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

Source: [An Erd\\"os--R\\'ev\\'esz type law of the iterated logarithm for order statistics of a stationary Gaussian process](<https://www.sota2.com/research/paper/68fa54a21e209131ce98d1fb>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T13:06:29.436Z

Authors: K. D\\k{e}bicki, K.M. Kosi\\'nski

Published: 2016-06-05T12:33:55.000Z

## 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 $h\_p(t)=(1/z\_p(t))p\\log\\log t$.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1606.01502>)

[PDF](<https://arxiv.org/pdf/1606.01502.pdf>)