International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 1, Pages 191-202
doi:10.1155/S0161171291000182

On the weak law of large numbers for normed weighted sums of I.I.D. random variables

André Adler1 and Andrew Rosalsky2

1Department of Mathematics, Illinois Institute of Technology, Chicago 60616, Illinois, USA
2Department of Statistics, University of Florida, Gainesville, Florida, USA

Received 14 March 1990

Copyright © 1991 André Adler and Andrew Rosalsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For weighted sums ∑j=1najYj of independent and identically distributed random variables {Yn,n≥1}, a general weak law of large numbers of the form (∑j=1najYj−νn)/bn→P0 is established where {νn,n≥1} and {bn,n≥1} are statable constants. The hypotheses involve both the behavior of the tail of the distribution of |Y1| and the growth behaviors of the constants {an,n≥1} and {bn,n≥1}. Moreover, a weak law is proved for weighted sums ∑j=1najYj indexed by random variables {Tn,n≥1}. An example is presented wherein the weak law holds but the strong law fails thereby generalizing a classical example.