The Annals of Probability

Maxima of Partial Samples in Gaussian Sequences

Yashaswini Mittal

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Abstract

Let $\{X_n, n \geqq 0\}$ be a Gaussian sequence with $EX_i \equiv 0; V(X_i) \equiv 1$ and $EX_iX_j = r_{ij}$. Define $M_n = \max_{0\leqq i \leqq n}X_i$ and $m_{n,k} = \max (X_i; i \in G_n)$ where $G_n = (t_1, \cdots, t_{n'})$ is a subset of $(0, 1, 2, \cdots, n)$ and $n' = \lbrack n/k\rbrack$ for some integer $k \geqq 1, \lbrack x\rbrack$ being the integral part of $x$. We show that $P\{c_n(M_n - m_{n, k}) \leqq x\} \rightarrow (1 + (k - 1)e^{-x})^{-1}$ as $n \rightarrow \infty$ for all $x \geqq 0$ where $c_n = (2 \log n)^\frac{1}{2}$, if the sequence is "moderately dependent," namely if \begin{equation*}\tag{1}(i) \sup_{ij} |r_{ij}| < \delta < 1\end{equation*} $$(ii) |r_{ij}| \leqq \rho(i - j) \text{for} |i - j| > N_0 \text{such that} \rho_n\log n = o(1).$$ Somewhat surprisingly the same result holds even though the sequence is "strongly dependent," namely if \begin{equation*}\tag{2}(i) r_{ij} = r(i - j); r_n \text{convex for} n \geqq 0 \text{and} r_n = o(1)\end{equation*} $$(ii) (r(n) \log n)^{-1} \text{is monotone and} o(1).$$

Article information

Source
Ann. Probab., Volume 6, Number 3 (1978), 421-432.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995528

Digital Object Identifier
doi:10.1214/aop/1176995528

Mathematical Reviews number (MathSciNet)
MR482950

Zentralblatt MATH identifier
0378.60037

JSTOR
links.jstor.org

Keywords
6030 6050 6285 Maxima Gaussian sequences strong and weak correlations partial samples logistic distribution

Citation

Mittal, Yashaswini. Maxima of Partial Samples in Gaussian Sequences. Ann. Probab. 6 (1978), no. 3, 421--432. doi:10.1214/aop/1176995528. https://projecteuclid.org/euclid.aop/1176995528


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