Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z+d (d ≥ 2)--the positive d-dimensional lattice points with partial ordering ≤, let {Xk, k ∈ Z+d} be i.i.d. random variables with mean 0, and set Sn = Σk ≤ n Xk, n ∈ Z+d. We establish precise asymptotics for Σn|nr',p-2P(|Sn| ≥ e|n|1/p), and for Σ(log |n|)δ/nP(|Sn| ≥ e√|n|log|n|), (0 ≤ δ ≤ 1) as e c→ 0, and for Σ{n:|n| ≥ 3}1/|n|log|n| P(|Sn| ≥ e√|n|log log|n|) as ec→√2(d-1)EX2.

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