Summing subsequences of random variables
Mark Schwartz · Rocky Mountain Journal of Mathematics · 1987
Given an increasing sequence N of positive integers and k ^ 1, call any one to one correspondence z : N -» N* an ordering (or numbering) of N onto N*.Let (X") be a sequence of random variables satisfying sup"E \X"\ (log + \X n \) k ~x < oo.Then there exists a subsequence N 0 = (/») such that, for any further subsequence N x -(i Jn ) and any ordering r satisfying | r(ij n ) | ^ /" for all « ^ 1, we have (J r -i (s) ) converges Cesàro a.s.for s e N*.