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June, 1973 A Convergence Theorem for Extreme Values from Gaussian Sequences
Roy E. Welsch
Ann. Probab. 1(3): 398-404 (June, 1973). DOI: 10.1214/aop/1176996934


Let $\{X_n, n = 0, \pm 1, \pm 2, \cdots\}$ be a stationary Gaussian stochastic process with means zero, variances one, and covariance sequence $\{r_n\}$. Let $M_n = \max \{X_1, \cdots, X_n\}$ and $S_n =$ second largest $\{X_1, \cdots, X_n\}$. Limit properties are obtained for the joint law of $M_n$ and $S_n$ as $n$ approaches infinity. A joint limit law which is a function of a double exponential law is known to hold if the random variables $X_i$ are mutually independent. When $M_n$ alone is considered Berman has shown that a double exponential law holds in the case of dependence provided either $r_n \log n \rightarrow 0$ or $\sum^\infty_{n=1} r_n^2 < \infty$. In the present work it is shown that the above conditions are also sufficient for the convergence of the joint law of $M_n$ and $S_n$. Weak convergence properties of the stochastic processes $M_{\lbrack nt\rbrack}$ and $S_{\lbrack nt \rbrack}$ with $0 < a \leqq t < \infty$ are also discussed.


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Roy E. Welsch. "A Convergence Theorem for Extreme Values from Gaussian Sequences." Ann. Probab. 1 (3) 398 - 404, June, 1973.


Published: June, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0258.62030
MathSciNet: MR350837
Digital Object Identifier: 10.1214/aop/1176996934

Primary: 62G30
Secondary: 60G15 , 62E20

Keywords: extreme-value theory , Gaussian processes , order statistics , weak convergence

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 3 • June, 1973
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