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Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables

Authors:	Jiang Chaowei Yang Xiaorong

Institution:	(1) Dept. of Math., Zhejiang Univ., Hangzhou, 310027, China;(2) Hangzhou Foreign Language School, Hangzhou, 310023, China

Abstract:	In the case of Z ₊ ^d (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {X _k, k ∈ Z ₊ ^d } i.i.d. random variables with mean 0, S _n = ∑_k≤n X _k and V _n ² = ∑_j≤n X _j ² , the precise asymptotics for $\sum\nolimits_n {\frac{1}{{\left\| n \right\|(\log \left\| n \right\|)^d }}P\left( {\left\| {\frac{{S_n }}{{V_n }}} \right\| \geqslant \varepsilon \sqrt {\log \log \left\| n \right\|} } \right)}$ and $\sum\nolimits_n {\frac{{(\log \left. n \right\|)\delta }}{{\left\| n \right\|(\log \left\| n \right\|)^{d - 1} }}P\left( {\left\| {\frac{{S_n }}{{V_n }}} \right\| \geqslant \varepsilon \sqrt {\log n} } \right)}$ , as ɛ ↘ 0, is established. Supported by the NNSF of China (10471126).

Keywords:	multidimensionally indexed random variable precise asymptotics self-normalized sum Davis law of large numbers law of iterated logarithm
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