- Email: [email protected]
Spectral analysis for processes with almost periodic covariances
Journal of Statistical Planning and Inference 140 (2010) 3608–3612
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Spectral analysis for processes with almost periodic covariances Murray Rosenblatt Department of Mathematics, University of California, San Diego, USA
a r t i c l e i n f o
abstract
Available online 22 May 2010
The structure of random processes with almost periodic covariances is described from a spectral perspective. Under appropriate conditions methods for spectral estimation are described for such processes which are neither stationary nor locally stationary. Some spectral mass is then located off the main diagonal in this spectral plane. A method for estimating the support of the spectral mass is described in the Gaussian case. A number of open questions are mentioned. & 2010 Elsevier B.V. All rights reserved.
Keywords: Almost periodic covariance Spectral estimation Periodogram Frequency shift Gaussian
1. Introduction There is by now an extensive literature on spectral analysis for stationary processes that Parzen (1957a, b) and many others (see Grenander and Rosenblatt, 1957) have contributed to. The same types of spectral estimates have been used adaptively in the analysis of processes referred to as locally stationary. Here we consider another class of nonstationary processes for which appropriately designed spectral estimates can still be used effectively for estimation and resolution. They are the harmonizable processes with almost periodic covariance function. The usual type of spectral estimate designed for a stationary or locally stationary process may there no longer be consistent and so may lead to erroneous conclusions. A process {xn} is harmonizable in the sense of Loe ve if it has a mean square Fourier representation Z p einl dZðlÞ, xn ¼ p
with the covariance of the random spectral function ZðlÞ covðZðlÞ,ZðmÞÞ ¼ Fðl, mÞ and rðn,mÞ ¼ covðxn ,xm Þ ¼
Z pZ p p
p
einlimm dFðl, mÞ,
Z pZ p p
p
jdFðl, mÞj o1:
{xn} is a harmonizable process with almost periodic covariance function if for each ðs, tÞ there are functions aj ðs, tÞ and real P j jaj ðs, tÞj o 1 such that X rðs þ t, t þ tÞ ¼ aj ðs, tÞexpðilj ðs, tÞtÞ,
lj ðs, tÞ with
j
Tel.: +1 858 534 2641.
E-mail address: [email protected] 0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.04.027
M. Rosenblatt / Journal of Statistical Planning and Inference 140 (2010) 3608–3612
3609
with the convergence uniform in s, t. An almost periodic covariance function of a harmonizable process implies that F has complex mass on a countable number of diagonal lines l ¼ m þb, b =bj, j =y, 1,0,1,y . We assume xn is a real valued process implying that if l ¼ m þ b is a line of spectral support, so is l ¼ mb. Then if F on l ¼ m þb is absolutely continuous with respect to Lebesgue measure with spectral density fb ðmÞ, fb ðmÞ ¼ f b ðm þbÞ ¼ f b ðmÞ ¼ fb ðmbÞ: It also follows that jfb ðmÞj2 rf0 ðmÞf0 ðm þ bÞ: Let L be the set of b values corresponding to lines with non-trivial spectral support. Note that the stationary case is that in which L ¼ f0g. ½p, p2 can be regarded as a torus where ðp, lÞ is identified with ðp, lÞ and ðl, pÞ with ðl,pÞ. Then l ¼ m þb,p r m r pb with l ¼ m þb2p, pbr m r p can be considered a closed curve on the torus. If p 4 b 40 fb ðmÞ is defined from m ¼ p to pb with fb2p ðmÞ from m ¼ pb to p. If p o bo 0 fb ðmÞ is defined from m ¼ pb to p with fb þ 2p ðmÞ from m ¼ p to m ¼ pb as its continuation. We can take the b’s in the range ½p, pÞ as long as we set fb ðmÞ ¼ fb2p ðmÞ for pb o m o p if p 4 b4 0 and fb ðmÞ ¼ fb þ 2p ðmÞ from m ¼ p to pb if p o b o0. See Lii and Rosenblatt (2002, 2006) and Hurd and Miamee (2007) for related discussion. A simple example of a process with almost periodic covariance function is given by the moving average process xn ¼
m X
aj ðnÞxnj ,
Exn 0,
ð1Þ
j¼0
with the deterministic sequences aj ðnÞ, j ¼ 0,1, . . . ,m almost periodic in n and the process fxn g a white noise process. In meteorological, astronomical, or engineering application periodic or almost periodic covariances often draw attention (see Lund et al., 1995). 2. Spectral estimates Let Fn ðlÞ denote the finite Fourier transform n=2 X
Fn ðlÞ ¼
xt eitl ,
t ¼ n=2
where it is understood that n is even. The periodogram is In ðl, mÞ ¼
1 Fn ðlÞFn ðlÞ: 2pðn þ1Þ
Assume that Kn(y) is a nonnegative continuous symmetric weight function of finite support such that 1 Let Kn ðyÞ ¼ b1 n Kðbn yÞ and bn k0, nbn -1 as n-1. Consider an estimate of fw(y) Z p f^ w ðyÞ ¼ In ðm þ w, mÞKn ðmyÞ dm:
R1 1
KðyÞ dy ¼ 1.
p
Assumption 1. fXk g, EX k 0, is harmonizable with almost periodic covariance function. The set L of b values with nontrivial spectral support is finite. The functions fb ðmÞ, b 2 L, are continuously differentiable on l ¼ ðb þ mÞ mod 2p on the torus ½p, p2 . Theorem 1. If Assumption 1 is satisfied XZ logðnÞ , Ef^ w ðyÞ ¼ fb ðmÞKn ðmZÞ dm sincðyðb,wÞÞ þ O n b2L where sincðyÞ ¼ sin
n þ1 nþ1 y y 2 2
and yðb,wÞ bw mod 2p: The estimate (2) holds uniformly in w and y. Notice that Ef^ w ðyÞ ¼ O
logðnÞ n
ð2Þ
3610
M. Rosenblatt / Journal of Statistical Planning and Inference 140 (2010) 3608–3612
if w 2 = L. If w 2 L and the fb’s are continuously differentiable up to 2nd order, Z 1 logðnÞ 00 Ef^ w ðyÞ ¼ fw ðyÞ þ b2n þfw ðyÞ u2 KðuÞ du þ O 2 n The estimates f^ ðyÞ are then asymptotically unbiased. w
Assumption 2. Let Ex4n o 1 for all n with the 4th cumulant rn1 ,n2 ,n3 ,n4 ¼ Eðxn1 xn2 xn3 xn4 Þrn1 ,n2 rn3 ,n4 rn1 ,n3 rn2 ,n4 rn1 ,n4 rn2 ,n3 satisfying an inequality jrn1 ,n2 ,n3 ,n4 jr cn2 n1 ,n3 n1 ,n4 n1 , P with a,b,c ca,b,c o 1. Let 0
yð1Þ ¼ ðl þ blÞ mod 2p, yð2Þ ¼ ðm0 þ m þ b0 Þ mod 2p, yð3Þ ¼ ðm0 þ blÞ mod 2p, 0
yð4Þ ¼ ðl þ m þb0 Þ mod 2p: Theorem 2. Under Assumptions 1 and 2 X Z pZ p fb ðm0 þ w0 Þfb0 ðm0 Þsincðyð1ÞÞsincðyð2ÞÞKn ðmyÞ Kn ðm0 y0 Þ dm dm0 covðf^ w ðyÞ, f^ w0 ðyÞÞ ¼ b,b0 2L
þ
p
p
X Z pZ p p
b,b0 2L
p
fb ðm0 Þfb0 ðm0 þ w0 Þ sincðyð3ÞÞsincðyð4ÞÞKn ðmyÞ Kn ðm0 y0 Þ dm dm0 þO
logðnÞ : n
Corollary 1. Under the assumptions of Theorem 2, ( X 2p covðf^ w ðyÞ, f^ w0 ðy0 ÞÞ ¼ dðZ0 Z þ w0 w þ bÞdðZ0 Zb0 Þfb ðZ0 þw0 Þfb0 ðZ0 Þ: nbn b,b0 2L )Z X 1 þ dðZ0 þ Z þ wbÞdðZ0 þ Z þ w0 þ b0 Þfb ðZ0 Þfb0 ðZ0 þ w0 Þ K 2 ðuÞ du þ o : nbn b,b0 2L If w ¼ w0 , Z ¼ Z0 2p s ðf^ w ðZÞÞ ¼ nbn 2
( 2
f0 ðZÞ þ
X
0
)Z
dð2Z þwbÞdð2Z þ w þb Þfb ðZÞfb ðZ þ wÞ
b,b0 2L
1 K 2 ðuÞ du þo : nbn
Here (
dðxÞ ¼
0
if xc0 mod 2p,
1
otherwise:
Theorem 1 and the Corollary imply that f^ w ðyÞ is a consistent estimate of fw(y). One should note that f^ 0 ðyÞ is an example of the classical spectral estimate used in the analysis of stationary or locally stationary processes. If the process has an almost periodic covariance function and is harmonizable but is not stationary f^ 0 ðyÞ cannot be used to effectively estimate fb(y) for bc0 mod 2p. One must use an estimate like f^ b ðyÞ. Here, however, one must note a basic difficulty. All is well if one knows that the b 2 L a priori. However, this would generally not be the case. A natural question is that of estimation of the b 2 L. And what happens if one uses an estimate of b to effect an estimate of fb ðmÞ? This will be considered in the case of a Gaussian process with almost periodic covariance function under appropriate assumptions. 3. Gaussian processes and estimation of the b’s in L There are a few obvious remarks that can be made about Gaussian processes before considering estimation of b 2 L. Let us first note that the only stationary Gaussian martingale difference are the white noise processes, that is, processes with constant spectral density. For E(xnxm)= rn m =0 if n 4 m and so the spectral density f ðlÞ of the process satisfies R ikl R e dFðlÞ ¼ 0 for k 40. But then eikl dFðlÞ ¼ 0 for jkj 4 0 implying an absolutely continuous spectrum with spectral 0 density f ðlÞ ¼ F ðlÞ constant.
M. Rosenblatt / Journal of Statistical Planning and Inference 140 (2010) 3608–3612
3611
Now consider the martingale difference condition for a Gaussian process with almost periodic covariance function. As R P 0 before E(XnXm) =0 if n 4m. But EðXm þ k Xm Þ ¼ b2L eiðm þ kÞb eikm dF b ðmÞ ¼ 0 if k4 0. Multiplying by eiðm þ kÞb for some b0 2 L and summing over m =1 to N with factor 1/N, in the limit as N-1 we have Z p eikm dF b0 ðmÞ ¼ 0 ð3Þ p
for k 40 for every b0 2 L. In particular for b= 0 Z p eikm dF 0 ðmÞ ¼ 0 p
for k4 0 so by the same argument as that given for a stationary process, F0 ðmÞ is absolutely continuous with f0 ðmÞ ¼ F00 ðmÞ constant. Also jDFb ðmÞj2 r DF0 ðmÞDF0 ðm þ bÞ implying that Fb ðmÞ is absolutely continuous with fb ðmÞ ¼ Fb0 ðmÞ. Further, (3) implies that X cb ðkÞeikm , b 2 L, fb ðmÞ ¼
ð4Þ
kZ0
with c b ðkÞ ¼ cb ðkÞ for k Z0. The necessary and sufficient conditions are then f0 ðmÞ constant and (4) for b 2 L, ba0. The harmonizable property coupled with that of having the covariance function almost periodic is basically a second order moment property. However, in the case of a Gaussian process with these properties, they are also retained by the square of the process. We remark now on how the b 2 L can be estimated for a Gaussian process satisfying Assumption 1 and some additional conditions. They amount to Assumption 3. For all the b 2 L, Z p fb ðmÞ dma0: p
P Also there is a function cðjÞ Z 0 with j integral such that jcb ðjÞj rcðjÞ for all b 2 L with j cðjÞ o 1. Rp It should be noted at this time that there are processes such that p fb ðmÞ dm ¼ 0 for some b 2 L. The procedure used to detect the b 2 L is based on the integrated periodogram Un ðbÞ ¼
Z p p
In ðb þ m, mÞ dm ¼
n=2 X 1 X 2 eibk n þ 1 k ¼ n=2 k
which is a periodogram of the square X2k of the process. The periodogram of X2k . Before fully describing the method of estimating the b’s in L, the mean and fluctuation of Un(b) will be described. This will suggest that the method of estimating the b’s is a plausible one. Some detailed estimates leading to Theorem 1 of Lii and Rosenblatt (2002) imply Theorem 3. Under Assumption 1, XZ p sinððn þ 1Þ=2Þyðb, bÞ logðnÞ þO fb ðmÞ dm EU n ðbÞ ¼ ððn þ 1Þ=2Þyðb,wÞ n b2L p uniformly in b. Let e 4 0 be any fixed positive number. Notice that the mean EUn(b) tends to zero uniformly outside of the intervals Rp ðbe,b þ eÞ with b 2 L for which p fb ðmÞ dma0. A bound for the fluctuation of the process Un ðbÞ can be given in the Gaussian case and Theorem 3 of Lii and Rosenblatt (2002) is restated here as Theorem 4. If {Xk} is a Gaussian sequence with almost periodic covariance function satisfying Assumptions 1 and 3, then ! 1=2 X logðnÞ cðjÞ lim supjUn ðbÞEU n ðbj r 25=2 q n-1 b n j with probability 1. q is the number of elements in L. Fix a level d 40. A procedure for estimating the b 2 L with Z p 4d f ð m Þ d m b p
ð5Þ
3612
M. Rosenblatt / Journal of Statistical Planning and Inference 140 (2010) 3608–3612
will be described. It will give estimates wn(b) of the b 2 L satisfying (5) with njwn ðbÞbj oan ¼ n1=5 except for a set of probability less than O(log(n)/n2). Split the range of wðp rw r pÞ into blocks of length c(n)/n (with limn-1 cðnÞ ¼ 1 and c(n)=o(n) as n-1). Compute the maximum of jUn ðbÞj in each block. Discard the blocks with maxima less than d þ ðlogðnÞ=nÞ1=4 . As n-1, the values wn(b) where the remaining maxima occur satisfying (5) are such that njwn ðbÞbj oan ¼ n1=5 for b 2 L with the exception of a set of probability less than O(log(n)/n2). Theorem 5. Let {Xk} be a Gaussian process satisfying Assumptions 1 and 3. Set d 40. There are estimates wn(b) of b 2 L such that njwn ðbÞbj on1=5 except for a set of probability O(log(n)/n2) for n sufficiently large. If one can get a result analogous to Theorem 4 for non-Gaussian processes, it is clear that the one can get similar results on estimates wn(b) of b 2 L. It is also clear that even in the Gaussian case it would be of interest to get a procedure that Rp would take care of the case exempted, b 2 L for which one might have p fb ðlÞ dl ¼ 0. Rp The naive procedure of estimating fb ðlÞ when j p fb ðlÞ dlj4 d by fwn ðbÞ ðlÞ under Assumptions 1 and 3 in the Gaussian case actually is effective. The estimates for Ef wn ðbÞ ðlÞfb ðlÞ and s2 ðfwn ðbÞ ðlÞÞ are comparable to what one obtains in the case that b is known. What happens in the case of a non-Gaussian process of type (1) to lim supjUn ðbÞEU n ðbÞj
n-1 b
is not clear. One suspects that results are likely to depend on the tail behavior of the x distribution. References Grenander, U., Rosenblatt, M., 1957. Statistical Analysis of Stationary Time Series. J. Wiley. Hurd, H., Miamee, A., 2007. Periodically Correlated Random Sequences. J. Wiley. Lii, K.-s., Rosenblatt, M., 2002. Spectral analysis for harmonizable processes. Ann. Statist. 30, 258–297. Lii, K.-s., Rosenblatt, M., 2006. Estimation for almost periodic processes. Ann. Statist. 34, 1115–1139. Lund, R., Hurd, H., Bloomfield, P., Smith, R., 1995. Climatological time series with periodic correlation. J. Climate 8, 2787–2809. Parzen, E., 1957a. On consistent estimates of the spectrum of a stationary time series. Ann. Math. Statist. 28, 329–348. Parzen, E., 1957b. On choosing an estimate of the spectral density function of a stationary time series. Ann. Math. Statist. 28, 921–932.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Spectral analysis for processes with almost periodic covariances Murray Rosenblatt Department of Mathematics, University of California, San Diego, USA
a r t i c l e i n f o
abstract
Available online 22 May 2010
The structure of random processes with almost periodic covariances is described from a spectral perspective. Under appropriate conditions methods for spectral estimation are described for such processes which are neither stationary nor locally stationary. Some spectral mass is then located off the main diagonal in this spectral plane. A method for estimating the support of the spectral mass is described in the Gaussian case. A number of open questions are mentioned. & 2010 Elsevier B.V. All rights reserved.
Keywords: Almost periodic covariance Spectral estimation Periodogram Frequency shift Gaussian
1. Introduction There is by now an extensive literature on spectral analysis for stationary processes that Parzen (1957a, b) and many others (see Grenander and Rosenblatt, 1957) have contributed to. The same types of spectral estimates have been used adaptively in the analysis of processes referred to as locally stationary. Here we consider another class of nonstationary processes for which appropriately designed spectral estimates can still be used effectively for estimation and resolution. They are the harmonizable processes with almost periodic covariance function. The usual type of spectral estimate designed for a stationary or locally stationary process may there no longer be consistent and so may lead to erroneous conclusions. A process {xn} is harmonizable in the sense of Loe ve if it has a mean square Fourier representation Z p einl dZðlÞ, xn ¼ p
with the covariance of the random spectral function ZðlÞ covðZðlÞ,ZðmÞÞ ¼ Fðl, mÞ and rðn,mÞ ¼ covðxn ,xm Þ ¼
Z pZ p p
p
einlimm dFðl, mÞ,
Z pZ p p
p
jdFðl, mÞj o1:
{xn} is a harmonizable process with almost periodic covariance function if for each ðs, tÞ there are functions aj ðs, tÞ and real P j jaj ðs, tÞj o 1 such that X rðs þ t, t þ tÞ ¼ aj ðs, tÞexpðilj ðs, tÞtÞ,
lj ðs, tÞ with
j
Tel.: +1 858 534 2641.
E-mail address: [email protected] 0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.04.027
M. Rosenblatt / Journal of Statistical Planning and Inference 140 (2010) 3608–3612
3609
with the convergence uniform in s, t. An almost periodic covariance function of a harmonizable process implies that F has complex mass on a countable number of diagonal lines l ¼ m þb, b =bj, j =y, 1,0,1,y . We assume xn is a real valued process implying that if l ¼ m þ b is a line of spectral support, so is l ¼ mb. Then if F on l ¼ m þb is absolutely continuous with respect to Lebesgue measure with spectral density fb ðmÞ, fb ðmÞ ¼ f b ðm þbÞ ¼ f b ðmÞ ¼ fb ðmbÞ: It also follows that jfb ðmÞj2 rf0 ðmÞf0 ðm þ bÞ: Let L be the set of b values corresponding to lines with non-trivial spectral support. Note that the stationary case is that in which L ¼ f0g. ½p, p2 can be regarded as a torus where ðp, lÞ is identified with ðp, lÞ and ðl, pÞ with ðl,pÞ. Then l ¼ m þb,p r m r pb with l ¼ m þb2p, pbr m r p can be considered a closed curve on the torus. If p 4 b 40 fb ðmÞ is defined from m ¼ p to pb with fb2p ðmÞ from m ¼ pb to p. If p o bo 0 fb ðmÞ is defined from m ¼ pb to p with fb þ 2p ðmÞ from m ¼ p to m ¼ pb as its continuation. We can take the b’s in the range ½p, pÞ as long as we set fb ðmÞ ¼ fb2p ðmÞ for pb o m o p if p 4 b4 0 and fb ðmÞ ¼ fb þ 2p ðmÞ from m ¼ p to pb if p o b o0. See Lii and Rosenblatt (2002, 2006) and Hurd and Miamee (2007) for related discussion. A simple example of a process with almost periodic covariance function is given by the moving average process xn ¼
m X
aj ðnÞxnj ,
Exn 0,
ð1Þ
j¼0
with the deterministic sequences aj ðnÞ, j ¼ 0,1, . . . ,m almost periodic in n and the process fxn g a white noise process. In meteorological, astronomical, or engineering application periodic or almost periodic covariances often draw attention (see Lund et al., 1995). 2. Spectral estimates Let Fn ðlÞ denote the finite Fourier transform n=2 X
Fn ðlÞ ¼
xt eitl ,
t ¼ n=2
where it is understood that n is even. The periodogram is In ðl, mÞ ¼
1 Fn ðlÞFn ðlÞ: 2pðn þ1Þ
Assume that Kn(y) is a nonnegative continuous symmetric weight function of finite support such that 1 Let Kn ðyÞ ¼ b1 n Kðbn yÞ and bn k0, nbn -1 as n-1. Consider an estimate of fw(y) Z p f^ w ðyÞ ¼ In ðm þ w, mÞKn ðmyÞ dm:
R1 1
KðyÞ dy ¼ 1.
p
Assumption 1. fXk g, EX k 0, is harmonizable with almost periodic covariance function. The set L of b values with nontrivial spectral support is finite. The functions fb ðmÞ, b 2 L, are continuously differentiable on l ¼ ðb þ mÞ mod 2p on the torus ½p, p2 . Theorem 1. If Assumption 1 is satisfied XZ logðnÞ , Ef^ w ðyÞ ¼ fb ðmÞKn ðmZÞ dm sincðyðb,wÞÞ þ O n b2L where sincðyÞ ¼ sin
n þ1 nþ1 y y 2 2
and yðb,wÞ bw mod 2p: The estimate (2) holds uniformly in w and y. Notice that Ef^ w ðyÞ ¼ O
logðnÞ n
ð2Þ
3610
M. Rosenblatt / Journal of Statistical Planning and Inference 140 (2010) 3608–3612
if w 2 = L. If w 2 L and the fb’s are continuously differentiable up to 2nd order, Z 1 logðnÞ 00 Ef^ w ðyÞ ¼ fw ðyÞ þ b2n þfw ðyÞ u2 KðuÞ du þ O 2 n The estimates f^ ðyÞ are then asymptotically unbiased. w
Assumption 2. Let Ex4n o 1 for all n with the 4th cumulant rn1 ,n2 ,n3 ,n4 ¼ Eðxn1 xn2 xn3 xn4 Þrn1 ,n2 rn3 ,n4 rn1 ,n3 rn2 ,n4 rn1 ,n4 rn2 ,n3 satisfying an inequality jrn1 ,n2 ,n3 ,n4 jr cn2 n1 ,n3 n1 ,n4 n1 , P with a,b,c ca,b,c o 1. Let 0
yð1Þ ¼ ðl þ blÞ mod 2p, yð2Þ ¼ ðm0 þ m þ b0 Þ mod 2p, yð3Þ ¼ ðm0 þ blÞ mod 2p, 0
yð4Þ ¼ ðl þ m þb0 Þ mod 2p: Theorem 2. Under Assumptions 1 and 2 X Z pZ p fb ðm0 þ w0 Þfb0 ðm0 Þsincðyð1ÞÞsincðyð2ÞÞKn ðmyÞ Kn ðm0 y0 Þ dm dm0 covðf^ w ðyÞ, f^ w0 ðyÞÞ ¼ b,b0 2L
þ
p
p
X Z pZ p p
b,b0 2L
p
fb ðm0 Þfb0 ðm0 þ w0 Þ sincðyð3ÞÞsincðyð4ÞÞKn ðmyÞ Kn ðm0 y0 Þ dm dm0 þO
logðnÞ : n
Corollary 1. Under the assumptions of Theorem 2, ( X 2p covðf^ w ðyÞ, f^ w0 ðy0 ÞÞ ¼ dðZ0 Z þ w0 w þ bÞdðZ0 Zb0 Þfb ðZ0 þw0 Þfb0 ðZ0 Þ: nbn b,b0 2L )Z X 1 þ dðZ0 þ Z þ wbÞdðZ0 þ Z þ w0 þ b0 Þfb ðZ0 Þfb0 ðZ0 þ w0 Þ K 2 ðuÞ du þ o : nbn b,b0 2L If w ¼ w0 , Z ¼ Z0 2p s ðf^ w ðZÞÞ ¼ nbn 2
( 2
f0 ðZÞ þ
X
0
)Z
dð2Z þwbÞdð2Z þ w þb Þfb ðZÞfb ðZ þ wÞ
b,b0 2L
1 K 2 ðuÞ du þo : nbn
Here (
dðxÞ ¼
0
if xc0 mod 2p,
1
otherwise:
Theorem 1 and the Corollary imply that f^ w ðyÞ is a consistent estimate of fw(y). One should note that f^ 0 ðyÞ is an example of the classical spectral estimate used in the analysis of stationary or locally stationary processes. If the process has an almost periodic covariance function and is harmonizable but is not stationary f^ 0 ðyÞ cannot be used to effectively estimate fb(y) for bc0 mod 2p. One must use an estimate like f^ b ðyÞ. Here, however, one must note a basic difficulty. All is well if one knows that the b 2 L a priori. However, this would generally not be the case. A natural question is that of estimation of the b 2 L. And what happens if one uses an estimate of b to effect an estimate of fb ðmÞ? This will be considered in the case of a Gaussian process with almost periodic covariance function under appropriate assumptions. 3. Gaussian processes and estimation of the b’s in L There are a few obvious remarks that can be made about Gaussian processes before considering estimation of b 2 L. Let us first note that the only stationary Gaussian martingale difference are the white noise processes, that is, processes with constant spectral density. For E(xnxm)= rn m =0 if n 4 m and so the spectral density f ðlÞ of the process satisfies R ikl R e dFðlÞ ¼ 0 for k 40. But then eikl dFðlÞ ¼ 0 for jkj 4 0 implying an absolutely continuous spectrum with spectral 0 density f ðlÞ ¼ F ðlÞ constant.
M. Rosenblatt / Journal of Statistical Planning and Inference 140 (2010) 3608–3612
3611
Now consider the martingale difference condition for a Gaussian process with almost periodic covariance function. As R P 0 before E(XnXm) =0 if n 4m. But EðXm þ k Xm Þ ¼ b2L eiðm þ kÞb eikm dF b ðmÞ ¼ 0 if k4 0. Multiplying by eiðm þ kÞb for some b0 2 L and summing over m =1 to N with factor 1/N, in the limit as N-1 we have Z p eikm dF b0 ðmÞ ¼ 0 ð3Þ p
for k 40 for every b0 2 L. In particular for b= 0 Z p eikm dF 0 ðmÞ ¼ 0 p
for k4 0 so by the same argument as that given for a stationary process, F0 ðmÞ is absolutely continuous with f0 ðmÞ ¼ F00 ðmÞ constant. Also jDFb ðmÞj2 r DF0 ðmÞDF0 ðm þ bÞ implying that Fb ðmÞ is absolutely continuous with fb ðmÞ ¼ Fb0 ðmÞ. Further, (3) implies that X cb ðkÞeikm , b 2 L, fb ðmÞ ¼
ð4Þ
kZ0
with c b ðkÞ ¼ cb ðkÞ for k Z0. The necessary and sufficient conditions are then f0 ðmÞ constant and (4) for b 2 L, ba0. The harmonizable property coupled with that of having the covariance function almost periodic is basically a second order moment property. However, in the case of a Gaussian process with these properties, they are also retained by the square of the process. We remark now on how the b 2 L can be estimated for a Gaussian process satisfying Assumption 1 and some additional conditions. They amount to Assumption 3. For all the b 2 L, Z p fb ðmÞ dma0: p
P Also there is a function cðjÞ Z 0 with j integral such that jcb ðjÞj rcðjÞ for all b 2 L with j cðjÞ o 1. Rp It should be noted at this time that there are processes such that p fb ðmÞ dm ¼ 0 for some b 2 L. The procedure used to detect the b 2 L is based on the integrated periodogram Un ðbÞ ¼
Z p p
In ðb þ m, mÞ dm ¼
n=2 X 1 X 2 eibk n þ 1 k ¼ n=2 k
which is a periodogram of the square X2k of the process. The periodogram of X2k . Before fully describing the method of estimating the b’s in L, the mean and fluctuation of Un(b) will be described. This will suggest that the method of estimating the b’s is a plausible one. Some detailed estimates leading to Theorem 1 of Lii and Rosenblatt (2002) imply Theorem 3. Under Assumption 1, XZ p sinððn þ 1Þ=2Þyðb, bÞ logðnÞ þO fb ðmÞ dm EU n ðbÞ ¼ ððn þ 1Þ=2Þyðb,wÞ n b2L p uniformly in b. Let e 4 0 be any fixed positive number. Notice that the mean EUn(b) tends to zero uniformly outside of the intervals Rp ðbe,b þ eÞ with b 2 L for which p fb ðmÞ dma0. A bound for the fluctuation of the process Un ðbÞ can be given in the Gaussian case and Theorem 3 of Lii and Rosenblatt (2002) is restated here as Theorem 4. If {Xk} is a Gaussian sequence with almost periodic covariance function satisfying Assumptions 1 and 3, then ! 1=2 X logðnÞ cðjÞ lim supjUn ðbÞEU n ðbj r 25=2 q n-1 b n j with probability 1. q is the number of elements in L. Fix a level d 40. A procedure for estimating the b 2 L with Z p 4d f ð m Þ d m b p
ð5Þ
3612
M. Rosenblatt / Journal of Statistical Planning and Inference 140 (2010) 3608–3612
will be described. It will give estimates wn(b) of the b 2 L satisfying (5) with njwn ðbÞbj oan ¼ n1=5 except for a set of probability less than O(log(n)/n2). Split the range of wðp rw r pÞ into blocks of length c(n)/n (with limn-1 cðnÞ ¼ 1 and c(n)=o(n) as n-1). Compute the maximum of jUn ðbÞj in each block. Discard the blocks with maxima less than d þ ðlogðnÞ=nÞ1=4 . As n-1, the values wn(b) where the remaining maxima occur satisfying (5) are such that njwn ðbÞbj oan ¼ n1=5 for b 2 L with the exception of a set of probability less than O(log(n)/n2). Theorem 5. Let {Xk} be a Gaussian process satisfying Assumptions 1 and 3. Set d 40. There are estimates wn(b) of b 2 L such that njwn ðbÞbj on1=5 except for a set of probability O(log(n)/n2) for n sufficiently large. If one can get a result analogous to Theorem 4 for non-Gaussian processes, it is clear that the one can get similar results on estimates wn(b) of b 2 L. It is also clear that even in the Gaussian case it would be of interest to get a procedure that Rp would take care of the case exempted, b 2 L for which one might have p fb ðlÞ dl ¼ 0. Rp The naive procedure of estimating fb ðlÞ when j p fb ðlÞ dlj4 d by fwn ðbÞ ðlÞ under Assumptions 1 and 3 in the Gaussian case actually is effective. The estimates for Ef wn ðbÞ ðlÞfb ðlÞ and s2 ðfwn ðbÞ ðlÞÞ are comparable to what one obtains in the case that b is known. What happens in the case of a non-Gaussian process of type (1) to lim supjUn ðbÞEU n ðbÞj
n-1 b
is not clear. One suspects that results are likely to depend on the tail behavior of the x distribution. References Grenander, U., Rosenblatt, M., 1957. Statistical Analysis of Stationary Time Series. J. Wiley. Hurd, H., Miamee, A., 2007. Periodically Correlated Random Sequences. J. Wiley. Lii, K.-s., Rosenblatt, M., 2002. Spectral analysis for harmonizable processes. Ann. Statist. 30, 258–297. Lii, K.-s., Rosenblatt, M., 2006. Estimation for almost periodic processes. Ann. Statist. 34, 1115–1139. Lund, R., Hurd, H., Bloomfield, P., Smith, R., 1995. Climatological time series with periodic correlation. J. Climate 8, 2787–2809. Parzen, E., 1957a. On consistent estimates of the spectrum of a stationary time series. Ann. Math. Statist. 28, 329–348. Parzen, E., 1957b. On choosing an estimate of the spectral density function of a stationary time series. Ann. Math. Statist. 28, 921–932.