Course in Probability Theory

7 .1 LIAPOUNOV'S THEOREM I 213and put for all x :fn(x)=f(x-xn-2), gn(x)=f(x-xn+2) .Thus we have, denoting by IB the indicator function of B C ~%l?' :I[xn +1,oo) :S f n (x) xn-1}+Or n3S nNow an elementary estimate yields for x -+ +oo :1 °° T{N > x}2 / dy1 x 22n f exp C2nx exp 2 ,(see Exercise 4 of Sec . 7 .4), and a quick computation shows further thatx2GJ'{N > x+ 1} =exp - 2 (1 +o(1)) ,Thus (24) may be written asXP{Sn > xnSn} =exp - (1+0(1)) + O 2 (S3"nSuppose n is so large that the o(1) above is strictly less thanvalue ; in order to conclude (23) it is sufficient to have[4 (1rnS3- = o exp +,G) ,nrnn --)~ oo .This is the sense of the condition (20), and the theorem is proved .E in absolute