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Some remarks on the ordinal strong law of large numbers. (Ukrainian, English) Zbl 1124.60005

Teor. Jmovirn. Mat. Stat. 72, 84-92 (2005); translation in Theory Probab. Math. Stat. 72, 93-102 (2006).
Let \(B\) be a Banach space with the norm \(\| \cdot\| \), let \(X_i,i=1,2,\dots, EX_i=0,\) be a sequence of i.i.d. random elements with values in the space \(B\), and let \(S_n=\sum_{i=1}^{n}X_i\). The sequence \(X_i,i=1,2,\dots\), satisfies the ordinal law of large numbers if \(\lim_{n\to\infty}(S_n/n)=0\) a.e. The sequence \(X_i,i=1,2,\dots\), satisfies the law of large numbers in the norm if \(\lim_{n\to\infty}(\| S_n\|/n)=0\) a.e. The author proves that the ordinal law of large numbers and the law of large numbers in the norm are equivalent for Banach lattices that do not contain uniformly the space \(l_1^n\).

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
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