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

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 $h_p(t)=(1/z_p(t))p\log\log t$.


Introduction and Main Results
Let X = {X(t) : t ∈ R + } be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, EX(t) = 0 and EX 2 (t) = 1. Suppose that the correlation function of X, r(t) = EX(t)X(0), satisfies the following regularity assumptions: r * (s) = sup t≥s |r(t)| < 1 for each s > 0, r(t) = O(t −2λ ) as t → ∞ for some λ > 0. (2) The analysis of extremes of Gaussian stochastic processes has a long history. The celebrated double sum method, primarily developed by Pickands, e.g., [8], and extended by seminal works of Piterbarg, e.g., [10] or monograph [9], plays central role in the extreme value theory of Gaussian processes. The technique developed there appeared to be an universal method, which may deliver answers also to classes of non-Gaussian processes, see for example, recent contributions of [5,6].
Laws of the iterated logarithm take important place in this theory, providing properties of extremal behavior of stochastic processes on large-time scale. One of important contributions in this domain is a result on the process ξ = {ξ(t) : t ≥ 0}, defined via ξ(t) = sup{s : 0 ≤ s ≤ t, X(s) ≥ (2 log s) 1/2 }. In particular, the law of the iterated logarithm implies that, see [11,12], lim sup t→∞ (ξ(t) − t) = 0 a.s.
Interestingly, under the above regularity assumptions, Shao [12] gave the lower bound of ξ(t) and obtained an Erdös-Révész type law of the iterated logarithm, that is, lim inf t→∞ log (ξ(t)/t) where H α is the Pickands' constant defined by H α = lim T →∞ T −1 Ee sup t∈[0,T ] ( √ 2B α/2 (t)−t α ) , with B α/2 = {B α/2 (t) : t ≥ 0} denoting fractional Brownian motion with Hurst index α/2 ∈ (0, 1], i.e., a continuous, centered Gaussian process with covariance function Equation (3) shows that for any t big enough there exists an s in [t − t(log t) (α−2)/(2α) · log 2 t, t] such that, almost surely, X(s) ≥ (2 log s) 1/2 and that the length of the interval t(log t) (α−2)/(2α) · log 2 t is smallest possible. Moreover, the bigger the parameter α is, the wider the interval will be. In this paper, we derive a counterpart of Shao's result for the order statistics process X r:n . Namely, for any n ≥ 0, we consider X 1 , . . . , X n , n mutually independent copies of X and denote by X r:n = {X r:n (t) : t ≥ 0} the rth smallest order statistics process, that is, for each t ≥ 0, 1 ≤ r ≤ n, Our first contribution is the theorem that extends classical findings of Qualls and Watanabe [11].
For all functions f that are positive and non-decreasing on some interval [T, ∞), T > 0, it follows that as the integral De ֒ bicki et al. [1,Theorem 2.2], see also [3], gave the expression for the asymptotic behavior of the probability in I f , namely is the distribution function of unit normal law, α/2 , 1 ≤ i ≤ n, are mutually independent fractional Brownian motions. H α,k is the generalized Pickands' constant introduced in [2]; see also [1]. Therefore, Theorem 1 provides a tractable criterion for settling the dichotomy of P (E f ).
Since I fp = ∞ for p ≥ 0, Theorem 1 implies that The second contribution of this paper is an Erdös-Révész type of law of the iterated logarithm for the process ξ p . Since then it follows that Theorem 2 shows that for t big enough, there exists an s in [t−h p (t), t] (as well as in [t, t+h p (t)] by (7)) such that X r:n (s) ≥ f p (s) and that the length of the interval h p (t) is smallest possible. One can retrieve (3)-(4) by setting n = 1, and p = 2−rα 2α + 1 = 2+α 2α . Theorem 2 not only generalizes Shao [12, Theorem 1.1], it also unveils the lacking so far structure of the lower bound of ξ p (t) by relating it, via h p (t), to the asymptotics of the tail distribution of the supremum of the underlying process evaluated at f p (t); in (3) t(log t) (α−2)/(2α) is of the same asymptotic order as the reciprocal of P sup s∈[0,1] X(s) > (2 log t) 1/2 . This shines new light on this type of results, which appear to be intrinsically connected with Gumbel limit theorems; see, e.g., [7], where the function h p (t) plays crucial role. We shall pursue this elsewhere.
The paper is organized as follows. In Section 2 we provide a collection of basic results on order statistics of stationary Gaussian processes, used throughout the paper, and prove auxiliary lemmas, which constitute building blocks of the proofs of the main results. These are given in the final part of the paper, Section 3.

Auxiliary Lemmas
We begin with some auxiliary lemmas that are later needed in the proofs. The following lemma is the general form of the Borel-Cantelli lemma; cf. [13].
The following two lemmas constitute useful tools for approximating the supremum of X r:n on a fixed interval by its maximum on a grid with a sufficiently dense mesh. Lemma 2. There exist positive constants K, c and u 0 such that for each θ > 0 and u ≥ u 0 .
Proof. Note that, by stationarity, there exists a constant K, that may vary from line to line, such that, for sufficiently large u, The last inequality follows from (5)  Lemma 3. For any θ > 0, as u → ∞, (1)).
The next lemma follows directly from [4, Theorem 2.4] and is a generalization of the classical Berman's inequality to order statistics.
Then, for any u 1 , . . . , u d > 0, for some positive constant C n,r depending only on n and r, Lemma 5. Under the conditions of Theorem 2, for any ε ∈ (0, 1), there exist positive constants K and ρ depending only on ε, α and λ such that Proof. Let, for any i ≥ 0 and ε ∈ (0, 1), For some θ > 0, define grid points in the interval I i , as follows Since f p is an increasing function, it easily follows that, with For any 1 ≤ l ≤ n and i ≥ 0, let X l,i be an independent copy of the process X l . Define a sequence of processes Now using Lemma 4 we find that Estimate of P 1 .
Since X r:n is a stationary process, from Equation 5 combined with Lemma 3, for any ε ∈ (0, 1), sufficiently large θ and S, Estimate of P 2 .

5
Finally, since the integrand in the definition ofÃ (r) si,usj,v is continuous and bounded on [0, r * (ε)], there exists a generic constant K not depending on S and T , which may differ from line to line, such that Therefore, for sufficiently large S, We can bound the first sum from the above by The second sum is bounded from above by Hence, for some positive constant ρ, depending only on ε, α and λ, which finishes the proof.  Proof. Let, for any i ≥ 0, a i = S + i so that y i = f p (a i ). Define grid points in the interval (a i , a i+1 ] as follows i /y i . Similarly as in the proof of Lemma 5, using Lemma 4 we have ai,uaj,v is as in (9).

Note that, by Lemma 3 combined with Equation 5
, provided that S is sufficiently large.
Completely similar to the estimation of P 2 in the proof of Lemma 5, we can arrive that there exist positive constants K and ρ, independent of S and T , such that, for sufficiently large S, The following lemma is a straightforward modification of Lemma 3.1 and 4.1 of Watanabe [14] and Qualls and Watanabe [11,Lemma 1.4].
Lemma 7. If Theorem 1 is true under the additional condition that for large t, it is true without the additional condition.

Proofs of the main results
Proof of Theorem 1. Note that the case I f < ∞ is straightforward and does not need any additional knowledge on process X r:n apart from the assumption of stationarity. Indeed, for sufficiently large T , and the Borel-Cantelli lemma completes this part of the proof since f is an increasing function. Now let f be any increasing function such that I f ≡ ∞. With the same notation as in Lemma 5 with f instead of f p , we find that, for any S > 0, where, recall, s i,u = S + i(1 + ε) + uθx , θ, ε > 0. Furthermore, for sufficiently large S and θ, cf. estimation of P 1 , The first limit is zero as a consequence of (15), and the second limit will be zero because of the asymptotic independence of the events E k . Indeed, there exist positive constants K and ρ, such that for any n > m, by the same calculations as in the estimate of P 2 in Lemma 5 after realizing that, by Lemma 7, we might restrict ourselves to the case when (14) holds. Therefore, P (E c i i.o.) = 1, which finishes the proof.

Proof of Theorem 2
Step 1. Let p > 1, then, for every ε ∈ (0, 1 4 ), Then by Lemma 5, where the last inequality follows by the fact that h p (t) = o(t), so that S k ∼ T k . Note that as k → ∞ (16) Hence, by the Borel-Cantelli lemma, Since ξ(t) is a non-decreasing random function of t, for every T k ≤ t ≤ T k+1 , we have For p > 1 elementary calculus implies which finishes the proof of this step.
Step 2. Let p > 1, then, for every ε ∈ (0, 1 4 ), Proof. As in the proof of the lower bound, put It suffices to show P (B n i.o.) = 1, that is Let a k i = S k + i and define grid points in the interval [a k i , a k i+1 ] as follows Clearly, for m ≥ 1, Putŷ k i = y k i − θ k i /y k i . Then, by Lemma 2, for some constants K independent of S and T , which may vary between (and among) lines, provided m is large enough. Therefore, To finish the proof of (18), we only need to show that Similarly to (16), we have Now from Lemma 6 it follows that for every k sufficiently large. Hence, Applying Lemma 4, we get for 0 ≤ t < k where, similarly to the proof of Lemma 5, It is easy to see that, so that, for 0 ≤ t < k and k large enough, and assuming without loss of generality that λ < 2, Hence we have, Step 3. If p ∈ (0, 1], then, for every ε ∈ (0, 1 4 ), Proof. Put T k = exp(k 1/p ), S k = T k exp −(1 + 2ε) 2 h p (T k ) . Proceeding the same as in the proof of (17), one can obtain that lim inf k→∞ log (ξ p (T k )/T k ) h p (T k )/T k ≥ −(1 + 2ε) 2 a.s.