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

摘要

Consider Z_+~d (d≥2)-the positive d-dimensional lattice points with partial ordering ≤, let {X_k, k∈Z_+~d} be i.i.d. random variables with mean 0, and set S_n = Σ_(k≤n)X_k, n∈Z_+~d. We establish precise asymptotics for Σ_n|n|~(r/p-2)P(|S-n|≥ε|n|~(1-p)), and for Σ_n(log|n|~δ)/|n|P(|S-n|≥ε|n|log|n|~(1/2)), (0≤δ≤1) as δ→0, and for Σ_{n:|n|≥3}1/|n|log|n|P(|S-n|≥ε|n|log log|n|~(1/2)) as δ→2(d-1)EX~2~(1/2).