Abstract:
The strong form of the Borel-Cantelli lemma is a variant of the strong law of large numbers for sums of the indicators of events. These sums are centered at the mean and normalized by some function from sums of probabilities of events. The series from probabilities is assumed to be divergent. In this paper, we derive new strong forms of the Borel-Cantelli lemma with smaller normalizing sequences than it was before. Conditions on variations of increments of indicators become stronger. We give examples in which these conditions hold.
Keywords:the Borel-Cantelli lemma, the quantitative Borel-Cantelli lemma, strong forms of the Borel-Cantelli lemma, suns of indicators of events, strong law of large numbers, almost surely convergence.
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