Extremes of order statistics of stationary processes

Abstract

Let {X i (t), t ≥ 0}, 1 ≤ i ≤ n be independent copies of a stationary process {X(t), t ≥ 0}. For given positive constants u, T , define the set of rth conjunctions C r,T,u := {t ∈ [0, T ] : X r:n (t) > u} with X r:n (t) the rth largest order statistics of X i (t), t ≥ 0, 1 ≤ i ≤ n. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions C r,T,u is not empty. Imposing the Albin's conditions on X, in this paper we obtain an exact asymptotic expansion of this probability as u tends to infinity. Furthermore, we establish the tail asymptotics of the supremum of the order statistics processes of skew-Gaussian processes and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes. As a by-product we derive a version of Li and Shao's normal comparison lemma for the minimum and the maximum of Gaussian random vectors.

Extremes of order statistics of self-similar processes

“…In the following, we only present the proof for the case that r = n − 1 and m = 1, the other cases follow by similar arguments. Note that we have by Lemma 3.1 in [19]…”

“…Note that the correlation functions of the above centered stationary Gaussian processes have regular varying tails at zero. Using the well-known results for stationary Gaussian processes (see, e.g., [21,6]) and the Lamperti's propositions in [5], it is easy to show that the above two self-similar Gaussian processes satisfy the conditions of The generalized self-similar skew-Gaussian processes: Recently, the skew-Gaussian processes have received a lot of attentions from both theoretical and applicable fields; see, e.g., [1,9,19]. Next, we will consider this non-Gaussian self-similar process and establish the tail asymptotic results by using our theorems in Section 3.…”

“…For smooth Gaussian random fields approximations of p n,T (u) are discussed for instance in [8,16,24]; results for non-Gaussian random fields can be found for instance in [10]. The recent contributions [17,19] derived asymptotic expansions of p r,T (u) Date: February 23, 2015.…”

“…Motivated by [5,19] and the tractability of self-similar processes, in this paper, we shall investigate the asymptotic behaviour of p r,T (u) as u → ∞ and T fixed. The main methodology employed here is from Patrik Albin [5].…”

Limit laws on extremes of nonhomogeneous Gaussian random fields

“…We refer to McCormick (1980), Konstant and Piterbarg (1993) and Piterbarg (1996) for further extensions to Gaussian processes and fields; Leadbetter and Rootzén (1982) and Albin (1990) for stationary non-Gaussian processes. For more related extensions, we refer to Dȩbicki, Hashorva, Ji and Ling (2015) and the reference therein.…”