Journal of Multivariate Analysis MV1618 journal of multivariate analysis 58, 182188 (1996) article no. 0046 A Note on the Asymptotic Normality of Sample Autocorrelations for a Linear Stationary Sequence* Shuyuan He Department of Probability and Statistics, Peking University, Beijing 100871, China We consider a stationary time series [Xt] given by Xt=k=& kZt&k, where [Zt] is a strictly stationary martingale difference white noise. Under assumptions { 2 that the spectral density f(*)of[Xt] is squared integrable and m |k|m k Ä0 for some {>1Â2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memory ARIMA( p, d, q) sequence, the condition { 2 m |k|m k Ä0 for some {>1Â2 is equivalent to the squared integrability of f(*). View metadata,This result citation extends and Theoremsimilar papers 4.2 of Cavazos-Cadenaat core.ac.uk [5], which were derived under brought to you by CORE 2 the condition m|k|m k Ä0. 1996 Academic Press, Inc. provided by Elsevier - Publisher Connector 1. INTRODUCTION Let [Xt] be a linear stationary time series given by 2 Xt= : kZt&k with 0< : k<, (1.1) k=& k=& 2 2 where [Zt] is a sequence of white noise: EZt=0, EZt =_ , EZt Zs=0 2 for t{s. Then [Xt] has the spectral density f(*)=(_ Â2?)|k=& k 2 2 _exp (ik*)| , the autocovariance function #(k)=_ j=& jj+k and the autocorrelation function \(k)=#(k)Â#(0). The sample autocovari- ance function based on X1 , X2 , ..., Xn is defined by #^ ( k )=#^ (&k)= n&k n t=1 (Xt+k&X n)(Xt&X n)Ân,0kn&1, with X n=t=1 XtÂn. The sample autocorrelation function is defined by \^ (k)=#^ ( k ) Â#^ (0), 1|k|n&1. Let brs= : [\(k+r)+\(k&r)&2\(r) \(k)] k=1 _[\(k+s)+\(k&s)&2\(s) \(k)]. (1.2) Received December 16, 1994 AMS 1991 subject classifications: 60B12; 62M10. Key words and phrases: autocorrelation, central limit theorem, martingale difference, ARIMA model. * Research partially supported by NSFC. 182 0047-259XÂ96 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. File: 683J 161801 . By:CV . Date:21:08:96 . Time:14:54 LOP8M. V8.0. Page 01:01 Codes: 4156 Signs: 1847 . Length: 50 pic 3 pts, 212 mm CLT OF SAMPLE AUTOCORRELATIONS 183 Suppose that f(*) is squared integrable and [Zt] is an independent and identically distributed (iid) white noise. Under the assumptions that 4 2 EZt < or m|k|m k Ä0, Cavazos-Cadena [5, Theorem 4.1, 4.2] proved, for each positive integer h, n1Â2(\^ (1)&\(1), \^ (2)&\(2), ..., \^ (h)&\(h))ON(0, B), as n Ä. (1.3) Where `` O '' means convergence in distribution and N(0, B) stands for the h-dimensional normal distribution with mean zero and covariance matrix B=(brs)h_h . Result (1.3) was established by Bartlett [3] under the condi- 4 tions: EZt < and k=& |k|<, and by Anderson and Walker [1] 2 under the conditions: k=& |k|< and k=& |k| k<. All their conditions imply the squared integrability of f(*). It is known that an equivalent condition for f(*) being squared 2 integrable is k=0 #k<. Another equivalent condition is brr<, for all r1 (see Theorem 2.1). From this point of view, we may not expect result (1.3) without the squared integrability of f(*). 2 The condition m|k|m k Ä0 is weaker than the condition used in Anderson and Walker [1], but does not include the stationary long memory (not the intermediate memory) sequence Xt=k=0 kZt&k defined by the ARIMA(0, d, 0) model d (1&B) Xt=Zt, t=0, 1, 2, ..., (1.4) where [Zt] is an iid white noise and d # (0, 0.25) (here, the condition d # (0, 0.25) ensures the squared integrability of f(*)). Indeed, the fact d&1 kCk for some constant C (see [4, pp. 467]) implies that lim m : 2C lim m x2d&2 dx=. (1.5) k | m Ä |k|m m Ä m In the following section we will prove (1.3) under conditions that [Zt] is a strictly stationary martingale difference white noise and for some {>1Â2, { 2 m : k Ä0, as m Ä , (1.6) |k|m and for a long memory ARIMA( p, d, q) (with d #(0,1Â2)) sequence, (1.6) is equivalent to the squared integrability of the spectral density f(*). 2. THE RESULTS AND PROOFS Theorem . 2 1 Let brs be defined by (1.2). Then the condition brr<, for all r1, implies the squared integrability of f(*). File: 683J 161802 . By:CV . Date:21:08:96 . Time:14:54 LOP8M. V8.0. Page 01:01 Codes: 2909 Signs: 1947 . Length: 45 pic 0 pts, 190 mm 184 SHUYUAN HE Proof. Using Bartlett's formula (see [2, pp. 489]), we have ? &2 2 2 brr=8?# (0) (\(r)&cos(r*)) f (*) d*, r1. (2.1) |0 &1 Let *0=cos \(1). From b11<, it is seen that we only need to prove 2 the integrability of f (*) in an open neighborhood of *0 . But this is ensured by the fact &1 |\(r)|=#0 : kk&r } k } =& &1 #0 : k k&r {}k r } 2 + : k k&r Ä0, as r Ä , } k r }= > 2 and cos(r*0)=cos((r&1) *0) cos(*0)&sin((r&1) *0) % _sin(*0) Ä0, as r Ä . Theorem . 2 2 Let [Xt] be defined by (1.1) with the spectral density f(*) being squared integrable and (1.6) being valid for some {>1Â2. Suppose that [Zt] is a strictly stationary martingale difference white noise with variance _2 and satisfies one of the following conditions: 2 2 (a) E(Zt | Zs; s<t)=_ a.s. for all t; 2 2 4 2 (b) [Zt] is ergodic, EZt Zs =_ for all s{t, and EZt Zt&j Zt&k=0 for any j, k0, j{k. Then for any positive integer h the result (1.3) is true. This theorem can be shown by following the procedure of proving Theorem 4.2 in Cavazos-Cadena [5], after proving (7.4) and (7.14) in that procedure via our Lemma 2.1 and Lemma 2.2. Lemma 2.1 Under the conditions of Theorem 2.2, for any positive integer h, 1Â2 n |D(n; h)| wÄp 0, as n Ä , (2.2) where ``wÄp '' means convergence in probability and n n 2 2 D(n; h)= : jj+h : Zt&jÂn& : Zt Ân . j \t t + =& =1 =1 File: 683J 161803 . By:CV . Date:21:08:96 . Time:14:54 LOP8M. V8.0. Page 01:01 Codes: 2519 Signs: 1230 . Length: 45 pic 0 pts, 190 mm CLT OF SAMPLE AUTOCORRELATIONS 185 2 Proof. Let aj=j j+k and !t=Zt . Then n n nD(n; h)= : aj : !t&j& : aj : !t= : anj !j , (2.3) j=& t=1 j=& t=1 j=& where n t=1 at&j&t=& at, for 1 j n, anj= n {t=1 at&j, otherwise. &k &k0 1&2 Let k0 be an integer such that 1&2 >0.5Â{. Define a(k)=n and { A( y)= : |jaj |Ây, B( y)=y : |aj| for y>0. (2.4) | j|y | j|y 2 We have limn a(k)=,1kk0. Using j=& |aj|j=& j < and Kronecker lemma (see [6, pp. 31]), we get A( y) Ä 0 and { 2 2 B( y)y |k|y (k+k+h)Ä0, as y Ä . Now, the fact that a(1)=n1Â2, n&1Â2a(k)=n1Â2&2&kn{(1&2&k+1)= { 1Â2 { a (k&1) and n a (k0) imply n 0 &1Â2 &1Â2 n : |anj |=n : |anj |+ : |anj |+ : |anj | j {j j j = =& =1 =& =1 &1Â2 1Â2 2n : |jaj |+2n : |aj| | j|n |j|>n &1Â2 &1Â2 2 n : | jaj |+n : | jaj | { 0<| j|a(1) a(1)<| j|a(2) &1Â2 &1Â2 +}}}+n : |jaj |+n a(k 0&1)<| j|a(k0) _ : |jaj |+B(n) a k j n = ( 0)<| | &1 &1Â2 2 a (1) : | jaj |+n a(2) : |aj | { j a a j a 0<| | (1) (1)<| | (2) &1Â2 1Â2 +}}}+n a(k0) : |aj|+n a(k0&1)<| j|a(k0) _ : |aj|+B(n) a k j n = ( 0)<| | 2[A(a(1))+B(a(1))+ } }} +B(a(k0&1)) +B(a(k0))+B(n)]Ä0, as n Ä . (2.5) 2 Now, (2.2) follows from Markov's inequality and E!t=_ . File: 683J 161804 . By:CV . Date:21:08:96 . Time:14:54 LOP8M. V8.0. Page 01:01 Codes: 2512 Signs: 983 . Length: 45 pic 0 pts, 190 mm 186 SHUYUAN HE Lemma 2.2. Under the conditions of Theorem 2.2, for any positive integer h, as n Ä (n) (n) (n) (W 1 , W 2 , ..., W h ) O (W1 , W2 , ..., Wh), as n Ä , (2.6) (n) &1Â2 n where W k =n j=1 ZjZj+kÂ_ and W1 , W2 , ..., Wh are iid random variables with common distribution N(0, 1). h Proof. For any real a1 , a2 , ..., ah , let Ut=j=1 ajZtZt&j. Then [Ut; t1] is a strictly stationary martingale difference with mean zero and 2 h 2 4 variance _u=j=1 aj _ . Let Ft=_[Zs; st] and U be the invariant - _ field of [Zt]. Then, U/Ft for all t. Hence, the ergodic theorem implies n 2 2 2 lim : E(Ut Ân | Ft&1)=E(Ut | U)=_u a.s., (2.7) n Ä t=1 and for any =>0, n 1 2 1Â2 lim : E(Ut I[|Ut |>n =] | Ft&1)=0 a.s.