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

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.