Limiting distribution for the maximal standardized increment of a random walk

作者：Kabluchko Zakhar^*; Wang Yizao

来源：Stochastic Processes and Their Applications, 2014, 124(9): 2824-2867.

DOI：10.1016/j.spa.2014.03.015

• Extreme value theory
• Increments of random walks
• Erdos-Renyi law
• Large deviations
• Moderate deviations
• Multiscale scan statistic
• Cramer series
• Gumbel distribution
• Double sum method
• Subgaussian distributions
• Changepoint detection

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<j <= n) S-j - S-i/root j - i
We prove that for an appropriate normalizing sequence a(n), the random variable M-n(2) - a(n) converges to the Gumbel extreme-value law exp{-e(-cx)}. The behavior of M-n depends strongly on the distribution of the X-k's. We distinguish between four cases. In the superlogarithmic case we assume that phi(t) < t(2)/2 for every t > 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.

• 出版日期2014-9