Recursiveness in Hilbert Spaces and Application to Mixed ARMA(p, q) Process

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1 International Journal of Contemporary Mathematical Sciences Vol. 9, 2014, no. 15, HIKARI Ltd, Recursiveness in Hilbert Spaces and Application to Mixed ARMA(p, q) Process R. Ben Taher, M. Rachidi and M. Samih Equip of DEFA - Department of Mathematics and Informatics Faculty of Sciences, University of My Ismail B.P. 4010, Beni M hamed, Meknes - Morocco Copyright c 2014 R. Ben Taher, M. Rachidi and M. Samih. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, the gap between the nonhomogeneous linear recurrence relation and the mixed ARMA(p; q) process in Hilbert spaces is provided. We focus ourselves on the case when the ARMA(p; q) process is of Fibonacci type. Notably, involving properties of Fibonacci sequences, we reach to establish the new formulations of some characteristics of the ARM A(p, q) process. For purpose of illustration, some examples are explored. Mathematics Subject Classification: Primary: 37H10, 40A99; Secondary: 37M10, 62M10, 65Q05 Keywords: ARMA process, Fibonacci sequences, Hilbert space, nonhomogeneous linear recurrence relation, stochastic process 1 introduction Let (Ω, A, P ) be a probability space and consider the stochastic process {X t ; t Z} of a mixed ARM A(p, q) process, satisfying the following difference equation, X t+1 = a 0 X t + + a p 1 X t p+1 + C t+1, (1)

2 716 R. Ben Taher, M. Rachidi and M. Samih such that C t+1 = A t+1 + b 0 A t + + b q 1 A t q+1, where a 0, a 1,, a p 1 R are the auto-regressive parameters, b 0, b 2,, b q 1 R are the auto- regressive moving average parameters and {A t ; t Z} is a stochastic process describing the white noise (see [4], for example). The mixed ARMA(p, q) process are currently used in the methods of prediction and time series (for more details see [4], [5], for example). For discrete indices t = n and ω Ω, we set x n = X n (ω) and c n = C n (ω). It ensues from Equation (1) that the sequence {x n } n 0 satisfies the following nonhomogeneous linear recursive equation, x n+1 = a 0 x n + + a p 1 x n p+1 + c n+1, for every n p 1, (2) where x 0,, x p 1 are the initial data. Solutions of Equation (2) are studied in the literature when c n is polynomial or factorial polynomial (see [2],[7], [8], [9], for example). Recently, two methods have been proposed, for studying solutions of Equation (2) in [2], [7]. More precisely, in [7] a matrix theoretical approach for solving (2) is developed for a general {c n } n 0. Beside in [2], a linearization process is introduced, when {c n } n 0 verifies a nonhomogeneous linear recursive relation. When c n = 0, for every n, it was proved in [11] that solutions of the homogeneous part of (2) are, x <h> n = ρ(n, p)s 0 + ρ(n 1, p)s ρ(n p + 1, p)s p 1, (3) for n p, where S k = a p 1 x k + + a k x p 1 (0 k p 1) and ρ(n, p) = k 0 +2k 1 + +pk p 1 =n p with ρ(p, p) = 1 and ρ(n, p) = 0 if n p 1. (k k p 1 )! a k 0 0 a k p 1 p 1, (4) k 0! k p 1! On the other hand, the closed relationship between stochastic process and the recursive relation (2), has been explored in [10], [11]. Moreover, it was established that some probabilistic properties are related both to recursiveness and Expression (4) in [13], [14]. Particularly, it was shown that Expression (4) is also a special case of Philippou polynomials (see [1], [3]). In this paper, we are interested in studying a theoretical approach of the mixed ARM A(p, q) (stochastic) process defined by (1) and the nonhomogeneous linear recurrence relation (2) in a real Hilbert space. We extend the matrix method of [7] to the case of a real Hilbert space (Section 2). We apply this extension in order to obtain a general solution of (2). Using the extension of the linearization method of [2] to the case of real Hilbert spaces, we study the case of ARMA(p, q) process of Fibonacci type defined by C t+1 = A t+1 = b 0 A t + b 1 A t b q 1 A t q+1 (Section 3). Properties concerning the computation of some characteristics of the ARM A(p, q) model, via the approach of linear recurrence relations, are explored (Section 4). Final

3 Recursiveness in Hilbert spaces and application 717 remarks on the operator methods and recursiveness are discussed in the last section. Throughout of this paper H will denote a real Hilbert space. 2 Equation (2) in a Hilbert space In this section, we extend the fundamental results established in [2], [7] upon the matrix method and the linearization process for solving Equation (2). To this aim, we recall first the following known result established in [7]. Proposition 2.1 Let {v n } n 0 be a real Fibonacci sequence given by v n+1 = a 0 v n + + a p 1 v n p+1 for n 0, where v 0, v 1,, v p 1 are specified initial data. Let A = (α i,j ) 1 i,j p be the matrix defined by α 1j = a j 1, for every 1 j p, and α ij = δ i 1,j, for every 2 i p, 1 j p. Then, for n 1, the entries a (n) is (0 i, s p 1) of the matrix powers A n are given by a (n) is = v (s) n i, where {v(s) n } n 0 are p copies of {v n } n 0, whose initial conditions are v (s) j = δ s,j (0 j p 1, 0 s p 1). Here, δ s,j represents the Kronecker symbol. In view of (3), we infer that for every s (0 s p 1) the sequence {v n+1} (s) n 0 can be formulated as follows. For 0 n p 1 we have v n (s) = δ n,s and v (s) n = a p s+1 ρ(n, p)+a p 1 ρ(n 1, p)+ +a p 1 ρ(n s+1, p), for n p, (5) where the ρ(n, p) are given by Expression (4). Consider {T n } n 0 and {C n } n 0 two sequences of H satisfying (2). For n p 1, we set Z n = t (T n,, T n p+1 ) H p and D n = t (C n, 0,, 0) H p, where t Z means the transpose of Z. It s easy to show that (2) is equivalent to the matrix equation Z n+1 = AZ n +D n+1, for n p 1, where A = (α i,j ) 1 i,j p is the matrix exhibited in Proposition 2.1. Thus, we obtain Z n = A n p+1 Z p 1 + n A n k D k, for every n p. Hence, similarly to the real case (see [2], [7]), we k=p establish that the solution of Equation (2) is, p 1 T n = v n p+1t (s) p s 1 + s=0 n k=p The sequence {T n <h> } n 0 defined by T n <h> v (0) n k C k, for every n 0. (6) = p 1 s=0 v (s) n p+1t p s 1 is a solution of the homogeneous part of Equation (2). In addition, a particular solution {T n <p> } n 0 of (2) is T n <p> = n v (0) n k C k = p 1 v n p+1t (s) p s 1 + T n, where k=p s=0

4 718 R. Ben Taher, M. Rachidi and M. Samih T <p> n = 0 for n = 0, 1,, p 1. The sequence {T <p> n } n 0 is called the fundamental particular solution of (2). Thence, in light of Proposition 2.1 and Equation (6), we get the following general result, Theorem 2.2 Let {T n } n 0 be a solution of (2) in a Hilbert space H. Then, for n 0, we have T n = T <h> n + T <p> n = T <h> n p 1 s=0 v (s) n p+1t <p> p s 1 + T <p> n, (7) where {T n <p> } n 0 is the fundamental particular solution of (2) and {T n <h> } n 0 is a solution of the homogeneous part of (2) with initial data T 0,, T p 1. Now let consider the case when {C n } n 0 is a linear recursive sequence. Set {W n } n 0 a recursive sequence in H, whose initial data are W 0,, W s 1 and W n+1 = b 0 W n + + b s 1 W n s+1, for n s 1, where b 0,, b s 1 are fixed in R. Suppose that C n = W n then, for every n (n p+s 1), Equation (2) yields W n j = T n j p 1 k=0 a kt n k j 1, for 0 j s 1. And the substitution of these expressions in (2), implies that T n (n p + s 1) satisfies the following linear recurrence equation of order p + s, p 1 s 1 s 1 p 1 T n+1 = a j T n j + b j T n j b j a k T n j k 1. (8) k=0 As a result, we get the following extension of the Linearisation Process of [2] (see Theorem 2.1 of [2]), Theorem 2.3 (Linearization Process). Let {T n } n 0 be a sequence of H solution of (2) and suppose that C n = W n, where {W n } n 0 is a homogeneous linear recursive sequence of order s. Then, the sequence {T n } n 0 satisfies a homogeneous linear recursive equation of order m = r + s in H, with initial data T 0,, T p+s 1 and whose coefficients are those of the polynomial p(z) = p 1 (z)p 2 (z), where p 1 (z) = z p p 1 a jz p j 1 and p 2 (z) = z s s 1 b jz s j 1. Remark 2.4 For E = R, it is well known that the solution of (2) can be written under the form T n = T n <h> + T n <p>, where {T n <h> } n 0 and {T n <p> } n 0 are the solutions of the homogeneous part and the particular solution of (2) respectively. It is easy to verify that this property is still valid for sequences (2) in H. Following the similar method of [11], we can bring out that the particular solution of (2) is T n <p> = T n T n <h>, for n 0.

5 Recursiveness in Hilbert spaces and application Study of the ARMA(p, q) process equations 3.1 General setting. In this section we are concerned by the Hilbert space H = L 2 (Ω, A, P ), where (Ω, A, P ) is a probability space. That is, H is the collection of the real-valued random variables (r.r.v. for short) X with finite variance, viz., E(X 2 ) = X 2 (ω)p (dω) = X 2 dp < +. Here H is equipped with the usual inner product < X, Y >= E(XY ) = XY dp. Expressions (5)-(6) and Theorem 2.2, allow us to formulate the result. Theorem 3.1 Let {X t H; t Z} be an ARMA(p, q) process and {A t H; t Z} its associated white noise. Then, for every t p, we have [ X t = ρ(t p + 1, p)x p 1 + [a p 2 ρ(t, p) + a p 1 ρ(t 1, p)]x p (9) p 1 s a p s+j 1 ρ(t j, p)]x p s [ a j ρ(t j, p)]x 0 + t ρ(t k, p)c k, where X 0, X 1,, X p 1 are the r.r.v of the initial data, the ρ(n, p) are given by (4) and C t = A t + b 0 A t b q 1 A t q. Similarly, for t 0, we set Y p t 1 = X t and b 0 = a 0 a p 1,, b j = ap j 1 a p 1,, b p 2 = a p 2 a p 1, b p 1 = 1 a p 1. Thus, we have X t = 0 X X s 1 X s p 1 X p 1 + Γ( t), (10) where 0 = ρ(t p + 1, p), 1 = b p 2 ˆρ( t + p 1, p) + b p 1 ˆρ( t + p 2, p), s 1 = s b p s+j 1 ˆρ( t + p 1 j, p), p 1 = p 1 b j ˆρ( t + p 1 j, p) and Γ( t) = t k=p ˆρ( t + p 1 k, p)[ C k+2p 1 ], with ˆρ(n, p) = k 0 +2k 1 + +pk p 1 =n p a p 1 k=p (k k p 1 )! b k 0 0 b k p 1 p 1. k 0! k p 1! The combinatorial expressions (9)-(10) show that every r.r.v. X t (for t p) can be expressed in terms of the initial r.r.v. X 0, X 1,, X p 1 and A t (t Z) the r.r.v. of the white noise. As far as we know (9)-(10) are not current in the literature on the ARM A(p, q) models.

6 720 R. Ben Taher, M. Rachidi and M. Samih Example - Study of the ARMA(2, q) process. Consider {X t } t Z an ARMA(2, q) process defined by the random vector of initial data (X 0, X 1 ), and X t = a 0 X t 1 + a 1 X t 2 + C t, (11) where C t+1 = A t+1 + b 1 A t + + b q A t q+1. Let P (z) = z 2 a 0 z a 1 = (z λ 1 )(z λ 2 ) be the characteristic polynomial of the homogeneous part of this process, where λ 1 0, λ 2 0 are in R. For studying the process (11), we discuss the two cases λ 1 = λ 2 and λ 1 λ 2. For λ 1 = λ 2, a direct computation shows that the Binet formula of ρ(t, 2) takes the form ρ(t, 2) = λ t 2 ( 1 + t). Thereby, for every t 2, we have X t = (t 2)λ t 3 X 1 + (2λ t 1 ( 1 + t) (t 2)λ t 1 )X 0 + t λ t k 2 (t k 1)C k. In the same way, for t 0 we obtain ρ( t, 2) = ( 1 λ ) t 2 ( 1 t) and thus X t = ( 1 λ ) t 1 X 0 + [2( 1) t ( 1 λ ) t ( t) + ( 1) t ( 1 λ ) t ( 1 t)]x 1 + Γ(t), where Γ(t) = 1 t k=2 λ t k+1 ( t k)c 3 k. For λ 1 λ 2, a similar calculation leads to the Binet formula of ρ(n, 2) is ρ(n, 2) = λn 1 1 λ n 1 2. Hence, for t > 0, we get, λ 1 λ 2 X t = λt 2 1 λ2 t 2 X 1 + [(λ 1 + λ 2 )( λt 1 1 λ t 1 2 λ t 2 1 λ t 2 2 ) + λ 1 λ 2 ]X 0 + Γ(t), λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 where Γ(t) = t k=2 Whence, where Γ(t) = 1 t k=2 and Λ 2 (t, λ j ) = λ 1+λ 2 λ t k 1 1 λ t k 1 2 λ 1 λ 2 C k. For t 0, we have k=2 1 2 λ ρ( t, 2) = ( )λ t 1 t 1 1. (λ 1 λ 2 ) t 2 λ 2 λ 1 X t = Λ 1 (t)x 0 + Λ 2 (t)x 1 + Γ(t), λ t k 2 λ t k 1 1 λ 2 λ 1 ( (λ 1 λ 2 [ λ t ) t 2 λ t 1 (λ 1 λ 2) t k )C 3 k, Λ 1 (t) = 1 1 λ 2 λ 1 ] + ( (λ 1 λ 2 )( λ t 1 ) t 1 (λ 1 λ 2 [ λ t 1 ) t 2 2 λ t 1 1 λ 2 λ 1 ], 2 λ t 1 1 λ 2 λ 1 ). 3.2 The ARMA(p, q) process of Fibonacci type and linearization process. We say that an ARMA(p, q) process is of Fibonacci type if in (1) the C t+1 takes the form C t+1 = A t+1 = b 0 A t + + b q 1 A t q+1, where {A t } t Z is the

7 Recursiveness in Hilbert spaces and application 721 r.r.v. of white noise. That is to say, {A t } t Z satisfies a linear recursive relation of order q and with initial data A 0,..., A q 1. With the aid of Theorem 2.2 and (8), the process {X t } t Z satisfies the following linear recursive equations of order r = p + q, r 1 1 X t+1 = (a 0 + b 0 )X t + (a j + b j c j )X t j + r 2 1 j=r 1 v j X t j p+q 1 j=r 2 c j X t j, (12) where c j = k+s=j;k 1,s 0 b k 1a s and r 1 = min(p, q), r 2 = max(p, q) with v j = a j c j for p > q, v j = b j c j for p < q and v j = 0 for p = q. Theorem 3.2 Let {X t H; t Z} be an ARMA(p, q) process of Fibonacci type and {A t H; t Z} its associated white noise, with initial data X 0,, X p 1 and A 0,, A q 1. Then, the process {X t } t Z satisfies a homogeneous linear recursive of order r = p + q in H, whose coefficients are those of the polynomial p(z) = p 1 (z)p 2 (z), where p 1 (z) = z p p 1 a jz p j 1 and p 2 (z) = z q q 1 b jz q j 1. For reason of simplicity we suppose that p q, since the same argumentation still valid in the case of p q. It s easy to see that Expression (12) takes the linear form X t+1 = p+q 1 w j X t j, such that the coefficients w j, derived from Theorem 2.3, are given by w 0 = a 0 +b 0, w j = a j +b j c j for 1 j q 1, w j = a j c j for q j p 1 (13) and w j = c j for p j p + q 1, and its initial data are R j = X j for 0 j p 1 and R j = X j + A j for p j p + q 1. (14) Theorems 2.3, 3.2 and (3) yield, Theorem 3.3 Under the data of Theorem 3.2, the system {X t ; t Z} obey to a linear recursive equation of order r = p + q, whose coefficients are given by (13) and its initial data of r.r.v. by (14). Moreover, for every t p + q, we have X t = ρ(t, p + q)s 0 + ρ(t 1, p + q)s ρ(t p q + 1, p + q)s p+q 1, where S k = w p+q 1 X k + + w k X p+q 1 (0 k p + q 1) and as in (4) the ρ(n, p + q) are, ρ(n, p + q) = k 0 +2k 1 + +(p+q)k p+q 1 =t p q with ρ(n, n) = 1 and ρ(n, s) = 0 if n s 1. (k k p+q 1 )! w k 0 0 w k p+q 1 p+q 1, k 0! k p+q 1!

8 722 R. Ben Taher, M. Rachidi and M. Samih Similarly, for t 0, by setting Y p+q t 1 = X t and ŵ 0 = w 0 w p+q 1,, ŵ j = w p j 1 w p+q 1,, ŵ p+q 2 = w p+q 2 w p+q 1, ŵ p+q 1 = 1 w p+q 1, we obtain X t = ˆ ρ(p + q t 1, p + q)ŝ0+ ˆρ(p+q t 2, p+q)ŝ1+ + ˆρ( t, p+q)ŝp+q 1, where Ŝk = ŵ p+q 1 X p+q k ŵ k X 0 (0 k p + q 1) and ˆρ(n, p + q) = k 0 +2k 1 + +(p+q)k p+q 1 =t p q with ˆρ(n, n) = 1 and ˆρ(n, s) = 0 if n s 1 (k k p+q 1 )! ŵ k 0 0 ŵ k p+q 1 p+q 1, k 0! k p+q 1! Example - Consider an ARMA(2, 2) process {X t ; t Z} such that P 1 (z) = (z λ 0 )(z λ 1 ) and P 2 (z) = (z λ 2 )(z λ 3 ), where λ 0, λ 1, λ 2, λ 3 are real numbers satisfying λ j λ k for j k. Hence, we have P (z) = (z λ 0 )(z λ 1 )(z λ 2 )(z λ 3 ). Then, for every n 0, the Binet formula of ρ(n, r) is ρ(n, r) = Γ n (λ j )+Θ n (λ j ), where Γ n (λ j ) = (λ 1 λ 2 )(λ 1 λ 3 )(λ 2 λ 3 )λ n 1 0 (λ 0 λ 2 )(λ 0 λ 3 )(λ 2 λ 3 )λ n 1 1 (λ 0 λ 1 )(λ 0 λ 2 )(λ 0 λ 3 )(λ 1 λ 2 )(λ 1 λ 3 )(λ 2 λ 3 ) 2 (λ 0 λ 1 )(λ 0 λ 2 )(λ 1 λ 2 )λ n 1 3 and Θ n (λ j ) = (λ 1 λ 2 )(λ 0 λ 3 )(λ 1 λ 3 )λ n 1 (λ 0 λ 1 )(λ 0 λ 2 )(λ 0 λ 3 )(λ 1 λ 2 )(λ 1 λ 3 )(λ 2 λ 3. Therefore, for ) t 3, we obtain, X t = ρ(t, 4)S 0 + ρ(t 1, 4)S 1 + ρ(t 2, 4)S 2 + ρ(t 3, 4)S 3, with S 0 = ω 3 X 0 + ω 2 X 1 + ω 1 X 2 + ω 0 X 3, S 1 = ω 3 X 1 + ω 2 X 2 + ω 1 X 3, S 2 = ω 3 X 2 + ω 2 X 3, S 1 = ω 3 X 3, where ω 0 = λ 0 + λ 1 + λ 2 + λ 3, ω 1 = 4 i j,i,j=1 λ iλ j, ω 2 = 4 i j k,i,j,k=1 λ iλ j λ k and ω 3 = λ 0 λ 1 λ 2 λ 3. 4 More on some characteristics of an ARMA(p, q) process In this section we are also concerned by the Hilbert space H = L 2 (Ω, A, P ), where (Ω, A, P ) is a probability space. Consider an ARMA(p, q) process (1), whose white noise A t (t Z) is with zero mean E(A t ) = 0 and constant variance var(a t ) = σ 2. Thus the sequence of moments {E(X t )} t Z satisfies the recursive equation E[X t+1 ] = a 0 E[X t ]+ +a p 1 E[X t p+1 ]. The Fibonacci sequence s properties, specially the Binet Formula, permits us to get. Proposition 4.1 Under the preceding data, suppose that λ 1, λ 2,, λ r are the distinct roots of the polynomial P (z) = z p a 0 z p 1 a p 1, of multiplicities m 1,, m p (respectively). Then, we have E[X t ] = r i=1 P i(t)λ t (Binet formula), for every t 0, where each P i (z) is a polynomial of degree m i 1 (1 i r).

9 Recursiveness in Hilbert spaces and application 723 Following (3)-(4), we manage to have the combinatorial form of E[X t ]. Proposition 4.2 Under the data of Proposition 4.1, we have E[X t ] = p 1 (t)e[x p 1 ] + + p s+1 (t)e[x p 1s+ ] (t)e[x 0 ]. for every t p, where p 1 (t) = ρ(t p+1, p), p s+1 (t) = s a p s+j 1ρ(t j, p) and 0 (t) = p 1 a jρ(t j, p) Furthermore, we study others characteristics of X t relying on the precedent results. We begin by the simple case when the {X t ; t Z} are mutually independent, the A t are also mutually independent and X t j is independent of A t for j, t 0. Thus, we have var[x t+1 ] = a 2 0var[X t ]+ +a 2 p+1var[x t p+1 ]+ (1 + b b q 1 )σ 2, where t Z. Therefore, the sequence {var[x t ]} t 0 is a sequence (2) of characteristic polynomial Q(z) = (z 1)(z p a 2 0z p 1 a 2 p 1), whose roots are 1, λ 2 1, λ 2 2,, λ 2 r, with multiplicities are 1, m 1,, m p (respectively). A direct application of the Binet formulas leads to derive the following result. Proposition 4.3 Under the data of Proposition 4.1, suppose that the X t are mutually independent, the A t mutually independent and X t i is independent of A t for t 0. Let λ 1, λ 2,, λ r be the distinct roots of the polynomial P (z) = z p a 0 z p 1 a p 1, of multiplicities m 1,, m p (respectively).then, we have var[x t ] = r i=1 P i(t)λ 2t +K, where K R and each P i (z) (1 i r) is a polynomial of degree m i 1, whose coefficients are arisen from the initial conditions var[x 0 ],, var[x p 1 ], var[x p ], either K. Now, we are going to discuss the general case without setting the independence condition of {X t ; t Z}. To this aim, we release the combinatorial expression of var[x t ] from Proposition 4.1, thereby we manage to have the following result. Proposition 4.4 Under the data of Proposition 4.1, suppose that the A t are mutually independent and X t i is independent of A t for t 0. Then, for every t p, we have p 1 var[x t ] = where α ij = i=0 ( i ( p 1 ) 2 ρ(t j, p) var[xi 2 ] + 2 i=p 1,j=p i<j,(i=1,j=2) α ij cov(x p i, X p j ), ) ( k=0 a j ) p i+k 1ρ(t k, p) h=0 a p j+h 1ρ(t h, p). When the r.r.v. X t (t Z) are mutually independent, we have the following corollary.

10 724 R. Ben Taher, M. Rachidi and M. Samih Corollary 4.5 Under the hypothesis of Proposition 4.4, suppose that {X t ; t Z} are mutually independent. Then, we have p 1 var[x t ] = i=0 ( p 1 ) 2 ρ(t j, p) var[xi 2 ]. We point out that we present, in Corollary 4.5, another expression of var[x t ]. 5 Operator methods and recursiveness In the real case, difference equations of Fibonacci type are also related to operator methods for studying the ARM A(p, q) process. That is, the equation φ(b)x t = θ(b)a t, where B is the auto-regressive operator, shows that X t = ψ(b)a t and A t = Π(B)X t, where ψ(b) = φ(b) 1 θ(b) = 1 + j=1 ψ jb j and Π(B) = θ(b) 1 φ(b) = 1 + j=1 π jb j such that j=1 ψ2 j < + and j=1 π2 j < +. For computing the ψ j, in terms of the a j (0 j p 1) and b j (0 j q 1), we use the equation φ(b)ψ(b) = θ(b). A straightforward computation shows that ψ 1 = a 0 b 0, ψ j = j 1 k=0 a kψ j k+1 + a j 1 b j 1 (1 < j < p 1), ψ p = p 1 k=0 a kψ p k+1 + a p 1 b p 1 ɛ p,q (with ɛ p,q = 1 for p q and ɛ p,q = 0 otherwise), and ψ n+1 = a 0 ψ n + + a p 1 ψ n p+1, for n max(p, q) 1. (15) Similarly, for exhibiting the π j in terms of the a j (0 j p 1) and b i (0 i q 1), a direct computation implies that π 1 = a 0 b 0, π j = j 1 k=0 b kπ j k+1 + a j 1 b j 1 (1 < j < q 1), π q = p 1 k=0 a kψ p k+1 + ɛ p,q a p 1 b p 1 (with ɛ p,q = 1 for p q and ɛ p,q = 0 otherwise), and π n+1 = b 0 π n + + b q 1 π n p+1, for n max(p, q) 1. (16) The characteristic polynomial of (15) and (16) are P 1 (z) = z p a 0 z p 1 a p 1 and P 2 (z) = z q b 0 z q 1 b q 1. It s easy to see that P 1 (z) = z p φ(1/z) and P 2 (z) = z q θ(1/z). Therefore, the stationary and invertible ARMA(p, q) process can be described as follows. Proposition 5.1 Let {X t ; t Z} be an ARMA(p, q) process, with associated white noise is {A t ; t Z}. Then, the following assertions are equivalent: (i) {X t ; t Z} is stationary (respectively invertible), (ii) For every root λ of P 1 (z) (respectively P 2 (z)) we have λ < 1. Moreover, we have lim n + ψ n = 0 and lim n + π n = 0. The last affirmation of Proposition 5.1 is an immediate consequence of the Binet formula of (15) and (16) (see [6], [9], for example). Moreover, if we

11 Recursiveness in Hilbert spaces and application 725 suppose that a j (0 j p 1) and b j (0 j q 1) are nonnegative, we can derive the asymptotic behavior of the two sequences {ψ n } n 0 and {π n } n 0, by using results of [6], [11]. References [1] D. L. Antzoulakos and A. N. Philippou, Multivariate Fibonacci polynomials of order k and the multiparameter negative binomial distribution of the same order, Applications of Fibonacci Numbers, vol. 3, Eds. G.E. Bergum, A.N. Philippou, A.F. Horadam, Kluwer Academic Publishers, Dordrecht, 1990, pp [2] R. Ben Taher, M. Mouline and M. Rachidi, Solving nonhomogeneous recurrence relations of order r by a linearization method and application to polynomial and factorial polynomial cases,fibonacci Quart. 40 (2002), p [3] R. Ben Taher and M. Rachidi, Linear recurrence relations in the algebra of matrices and applications, Linear Algebra and its Applications, Vol. 330 (1-3) (2001), p [4] P. J. Browcwell and R.A. Davis, Time series : Theory and applications, Springer-Verlag (1987 and 1991). [5] A. Charpentier, Cours de séries temporelles. Théorie et applications. EN- SAE - Université Dauphine. Paris (2003). [6] F. Dubeau, W. Motta, M. Rachidi and O. Saeki, On weighted r- generalized Fibonacci sequences, Fibonacci Quart (1997), p [7] B. El Wahbi, M. Mouline and M. Rachidi, Solving nonhomogeneous recurrence relations by matrix methods, Fibonacci Quart. 40 (2002), p [8] A. F. Horadam, Falling Factorial Polynomials of Generalized Fibonacci Type, Applications of Fibonacci Numbers, Eds G.E. Bergum, A.N. Philippou and A.F. Horadam, Dordrecht : Kluwer (1990), p [9] W. G. Kelly and A. C. Peterson, Difference Equations : An introduction with Applications, San Diego - Academic Press (1991).

12 726 R. Ben Taher, M. Rachidi and M. Samih [10] M. Mouline and M. Rachidi, Suites de Fibonacci généralisées et chaines de Markov, Real Academia de Ciencias Exatas, Fisicas y Naturales de Madrid 89,1-2, (1995), p [11] M. Mouline and M. Rachidi, Application of Markov chaines properties to r-generalized Fibonacci sequences, Fibonacci Quart. 37-1, (1999), p [12] A. M. Ostrowski, Solutions of equations in Euclidean and Banach spaces. New York and London : Academic Press, [13] A. N. Philippou, A note on the Fibonacci sequence of order k and the multinomial coefficients, Fibonacci Quart. 21, (1983), p [14] A. N. Philippou and A. A. Muwafi, Waiting for the k-th consecutive successs and the Fibonacci sequences of order k, Fibonacci Quart. 20 (1982), p Received: June 5, 2014; Published: December 1, 2014