International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 1, Pages 191-202
On the weak law of large numbers for normed weighted sums of I.I.D. random variables
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.
For weighted sums of independent and identically distributed random variables
, a general weak law of large numbers of the form is established where
and are statable constants. The hypotheses involve both the behavior of the tail of the
distribution of and the growth behaviors of the constants and . Moreover, a weak
law is proved for weighted sums indexed by random variables . An example is presented
wherein the weak law holds but the strong law fails thereby generalizing a classical example.