Tsung-Lin Cheng and Yuan-Shih Chow. Taipei115,Taiwan R.O.C.

Transcription

1 A Generalization and Alication of McLeish's Central Limit Theorem by Tsung-Lin Cheng and Yuan-Shih Chow Institute of Statistical Science Academia Sinica Taiei5,Taiwan R.O.C. Abstract. A stable convergence theorem for arrays of random variables is established. Moreover, interchangeable sequence of random variables rovides not only an alication but a good examle for the main result. Introduction. The central limit theorem (C.L.T.) for deendent random variables has been discussed for many authors. A list of references before 980 is given by Hall and Heyde (980). Esecially, they (Lemma 3., Hall and Heyde 980) generalize McLeish's Theorem 2.(974) by introducing Renyi's(963) stable convergence. However, their comment \Theorem 3. is a trivial consequence of the convergence theorem for characteristic functions" is not correct. The roof of necessity is not easy at all. Fortunately, the suciency is trivial and more useful. We'll state some related denitions of stable convergence for the following discussions. Let ( F P) be a robability sace, G be a sub--algebra of F, andy n be r.v.s. Denition.( Renyi, 963 ) Let Y n d! Y. If for every B 2Gthere is a countable, dense set of oints x, such that lim n P (Y n x B)=Q B (x) exists.

2 We say that Y n stably! F Y (or Y )ong, denoted by If G = f g, then! is just d!. Y n! Y: If G = F, then! is the usual Y n stably! F Y (we dro the F), denoted by s!. For examle, let X and X 0 be indeendent, identically distributed. Set Z 2n = X Z 2n+ = X 0. Then P (Z 2n y X )! P (X y ^ ) P (Z 2n+ y X )! P (X y)p (X ): Hence it is not true that Z n s! X however Z n d! X. Denition. For L -r.v.s Z n and Z, R A Z n! R A Z for every A 2Gi Z n! Z weakly in L ( G P) i (W G) Z n! Z. (W F) Z n! Z is denoted by Z W n! Z. Lemma. Y n! Y i there exists a r.v. Y 0 on ( I GB P, where I =(0 ), B is the class of all Borel sets in I and is the Lebesque measure on B, with the same distribution as Y such that for all real t (L..)E(e ityn A )! E(e ity 0 AI ), for all A 2G: (i.e., e ityn (w G)! e ity 0 )and (L..2)E(e ity 0 AI ) is continuous at t =0for all A 2G. The roof of Lemma is not trivial, esecially the necessity. For the requirement of sace and usefullness, We just give the roof of suciency. Proof of Suciency. Let (L..) and (L..2) hold. Let A =. By Levy's Continuity Theorem, Y n d! Y 0.For A 2Gwith P (A) > 0 P ((Y n y)a)! P 0 ((Y 0 y)(ai)) for a conuntable dense sert of oints y, by Levy's Continuity Theorem again. This comlete 2

3 the roof. The Suciency of Lemma can be alied to the following lemma which generalizes the results of McLeish (974) and Hall and Heyde (980). Let fx n j F n j j k n g be an array ofl -r.v.s on ( F P)(F n j F n j+ F, X n j is F n j -measurable). Dene T n (t) = Q k n j= ( + itx n j ), for t 2 R, where i = ;. Lemma 2. Let 2 be a ositive G-measurable r.v. where G is a -subalgebra of F. Suose that (L.2.) Xn max jx n i j ik n (L.2.2) Un 2 P ik n Xn i 2! 0! 2 (L.2.3) T n T n (t) (w G)! for every real t (L.2.3') ft n n g is u.i.. Then S n! Z, where E(e itz )=E(e ; 2 2 t 2 ) and for A 2G E(e itsn A )! E(e ( ;2 t 2 2 ) A ): () Remark. (i). If G = f g, then Lemma 2 reduces to McLeish's Theorem 2., 974. (ii). If G = F, then (L.2.3) imlies (L.2.3'), i.e. for any L -random variables Y Y n,if R A Y n! R A Y for all A 2Fthen (Y n n ) is u.i. (Proosition IV 2.2., Neveu 965). Hence if G = F, then Lemma 2 imlies Hall and Heyde's Lemma 3.. We'll see the roof in the Aendix. According to the above remarks, we conjecture that if there exists an examle of G that is neither f g nor F? Is there any alication for Lemma 2? In this aer, we rovide an answer (of course not unique). 2 Alication and Examle. By Theorem 7.3.2, Chow and Teicher 997, it is known that any innite sequence of interchangeable random variables is conditional i.i.d. given some -algebra. (The - algebra of ermutable events, which is dened below. ) Based on Lemma 2, we can rove 3

4 a more general form of Theorem 9.2., Chow and Teicher in an alternative way. Denition 2.(.232, Chow and Teicher 997) A maing =( 2 ) from the set N of all ositive integers onto itself is called a nite ermutation if is one-to-one and n = n for all but nite distinct integers. Let Q denote the set of all nite ermutations and let B be the class of Borel subsets of R = R R and X =(X X 2 )asequenceon( F P). Dene X =(X X 2 ), for =( 2 ). Then S = fx ; (B) :B 2B P(X ; (B)(X) ; (B)) = 0 all 2 Qg is called the -algebra of nite ermutable events (of X), denotes the symmetric difference. Moreover we call S the -algebra of nite ermutable events. Due to Examle 7.3., Chow and Teicher 997, we get Lemma 3. Let fy n n g be a nonnegative interchangeable sequence of random variables. Suose that EY <, then n nx j= Y j! E(Y js): The following theorem is an alication of lemma 2. Theorem. Let fx n g be a sequence of interchangeable random variables with EX =0, E(X js) =0a.s. and EX 2 <. (L.2.3) and Then n nx (s S) X k! Z where E(e itz A )=E(e ; 2 t2 E(X 2 js) A ), for any A 2S. Proof: Since fx n g are interchangeable, so are fx 2 ng. By assumtion EX 2 < and Lemma 3, we have n nx X 2 k! E(X 2 js 2 ) 4

5 which satises (L.2.2). P (max j P X j jn n j >)=P ( n X 2 j n (jx j j> n) > 2 ) j= EX 2 2 (jx j> n)! 0 as n!,which roves (L.2.). Since X n are conditional indeendent given S, for any A 2Sand t 2<, E(T n (t) A )=E(E(T n (t)js) A )=E nq E( + itx k = njs)a = E(A )=P (A) which imlies (L.2.3). Moreover, since EX 2 <, T n is u.i. and this fact leads to (L.2.3'). By Lemma, the roof is done. Remark. The necessity of Theorem 9.2., Chow and Teicher 997, is a secial case of the above theorem. Therefore, Lemma 2 rovides as a owerful tool in roving CLT for interchangeable sequences. However, the suciency of Chow andteicher's Theorem 9.2. is interesting. The stable convergnece for interchangeable sequences reduces to the ususal CLT according to the condition Cov(X X 2 )=0=Cov(X 2 X2 2): We end this aer by an examle. Examle. Let be a random variable such thatfy n g is a sequence of comlex valued random variables and is indeendent, uniformly distributed on T = fz 2C: jzj = g given, where C denotes the set of all comlex numbers. Dene f(z) = ; ( +=(2)arg z) forz 2C, where (x) =P (N(0 ) <x). Then 2 f(y n ) is indeendent, normally distributed, given. Let fw n g be a sequence of indeendent standard normal random variables given such that fw n g and fy n g are indeendent given. 5

6 Dene X n = ; 2 f(y n )+W n.thenfx n g are conditional i.i.d given. Moreover EX = 0 and EX 2 <. In this case, for any A 2 () Moreover, since EX 2 <, T n(t) is u.i.. By Corollary 3, n G = (). E(T n A )=E(E(T n (t)j) A ) = E( Q n E( + itx k = nj) A )=E( A ): n P X k! Z, where Ee itz A = Ee 2 2 t 2 A for any A 2 () and Aendix. Proof of Lemma 2. Dene log( + ix) =ix + x2 2 ; (x) j(x)j jxj3 for jxj : Hence e ix =(+ix)exf; 2 x2 + (x)g. Put I n = e itsn Then and W n = exf t2 2 U 2 n + P i (tx n i )g. I n = T n W n = T n (W n ; ex(; 2 t 2 =2)) + T n exf; 2 t 2 =2g: (2) Note that, by Lemma, () =) Now we are going to establish (). S n! Z. (L:2:3) =) ET n A! E A A 2G=) ET n W! EW (3) for every bounded G-measurable random variable W (by the usual aroximation). By (3), E(T n A exf; 2 t 2 g) n!! E(exf; 2 t 2 =2g A ): By (2), for () we need only to rove E(T n (W n ; ex(; 2 t 2 =2)) A )! 0: (4) 6

7 On the set [jtxn j], j P (tx n i )jjt 3 j P i i (L.2.2.). By (L.2.) again, on [jtjxn > ], jx n i j 3 jtj 3 X n U 2 n! 0, by (L.2.) and j X i (tx n i )j! 0: (5) (5) and (L.2.2) imly W n! ex(; 2 t 2 =2). By (L:2:3 0 ), ft n n g is tight and then T n (W n ; ex(; 2 t 2 =2))! 0. Since T n (W n ; ex(; 2 t 2 =2)) = I n ; T n ex(; 2 t 2 =2) is u.i. by (L:2:3 0 ). Hence EjT n (W n ; ex(; 2 t 2 =2))j!0which imlies (:3) and, as a consequence, () holds. This comletes the roof of Lemma. References. Aldous, D.J. and Eagleson, G.K.(978). On mixing and stability of limit theorems. Ann. Probab. V.6, Chow, Y.S. and Teicher, H.(997). Probability theory. 3rd edition, Sringer-Verlag. Hall, P. and Heyde, C.C. (980). Martingale limit theorey and its alication. Academic Press. Janson, S.(988). Some airwise indeendent sequences for which the Central Limit Theorem fails, Stochastics, 23(988), Krajka, A.(998). On an examle of sums of airwise indeendent random variables for which the central limit theorem hold. Yokohama Math. Journal, V.45, McLeish, D.L.(974). Deendent central limit theorem and invariance rinciles, Ann. Probab., V.2, Renyi, A. (963). On stable sequences of events. Sankhya Ser. A 25,