Sankhya: The Indian Journal of Statistics

1992, Volume 54, Series A, Pt. 2 , pp. 215--231

CESÀRO UNIFORM INTEGRABILITY AND THE STRONG LAW OF LARGE NUMBERS

By

TAPAS K. CHANDRA and A. GOSWAMI, *Indian Statistical Institute*

SUMMARY. This paper studies some consequences of conditions like the uniform integrability and Cesàro uniform integrability (introduced by Chandra (1989)) in the context of strong laws of large numbers (SLLNs). Extending certain arguments of Etemadi (1981), Csörgo, Tandori and Totik (1983), we relax the condition of ‘identical distribution and/or independence’ in Etemadi’s extension of Kolmogorov’s SLLN and in the classical SLLN’s of Markov and Cantelli. We also extend the recent SLLN of Landers and Rogge (1986) and a result of Calderon (1983) related to the classical SLLN of Marcinkiewicz and Zygmund.

*AMS(1980) subject classification.* 60F99.

*Key words and phrases.* $L_p$ version of Berry-Essen theorem, non linear statistics linear process.