Let $\{X_n, n \geqq 1\}$ be a real-valued, stationary Gaussian sequence with mean zero and variance one. Let $M_n = \max_{1\leqq i\leqq n} X_i, r_n = E(X_{n+1}X_1); c_n = (2 \ln n)^{\frac{1}{2}}$ and $b_n = c_n - \frac{1}{2}\lbrack\ln (4\pi \ln n)\rbrack/c_n$. Define $U_n = 2c_n(M_n - c_n)/\ln\ln n$ and $V_n = c_n(M_n - b_n)$. If $r_n = O(1/\ln n)$ as $n \rightarrow \infty$ then (i) $p(\lim \inf_{n\rightarrow\infty} U_n = -1) = p(\lim \sup_{n\rightarrow\infty} U_n = 1) = 1$, and (ii) $E\{\exp(tV_n)\} \rightarrow E\{\exp (tX)\}$ as $n \rightarrow \infty$ for all $t$ sufficiently small where $X$ is a random variable with distribution function $e^{-e^{-x}}; -\infty < x < \infty$.
Publié le : 1974-04-14
Classification:
Maxima,
Stationary Gaussian sequence,
law of iterated logarithms,
moment generating functions,
60G10,
60G15,
60G17,
60F15,
60F20,
62F30,
62E20
@article{1176996705,
author = {Mittal, Yash},
title = {Limiting Behavior of Maxima in Stationary Gaussian Sequences},
journal = {Ann. Probab.},
volume = {2},
number = {6},
year = {1974},
pages = { 231-242},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996705}
}
Mittal, Yash. Limiting Behavior of Maxima in Stationary Gaussian Sequences. Ann. Probab., Tome 2 (1974) no. 6, pp. 231-242. http://gdmltest.u-ga.fr/item/1176996705/