Extremes of a Type of Locally Stationary Gaussian Random Fields with Applications to Shepp Statistics

Zhongquan Tan () and Shengchao Zheng
Additional contact information
Zhongquan Tan: Jiaxing University
Shengchao Zheng: Jiaxing University

Journal of Theoretical Probability, 2020, vol. 33, issue 4, 2258-2279

Abstract: Abstract Let $$\{Z(\tau ,s), (\tau ,s)\in [a,b]\times [0,T]\}$$ { 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 $$\sigma ^{2}(\tau ,s)$$ σ 2 ( τ , s ) satisfying $$\sigma ^{2}(\tau ,s)=\sigma ^{2}(\tau )$$ σ 2 ( τ , s ) = σ 2 ( τ ) . We first derive the exact tail asymptotics (as $$u \rightarrow \infty $$ u → ∞ ) for the probability that the maximum $$M_H(T) = \max _{(\tau , s) \in [a, b] \times [0, T]} [Z(\tau , s) / \sigma (\tau )]$$ M H ( T ) = max ( τ , s ) ∈ [ a , b ] × [ 0 , T ] [ Z ( τ , s ) / σ ( τ ) ] exceeds a given level u, for any fixed $$0 0$$ T > 0 ; and we further derive the extreme limit law for $$M_{H}(T)$$ 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; Primary 60G15; Secondary 60G70 (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-019-00953-6 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:33:y:2020:i:4:d:10.1007_s10959-019-00953-6

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-019-00953-6

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().