EXPLICIT REPRESENTATION OF FINITE PREDICTOR COEFFICIENTS AND ITS APPLICATIONS
BY AKIHIKO INOUE AND YUKIO KASAHARA
Abstract. We consider the finite-past predictor coefficients of stationary time series, and establish an explicit representation for them, in terms of the MA and AR coefficients. The proof involves the alternate iteration of projection operators associated with the infinite past and the infinite future. We provide several applications, which include rates of convergence of the finite predictor coefficients, an equality of Baxter-type for long memory processes, and a simple representation of the partial autocorrelation functionα(·). We use the last result to obtain the precise asymptotic behavior of α(·) with remainder, for the fractional ARIMA processes.
1. Introduction
Let {Xk} = {Xk : k ∈ Z} be a real, zero-mean, weakly stationary process, defined on a probability space (Ω,F, P ), which we shall simply call a stationary process. We denote by H the real Hilbert space spanned by {Xk : k ∈ Z} in L2(Ω,F, P ). The norm of H is given by Y := E[Y2]1/2. For n∈ N, we denote by H[−n,−1] and H(−∞,−1] the subspaces of H spanned by {X−n, . . . , X−1} and {Xk : k≤ −1}, respectively. We write P[−n,−1] and P(−∞,−1] for the orthogonal
projection operators of H onto H[−n,−1]and H(−∞,−1], respectively. The projection P[−n,−1]X0 (respectively, P(−∞,−1]X0) stands for the best linear predictor of the future value X0 based on the finite past{X−n, . . . , X−1} (respectively, the infinite past {Xk : k ≤ −1}), and its mean square prediction error is given by σn2 := X0− P[−n,−1]X02 (respectively, σ2:=X0− P(−∞,−1]X02).
For nondeterministic{Xk} (see Section 2.1), the finite predictor coefficients φn,j are the uniquely determined ones in
P[−n,−1]X0= n j=1 φn,jX−j. (1.1) Date: May 4, 2004.
1991 Mathematics Subject Classification. Primary 60G25; secondary 62M20, 62M10.
Key words and phrases. Predictior coefficients, AR coefficients, MA coefficients, prediction,
fractional ARIMA processes, long memory, Baxter’s inequality, partial autocorrelation functions.
As is well known, we can calculate the numerical values of φn,1, . . . , φn,nas well as the mean square prediction error σn2from the values γ(0), . . . , γ(n) of the autoco-variance function of{Xk}, using recursive algorithms such as the Durbin–Levinson algorithm (see Brockwell and Davis [5], Sections 3.4 and 5.2). The recursive meth-ods are of great practical importance in time series analysis. However, they are not necessarily effective in problems of theoretical character, in particular, those related to the asymptotic behavior as n→ ∞.
A classical problem of this type is the rate of convergence of σn2− σ2 ↓ 0 as n → ∞. See for instance Ginovian [7], where references to earlier work — by Grenander and Rosenblatt, Grenander and Szeg¨o, Baxter, Ibragimov and many others — are given. The arguments in these references are closely related to the theory of orthogonal polynomials as described in [9].
A new approach to a related problem was introduced by the first author [14]. For the partial autocorrelation function α(·) of a stationary process {Xk} with short or long memory, the asymptotic behavior of|α(n)| as n → ∞ was obtained using a representation of the mean square prediction error σn2in terms of the MA (moving-average) coefficients ck and the AR (autoregressive) coefficients ak (see Section 2.2 for the definitions of ck and ak). More precisely, after the precise behavior of σn2−σ2was obtained by arguments involving ckand ak, that of|α(n)| was derived from the result by a Tauberian argument, and in so doing the necessary Tauberian condition was verified using the representation of σn2. The partial autocorrelation α(n), which is equal to φn,n, is the correlation between X0 and Xn, adjusted for the intervening observations X1, . . . , Xn−1 (see (3.22) for definition). The partial autocorrelation function (PACF) is one of the core concepts in time series analysis. By the same approach but with extra complication, similar results on |α(n)| were obtained in [15] and [17] for the fractional ARIMA (autoregressive integrated moving-average) processes. The fractional ARIMA model is an important para-metric model including a class of long memory processes. It was independently introduced by Granger and Joyeux [8] and Hosking [12] (see Example 2.4).
The advantage of such approach, i.e., that via ckand ak, has become more appar-ent in [16] where a represappar-entation of the partial autocorrelation function α(·) itself, in terms of ck and ak, was derived for the first time. The representation enabled us to study the behavior of α(·) more directly without using Tauberian arguments.
As a result, proofs were simplified considerably, and results were improved in sev-eral ways. In particular, the asymptotic behavior of α(n) as n→ ∞, rather than that of |α(n)|, was obtained. For example, it was shown that, for the fractional ARIMA(p, d, q) processes with d∈ (−1/2, 1/2) \ {0}, α(n) ∼ d/n as n → ∞.
In this paper, our main interest is in the finite predictor coefficients φn,j which are among the most basic quantities in the prediction theory for {Xk}. After we establish an explicit representation of the type above for φn,j, i.e., that in terms of the MA coefficients ck and the AR coefficients ak, we give several applications of the representation.
For n ∈ N, we write H[−n,∞) for the subspace of H spanned by {Xk : k ≥ −n}, and P[−n,∞) for the orthogonal projection operator of H onto H[−n,∞). Our method for the proof of the representation of φn,j involves the alternate iteration of P(−∞,−1] and P[−n,∞) as well as Theorem 3.1 in [14] which concerns the key equality H(−∞,−1]∩H[−n,∞)= H[−n,−1]. By the method, we first establish a general theorem for the finite predictor coefficients (Theorem 2.2). The result yields the representation of φn,j in terms of ck and ak when Wiener’s prediction formula is available (Theorem 2.3). In applications, it is essential that φn,j is expressed in terms of absolutely convergent series made up of ck and ak. We derive such an expression (Theorem 2.7) under some conditions on ck and ak, that is, (A1) or (A2) in Section 2.3. The condition (A1) corresponds to short memory processes, while (A2) to long memory processes.
The first application of the representation of φn,jconcerns the rate of convergence of φn,j to its limit as n→ ∞. If we let n → ∞, then, under suitable conditions, φn,j converges to the infinite prediction coefficient φj in
P(−∞,−1]X0= ∞ j=1 φjX−j. (1.2)
The rate at which φn,j converges to φj is an interesting question. A textbook treatment of this problem can be found in Pourahmadi [20], Section 7.6; this book, as well as [5], provides excellent background of time series analysis and prediction theory for the present paper. Using the representation of φn,j, we give precise results on the rate of convergence for a class of intermediate memory processes (Theorem 3.7) as well as for a class of long memory processes which includes the fractional ARIMA(p, d, q) processes with d∈ (0, 1/2) (Theorem 3.3).
The second application of the representation of φn,j is related to the additional error P[−n,−1]X0−nj=1φjX−j that arises when we use the infinite predictor coefficients φj instead of the finite ones φn,j (see Baxter [1], page 138, and Cheng and Pourahmadi [6], page 118). There exists a known inequality that deals with this problem, and is commonly referred to as Baxter’s inequality (see [1]; see also Berg [3], [6] and Section 6.6.2 in [20]). It takes the form
n j=1 |φn,j− φj| ≤ M ∞ k=n+1 |φk| (1.3)
with some positive constant M . The original inequality (1.3) of Baxter was an assertion for short memory processes, and his proof in [1] used the theory of or-thogonal polynomials. By simple arguments based on the representation of φn,j, we prove (1.3) not only for short memory processes but also for long memory pro-cesses including the fractional ARIMA propro-cesses (Theorems 4.3 and 4.5). In the short memory case, the proof gives an M explicitly in terms of ak and ck.
The third application is a representation of the PACF α(·) in terms of ak and ck. The proof involves the representation of φn,j and the equality
φn,j− φn+1,j = φn,n+1−jα(n + 1), (1.4)
which is a part of the Durbin-Levinson algorithm (see, e.g., [5], (5.2.4), page 171). As stated above, we already have such a representation of α(·) in [16]. The repre-sentation of the finite predictor coefficients also gives this type of result since φn,n is nothing but α(n). Among these results, the new representation of α(·) (Theorem 5.4) is of special interest in view of its simplicity. We illustrate the usefulness by applying the result to two classes of processes, that is, a class of short memory pro-cesses and the fractional ARIMA model. For the latter, we derive the asymptotic behavior of α(n) with remainder (Theorem 5.6), which is a refinement of the results in [15, 16].
In Section 2, we prove the representation of the finite predictor coefficients φn,j. In Section 3, we apply it to the rate of convergence of φn,j. In Section 4, we prove inequalities of Baxter-type using the representation of φn,j. Finally in Section 5, we prove a new representation of the PACF α(·) using that of φn,j, and then apply it to derive the asymptotic behavior of α(·) for a class of short memory processes and the fractional ARIMA model.
2. Finite predictor coefficients
Let {Xn} = {Xn : n ∈ Z} be a stationary process; as stated in Section 1, this means that {Xn} is a real, zero-mean, weakly stationary process, defined on a probability space (Ω,F, P ). The autocovariance function γ(·) of {Xn} is defined by
γ(n) := E[XnX0], n∈ Z.
If there exists an even, nonnegative, and integrable function ∆(·) on [−π, π] such that
γ(n) = π
−π
einλ∆(λ)dλ, n∈ Z,
then ∆(·) is called a spectral density of {Xn}. As is well known, {Xn} is purely nondeterministic (PND) if and only if it has a positive spectral density such that π
−π| log ∆(λ)|dλ < ∞ (see Section 5.7 in [5] and Chapter II in [21]). In this paper,
we say that a stationary process{Xn} is long memory (respectively, short memory) if∞k=−∞|γ(k)| = ∞ (respectively, < ∞). See Beran [2], page 6, and [5], Section 13.2.
As we also stated in Section 1, we denote by H the closed real linear hull of {Xk : k ∈ Z} with respect to the norm Y := E[Y2]1/2. Then H is a real
Hilbert space with inner product (Y, Z) := E[Y Z]. For n, m ∈ Z with n ≤ m, we write H(−∞,n], H[n,∞), H[n,m] and H{n} for the closed subspaces of H spanned by {Xk : −∞ < k ≤ n}, {Xk : n ≤ k < ∞}, {Xk : n ≤ k ≤ m}, and Xn, respectively. Notice that H{n} = H[n,n]. For an interval I, we write PI for the orthogonal projection operator of H onto HI.
2.1. General result. Let Y ∈ H. For n ∈ N, P[−n,−1]Y is the best linear predictor of Y based on the observations X−n, . . . , X−1. If {Xk} is nondeterministic, that is, X0 ∈ H(−∞,−1], then X−n, . . . , X−1 are linearly independent, whence we can express the predictor P[−n,−1]Y uniquely in the form
P[−n,−1]Y = n j=1 φn,j(Y )X−j. (2.1)
In this section, we are concerned with the real coefficients φn,j(Y ). For n, k∈ N, we define the orthogonal projection operator Pk
n by Pnk:= P(−∞,−1], k = 1, 3, 5, . . . , P[−n,∞), k = 2, 4, 6, . . . . (2.2) 5
Lemma 2.1. Assume that {Xn} is nondeterministic. Let Y be an arbitrary
ele-ment of H. Then, for n, k ∈ N, there exist unique real coefficients φkn,1(Y ), . . . , φk
n,n(Y ) as well as Znk ∈ H(−∞,−n−1] for k odd and Znk ∈ H[0,∞) for k even, such
that PnkPnk−1· · · Pn1Y = n j=1 φkn,j(Y )X−j+ Znk.
Proof. We assume that k is odd. From Lemma 6.1 in [20] (Regression Lemma), it follows that
H(−∞,−1]= H(−∞,−n−1]+ H[−n,−1] (direct sum) (2.3)
(see the proof of Theorem 6.3 in [20]). Since X−n, . . . , X−1are linearly independent and Pk
nPnk−1· · · Pn1Y ∈ H(−∞,−1], the lemma for k odd follows. The case in which
k is even is proved in a similar fashion. It is natural to ask if φk
n,j(Y ) converges to φn,j(Y ) as k→ ∞. The next theorem
answers this question in the affirmative for PND {Xn} with spectral density ∆(·) such that
π
−π
∆(λ)−1dλ <∞. (2.4)
Theorem 2.2. Let n∈ N and j = 1, . . . , n. We assume that
{Xn} is purely nondeterministic and satisfies (2.4).
(2.5)
Then, for every Y ∈ H, we have φn,j(Y ) = limk→∞φk n,j(Y ).
Proof. By Theorem 3.1 in [14], the assumption (2.5) implies H(−∞,−1]∩ H[−n,∞)= H[−n,−1] (n = 1, 2, . . . ). (2.6)
This and von Neumann’s theorem (see, e.g., Theorem 9.20 in [20]) yield
s-lim m→∞P k nPnk−1· · · Pn1= P[−n,−1] (n = 1, 2, . . . ). (2.7) We put k:= Xk− P(−∞,k−1]Xk (k∈ Z). (2.8)
Then, from Lemma 2.1, we see that
(Pn2k+1Pn2k· · · Pn1Y, −1) = φ2k+1n,1 (Y )· −12.
By (2.7), the left-hand side tends to (P[−n,−1]Y, −1) as k → ∞. Thus an,1 := limk→∞φ2k+1n,1 (Y ) exists. In the same way, letting k→ ∞ in
we find the existence of an,2 := limk→∞φ2k+1n,2 (Y ). Repeating this argument, we see that an,j := limk→∞φ2k+1n,j (Y ) exists for all j = 1, . . . , n. Hence Zn := limk→∞Zn2k+1also exists in H, and we have
Zn = P[−n,−1]Y −n
j=1an,jX−j.
Since the right-hand side is in H[−n,−1], so is Zn. Moreover, Zn ∈ H(−∞,−n−1] since, for every k ≥ 1, Zn2k+1 belongs to the closed subspace H(−∞,−n−1]. Com-bining, Zn ∈ H[−n,−1] ∩ H(−∞,−n−1]. However, by (2.3), this implies Zn = 0. Thus P[−n,−1]Y = nj=1an,jX−j. By uniqueness, we obtain φn,j(Y ) = an,j = limk→∞φ2k+1n,j (Y ). Similarly, we have φn,j(Y ) = limk→∞φ2kn,j(Y ). Thus the theo-rem follows.
Remark 1. A stationary process{Xn} is said to be purely minimal if X0 does not belong to the closed linear span of{Xk : k ∈ Z, k = 0} in H (see Section 8.5 in [20]). The assumption (2.5) of Theorem 2.2 is equivalent to saying that{Xn} is purely minimal (see Makagon and Weron [19] and Salehi [23]; see also Theorem 8.10 in [20]).
Remark 2. The proofs above show that the conclusion of Theorem 2.2 holds under the following weaker assumption than (2.5):
{Xn} is nondeterministic and satisfies (2.6).
(2.9)
In fact, we have assumed (2.5) to ensure (2.9). The condition (2.5) holds in most interesting examples, and, unlike (2.9), we can easily check it.
Remark 3. From (2.6), we obtain
H(−∞,−1]∩ H[0,∞)⊂ H(−∞,−1]∩ H[−1,∞)= H{−1}, H(−∞,−1]∩ H[0,∞)⊂ H(−∞,0]∩ H[0,∞)= H{0},
while X−1and X0are linearly independent if{Xn} is nondeterministic. Thus, (2.9) implies that H(−∞,−1]∩H[0,∞) ={0}. Combining this and the arguments in Helson and Sarason [10], page 6, we see that, under (2.9),{Xn} must be PND, whence has a spectral density ∆(·). It is an open problem to characterize the condition (2.9) in terms of ∆(·).
2.2. Representation in terms of MA and AR coefficients. In this section, we assume that the stationary process{Xn} is purely nondeterministic. For n ∈ N
and m∈ N ∪ {0}, we can express the (m + 1)-step predictor P[−n,−1]Xmuniquely in the form P[−n,−1]Xm= n j=1 φmn,jX−j. (2.10)
We are concerned with representation of the real coefficients φmn,j, which we call the (m + 1)-step finite predictor coefficients. In the 1-step case m = 0, we have φ0n,j= φn,j by (1.1).
We consider the following outer function:
h(z) :=√2πexp 1 4π π −π eiλ+ z eiλ− zlog ∆(λ)dλ , z∈ C, |z| < 1. The function h(z) is holomorphic and has no zeros in |z| < 1. We define the MA coefficients cn by
h(z) =∞
n=0cnz
n, |z| < 1,
and the AR coefficients an by
−1/h(z) =∞
n=0anz
n, |z| < 1
(cf. Section 2 in [14]). Both{cn} and {an} are real sequences, and we have c0> 0 and ∞0 (cn)2 < ∞. The coefficients cn and an are actually those that appear in the following MA(∞) and AR(∞) representations, respectively, of {Xn} (under suitable condition such as (2.14) below for the latter):
Xn = n j=−∞ cn−jξj, n∈ Z, (2.11) n j=−∞ an−jXj+ ξn= 0, n∈ Z, (2.12)
where {ξk} is the innovation process given by ξk = k/k with k in (2.8); see, e.g., Chapter II in [21] for (2.11) and (4.9) in [14] for (2.12). By the assumption that{Xk} is PND, {ξk} forms a complete orthonormal system of H such that, for every n∈ Z, the closed linear span of {ξk:−∞ < k ≤ n} in H is equal to H(−∞,n]. Notice that the sums in (2.11) and (2.12) may not converge absolutely in H.
We put bmj := m k=0 ckaj+m−k, m, j∈ N ∪ {0}.
In particular, b0j = c0aj. For n ∈ N and m, j ∈ N ∪ {0}, we define bm k (n, j) recursively by bm1 (n, j) = bmj , bmk+1(n, j) =∞ m1=0 bmn+1+m 1b m1 k (n, j), k = 1, 2, . . . . (2.13)
From the proof of Theorem 2.3 below, we see that, under the condition
∞
n=0
|an| < ∞
(2.14)
which ensures the absolute convergence of the sums in (2.12), the sums in (2.13) also converge absolutely. We put, for m∈ N ∪ {0}, n ∈ N, and j = 1, 2, . . . , n,
gkm(n, j) :=
bm
k(n, j), k = 1, 3, . . . ,
bmk(n, n + 1− j), k = 2, 4, . . . .
We write ∞− for the improper sum: ∞− = limM→∞M. The following theorem gives an explicit representation of the (m + 1)-step finite predictor coeffi-cients φmn,j in (2.10), in terms of the MA and AR coefficients, under the absolute convergence of the sums in (2.12).
Theorem 2.3. We assume that the AR coefficients anof a purely nondeterministic stationary process {Xn} satisfy (2.14). Then we have φmn,j = ∞−k=1gmk (n, j) for n∈ N, m ∈ N ∪ {0} and j = 1, . . . , n, that is,
P[−n,−1]Xm= n j=1 ∞− k=1g m k (n, j) X−j.
Proof. For m ∈ N ∪ {0} and n ∈ N, we have the following Wiener prediction formulas (cf. [14, Theorem 4.4]): P(−∞,−1]Xm=∞ j=1b m j X−j, (2.15) P[−n,∞)X−n−1−m=∞ j=1b m j X−n−1+j, (2.16)
the sums converging absolutely in H. Recall Pnk from (2.2). From (2.15), we have
Pn1Xm=n j=1g m 1 (n, j)X−j+∞ m1=0b m n+1+m1X−n−1−m1.
From this and (2.16), it follows that
Pn2Pn1Xm= n j=1 gm1(n, j)X−j+ ∞ m1=0 bmn+1+m1 ∞ j=1 bm1 j X−n−1+j = n j=1 {gm 1 (n, j) + gm2(n, j)}X−j+ ∞ m1=0 bmn+1+m 1 ∞ m2=0 bm1 n+1+m2Xm2. 9
Similarly, Pn3Pn2Pn1Xm= n j=1 {gm 1(n, j) + g2m(n, j)}X−j + ∞ m1=0 bmn+1+m 1 ∞ m2=0 bm1 n+1+m2 ∞ j=1 bm2 j X−j = n j=1 {gm 1(n, j) + g2m(n, j) + gm3 (n, j)}X−j + ∞ m1=0 bmn+1+m1 ∞ m2=0 bm1 n+1+m2 ∞ m3=0 bm2 n+1+m3X−n−1−m3.
Repeating this argument, we see that φk
n,j(Xm) in Lemma 2.1 with Y = Xm are
given by φkn,j(Xm) =kl=1glm(n, j). The condition (2.14) implies ∞0 (an)2<∞, whence (2.4) (cf. [14], Proposition 4.2). Thus the theorem follows from Theorem 2.2.
2.3. Representation by absolutely convergent series. In later applications, it is essential to express the finite predictor coefficients φn,j in (1.1) by an absolutely convergent series made up of ak and ck. In this section, we first give such an expression for bm
k (n, j). In the 1-step case m = 0, the result gives the desired
representation for φn,j.
We write R0 for the class of slowly varying functions at infinity: the class of positive, measurable (·), defined on some neighborhood [A, ∞) of infinity, such that limx→∞ (λx)/ (x) = 1 for all λ > 0 (see Chapter 1 in [4] for background).
Throughout this section, we assume that the stationary process {Xn} satisfies one of the following conditions (A1) and (A2):
(A1) {Xn} is purely nondeterministic, and {an} and {cn} satisfy, respectively, (2.14) and ∞ n=0 |cn| < ∞. (2.17)
(A2) {Xn} is purely nondeterministic, and, for d ∈ (0, 1/2) and (·) ∈ R0,{cn} and{an} satisfy, respectively,
cn∼ n−(1−d) (n), n→ ∞, (2.18) an ∼ n−(1+d) 1 (n)· d sin(πd) π , n→ ∞. (2.19)
It should be noticed that (2.19) implies (2.14). By (2.11), the autocovariance function γ(·) has the expression
γ(n) = ∞ k=0 c|n|+kck, n∈ Z. (2.20)
Hence, (2.17) implies that ∞ n=0|γ(n)| ≤ ∞ k=0|ck| 2 <∞.
Thus{Xn} is short memory under (A1). On the other hand, by (2.20) and Propo-sition 4.3 in [13], (2.18) implies that
γ(n)∼ n−(1−2d) (n)2B(d, 1− 2d), n→ ∞. (2.21)
Since 0 < 1− 2d < 1, we see that {Xn} is long memory under (A2). We remark that, under suitable conditions, (2.18), (2.19) and (2.21) are equivalent (cf. [14], Theorem 5.1).
Example 2.4. For d∈ (−1/2, 1/2) and p, q ∈ N ∪ {0}, a stationary process {Xn}
is said to be a fractional ARIMA(p, d, q) process if it has a spectral density ∆(·) of the form ∆(λ) = 1 2π |θ(eiλ)|2 |φ(eiλ)|2|1 − e iλ|−2d, −π ≤ λ ≤ π,
where φ(z) and θ(z) are polynomials with real coefficients of degrees p, q, respec-tively. We assume that φ(z) and θ(z) have no common zeros, and that neither φ(z) nor θ(z) has zeros in the closed unit disk {z ∈ C : |z| ≤ 1}. We also assume without loss of generality that θ(0)/φ(0) > 0. Then the outer function h(·) is given by h(z) = (1− z)−dθ(z)/φ(z) (see, e.g., Section 2 in [15]). If 0 < d < 1/2, then {Xn} satisfies (A2) for some constant function (·) (see Corollary 3.1 in [18]). If
d = 0, then {Xn} is also called an ARMA(p, q) process (see [5], Chapter 3), and both{cn} and {an} decay exponentially, whence (A1) is satisfied.
We put Bn:= ∞ v=0 |cvan+v|, n∈ N ∪ {0}.
For n, k, u, v∈ N ∪ {0}, we define Dk(n, u, v) recursively by D0(n, u, v) := δuv, Dk+1(n, u, v) :=∞ w=0Bn+v+wDk(n, u, w). 11
We have, for example, D3(n, u, v) = ∞ v1=0 ∞ v2=0 Bn+v+v1Bn+v1+v2Bn+v2+u.
By the Fubini–Tonelli theorem, we have Dk(n, u, v) = Dk(n, v, u).
Lemma 2.5. The conditions (A1) and (A2) imply, for k, n, v∈ N ∪ {0}, ∞ u=0 Dk(n, u, v) <∞ and ∞ u=0 Dk(n, u, v)2<∞,
respectively. In particular, we have Dk(n, u, v) <∞ for k, n, u, v ∈ N ∪ {0}. Proof. First we assume (A1). Then
∞ m=0Bm≤ ∞ u=0|cu| ∞ u=0|au| <∞. This and the nonnegativity of Bmimply, for example,
∞ u=0 D3(n, u, v) = ∞ u=0 ∞ v1=0 ∞ v2=0 Bn+v+v1Bn+v1+v2Bn+v2+u ≤ ∞ m=0Bm 3 <∞. The general case can be proved in the same way.
Next we assume (A2). The proof in this case is the same as that of Lemma 2.1 in [16]. By (A2) and Proposition 4.3 in [13], we have Bn = O(n−1) as n → ∞. Therefore, for n ∈ N, fu → ∞v=0Bn+u+vfv defines a bounded linear operator on l2 (see Chapter IX in [11]). Since Dk+1(n, u, v) =wBn+u+wDk(n, w, v), we obtain the desired result by induction on k.
We put βn:= ∞ v=0 cvav+n, n = 0, 1, . . . . (2.22)
In view of Lemma 2.5, we may define δk(n, u, v) recursively by, for k, n, u, v ∈
N∪ {0}, δ0(n, u, v) = δuv, δk+1(n, u, v) =∞ w=0βn+v+wδk(n, u, w). (2.23)
By Lemma 2.5 and the Fubini theorem, we have δk(n, u, v) = δk(n, v, u). The following theorem expresses bmk(n, j) by an absolutely convergent series.
Theorem 2.6. We assume either (A1) or (A2). Then, for n, k ∈ N and m, j ∈ N∪ {0}, bmk(n, j) = m v=0 cm−v ∞ u=0 aj+uδk−1(n + 1, u, v), (2.24)
the sum converging absolutely.
Proof. By Lemma 2.5 and (3.4), we have ∞
u=0|aj+u|Dk−1(n + 1, u, v)
≤ {supuDk−1(n + 1, u, v)}∞
u=0|aj+u| < ∞.
(2.25)
Thus, the right-hand side of (2.24), which we denote by Bm
k (n, j), converges
abso-lutely. To prove the proposition, it is enough show that Bm
k (n, j) satisfies the same
recursion as (2.13). First we have B1m(n, j) =m v=0cm−v ∞ u=0aj+uδuv = m v=0cm−vaj+v= b m j ,
as desired. Next, the Fubini–Tonelli theorem and (2.25) yield, for k≥ 1,
∞ u=0 aj+uδk(n + 1, u, v) = ∞ u=0 aj+u ∞ w=0 ∞ m1=w cm1−wan+1+v+m1 δk−1(n + 1, u, w) = ∞ m1=0 an+1+v+m1 m1 w=0 cm1−w ∞ u=0 aj+uδk−1(n + 1, u, w) = ∞ m1=0 an+1+v+m1Bm1 k (n, j), so that Bk+1m (n, j) = m v=0 cm−v ∞ m1=0 an+1+v+m1Bm1 k (n, j) = ∞ m1=0 m v=0cm−van+1+m1+v Bm1 k (n, j) = ∞ m1=0 bmn+1+m1Bm1 k (n, j). Thus Bm k (n, j) satisfies (2.13).
For later applications, we consider the case m = 0 separately. We put
dk(n, j) := δk(n, 0, j), n, k, j∈ N ∪ {0}. 13
Then, by (2.23), dk(n, j) satisfies the following recursion: for k, n, j∈ N ∪ {0}, d0(n, j) = δj0, dk+1(n, j) =∞ v=0βn+j+vdk(n, v). (2.26)
More explicitly, dk(n, j) are given by, for n, j∈ N ∪ {0}, d1(n, j) = βn+j, d2(n, j) = ∞ v1=0 βn+j+v1βn+v1, and, for k = 3, 4, . . . , dk(n, j) = ∞ v1=0 · · · ∞ vk−1=0 βn+j+vk−1βn+vk−1+vk−2· · · βn+v2+v1βn+v1,
the sums converging absolutely. We put
bk(n, j) := b0k(n, j), gk(n, j) := g0k(n, j)
for (k, n, j) for which the right-hand sides are defined. Then, for n ∈ N and j = 1, 2, . . . , n, we have gk(n, j) = bk(n, j), k = 1, 3, . . . , bk(n, n + 1− j), k = 2, 4, . . . . (2.27)
By Theorem 2.3 and Proposition 2.6, we immediately obtain the following final form of the representation of the 1-step finite predictor coefficients φn,j.
Theorem 2.7. We assume either (A1) or (A2). Then, for n∈ N and j = 1, . . . , n,
we have φn,j =∞−k=1gk(n, j) with (2.27) and b1(n, v) = c0av, v≥ 0, bk(n, v) = c0 ∞ u=0 av+udk−1(n + 1, u), k≥ 2, v ≥ 0, the sum on the right-hand side converging absolutely.
Remark 4. Under either (A1) of (A2), the sum∞k=1gk(n, j) converges absolutely for n large enough and j = 1, . . . , n. See Propositions 3.4 and 4.4 below.
3. Rates of convergence of finite predictor coefficients
If the stationary process {Xn} is PND and satisfies (2.14), then we have the Wiener prediction formula (2.15) with m = 0 or (1.2) with
φj= c0aj, j∈ N. (3.1)
We call φj the infinite predictor coefficients. It holds that
lim
n→∞φn,j = φj, j ∈ N
(cf. Theorem 7.14 in [20]). In this section, we investigate the rate at which φn,j converges to φj as n→ ∞. Notice that, by (2.13), (2.27) and (3.1), we have
φj= b1(n, j) = g1(n, j), n∈ N, j = 1, . . . , n. (3.2)
Thus φjis the first term of the series∞−k=1gk(n, j) in Theorem 2.7 expressing φn,j. This suggests the usefulness of the expression for our purpose.
3.1. Long memory processes. In this section, we assume that the stationary process{Xn} satisfies (A2) in Section 2.3. Thus {Xn} is long memory.
For u≥ 0, we put f1(u) := 1 π(1 + u), f2(u) := 1 π2 ∞ 0 ds1 (s1+ 1)(s1+ 1 + u), and, for k = 3, 4, . . . , fk(u) := 1 πk ∞ 0 dsk−1· · · ∞ 0 ds1 1 (sk−1+ 1)× k−2 m=1 1 (sm+1+ sm+ 1) ×(s 1 1+ 1 + u) (see [15], Section 3; see also [14], Section 6).
Lemma 3.1. (i) ∞k=1f2k(0)x2k = (π−1arcsin x)2 for|x| < 1; (ii) ∞k=1f2k−1(0)x2k−1= π−1arcsin x for|x| < 1.
Proof. Let j ≥ 1. We easily see that f1+j(u) = 0∞f1(s + u)fj(s)ds for u ≥ 0. Hence, we have, for u≥ 0,
f2+j(0) = ∞ 0 f1(s)fj+1(s)ds = ∞ 0 dsf1(s) ∞ 0 f1(u + s)fj(u)du = ∞ 0 ∞ 0 f1(s + u)f1(s)ds fj(u)du = ∞ 0 f2(u)fj(u)du. Repeating this argument, we obtain
∞ 0
fi(u)fj(u)du = fi+j(0), i, j∈ N. (3.3)
Thus, the assertion (i) follows from Lemma 6.5 in [14], while (ii) from Lemma 3.4 in [16].
Recall dk(n, u) from Section 2.3.
Proposition 3.2. (i) For r∈ (1, ∞), there exists N ∈ N such that 0 < dk(n, u)≤ fk(0){r sin(πd)}
k
n , u∈ N ∪ {0}, k ∈ N, n ≥ N. (3.4)
(ii) For k∈ N and u ∈ N ∪ {0}, dk(n, u)∼ n−1fk(0) sink(πd) as n→ ∞. Proof. Let r > 1. Recall βn from (2.22). The condition (A2) implies
βn∼sin(πd)
π n
−1, n→ ∞
(3.5)
(see Proposition 4.3 in [13]). Thus, for n large enough,
0 < β[ns]+n+u≤ r
1/2sin(πd)
π([ns] + n + u), s≥ 0, u ∈ N ∪ {0}. Since we have, for n large enough,
1
[ns] + n + u ≤ r1/2
n(s + 1), s≥ 0, u ∈ N ∪ {0}, there exists N1∈ N such that
0 < β[ns]+n+u≤ r sin(πd) π(s + 1)n
−1, s≥ 0, u ∈ N ∪ {0}, n ≥ N
1. (3.6)
In the same way, we can choose N2 so that 0 < β[ns2]+[ns1]+n≤ r sin(πd)
π(s2+ s1+ 1)n
−1, s
1, s2≥ 0, n ≥ N2. (3.7)
Therefore, we have, for n≥ N := max(N1, N2), 0 < d3(n, u) = ∞ 0 ds2 ∞ 0 ds1· β[s2]+n· β[s2]+[s1]+n· β[s1]+n+u = n2 ∞ 0 ds2 ∞ 0 ds1· β[ns2]+n· β[ns2]+[ns1]+n· β[ns1]+n+u ≤ {r sin(πd)}3 n 1 π3 ∞ 0 ds2 ∞ 0 ds1 1 (s2+ 1)(s2+ s1+ 1)(s1+ 1) = {r sin(πd)} 3 n f3(0),
which implies (3.4) with k = 3. Notice that N is independent of the choice k = 3. We can prove (3.4) for general k and the same N in a similar fashion.
We also prove (ii) only for k = 3; the general case can be treated in the same way. By (3.5), we have lim n→∞nβ[ns]+n+u= sin(πd) π(s + 1), s≥ 0, u ∈ N ∪ {0}, (3.8) lim n→∞nβ[ns2]+[ns1]+n= sin(πd) π(s2+ s1+ 1), s1, s2≥ 0. (3.9)
By (3.6)–(3.9) and the dominated convergence theorem, we obtain
lim n→∞ ∞ 0 ds2 ∞ 0 ds1· nβ[ns2]+n· nβ[ns2]+[ns1]+n· nβ[ns1]+n+u = sin 3(πd) π3 ∞ 0 ds2 ∞ 0 ds1 1 (s2+ 1)(s2+ s1+ 1)(s1+ 1). This implies limnnd3(n, u) = sin3(πd)f3(0) or (ii) with k = 3, as desired.
The following theorem gives the rate for long memory processes, at which φn,j converges to φj. It applies, in particular, to the fractional ARIMA(p, d, q) processes with 0 < d < 1/2.
Theorem 3.3. We assume (A2). Then we have, for j ∈ N,
lim
n→∞n{φn,j− φj} = d
2∞
u=j
φu.
Proof. Let r > 1 be chosen so that 0 < r sin(πd) < 1. By Lemma 3.1, ∞
k=1fk(0){r sin(πd)} k<∞.
(3.10)
Let N be as in Proposition 3.2 (i). Then, for n≥ N and j = 1, . . . , n, n∞ u=0an−j+u ∞ k=1d2k−1(n, u) ≤ ∞ u=0 |an−j+u| ∞ k=1 nd2k−1(n, u) ≤∞ k=1f2k−1(0){r sin(πd)} 2k−1 ∞ u=n−j|au|, so that lim n→∞n ∞ u=0an−j+u ∞ k=1d2k−1(n, u) = 0.
Proposition 3.2, Lemma 3.1, and the dominated convergence theorem yield
lim n→∞n ∞ u=0aj+u ∞ k=1d2k(n, u) = ∞ k=1f2k(0) sin 2k(πd) ∞ u=0aj+u= d 2∞ u=jau.
Therefore, by Theorem 2.7 and (3.2), we have
lim n→∞n{φn−1,j− φj} = lim n→∞ n∞ k=1b2k+1(n− 1, j) + n ∞ k=1b2k(n− 1, n − j) = lim n→∞c0n ∞ u=0 aj+u ∞ k=1 d2k(n, u) + lim n→∞c0n ∞ u=0 an−j+u ∞ k=1 d2k−1(n, u) = c0d2∞ u=jau= d 2∞ u=jφu, as desired.
From the proof of Theorem 3.3, we also obtain the following proposition.
Proposition 3.4. We assume (A2). Let N be as in the proof of Theorem 3.3.
Then we have ∞k=1|gk(n, j)| < ∞ for n ≥ N and j = 1, . . . , n. 17
3.2. Intermediate memory processes. There are several possible setups for short-memory processes, to which we can apply Theorem 2.7 to obtain the rate of convergence of φn,j. Among them, we consider that of so called “intermediate memory” processes (see [5], page 520). Our class is defined in the following way:
(A3) the stationary process{Xn} is purely nondeterministic, and {an}, {cn} and the autocovariance function γ(·) satisfy the following conditions:
(i) cn≥ 0 for all n ≥ 0;
(ii) {cn} is eventually decreasing to zero; (iii) {an} is eventually decreasing to zero;
(iv) for p∈ (1, ∞) and ∈ R0, γ(n)∼ n−p (n) as n→ ∞.
This class of stationary processes was considered in [16]. It is suited for asymp-totic analysis. The condition (iv) implies that the stationary processes satisfying (A3) are short memory.
Example 3.5. Let p > 1. If the autocovariance function γ(·) of a stationary process{Xn} is of the form γ(n) = 1/(1 +|n|)p, then it satisfies (A3). See Example
in Section 7 of [14].
Throughout this section, we assume that the stationary process {Xn} satisfies (A3). We define a constant D by
D :=∞
k=−∞γ(k),
which is positive by (A3)(i) and (2.20). By Theorem 5.3 in [14] (see also the proof of Theorem 6.7 in [14]), (A3) implies the following asymptotics:
cn∼ n−p (n)D−1/2, n→ ∞, (3.11) an ∼ n−p (n)D−3/2, n→ ∞, (3.12) βn∼ n−p (n)D−1, n→ ∞. (3.13)
Thus, in particular,{Xn} satisfies (A1).
We need the following estimates to obtain the rate of convergence for φn,j.
Proposition 3.6. (i) For r∈ (1, ∞), there exists N ∈ N such that 0 < ndk(n, u)
(nβn)k ≤ (rπ) kf
k(0), u∈ N ∪ {0}, k ∈ N, n ≥ N.
(3.14)
Proof. Let r > 1. By (3.13) and the Uniform Convergence Theorem ([4], Theorem 1.5.2), we see that, for n large enough,
0 < β[ns]+n+u βn ≤ r 1/2 n [ns] + n + u p , s≥ 0, u ∈ N ∪ {0}. Since we have, for n large enough,
n [ns] + n + u ≤ r1/2 (s + 1)p ≤ r1/2 s + 1, s≥ 0, u ∈ N ∪ {0}, there exists N1∈ N such that
0 < β[ns]+n+u
βn ≤
r
1 + s, s≥ 0, u ∈ N ∪ {0}, n ≥ N1. (3.15)
In the same way, we may take N2∈ N so large that 0 < β[ns2]+[ns1]+n
βn ≤
r
1 + s2+ s1, s1, s2≥ 0, n ≥ N2. (3.16)
Therefore, we have, for n≥ N := max(N1, N2), 0 < d3(n, u) n2βn3 = 1 n2β3n ∞ 0 ds2 ∞ 0 ds1· β[s2]+n· β[s2]+[s1]+n· β[s1]+n+u = ∞ 0 ds2 ∞ 0 ds1·β[ns2]+n βn · β[ns2]+[ns1]+n βn · β[ns1]+n+u βn ≤ (rπ)3f 3(0),
which implies (3.14) with k = 3. Notice that N is independent of the choice k = 3. In the same way, we can prove (3.14) for general k and the same N .
By (3.13), lim n→∞ β[ns]+n+u βn = 1 (1 + s)p, s≥ 0, u ∈ N ∪ {0}. (3.17)
By (3.15), (3.17) and the dominated convergence theorem, we have d2(n, u) nβ2n = ∞ 0 ds·β[ns]+n βn · β[ns]+n+u βn → ∞ 0 1 (1 + s)2pds = 1 2p− 1 as n→ ∞. Thus (ii) follows.
The following theorem gives the rate for the present class of{Xn}, at which φn,j converges to φj.
Theorem 3.7. We assume (A3). Then we have, for j ∈ N,
φn,j− φj ∼ n−(2p−1) (n)2 1 2p− 1 D−5/2c0+ D−2∞ u=jφu , n→ ∞.
Proof. Let r > 1 and N be as in Proposition 3.6 (i). Let t∈ (0, 1). Since (3.13) implies limn→∞nβn = 0, there exists an integer N such that
0 < nβnrπ < t, n≥ N. 19
Then we have, for l≥ 3, n ≥ max(N, N) and u≥ 0, 0 < dl(n, u) nβn2 < (nβn) l−2(rπ)lf l(0) < (nβn)(rπ/t)3tlfl(0), so that, for j = 1, . . . , n, 1 nβn2 ∞ u=0an−j+u ∞ k=2d2k−1(n, u) ≤∞ u=0|an−j+u| ∞ k=2 d2k−1(n, u) nβn2 ≤ (nβn)× (rπ/t)3 ∞ k=2f2k−1(0)t 2k−1 ∞ u=0|au|.
Since (3.13) implies nβn2 ∼ n−(2p−1) (n)2D−2, we obtain lim n→∞ 1 n−(2p−1) (n)2 ∞ u=0 an−j+u ∞ k=2 d2k−1(n, u) = 0 (j∈ N). (3.18)
In the same way, we have, for n large enough,
1 nβn2 ∞ u=0aj+u ∞ k=2d2k(n, u) ≤∞ u=0|aj+u| ∞ k=2 d2k(n, u) nβ2n ≤ (nβn)× (rπ/t)3 ∞ k=2f2k(0)t 2k ∞ u=j|au|, so that lim n→∞ 1 n−(2p−1) (n)2 ∞ u=0 aj+u ∞ k=2 d2k(n, u) = 0 (j∈ N). (3.19)
Proposition 3.6 and the dominated convergence theorem yield
lim n→∞ 1 nβn2 ∞ u=0 aj+ud2(n, u) = lim n→∞ 1 n−(2p−1) (n)2 ∞ u=0 aj+ud2(n, u) = D −2 2p− 1 ∞ u=j au. (3.20)
As in the proof of Proposition 3.6, we have
1 nanβn ∞ u=0 an−j+ud1(n, u) = 1 nanβn ∞ 0 an−j+[s]· βn+[s]ds = ∞ 0 an−j+[ns] an · β[ns]+n βn ds→ ∞ 0 1 (1 + s)2pds, n→ ∞ or, by (3.12) and (3.13), lim n→∞ 1 n−(2p−1) (n)2 ∞ u=0 an−j+ud1(n, u) = D −5/2 2p− 1. (3.21)
Combining (3.18)–(3.21) and Theorem 2.7, we obtain 1 n−(2p−1) (n)2{φn−1,j− φj} = 1 n−(2p−1) (n)2 ∞ k=1b2k+1(n− 1, j) + ∞ k=1b2k(n− 1, n − j) = c0 n−(2p−1) (n)2 ∞ u=0 aj+u ∞ k=1 d2k(n, u) + ∞ u=0 an−j+u ∞ k=1 d2k−1(n, u) → c0 2p− 1 D−5/2+ D−2∞ u=jau = 1 2p− 1 D−5/2c0+ D−2∞ u=jφu , as required.
For nondeterministic stationary process{Xn}, the partial autocorrelation func-tion (PACF) α(·) is defined by α(1) := γ(1)/γ(0) and
α(n) := (Xn− P[1,n−1]Xn, X0− P[1,n−1]X0) Xn− P[1,n−1]Xn · X0− P[1,n−1]X0
, n = 2, 3, . . . (3.22)
(see Sections 3.4 and 5.2 in [5] for background). By Theorem 1.8 in [16], (A3) implies α(n)∼ n−p (n)D−1 as n→ ∞. Since p > 1, this yields
∞
n=1
|α(n)| < ∞. (3.23)
The implication (A3) ⇒ (3.23) is also verified by Theorem 5.5 below since (3.12) implies an = O(n−(p+1)/2) as n → ∞. In view of (3.23) and Theorem 3.7, we have provided a counterexample to Theorem 7.23 in [20] which seems to mistakenly assert that, for each j, φn,jconverges to φj exponentially if and only if (3.23) holds. In fact, by Theorem 2.3 in [1] with m = 0 (see also Proposition 4.1 below), (3.23) holds under (A1) which is weaker than (A3).
4. Baxter’s inequality
We are concerned with inequalities of Baxter-type which are related to the norm convergence of φn,j to φj.
4.1. Baxter’s inequality for short memory processes. In this section, we give a short and easy proof to the original inequality (1.3) of Baxter, i.e., the case λ = 0 of Theorem 2.2 in [1]. The proof is based on the representation of φn,j (Theorem 2.7), which is available since the assumption in [1] is the same as (A1) in Section 2.3 by the following proposition.
Proposition 4.1. For a stationary process {Xn}, the following conditions are
equivalent:
(i) {Xn} satisfies (A1);
(ii) {Xn} has a positive continuous spectral density and satisfies (2.17); (iii) {Xn} has a positive continuous spectral density and satisfies (2.14);
(iv) {Xn} is short memory, i.e., ∞k=−∞|γ(k)| < ∞, and has a positive continu-ous spectral density.
Proof. Notice that if {Xn} has a positive continuous spectral density ∆(·) on [−π, π], then it is PND since−ππ | log ∆(λ)|dλ < ∞ holds.
Suppose (i). We write h(eiλ) for the nontangential limit of h(z), i.e.,
h(eiλ) = lim
r→1−0h(re
iλ) =∞ n=0cne
inλ, −π ≤ λ ≤ π.
Since (2.17) implies the continuity of h(eiλ), the spectral density ∆(λ) is also con-tinuous by the equality ∆(λ) = 2π|h(eiλ)|2. Letting r→ 1 − 0 in
∞ n=0cnr neinλ ∞ n=0anr neinλ=−1, −π ≤ λ ≤ π, we obtain ∞ n=0cne inλ ∞ n=0ane inλ=−1, −π ≤ λ ≤ π.
This implies that h(eiλ), whence ∆(λ), has no zeros on [−π, π]. Thus ∆(·) is
positive, whence (ii) and (iii) follow.
Suppose (iii). In the same way as above, we have 1 ∆(λ) = 1 2π ∞ n=0ane inλ2, −π ≤ λ ≤ π.
This implies ∞n=0aneinλ = 0 for every λ ∈ [−π, π]. By Wiener’s theorem for
absolutely convergent Fourier series (cf. Lemma 11.6 in [22]), we obtain (2.17) (cf. [3], page 493). Thus (i) follows. The proof of the implication (ii) ⇒ (i) is similar.
By (2.20), (ii) implies (iv). Conversely, we assume (iv). Then we have (2.14), whence (iii), by (3.1) and the arguments in [1], pp. 139–140, which involve the Wiener–L´evy theorem. This completes the proof.
We define A(j) :=∞ u=j|au|, F (j) := ∞ v=0|cv| A(j), j = 0, 1, . . . . (4.1)
Then F (j) decreases to zero as j→ ∞ under (A1). Recall dk(n, j) from (2.26).
Proof. Let n∈ N. We use induction on k. Since d1(n, u) = βn+u, we have ∞ u=0|d1(n, u)| = ∞ u=0|βn+u| ≤ ∞ v=0|cv| ∞ u=0|an+u+v| ≤ F (n).
We assume∞u=0|dk(n, u)| ≤ F (n)k for k∈ N. Then
∞ u=0 |dk+1(n, u)| ≤ ∞ v=0 |dk(n, v)| ∞ u=0 |βn+v+u| ≤ F (n) ∞ v=0 |dk(n, v)| ≤ F (n)k+1.
Thus the inequality also holds for k + 1.
Here is Baxter’s inequality with M given explicitly.
Theorem 4.3. We assume one, whence all, of (i)–(iv) in Proposition 4.1. Then,
for every L∈ (1, ∞), there exists N ∈ N such that (1.3) holds for n ≥ N with M = ∞ v=0|cv| ∞ u=1|au| L.
Proof. By Theorem 2.7, Lemma 4.2 and (3.2), we have, for n = 1, 2, . . . ,
n j=1 |φn,j− φj| ≤ c0 n j=1 ∞ u=0 |au+j| ∞ k=1 |d2k(n + 1, u)| + c0 n j=1 ∞ u=0 |au+n+1−j| ∞ k=1 |d2k−1(n + 1, u)| = c0∞ k=1 ∞ u=0|dk(n + 1, u)| n j=1|au+j| ≤ c0A(1)∞ k=1 ∞ u=0|dk(n + 1, u)| ≤ c0A(1) ∞ k=1F (n + 1) k.
Choose N ∈ N so that 0 < 1/{1 − F (N + 1)} < L. If n ≥ N, then F (n + 1) ≤ F (N + 1) < 1. Therefore, ∞ k=1 F (n + 1)k= F (n + 1) 1− F (n + 1) ≤ F (n + 1) 1− F (N + 1) ≤ LF (n + 1). However, by (3.1) and (4.1), we have
c0F (n + 1) = ∞
v=0|cv|
∞
j=n+1|φj|.
Thus the theorem follows.
From the proof of Theorem 4.3, we also obtain the next proposition.
Proposition 4.4. We assume (A1). Choose N ∈ N so that F (N + 1) < 1. Then
we have ∞k=1|gk(n, j)| < ∞ for n ≥ N and j = 1, . . . , n. 23
4.2. Baxter’s inequality for long memory processes. In this section, we prove Baxter’s inequality for long memory stationary processes. The fractional ARIMA(p, d, q) processes with 0 < d < 1/2 are among them.
Theorem 4.5. We assume (A2). Then there exists a positive constant M such
that (1.3) holds for all n∈ N.
Proof. Let r > 1 be chosen so that 0 < r sin(πd) < 1. Then we have (3.10). By Proposition 3.2 and (2.19), we may take a positive integer N such that both (3.4) and an > 0 hold for n≥ N. Pick δ ∈ (0, d). By (2.19) and Theorem 1.5.6 (iii) in [4] (Potter-type bounds), we may assume that
am/an ≤ 2 max{(n/m)1+d−δ, (m/n)1+d+δ}, m, n≥ N. (4.2)
As in the proof of Theorem 4.3, we have, for n≥ N + 3,
n−1 j=1 |φn−1,j− φj| ≤ c0 ∞ k=1 ∞ u=0 dk(n, u) n−1 j=1 |au+j| = c0{G1(n) + G2(n)}, where G1(n) := ∞ k=1 ∞ u=0 dk(n, u) N +1 j=1 |au+j|, G2(n) := ∞ k=1 ∞ u=0 dk(n, u) n−1 j=N +2 au+j. For n≥ N + 3, we have G1(n)≤ n−1∞ k=1fk(0){r sin(πd)} k N +1 j=1 ∞ u=0 |au+j|, and G2(n)≤ ∞ k=1 ∞ 0 du· dk(n, [u]) n N a[u]+[s]+2ds = nan ∞ k=1 ∞ 0 du· ndk(n, [nu]) 1 N/n a[nu]+[ns]+2 an ds. By (4.2), we have, for u > 0, n≥ N + 3, and N/n ≤ s ≤ 1,
a[nu]+[ns]+2 an ≤ 2 max n [nu] + [ns] + 2 1+δ+d , n [nu] + [ns] + 2 1−δ+d ≤ 2 max{(u + s)−(1+d+δ), (u + s)−(1+d−δ)}.
Hence, by (3.4), G2(n)≤ 2 1 0 ds ∞ 0 (u + s)−(1+d+δ)+ (u + s)−(1+d−δ) du × nan ∞ k=1fk(0){r sin(πd)} k ≤ 2 1 (δ + d)(1− d − δ)+ 1 (d− δ)(1 − d + δ) × nan ∞ k=1fk(0){r sin(πd)} k.
Combining these estimates, we obtain
lim sup n→∞ nd (n)n−1 j=1|φn−1,j− φj| <∞.
Since ∞k=nφk = c0k=n∞ ak ∼ c0sin(πd)/{πnd (n)} as n → ∞, the theorem
follows.
5. Partial autocorrelation functions
5.1. Representation in terms of MA and AR coefficients. In this section, we assume that the stationary process{Xn} satisfies either (A1) or (A2) in Section 2.3. The aim of this section is to prove an explicit representation of the PACF α(·) in terms of the MA and AR coefficients. The formula is much simpler than the earlier one in [16]. The proof is based on Theorem 2.7.
Recall βn from (2.22). For n∈ N ∪ {0}, we define α1(n) := βn and, for k = 3, 5, . . . , αk(n) := ∞ v1=0 · · · ∞ vk−1=0 βn+v1βn+1+v1+v2· · · βn+1+vk−2+vk−1βn+1+vk−1. As in the case of dk(n, j) in Section 2.3, the sums converge absolutely. We have
α2k+1(n) = ∞ v=0 βn+vd2k(n + 1, v), n, k∈ N ∪ {0}. (5.1)
The following proposition is the key to the proof of the main result in this section (i.e., Theorem 5.4 below).
Proposition 5.1. For n, j∈ N ∪ {0} and k ∈ N, we have
d2k(n, j)− d2k(n + 1, j) = k l=1 α2k−2l+1(n)d2l−1(n, j). (5.2)
Proof. Let n, j∈ N ∪ {0}. We use mathematical induction on k. Since α1(n) = βn and d1(n, j) = βn+j, we have
d2(n, j) =∞
v=nβvβv+j =
∞
v=nα1(v)d1(v, j),
which implies (5.2) with k = 1.
We assume that (5.2) holds for k∈ N. By (2.26) and the Fubini–Tonelli theorem, we have d2k+2(n, j) =∞ v2=0 ∞ v1=n βv2+v1βj+v1 d2k(n, v2), whence d2k+2(n, j)− d2k+2(n + 1, j) = I + II, where
I =∞ v2=0 ∞ v1=0βn+v2+v1βn+j+v1 [d2k(n, v2)− d2k(n + 1, v2)] , II =∞ v2=0βn+v2βn+jd2k(n + 1, v2).
Then, by (2.26), (5.2), and the Fubini–Tonelli theorem,
I =∞ v2=0 ∞ v1=0βn+v2+v1βn+j+v1 k l=1α2k−2l+1(n)d2l−1(n, v2) =k l=1α2k−2l+1(n) ∞ v1=0 βn+j+v1∞ v2=0 βn+v1+v2d2l−1(n, v2) =k l=1α2k−2l+1(n)d2l+1(n, j) = k+1 l=2 α2(k+1)−2l+1(n)d2l−1(n, j), while, by (5.1), we have II =∞ v2=0βn+v2d2k(n + 1, v2) βn+j= α2k+1(n)d1(n, j). Thus we obtain (5.2) with k replaced by k + 1, as desired.
We derive two kinds of difference equations for bk(n, j) from Proposition 5.1.
Proposition 5.2. For n, k∈ N and j ∈ N ∪ {0}, we have
b2k+1(n, j)− b2k+1(n + 1, j) = k l=1 α2k−2l+1(n + 1)b2l(n, j). (5.3)
Proof. From Theorem 2.7 and Proposition 5.1, we see that b2k+1(n, j)− b2k+1(n + 1, j) = c0∞ u=0aj+u{d2k(n + 1, u)− d2k(n + 2, u)} = c0∞ u=0aj+u k l=1α2k−2l+1(n + 1)d2l−1(n + 1, j) =k l=1α2k−2l+1(n + 1)c0 ∞ u=0aj+ud2l−1(n + 1, j) =k l=1α2k−2l+1(n + 1)b2l(n, j).
Thus the proposition follows.
Proposition 5.3. For n, k∈ N, and j = 1, . . . , n, we have
b2k(n, n + 1− j) − b2k(n + 1, n + 2− j) = k l=1 α2k−2l+1(n + 1)b2l−1(n, n + 1− j). (5.4)
Proof. Since d1(n + 1, u) = α1(n + 1 + u) and b1(n, j) = c0aj, we see from Theorem 2.7 that b2(n, n + 1− j) = c0 ∞ u=0 an+1−j+ud1(n + 1, u) = ∞ u=n+1 α1(u)b1(n, u− j). Similarly, we have b2(n + 1, n + 2− j) = c0 ∞ u=0 an+2−j+ud1(n + 2, u) = ∞ u=n+2 α1(u)b1(n, u− j). Thus (5.4) holds for k = 1.
Suppose that k≥ 2. Then, from (2.26) and Theorem 2.7, b2k(n, n + 1− j) = c0 ∞ v=0 an+1−j+v ∞ u=0 βn+1+v+ud2(k−1)(n + 1, u), b2k(n + 1, n + 2− j) = c0 ∞ v=1 an+1−j+v ∞ u=0 βn+1+v+ud2(k−1)(n + 2, u). Hence b2k(n, n + 1− j) − b2k(n + 1, n + 2− j) = I + II with I = c0 ∞ v=0 an+1+v−j ∞ u=0 βn+1+v+ud2(k−1)(n + 1, u)− d2(k−1)(n + 2, u), II = c0an+1−j ∞ u=0 βn+1+ud2(k−1)(n + 2, u). By (2.26) and Theorem 2.7, I is equal to
c0 ∞ v=0 an+1−j+v ∞ u=0 βn+1+v+u k−1 l=1 α2(k−1)−2l+1(n + 1)d2l−1(n + 1, v) = k−1 l=1 α2(k−1)−2l+1(n + 1)c0 ∞ v=0 an+1−j+v ∞ u=0 βn+1+v+ud2l−1(n + 1, u) =k−1 l=1 α2(k−1)−2l+1(n + 1)b2l+1(n, n + 1− j) =k l=2α2k−2l+1(n)b2l−1(n, n + 1− j),
while, by (5.1) and the equality b1(n, n + 1− j) = can+1−j, we have II = α2k−1(n + 1)b1(n, n + 1− j).
Thus we obtain (5.4), as desired.
Here is the representation of the PACF α(·) in terms of the MA and AR coeffi-cients.
Theorem 5.4. We assume either (A1) or (A2). Then α(n) =∞−k=1α2k−1(n) for n = 2, 3, . . . .
Proof. We have b1(n, j)− b1(n + 1, j) = c0aj− c0aj= 0 for n∈ N and j = 1, . . . , n. This, together with Theorem 2.7 and Propositions 5.2 and 5.3, yields
φn,j− φn+1,j =∞− k=1{gk(n, j)− gk(n + 1, j)} =∞− k=1{b2k+1(n, j)− b2k+1(n + 1, j) +b2k(n, n + 1− j) − b2k(n + 1, n + 2− j)} =∞− k=1 k l=1 α2k−2l+1(n + 1){b2l(n, j) + b2l−1(n, n + 1− j)} =∞− l=1 {b2l(n, j) + b2l−1(n, n + 1− j)} ∞− k=lα2k−2l+1(n + 1) = ∞− k=1α2k−1(n + 1) φn,n+1−j. Combining this and (1.4), we obtain the theorem.
5.2. Asymptotics for short memory processes. In this section, we apply The-orem 5.4 to the partial autocorrelation functions of a class of short memory pro-cesses.
Theorem 5.5. Let p > 1 and let {Xn} be a purely nondeterministic stationary
process. We assume (2.17) and
an= O(n−p) (n→ ∞). (5.5)
Then the partial autocorrelation function α(·) satisfies α(n) = O(n−p) (n→ ∞). (5.6)
Clearly, (5.5) implies (2.14), so that{Xn} in Theorem 5.5 satisfies (A1). Proof. By (5.5), there exists L1∈ (0, ∞) such that
|an| ≤
L1
np (n = 1, 2, . . . ).
Recall βn from (2.22) and α2k+1(·) from the previous section. It holds that |α1(n)| = |βn| ≤ ∞ v=0 |cv| L1 (n + v)p ≤ L2 np (n = 1, 2, . . . )
with L2 := L1∞k=0|ck|. Let A(·) and F (·) be as in (4.1). By (5.1) and Lemma 4.2, we have, for n, k∈ N, |α2k+1(n)| ≤ ∞ v=0 |βn+vd2k(n + 1, v)| ≤ L2 npF (n + 1) 2k.
