Summability methods for independent identically distributed random variables

Abstract

In this paper, we present certain theorems concerning the Cesaro ( C , α ) (C,\alpha ) , Abel ( A ) (A) , Euler ( E , q ) (E,q) and Borel ( B ) (B) summability of Σ Y i \Sigma {Y_i} , where Y i = X i − X i − 1 , X 0 = 0 {Y_i} = {X_i} - {X_{i - 1}},{X_0} = 0 and X 1 , X 2 , ⋯ {X_1},{X_2}, \cdots are i.i.d. random variables. While the Kolmogorov strong law of large numbers and the Hartman-Wintner law of the iterated logarithm are related to ( C , 1 ) (C,1) summability and involve the finiteness of, respectively, the first and second moments of X 1 {X_1} , their analogues for Euler and Borel summability involve different moment conditions, and the analogues for ( C , α ) (C,\alpha ) and Abel summability remain essentially the same.