Course in Probability Theory

2 8 0 1 RANDOM WALKIt is clear from the preceding theorem that the key to recurrence is thevalue 0, for which we have the criterion below .Theorem 8 .3 .2 .If for some E > 0 we have(4) 1: -,:;P { IS, I< E} < oc,nthen(5) -:flIS .< E i .o .} = 0(for the same c) so that 0 ' N . If for every E > 0 we have(6) EJ ?;D{IS,I < E} = oo,nthen(7) ?;?If IS, I < E i .o .} = 1for every E > 0 and so 0 E 9I .Remark . Actually if (4) or (6) holds for any E > 0, then it holds forevery c > 0 ; this fact follows from Lemma 1 below but is not needed here .PROOF . The first assertion follows at once from the convergence part ofthe Borel-Cantelli lemma (Theorem 4 .2 .1) . To prove the second part considerF = lim inf{ I S, I > E} ;nnamely F is the event that IS n I < E for only a finite number of values of n .For each co on F, there is an m(c)) such that IS, (w)I > E for all n > m(60) ; itfollows that if we consider "the last time that IS, I < E", we have0C~'(F) =7{ISmI< E ; IS, I > E for all n > m + 1} .M=0Since the two independent events ISm I < E and I S n - Sm I > 2E together implythat IS, I > e, we have1 > 1' (F) >am=1ISmI< E},'~P{IS, - SmI > 2E for all n > m+ 1}M=1{ISmI< E}P2E,1(0)