Extremes of a Type of Locally Stationary Gaussian Random Fields with Applications to Shepp Statistics
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Extremes of a Type of Locally Stationary Gaussian Random Fields with Applications to Shepp Statistics Zhongquan Tan1
· Shengchao Zheng1
Received: 31 March 2019 / Revised: 28 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Let {Z (τ, s), (τ, s) ∈ [a, b]×[0, T ]} with some positive constants a, b, T be a centered Gaussian random field with variance function σ 2 (τ, s) satisfying σ 2 (τ, s) = σ 2 (τ ). We first derive the exact tail asymptotics (as u → ∞) for the probability that the maximum M H (T ) = max(τ,s)∈[a,b]×[0,T ] [Z (τ, s)/σ (τ )] exceeds a given level u, for any fixed 0 < a < b < ∞ and T > 0; and we further derive the extreme limit law for M H (T ). As applications of the main results, we derive the exact tail asymptotics and the extreme limit laws for Shepp statistics with stationary Gaussian process, fractional Brownian motion and Gaussian integrated process as inputs. Keywords Extremes · Locally stationary Gaussian random fields · Shepp statistics · Exact tail asymptotics · Extreme limit law Mathematics Subject Classification (2010) Primary 60G15; Secondary 60G70
1 Introduction Let B(t) be a standard Brownian motion and define the Shepp statistics as Z (s) = sup B(s + τ ) − B(s), τ ∈[0,1]
for s ≥ 0. Since in various theoretical and applied problems, the Shepp statistics appears as the limit process due to the central limit theorem, vast interest has been
This work was supported by Natural Science Foundation of Zhejiang Province of China (No. LY18A010020) and National Science Foundation of China (No. 11501250).
B 1
Zhongquan Tan [email protected] College of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing 314001, People’s Republic of China
123
Journal of Theoretical Probability
paid to the analysis of Shepp statistics, see e.g., [2,5,9,15–17,23,24], [8], [18] and [19]. A natural extension of the Shepp statistics is the following process: Z H (s) = sup B H (s + τ ) − B H (s), τ ∈[0,1]
where B H (t) is a standard fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1). This type of Shepp statistics has been extensively studied in [4,6,21] and [20]. Noting that the variance of the increment B H (s + τ ) − B H (s) is τ 2H , another type of natural extension of the Shepp statistics is Z H (s) = sup
τ ∈[a,b]
B H (s + τ ) − B H (s) , 0 < a < b < ∞. τH
[22] investigated the extreme M H (T ) = sup0≤s≤T Z H (s) and derived the following results. Theorem 1.1 For H ∈ (0, 1), 0 < a < b < ∞ and any T > 0, it holds that P
max Z H (s) > u
s∈[0,T ]
=
2 T H2H
1/H 2 1 (1/a − 1/b)u H (u)(1 + o(1)), (1) 2
as u → ∞, and −x lim max P aT max Z H (s) − bT ≤ x − exp(−e ) = 0, T →∞ x∈R s∈[0,T ]
(2)
where aT =
√
2 ln T , bT = aT + aT−1 1 1 1 2 − ln ln T + ln 2− H H2H × (1/a − 1/b)(2π )−1/2 . H 2
(u) and H2H denote the tail distribution function of standard normal variable and Pickands constant (see definition in the next section ), respectively. The Shepp statistics Z H (s) is a non-G
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