Course in Probability Theory

236 1 CENTRAL LIMIT THEOREM AND ITS RAMIFICATIONSIn the case of a single sequence of independent and identically distributedr .v .'s {Xj, j > 1 } with mean 0, variance 62 , and third absolute moment y < oc,the right side of (1) reduces tony Aoy 1Ao (n j2 ) 3/2 =9 3 n 1/2'H. Cramer and P . L. Hsu have shown that under somewhat stronger conditions,one may even obtain an asymptotic expansion of the form :H1(x) H2(x) H3(x)F" (x) _ ~(x)+ n1/2 + n + n3/2 + . . .where the H's are explicit functions involving the Hermite polynomials . Weshall not go into this, as the basic method of obtaining such an expansion issimilar to the proof of the preceding theorem, although considerable technicalcomplications arise . For this and other variants of the problem see Cramer[10], Gnedenko and Kolmogorov [12], and Hsu's paper cited at the end ofthis chapter .We shall give the proof of Theorem 7 .4 .1 in a series of lemmas, in whichthe machinery of operating with ch .f.'s is further exploited and found to beefficient .Lemma 1 . Let F be a d .f., G a real-valued function satisfying the conditionsbelow :Set(i) lim1,_ 00 G(x) = 0, lima ,+. G(x) = 1 ;(ii) G has a derivative that is bounded everywhere : supx jG'(x)i < M .1(2) = 2M sup I F(x) - G(x)l .Then there exists a real number a such that we have for every T > 0 :( T° 1 - cosx(3) 2MTA { 3 2 dx - nl / o xcos TxJ {F(x + a) - G(x + a)} dxX 2PROOF . Clearly the A in (2) is finite, since G is everywhere bounded by(1) and (ii) . We may suppose that the left member of (3) is strictly positive, forotherwise there is nothing to prove ; hence A > 0 . Since F - G vanishes at+oc by (i), there exists a sequence of numbers {x„ } converging to a finite limit