An Erdös--Révész 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 $f$, $\mathbb{P}({\mathcal{E}}_f)=\mathbb{P}(X_{r:n}(t)>f(t)\text{i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\mathbb{P}({\sup }_{s\in [0,1]}{X}_{r:n}(s)>f_p(t))=\mathcal{C}(t{\log }^{1-p}t{)}^{-1}$, $\mathcal{C}>0$, $\mathbb{P}({\mathcal{E}}_{f_p})=1_{\{p\ge 0\}}$. 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 ${\mathrm{limsup}}_{t\to \infty }({\xi }_p(t)-t)=0$ a.s.. Complementary, we prove an Erdös-Révész type law of the iterated logarithm lower bound on ${\xi }_p(t)$, i.e., ${\mathrm{liminf}}_{t\to \infty }({\xi }_p(t)-t)/h_p(t)=-1$ a.s., $p>1$, ${\mathrm{liminf}}_{t\to \infty }\log ({\xi }_p(t)/t)/(h_p(t)/t)=-1$ a.s., $p\in (0,1]$, where $h_p(t)=(1/z_p(t))p\log \log t$.

References in corpus (2)

• On the Supremum of gamma-reflected Processes with Fractional Brownian Motion as Input
• On the Probability of Conjunctions of Stationary Gaussian Processes

Cited by in corpus (2)

• An Erdös-Révész type law of the iterated logarithm for reflected fractional Brownian motion
• Sample path properties of reflected Gaussian processes