Abstract
If X1, X2,... are independent random variables with zero expectation and finite variances, the cumulative sums Sn are, on the average, of the order of magnitude Sn, where Sn2 = E(Sn2). The occasional maxima of the ratios Sn/Sn are surprisingly large and the problem is to estimate the extent of their probable fluctuations.
Specifically, let Sn* = (Sn - bn)/an, where {an} and {bn}, two numerical sequences. For any interval I, denote by p(I) the probability that the event Sn* ε I occurs for infinitely many n. Under mild conditions on {an} and {bn}, it is shown that p(I) equals 0 or 1 according as a certain series converges or diverges. To obtain the upper limit of Sn/an, one has to set bn = ± ε an, but finer results are obtained with smaller bn. No assumptions concerning the under-lying distributions are made; the criteria explain structurally which features of {Xn} affect the fluctuations, but for concrete results something about P{Sn>an} must be known. For example, a complete solution is possible when the Xn are normal, replacing the classical law of the iterated logarithm. Further concrete estimates may be obtained by combining the new criteria with some recently developed limit theorems.
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Selected References
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