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

Let $$\{X(t):t\in \mathbb R_+\}$$ { X ( t ) : t ∈ R + } be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, $$\mathbb E X(t) = 0, \mathbb E X^2(t) = 1$$ E X ( t ) = 0 , E X 2 ( t ) = 1 and correlation function satisfying (i) $$r(t) = 1 - C|t|^{\alpha } + o(|t|^{\alpha })$$ r ( t ) = 1 - C | t | α + o ( | t | α ) as $$t\rightarrow 0$$ t → 0 for some $$0\le \alpha \le 2$$ 0 ≤ α ≤ 2 and $$C>0$$ C > 0 ; (ii) $$\sup _{t\ge s}|r(t)| 0$$ s > 0 and (iii) $$r(t) = O(t^{-\lambda })$$ r ( t ) = O ( t - λ ) as $$t\rightarrow \infty $$ t → ∞ for some $$\lambda >0$$ λ > 0 . For any $$n\ge 1$$ n ≥ 1 , consider n mutually independent copies of X and denote by $$\{X_{r:n}(t):t\ge 0\}$$ { X r : n ( t ) : t ≥ 0 } the rth smallest order statistics process, $$1\le r\le n$$ 1 ≤ r ≤ n . We provide a tractable criterion for assessing whether, for any positive, non-decreasing function $$f, \mathbb P(\mathscr {E}_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text { i.o.})$$ f , P ( E f ) = P ( X r : n ( t ) > f ( t ) i.o. ) equals 0 or 1. Using this criterion we find, for a family of functions $$f_p(t)$$ f p ( t ) such that $$z_p(t)=\mathbb P(\sup _{s\in [0,1]}X_{r:n}(s)>f_p(t))=O((t\log ^{1-p} t)^{-1})$$ z p ( t ) = P ( sup s ∈ [ 0 , 1 ] X r : n ( s ) > f p ( t ) ) = O ( ( t log 1 - p t ) - 1 ) , that $$\mathbb P(\mathscr {E}_{f_p})= 1_{\{p\ge 0\}}$$ P ( E f p ) = 1 { p ≥ 0 } . Consequently, with $$\xi _p (t) = \sup \{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}$$ ξ p ( t ) = sup { s : 0 ≤ s ≤ t , X r : n ( s ) ≥ f p ( s ) } , for $$p\ge 0$$ p ≥ 0 we have $$\lim _{t\rightarrow \infty }\xi _p(t)=\infty $$ lim t → ∞ ξ p ( t ) = ∞ and $$\limsup _{t\rightarrow \infty }(\xi _p(t)-t)=0$$ lim sup t → ∞ ( ξ p ( t ) - t ) = 0 a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on $$\xi _p(t)$$ ξ p ( t ) , namely, that $$\liminf _{t\rightarrow \infty }(\xi _p(t)-t)/h_p(t) = -1$$ lim inf t → ∞ ( ξ p ( t ) - t ) / h p ( t ) = - 1 a.s. for $$p>1$$ p > 1 and $$\liminf _{t\rightarrow \infty }\log (\xi _p(t)/t)/(h_p(t)/t) = -1$$ lim inf t → ∞ log ( ξ p ( t ) / t ) / ( h p ( t ) / t ) = - 1 a.s. for $$p\in (0,1]$$ p ∈ ( 0 , 1 ] , where $$h_p(t)=(1/z_p(t))p\log \log t$$ h p ( t ) = ( 1 / z p ( t ) ) p log log t .