The Annals of Probability

Weak Convergence for the Maxima of Stationary Gaussian Processes Using Random Normalization

William P. McCormick

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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.

Article information

Source
Ann. Probab., Volume 8, Number 3 (1980), 483-497.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176994723

Digital Object Identifier
doi:10.1214/aop/1176994723

Mathematical Reviews number (MathSciNet)
MR573289

Zentralblatt MATH identifier
0434.60033

JSTOR
links.jstor.org

Subjects
Primary: 60G10: Stationary processes
Secondary: 60G15: Gaussian processes 60F99: None of the above, but in this section

Keywords
Maxima stationary Gaussian processes limit distribution

Citation

McCormick, William P. Weak Convergence for the Maxima of Stationary Gaussian Processes Using Random Normalization. Ann. Probab. 8 (1980), no. 3, 483--497. doi:10.1214/aop/1176994723. https://projecteuclid.org/euclid.aop/1176994723


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