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Precise rates in the law of the iterated logarithm under dependence assumptions

Authors:

Wei Dong Liu Zheng Yan Lin

Affiliation:

(1) Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China

Abstract:

Let {X,X ₁,X ₂,...} be a strictly stationary φ-mixing sequence which satisfies EX = 0, EX ²(log₂ |X|)² < ∞ and φ(n) = O(1/(log n)^T) for some T > 2. Let S _n = Σ_k=1ⁿ X _k and a _n = O(√n/(log₂ n)^γ) for some γ > 1/2. We prove that

$mathop {lim}limits_{varepsilon searrow sqrt 2 } sqrt {varepsilon ^2 - 2} sumlimits_{n = 3}^infty {frac{1} {n}P(|S_n | geqslant varepsilon sqrt {ES_n^2 log_2 n} + a_n ) = sqrt 2 .}$

. The results of Gut and Spătaru (2000) are special cases of ours. Research supported by National Natural Science Foundation of China (No. 10571159)

Keywords:

law of the iterated logarithm mixing sequences strong approximation

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