Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
10527358 | Stochastic Processes and their Applications | 2014 | 44 Pages |
Abstract
Let X1,X2,⦠be independent identically distributed (i.i.d.) random variables with EXk=0, V arXk=1. Suppose that Ï(t)âlogEetXk<â for all t>âÏ0 and some Ï0>0. Let Sk=X1+â¯+Xk and S0=0. We are interested in the limiting distribution of the multiscale scan statisticMn=max0â¤i0. In this case, we show that the main contribution to Mn comes from the intervals (i,j) having length lâjâi of order a(logn)p, a>0, where p=q/(qâ2) and qâ{3,4,â¦} is the order of the first non-vanishing cumulant of X1 (not counting the variance). In the logarithmic case we assume that the function Ï(t)â2Ï(t)/t2 attains its maximum mâ>1 at some unique point t=tââ(0,â). In this case, we show that the main contribution to Mn comes from the intervals (i,j) of length dâlogn+alogn, aâR, where dâ=1/Ï(tâ)>0. In the sublogarithmic case we assume that the tail of Xk is heavier than exp{âx2âε}, for some ε>0. In this case, the main contribution to Mn comes from the intervals of length o(logn) and in fact, under regularity assumptions, from the intervals of length 1. In the remaining, fourth case, the Xk's are Gaussian. This case has been studied earlier in the literature. The main contribution comes from intervals of length alogn, 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 Cramér, Bahadur, Ranga Rao, Petrov and others, and a careful extreme value analysis of the random field of standardized increments by the double sum method.
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Authors
Zakhar Kabluchko, Yizao Wang,