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A STRONG LAW FOR COMPACTLY UNIFORMLY INTEGRABLE SEQUENCES OF INDEPENDENT RANDOM ELEMENTS IN BANACH SPACES

For a compactly uniformly integrable sequence of inde- pendent random elements {Vn,n 1} in a real separable Banach space, conditions are provided for the strong law of large numbers P n=1 (Vi EVi)/bn ! 0 almost certainly to hold where {bn,n 1} is a sequence of positive constants. The main result is general enough to include as special cases a strong law of Adler, Rosalsky, and Taylor (3) for compactly uniformly integrable sequences and a strong law of Taylor and Wei (11) for uniformly tight sequences. Illustrative examples are provided which show that the main result is sharp or which show how it improves upon or is dierent from other results in the literature.