Probability distributions related to the law of the iterated logarithm.
Let W(t) denote a standard Wiener process for 0 </= t < infinity. We compute the probability that W(t) >/= t((1/2))A(t) for some t >/= 1 (or for some t >/= 0) for a certain class of functions A(t), including functions which are approximately (2 log log t)((1/2)) as t --> infinity. We also give an invariance principle which states that this probability is the limit as m --> infinity of the probability that s(n) >/= n((1/2))A(n/m) for some n >/= m (or for some n >/= 1), where s(n) is the sum of n independent and identically distributed random variables with mean 0 and variance 1.
Robbins
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COLUMBIA UNIVERSITY AND STANFORD UNIVERSITY.
Siegmund
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