An Erd\"os--R\'ev\'esz type law of the iterated logarithm for order statistics of a stationary Gaussian process

Abstract

Let $\{X(t):t\in\mathbb R_+\}$ be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, $\mathbb E X(t) = 0$, $\mathbb E X^2(t) = 1$ and correlation function satisfying (i) $r(t) = 1 - C|t|^{\alpha} + o(|t|^{\alpha})$ as $t\to 0$ for some $0\le\alpha\le 2, C>0$, (ii) $\sup_{t\ge s}|r(t)|<1$ for each $s>0$ and (iii) $r(t) = O(t^{-\lambda})$ as $t\to\infty$ for some $\lambda>0$. For any $n\ge 1$, consider $n$ mutually independent copies of $X$ and denote by $\{X_{r:n}(t):t\ge 0\}$ the $r$th smallest order statistics process, $1\le r\le n$. We provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\mathbb P(\mathscr E_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text{i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\mathbb P(\sup_{s\in[0,1]}X_{r:n}(s)>f_p(t))=\mathscr C(t\log^{1-p} t)^{-1}$, $\mathscr C>0$, $\mathbb P(\mathscr E_{f_p})= 1_{\{p\ge 0\}}$. Consequently, with $\xi_p (t) = \sup\{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}$, for $p\ge 0$, $\lim_{t\to\infty}\xi_p(t)=\infty$ and $\limsup_{t\to\infty}(\xi_p(t)-t)=0$ a.s.. Complementary, we prove an Erd\"os-R\'ev\'esz type law of the iterated logarithm lower bound on $\xi_p(t)$, i.e., $\liminf_{t\to\infty}(\xi_p(t)-t)/h_p(t) = -1$ a.s., $p>1$, $\liminf_{t\to\infty}\log(\xi_p(t)/t)/(h_p(t)/t) = -1$ a.s., $p\in(0,1]$, where $h_p(t)=(1/z_p(t))p\log\log t$.