球面高斯随机场像集和图集的Hausdorff测度  

Hausdorff Measures of Image Set and Graph Set of Spherical Gaussian Random Fields

作　　者：桑利恒 陈振龙[1] Li Heng SANG;Zhen Long CHEN(School of Statistics and Mathematics,Zhejiang Gongshang University,Hangzhou310018,P.R.China;School of Mathematics and Finance,Chuzhou University,Chuzhou 239000,P.R.China)

机构地区：[1]浙江工商大学统计与数学学院,杭州310018 [2]滁州学院数学与金融学院,滁州239000

出　　处：《数学学报（中文版）》2023年第5期951-970,共20页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金项目(11971432);浙江省自然科学基金项目(LY21G010003,LQ22G010001);安徽省高校优秀青年人才基金项目(GXYQ202058);滁州学院科研启动基金项目(2021QD11)。

摘　　要：本文考虑了单位球面S2上取值于Rd中的球面高斯随机场T={T(t),t∈S^(2)}的像集和图集的Hausdorff测度,其中指数α满足2 <α <4.对S^(2)中每个紧集D,获得了当4 <(α-2)d时,以概率1,有0 <Φ-m(T(D))<∞,其中Φ-m(·)为Hausdorff测度,并且Φ(r)=r4/α-2loglog1/r.同时,也获得了图集的Hausdorff测度,即以概率1,有0 <Φ-m(Gr(T(D)))<∞,其中当(α-2)d <4时,Φ(r)=r^(2+(1-α-2/2)d)(log log1/r)((α-2)d/4);当(α-2)d>4时,Φ(r)=r^(4)/α-2log lor1/r.本文将欧氏空间相关结果推广到了球面情况.This paper is concerned with Hausdorff measures of the image and the graph for spherical Gaussian random fields T={T(t),t∈S^(2)}valued in Rd on the unit sphere S^(2) with 2<α<4.For each compact set D on S^(2),we show that when 4<(α-2)d,with probability one,0<Φ-m(T(D))<∞,where Φ-m denotes the Hausdorff measure with Hausdorff function d(r)=ra-2 loglog 1.Meanwhile,we obtain the Hausdorff measures of the graph,that is,with probability one,0<Φ-m(Gr(T(D)))<∞ for(α-2)d<4 where 0(r)=r^(2)+(1-42)a(log log 1)2/a(α-2)d and for(α-2)d>4 where d(r)=ra-2 log log 1.This paper extends the corresponding results in Euclidean space to the sphere case.

关 键 词：球面高斯随机场 像集 图集 HAUSDORFF测度 

分 类 号：O211.6[理学—概率论与数理统计]