Abstract: The strong form of the Borel–Cantelli lemma is a version of the strong law of large numbers for sums of indicators of events. These sums are centered at the mean and are normalized by some function of sums of probabilities of events. The series of these probabilities is assumed to be divergent. In this paper, we derive new strong forms of the Borel–Cantelli lemma with smaller normalizing sequences than those in earlier studies. The constraints on variances in the increments of the sums of event indicators become stronger. We give examples in which these constraints hold.