摘要
Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in ℤ<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n ∈ ℤ<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = Σ<SUB> m【j≤n </SUB>ζ<SUB> j </SUB>, σ<SUP>2</SUP>(|n−m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t 】 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 【 α 【 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 {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in ℤ<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n ∈ ℤ<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = Σ<SUB> m<j≤n </SUB>ζ<SUB> j </SUB>, σ<SUP>2</SUP>(|n−m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t > 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 < α < 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.
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
