3 we now concentrate on proving 927 write n itktfsn 1

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(3) We now concentrate on proving (9.27). Write n I)t>k(tfsn)- 1) + t 2 /2 k=l ( itX js t 1 it 2 2) e k n-1-i-Xk--(-)Xk . k=l Sn 2 Sn Let 0 represent what is inside the expectation on the previous line . We get the decomposition n = L {E(•)l[IXkl/sn:::f) + £(·)11Xkl/sn>fJ} k=l =I +II. We now show I and I I are small. For I we have using (9 . 6) with n = 2: where we used the fact that
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9.8 The Lindeberg-Feller CLT 319 Now we show why I I is small. We have (from (9.6) with n = 2) by the Lindeberg condition (9.23). This completes the proof of (9.27) and hence the theorem is proved. 0 We next present a sufficient condition for the Lindeberg condition (9.23) called the Liapunov condition which is relatively easy to verify. Corollary 9.8.1 (Liapunov Condition) Let {Xk, k 2: 1} be an independent se- quence of random variables satisfying E(Xk) = 0, Var(Xk) = uf < oo, s; = Lk=l uf . /!for some 8 > 0 then the Lindeberg condition (9.23) holds and hence the CLT. Remark. A useful special case of the Liapunov condition is when 8 = 1: Proof. We have 0
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320 9. Characteristic Functions and the Central Limit Theorem Example: Record counts are asymptotically normal. We now to examine the weak limit behavior of the record count process. Suppose {Xn, n ::=:: 1} is an iid sequence of random variables with common continuous distribution F, and define n h = 1[Xk is a record)• f.ln = L 1k. i=l So f.ln is the number of records among X 1, ... , X n. We know from Chapter 8 that asn oo Here we will prove To check this, recall Thus So Now and therefore f.ln a.s. 1 logn f.ln -logn ::::}N(0,1). 1 1 Var(h) = k- k 2 . Lk=l El1k- £(1k)l 3 < Lk=l 0 + sg - (log n )3/2 logn 0 (logn)3/2 .
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9.9 Exercises 321 So the Liapunov condition is valid and thus J.ln- E(J..tn) => N(O, 1). JVar(J..tn) Note Jvar(J..tn) "' sn "' jlog n and E(J..tn) -logn _ Lk=tl-logn "'_Y_--+ 0 jlog n - Jlog n Jlog n ' where y is Euler's constant. So by the convergence to types theorem J.ln -logn JiOiii" => N(O, 1). D 9. 9 Exercises 1. Triangular arrays. Suppose for each n, that {Xk.n. 1 :::: k :::: n} are inde- pendent and define Sn = Lk=l xk.n· Assume E(Xk.n) = 0 and Var(Sn) = 1,and n LE (1Xk,nl 2 1(1xk.nl>t])--+ 0 k=l as n --+ oo for every t > 0. Adapt the proof of the sufficiency part of the Lindeberg-Feller CLT to show Sn => N(O, 1). 2. Let {Xn, n 0} be a sequence of random variables. (a) Suppose {Xn. n 0} are Poisson distributed random variables so that for n 0 there exist constants An and e-An)...k P[Xn=k]=--n, k! Compute the characteristic function and give necessary and sufficient con- ditions for Xn => Xo. (b) Suppose the {Xn} are each normally distributed and E(Xn) = J.ln E R, Var(Xn) =a;. Give necessary and sufficient conditions for Xn => Xo.
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322 9. Characteristic Functions and the Central Limit Theorem 3. Let {Xt. k 1} be independent with the range of Xk equal to {±1, ±k} and 1 1 P[Xk = ±1] = 2(1- k 2 ), 1 P[Xk = ±k] = 2k 2 By simple truncation, prove that Sn I Jn behaves asymptotically in the same way as if Xk = ±1 with probability 1/2. Thus the distribution of Sn/Jn tends to N(O, 1) but Var(Sn/Jn)--+ 2.
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