finity. We shall determine the almost everywhere constant lim sup Tn in terms of the moment generating function of X1 and the radius of convergence of E xf(n), denoted r. If we define Nn as the number of consecutive successes beginning at trial n + 1 in a sequence of Bernoulli trials with success probability p, we let Un- N/f(n). It is shown that, almost surely, lim sup Un = log, r. This result will be compared with lim sup Tn for the case of Bernoulli trials. The author is indebted to Professor D. J. Newman for drawing his attention to this problem.
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L. A. Shepp (1964) studied this question.