On Chung-Teicher type strong law for arrays of vector-valued random variables
Sufficient conditions for the strong law of large numbers (SLLN) on a separable Banach space \(B\) are usually given under the geometric assumption that \(B\) is of Rademacher type \(p\) for some \(1<p\leq2\). The author considers arrays \(\{X_{ni}:\,1\leq i\leq k_n,\,n\geq1\}\) of rowwise independent \(B\)-valued random variables and studies the equivalence of a weak law of large numbers (WLLN) to its strong version \(\sum_{i=1}^{k_n}a_{ni}X_{ni}\to0\) almost surely, for some weights \(a_{ni}\in\mathbb R\setminus\{0\}\). The presented sufficient conditions for equivalence of WLLN and SLLN are of Chung type or Chung-Teicher type. Although the results are formulated for general separable Banach spaces \(B\), the assumptions made have some relationship to the Rademacher type \(p\) condition. Consequently, sufficient conditions for SLLN on Rademacher type \(p\) spaces are derived. For the Chung-Teicher type SLLN an essential simplification is made when assuming the random variables to be stochastically dominated and the weights to build a Toeplitz array.
scientific article; zbMATH DE number 2168624