Course in Probability Theory

248 1 CENTRAL LIMIT THEOREM AND ITS RAMIFICATIONSThis requires just a rough estimate entailing a suitable choice of c . By (8)applied to {-X,) and (15), for almost every w the following two assertionsare true :(i) S,jk+ , ( cv) - S„ k. (w) > (p(1 - (3/2), tk ) for infinitely many k ;(ii) Silk (w) > -cp(2, silk ) for all sufficiently large k .For such an w, we have thenS(17) S„ k .+) (w) > 2 , tk - cp(2, sil k ) for infinitely many k .Using (9) and log log t2 log log Snk+l, we see that the expression in the rightside of (17) is asymptotically greater thanCS\ 11 - 2 1 -\\ / C2- C cp(1, Snk+i )> (P(1 - S, Snk+3 ),provided that c is chosen sufficiently large . Doing this, we have thereforeproved that(18) '7(Enk+l _ i .o .) = 1,which certainly implies (16) .4° . The truth of (8) and (16), for each fixed S, 0 < S 1} . Recall that (3) is more than sufficient to ensure the validityof the central limit theorem, namely that S n /s n converges in dist . to c1. Thusthe law of the iterated logarithm complements the central limit theorem bycircumscribing the extraordinary fluctuations of the sequence {S n , n > 1} . Animmediate consequence is that for almost every w, the sample sequence S, (w)changes sign infinite&v often . For much more precise results in this directionsee Chapter 8 .In view of the discussion preceding Theorem 7 .3.3, one may wonderabout the almost everywhere bounds formax Sin , max JSmJ, and so on .1