2014/03/28 by Krzysztof Dȩbicki, Krzysztof Debicki, Debicki, Krzysztof +6 · 2 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Financial Risk and Volatility Modeling #Methodology (stat.ME) #Other Statistics (stat.OT) #Probability (math.PR) #Statistical Methods and Inference #math.PR #stat.ME #stat.OT
paper · pdf · doi:10.48550/arxiv.1403.7354
20 pages, revised version
openalex publication_date 2014/03/28 · arxiv created 2014/08/06 · arxiv updated 2014/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \Xi(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 Cr,T,u:= \t∈ [0,T]: Xr:n(t) > u\ with Xr:n(t) the rth largest order statistics of X1(t), … , Xn(t), t≥ 0. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions Cr,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. Further, we establish the tail asymptotics of the supremum of a generalized skew-Gaussian process 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.