Limsup results and LIL for partial sum processes of a Gaussian random field |
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Authors: | Yong-Kab Choi Miklós Csörgő |
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Affiliation: | (1) Present address: 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 |
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Abstract: | Let {ξ
j
; j ∈ ℤ+
d
be a centered stationary Gaussian random field, where ℤ+
d
is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝd, having nonnegative integer coordinates. For each j = (j
1
, ..., jd) in ℤ+
d
, we denote |j| = j
1
... j
d
and for m, n ∈ ℤ+
d
, define S(m, n] = Σ
m<j≤n
ζ
j
, σ2(|n−m|) = ES
2
(m, n], S
n
= S(0, n] and S
0
= 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.
Research supported by NSERC Canada grants at Carleton University, Ottawa |
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Keywords: | stationary Gaussian random field regularly varying function large deviation probability law of the iterated logarithm |
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