CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS

and let SN = Ek-l aN,k(Xk - EXk). Early work in probability dealt with the convergence (almost everywhere and in probability) to zero of the sequence AN. More recent work has dealt with the convergence to zero of sequences of the form SN under various assumptions on the coefficients aN,k and the distributions of the XN's. In most cases the assumptions made about the XN's have been not much stronger or weaker than the assumption of a finite upper bound on their ,yth absolute moments for some 'y > 1. The classical result giving exponential convergence rates in the law of large numbers was established by Cram6r [6] (see also [4]) and states that if the XN'S are identically distributed, and if their common moment generating function is finite in some interval about the origin, then for each e > 0 there exists 0 'e 2p . Baum, Katz, and Read [3] investigated this ex

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