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On complete convergence in mean for double sums of independent random elements in Banach spaces. (English) Zbl 1386.60018

Let \(X\) be a real separable Banach space and \(\{T_{m,n}: \, m,n\geq1\}\) be a double array of random elements with values in \(X\). In this paper, the notion of \(T_{m,n}\) converging completely to 0 in means of order \(p, \, p>0\) is studied. It is noted that this notion is stronger than: (i) \(T_{m,n}\) converging completely to 0 and (ii) \(T_{m,n}\) converging to 0 in means of order \(p\) as \(\max\{m, n\}\to\infty\). When \(X\) is of Rademacher type \(p, \, 1 \leq p \leq 2\), for a double array of independent mean 0 random elements \(\{V_{m,n}: \, m \geq 1, n \geq 1\}\) in \(X\) and a double array of constants \(\{b_{m,n}: \, m \geq 1, n \geq 1\}\), conditions are provided under which \(\max_{1\leq k\leq m,\,1\leq l \leq n } \| \sum_{i=1}^k \sum_{j=1}^l V_{i,j}\| /b_{m,n}\) converges completely to 0 in means of order \(p\). Moreover, these conditions are shown to provide an exact characterization of Rademacher type \(p\), \(1 \leq p \leq 2,\) Banach spaces. At the end of the paper, some illustrative examples are presented.

MSC:

60B11 Probability theory on linear topological spaces
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
60F25 \(L^p\)-limit theorems
40B05 Multiple sequences and series
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