Abstract

It will be shown and induced that the d-dimensional indices in the Banach spaces version conditions n(EXnp/|nα|p)< are sufficient to yield limmin1jd(nj)(1/|nα|)knj=1d(1(kj1)/nj)Xk=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.