© Hindawi Publishing Corp. ON A CERTAIN CLASS OF NONSTATIONARY SEQUENCES IN HILBERT SPACE

We study the functions of correlation K ( n , m ) = 〈 X ( n ) , X ( m ) 〉 of certain sequences: X ( n ) = T n x 0 , x 0 ∈ H where T is a contraction in Hilbert space H . By using the spectral methods of the nonunitary operators we give the 
general form of K ( n , m ) and its asymptotic behaviour lim p → + ∞ K ( n + p , m + p ) .


Introduction. Let X(n) (n ∈ A = IN
or Z) be a sequence of elements of a separable Hilbert space H.The function of correlation of X(n) is given by formula K(n, m) = X(n), X(m) . (1.1) If the function of correlation depends only on the difference of arguments, that is, K(n, m) = K(n−m), one calls that X(n) is stationary.Kolmogorov (see [4]) showed that if X(n) is stationary and A = Z, then where U is a unitary operator acting in the subspace H X which is defined as the closed linear envelope of X = {X(n); n ∈ Z}.This representation as well as the spectral theory of the monoparametric groups of unitary operators allowed to find the general form of the function K(n, m) in the stationary case.More exactly, one has (see [4]) e i(n−m)λ dF X (λ), (1.3) where F X is real function, continuous on the left and nondecreasing on [−π ; +π] such that F X (−π) = 0.This function is called spectral function of X(n).
In this paper, we are interested in some sequences of the form where T is a linear contraction ( T ≤ 1) in H.Such sequences are called linearly representable and were introduced by Yansevitch [8,9].They represent a natural generalization of the sequences of the form (1.2).But they were especially introduced like the analogue of the processes of the form Y (t) = e itA y 0 , (1.5) where A is a dissipative ((A−A * )/i ≥ 0) operator in H.The correlation theory of these processes constituted a remarkable field of application for the spectral theory of nonselfadjoint operators [2,3,5,10].
Necessary and sufficient criteria in terms of function of correlation for linear representability (1.4) are established by the following theorem [8].
for every (λ n ) N n=0 and (µ m ) M m=0 in the field of complex numbers.
Definition 1.2.Let X(n) = T n x 0 be a linearly representable sequence.The difference of correlation of X(n) is the function It is clear that in the stationary case, W (n,m) = 0.
Formula (1.7) implies that, for every natural p ≥ 1, what gives, for p → +∞, Consequently, the study of linearly representable sequences can be carried out in two stages.
(a) To find the limit lim p→+∞ K(n + p, m + p).(b) To give the explicit expression of the quantity Φ k (n).In [8], the case when dim(I − T * T )H = 1 was considered and the spectrum of T is made up only of eigenvalues In this case, one has [8] where Γ is a closed contour containing all the spectrum of T .
Let T be a simple contraction (i.e., there is no invariant for T and T * subspace in which, T induces a unitary operator) with spectrum σ (T ) on the circle unit.It is known [1] that there exists an increasing function α on the interval [0,l] (l 0) such that (1.13) Definition 1.3.Say that X(n) = T n x 0 belongs to the class D (r ) [α] if T is a contraction such that (1.13) holds and dim(I − T * T )H ≤ r .
Throughout this paper, we will suppose that α is a continuous function.In this case, we will prove the following results.
Theorem 1.4.Let T n x 0 be an element of class D (r )  [α].Assume that T is simple.Then, where Γ is any closed contour containing all the spectrum of T , F(n − m) is a Hermitian nonnegative function which equals zero in the case when dim(I − T * T )H = 1, and K ∞ (n − m) is defined by the spectrum of T .Moreover, if T has a singular spectrum or the measurement of the intersection of its spectrum with the unit circle is null, then (1.14), then there exists a linearly representable sequence X(n) = T n x 0 such that X(n) ∈ D (r ) [α] and the function of correlation of X(n) equals K(n, m).
Throughout this paper, H is a separable Hilbert space and ⊕ denotes orthogonal sum.

On the structure of lim
Moreover, if T is invertible, then Proof.The existence and positivity of R are a consequence of the fact that the sequence A n is a decreasing and bounded sequence of positive operators.Formulas (2.1) and (2.2) are verified easily. (2.3) Consider now the sequence (2.5) A direct calculation shows that σ (T ) = {e −iα(x) : x ∈ [0,l]} and Hence, the sequence Ψ (x, n) is an element of the class D (1) [α].
(2.16) Hence, (2.17) Let By using the theorem of Lebesgue about dominated convergence, one can show that I 2 = I 3 = 0. Thus, is an absolutely continuous function in x.Moreover, since operator T is a contraction, then (2.23) It is known [6] that if the X(n) = T n x 0 ∈ D (1) [α] and T is simple, then T = U −1 T U where U is a unitary operator from H into L 2 [0;l] .Hence, from Theorem 2.4, the following theorem follows.
Theorem 2.5.Let X(n) = T n x 0 ∈ D (1) [α].Suppose that T is simple.Then, there exists an increasing function β on [0,l] such that (2.24) [0;l] (r times), with scalar product: ..,g r . (2.25) In this space, define the operator T (r ) = T ⊕•••⊕ T as follows: Every sequence of the form (2.27) is an element of class D (r ) [α] (see [1]).The following relations hold immediately: (2.28) Theorem 2.6.Let X(n) = T n x 0 ∈ D (r )  [α].Suppose that T is simple.Then, there exists r increasing functions {β j } r j=1 on [0,l] and there exists a Hermitian nonnegative function F(n− m) such that (2.29) Proof.According to [1], there exists a unitary operator B defined in a Hilbert space M such that operator T is unitarily equivalent to the restriction of operator B(r where U is a unitary operator from H into Θ.Thus, (2.31) satisfies all conditions of Theorem 2.6.Finally, one has (2.32) To complete the demonstration, it is enough to notice that (2.33) We now will see two situations where If T is simple and the measurement of the intersection of its spectrum with the circle unit is null, then Proof.One has Under these assumptions, one has according to [7] Theorem 2.8.Let T n x 0 be an element of class D (r ) [α] and let Proof.Under these assumptions, operator T is unitarily equivalent to operator T (r ) (see [1]).Thus, K ∞ (n − m) = K ∞ (n − m).But in this case, the characteristical function S(λ) of operator T (r ) satisfies the following relations (see [1,7]): where is a singular function.Thus det S(λ) is an interior function and according to [1], for every Ψ = ( Ψ 1 ,..., Ψ r ) ∈ L r 2 , lim p→+∞ T n+p x 0 2 = 0.By using the same reasoning that in Proposition 2.7, one shows that

General form of K(n, m)
Theorem 3.1.Let T n x 0 be an element of class D (r )  [α].Assume that T is simple.Then, where Proof.Using the same reasoning that in Theorem 2.6, one can affirm that where F(n− m) satisfies the conditions of Theorem 3.1.According to (1.11), One has (see [1]) where e k (k = 1,...,r ) is the canonical basic in C r .Thus, Since T is bounded, then where Γ is a closed contour containing all the spectrum of T .Consequently, A direct calculation shows that the form Φ k (n) is as in (3.1).
Theorem 3.1 admits the following reciprocal.
Theorem 3.2.If a function K(n, m) admits the representation (3.1), then there exists a linearly representable sequence X(n) = T n x 0 such that X(n) ∈ D (r ) [α] and the function of correlation of X(n) equals K(n, m).
Proof.Since F(n − m) is a Hermitian nonnegative function, there exists (see [4]) a unitary operator S defined in a Hilbert space M such that By the functions α and { Ψ 0k } r k=1 appearing in representation (3.1), construct, in the space L r 2 , the sequence where operator T is defined in L 2 [0;l] by formula (2.5).Let H denotes the Hilbert space L 2 r ⊕ M with scalar product: In this space, define the operator T = T (r ) ⊕ S by T (g + y M ) = T (r )(g) + S(y M ).Operator T is a contraction and dim(I − T * T )H = r .Thus, the sequence X r ) [α] whose function of correlation equals the given function K(n, m).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning .22)That means that the sequence V p (p ≥ 1) of total variation of L 0 (p, x) on [0,l] is bounded.Moreover, function e iτα(x) is continuous.Thus, x) dK 0 (x).

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation