, and to show that the latter automatically implies the strong law of large numbers. No dependence assumption is made, but only the existence of a certain moment of individual random variables. Applications and improvements are given in the case of a martingale difference, \(\rho\)-mixing sequences, the so-called “mixingales”, logarithmically weighted sums, and sequences with superadditive moment function.

Reviewer: Gheorghe Stoica (Saint John)

MSC:

60F15

Strong limit theorems

Keywords:

Hájek-Rényi maximal inequality; strong law of large numbers; cumulative sums