A Generalization and Application of McLeish's Central Limit Theorem by Tsung-Lin Cheng and Yuan-Shih Chow Institute of Statistical Science Academia Sinica Taipei 115, Taiwan R.O.C.
[email protected]
Abstract. A stable convergence theorem for arrays of random variables is established.
Moreover, interchangeable sequence of random variables provides not only an application but a good example for the main result.
1 Introduction. The central limit theorem (C.L.T.) for dependent random variables has been discussed for many authors. A list of references before 1980 is given by Hall and Heyde (1980). Especially, they (Lemma 3.1, Hall and Heyde 1980) generalize McLeish's Theorem 2.1(1974) by introducing Renyi's(1963) stable convergence. However, their comment \Theorem 3.1 is a trivial consequence of the convergence theorem for characteristic functions" is not correct. The proof of necessity is not easy at all. Fortunately, the suciency is trivial and more useful. We'll state some related de nitions of stable convergence for the following discussions. Let ( ; F ; P ) be a probability space, G be a sub--algebra of F , and Yn be r.v.s.
De nition.( Renyi, 1963 ) Let Yn !d Y . If for every B 2 G there is a countable, dense set of points x, such that
lim n P (Yn x; B ) = QB (x) exists . 1
We say that Yn stably ! FY (or Y ) on G , denoted by s;G Yn ! Y: (
)
s;G ) d. If G = f;; g, then (! is just ! If G = F , then (s;!G) is the usual Yn stably ! FY (we drop the F ), denoted by !s .
For example, let X and X 0 be independent, identically distributed. Set Z2n = X; Z2n+1 = X 0. Then
P (Z n y; X ) ! P (X y ^ ) P (Z n y; X ) ! P (X y)P (X ): 2
2 +1
Hence it is not true that Zn !s X ; however Zn !d X .
De nition 1. For L -r.v.s Zn and Z , RA Zn ! RA Z for every A 2 G i Zn ! Z weakly in L ( ; G ; P ) i Zn W;!G Z . W Zn W;!F Z is denoted by Zn ! Z. 1
(
)
1
(
)
s;G Lemma 1. Yn ! Y i there exists a r.v. Y 0 on ( I; G B; P , where I = (0; 1), 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.1.1)E (eitYn 1A ) ! E (eitY 0 1AI ), for all A 2 G :
(i.e., eitYn (w;!G) eitY 0 ) and (L.1.2)E (eitY 0 1AI ) is continuous at t = 0 for all A 2 G . The proof of Lemma 1 is not trivial, especially the necessity. For the requirement of space and usefullness, We just give the proof of suciency. Proof of Suciency. Let (L.1.1) and (L.1.2) hold. Let A = . By Levy's Continuity Theorem, Yn !d Y 0. For A 2 G with P (A) > 0; P ((Yn y)A) ! P 0((Y 0 y)(A I )) for a conuntable dense sert of points y, by Levy's Continuity Theorem again. This complete 2
the proof. The Suciency of Lemma 1 can be applied to the following lemma which generalizes the results of McLeish (1974) and Hall and Heyde (1980). Let fXn;j ; Fn;j ; 1 j kng be an array of L1-r.v.s on ( ; F ; P ) (Fn;j Fn;j+1 F , Xn;j p n is Fn;j -measurable). De ne Tn(t) = Qkj=1 (1 + itXn;j ), for t 2 R, where i = ;1.
Lemma 2. Let 2 be a positive G -measurable r.v. where G is a -subalgebra of F . Suppose that p (L.2.1) Xn 1max j X j ! 0 n;i ikn 2 (L.2.2) Un2 Pikn Xn;i !p 2 (L.2.3) Tn Tn(t) (w;!G) 1 for every real t (L.2.3') fTn; n 1g is u.i.. s;G ) Then Sn (! Z , where E (eitZ ) = E (e; 21 2t2 ) and for A 2 G
E (eitSn 1A ) ! E (e ;2 t 1A ): (
2 2
(1)
)
Remark. (i). If G = f;; g, then Lemma 2 reduces to McLeish's Theorem 2.1, 1974. (ii). If G = F , then (L.2.3) implies (L.2.3'), i.e. for any L -random variables Y; Yn , if R R A Yn ! A Y for all A 2 F then (Yn ; n 1) is u.i. (Proposition IV 2.2., Neveu 1965). Hence if G = F , then Lemma 2 implies Hall and Heyde's Lemma 3.1. We'll see the proof 1
in the Appendix. According to the above remarks, we conjecture that if there exists an example of G that is neither f;; g nor F ? Is there any application for Lemma 2? In this paper, we provide an answer (of course not unique).
2 Application and Example. By Theorem 7.3.2, Chow and Teicher 1997, it is known that any in nite sequence of interchangeable random variables is conditional i.i.d. given some -algebra. (The algebra of permutable events, which is de ned below. ) Based on Lemma 2, we can prove 3
a more general form of Theorem 9.2.1, Chow and Teicher in an alternative way. De nition 2.(p.232, Chow and Teicher 1997) A mapping = (1; 2; ) from the set N of all positive integers onto itself is called a nite permutation if is one-to-one and n = n for all but nite distinct integers. Let Q denote the set of all nite permutations and let B1 be the class of Borel subsets of R1 = R R and X = (X1; X2; ) a sequence on ( ; F ; P ). De ne X = (X1 ; X2 ; ), for = (1; 2; ). Then
S = fX ; (B ) : B 2 B1; P (X ; (B )(X ); (B )) = 0; all 2 Qg 1
1
1
is called the -algebra of nite permutable events (of X ), denotes the symmetric difference. Moreover we call S the -algebra of nite permutable events. Due to Example 7.3.1, Chow and Teicher 1997, we get Lemma 3. Let fYn ; n 1g be a nonnegative interchangeable sequence of random variables. Suppose that EY1 < 1, then n 1X p Y j ! E (Y1 jS ): n j =1
The following theorem is an application of lemma 2.
Theorem. Let fXn g be a sequence of interchangeable random variables with EX = 0, E (X jS ) = 0 a.s. and EX < 1. (L.2.3) and Then 1
1
2 1
p1n
n X k=1
Xk s;!S Z; (
)
where E (eitZ 1A ) = E (e; 21 t2E(X12jS)1A ), for any A 2 S .
Proof: Since fXn g are interchangeable, so are fXn g. By assumption EX < 1 and 2
Lemma 3, we have
n 1X X 2 !p E (X 2jS );
nk
=1
k
1
4
2
2 1
which satis es (L.2.2). n
pXnj j > ) = P ( P Xnj 1 jXj j>pn > ) P (max j j n j 2 EX 1 jX1j>pn ! 0; 2
(
=1
1
2 1 (
)
2
)
as n ! 1, which proves (L.2.1). Since Xn are conditional independent given S , for any A 2 S and t 2 1], X
j (tXn;i)j !p 0: i
(5)
(5) and (L.2.2) imply Wn !p exp(;2t2=2). By (L:2:30), fTn; n 1g is tight and then Tn(Wn ; exp(;2t2=2)) !p 0. Since Tn(Wn ; exp(;2t2=2)) = In ; Tnexp(;2t2=2) is u.i. by (L:2:30). Hence E jTn(Wn ; exp(;2t2=2))j ! 0 which implies (1:3) and, as a consequence, (1) holds. This completes the proof of Lemma 1.
References. Aldous, D.J. and Eagleson, G.K.(1978). On mixing and stability of limit theorems. Ann. Probab. V.6, p325-331. Chow, Y.S. and Teicher, H.(1997). Probability theory. 3rd edition, Springer-Verlag. Hall, P. and Heyde, C.C. (1980). Martingale limit theorey and its application. Academic Press. Janson, S.(1988). Some pairwise independent sequences for which the Central Limit Theorem fails, Stochastics, 23(1988), p439-448. Krajka, A.(1998). On an example of sums of pairwise independent random variables for which the central limit theorem hold. Yokohama Math. Journal, V.45, p87-96. McLeish, D.L.(1974). Dependent central limit theorem and invariance principles, Ann. Probab., V.2, p620-628. Renyi, A. (1963). On stable sequences of events. Sankhya Ser. A 25, 293-302.
7
