ON COVARIANCE GENERATING FUNCTIONS AND SPECTRAL DENSITIES OF PERIODICALLY CORRELATED AUTOREGRESSIVE PROCESSES

Periodically correlated autoregressive nonstationary processes of finite order are considered. The corresponding Yule-Walker equations are applied to derive the generating functions of the covariance functions, what are called here the periodic covariance generating functions. We also provide closed formulas for the spectral densities by using the periodic covariance generating functions, which is a new technique in the spectral theory of periodically correlated processes.


Introduction
In this work, we will consider periodically correlated autoregressive processes of finite order p ≥ 1 (PCAR(p)).The aim is to develop a technique for analytic evaluation of the spectral densities.The work of Nematollahi and Soltani [6] treated the case of p = 1.Their work exhibits the complexities in deriving closed forms for the spectral densities even for p = 1.Moreover, it seems that their approach cannot even be extended for the case of p > 1.Our approach, presented in this article, is new; it is based on employing the periodic covariance generating function (PCGF), which is the generating function of the covariance function.The PCGF for PCAR (p) has not yet been produced, to the best of our knowledge.Of course the significance of generating functions has been frequently realized in queueing, Markov chains, and most fields in engineering, but rarely in time series.We will derive the PCGF by using Yule-Walker equations for PCAR(p).Through some examples we will demonstrate the details of our method of deriving the PCGF.
The authors learned from a referee that Sakai [8] derived the formula for the spectral density matrix of periodic ARMA processes.The spectral part of this work is an alternative to other methods, but it is new in deriving the spectral density of PCAR(p) processes.
This paper is organized as follows.Preliminaries are given in Section 2. The PCGF for PCAR(p) is derived in Section 3. Closed forms for the spectral densities of PCAR(p) are established in Section 4. Some examples are also provided, including a new derivation for the Nematollahi and Soltani [6] formula for the spectral density.
Concerning the literature on periodically correlated processes, periodically correlated AR or ARMA processes, we refer the reader to the works of Miamee [4], Pourahmadi and Salehi [7], Soltani and Parvardeh [10], and the references therein.For numerical computation of autocovariance function of periodically correlated ARMA processes, see Shao and Lund [9].

Preliminaries
A centered discrete-time second-order process X = {X t , t ∈ Z}, Z is the set of integers, is said to be a PCAR(p) series if it is generated by the following model: where c i (t), i = 1,..., p, are periodic functions in t with period T, and the process {Z t } is periodic white noise PWN(0, σ 2 t ,T), which means (i) EZ t = 0, (ii) σ 2 t+T = σ 2 t > 0, and (iii) EZ t Z s = 0, for t = s.The smallest integer T satisfying (2.1) is called the period.Note that if T = 1, (2.1) reduces to an ordinary AR model.In this work, we assume T ≥ 1 and it stands for the period.The simplest way to justify the model in (2.1) is to apply the connection between PC and multivariate stationary processes, Gladyshev [2].Indeed, let ), where Y k t = X tT+k , and Z tT+k , t ∈ Z, k = 0,...,T − 1.Then model (2.1) can be restated in the multivariate AR model where p * = [p/T], [x] denotes the smallest integer greater than or equal to x.The T × T coefficients Φ i , i = 1,..., p * , are given in [9].Let Φ(z) where σ 2 m = EZ 2 T−m and c 0 (s) = −1, s = 1,2,...,T.Furthermore, for l = 0,1,... and t,v = 0,...,T − 1.

A method to evaluate R t (k)
We apply the generating function method, see [3], for solving the system of recurrence equations given by (2.5).Let us first introduce the following notations and conventions: u and u 0 are the quotient and reminder in division of u by T, that is,
where M l (x,s) = − p u=0 c uT+l (s)x uT+l , for l = 0,1,..., p 0 , and (3.4) Our aim is to specify the periodic covariance generating functions (PCGFs) A i (x),i = 0,...,T − 1, then the coefficients a i (k), so R i (k), can be easily determined.For solving the system of equations in Theorems 3.2 and 3.9, as we will see in the following, we need to know the vector e(x) = (e T (x),e T−1 (x),...,e 1 (x)) .
Proof.It is clear that the vector e(x) is fully specified by {a i (k); k = 0,..., p − 1, i = 0,...,T − 1}, which is the unique solution to the linear equations given above.By using the first equation given by (2.5) and the fact that R T−(m+i we will obtain the first system of equations given above.The second system of equation can be easily deduced by using the second equations given by (2.5) and the fact that The proof is complete.
Theorem 3.2.The periodic covariance generating functions A i (x), i = 0,...,T − 1, are given by where B(x) is the T × T matrix formed by first T columns of and each U j,r is the matrix formed from a (2T − 1) × (2T − 1) identity matrix, I, in this way that the jth column of U j,r is the sum of rth and jth columns of I, and other columns of U j,r are the same as those of I.

Z. Shishebor et al. 5
Proof.It easily follows from (2.5) that for m = 0,1,...,T − 1, So the system of equations given above can be written as Since A i (x), i = 0,1,...,T − 1, are periodic in i with period T, the system of equations given above can also be written as follows: where a(x) = (A T (x),A T−1 (x),...,A 1 (x),A T (x),...,A 2 (x)) .To obtain (3.6) from (3.10), first form S(x) as in (3.7) and let B(x) be the matrix formed by first T columns of S(x).
Remark 3.3.According to the causality assumption, the covariance functions are welldefined uniquely.Thus for each x, |x| ≤ 1, the system of equations in (3.6) possesses a unique solution for a(x), giving that B(x) is invertible.
Example 3.4.Let X be a causal PCAR(1) process with period T = 1, with c 1 (t) = c and σ 2 t = σ 2 .In this case, the matrix B(x) reduces to so the model is causal if and only if |c| < 1.Also, it is easily shown that EX 2 t = σ 2 /(1 − c 2 ) and so and hence for k ∈ Z.This is in adaptation with the usual stationary autoregressive models of order one.
Example 3.6.Let X be a causal PCAR(2) process with period T, T ≥ 3.In this case, the equations in Lemma 3.1 reduces to and the elements of the matrices D(x) and B(x) are given by (3.22) Now (3.6) can be solved for a(x) by using a mathematical software.
Example 3.7.Let X be a causal PCAR(2) with period T = 1 (stationary case), with c 1 (t) = c 1 , c 2 (t) = c 2 .In this case, the formula (3.6) takes the simple form The covariance generating function for AR( 2) is defined in [1, Page 103] as Therefore, in our settings, G(x) and A(x) are related through where A(x) = ∞ k=0 a(k)x k , and its closed form is given by (3.23).It can be readily verified that A(x) in (3.23) and G(x) in (3.24) satisfy (3.25) in view of the following boundary conditions: In the following, we consider the simple case T = p = 2, which will clarify the proof of Theorem 3.2.
Z. Shishebor et al. 11 In the following, we will examine the structure of the PCGF {A l (x), l = 0,...,T − 1} in order to establish its relation with the spectral density of the process.Interestingly, where Let us call A j l (x) the "jth partition" of A l (x).For T = 2, these partitions can be specified from the corresponding PCGF A l (x) through It is not clear if this is the case for T > 2. Nevertheless, the following theorem indicates that these partitions are solutions to T linear systems each consisting of T linear equations.
Let us recall that based on our notation every u uniquely can be written as u = u T + u 0 .
Remark 3.10.Note that clearly the argument kT + j + p − l 0 is nonnegative and takes its maximum whenever k = j + p − l − 1.In this case, we have kT So all e j,T−m (x) in (3.41) are completely determined using Lemma 3.1.
The classification in equation (3.40) is crucial in finding the partitions.Indeed, since A l+kT s+k T (x) = A l s (x), for all k,k ∈ Z, the classification in (3.40) takes places, which enables one to solve each system separately for any r = 0,1,...,T − 1, as in Theorem 3.2.

A characterization for the spectral density matrix of a PCAR(p) process
The spectral density of a PC process was introduced in [2], if it exists, it is a Hermitian nonnegative definite T × T matrix of complex functions on [0,2π), for which where f k (λ) and f jk (λ), j,k = 0,1,...,T − 1, are related through In this section, we characterize f(λ) of a PCAR(p) process.As we mentioned in Section 2, corresponding to every PC process X t , t ∈ Z, with period T, the T-dimensional random sequence ), where Y k t = X tT+k , t ∈ Z, k = 0,...,T − 1, is stationary in the wide sense, and from the causality, the spectral distribution matrix of the process Y t has a uniformly continuous spectral density matrix h(λ) = [h jk (λ)] j,k=0,1,...,T−1 .Moreover, where Z. Shishebor et al. 13 for j,k = 0,1,...,T − 1, 0 ≤ λ < 2π, where and A j l are the partitions given in Theorem 3.9.Proof.Note that for j,k = 0,1,...,T − 1, Consequently, for 0 ≤ λ < 2π and j ≤ k, we obtain that Using (4.8) and (4.9), we arrive at (4.5).
The following theorem gives the spectral density matrix f(λ) in terms of the partitions of PCGF {A l (x), l = 0,...,T − 1}, which are completely determined in Theorem 3.9.
Theorem 4.2.The spectral density of a PCAR(p) process is a Hermitian nonnegative definite T × T matrix of functions where Proof.It was proved by Gladyshev [2] that where f(λ) is the spectral density matrix of the {X t }, and V(λ) is a unitary matrix depending on λ with elements