Course in Probability Theory

7.2 LINDEBERG-FELLER THEOREM 1 217We apply the lemma to (1) as follows . For each m > 1, we havelim m f x 2 d F n jn(x) = 0 .+oo j1 = IXI>1/mIt follows that there exists a sequence {>7n }decreasing to 0 such thatfj=1XI > 1/nx 2 dF n j (x) --)~ 0 .Now we can go back and modify the definition in (3) by replacing 77 withi) n . As indicated above, the cited corollary becomes applicable and yields theconvergence of [S ;, - ~~(S;,)]/sn in dist . to (Y, hence also that of Sn/s' asremarked .Finally we must go from Sn to S n . The idea is similar to Theorem 5 .2 .1but simpler . Observe that, for the modified Xn j in (3) with 27 replaced by 17n,we have~P{Sn :A Sn } -