On inequalities for conditional probabilities of unions of events and the conditional Borel—Cantelli lemma

Abstract

We derive new upper and lower bounds for conditional (given σ-field A) probabilities of unions of events. These bounds are sharp. Taking of expectations from left-hand and right-hand sides of such inequalities may yield better bounds than a direct estimation of probabilities of events. An example is given. We also derive new generalizations of the conditional Borel—Cantelli lemma. Taking of expectations, one can obtain new variants of the Borel—Cantelli lemma under conditions that differ from classical ones. Refs 11.