International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 1, Pages 33-36
doi:10.1155/S0161171295000032

On complete convergence of the sum of a random number of a stable type P random elements

André Adler1 and Andrey Volodin2

1Department of Mathematics, Illinois Institute of Technology, Chicago 60616, IL, USA
2Research Institute of Mathematics, Kazan University, Tatarstan, Kazan 420008, Russia

Received 1 September 1992; Revised 10 March 1993

Copyright © 1995 André Adler and Andrey Volodin. 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

Complete convergence for randomly indexed normalized sums of random elements of the form ∑j=1TnXj/ϕ(Tn) is established. The random elements {Xn} belong to a type p stable space and are assumed to be independent, but not necessarily identically distributed. No assumptions are placed on the joint distributions of the stopping times {Tn}.