On the strong law of large numbers for delayed sums and random fields (Q626026)

In this paper, the authors discuss strong laws of large numbers for sums of independent, identically distributed (i.i.d.), real-valued random variables, taken over moving blocks of an increasing size. For example, let \(\{X,X_j,j\geq 1\}\) be an i.i.d. sequence indexed by \(\mathbb{Z}_+\) and consider the delayed sums \(T_{n,n+k} =\sum_{j=n+1}^{n+k} X_j\) with \(n,k\geq 1\). Let \(L\) be a slowly varying function, ultimately nondecreasing to \(+\infty\), and let \(L_1^{\#}\) denote the de Bruijn conjugate of \(L_1=1/L\). Then, as a first result, a new proof is given for the fact that \[ \frac{T_{n,n+n/L(n)}}{n/L(n)}\to 0\;\;a.s. \;\hbox{as}\;\;n\to\infty \;\;\Longleftrightarrow\;\;E| X| L_1^\#(| X| )<\infty\;\;\hbox{and}\;\;EX=0 \] (cf. also \textit{N. H. Bingham } and \textit{C. M. Goldie} [Math. Z. 199, No. ~3, 443--454 (1988; Zbl 0631.60038)]). The authors' main results then prove corres\-ponding equivalences for i.i.d. random fields indexed by \(\mathbb{Z}_+^d\) with \(d\geq 2\), where they confine themselves to the case when the slowly varying functions are the same in each coordinate. For \(d=2\), however, two additional theorems are presented in which the right-upper edges of the moving rectangles are allowed to expand at different rates.