A LINEAR FILTER FOR THE OPERATORIAL PREDICTION OF A PERIODICALLY CORRELATED PROCESS

PREDICTION OF A PERIODICALLY CORRELATED PROCESS

ILIE VALUS¸ESCU

For a periodically Γ-correlated process{fn}in anL(E)-right moduleH, a linear operatorial Wiener filter for prediction is obtained in terms of the coefficients of the maximal outer function of an attached stationary process. Also, an evaluation of the prediction error, which in this case is a periodic function, is given in terms of the same coefficients.

AMS 2000 Subject Classification: 47A20, 47N30, 60G25.

Key words: complete correlated actions, operator model, maximal outer function, linear predictor.

1. PRELIMINARIES

Let E be a separable Hilbert space, L(E) the C^∗-algebra of all linear bounded operators onE and Ha rightL(E)-module. Anaction ofL(E) onH is a map from L(E)× Hinto H given by (A, h)→ hA. For convenience, the action of A on h will be written as Ah, but will be understood as hA, in the sense of the right L(E)-module.

A map Γ fromH × H intoL(E) given by

(1.1) (f, g)→Γ[f, g]

such that

(i) Γ[h, h]≥0, Γ[h, h] = 0 impliesh= 0;

(ii) Γ[g, h] = Γ[h, g]^∗;

(iii) Γ[ΣiAihi,ΣjBjgj] = Σi,jA^∗_iΓ[hi, gj]Bj,

for finite sums of actions of L(E) onH, is called a correlation of the action of L(E) on H.

A triple{E,H, Γ}whereΓ has the above properties was called in [5] the correlated action of L(E) on H. For prediction purposes, the Hilbert spaceE will be called the parameter space, and the right L(E)-module H will be the state space. As was seen in [5], to each correlated action {E,H, Γ}, by the

REV. ROUMAINE MATH. PURES APPL.,54(2009),1, 53–67

positively definite Aronsjain reproducing kernel (1.2) hγ_λ1, γ_λ2i= Γ[h2, h1]a1, a2

E, whereλi = (ai, hi)∈ E × H, we can attach a Hilbert space K – the measuring space.

It is easy to see thatL(E,K) is a rightL(E)-module and Γ :L(E,K)× L(E,K)→ L(E)

given by Γ[V₁, V₂] = V₁^∗V₂, is a correlation of the action AV := V A of L(E) on L(E,K). This correlated action {E,L(E,K), Γ} was called the operator model. In [6] it was proved that an abstract correlated action {E,H, Γ} can be imbedded into the operator model by considering for each h ∈ H the operatorV_h fromL(E,K) given by

(1.3) V_ha=γ_(a,h).

The imbedding h→Vh is unique up to a unitary equivalence.

The correlated action{E,H, Γ}is said to be acomplete correlated action, if the imbedding h→Vh ofH intoL(E,K) is surjective.

A Γ-stationary process in H is a sequence {f_t}_t∈G in H, where G is a locally compact group or hypergroup, such that Γ[f_t, f_s] only depends on the differences−t, not separately onsandt. The function Γ :G→ L(E) given by

(1.4) Γ(t) = Γ[f₀, f_t]

is called thecorrelation function of the process {f_t}_t∈G.

Operator valued Γ-stationary processes in complete correlated actions, where G isZorR, were studied in, e.g., [4]–[7], and prediction problems were solved. An appropriate Wiener filter for prediction and an estimation for the prediction error operator, have been obtained.

The previous results can be extended to the matrix valued case with the aim to apply them in the study of periodically Γ-correlated processes.

LetT ≥2 be a positive integer and

(1.5) H^T =H × H × · · · × H

the Cartesian product ofT copies of the rightL(E)-moduleH. An elementX of H^T will be seen as a line vector (h₁, . . . , h_T). On H^T it is possible to have the action of L(E) on the components with the same operatorA ∈ L(E), or on each component with differentAi ∈ L(E). Also, we can consider the action of L(E)^T×T onH^T, taking

(1.6) A(h1, . . . , hT) := (h1, . . . , hT)A

for each matrixA= (Aij)^T_i,j=1fromL(E)^T×T, in the sense of the right module.

It is easy to see thatH^T is anL(E)^T×T-right module and the action ofL(E) on H^T is a particular case of the action ofL(E)^T×T onH^T for the case of diagonal matrices with the same operator, or different operators on the diagonal.

Having the action of L(E)^T×T on H^T, various correlations of this ac- tion can be constructed. For our goal we are interested in two operatorial correlations on H^T, namely,

(1.7) Γ₁[X, Y] =

T−1

X

k=0

Γ[x_k, y_k], where X= (x0, x1, . . . xT−1),Y = (y0, y1, . . . , yT−1) , and (1.8) Γ_T[X, Y] = Γ[xi, yj]

i,j∈{0,1,...,T−1}.

Using (1.6), it is easy to verify properties (i)–(iii) of a correlation of the action of L(E), respectively ofL(E)^T×T, for Γ₁ and Γ_T.

So, starting with the correlated action {E,H,Γ} of L(E) on H, we ob- tain the correlated actions {E,H^T,Γ1} and {E,H^T,ΓT} of L(E), respectively L(E)^T×T, on H^T. Remark that the correlation Γ1 is the trace of the matrix given by the correlation Γ_T.

In [11], a unique (up to a unitary equivalence) imbedding of H^T into L(E,K)^T was considered defining the operator W_X = (Vx1, . . . , Vx_T) forX = (x₁, . . . , x_n)∈ H^T. So, the correlated actions{E,H^T,Γ₁}and{E,H^T,Γ_T}can be seen in appropriate operator models as{E,L(E,K)^T,Γ₁}and{E,L(E,K)^T, ΓT}, where forWi= (V_i¹, . . . , V_i^T) fromL(E,K)^T we have

(1.9) Γ1[W1, W2] =

T

X

k=1

Γ[V₁^k, V₂^k] =

T

X

k=1

(V₁^k)^∗V₂^k, respectively,

(1.10) Γ_T[W₁, W₂] = Γ[V₁^j, V₂^k]T

j,k=1= (V₁^j)^∗V₂^kT j,k=1.

AnotherL(E)^T×T-right module which will be considered in the study of periodically correlated processes will be (H^T)^T with an appropriate correlation of the action of L(E)^T×T.

It is known [6] that for each stationary process there exists an operator valued semispectral measureF on the torusTsuch that its correlation function has the integral form

(1.11) Γ(n) =

Z 2π

0

e^−intdF(t).

Also [6] to each semispectral measure F it is attached a maximalL²-bounded outer function Θ(λ) such thatFΘ≤F.

Under a Harnack type domination of the semispectral measure of a sta- tionary process, an operatorial linear Wiener filter was obtained [7] in terms

of its maximal outer function coefficients. Namely, if there exists a positive constantc such that

(1.12) c·dt≤F ≤c⁻¹·dt,

then the maximal function of the process is a bounded invertible function {E,E,Θ(λ)} with a bounded inverse Ω(λ). As a consequence, the predictiable part ˆfn+1 of fn+1 can be obtained as

(1.13) fˆn+1 =

∞

X

j=0

Ajfn−j,

where the Wiener filter{A_j}for prediction is given by the operator coefficients of Θ(λ) and its inverse Ω(λ) as

(1.14) A_j =

j

X

p=0

Θ_p+1Ωj−p.

2. PERIODICALLY Γ-CORRELATED PROCESSES

Aperiodically Γ-correlated process is a sequence {f_t}_t∈G inHwith the property that there exists a positive T such that

(2.1) Γ[ft+T, fs+T] = Γ[ft, fs]

for any t, s∈G. The smallest T which verifies (2.1) is called theperiod of the periodically Γ-correlated process {f_t}_t∈G.

In this paper the discrete case G =Z is only investigated. Here T is a positive integer. For T = 1 the process is stationary Γ-correlated.

Let us remark that thecorrelation function of a periodically Γ-correlated process {f_t}_t∈Z with period T ≥2 is the function Γ :Z×Z→ L(E) given by (2.2) Γ(t, s) = Γ[ft, fs].

This is a periodic function with the same period T.

For a periodically Γ-correlated process{f_t}we also consider the covari- ance function defined as

(2.3) B(n, t) = Γ(n+t, n).

Conversely, knowing the covariance function of a periodically Γ-correlated process, the corresponding correlation function is obviously given

(2.4) Γ(s, n) =B(n, s−n).

TheL(E)-valued functionB(n, t) is a periodic function of the first argu- mentn, with the same periodT, and has a Fourier representation of the form

(2.5) B(n, t) =

T−1

X

k=0

B_k(t) exp(2πikn/T),

where B_k(t) are L(E)-valued coefficients fork= 0,1, . . . , T −1. For conveni- ence, B_k(t) can be extended toZon account of the equationB_k(t) =B_k+T(t).

A simple computation on (2.5) shows that

(2.6) Bl(t) = 1

T

T−1

X

n=0

B(n, t) exp(−2πinl/T).

The correlation function and the covariance function are positively defi- nite functions. In [11], a generalization of Gladyshev’s theorem [1] to the complete correlated actions case was proved. It can be stated as follows

Theorem 1 (Gladyshev). A functionB(n, t) defined by (2.5) is the co- variance function of some periodically Γ-correlated process in H with period T, if and only if the T ×T matrix valued function

(2.7) B(t) = B_jk(t)T−1

j,k=0

is the operator correlation function of some stationary Γ_T-correlated process from H^T, where

(2.8) B_jk(t) =Bk−j(t) exp(2πijt/T) and the Γ_T-correlation on H^T is given by (1.8).

As in the stationary case, prediction spaces can be attached to a perio- dically Γ-correlated process. So, thepast and present of{f_t}is the submodule of H given by

(2.9) H^f_n=

X

k

A_kf_k; A_k∈ L(E); k≤n

,

the remote past is

(2.10) H^f_−∞=\

n

H_n^f

while the space generated by the process is

(2.11) H^f_∞=

X

k

A_kf_k; A_k∈ L(E)

.

By means of the imbedding h → Vh of H into L(E,K), the past and present at moment tbecomes a subspace in the measuring spaceK, namely,

(2.12) K^f_t = _

k≤t

V_fkE,

or

(2.13) K^f_t = _

h∈H^f_t

V_hE,

the remote past

(2.14) K_−∞^f =\

n

K^f_n,

and the space generated by the process

(2.15) K^f_∞=

∞

_

−∞

V_ftE.

Similar submodules inH^T can be attached to a process{X_n} ⊂ H^T. We namely have

(2.16) H_n^X =

X

k

A_kX_k; A_k∈ L(E)^T×T, k≤n

,

(2.17) H_−∞^X =\

n

H_n^X

and

(2.18) H_∞^X =

X

k

A_kX_k; A_k ∈ L(E)^T×T

.

Using the imbedding X → W_X of H^T intoL(E,K)^T, the corresponding sub- spaces from K^T will be

(2.19) K_n^X = _

k≤n

W_XkE,

(2.20) K_−∞^X =\

n

K_n^X

and

(2.21) K_∞^X =

∞

_

−∞

WXnE.

3. A LINEAR WIENER FILTER FOR PREDICTION As we have seen, imbedding the right L(E)-module H (the state space) into L(E,K), where K is the measuring space, the study of some processes from H can be done in the operator model L(E,K), which is a right L(E)- module. Also L(E,K)^T is a right L(E)^T×T-module or, in particular, a right L(E)-module. We have considered Γ_T as a correlation of the action ofL(E)^T×T on L(E,K)^T, and Γ₁ as a correlation of the action of L(E) on L(E,K)^T. The construction can be continued to

L(E,K)^TT

, which becomes a right L(E)^T×T-module. In what follows the elements from

L(E,K)^TT

will be writ- ten in capital boldface characters, to avoid the confusion with the elements from L(E,K)^T which are only designed by capital letters.

An elementZ= (W1, . . . , WT) is a line vector with W_k∈ L(E,K)^T, and W_k= (V_k¹, V_k², . . . , V_k^T) withV_k^j ∈ L(E,K).

A correlation of the action ofL(E)^T×T on

L(E,K)^TT

is defined as (3.1) Γ_T[Z1,Z2] = Γ1[W1j, W_2k]T

j,k=1,

whereW1j = (V_1j¹, . . . , V_1j^T) andW_2k= (V_2k¹, . . . , V_2k^T) are elements ofL(E,K)^T. Let{f_n}be an arbitrary process inHand Ethe operator of multiplying by e^−2πi/T. On account of the construction of the measuring space K and the action of L(E) on H(respectively L(E,K)) as rightL(E)-module, to each {f_n}_n∈Z fromH we can attachT sequences inH^T of the form

(3.2) X_n^k= E^knf_n, E^k(n+1)f_n+1, . . . , E^k(n+T−1)f_n+T−1 , where k∈ {0,1, . . . , T−1}.

Using the imbeddingsh →V_h and X →W_X of H intoL(E,K), respec- tivelyH^T intoL(E,K)^T, we obtainT sequences in L(E,K)^T by taking

(3.3) Z_n^k=W_Xk

n, k∈ {0,1, . . . , T −1}, and a sequence in

L(E,K)^TT

, by taking

(3.4) Zn= 1

√

T Z_n⁰, Z_n¹, . . . , Z_n^T−1 .

Using Gladyshev’s theorem, in what follows a linear predictor for periodi- cally Γ-correlated processes in complete correlated actions will be obtained, thus generalizing the scalar case [3]. To do this, we will first see that the process attached by (3.4) to a periodically Γ-correlated process {f_n} from H is an explicit form of an attached stationary process.

Theorem 2. A process {f_n} from H is periodically Γ-correlated with period T if and only if {Z_n}_n∈Z from

L(E,K)^TT

attached by (3.4) is a sta- tionary Γ_T-correlated process.

Proof. If {f_n} from H is periodically Γ-correlated with period T, then for each entry of the matrix

Γ_T[Z_m,Z_n] =1

TΓ₁[Z_m^j , Z_n^k]T−1 j,k=0

we have

Γ1[Z_m^j , Z_n^k] = Γ1[W_X^j

m, W_Xk n] =

T−1

X

p=0

Γ[E^j(m+p)V_fm+p, E^k(n+p)V_fn+p] =

=

T−1

X

p=0

V_f^∗_m+pV_fn+pE−j(m+p)+k(n+p)=

=V_f^∗_mV_fnE^−jm+kn+

T−1

X

p=1

V_f^∗_m+pV_fn+pE−j(m+p)+k(n+p)=

= Γ[V_fm, V_fn]E^−jm+kn+

T−1

X

p=1

Γ[V_fm+p, V_fn+p]E−j(m+p)+k(n+p) =

= Γ[V_fm+T, V_fn+T]E^{−j(m+T)+k(n+T)}+

T−1

X

p=1

Γ[V_fm+p, V_fn+p]E−j(m+p)+k(n+p)=

=

T

X

p=1

Γ[V_fm+p, V_fn+p]E−j(m+p)+k(n+p)=

=

T−1

X

s=0

Γ[V_fm+s+1, V_fn+s+1]E−j(m+s+1)+k(n+s+1)=

=

T−1

X

s=0

Γ[E^j(m+1+s)V_fm+1+s, E^k(n+1+s)V_fn+1+s] =

= Γ₁[W_Xj

m+1, W_Xk

n+1] = Γ₁[Z_m+1^j , Z_n+1^k ].

This implies that Γ_T[Z_m,Z_n] = Γ_T[Z_m+1,Z_n+1], i.e., the process {Z_n}_n∈Z is stationary Γ_T-correlated in

L(E,K)^TT

.

Conversely, if {Z_n}_n∈Z is stationary ΓT-correlated, then each entry of the Γ_T-correlation matrix verifies the relation

1

TΓ₁[Z_m^j , Z_n^k] = 1

TΓ₁[Z_m+1^j , Z_n+1^k ].

For j=k= 0 we have

Γ1[W_Xm⁰ , W_Xn⁰] = Γ1[W_X⁰

m+1, W_X⁰

n+1],

T−1

X

p=0

Γ[V_fm+p, V_fn+p] =

T−1

X

s=0

Γ[V_fm+1+s, V_fn+1+s], hence, by reducing the similar terms,

Γ[V_fm, V_fn] = Γ[V_fm+T, V_fn+T],

i.e., the process {f_n}n∈Z is periodically Γ-correlated with periodT. We will see that for a periodically Γ-correlated process{f_n}_n∈Z, the sta- tionary Γ_T-correlated process{Z_n}_n∈Zdefined by (3.4) verifies the conditions of Gladyshev’s theorem. Indeed, by (2.6), the coefficients of the ΓT-correlation matrix function (2.7) have the form

B_jk(t) = 1

TΓ₁[Z_t^j, Z₀^k] = 1

TΓ₁[W_X^j

t, W_Xk 0] =

= 1 T

T−1

X

p=0

Γ[Vft+p, Vfp]E^−j(t+p)+kp = 1 T

T−1

X

p=0

Γ(t+p, p)E^−j(t+p)+kp =

=E^−jt1 T

T−1

X

p=0

B(p, t)E^p(k−j)=E^−jtBk−j(t).

So, B_jk(t) =Bk−j(t) exp(2πijt/T) and condition (2.8) is verified.

According to the definition of thepast and present given in the previous section, the past and present of the process{Z_n}_n∈Zfrom

L(E,K)^TT

will be a subspace from (K^T)^T given by

(3.5) K_n^Z= _

k≤n

ZkE, where the elements are of the form

(3.6) Z=X

k≤n

A_kZ_ka_k,

while the action of A_k ∈ L(E)^T×T is understood in the sense of the right L(E)^T×T-module

L(E,K)^TT

.

Also, for the process{Z_n}from

L(E,K)^TT

another “past and present” denoted by K^Z_n can be considered in K^T, namely, the linear span of the finite sums of the form

(3.7) X

k≤n

A_kZ_k^ja_k, 0≤j≤T −1,

or

(3.8) K^Z_n = _

k≤n

Z_k^jE.

Due to the particular form of the stationary ΓT-correlated process at- tached to a periodically Γ-correlated process, the geometry of the past and present spaces can be described as follows.

Theorem 3. The past and present of {Z_n}_n∈Z has the structure

(3.9) K_n^Z = K_n^ZT

and

(3.10) K_n^Z=K^f_n× K^f_n+1× · · · × K_n+T^f ₋₁.

Proof. For each finite linear combination fromK_n^Z, in (K^T)^T we have X

k≤n

A_kZ_ka_k = 1

√T X

k≤n

A_k(Z_k⁰, Z_k¹, . . . , Z_k^T−1)a_k =

= 1

√T X

k≤n

A^ij_kT−1

i,j=0(Z_k⁰, Z_k¹, . . . , Z_k^T−1)a_k=

= 1

√ T

X

k≤n

T−1

X

j=0

Z_k^jA^j0_k ak,

T−1

X

j=0

Z_k^jA^j1_k ak, . . . ,

T−1

X

j=0

Z_k^jA^j,T−1_k ak

.

It follows that each component of the linear combination considered is of the form P

k≤n T−1

P

j=0

A^ji_kZ_k^jak, i.e., it belongs toK^Z_n. Therefore, K_n^Z⊂(K^Z_n)^T. Conversely, each linear combination from (K^Z_n)^T has the formZ= (Z⁰, Z¹, . . . , Z^T−1), where Zⁱ = P

k≤n T−1

P

j=0

A^ji_kZ_k^ja_k. It follows that Z = P

k≤n

A_kZ_ka_k ∈ K_n^Z. Therefore, (K^Z_n)^T ⊂K_n^Z and (3.9) is proved.

To prove (3.10), let us note that from the definition (3.3) of a component Z_m^k, wherem≤n, we have

Z_m^ka=W_Xk

ma= (E^kmV_fma, E^k(m+1)V_fm+1a, . . . , E^k(m+T−1)V_fm+T−1a) as elements from K^f_n× K^f_n+1× · · · × K^f_n+T−1. Therefore,

K^Z_n ⊂ K^f_n× K^f_n+1× · · · × K_n+T^f ₋₁ ⊂ K^T.

Conversely, a linear combination of the form Z =

T−1

P

j=0

E^−j(m+k)Zm^j a, where Zm^j are the components of Z_m has the form

Z=

T−1

X

j=0

W_X^j

mE^−j(m+k)a=

T−1

X

j=0

Vf_m+lEj(m+l−j(m+k)

aT−1 l=0 =

=

T−1

X

j=0

V_fm+lE^j(l−k)aT−1 l=0 =

V_{f m+l}a

T−1

X

j=0

E^j(l−k) T−1

l=0

= V_{f m+l}a·T δ_lkT−1 l=0

for each m≤nand 0≤k≤T−1. Therefore for each 0≤k≤T−1 we have {0} × · · · × {0} × K^f_n+k× {0} × · · · × {0} ⊂ K^Z_n,

consequently

K^f_n× K^f_n+1× · · · × K_n+T^f ₋₁ ⊂ K_n^Z. This completes the proof.

In what follows, as in [6] we suppose that the L(E)^T×T-valued semi- spectral measure attached to the ΓT-correlation function of the process {Z_n} satisfies a Harnack type boundedness condition (1.12). Then the invertible maximal matrix function

(3.11) Θ(λ) = Θ_ij(λ)T−1

i,j=0

has a bounded inverse

(3.12) Ω(λ) = Ωij(λ)T−1

i,j=0

and the predictable part of Zn+1 can be obtained as

(3.13) Zˆn+1 =

∞

X

k=0

A_kZn−k,

where the Wiener filter

(3.14) A_k= A^ij_kT−1

i,j=0

for prediction is given in terms of the coefficients of its maximal function in a similar way as in (1.14).

Theorem 4. If {f_n}_n∈Z is a periodically Γ-correlated process and the predictable part of the attached stationary Γ_T-correlated process {Z_n}_n∈Z is given by (3.13) and (3.14), then the predictable part of {f_n}n∈Z is

(3.15) fˆn+1=

∞

X

k=0

C_kfn−k,

where C_k =

T−1

P

j=0

√1

TA^j0_k E^j(n−k).

Proof. Let us consider the predictable part Zˆ_n+1 = ( ˆZ_n+1⁰ ,Zˆ_n+1¹ , . . . ,Zˆ_n+1^T−1) =P_KZ

nZ_n+1=P_KZ

n(Z_n+1⁰ , Z_n+1¹ , . . . , Z_n+1^T−1).

Considering the zero component of ˆZ_n+1 and using Theorem 3, we get Zˆ_n+1⁰ =P_K^Z

nZ_n+1⁰ =P_K^Z

n(Vfn+1, Vfn+2, . . . , Vfn+T) =

= (P_Kf

nV_fn+1, . . . , P_Kf

n+T−1V_fn+T).

On the other hand, using (3.13), we have Zˆn+1 =

∞

X

k=0

AkZn−k =

∞

X

k=0

A^ij_k 1

√

T(Z_n−k⁰ , Z_n−k¹ , . . . , Z_n−k^T−1) =

= 1

√ T

∞

X

k=0

T−1

X

j=0

Z_n−k^j A^j0_k ,

T−1

X

j=0

Z_n−k^j A^j1_k , . . . ,

T−1

X

j=0

Z_n−k^j A^j,T_k ⁻¹

=

= 1

√ T

∞

X

k=0 T−1

X

j=0

Z_n−k^j A^j0_k , . . . , 1

√ T

∞

X

k=0 T−1

X

j=0

Z_n−k^j A^j,T_k ⁻¹

=

= Zˆ_n+1⁰ ,Zˆ_n+1¹ , . . . ,Zˆ_n+1^T−1 . It follows that

Zˆ_n+1⁰ = 1

√ T

∞

X

k=0 T−1

X

j=0

Z_n−k^j A^j0_k = 1

√ T

∞

X

k=0 T−1

X

j=0

A^j0_k Z_n−k^j =

=

∞

X

k=0 T−1

X

j=0

√1

TA^j0_k W_X^j

n−k =

∞

X

k=0 T−1

X

j=0

√1

TA^j0_k E^j(n−k+s)Vfn−k+s

T−1 s=0 =

= ^∞

X

k=0 T−1

X

j=0

√1

TA^j0_k E^j(n−k+s)V_fn−k+s T−1

s=0

.

Therefore, VP

Hf

nfn+1 =P_Kf

nV_fn+1 =

∞

X

k=0 T−1

X

j=0

√1

TA^j0_k E^j(n−k)V_fn−k, where P_Hf

n is the “Γ-orthogonal projection” on the submoduleH^f_n of H.

So, the corresponding operatorial Wiener filter for the prediction of a periodically Γ-correlated process from His given by (3.15).

Remark, that in the periodic case the prediction error (3.16) ∆(n) = Γ[fn+1−fˆn+1, fn+1−fˆn+1]

is a periodic function, not an operator as in the stationary case. Therefore, we have

(3.17) ∆(n) =

T−1

X

k=0

∆_kexp(2πijk/T) and, conversely, the coefficients ∆_k can be obtained as

(3.18) ∆k= 1

T

T−1

X

j=0

∆(j) exp(−2πijk/T).

The next theorem gives a characterization of the prediction error for a periodically Γ-correlated process {f_n} in terms of the coefficients of the maximal function of the attached ΓT-correlated process {Z_n}.

Theorem 5. The prediction error ∆(n) of a periodically Γ-correlated process {f_n} has the form

(3.19) ∆(n) =

T−1

X

k=0

DkE^−k(n+1),

where the operator coefficients Dk ∈ L(E) are the entries of the zero line of the prediction error matrix of the attached stationary process {Z_n}, namely,

(3.20) D_k=

T−1

X

s=0

Θ^∗_s0Θ_sk,

where Θij = Θij(0) from the maximal function (3.11) of the process {Z_n}.

Proof.Let ∆ be the prediction error matrix of the stationary Γ_T-correlated process {Z_n}attached to {f_n}by (3.4). Then, for each n∈Z,

∆ = Γ_T[Z_n+1−Zˆ_n+1,Z_n+1−Zˆ_n+1] = ∆_ijT−1 i,j=0, where the operators ∆_ij are given by

∆ij = 1

TΓ1[Z_n+1ⁱ −Zˆ_n+1ⁱ , Z_n+1^j −Zˆ_n+1^j ].

It is known [6] that if Θ(λ) is the maximal function of a stationary process, then the prediction error is given by

(3.21) ∆ = Θ^∗(0)Θ(0).

From (3.11), putting Θij = Θij(0), we get

(3.22) ∆ij =

T−1

X

s=0

Θ^∗_siΘsj. On the other hand, from (3.18) we have

∆_k= 1 T

T−1

X

j=0

∆(j)E^kj = 1 T

T−1

X

j=0

Γ[fj+1−fˆj+1, fj+1−fˆj+1]E^kj =

= 1 T

T−1

X

j=0

V^∗

fj+1−fˆj+1V_f

j+1−fˆj+1E^kj = 1 TE^−k

T−1

X

j=0

V^∗

fj+1−fˆj+1V_f

j+1−fˆj+1E^k(j+1)=

= 1 TE^−k

T−1

X

j=0

Γ[V_f

j+1−fˆj+1, E^k(j+1)V_f

j+1−fˆj+1] =

=E^−k1

TΓ1[Z₁⁰−Zˆ₁⁰, Z₁^k−Zˆ₁^k] =E^−k∆_0k=E^−k

T−1

X

s=0

Θ^∗_s0Θ_sk. It follows from (3.17) that

∆(n) =

T−1

X

k=0

E^−k

T−1

X

s=0

Θ^∗_s0Θ_skE^−kn=

T−1

X

k=0

D_kE^−k(n+1),

where D_k =

T−1

P

s=0

Θ^∗_s0Θ_sk, and the proof is complete.

Acknowledgements.This work was partly supported by Grant ANCS 2-CEx06-11- 34/2006.

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Received 2 May 2008 Romanian Academy

“Simion Stoilow” Institute of Mathematics P.O. Box 1-764

010702 Bucharest, Romania Ilie.Valusescu@imar.ro