Statistics & Probability Letters 16 (1993) 301-304

North-Holland

16 March 1993

An L,-convergence theorem for heterogeneous mixingale arrays with trending moments

James Davidson London School uf Economics, London, UK

Received August 1991

Abstract: This paper gives a generalization of an L,-convergence theorem for dependent processes due to Andrews (1988). Among

the cases covered by this result are weak laws of large numbers for random sequences {X,) having moments tending to either

infinity or zero as t +m.

Keywords: Weak law of large numbers; mixingales; nonstationarity

Andrews (1988) combines a theorem on mar- tingale convergence due to Chow (1971) with techniques developed by McLeish (1975a, 1975b, 1977) to obtain a law of large numbers for mixin- gales. The purpose of the present note is to extend these results to allow for global hetero- geneity, including cases where the moments of a sequence are tending to either infinity or zero over time. One application where this extension is indispensable arises in the proof of the central limit theorem for near-epoch dependent func- tions of mixing processes with variances tending to zero; see Davidson (19931.

Let an array of pairs (X,,, FE,; - 00 < t < m, n > 1) be defined on a probability space (0, 9, P),

where the X,, are random variables and the Fnt are a-subfields of 9, increasing in T. The array will be called an L,-mixingale, for p > 1, if there exists an array of nonnegative constants (c,~), and

Correspondence to: James Davidson, Department of Eco- nomics, London School of Economics, Houghton Street, Lon-

don WC2 2AE, UK.

also a nonnegative sequence (J’,& such that 5, + 0 as m + ~0, and

II E(X,, 1 K.,-,> IL =s cnt5;, 3 (1)

IIKz-wn, ~~,t+maJ~CntLl+l~ (2)

hold for all t, n and m 2 0. The sequence ([,J is sometimes said to be of size A,, if [m = O(m -*) for A > A,,. In the case where [,,, = 0 for m > 0, the array becomes a martingale difference (m.d.1. The single-indexed case where X,,, = X,, cFni,t = <F, and c,,[ = c, for each II will be called a mixingale sequence.

The main result is the following.

Theorem 1. Let the array IX,,, .Fn7;11) be a L,- mixingale with respect to constant array of {cn,} such that

(a> lim sup 5 c,, < co, .+c= I=1

(b) lim sup 5 tit = 0, n--to2 t=1

(c) {Xnt/c,,} is uniformly integrable,

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Volume 16, Number 4 STATISTICS & PROBABILITY LETTERS 16 March 1993

where k, is an increasing, integer-ualued function ofnandk,-tmasn+co.Then

Observe that there is no restriction on the mixing size. Any positive rate of memory decay will suffice for the result. Also, every L,-mixingale for p > 1 is a L,-mixingale, so that the theorem has full generality with respect to choice of p.

This is a very general result for which some special cases are more familiar then others. The case where X,,, = Xt/a, where (a,) is a positive constant sequence, and Fnt = <Yt:, each n, is im- portant enough to deserve stating as a corollary.

Corollary 1. Suppose (X,, Ft}y is a L,-mixingale sequence with respect to constant sequence {b,};, and {a,}: is another positive constant sequence such that

(a)

(b)

(cl Then

t b, = O(a,>, t=l

2 bf = o(a,2), f=l

{X,/b,) is uniformly integrable.

limEa,‘kX, =O. Cl n-m

I I f=l

Notice that conditions (a) and (b) of Corollary 1 together require that a, ~CO, so that this condi- tion does not need to be separately asserted. It is easily verified that the conditions are observed when b, = ta for any (Y > - 1, by choosing a,, = n’+a for cr> -1, and a,=log n for a= -1. In particular, when b, = 1 for all t and a,, = n we have the result that for a uniformly integrable L,-mixingale of arbitrary size, lim,,,E 1 x,, I = 0 where x,, = n-‘C:=,X,. This is Andrews’ (1988) Theorem 1. More generally, choosing a,, = Cy= ,b, will automatically satisfy condition (a), and then condition (b) will also hold whenever b, = O(ta>. On the other hand, a case where the conditions fail is where b, = 1 and b, = C:::b,, t > 1. Then, b, = 0(2’), so that b, grows faster than t” for every (Y > 0. In this case condition (a) imposes the

requirement b, = O(a,), so that b,” = O(az>, con- tradicting condition (b).

Proof of Theorem 1. Following Andrews’ ap- proach, we prove an extension of the Chow (1971) theorem for martingale differences (m.d.3) based on Andrews Lemma (Andrews, 1988, p. 465) which in turn draws on the proof of Theorem 2.22 of Hall and Heyde (1980). This shows gen- eral L,-convergence for 1 <p G 2. The result is then extended to L,-mixingales using McLeish’s device of expressing the mixingale as a ‘telescop- ing sum’ of m.d.‘s.

Lemma 1. Suppose that {Xnt, &,I is a m.d. array satisfying conditions (a) and (b) of Theorem 1, and where the array { 1 X,,t/~,t 1 “} is uniformly inte- grable, 1 <p < 2. Then

Ilk, II lim 2 X,, =O.

/I /I n+m f-1 P

Proof. Uniform integrability implies that

SUP E( I X,,t/c,,t I ‘I( I &/c,t I > M)) -+ 0 n,t as M-m.

One may therefore find, for any E > 0, a constant B, < 03 such that

sup { II XJ( 1 XII, 1 > 4?Cn,)lIp/Cnt} G &. (3) n,t

Let

r,, = X,J( I Xnt I Q Bsnt ) and

Z,, =XJ( I X, I >Bccnt)~

so that

X,, = Y,, + z,,

= (r,, - E(rnl 1 &J-J)

+ (znt -JwL 1 K.t-A

since E(X,, I F&i) = 0. Then

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Volume 16, Number 4 STATISTICS & PROBABILITY LETTERS

Consider each of First, since p < 2,

these right-hand side terms.

< ( tgEY2)"z<BE( t&:tj”2. (5)

Second, by the conditional Jensen inequality,

(I 2 (Znt-E(ZntIST,,t-l)) t=1 II P

G 2 Il(z,t-~(Z,tl~,t-l))ll, t=1

k,, k,

6 2 c Il-LtlIp G 2E c cm. (6) I=1 t=1

It follows by condition (b) that 3N, 2 1 such

that for n >N,,

5 c$ < B,2~2. (7) t=1

Putting together (4) with (5) and (6) shows that for II > N,,

II ii 5 X,,, <BE, I=1 P

(8)

where B = 1 + 2X:: icnt < W, by (a). The lemma follows since F is arbitrary. q

Proof of Theorem 1 (continued). If (X,J is uni- formly integrable SO iS {E(X,, 1 Fn,t+k)). Fix j, and let

Y,,= 5 [E(xnt~.9,,t+j)-E(xnt~%,t+j-l)]~

t=1

The sequence (Y,,, 53’&+j1~p=, is a martingale, and by condition (c) the array

F n,t+j >

is uniformly integrable, and

Ll Yni- 0

16 March 1993

(9)

by Lemma 1. For it4 > 1,

M-l kn

C Kj= C E(Xnt I’K+M-1) j=l-M t=1

- 5 E(X,t I F_,-,), t= 1

and hence

5 x,, = “2’ Ynj t=1 j=l-M

k,,

+ ,;, [A-E(Xnt I~+,-,)]

(10)

The triangle inequality and the L,-mixingale property give

kr,

+ xE(X,,-E(X,,lSi,_ t=1

k,

+ 2 EIE(X,t I CM)I I= 1

M kn

G j_EMEIK, I + XM c cnt t=1

‘)I

(12)

According to the assumptions the second mem- ber on the right-hand side of (12) is O(M-“) for some 6 > 0, and given any E > 0 there exists M, such that 4’,,,,C~nlcnf < $E for M > ME. And by choosing n large enough the sum of 2M terms on the right-hand side of (12) can be made smaller than ;E for any finite M, by (9). So, by choosing M > M, we have E 1 CF- 1Xnt ) < e when n is large enough. The theorem is now proved since F is arbitrary. 0

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Volume 16, Number 4 STATISTICS & PROBABILITY LETTERS 16 March 1993

References

Andrews, D.W.K. (1988), Laws of large numbers for depend-

ent non-identically distributed random variables, Econo-

metric Theory 4, 458-467. Chow, Y.S. (1971), On the LO-convergence of nm’iPS,, 0 < p

< 2, Ann. Math. Statist. 42, 393-394. Davidson, J. (19931, The central limit theorem for globally

nonstationary near-epoch dependent functions of mixing

processes: the asymptotically degenerate case, to appear

in: Econometric Theory.

Hall, P. and CC. Heyde (19801, Martingale Limit Theory and its Application (Academic Press, New York).

McLeish, D.L. (1975a), A maximal inequality and dependent

strong laws, Ann. Probab. 3(5), 329-839. McLeish, D.L. (1975b), Invariance principles for dependent

variables, Z. Wahrsch. Venu. Gebeite 32, 165-178.

McLeish, D.L. (1977), On the invariance principle for nonsta-

tionary mixingales, Ann. Probab. S(4), 616-621.

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