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,0 k n&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.
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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 r 1 (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 k Ck for some constant C (see [4, pp. 467]) implies that
lim m : 2 C 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 r 1, implies the squared integrability of f(*).
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Proof. Using Bartlett's formula (see [2, pp. 489]), we have
? &2 2 2 brr=8?# (0) (\(r)&cos(r*)) f (*) d*, r 1. (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 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)= ,1 k k0. 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&k n{(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; t 1] is a strictly stationary martingale difference with mean zero and 2 h 2 4 variance _u= j=1 aj _ . Let Ft=_[Zs; s t] 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. (2.8) n Ä n t=1 Where I[A] is the indicator function of the even A. Using Theorem 8.1 of n 1Â2 2 Pollard [7], we get t=1 UtÂn O N(0, _u). Which implies (see [4, Proposition 6.3.1.]) n n n &1Â2 (n Â_) : ZtZt&1, : ZtZt&2, ..., : ZtZt&h O(W1, W2, ..., Wh). \ t=1 t=1 t=1 + (2.9) Now Lemma 2.2 follows from n n &1Â2 n E : ZtZt&k& : ZtZt+k } t t } =1 =1 k n &1Â2 n E : ZtZt&k + : ZtZt+k Ä0, as n Ä , {}t } }t n k }= =1 = & +1 for all fixed k 1. Now, we give a sufficient condition under which (1.6) holds. Theorem . . 2 3 Let [Xt] be defined by (1.1) with the spectral density function f(*). $ (a) If for some $>3Â4, lim supk Ä k (|k|+|&k|)< , then f(*) is squared integrable and (1.6) is true for some {>1Â2. File: 683J 161805 . By:CV . Date:21:08:96 . Time:14:54 LOP8M. V8.0. Page 01:01 Codes: 2567 Signs: 1245 . Length: 45 pic 0 pts, 190 mm CLT OF SAMPLE AUTOCORRELATIONS 187 3Â4 (b) If lim infk Ä k (|k|+|&k|)>0 and k k+h j j+h 0 for all k, j and h 1, then f(*) will not be squared integrable and 1Â2 2 lim supm Ä m |k| m k {0. Proof. It is clear that we only need to prove this Theorem for the case $ of 0=1 and k=1Â|k| , k=\1, \2, .... (a) For { #(1Â2, 2$&1) we have, dx m{ : 2 m{ Ä 0, as m Ä , k | 2$ |k| m m&1 x and 2 2 2 : #h=_ : : kk+h h h \ k + 2 2 0 2 2 =_ : : kk+h+ : kk+h+ : kk+h h \k k h k h + 2 1 <& =& 2 h dx dx (h&1)Â2 dx 2 2 _2 : +2 +2 +2 + $ | $ $ | 2$ | $ $ $ h {h 1 x h h x 1 x (h&x) (h&1) = 2 &$ 1&2$ 2 C0 : [h log h+h ] h 2 < , (2.10) where C0 is an universal constant. It is known that (2.10) is equivalent to the integrability of f 2(*). (b) For $ 3Â4, we have 1Â2 2 1Â2 dx lim inf m : k lim inf m 2$>0 m m | Ä |k| m Ä m x and 2 dx 2 : #2 : _2 : : _2 = . (2.11) h k k+h | 2$ h h \k + h { 1 (x+h) = 1 1 1 1 As an application of Theorem 2.2 and 2.3, let us consider the stationary long memory sequence [Xt] defined by the following ARIMA( p, d, q) model (see [4, pp. 469]): d (1&B) A(B) Xt=B(B) Zt, t=\1, \2, ..., File: 683J 161806 . By:CV . Date:21:08:96 . Time:14:54 LOP8M. V8.0. Page 01:01 Codes: 2285 Signs: 935 . Length: 45 pic 0 pts, 190 mm 188 SHUYUAN HE where B is the backward shift operator of time t, d<1Â2, p j q j A(z)=1& j=1 ajz {0 for |z| 1, B(z)=1& j=1 bjz {0 for |z|<1 and 2 [Zt] is a white noise with mean zero and variance _ . [Xt] has the spectral density _2 B(ei*) 2 f(*)= , (2.12) 2? }(1&ei*)d A(ei*)} and can be expressed as a linear sequence of [Zt]: Xt= k=0 kZt&k. Here, [k] is determined by Taylor expansion: B(z) k d = : kz ,|z|<1. (2.13) (1&z) A(z) k=0 k d k Let B(z)ÂA(z)= k=0 akz ,|z| 1 and 1Â(1&d) = k=0 bkz ,|z|<1. k Then there exists a \ # (0, 1) and a constant C>0 such that ak=o(\ ), as d&1 kÄ and |bk | Ck . It follows that for some constant C1 , k d&1 |k |= : bjak&j C1k , k=1, 2, .... }j } =0 Hence, d<0.25 implies the condition (a) of Theorem 2.3 and which is just the necessary and sufficient condition for the spectral density f(*)tobe squared integrable. ACKNOWLEDGMENTS The author thanks the referee for his helpful suggestions. REFERENCES [1] Anderson, T. W., and Walker, A. M. (1964). On the asymptotic distribution of the autocorrelations of a sample from a linear stochastic process. Ann. Math. Statist. 35, 12961303. [2] Anderson, T. W. (1971). The Statistical Analysis of Time Series. Wiley, New York. [3] Bartlett, M. S. (1946). On the theoretical specification and sampling properties of auto- correlated time-series. J. Roy. Statist. Soc. Supp. 8, 2741, 8597. [4] Brockwell, P. J., and Davis, R. A. (1987). Time Series: Theory and Methods. Springer- Verlag, New York. [5] Cavazos-Cadena, R. (1994). The asymptotic distribution of sample autocorrelations for a class of linear filters. J. Multivariate Anal. 48, 249274. [6] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. [7] Pollard, D. (1984). Convergence of Stochastic Processes. Springer-Verlag, New York. File: 683J 161807 . By:CV . Date:21:08:96 . Time:14:54 LOP8M. V8.0. Page 01:01 Codes: 2456 Signs: 1772 . 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