If X 1 , X 2 ,... are independent random variables with zero expectation and finite variances, the cumulative sums S n are, on the average, of the order of magnitude S n , where S n 2 = E ( S n 2 ). The occasional maxima of the ratios S n /S n are surprisingly large and the problem is to estimate the extent of their probable fluctuations. Specifically, let S n * = ( S n - b n )/ a n , where { a n } and { b n }, two numerical sequences. For any interval I , denote by p ( I ) the probability that the event S n * ε I occurs for infinitely many n . Under mild conditions on { a n } and { b n }, it is shown that p ( I ) equals 0 or 1 according as a certain series converges or diverges. To obtain the upper limit of S n /a n , one has to set b n = ± ε a n , but finer results are obtained with smaller b n . No assumptions concerning the under-lying distributions are made; the criteria explain structurally which features of { X n } affect the fluctuations, but for concrete results something about P { S n > a n } must be known. For example, a complete solution is possible when the X n 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.