Sankhya: The Indian Journal of Statistics

1989, Volume 51, Series A, Pt. 3, pp. 309--317

UNIFORM INTEGRABILITY IN THE CESÀRO SENSE AND THE WEAK LAW OF LARGE NUMBERS

By

TAPAS K. CHANDRA, *Indian Statistical Institute*

SUMMARY. It is shown here that L^{1} convergence holds in large number of cases where the WLLN holds; in fact, it is shown that the proof of the stronger fact is somewhat easier and more straightforward. In particular, several extensions and variations of the classical Khinchin WLLN are obtained.

*AMS (1985) subject classification.* Primary 60F05; secondary 60G42.

*Key words and phrases.* Uniform integrability, uniform integrability in the Cesaro sense, weak law of large numbers, martingale-difference sequence, $/phi$-mixing sequence.