Limit distributions for the maxima of stationary Gaussian processes

Abstract

Let { X n } be a stationary Gaussian sequence with E{ X 0} = 0, { X 2 0} = 1 and E{ X 0 X n } = r n n Let c n = (2ln n) built1 2 , b n = c n − 1 2 c -1 n ln(4π ln n), and set M n = max 0 ⩽ k⩽ n X k . A classical result for independent normal random variables is that P[c n(M n−b n)⩽x]→exp[-e -x] as n → ∞ for all x. Berman has shown that (1) applies as well to dependent sequences provided r n ln n = o(1). Suppose now that { r n } is a convex correlation sequence satisfying r n = o(1), ( r n ln n) -1 is monotone for large n and o(1). Then P[r n -1 2 (M n − (1−r n) 1 2 b n)⩽x] → Ф(x) for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for { M n }, there are others. In particular, the limit distribution is given below when r n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).