自相似随机过程的顺序统计量的极值  

Extremes of order statistics of self-similar processes

作　　者：凌成秀[1,2] 彭作祥[1] 

机构地区：[1]西南大学数学与统计学院,重庆400715 [2]Department of Actuarial Science,Faculty of Business and Economics,University of Lausanne,Lausanne 1015,Switzerland

出　　处：《中国科学：数学》2016年第8期1139-1148,共10页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11171275);中央高校基本科研业务费专项资金(批准号:XJDK2016C118和SWU115089)资助项目

摘　　要：设{X_i(t),t≥0}(1≤i≤n)为独立且与随机过程{X(t),t≥0}具有相同的任意有限维分布的随机过程.给定门限u>0和第r个上端顺序统计量X_(r:n),定义第r个关联点集为C_r(u):={t∈[0,1]:X_(r:n)(t)>u}.计算p_r(u)=P{C_r(u)≠φ}在大脑图像处理和数字交互系统等领域中有广泛的应用背景.本文考虑具有概率连续的样本轨道的实值自相似过程X,满足一定的Albin条件,当u→∞时p_r(u)的渐近式,同时得到X_(r:n)超过递增门限的平均逗留时间的渐近式.最后,这些理论结果应用到广义偏Gauss自相似过程(包括χ过程、双分式Brown运动和子分式Brown运动)等重要的自相似过程.Let {X_i（t）,t≥0}（1≤i≤n） be independent copies of a random process {X（t）,t≥0}.For a given positive threshold u,define the set of r-th conjunctions C_r（u） ：= {t ∈ [0,1] ：X_（r：n）（t） u} with X_（r：n）the r-th largest order statistics of X_i（1≤i≤n）.In numerical applications such as brain mapping and digital communication systems,what of interest is the approximation of p_r（u） = P{C_r（u）≠φ} as u → ∞.In this paper,we consider X to be a self-similar R-valued process with P-continuous sample paths.By imposing the Albin＇s conditions directly on X,we establish an exact asymptotic expansion of p_r（u） as u tends to infinity.As a by-product we derive the asymptotic tail behaviour of the mean sojourn time of Xr：n over an increasing threshold.Finally,our findings are applied to establishing the approximation of pr（u） with X a generalized skew-Gaussian self-similar process including χ processes,bi-fractional Brownian motions and sub-fractional Brownian motions.

关 键 词：自相似过程 顺序统计过程 平均逗留时间 偏Gauss过程 

分 类 号：O212.1[理学—概率论与数理统计]