Limit distributions for the maxima of stationary Gaussian processes

Y. Mittal and D. Ylvisaker

Stochastic Processes and their Applications, 1975, vol. 3, issue 1, 1-18

Abstract: Let {Xn} be a stationary Gaussian sequence with E{X0} = 0, {X20} = 1 and E{X0Xn} = rnn Let cn = (2ln n), bn = cn- c-1n ln(4[pi] ln n), and set Mn = max0 [less-than-or-equals, slant]k[less-than-or-equals, slant]nXk. A classical result for independent normal random variables is that P[cn(Mn-bn)[less-than-or-equals, slant]x]-->exp[-e-x] as n --> [infinity] for all x. Berman has shown that (1) applies as well to dependent sequences provided rnlnn = o(1). Suppose now that {rn} is a convex correlation sequence satisfying rn = o(1), (rnlnn)-1 is monotone for large n and o(1). Then P[rn-1/2(Mn - (1-rn)1/2bn)[less-than-or-equals, slant]x] --> F(x) for all x, where F is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {Mn}, there are others. In particular, the limit distribution is given below when rn is (sufficiently close to) [gamma]/ln n. We further exhibit a collection of limit distributions which can arise when rn 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).

Date: 1975
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