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Some strong laws of large numbers for sums of random elements. (English) Zbl 0780.60009

The starting point of the authors is the following result: If \(W_ n\), \(n \geq 1\), is a sequence of independent random elements in a real separable Rademacher type \(p\) \((1 \leq p \leq 2)\) Banach space \(X\) and \(0<b_ n \uparrow \infty\) is a sequence of constants, then the condition \[ \sum_ nb_ n^{-p}E \| W_ n \|^ p<\infty \tag{1} \] implies \[ \left\| b_ n^{-1}\sum^ n_{k=1}(W_ k-EW_ k)\right\| \to 0 \quad\text{a.s.} \tag{2} \] Among other results it is shown that, instead of Rademacher type \(p\), in this assertion one can assume that \(n=O(b_ n)\) and \(W_ n\), \(n\geq 1\), is compactly uniformly integrable. The paper contains many examples and historical comments.
Reviewer’s remark: The authors note that the implication \((1)\Rightarrow(2)\) is not valid without the assumption of Rademacher type \(p\) on \(X\). In fact it is known that if \(\sup_ nb_ n^{- 1}b_{2n}<\infty\), then the validity of \((1)\Rightarrow(2)\) implies that \(X\) is of Rademacher type \(p\) [cf. A. Shangua, Laws of large numbers in Banach space (in Russian), Candidate dissertation, Tbilisi (1981)].

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
60B11 Probability theory on linear topological spaces
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