Strong law of large numbers for 2-exchangeable random variables

The investigation of the role of independence in the classical SLLN leads to a natural generalization of the SLLN to the case where the random variables are 2-exchangeable; namely, let {Xi: i [greater-or-equal, slanted] 1} be a sequence of random variables such that all ordered pairs (Xi, Xj), i [not equal to] j, are identically distributed. Then we show, among other things, that where X is in general a non-degenerate random variable. This provids a unified treatment of the SLLN for both exchangeable and pairwise independent random variables. We also show that, under 2-exchangeability, to preserve the Glivenko-Cantelli Theorem - sometimes refered to as the fundamental theorem of statistics - it is necessary that the random variables be pairwise independent.