Abstract
We extend to random fields case, the results of Woyczynski, who proved Brunk's type strong law of large numbers (SLLNs) forđč-valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above-mentioned SLLN, related to results obtained by Acosta 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.
Highlights
We will study the limiting behavior of multiple sums of random vectors indexed by lattice points, so called random fields
The main aim of this paper is to prove a couple Brunk type strong laws of large numbers for independent B-valued random fields
2.34 and on the virtue of Lemma 2.9 and the Borell-Cantelli lemma, the proof will be completed if we show that for any λ > 0, Vk 2kâ1
Summary
We extend to random fields case, the results of Woyczynski, who proved Brunkâs type strong law of large numbers SLLNs for B-valued random vectors under geometric assumptions. We give probabilistic requirements for above-mentioned SLLN, related to results obtained by Acosta 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.
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