Shengchao Zheng | Searcharxiv

arXiv2023

On the maxima of nonstationary random fields subject to missing observations

Motivated by the papers of Mladenovc and Piterbarg (2006), Krajka (2011) and Pereira and Tan (2017), we study the limit properties for the maxima from nonstationary random fields subject to missing observations and obtain the weakly convergence and almost sure convergence results for these maxima. Some examples such as Gaussian random fields, $chi$-random fields and Gaussian order statistics fields are given to illustrate the obtained results.

math.PR↗

arXiv2019

Extremes of a type of locally stationary Gaussian random fields with applications to Shepp statistics

Let $\{Z(τ,s), (τ,s)\in [a,b]\times[0,T]\}$ 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 firstly derive the exact tail asymptotics for the maximum $M_{H}(T)=\max_{(τ,s)\in[a,b]\times[0,T]}Z(τ,s)/σ(τ)$ up crossing some level $u$ with any fixed $0 0$; 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 law for Shepp statistics with stationary Gaussian process, fractional Brownian motion and Gaussian integrated process as input.

math.PR↗