THE LAW OF LARGE NUMBERS AND THE ITO-NISIO THEOREM FOR VECTOR-VALUED RANDOM-FIELDS

Strong convergence results are obtained for vector-valued random field s. Substantial development of Banach-valued random fields and summabil ity results is needed to provide the framework for the major results s ince many plausible extensions fail for multi-indexed Banach-valued ra ndom variables. This development yields general convergence results fo r random fields in Banach spaces, including an Ito-Nisio theorem and s trong laws of large numbers.