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

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 ARMA(p, q) process. For purpose of illustration, some examples are explored. Mathematics Subject Classification: Primary: 37H10, 40A99; Secondary: 37M10, 62M10, 65Q05

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 ARM A(p, q) process of Fibonacci type defined by C t+1 = A t+1 = b 0 A t + b 1 A t−1 + · · · + 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 remarks on the operator methods and recursiveness are discussed in the last section.
Throughout of this paper H will denote a real Hilbert space.

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].
In view of (3), we infer that for every s (0 ≤ s ≤ p − 1) the sequence {v (s) n+1 } n≥0 can be formulated as follows. For 0 ≤ n ≤ p − 1 we have v (s) n = 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 k=p A n−k D k , for every n ≥ p. Hence, similarly to the real case (see [2], [7]), we establish that the solution of Equation (2) is, The sequence {T <h> n } n≥0 defined by T <h> n−p+1 T p−s−1 is a solution of the homogeneous part of Equation (2). In addition, a particular solution 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, where {T <p> n } n≥0 is the fundamental particular solution of (2) and {T <h> n } 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 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, 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) = Remark 2.4 For E = R, it is well known that the solution of (2) can be written under the form T n = T <h> n + T <p> n , where {T <h> n } n≥0 and {T <p> n } 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 <p> n = T n − T <h> n , for n ≥ 0.
3 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., 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 ARM A(p, q) process and {A t ∈ H; t ∈ Z} its associated white noise. Then, for every t ≥ p , we have 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 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.

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 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, where c j = k+s=j;k≥1,s≥0 b k−1 a 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 ARM A(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 j=0 a j z p−j−1 and 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 j=0 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

More on some characteristics of an ARM A(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 ARM A(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 The Fibonacci sequence's properties, specially the Binet Formula, permits us to get.
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[ 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 When the r.r.v. X t (t ∈ Z) are mutually independent, we have the following corollary.
We point out that we present, in Corollary 4.5, another expression of var[X t ].