Limsup Results and LIL for Partial Sum Processes of a Gaussian Random Field  

作　　者：Yong-Kab CHOI Mikls CSRG 

机构地区：[1]Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea [2]School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive,Ottawa, K1S 5B6, Canada

出　　处：《Acta Mathematica Sinica,English Series》2008年第9期1497-1506,共10页数学学报（英文版）

基　　金：NSERC Canada grants of Miklos Csorgo and Barbara Szyszkowicz at Carleton University,Ottawa,and by KRF-2003-C00098;NSERC Canada grants at Carleton University,Ottawa

摘　　要：Let {&#958;<SUB> j </SUB>; j &#8712; &#8484;<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where &#8484;<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space &#8477;<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in &#8484;<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n &#8712; &#8484;<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = &#931;<SUB> m【j&#8804;n </SUB>&#950;<SUB> j </SUB>, &#963;<SUP>2</SUP>(|n&#8722;m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that &#963;(|n|) can be extended to a continuous function &#963;(t) of t 】 0, which is nondecreasing and regularly varying with exponent &#945; at b &#8805; 0 for some 0 【 &#945; 【 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.Let {&#958;<SUB> j </SUB>; j &#8712; &#8484;<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where &#8484;<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space &#8477;<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in &#8484;<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n &#8712; &#8484;<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = &#931;<SUB> m&lt;j&#8804;n </SUB>&#950;<SUB> j </SUB>, &#963;<SUP>2</SUP>(|n&#8722;m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that &#963;(|n|) can be extended to a continuous function &#963;(t) of t &gt; 0, which is nondecreasing and regularly varying with exponent &#945; at b &#8805; 0 for some 0 &lt; &#945; &lt; 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.

关 键 词：stationary Gaussian random field regularly varying function large deviation probability law of the iterated logarithm 

分 类 号：O211.6[理学—概率论与数理统计]