# An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Journal of Theoretical Probability, Sep 2016

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 $t\rightarrow 0$ for some $0\le \alpha \le 2$ and $C>0$; (ii) $\sup _{t\ge s}|r(t)|<1$ for each $s>0$ and (iii) $r(t) = O(t^{-\lambda })$ as $t\rightarrow \infty$ 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 rth 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, for a family of functions $f_p(t)$ such that $z_p(t)=\mathbb P(\sup _{s\in [0,1]}X_{r:n}(s)>f_p(t))=O((t\log ^{1-p} t)^{-1})$, that $\mathbb P(\mathscr {E}_{f_p})= 1_{\{p\ge 0\}}$. Consequently, with $\xi _p (t) = \sup \{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}$, for $p\ge 0$ we have $\lim _{t\rightarrow \infty }\xi _p(t)=\infty$ and $\limsup _{t\rightarrow \infty }(\xi _p(t)-t)=0$ a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on $\xi _p(t)$, namely, that $\liminf _{t\rightarrow \infty }(\xi _p(t)-t)/h_p(t) = -1$ a.s. for $p>1$ and $\liminf _{t\rightarrow \infty }\log (\xi _p(t)/t)/(h_p(t)/t) = -1$ a.s. for $p\in (0,1]$, where $h_p(t)=(1/z_p(t))p\log \log t$.

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K. Dębicki, K. M. Kosiński. An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process, Journal of Theoretical Probability, 2016, 1-19, DOI: 10.1007/s10959-016-0710-8