Summary
Let {X N , N=0, ±1, ±2,...} be a stationary Gaussian stochastic process with means zero, variances one, and covariance sequence {r N }. Let Z N =max X k . Limit 1≦k≦N properties are obtained for Z N , as N approaches infinity. A double exponential limit law is known to hold if the random variables X iare mutually independent, that is r N ≡ 0, N ≢ 0. Berman has shown that the same law holds in the case of dependence, provided r N approaches zero “sufficiently fast”. Specifically sufficient conditions are that either lim r N log N=0, N→∞ or \(\sum\limits_{N = 1}^\infty {r^2 _N } < \infty \). In the present work, it is shown, however, that lim r N =0 is not sufficient. N=1 N→∞ A corresponding law is obtained for a separable, measurable version of a continuous parameter process. Sufficient conditions are obtained for the “strong laws of large numbers”,
, a.s. in both discrete, and continuous time.
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Pickands, J. Maxima of stationary Gaussian processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 7, 190–223 (1967). https://doi.org/10.1007/BF00532637
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DOI: https://doi.org/10.1007/BF00532637