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Strong and weak laws of large numbers for double sums of independent random elements in Rademacher type \(p\) Banach spaces. (English) Zbl 1112.60002

Summary: For a double array of independent random elements \(\{V_{mn}\), \(m\geq 1\), \(n\geq 1\}\) in a real separable Rademacher type \(p\) \((1\leq p\leq 2)\) Banach space, strong and weak laws of large numbers are established for the double sums \(\sum_{i=1}^m \sum_{j=1}^n V_{ij}\), \(m\geq 1\), \(n\geq 1\). For the strong law result, it is assumed that \(EV_{mn}=0\), \(m\geq 1\), \(n\geq 1\) and conditions are provided under which \(\sum_{i=1}^m \sum_{j=1}^n V_{ij}/mn\to 0\) almost surely as \(\max\{m,n\}\to\infty\). These conditions for the strong law are shown to completely characterize Rademacher type \(p\) Banach spaces. The weak law results provide conditions for \(\sum_{i=1}^m \sum_{j=1}^n (V_{ij}-c_{ijmn})/mn @>P>> 0\) as \(\max\{m,n\}\to\infty\) or for \(\sum_{i=1}^{T_m} \sum_{j=1}^{\tau_n} (V_{ij}- c_{ijmn})/mn @>P>>0\) as \(\max\{m,n\}\to\infty\) to hold where \(c_{ijmn}= E(V_{ij}/(\|V_{ij}\|\leq mn))\), \(i,j,m,n\geq 1\) and \(\{T_m\), \(m\geq 1\}\) and \(\{\tau_n\), \(n\geq 1\}\) are sequences of positive integer-valued random variables. Illustrative examples are provided.

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
60F05 Central limit and other weak theorems
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