Course in Probability Theory

4 .2 ALMOST SURE CONVERGENCE ; BOREL-CANTELLI LEMMA 1 81Theorem 4 .2 .5. The implication (6) remains true if the events {E„ } are pairwiseindependent .PROOF . Let I„ denote the indicator of E,,, so that our present hypothesisbecomes(8)dm :A n : L (Im 1 n ) = 1(I .)(%'(I n ) .Consider the series of r .v .'s : >n °_ 1 I n (w) . It diverges to +oo if and only ifan infinite number of its terms are equal to one, namely if w belongs to aninfinite number of the En 's . Hence the conclusion in (6) is equivalent to(9)00What has been said so far is true for arbitrary En 's . Now the hypothesis in(6) may be written as00E AI 1 ) = +00 -n=1Consider the partial sum Jk = En=1 I n .have for every A > 0 :Using Chebyshev's inequality, we(10) {I Jk - 0 (Jk)I < A6(Jk)} > 1 -U 2 (Jk)A2U2(Jk)= 1 - 1A 2 'where or 2 (J) denotes the variance of J . WritingPit = ((I,) = ~~( En ),we may calculate a 2 (Jk) by using (8), as follows :kn=1In + 21