Some Remarks on the Theory of Limit Theorems for Multi-Indexed Sums

Abstract

There exists an (unjustified) point of view that all limit theorems for multi-indexed sums are straightforward consequences of the corresponding classical results for cumulative sums and thus the case of multi-indices is non-independent. We shortly discuss this controversial claim. First we consider in more detail one of the results for multi-indexed sums that indeed follows easily from its classical counterpart (this is an example that supports the above mentioned point of view). Then we explain why this is not a universal phenomenon.