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On complete convergence in mean of normed sums of independent random elements in Banach spaces. (English) Zbl 1087.60009

Let \((T_n)\) be a sequence of random elements in a separable Banach space. For \(p>0\), \(T_n\) is said to converge completely to \(0\) in mean of order \(p\) if \(\sum_1^{\infty}E\| T_n\| ^p<\infty\). This mode of convergence was introduced by Y.S. Chow [Bull. Inst. Math., Acad. Sin. 16, No. 3, 177–201 (1988; Zbl 0655.60028)] for real-valued random variables. In the paper under review conditions are provided under which a normed sum of independent random elements in a Rademacher type \(p\) Banach space converges completely to \(0\) in mean of order \(p\). An example, showing that the conditions are sharp, is given. Moreover, these conditions provide an exact characterization of Rademacher type \(p\) Banach spaces. 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

Citations:

Zbl 0655.60028
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References:

[1] DOI: 10.1080/07362998708809104 · Zbl 0617.60028 · doi:10.1080/07362998708809104
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