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Strong laws of large numbers for \(\mathbb B\)-valued random fields. (English) Zbl 1180.60028

Summary: We extend to random fields case, the results of W. A. Woyczynski [Probab. Math. Stat. 1, 117–131 (1980; Zbl 0502.60006)], who proved H. D. Brunk’s [Duke Math. J. 15, 181–195 (1948; Zbl 0030.20003)] type strong law of large numbers (SLLNs) for \(\mathbb B\)-valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above-mentioned SLLN, related to results obtained by A. De Acosta [Ann. Probab. 9, 157–161 (1981; Zbl 0449.60002)] as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors.

MSC:

60F15 Strong limit theorems
60B11 Probability theory on linear topological spaces
60G60 Random fields
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References:

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