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.


60F15 Strong limit theorems
60B11 Probability theory on linear topological spaces
60G60 Random fields
Full Text: DOI EuDML


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