Necessary and sufficient conditions for complete convergence of double weighted sums of pairwise independent identically distributed random elements in Banach spaces
- 37 Downloads
This paper provides necessary and sufficient conditions for complete convergence of double weighted sums of pairwise independent identically distributed random elements in Banach spaces. The main theorem extends Theorem 1.2 in  to the double weighted sum setting. The sharpness of the main result is illustrated by showing that the main theorem can fail if we replace the identical distribution condition by a slightly weaker condition, even when the random elements are independent and uniformly almost surely bounded.
Key words and phrasescomplete convergence double array weighted sum pairwise independent random element Banach space
Mathematics Subject Classification60B11 60B12 60F15
Unable to display preview. Download preview PDF.
The authors are grateful to Professor Andrew Rosalsky for reading the first draft of the paper and giving very helpful comments.
- 2.P. Y. Chen and D. C. Wang, \(L^r\) convergence for \(B\)-valued random elements, Acta Math. Sin. (Engl. Ser.), 28 (2012), 857–868Google Scholar
- 5.O. Klesov, Limit theorems for multi-indexed sums of random variables, Probability Theory and Stochastic Modelling, 71, Springer-Verlag (Heidelberg, 2014)Google Scholar