Translator Disclaimer
June, 1980 Weak Convergence for the Maxima of Stationary Gaussian Processes Using Random Normalization
William P. McCormick
Ann. Probab. 8(3): 483-497 (June, 1980). DOI: 10.1214/aop/1176994723

Abstract

Let $\{X_k, k \geqslant 1\}$ be a stationary Gaussian sequence with $EX_1 = 0, EX^2_1 = 1$, and $EX_1X_{n+1} = r_n$. Let $c_n = (2 \ln n)^{\frac{1}{2}}, b_n = c_n - \ln(4\pi \ln n)/2c_n$ and set $M_n = \max_{1\leqslant k\leqslant n}X_k, \bar{X}_n = \frac{1}{n} \Sigma^n_{k=1}X_k$, and $s^2_n = \frac{1}{n} \Sigma^n_{k=1}(X_k - \bar{X}_n)^2$. If $r_n$ is not identically one and $(\ln n)/n\Sigma^n_{k=1}|r_k - r_n| = o(1)$, it is shown that \begin{equation*}\tag{1}\lim_{n\rightarrow\infty}P\big\{c_n\big(\frac{M_n - \bar{X}_n}{s_n} - b_n\big) \leqslant x\big\} = \exp\{-e^{-x}\}.\end{equation*} If we further assume $(r_n \ln n)^{-1} = o(1)$ then it is shown that \begin{equation*}\tag{2} \lim_{n\rightarrow\infty}P\big\{r^{-\frac{1}{2}}_n\big(\frac{M_n}{(1 - r_n)^{\frac{1}{2}}} - b_n\big) \leqslant x\big\} = (\frac{1 -\gamma}{2\pi})^{\frac{1}{2}}\int^x_{-\infty} e^{-\frac{(1 - \gamma)u^2}{2}}du\end{equation*} where $\gamma = F(\{o\})$ is the atom at zero of the spectral distribution associated with $r$. A version of these results for continuous time processes is also presented.

Citation

Download Citation

William P. McCormick. "Weak Convergence for the Maxima of Stationary Gaussian Processes Using Random Normalization." Ann. Probab. 8 (3) 483 - 497, June, 1980. https://doi.org/10.1214/aop/1176994723

Information

Published: June, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0434.60033
MathSciNet: MR573289
Digital Object Identifier: 10.1214/aop/1176994723

Subjects:
Primary: 60G10
Secondary: 60F99, 60G15

Rights: Copyright © 1980 Institute of Mathematical Statistics

JOURNAL ARTICLE
15 PAGES


SHARE
Vol.8 • No. 3 • June, 1980
Back to Top