Course in Probability Theory

226 1 CENTRAL LIMIT THEOREM AND ITS RAMIFICATIONSthe last relation from (1) and the one preceding it from a hypothesis of thetheorem . Thus for each 77 > 0, we have for all sufficiently large nJ x > 27 S' nx2 dF n j (x) = 0, 0 < j < k n - 1,where F,, j is the d .f. of Y, j . Hence Lindeberg's condition is satisfied for thedouble array {Y nj/s;,}, and we conclude that S;,/s n converges in d ist. t o (D .This establishes the theorem as remarked above .The next extension of the central limit theorem is to the case of a randomnumber of terms (cf. the second part of Sec . 5 .5) . That is, we shall deal withthe r .v . S vn whose value at c) is given by Svn (w) (co), whereS"( 60 ) X - ( )j=1as before, and {vn (c)), n > 1 } is a sequence of r.v .'s . The simplest case,but not a very useful one, is when all the r .v .'s in the "double family"{Xn , Vn , n > 1 } are independent . The result below is more interesting and isdistinguished by the simple nature of its hypothesis . The proof relies essentiallyon Kolmogorov's inequality given in Theorem 5 .3 .1 .Theorem 7 .3 .2 . Let {X j , j > 1 } be a sequence of independent, identicallydistributed r .v.'s with mean 0 and variance 1 . Let {v,,, n > 1} be a sequenceof r.v .'s taking only strictly positive integer values such that(3)where c is a constant : 0 < c < oc . Then S vn / vn converges in dist . to (D .PROOF . We know from Theorem 6 .4 .4 that S 7, / Tn- converges in d ist . toJ, so that our conclusion means that we can substitute v n for n there . Theremarkable thing is that no kind of independence is assumed, but only the limitproperty in (3) .Vnnnin pr .,First of all, we observe that in the result of Theorem 6 .4.4 wemay substitute [cn] (= integer part of cn) for n to conclude the convergencein dist . of S[un]/,/[cn] to c (why?) . Next we writeS Vn _ S[cn] S v, - S[cn]V n /[cn ] + f[cn ]The second factor on the right converges to 1 in pr ., by (3) . Hence a simpleargument used before (Theorem 4 .4 .6) shows that the theorem will be provedif we show thatVn(4)Sv n - S[ `n] -a 0 in pr .,/[cn ]