PERIODICITY IN DISTRIBUTION. I. DISCRETE SYSTEMS

We consider the existence of periodic in distribution solutions to the difference equations in a Banach space. A random process is called periodic in distribution if all its finitedimensional distributions are periodic with respect to shift of time with one period. Only averaged characteristics of a periodic process are periodic functions. The notion of the periodic in distribution process gave adequate description for many dynamic stochastic models in applications, in which dynamics of a system is obviously nonstationary. For example, the processes describing seasonal fluctuations, rotation under impact of daily changes, and so forth belong to this type. By now, a considerable number of mathematical papers has been devoted to periodic and almost periodic in distribution stochastic processes. We give a survey of the theory for certain classes of the linear difference equations in a Banach space. A feature of our treatment is the analysis of the solutions on the whole of axis. Such an analysis gives simple answers to the questions about solution stability of the Cauchy problem on +∞, solution stability of analogous problem on −∞, or of existence solution for boundary value problem and other questions about global behaviour of solutions. Examples are considered, and references to applications are given in remarks to appropriate theorems.


Bounded and periodic solutions of difference equations.
In this section, we construct an explicit representation of bounded or periodic solutions for abstract deterministic linear difference equation with a constant or periodic operator coefficient and bounded or periodic input. Then, general linear difference equations are studied too. The existence of bounded and periodic solutions is present for some difference equations with Lipschitz type nonlinearity and for the equation of Riccati type. Stability of solutions under bounded perturbation of operator coefficients is considered. are well defined and belong to ᏸ(B) (see, e.g., [23] or [16]). Moreover, P 2 − = P − , P 2 + = P + , and P − P + = P + P − = Θ. The operators P − and P + are called the spectral projectors or Riesz spectral projectors.

Simple linear equation.
First, we give basic theorem about the difference equation with one operator coefficient. The many applications lead to the equation of such kind. The following result has been proved in [9]. Let A ∈ ᏸ(B) be a fixed operator. Thus u(n) = u(0) =: u for every n ∈ Z. Therefore, given any element z ∈ B, there is a unique element u ∈ B such that A − λ 0 I u = z. (1.10) Thus, by the Banach theorem, the operator A − λ 0 I is invertible.

General linear equation with one operator coefficient.
We now consider a generalization of (1.5). To this generalization lead also some applications (see, e.g., [26,32,35]). We need the following condition. Condition 1.6. Let ω be a complex-valued function which is analytic in some neighbourhood of circle S.
(1.42) Therefore, the function AΦ is analytic for z ∈ K and, by closedness of the operator A, we deduce that (1.44) Thus, {c(n)} ∈ ᏹ, condition (ii) of the definition of class ᏹ is a consequence of the equality For a bounded sequence {y(n)}, set It is clear that The closedness of A implies that x(n) ∈ D(A) for n ∈ Z, and  (1.50) Let {w(n)} be a bounded solution of (1.27) with y(n) = v(n), n ∈ Z, such that (1.52) (a) It is easy to see that the existence of Theorem 1.3 is a simple consequence of Theorem 1.8.
(b) The equation  The operator-valued function Ω is defined by Condition 1.10 in some neighbourhood of the circle S. Theorem 1.11. The following statements are equivalent: has a unique bounded solution {x(n)} for every bounded sequence {y(n)}; (ii) for every z ∈ S, the operator has a continuous inverse.
Proof. The argument used to establish Theorem 1.8 can be adapted to get this result.

Equation with periodic coefficients.
Let p ∈ N and {A(n) : n ∈ Z} ⊂ ᏸ(B) be a sequence of bounded operators such that (1.61) Now, we first consider the equation It is a simple situation in which the previous results can be applied. Put The uniqueness is immediate. Remark 1.13. Condition (1.64) is equivalent, for every k ∈ {1, 2,...,p − 1}, to the following equality: ( Let a function ω satisfy Condition 1.6 and {a k : k ∈ Z} be its Laurent series coefficients. We now consider the equation where " " means the transition to column-vector. Let Proof. This is immediate. Exercise. Characterize the bounded operators A, C such that the equation  Proof. Let {y(n)} be a bounded sequence in B. Consider the following successive approximations: By induction, it is easy to see that, for m ≥ 1, Therefore, for every n ∈ Z, there is an element x(n) ∈ B such that in norm topology. By the closedness of A, we have This implies the following:   Theorem 1.20 (see [19] Proof. Let Y be a bounded sequence which satisfies (1.99). We consider a sequence {X m : m ≥ 0}, for n ∈ Z, given by (1.100) It can be easily checked that (1.101) By induction and by the following identity for n ≥ 1 (1.107) Relation (1.103) implies also the convergence of the series in norm B, n ∈ Z. Thus by the closedness of A, for every n ∈ Z, (4) If Y ∞ > 0, then the sequence {X(n) = Θ : n ∈ Z} is not a solution of (1.98).

Stability of solutions under perturbation of operator coefficients
Theorem 1.21 (see [11,12]). Let the operators satisfy the following conditions: Then the equation It is easy to verify that Therefore, for every n ∈ Z, there is an element (1.117) Taking the limit in j in both sides of equality (1.115), we conclude that x m is a solution of (1.113). By (1.112) and (1.113), it follows that (1.118) Theorem 1.22 (see [12]). Let operators from ᏸ(B) A(n), A m (n), n ∈ Z, m ≥ 1, (1.119) satisfy the following conditions for fixed p ∈ N: Let {y(n)} be a p-periodic sequence in B. Then the equation and, for every m greater than some m 0 ∈ N, the equation x − x m ∞ → 0, m → ∞. (1.122) Proof. By Theorem 1.12, the proof is analogous to that of Theorem 1.21 and is omitted. Some other results about stability of bounded solutions can be found in [10].

Theorem 1.24. Let A be an operator satisfying condition (1.6) and let {y(n)} be a bounded sequence in B.
For any n ∈ Z and c > 0, where z is a unique bounded solution to (1.5).
Proof. This follows from the proofs of Theorems 1.3 and 1.23.

Periodic random sequences in Banach spaces.
Strictly periodic in distribution random sequences, second-order periodic random sequences and their connection between themselves and with stationary sequences are the topics of this section. We discuss the basic properties of the periodic in distribution random sequences in a Banach space. The second-order periodic and weakly second-order periodic processes in a Hilbert space are treated. The results of Sections 1 and 2 are the basis to study the stochastic difference equations with periodic disturbances or periodic structure. The definition of the second-order periodic C-valued process has been introduced by Gladyshev [18], where spectral properties of second-order periodic process have also been considered. For Hilbert-valued process, it is natural to consider two definitions for the second-order periodicity which are identical in finite-dimensional space. For more details see [9]. Periodicity often arises in economic and geophysical time series, see the article by Troutman [40], where sufficient conditions for existence of the real periodic solution to difference equations are given. In Jiri [24,25], the methods for the statistical analysis of the periodic autoregressive processes in finite-dimensional space are given. Periodic in distribution and the second-order periodic random processes have been considered by Morozan [33]. The statistical problems including the estimation of correlation function for the second-order periodic processes have been investigated by Pagano [36]. Almost periodic in distribution processes with discrete time have also been considered [21,33]. Some methods of analysis of stationary random sequences in a Banach space can be found in [3,31,34].

Periodic in distribution random sequences.
Let (B, · ) be a complex separable Banach space with the zero element0 and let B * be its dual space. Let Ꮾ(B) be the Borel σ -algebra on B. We refer to [27] for the basic B-valued random variables or random elements theory. All random elements which arise in the next sections are considered on a complete probability space (Ω, Ᏺ, P). Further, all equalities, inequalities, and so on, with the random elements or random variables mean those almost surely.
is called a B-valued random process with discrete time or random sequence in B if, for each n ∈ Z, the mapping is a B-valued random element. For each ω ∈ Ω, the sequence {x(n, ω) : n ∈ Z} is called trajectory of the random process: be the set of all sequences with elements from B; and, for each j ∈ Z, let The σ -algebra in the space B Z is defined as For every random process {x(n) : n ∈ Z} there is a correspondent probability measure µ x on Ꮾ Z , which is defined by the equality (2.7) By the well-known Kolmogorov theorem, see, for instance, [17], the measure µ x is uniquely restored by collection of the finite-dimensional distributions where for every given I, the measure µ I x is a unique extension on The map θ(j) is one-to-one and the maps θ(j) and θ(j) −1 are measurable. For a random process {x(n) : n ∈ Z}, let µ θ(j)x (2.12) be the measure which corresponds to the process Let a number p ∈ N be fixed.
is periodic on Z of period p.

Remarks.
(1) A periodic of period p = 1 in distribution random processes is stationary.
(2) A stationary random process is periodic of period p in distribution for every p ∈ N. (2.17) are called periodically with period p connected or periodically connected, if the B mvalued random process is periodic of period p in distribution.
We refer to [9, Chapter 2] for basic properties of periodic in distribution processes and list only several that we need later.

Lemma 2.4. A process {x(n) : n ∈ Z} is periodic in distribution of period p if and only if
Proof. By the Kolmogorov theorem, it is sufficient to prove equality (2.19) only for finite-dimensional distributions (2.8); and, therefore, by uniqueness of an extension of a measure, it suffices to prove (2.19) for functions (2.10). See Definition 2.2.
is stationary.
Proof. We first show that if {x(n) : n ∈ Z} is p-periodic in distribution, then process (2.20) is stationary. Let where is given. By uniqueness of an extension of measure from semi-algebra to the σ -algebra generated by semi-algebra, (see, e.g., [22] and Definition 2.2) we have (2.23) Now, assume that (2.20) is stationary. Given any For the sets (2.27) Theorem 2.6. Let {x(n) : n ∈ Z} be a p-periodic in distribution random process in B, B 1 a complex separable Banach space, and m ∈ N. Suppose that a map Then, for every n 1 ,...,n m from Z, the B 1 -valued random process be given. We have, clearly, To prove the theorem, we use Definition 2.2 to show that By periodicity of the process {x(n) : n ∈ Z}, equality (2.34) holds for the sets Therefore, (2.34) holds by the uniqueness of extension of measure.
Corollary 2.7. Let {x(n) : n ∈ Z} be a p-periodic in distribution random process in B.
(i) The process (v) If the processes {x j (n) : n ∈ Z}, j = 1, 2 are periodically connected with period p, then the process for every measurable parallelepiped: (2.46) Then the function is p-periodic on Z.
Theorem 2.10. The following assertions are equivalent: Proof. (i) implies (ii). This follows directly from Theorem 2.8.
(ii) implies (i). It is enough to show that the measures µ(n 1 ,...,n m ; ·) and µ(n 1 + p,...,n m + p) are equal for every m ∈ N and {n 1 ,...,n m } ⊂ Z. Therefore, we may equivalently prove the equality of the corresponding characteristic functions, see [2]. However, by Theorem 2.8, it follows that for every g ∈ (B m ) * , (2.50) Re g j ,x n j + n Then the process (2.56) and, it follows from (2.56), that (2.58) Then, for every n ∈ Z, the series converges almost surely in B-norm and {y(n) : n ∈ Z} is p-periodic in distribution.
Theorem 2.14. Let {x t (n) : n ∈ Z}, t ∈ N, be a sequence of p-periodic in distribution processes and let {x(n) : n ∈ Z} be a process such that, for every n ∈ Z, (2.62) Proof. Let in distribution. Hence, for every (2.69) Let {ξ k : k ∈ Z} be a stationary process in C such that E|ξ 0 | < +∞. If the processes {x(n)} and {ξ k } are independent, then the processes are p-periodic in distribution.

Second-order periodic random process in Hilbert spaces.
In this section, we develop the concept of a second-order periodic random process and give some of its basic properties. Let (Ω, Ᏺ, P) be a complete probability space, and let (H, (·, ·)) be a complex separable Hilbert space with inner product (·, ·) and corresponding norm · .
Let x(k) and x(n) be H-valued random variables such that Then the Bochner integrals is a continuous bilinear mapping. By the Riesz theorem, there is a unique linear bounded operator S k,n such that The operator S k,n is called joint covariance of x(k) and x(n). It is known (see, e.g., [28,39]) that (i) S * k,n = S n,k ; (ii) the operator S n,n is selfadjoint nonnegative and nuclear; (iii) the following relations hold:   The process {x(n) : n ∈ Z} is called second-order (SO) p-periodic if (i) for all n ∈ Z, Ex(n + p) = Ex(n); (ii) for all {k, n} ⊂ Z, S k+p,n+p = S k,n . An SO 1-periodic process is second-order stationary.
(2) By Theorem 2.8, it follows that a p-periodic process x such that E x(k) 2 It can be verified that the joint covariance of vectors is the matrix Then the process {x(n) : n ∈ Z} is SO p-periodic.
Proof. It is enough to prove that, for every n ∈ Z,

Stationary and periodic solutions of difference equations.
This section contains the conditions for existence of stationary or periodic in distribution solutions to the linear difference equations in the spectral terms. It is proved that these conditions are stable under perturbations of input random process and under bounded perturbation of operator coefficients. The equations with unbounded operator coefficient and nonlinear equations are also considered.

Linear difference equations.
A stochastic process with discrete time in a complex separable Banach space B by definition is a collection of B-valued random elements {x(n) : n ∈ Z} defined on a complete probability space (Ω, Ᏺ, P).
Two processes {x(n) : n ∈ Z} and {y(n) : n ∈ Z} are said to be stochastic equivalent if ∀n ∈ Z : P x(n) = y(n) = 1. (3.1) Two processes which are stochastic equivalent are in fact almost surely equal. Further, we shall consider equivalent processes as equal.
Theorem 3.1 (see [7,9]). Let A ∈ ᏸ(B). The following statements are equivalent: (i) the equation therefore the B × C-valued process x(n), e iθ t n : n ∈ Z (3.5) is stationary. Hence the process is also stationary, moreover, Since, by (3.2), we have Therefore, for every z ∈ B, (3.10) has the solution u ∈ B. This solution is unique. Indeed, if v ∈ B, v ≠ u is also a solution for (3.10), then the process is a stationary solution for (3.2) which is not equal to {x(n) : n ∈ Z}.
Thus A − tI is a one-to-one mapping on B and t ∈ (C \ σ (A)).
(ii) implies (i). Let {y(n) : n ∈ Z} be a stationary process such that E y(0) < +∞. Then with notations from Section 1, we define the random element The series for x(n) converges almost surely in B-norm since by (ii), By Theorem 2.12, the process {x(n) : n ∈ Z} is stationary and E x(0) < +∞. It can be easily checked, by boundedness of A, that this process satisfies (3.2).

Remarks.
(1) Theorem 3.1 is not a direct consequence of the deterministic Theorem 1.3 and the moment condition of Theorem 3.1 is essential.

(3) If σ (A) ⊂ {z | |z| < 1} then the stationary solution of (3.2) is stable in the mean for n → +∞.
(4) If {y(n) : n ∈ Z} is a sequence of independent identically distributed B-valued random elements and Ᏺ n is the σ -algebra generated by y(k), k ≤ n for n ∈ Z, then the stationary solution of (3.2), in remark (3), is anticipative with respect to {Ᏺ n ,n ∈ Z} and, in remark (2), is nonanticipative with respect to {Ᏺ n ,n ∈ Z}.

15)
for every p-periodic process {y(n) : n ∈ Z}, such that Proof. The proof is analogous to that of Theorem 3.1 and is omitted.
has a unique p-periodic solution {x(n) : n ∈ Z}, such that for k ∈ Z. By Lemma 2.5, {y(n) : n ∈ Z} is p-periodic in B. It is evident that Let {x(n) : n ∈ Z} be a unique p-periodic solution for (3.19) with the above constructed process {y(k) : k ∈ Z}. By Lemma 2.5, the process {x(kp + j) : k ∈ Z} is stationary; and it can be easily checked that this process satisfies the equation (3.26) By Lemma 2.5, the process {z 0 (k) : k ∈ Z} is stationary, moreover, E z 0 (0) < +∞. By Theorem 3.1, there is a unique stationary B-valued process {x 0 (k) : k ∈ Z} satisfies the equation Now, we consider the process {x(n) : n ∈ Z} defined by is stationary and therefore, by Lemma 2.5, the process {x(n) : n ∈ Z} is p-periodic in B. The equality holds for the value n = kp + j, k ∈ Z, 0 ≤ j ≤ p − 2, by the definition of process {x(n) : n ∈ Z}. To prove (3.30) for values n = kp + p − 1, k ∈ Z, we have (3.31) We consider the equation which can be found in various applications, see [26,32,35]. The following existence stationary solution theorem is based on some modification of the method used in Section 1.   where We observe that U(k) ∈ ᏸ(B), k ∈ Z. Now let {y(n) : n ∈ Z} be a sequence of independent identically distributed random elements such that Ey(0) =0. Fixing n ∈ Z and taking the conditional expectation of (3.36) gives the σ -algebra generated by y(k), k ≠ n, we have Hence, it follows that U(n) = Θ. Indeed, let {u(j) : j ≥ 1} be a countable dense set in B. The random element y(n), by definition, is such that Then U(n)u(j) =0, j ≥ 1. Therefore, U(n) = Θ.

Thus we have
where t 0 , t 1 are the numbers corresponding to the sequence {c(k) : k ∈ Z} in the definition of ᏹ, see Section 1. By the definition of ᏹ, the series k∈Z Ac(k)z k , z ∈ K (3.42) also converges in the operator norm; and by the closedness of A, it follows that for all x ∈ B. Now, by (3.40), we obtain Therefore, for every y ∈ B, z ∈ K, the equation For z ∈ K, fixed (3.45) has a unique solution x for every y ∈ B. Indeed, if a solution of (3.45) is not unique, then there is u ∈ D(A), u ≠0, such that Now put y = u in (3.45). Then we find (3.50) The operator-valued function Φ − is analytic in K; and for every x ∈ B, the relation Φ − (z)x ∈ D(A) is true. Hence, the following expansion in Laurent series This series absolutely converges in K in operator norm. By definition of Φ − , and therefore (3.40) is true. Moreover, Hence, the function AΦ − also is analytic in K; and by the closedness of A, we have for every k ∈ Z, converge almost surely in B-norm. Since operator A is closed, it follows that for every n ∈ Z. By Theorem 2.12, process {x(n) : n ∈ Z} is stationary. It can be easily verified that relations (3.40) imply (3.32). Uniqueness is obvious.
We now take up an important extension of Theorem 3.1. Suppose that ω is a function which satisfies Condition 1.6 from Section 1 and let {a k : k ∈ Z} be the Laurent series coefficients for ω. for every stationary process in B{y(n) : n ∈ Z} with E y(0) < +∞; Proof. The proof is analogous to that of Theorem 3.6 and is omitted. See also the deterministic Theorem 1.8 for comparison. Just in the same way, we also get the following result. Theorem 3.8 (see [9]). Let A be a closed operator. The following statements are equivalent: (i) there is a sequence of operators {c(k) : k ∈ Z} ∈ ᏹ; and (3.59) has a unique p-periodic solution {x(n) : n ∈ Z} such that
Proof. (i) implies (ii). It can be established by the use of an argument similar to that of the first part of Theorem 3.1.
(ii) implies (i). Suppose that, for every λ ∈ S, operator (3.64) has an inverse defined on B. Then, for every λ ∈ S, there is r (λ) > 0 such that the operator has a bounded inverse for all z with |z − λ| < r (λ). Since S is compact in C, there is an annulus such that, for every z ∈ K, operator (3.65) has bounded inverse R(z). Now we prove that the function R is analytic in K. Let z 0 ∈ K be a fixed point. By definition of R, it follows that (3.67) Therefore, By equality (3.68), we obtain for every value z, such that be the Laurent series for R. Note that To establish this, note that for every n ∈ Z, we have (3.80) we have

By (3.72) and the identity
In addition, by identity Therefore, (3.86) Stationary solutions of a two-dimensional stochastic difference equation in a Banach space have been developed in [19].

Perturbed equation.
In many problems of applications, a given system is subject to deterministic and random influences that must be taken into account in the appropriate mathematical models. Usually, mathematical models are only approximation to real behavior of systems. Therefore, mathematical models must be stable under perturbation of its parameters. From the viewpoint of applications, stability is just as important as existence. We give such results in this section. From the mathematical viewpoint, many of such results are the theorems about the continuous dependence of solution in given norm from equation coefficients. Theorem 3.10 (see [11,12]). Let the operators satisfy the following conditions: Then for each stationary B-valued process {y(n) : n ∈ Z} with E y(0) < +∞, we have the equation and for every m greater than some m 0 ∈ N, the equation has a unique stationary solution {x(n) : n ∈ Z} and a unique solution {x m (n) : n ∈ Z}, respectively, for which and let P − and P + be the spectral projectors corresponding to the spectral sets σ − (A) and σ + (A), respectively. Then, for every n ∈ Z, we have where, for j ∈ Z, The series for x(n) converges almost surely in B-norm and E x(0) < +∞, moreover, Then by [27], for every n ∈ Z, there is a random element x m (n) such that   Then, the statement of Theorem 3.10 holds for the solutions of the following equations: We now return to the linear stochastic equation (3.19). Let {A(n) : n ∈ Z} ⊂ ᏸ(B); and, for a fixed p ∈ N, let (3.104) The proof of this statement follows along the lines of the proof of Theorem 3.10.
It is easily seen that the solution {x(n) : n ∈ Z} for (3.105) satisfies the equation (3.111) Then, using the above statement, we define {x(νp) : ν ∈ Z} as the solution for (3.110) and put we have the solution {x(n) : n ∈ Z} for (3.105). Now, using the approximating method of Theorem 3.10, it is easy to prove the existence of a solution for (3.106) for every m greater than some m 0 ∈ N.
The study of a random perturbation is complicated and more interesting. In the sequel, we investigate this question. The following result will be useful in applications. Theorem 3.14 (see [5, Chapter 10, Section 1], [9]). Let {A, C} ⊂ ᏸ(B) and let {(ε(n), y(n)) : n ∈ Z} be a stationary metrically transitive random process in C×B. Suppose that Then the equation has a unique stationary solution {x(n) : n ∈ Z}.
Proof. Let n ∈ Z be fixed. We show that the series converges almost surely in B-norm. It is obvious that Since ε is metrically transitive, we have, by the ergodic theorem, (see [5, Chapter X, almost surely. For any a > 1, it follows from (i) that ∞ j=1 P ln + y(n − j − 1) > j ln a < +∞. and hence lim sup Thus, the sequence of the partial sums of series for x(n) is the Cauchy sequence almost surely in B-norm. Hence, (see [27]) there is a random element x(n) which is the limit of these sums. By the boundedness of the operators A and C, we find (3.120) By Theorem 2.12, {x(n) : n ∈ Z} is stationary. In the same manner we get the uniqueness.
Analogous to Theorem 3.14, this result can be proved for p-periodic processes. Instead, we give the following theorem with the moment conditions. Theorem 3.15 (see [9]). Let operators {A, C} ⊂ ᏸ(B). Suppose that {ε(n) : n ∈ Z} is a sequence of independent identically distributed random variables in C, {y(n) : n ∈ Z} is a p-periodic in distribution random process in B which is independent from {ε(n) : Then there is a unique p-periodic in distribution process {x(n) : n ∈ Z} in B that satisfies (3.113) such that (3.121) Proof. Since, for every n ∈ Z, j ≥ 1, the series converges almost surely in B-norm and By Theorem 2.12, the process {x(n) : n ∈ Z} is p-periodic in distribution.

Nonlinear equation.
In this section, we consider some nonlinear equations, which are disturbed by stochastic processes, and give conditions for existence stationary or periodic in distribution solutions to these equations.

A nonlinearity satisfying Lipschitz condition.
Let p ∈ N be given. Theorem 3.16 (see [15]). Let G : Z × B → B be a function such that Let {y(n) : n ∈ Z} be a p-periodic in distribution process in B such that Now let n ∈ Z and x 0 be fixed. To prove that the sequence {x(n; n 0 ,x 0 ) : n 0 ≤ n} is the Cauchy sequence almost surely in B-norm, we obtain for n 1 < n 2 < n the following inequality: x n; n 1 ,x 0 − x n; n 2 ,x 0 = x n; n 2 ,x n 2 ; n 1 ,x 0 − x n; n 2 ,x 0 almost surely. In addition, almost surely. The series in (3.130) converges almost surely by periodicity of the process {y(n) : n ∈ Z} and the moment condition. By (3.130), it follows that n n 2 =−∞ λ n−n 2 E sup Thus there is (see [27]) a random element x(n; x 0 ) such that x n; x 0 = lim n 0 →−∞ x n; n 0 ,x 0 (3.134) in the B-norm almost surely. Now we prove that for every n ∈ Z, the element x(n; x 0 ) is independent from x 0 with probability one. Let z 0 ∈ B. Then x n; n 0 ,x 0 − x n; n 0 ,z 0 and therefore, P x n; x 0 = x n; z 0 = 1. (3.136) We, further, write x(n) instead of x(n; x 0 ). Using Let T : Ᏸ → B Z be the mapping defined by The mapping T is measurable and for mapping θ(p), we have Then w k+1 = G k, w k + u k+p ; w k+p+1 = G k + p,w k+p + u k+p (3.144) for all k ∈ Z. According to (ii), we have for j ≥ 1. Hence by inequality, that is analogous to (3.130), it follows that w k+1 =w k+p+1 , k∈ Z.
By Lemma 2.4, the process {x(n) : n ∈ Z} is p-periodic in distribution.
To prove the uniqueness, we consider a p-periodic in distribution process {z(n) : n ∈ Z} which is the solution of (3.126), such that E z(k) < +∞, 1 ≤ k ≤ p. Then Thus P x(n) = z(n), n ∈ Z = 1. This result is important for applications. By a fixed point theorem, the sequence, defined by (3.156) converges in L 2 ([0, 1]) to a unique solution of the integral equation If each function z n+1 from (3.156) is defined with the random error y(n) as in (3.155), and errors {y(n)} forms a stationary process, then the sequence {x(n)} also is stationary process and consequently does not converge to solution of (3.157). However, there exist methods for estimation of solution for (3.157) in this situation, see [4,13].

Equations near to linear.
We apply the techniques described above to study the existence stationary and periodic in distribution solution of nonlinear stochastic difference equations, which are near to linear equations. Of course, these conditions are only sufficient.
If A ∈ ᏸ(B) is an operator such that σ (A) ∩ S = ∅, let P − and P + be the spectral projectors corresponding to the spectral sets that lie inside and outside of S, respectively. (3.158) Let {y(n) : n ∈ Z} be a p-periodic in distribution process in B, such that Then the equation Proof. Let x 0 ∈ B. By Theorem 3.2, there is a unique p-periodic in distribution random process {x 1 (n) : n ∈ Z} in B, such that (3.161) Given for j ≥ 1 the p-periodic in distribution process {x j (n) : n ∈ Z}, by Theorem 3.2 there is a unique p-periodic in distribution solution {x j+1 (n) : n ∈ Z} of x j+1 (n + 1) = Ax j+1 (n) + G n, x j (n) − Ax j (n) + y(n), n ∈ Z, (3.162) in addition, E x j+1 (k) < +∞, 1 ≤ k ≤ p. We also have for any n ∈ Z and m ≥ 1. By (ii) of Theorem 3.19, we obtain From (3.164) and (3.165), we have that for every n ∈ Z, the sequence of the random elements {x m (n) : m ≥ 1} converges in the first-order mean [27] to a random element x(n) with E x(n) < +∞. According to Theorem 2.14, {x(n) : n ∈ Z} is p-periodic in distribution in B. In addition, from (3.162), we have (3.160).
To prove the uniqueness, we suppose that {z(n) : n ∈ Z} is a p-periodic in the distribution solution of (3.160). By Theorem 3.2, we have for any n ∈ Z and for the analogous representation of z(n). Then we obtain E x(n) − z(n) = 0, n∈ Z.  Suppose that a function f : Z × B 3 → B satisfies the following conditions: (i) f (n+ p, u, v, w) = f (n,u,v,w); (ii) the following inequality holds: for all n ∈ Z, {u, v, w, u j ,v j ,w j ,j = 1, 2} ⊂ B with a number L such that where {c(k)} are the Laurent series coefficients of the function Let {y(n) : n ∈ Z} be a p-periodic in distribution process such that Then the equation has the bounded inverse R(z). Suppose that b : B 2 → B is a function which satisfies the conditions: where {c(k)} are the Laurent series coefficients of R. Let {y(n) : n ∈ Z} be a stationary process such that E y(0) < +∞. (3.179) Then the equation has a unique stationary solution {x(n) : n ∈ Z} such that E x(0) < +∞.

Second-order stationary and periodic solutions.
Let B = H be a Hilbert space over R. The existence problems of SO stationary and SO periodic in distribution solutions for difference equations are more complicated than those from previous sections. Finite-dimensional stochastic equations with constant coefficients are studied in detail, see Arató [1]. The aim of this section is to prove criteria for the existence of SO stationary and SO periodic solutions for linear stochastic difference equations in a Hilbert space [6]. then x and y are said to be orthogonal to each other. We note that mutual correlation operator of orthogonal elements is the zero operator, and that orthogonality of mutually Gaussian elements implies their independence. Given sequence {y(n) : n ∈ Z} of random elements such that E y(n) 2 < +∞, n ∈ Z and m ∈ Z, let L y m be mean-square closure of linear combination  Since x(n) ∈ L y n−1 , we obtain that the correlation operator S x of the element x(0) satisfies the equality S x = AS x A * + S y , (3.187) where A * is the adjoint operator of A. From (3.187), for every n ∈ N, we have Hence, taking into account the properties of the correlation operators, the convergence of the following series is obtained A k e j 2 < +∞. (3.195) Sufficiency. If 1 ∈ (C \ σ (A)), the expectation Ex(0) is a unique solution of Ex(0) = AEx(0) + Ey(0), (3.196) for every Ey(0). The element ξ(n) has the correlation operator and for 1 ≤ m < k, the following equality holds. For the orthonormal basis {e j : j ≥ 1} in H consisting of the eigenvectors of the operator S y , we obtain, from representation (3.208), see [27]. The last series is convergent in the quadratic mean, with the correlation operator S x of the element x(0) being for every n ∈ Z.
To complete the proof of sufficiency, we need only to show uniqueness. Let {z(n) : n ∈ Z} be an SO stationary solution of (3.183) satisfying the conditions of Theorem 3.22. Then for every n ∈ Z, we have It is easy to see that and that x(n) = z(n) with probability one for each n ∈ Z. Proof. It is a direct consequence of the previous theorem and the known properties of the Gaussian random elements, see [38,9] for the full proof.  [30]. For the infinite-dimensional space, this is not true, see [9, Chapter 2, Section 2.7] for the details. where {x(n) ∈ L y n−1 : n ∈ Z} is a unique SO stationary solution to (3.183).
Proof. The proof follows the lines of the proof of Theorem 3.22 and is omitted. Proof. See the proof of Theorem 3.22.
(3.241) Let x 0 (k) := E(x(kp)/ξ(j), j ≤ k − 1), k ∈ Z. Then x 0 (k) ∈ L ξ k−1 , k ∈ Z, and from (3.241), we have (3.242) The process {x 0 (k) : k ∈ Z} is a unique solution for (3.242  Theorem 3.26 admits the following generalization. Let p ∈ N ∪{0} and let ᏼ be the class of all H-valued SO p-periodic processes {y(n) : n ∈ Z}, such that Ey(n) =0, n ∈ Z; the random elements y(k) and y(n) are orthogonal if |k − n| > p. Proof. The proof is similar to that of Theorem 3.26 and so is omitted.
The following sufficient condition will be useful in the applications. Suppose that the spectrum of operator B 0 consists of two parts σ − and σ + , such that sup |z| : z ∈ σ − < 1, inf |z| : z ∈ σ + > 1. Proof. See the proof of Theorems 3.1, 3.26, and see [6] for more details.