Best possible sufficient conditions for strong law of large numbers for multi-indexed orthogonal random elements (Q925373)

Summary: It will be shown and induced that the \(d\)-dimensional indices in the Banach spaces version conditions \(\sum_n(E\|X_{\mathbf n}\|^p/|{\mathbf n}^\alpha|^p)<\infty\) are sufficient to yield \[ \lim_{\min_{1\leq j\leq d}(n_j)\to\infty}(1/|{\mathbf n}^\alpha|)\sum_{{\mathbf k}\leq{\mathbf n}}\prod_{j=1}^d(1-(k_j-1)/n_j)X_k=0 \] a.s. for arrays of James-type orthogonal random elements. Particularly, it will be shown also that there are the best possible sufficient conditions for multi-indexed independent real-valued random variables.