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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机译：如果X1，X2，...是具有零期望和有限方差的独立随机变量，则平均总和Sn的数量级为Sn，其中Sn 2 = E（Sn 2 ）。比率Sn / Sn的偶然最大值非常大，问题在于估计其可能的波动程度。具体来说，令Sn * =（Sn-bn）/ an，其中{an}和{bn}，两个数字序列。对于任何间隔I，用p（I）表示事件 Sn * ε I 对于无限多个 n < / em>。在{ an }和{ b n }的温和条件下，显示 p （ I ）等于0或1，因为某些级数收敛或发散。要获得 S n / a n 的上限，必须设置 b n < / em> =±ε a n ，但使用较小的 b n 可获得更好的结果。没有关于基础分布的假设；该标准从结构上解释了{ X n }的哪些特征会影响波动，但是对于具体结果，有关 P { S n a n }必须是已知的。例如，当 X n 正常时，有可能采用完整的解决方案来代替迭代对数的经典定律。通过将新标准与一些最近开发的极限定理相结合，可以获得进一步的具体估计。
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