A note on the Borel-Cantelli lemma

Abstract

It is often desirable to find the probability that for a given sequence E, E, of events n infinite number of E occur.If the E re inde- pendent events, then by the Borel-Cntelli lemma this probability is 0 or 1 ccording as P (E) converges or diverges.This and similar situations are perhaps more easily understood when trans- lated into the lnguge of random vribles.Let X be the random vrible denoting the number of E,..., En which occur.Then EX -'=<_<_ P(E) nd EX '_<_,;.<_P(E n E;.).An infinite number of E occur if nd only if lim Xn .It is possible to sharpen the Borel-Cantelli lemm by considering the random variables X/EX.For if the events E re independent nd P (E) diverges, then it follows from strong lw of large numbers (cf.Love [5, p. 238]) that lim X/EX 1 with probability 1.

Locations

• Illinois Journal of Mathematics - View - PDF