Let X1,X2,… be independent identically distributed (i.i.d.) random variables with EXk=0, V arXk=1. Suppose that φ(t)≔logEetXk<∞ for all t>−σ0 and some σ00. Let Sk=X1+⋯+Xk and S0=0. We are interested in the limiting distribution of the multiscale scan statisticMn=max0≤i<j≤nSj−Sij−i. We prove that for an appropriate normalizing sequence an, the random variable Mn2−an converges to the Gumbel extreme-value law exp{−e−cx}. The behavior of Mn depends strongly on the distribution of the Xk’s. We distinguish between four cases. In the superlogarithmic case we assume that φ(t)<t2/2 for every t>0. In this case, we show...</j≤nsj−sij−i.></∞>