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 \(t\rightarrow 0\) 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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1 Introduction and Main Results
Let \(X=\{X(t):t\in \mathbb R_+\}\) be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, \(\mathbb EX(t) = 0\) and \(\mathbb EX^2(t) = 1\). Suppose that the correlation function of \(X, r(t) = \mathbb EX(t) X(0)\), satisfies the following regularity assumptions:
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 \(\xi =\{\xi (t):t\ge 0\}\), defined via \(\xi (t) = \sup \{s:0\le s\le t, X(s) \ge (2\log s)^{1/2}\}\). In particular, the law of the iterated logarithm implies that, see [11, 12],
Interestingly, under the above regularity assumptions, [12] gave the lower bound of \(\xi (t)\) and obtained an Erdös–Révész type law of the iterated logarithm, that is,
where \(\mathcal H_{\alpha }\) is the Pickands’ constant defined by \(\mathcal H_{\alpha }=\lim _{T\rightarrow \infty } T^{-1} \mathbb {E}e^{\sup _{t\in [0,T]}( \sqrt{2}B_{\alpha /2}(t)-t^{\alpha })}\), with \(B_{\alpha /2}=\{B_{\alpha /2}(t):t\ge 0\}\) denoting fractional Brownian motion with Hurst index \(\alpha /2\in (0,1]\), i.e., a continuous, centered Gaussian process with covariance function
Equation (3) shows that for any t big enough there exists an s in \([t-t(\log t)^{(\alpha -2)/(2\alpha )}\cdot \log _2 t, t]\) such that, almost surely, \(X(s)\ge (2\log s)^{1/2}\) and that the length of the interval \(t(\log t)^{(\alpha -2)/(2\alpha )}\cdot \log _2 t\) is smallest possible. Moreover, the bigger the parameter \(\alpha \) 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 \(n\ge 0\), we consider \(X_1,\ldots ,X_n, n\) mutually independent copies of X and denote by \(X_{r:n}=\{X_{r:n}(t):t\ge 0\}\) the rth smallest order statistics process, that is, for each \(t\ge 0, 1\le r\le n\),
Our first contribution is the theorem that extends classical findings of Qualls and Watanabe [11].
Theorem 1
For all functions f that are positive and non-decreasing on some interval \([T,\infty ), T>0\), it follows that
as the integral
[1, Theorem 2.2], see also [3], gave the expression for the asymptotic behavior of the probability in \(\mathscr {I}_f\), namely
where \(\hat{r} = n-r+1, \Psi (u)=1-\Phi (u)\) and \(\Phi (u)\) is the distribution function of unit normal law,
and \(B_{\alpha /2}^{(i)}, 1\le i\le n\), are mutually independent fractional Brownian motions. \(\mathcal H_{\alpha , k}\) is the generalized Pickands’ constant introduced in [2]; see also [1]. Therefore, Theorem 1 provides a tractable criterion for settling the dichotomy of \(\mathbb P\left( \mathscr {E}_f \right) \).
For instance, let
One easily checks that, as \(u\rightarrow \infty \),
Hence, for any \(p\in \mathbb R\),
Furthermore,
Next, consider the process \(\xi _p=\{\xi _p(t):t\ge 0\}\) defined as
Since \(\mathscr {I}_{f_p} = \infty \) for \(p\ge 0\), Theorem 1 implies that
Let, cf. (6),
The second contribution of this paper is an Erdös–Révśz type of law of the iterated logarithm for the process \(\xi _p\).
Theorem 2
If \(p>1\), then
If \(p\in (0,1]\), then
Now, let us complementary put \(\eta _p = \{\eta _p(t):t\ge 0\}\), where
Since
and
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 \(X_{r:n}(s)\ge f_p(s)\) and that the length of the interval \(h_p(t)\) is smallest possible. One can retrieve (3)–(4) by setting \(n=1\), and \(p= \frac{2-\hat{r}\alpha }{2\alpha } + 1 = \frac{2+\alpha }{2\alpha }\). Theorem 2 not only generalizes [12, Theorem 1.1], it also unveils the lacking so far structure of the lower bound of \(\xi _p(t)\) 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) \(t(\log t)^{(\alpha -2)/(2\alpha )}\) is of the same asymptotic order as the reciprocal of \(\mathbb P\left( \sup _{s\in [0,1]} X(s) > (2\log t)^{1/2} \right) \). 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 Sect. 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, Sect. 3.
2 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].
Lemma 1
Consider a sequence of events \(\{E_k:k\ge 0\}\). If
then \(\mathbb P\left( E_n\text { i.o.} \right) = 0\). Whereas, if
then \(\mathbb P\left( E_n\text { i.o.} \right) = 1\).
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 \(\theta >0\) and \(u\ge u_0\).
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) and the classical result of [7, Lemma 12.2.5], where the constant \(c>0\) is given therein. \(\square \)
The proof of the following lemma follows line-by-line the same reasoning as the proof of [1, Theorem 2.2], and thus we omit it.
Lemma 3
For any \(\theta >0\), as \(u\rightarrow \infty \),
The next lemma follows directly from [4, Theorem 2.4] and is a generalization of the classical Berman’s inequality to order statistics.
Lemma 4
For some \(n,d\ge 1\), and any \(1\le l\le n\) let \(\{\xi _l^{(0)}(i): 1\le i \le d\}\) and \(\{\xi _l^{(1)}(i): 1\le i \le d\}\) be a sequence of \(\mathcal N(0,1)\) variables and set \(\sigma ^{(\kappa )}_{il,jk} = E{\xi _l^{(\kappa )}(i)\xi _{k}^{(\kappa )}}(j), \kappa =0,1\). For any \(1\le r\le n\) and \(1\le i\le d\), let \(\xi _{r:n}^{(\kappa )}(i)\) be the rth order statistic of \(\xi _{1}^{(\kappa )}(i),...,\xi _{n}^{(\kappa )}(i)\). Suppose that, for any \(1\le i,j\le d, 1\le l,k\le n, \kappa =0,1\),
for some \(\sigma _{ij}^{(\kappa )}\). Now define
Then, for any \(u_1,\ldots ,u_d>0\), for some positive constant \(C_{n,r}\) depending only on n and r,
Lemma 5
Under the conditions of Theorem 2, for any \(\varepsilon \in (0,1)\), there exist positive constants K and \(\rho \) depending only on \(\varepsilon , \alpha \) and \(\lambda \) such that
for any \(T-1\ge S\ge K\).
Proof
Let, for any \(i\ge 0\) and \(\varepsilon \in (0,1)\),
For some \(\theta >0\), define grid points in the interval \(I_i\), as follows
Since \(f_p\) is an increasing function, it easily follows that, with \(T(S,\varepsilon )=[(T-S-1)/(1+\varepsilon )]\),
For any \(1\le l\le n\) and \(i\ge 0\), let \(X_{l,i}\) be an independent copy of the process \(X_l\). Define a sequence of processes \(Y_l=\{Y_l(t):t\in \cup _i I_i\}\) as \(Y_l(t) = X_{l,i}(t)\), if \(t\in I_i\). Let \(Y_{r:n}=\{Y_{r:n}(t):t\ge 0\}\) be the rth order statistic of \(Y_1,\ldots , Y_n\). Put
and note that
Now using Lemma 4 we find that
Estimate of \(P_1\).
Since \(X_{r:n}\) is a stationary process, from Eq. 5 combined with Lemma 3, for any \(\varepsilon \in (0,1)\), sufficiently large \(\theta \) and S,
Estimate of \(P_2\).
Noting that, for any \(0\le i<j, 0\le u\le L_i, 0\le v\le L_j\);
we have
Without loss of generality assume that \(\lambda < 2\). From (2) it follows that there is \(s_0\) such that for every \(s>s_0\),
Finally, since the integrand in the definition of \(\tilde{A}_{s_{i,u}s_{j,v}}^{(r)}\) is continuous and bounded on \([0,r^*(\varepsilon )]\), 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 \(\rho \), depending only on \(\varepsilon , \alpha \) and \(\lambda \),
which finishes the proof. \(\square \)
Lemma 6
Under the conditions of Theorem 2, for any \(\varepsilon \in (0,1)\), there exist positive constants K and \(\rho \) depending only on \(\varepsilon , \alpha \) and \(\lambda \) such that
for any \(T-1\ge S\ge K\), where \(y_i = f_p(S+i)\) and \(\theta _i = y_i^{-\frac{8}{\alpha }}\).
Proof
Let, for any \(i\ge 0, a_i = S + i\) so that \(y_i = f_p(a_i)\). Define grid points in the interval \((a_i, a_{i+1}]\) as follows
Finally, put \(\hat{y}_i = y_i - \theta _i^{\frac{\alpha }{4}}/y_i\). Similarly as in the proof of Lemma 5, using Lemma 4 we have
where \(\tilde{A}_{a_{i,u}a_{j,v}}^{(r)}\) is as in (9).
Estimate of \(P_1'\).
Note that, by Lemma 3 combined with Eq. 5,
provided that S is sufficiently large.
Estimate of \(P_2'\).
Noting that, for \(j\ge i+2\), and any \(0\le u\le L_i, 0\le v\le L_j\);
we have
Since the integrand in definition of \(\tilde{A}_{a_{i,u}a_{j,v}}^{(r)}\) is continuous and bounded on \([0,r^*(1)]\), there exists a constant K such that
On the other hand, by (1), there exist positive constants \(s_0<1\), such that, for every \(0\le s\le s_0\),
Hence,
Therefore, by (11)–(13) we obtain
Completely similar to the estimation of \(P_2\) in the proof of Lemma 5, we can arrive that there exist positive constants K and \(\rho \), independent of S and T, such that, for sufficiently large S,
\(\square \)
The following lemma is a straightforward modification of Lemmas 3.1 and 4.1 of [14] and [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.
3 Proofs of the Main Results
Proof of Theorem 1
Note that the case \(\mathscr {I}_f <\infty \) 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 \(\mathscr {I}_f\equiv \infty \). With the same notation as in Lemma 5 with f instead of \(f_p\), we find that, for any \(S>0\),
where, recall, \(s_{i,u}= S + i(1+\varepsilon ) + u \theta x_i^{-2/\alpha }, L_i=[1/(\theta x_i^{-2/\alpha })], \theta ,\varepsilon >0\). Furthermore, for sufficiently large S and \(\theta \), cf. estimation of \(P_1\),
Let \(E_i=\{\max _{1\le u\le L_i} X_{r:n}(s_{i,u}) \le x_i\}\), and note that
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 \(\rho \), 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, \(\mathbb P\left( E_i^c \text { i.o.} \right) =1\), which finishes the proof. \(\square \)
Proof of Theorem 2
Step 1. Let \(p>1\), then, for every \(\varepsilon \in (0,\frac{1}{4})\),
\(\square \)
Proof
Let \(\{T_k:k\ge 1\}\) be a sequence such that \(T_k\rightarrow \infty \), as \(k\rightarrow \infty \). Put \(S_k =T_k -(1+2\varepsilon )^2 h_p(T_k)\). Then by Lemma 5,
where the last inequality follows by the fact that \(h_p(t)=o(t)\), so that \(S_k\sim T_k\). Note that as \(k\rightarrow \infty \)
Now take \(T_k = \exp (k^{1/p})\). Then
Hence, by the Borel–Cantelli lemma,
Since \(\xi (t)\) is a non-decreasing random function of t, for every \(T_k\le t\le T_{k+1}\), we have
For \(p >1\) elementary calculus implies
so that
which finishes the proof of this step. \(\square \)
Step 2. Let \(p>1\), then, for every \(\varepsilon \in (0,\frac{1}{4})\),
Proof
As in the proof of the lower bound, put
Let
It suffices to show \(\mathbb P\left( B_n \text { i.o.} \right) =1\), that is
Let \(a_i^k = S_k + i\) and define grid points in the interval \([a_i^k,a_{i+1}^k]\) as follows
Put
Clearly, for \(m\ge 1\),
Put \(\hat{y}_i^k = y_i^k-\theta _i^k/y_i^k\). 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,
and
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 \(0\le t<k\)
where, similarly to the proof of Lemma 5,
where
It is easy to see that,
so that, for \(0\le t<k\) and k large enough, and assuming without loss of generality that \(\lambda <2\),
Therefore,
Hence we have,
Now (19) follows from (21), (22) and (20) and the general form of the Borel–Cantelli lemma. \(\square \)
Step 3. If \(p\in (0,1]\), then, for every \(\varepsilon \in (0,\frac{1}{4})\),
and
Proof
Put
Proceeding the same as in the proof of (17), one can obtain that
On the other hand it is clear that
since
This proves (23).
Let
Noting that
along the same lines as in the proof of (18), we also have
which proves (24). \(\square \)
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Acknowledgments
K. Dębicki was partially supported by National Science Centre Grant No. 2015/17/B/ST1/01102 (2016-2019). Research of K. Kosiński was conducted under scientific Grant No. 2014/12/S/ST1/00491 funded by National Science Centre.
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Dębicki, K., Kosiński, K.M. An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process. J Theor Probab 31, 579–597 (2018). https://doi.org/10.1007/s10959-016-0710-8
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DOI: https://doi.org/10.1007/s10959-016-0710-8