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

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 ² _n = ∑ _j≤n X ² _j , 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.