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

2. Preliminaries

A first important observation is that inequalities which do not depend on the (partial) order of the index set Z₊d, such as the triangle inequality, moment inequalities for sums, and so on, remain valid “automatically”. Namely, such relations only depend on the fact that, if   are random variables and   their partial sums, then   is simply a sum of   random variables.
The following quantities and their asymptotic behaviour turn out to be crucial. Let

The following asymptotics hold:
equation(2.1)

and
equation(2.2)

see [12, Chapter XVIII] (for the case d=2; the general case is analogous) and [30, Chapter 12]. (The quantity d(j) has no pleasant asymptotics; e.g.,  , and  .)
Another important observation is that, since all terms in the sums we consider are nonnegative, we may change the order of summation, in particular as follows (cf. also e.g. [7] and [8]),
equation(2.3)

More importantly, whenever the functions involving   only depend on the value of  , the second summation can be simplified further. For example, for the sum in Theorem 1 we have
equation(2.4)

This observation should be kept in mind throughout.
We also need the following lemmas, the first three of which generalize Gut and Spătaru [10].
Lemma 2.1. Assume thatE[|X|β(log(1+|X|))d⁻¹]<∞, and setb(ε)=ε⁻βp^/(β⁻p⁾, where 1⩽p<β<α. For any constanta>0,

Proof. We first note that k>b(ε) if and only if k<εβkβ^/p. It follows that

Lemma 2.2. Letr⩾2, assume thatE[|X|r(log(1+|X|))d⁻¹]<∞, and setρ(ε)=ε⁻²p^/(2−p⁾, where 1⩽p<2. For any constanta>0,

Proof. Modifying the previous proof we first note that k>ρ(ε) if and only if  . It follows (omitting some of the steps) that

Lemma 2.3. Assume that and setc(ε)=eM^/ε², whereM>1. Let 0⩽δ⩽1.For any constanta>0,