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=∑k⩽nXk, n∈Z+d. We establish precise asymptotics for ∑n|n|r/p−2P(|Sn|⩾ε|n|1/p), and for ∑n(log |n|)δ|n| P(|Sn|⩾ε|n| log |n|), (0⩽δ⩽1) as ε↘0, and for ∑{n:|n|⩾3}1|n| log |n|P(|Sn|⩾ε|n| log log |n|) as ε↘2(d−1)EX2.