The law of large numbers for product partial sum processes indexed by sets (Q1323847)

Let \(\{X_ i: i\in N^{d_ 1}\}\) and \(\{Y_ j: j\in N^{d_ 2}\}\) be independent families of i.i.d. integrable random variables, where \(N=\{1,2,\ldots\}\). Let \({\mathcal A}\) be a sub-family of \({\mathcal B}([0,1]^ d)\), where \(d=d_ 1+d_ 2\). Denote \(| i|=\max_ k i_ k\) for \(i=(i_ 1,\ldots,i_{d_ 1})\), \(| j|=\max_ m j_ m\) for \(j=(j_ 1,\ldots,j_{d_ 2})\) and \[ \begin{multlined} \mathbb{C}_{nij}=\{(x_ 1,\ldots,x_ d)\in R^ d: (i_ k-1)/n\leq x_ k\leq i_ k/n,\;k=1,\ldots,d_ 1;\\ (j_ m-1)/n\leq x_{d_ 1+m}\leq j_ m/n,\;m=1,\ldots,d_ 2\}.\end{multlined} \] Assume that \(\{\lambda_ n\}\) is a sequence of positive Borel measures on \([0,1]^ d\), satisfying conditions \(\lambda_ n(\mathbb{C}_{nij})\leq cn^{-d}\) for some \(c<\infty\); \(\sup_{A\in{\mathcal A}}\mid\lambda_ n(A)-\lambda(A)\mid\to 0\) as \(n\to\infty\), where \(\lambda\) is the Lebesgue measure. If some metric entropy condition or smooth boundary condition is fulfilled, then \[ \sup_{A\in{\mathcal A}}\biggl|\sum_{| i|\leq n,| j|\leq n} X_ iY_ j\lambda_ n(A\cap{\mathbb{C}}_{nij})- E(X)E(Y)\lambda(A)\biggr|\to 0\quad \text{(with probability 1)} \] as \(n\to\infty\). Some interesting corollaries are proved.