The law of large numbers for partial sum processes indexed by sets (Q1822128)

Let \(\{X_ j:\) \(j\in {\mathbb{N}}^ d\}\) be i.i.d. with \(EX_ j=0\) and for subsets A of \([0,1]^ d\) define \(S_ n(A)=\sum_{| j| \leq n}X_ j I(j/n\in A)\). The problem of convergence of \(S_ n(A)\) to 0 uniformly for A in a certain given class \({\mathcal A}\) is studied. It is shown that this property is independent of the particular distribution of the \(X_ j\), and necessary and sufficient conditions are given in terms of the size of \({\mathcal A}\) measured by metric entropy.