Annals of Probability

Almost Sure Limit Points of Maxima of Stationary Gaussian Sequences

H. Vishnu Hebbar

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Abstract

Let $\{X_n, n \geqslant 1\}$ be a discrete-parameter stationary Gaussian process with $E(X_i) = 0, E(X^2_i) = 1$ for all $i$ and $E(X_iX_{i+n}) = r(n)$. Let $M_n =$ maximum$(X_1, X_2 \cdots X_n)$. Under the condition that either $(\log n)^{1+\gamma}r(n) = O(1)$ as $n \rightarrow \infty$, for some $\gamma > 0$ or $\sum^\infty_{j=1}r^2(j) < \infty$, the set of all almost sure limit points of the vector sequence $$\big\{\frac{M_{1,n} - b_n}{a_n}, \frac{M_{2,n} - b_n}{a_n}, \cdots \frac{M_{p,n} - b_n}{a_n}\big\}$$ is obtained, where $(M_{j,n}), j = 1,2 \cdots p$ are independent copies of $(M_n); a_n = (\log\log n)(2 \log n)^{-\frac{1}{2}}$ and $b_n = (2 \log n)^{\frac{1}{2}}$.

Article information

Source
Ann. Probab., Volume 8, Number 2 (1980), 393-399.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176994788

Digital Object Identifier
doi:10.1214/aop/1176994788

Mathematical Reviews number (MathSciNet)
MR566605

Zentralblatt MATH identifier
0428.60041

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G15: Gaussian processes 60G10: Stationary processes

Keywords
Maxima stationary Gaussian sequence limit point

Citation

Hebbar, H. Vishnu. Almost Sure Limit Points of Maxima of Stationary Gaussian Sequences. Ann. Probab. 8 (1980), no. 2, 393--399. doi:10.1214/aop/1176994788. https://projecteuclid.org/euclid.aop/1176994788


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