TY - JOUR

T1 - On the Strong Form of the Borel–Cantelli Lemma

AU - Frolov, A. N.

N1 - Publisher Copyright: © 2022, Pleiades Publishing, Ltd.

PY - 2022/3

Y1 - 2022/3

N2 - 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.

AB - 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.

KW - almost sure convergence

KW - Borel–Cantelli lemma

KW - quantitative Borel–Cantelli lemma

KW - strong form of the Borel–Cantelli lemma

KW - strong law of large numbers

KW - sums of indicators of events

UR - http://www.scopus.com/inward/record.url?scp=85131880715&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/2f906049-76b1-381d-a220-62d2c6323562/

U2 - 10.1134/S1063454122010058

DO - 10.1134/S1063454122010058

M3 - Article

AN - SCOPUS:85131880715

VL - 55

SP - 64

EP - 70

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -