Let X-1, X-2, ... be independent identically distributed (i.i.d.) random variables with EXk = 0, VarX(k) = 1. Suppose that phi(t) := log Ee(tXk) < infinity for all t > -sigma(0) and some sigma(0) > 0. Let S-k = X-1 + ... +X-k and S-0 = 0. We are interested in the limiting distribution of the multiscale scan statistic M-n = max (0 <= i 0. In this case, we show that the main contribution to M-n comes from the intervals (i, j) having length l := j - i of order a (log n)(p), a > 0, where p = q/(q - 2) and q is an element of {3, 4,...} is the order of the first non-vanishing cumulant of X-1 (not counting the variance). In the logarithmic case we assume that the function psi(t) := 2 phi(t)/t(2) attains its maximum m(*) > 1 at some unique point t = t(*) is an element of (0, infinity). In this case, we show that the main contribution to M-n comes from the intervals (i, j) of length d(*) log n + a root log n, a is an element of R, where d(*) = 1/phi(t(*)) > 0. In the sublogarithmic case we assume that the tail of X-k is heavier than exp{-x(2-epsilon)}, for some epsilon > 0. In this case, the main contribution to M-n comes from the intervals of length o(log n) and in fact, under regularity assumptions, from the intervals of length 1. In the remaining, fourth case, the X-k's are Gaussian. This case has been studied earlier in the literature. The main contribution comes from intervals of length a log n, a > 0. We argue that our results cover most interesting distributions with light tails. The 'proofs are based on the precise asymptotic estimates for large and moderate deviation probabilities for sums of i.i.d. random variables due to Cramer, Bahadur, Ranga Rao, Petrov and others, and a careful extreme value analysis of the random field of standardized increments by the double sum method. (C) 2014 Elsevier B.V. All rights reserved.