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http://ijmms.hindawi.com © Hindawi Publishing Corp.

PERIODICITY IN DISTRIBUTION.

I. DISCRETE SYSTEMS

A. YA. DOROGOVTSEV

Received 5 January 2001 and in revised form 1 October 2001

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 finite-dimensional 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 pro-cesses. 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 exis-tence 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.

2000 Mathematics Subject Classification: 34-02, 34F05, 34K50, 60-02, 60H10, 37L55, 93E03.

1. Bounded and periodic solutions of difference equations. In this section, we construct an explicit representation of bounded or periodic solutions for abstract de-terministic 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. Sta-bility of solutions under bounded perturbation of operator coefficients is considered.

1.1. Notations. Let(B,·)be a complex Banach space with a zero element ¯0 and

(B)be a Banach algebra of all continuous linear mappingsA:B→Bwith a unity elementIand zero elementΘ.

For an operatorAfromᏸ(B), letσ (A)andρ(A)be the spectrum and resolvent set ofA, respectively.

Suppose thatσ (A)is the union of two spectral setsσandσ+. IfΓis a simple cycle which lies inC\σ (A)and envelopsσ−, then the operators

P−:= −2π i1

Γ(A−λI)

−1dλ, P

(2)

are well defined and belong toᏸ(B)(see, e.g., [23] or [16]). Moreover,P2

−=P−,P+2=P+, andP−P+=P+P−=Θ. The operatorsP− andP+are calledthe spectral projectorsor

Riesz spectral projectors.

Definition1.1. The sequence{y(n):n∈Z}withy(n)∈B,n∈Zisboundedif

y∞:=sup n∈Z

y(n)<+∞. (1.2)

Definition 1.2. For fixed p∈N, the sequence {y(n):n∈Z}with y(n)∈B, n∈Z, isp-periodicif

y(n+p)=y(n), n∈Z. (1.3)

Define

S:=z∈C:|z| =1. (1.4)

1.2. Simple linear equation. First, we give basic theorem about the difference equa-tion with one operator coefficient. The many applicaequa-tions lead to the equaequa-tion of such kind. The following result has been proved in [9]. LetA∈(B)be a fixed operator.

Theorem1.3. The following statements are equivalent:

(a) the equation

x(n+1)=Ax(n)+y(n), n∈Z (1.5)

has a unique bounded solution {x(n): n∈ Z} for every bounded sequence {y(n):n∈ Z};

(b) for spectrumσ (A), we have

σ (A)∩S= ∅. (1.6)

Proof. (a) implies (b). Letλ0SandzB. Let{x(n):nZ}be a unique bounded solution of (1.5) that corresponds to the bounded sequence{−λn0z:n∈Z}. Put

u(n):=x(n)λ−0n, n∈Z. (1.7) From formula (1.5), it is clear that the equation

λ0u(n+1)=Au(n)−z, n∈Z (1.8)

has a unique bounded solution{u(n):n∈Z}for everyz∈B. Consequently,

λ0u(n+1)−u(n)=Au(n)−u(n−1), n∈Z. (1.9)

Thusu(n)=u(0)=:ufor everyn∈Z. Therefore, given any elementz∈B, there is a unique elementu∈Bsuch that

A−λ0Iu=z. (1.10)

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(b) implies (a). Let

σ−:=σ (A)∩z∈C:|z|<1, σ+:=σ (A)\σ−, (1.11)

andP−,P+are the corresponding spectral projectors. It is easy to show that

L:=

j=0

AP−j+ 1

j=−∞

AP+j<+∞, (1.12)

where(AP−)0:=P−(see, e.g., [16,23]). If{y(n):n∈Z}is a bounded sequence inB, then the sequence

x(n):= j∈Z

G(j)y(n−1−j), n∈Z, (1.13)

where

G(j):=   

APj, j≥0;

−AP+j, j≤ −1, jZ

(1.14)

is also bounded. Moreover, for everyn∈Z, we have

Ax(n)=AP−+AP+x(n)

=

j=0

APj+1y(n−1−j)−

−1

j=−∞

AP+j+1y(n−1−j)

=x(n+1)−P−y(n)−P+y(n)=x(n+1)−y(n).

(1.15)

Now, we prove that the solution{x(n):n∈Z}for (1.5) is unique. Let{u(n):n∈Z}

be a solution of (1.5) which corresponds to{y(n):n∈Z}. Then, the sequence

v(n):=x(n)−u(n):n∈Z (1.16)

satisfies the equation

v(n+1)=Av(n), n∈Z, (1.17)

which is equivalent to the system

v−(n+1)=AP−v−(n), v+(n+1)=AP+v+(n); n∈Z (1.18)

with

v−(n):=P−v(n), v+(n):=P+v(n), n∈Z. (1.19)

From this we have, form≥1,

v−(n)=AP−mv−(n−m)≤AP−msup k∈Z

v−(k),

v+(n)=AP+−mv+(n+m)≤AP+−msup k∈Z

v+(k). (1.20)

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Let{A0, A1} ⊂(B)and{x(n)}:= {x(n):n∈Z}.

Corollary1.4. IfA0A1=A1A0holds, then the following statements are equivalent: (a) the equation

x(n+1)=A0x(n)+A1x(n−1)+y(n), n∈Z (1.21)

has a unique bounded solution{x(n)}for every bounded sequence{y(n)};

(b) for everyλ∈S, the operatorλ2IλA0A1has a bounded inverse.

Proof. (a) implies (b). The argument used to establish the first part ofTheorem 1.3 can be adapted easily to get this result.

(b) implies (a). It is a direct consequence ofTheorem 1.3for the equation

x(n+1) x(n)

=

A0 A1 I Θ

·

x(n) x(n−1)

+

I Θ Θ Θ

·

y(n) y(n−1)

, n∈Z. (1.22)

1.3. Periodic solutions. We give the condition under which (1.5), with ap-periodic sequence{y(n):n∈Z}, has a uniquep-periodic solution. Put

Sp:=z∈C:zp=1. (1.23)

Theorem1.5. The following statements are equivalent:

(a) equation (1.5) has a uniquep-periodic solution{x(n)}for everyp-periodic input {y(n)};

(b) for spectrumσ (A), we have

σ (A)∩Sp= ∅. (1.24)

Proof. It is immediate. A complete proof ofTheorem 1.5can be found in [9, Chap-ter 1, Section 1.1].

1.4. 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 ωbe a complex-valued function which is analytic in some neighbourhood of circleS.

UnderCondition 1.6, the functionωmay be written as the Laurent series:

ω(z)= n∈Z

anzn, n∈Z, (1.25)

where

an= 1 2π i

S

ω(z)z−n−1dz, nZ;

r=lim sup n→−∞

an

1/|n|, R=lim sup n→+∞

an 1/n−1

.

(1.26)

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LetAbe a closed linear operator with dense domainD(A)inB. We consider the equation

k∈Z

akx(n+k)=Ax(n)+y(n), n∈Z, (1.27)

where{y(n)}is a bounded sequence inB.

Definition1.7. Sequence{x(n)}is called asolutionof (1.27), if (i) for alln∈Z,x(n)∈D(A);

(ii) sequence{x(n)}satisfies (1.27).

Letᏹbe the class of all sequences{c(n)} ⊂(B)satisfying the conditions: (i) for alln∈Z,c(n):B→D(A);

(ii) for alln∈Z,Ac(n)∈(B);Ac(n)=c(n)AonD(A);

(iii) lim supn→∞c(n)1/n<1, lim supn→−∞c(n)1/|n|<1; lim supn→∞Ac(n)1/n <1, lim supn→−∞Ac(n)1/|n|<1.

Theorem1.8. If (1.27) has a unique bounded solution{x(n)}for every bounded sequence{y(n)}, then

σ (A)∩ω(S)= ∅. (1.28)

If an operatorAsatisfies condition (1.28), then (1.27) has a unique bounded solution {x(n)}for every bounded sequence{y(n)}; and there exists a sequence{c(n)} ∈

such that,

x(n)= k∈Z

c(k−n)y(k), n∈Z. (1.29)

Proof. (1) Suppose that (1.27) has a unique bounded solution{x(n)}for every bounded input{y(n)}. Let v∈B, λ0∈S. For a unique bounded solution{x(n)}

of (1.27) with

y(n)= −λn

0v:n∈Z

, (1.30)

we have

k∈Z

akλk0u(n+k)=Au(n)−v, (1.31)

whereu(n):=x(n)λ−0n∈D(A),n∈Z. The bounded solution{u(n)}of (1.31) is also unique. Therefore,

k∈Z

akλk0

u(n+k)−u(n−1+k)=Au(n)−u(n−1), n∈Z; (1.32)

and by uniqueness of solution for (1.31), we conclude that

∃u∈D(A):u(n)=u(0)=u, n∈Z. (1.33)

Thus

∀v∈B, !u∈D(A):A−ωλ0Iu=v; (1.34)

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(2) From assumption (1.28) and the closedness of the setσ (A), it follows that there exists an annulus

K:=z∈C:t0<|z|< t1, (1.35)

with somer < t0<1< t1< Rsuch that

σ (A)∩ω(K)= ∅. (1.36)

For the operator-valued function

Φ(z):=ω(z)I−A−1, z∈K, (1.37)

we have, for everyz∈K,

Φ(z)∈(B), Φ(z):B →D(A) (1.38)

and the Laurent series

Φ(z)= n∈Z

c(n)zn, zK, (1.39)

where

c(n)= 1

2π i

SΦ

(z)z−n−1dz, nZ, (1.40)

see [23]. From the definition of the functionΦ, we have, for everyz∈KonB

ω(z)I−AΦ(z)=I, (1.41)

or

AΦ(z)=ω(z)Φ(z)−I. (1.42)

Therefore, the functionAΦis analytic forz∈Kand, by closedness of the operatorA, we deduce that

Ac(n)=2π i1

S

AΦ(z)z−n−1dz, n∈Z, (1.43) From (1.42), it follows that

Ac(n)=2π i1

S

ω(λ)Φ(λ)−Iλ−n−1

=2π i1

S

k∈Z

m∈Z

akc(m)λk+m−n−1dλ−Iδn,0

=

k∈Z

akc(n−k)−Iδn,0, n∈Z.

(1.44)

Thus,{c(n)} ∈ᏹ, condition (ii) of the definition of classᏹis a consequence of the equality

AΦ(z)=ω(z)Φ(z)−I=Φ(z)A, z∈K (1.45)

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For a bounded sequence{y(n)}, set

x(n):= k∈Z

c(k−n)y(k), n∈Z. (1.46)

It is clear that

x∞≤

k∈Z

c(k)·y∞. (1.47)

The closedness ofAimplies thatx(n)∈D(A)forn∈Z, and

Ax(n)= k∈Z

Ac(k−n)y(k), n∈Z. (1.48)

From (1.48) and (1.44), it follows that

k∈Z

akx(n+k)−Ax(n)=

k∈Z

ak

m∈Z

c(m−n−k)y(m)− m∈Z

Ac(m−n)y(m)

=

m∈Z

k∈Z

akc(m−n−k)−Ac(m−n)  y(m)

=

m∈Z

Iδm−n,0y(m)=y(n), n∈Z.

(1.49)

Thus{x(n)}is a bounded solution of (1.27).

To conclude the proof, we have to show that solution{x(n)}is unique. Let{u(n)}

be a bounded solution of (1.27). Then the sequence{v(n):=x(n)−u(n)}is a bounded solution of the equation

k∈Z

akv(n+k)=Av(n), n∈Z. (1.50)

Let{w(n)}be a bounded solution of (1.27) withy(n)=v(n),n∈Z, such that

w(n)= k∈Z

c(k−n)v(k), n∈Z, (1.51)

where{c(n)} ∈ᏹ. Using (1.50), we obtain

v(n)= k∈Z

akw(n+k)−Aw(n)

=

k∈Z

ak

m∈Z

c(m−n−k)v(m)−A m∈Z

c(m−n)v(m)

=

k∈Z

ak

m∈Z

c(m)v(n+k+m)−A m∈Z

c(m)v(n+m)

=

m∈Z

c(m)

k∈Z

akv(n+k+m)−Av(n+m)  =¯0.

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1.4.1. Examples. We want to illustrate by means of simple examples howTheorem 1.8may be applied.

(a) It is easy to see that the existence ofTheorem 1.3is a simple consequence of Theorem 1.8.

(b) The equation

x(n+1)2x(n)+x(n−1)=Ax(n)+y(n), n∈Z (1.53)

has a unique bounded solution for every bounded input{y(n)}, if and only if

σ (A)∩[−4,0]= ∅. (1.54)

(c) The equation

1 2

x(n+1)−x(n−1)=Ax(n)+y(n), n∈Z (1.55)

has a unique bounded solution for every bounded input{y(n)}, if and only if

σ (A)∩is|s∈[−1,1]= ∅. (1.56)

(d) For (1.5), we have fromTheorem 1.8the generalization ofTheorem 1.3to the unbounded operatorA.

Remark 1.9. The particular cases ofTheorem 1.8 can be improved in different ways [9].

Theorem 1.8can be generalized in the following way.

Condition1.10. Let{U (n):n∈Z} ⊂(B)be a sequence of operators such that

lim sup n→−∞

U (n)1/|n|<1, lim sup n→∞

U (n)1/n<1. (1.57)

Define

(z):= n∈Z

U (n)zn. (1.58)

The operator-valued functionΩis defined by Condition 1.10in some neighbour-hood of the circleS.

Theorem1.11. The following statements are equivalent:

(i) the equation

k∈Z

U (k)x(n+k)=Ax(n)+y(n), n∈Z (1.59)

has a unique bounded solution{x(n)}for every bounded sequence{y(n)};

(ii) for everyz∈S, the operator

A−(z) (1.60)

has a continuous inverse.

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1.5. Equation with periodic coefficients. Letp∈Nand{A(n):n∈Z} ⊂(B)be a sequence of bounded operators such that

A(n+p)=A(n), n∈Z. (1.61)

Now, we first consider the equation

x(n+1)=A(n)x(n)+y(n), n∈Z, (1.62)

where{y(n)}is a bounded sequence inB. It is a simple situation in which the previous results can be applied. Put

D:=A(p−1)A(p2)···A(1)A(0). (1.63)

Theorem1.12(see [14]). The following statements are equivalent:

(i) equation (1.62) has a unique bounded solution{x(n)}for every bounded se-quence{y(n)};

(ii) for spectrumσ (D), we have

σ (D)∩S= ∅. (1.64)

Proof. (i) implies (ii). Let{x(n)} be a unique solution of (1.62) for a bounded sequence{y(n)}. Iterating (1.62), we find that

x(n+1)p=Dx(np)+z(n), (1.65)

where

z(n):= p−2

j=0

A(p−1)···A(j+1)y(np+j)+y(np+p−1). (1.66)

The following statements hold: (1) sequence{z(n)}is bounded;

(2) for every bounded sequence{z(n)}there exists a bounded sequence{y(n)}

such that equality (1.66) holds;

(3) the uniqueness of the solution (1.62) is equivalent to that of solution of (1.65). Therefore, if (1.62) has a unique bounded solution{x(n)}for every bounded se-quence{y(n)}, then the equation

u(n+1)=Du(n)+z(n), n∈Z (1.67)

has a unique bounded solution{u(n)}for every bounded sequence{z(n)}. By use of Theorem 1.3, we have (1.64).

(ii) implies (i). Given a bounded sequence{y(n)}, we define the bounded sequence

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bounded solution{u(n)}. Sequence{x(n)}, defined by

x(np):=u(n);

x(np+1):=A(0)u(n)+y(np);

x(np+k):=A(k−1)···A(0)u(n)

+ k−2

j=0

A(k−1)···A(j+1)y(np+j)+y(np+k−1), 2≤k≤p−1, nZ,

(1.68)

is bounded. This sequence{x(n)}is a solution of (1.62) since

A(np)x(np)=A(0)x(np)

=x(np+1)−y(np);

A(np+1)x(np+1)=A(1)x(np+1)

=x(np+2)−y(np+1); A(np+k)x(np+k)=A(k)x(np+k)

=x(np+k+1)−y(np+k), 1≤k≤p−1.

(1.69)

The uniqueness is immediate.

Remark1.13. Condition (1.64) is equivalent, for everyk∈ {1,2, . . . , p1}, to the following equality:

σA(k−1)···A(0)A(p−1)A(p2)···A(k)∩S= ∅. (1.70)

This follows from the relation

σ (AC)\{0} =σ (CA)\{0} (1.71)

which holds for every{A, C} ⊂(B). Indeed, ifz≠0 and operatorAC−zI has an inverseE, then operatorz−1(CEA−I)is inverse toCA−zI.

Let a functionωsatisfyCondition 1.6and{ak:k∈Z}be its Laurent series coeffi-cients. We now consider the equation

k∈Z

akx(n+k)=A(n)x(n)+y(n), n∈Z (1.72)

for a bounded sequence{y(n)}inB. Equation (1.72) can be rewritten as an equa-tion fromTheorem 1.12by the following way. LetBpbe a Banach space of column-vectorsuwith coordinatesu(1), u(2), . . . , u(p)fromBand with norm

u:= max 1≤k≤p

u(k). (1.73)

Denote by

x(n):=x(np), x(np+1), . . . , x(np+p−1),

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where “” means the transition to column-vector. Let

A:=          

A(0) Θ Θ ··· Θ

Θ A(1) Θ ··· Θ

Θ Θ A(2) ··· Θ

..

. ... ... . .. ... Θ Θ Θ ··· A(p−1)

          ,

U (j):=

         

Iajp Iajp+1 Iajp+2 ··· Iajp+p−1 Iajp−1 Iajp Iajp+1 ··· Iajp+p−2 Iajp−2 Iajp−1 Iajp ··· Iajp+p−3

..

. ... ... . .. ...

Iajp−p+1 Iajp−p+2 Iajp−p+3 ··· Iajp           (1.75)

forj∈Z. Then{A, U (j)} ⊂(Bp). It is a direct matter to verify that the problem of existence of a bounded solution for (1.72) inBis equivalent to the problem of existence of a bounded solution for the equation

j∈Z

U (j) x(n+j)=Ax(n) +y(n), n∈Z (1.76)

inBp.

Lemma1.14. IfCondition 1.6for functionωholds, then

lim sup n→−∞

U (n)1/|n|<1, lim sup n→∞

U (n)1/n<1. (1.77)

Proof. This result can be established by direct calculation.

Put

Φj(z):=I

n∈Z

anp+jzn, 0≤j≤p−1. (1.78)

Lemma1.15. Givenzof some neighbourhood ofS, the following equalities hold

Ω(z)= n∈Z

U (n)zn=

         

Φ0(z) Φ1(z) Φ2(z) ··· Φp−1(z) zΦp−1(z) Φ0(z) Φ1(z) ··· Φp−2(z) zΦp−2(z) zΦp−1(z) Φ0(z) ··· Φp−3(z)

..

. ... ... . .. ...

zΦ1(z) zΦ2(z) zΦ3(z) ··· Φ0(z)

          . (1.79)

Proof. This is immediate.

Lemma1.16. Equation (1.72) has a unique bounded solution for every bounded

se-quence{y(n)}if and only if the operator

A(z) (1.80)

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Proof. This is a direct consequence ofTheorem 1.12and Lemmas1.14and1.15.

Lemma 1.16is not the final result for (1.72), since the analysis of spectrum of op-eratorA(z)is not simple even for particular cases.

Exercise. Characterize the bounded operatorsA,Csuch that the equation

x(n+1)=(A+Csinn)x(n)+y(n), n∈Z (1.81)

has a unique bounded solution{x(n):n∈Z}for every bounded sequence{y(n): n∈Z}.

1.6. Nonlinear equation. We now consider an example of nonlinear difference equa-tion which satisfied the Lipschitz condiequa-tion nonlinearity. This equaequa-tion is connected with (1.27).

Theorem 1.17. LetA be a closed linear operator with dense domainD(A) in B and a functionωsatisfyingCondition 1.6with Laurent series coefficients{ak}. Let the sequence{c(n)} ∈corresponds to the operatorAand the functionωinTheorem 1.8. Iff:B→Bsuch that

f (u)f (v)Luv, {u, v} ⊂B (1.82)

withL >0, and

Lmax

n∈Z

c(n), n∈Z

Ac(n)

<1, σ (A)∩ω(S)= ∅, (1.83)

then the equation

k∈Z

akx(n+k)=Ax(n)+fx(n)+y(n), n∈Z (1.84)

has a unique bounded solution for every bounded sequence{y(n)}.

Proof. Let{y(n)}be a bounded sequence inB. Consider the following successive approximations:

x0(n)=¯0, n∈Z, (1.85)

and form≥0

k∈Z

akxm+1(n+k)=Axm+1(n)+fxm(n)+y(n), n∈Z. (1.86)

ByTheorem 1.8, for everym≥0, (1.86) has a unique bounded solution{xm(n)} ⊂ D(A)and

xm+1(n)=

k∈Z

c(k−n)fxm(k)

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By induction, it is easy to see that, form≥1,

xm+1(n)−xm(n)≤Lq1xm−xm−1≤Lq1mf¯0+y∞, (1.88)

where

q1:= k∈Z

c(k). (1.89)

Therefore, for everyn∈Z, there is an elementx(n)∈Bsuch that

xm(n) →x(n), n → ∞ (1.90)

in norm topology. By the closedness ofA, we have

Axm+1(n)=

k∈Z

Ac(k−n)fxm(k)+y(k), n∈Z, m≥1. (1.91)

This implies the following:

Axm+1(n)Axm(n)

Lq2mf (¯0)+y∞, m≥1 (1.92)

for everyn∈Zand

q2:= k∈Z

Ac(k). (1.93)

Hence, for everyn∈Z, there is an elementu(n)∈Bsuch that

Axm(n) →u(n), m → ∞. (1.94)

Then the closedness ofAimplies that

∀n∈Z:x(n)∈D(A), Ax(n)=u(n). (1.95)

It is clear that the sequence{x(n)}is bounded. We can apply (1.87) and (1.91) for m→ ∞to obtain

x(n)= k∈Z

c(k−n)fx(k)+y(k), n∈Z;

Ax(n)= k∈Z

Ac(k−n)fx(k)+y(k), n∈Z. (1.96)

By (1.44), we have (1.86).

1.7. Nonlinear operator equation

Definition1.18. A sequenceY:= {Y (n):nZ} ⊂(B)isboundedif

Y∞:=sup n∈Z

(14)

Let Abe a densely defined operator in B with domain D(A). In this section, we consider one example of the essentially nonlinear difference equation

AX(n)=X(n)X(n+1)−X(n−1)+Y (n), n∈Z (1.98)

for an operator sequence{X(n)}and the given bounded sequence of operators{Y (n)}.

Definition1.19. A sequenceX:={X(n)}of operators is called asolutionof (1.98), if

(i) for alln∈Z,X(n):B→D(A),AX(n)∈(B); (ii) equation (1.98) holds.

Theorem 1.20(see [19]). LetAbe a densely defined operator in B with domain D(A). If there is an operatorW∈(B)such that

(i) W:B→D(A),AW=IonB;

(ii) for a sequenceY, we have the inequality

8W2Y

∞≤1. (1.99)

Then (1.98) has a bounded solution.

Proof. LetYbe a bounded sequence which satisfies (1.99). We consider a sequence {Xm:m≥0}, forn∈Z, given by

X0(n):=W Y (n),

Xm(n):= m−1

k=0

W Xm−k−1(n)

Xk(n+1)−Xk(n−1)

, m≥1. (1.100)

It can be easily checked that

Xm2W m−1

k=0

Xm−k−1Xk, m≥1; X0

∞≤ W·Y∞.

(1.101)

By induction and by the following identity forn≥1 n

k=0

C2nn−−k2kC2kk 1 k+1=

1 2C

n+1

2(n+1), (1.102) (see, e.g., [37, Chapter 3, Problems, 11.(a)]) we have

Xm 1 m+1C

m

2m2mW2m+1Ym∞+1, m≥0. (1.103) Forn≥2, the number

1 nC

n−1

2n−2 (1.104)

is called thenth Catalan number. Then, by (1.99) and (1.103), the series

X(n):=

k=0

(15)

converges in the operator norm for everyn∈Z, since

Cm 2m

22m

π m, m → ∞. (1.106)

By definition, for everym≥0,n∈Z,N≥1, andz∈B,

Xm(n)z∈D(A), N

k=0

Xm(n)z∈D(A),

A

 N

m=0

Xm(n)z

=Y (n)z+ N

m=1 m1

k=0

Xm−k−1(n)Xk(n+1)−Xk(n−1)z.

(1.107)

Relation (1.103) implies also the convergence of the series

Y (n)z+

m=1

 m−1

k=0

Xm−k−1(n)

Xk(n+1)−Xk(n−1) 

z (1.108)

in normB,n∈Z. Thus by the closedness ofA, for everyn∈Z, X(n)z∈D(A),

AX(n)=Y (n)z+

m=1

 m−1

k=0

Xm−k−1(n)Xk(n+1)−Xk(n−1) 

z. (1.109)

Hence

AX(n)=X(n)X(n+1)−X(n−1)+Y (n), n∈Z. (1.110)

Remarks. (3) The proof ofTheorem 1.20contains the successive approximations for the solution of (1.98).

(4) IfY∞>0, then the sequence{X(n)=Θ:n∈Z}is not a solution of (1.98).

1.8. Stability of solutions under perturbation of operator coefficients

Theorem1.21(see [11,12]). Let the operators

A;Am(n), n∈Z, m≥1

(B) (1.111)

satisfy the following conditions:

(i) σ (A)∩S= ∅;

(ii) δm:=sup{Am(n)−A |n∈Z} →0,m→ ∞. Let{y(n)}be a bounded sequence inB.

Then the equation

x(n+1)=Ax(n)+y(n), n∈Z, (1.112)

and, for everymgreater than somem0∈N, the equation

xm(n+1)=Am(n)xm(n)+y(n), n∈Z (1.113)

have bounded solutions{x(n)}and{xm(n)}, respectively, and xxm

(16)

Proof. Equation (1.112) has a unique bounded solution byTheorem 1.3. Letm0∈ Nsuch thatLδm<1 form≥m0, see the proof ofTheorem 1.3. To prove the existence of the solution of (1.113) for m≥m0, we construct byTheorem 1.3the following successive approximations:

x0

m(n):=¯0, n∈Z;

xjm+1(n+1)=Axmj+1(n)+Am(n)−Axmj(n)+y(n), n∈Z, j≥0.

(1.115)

It is easy to verify that

j

m≤Lδmjm−1, j≥1, (1.116)

where∆jm := xmj+1−xmj∞, j≥1. Therefore, for everyn∈Z, there is an element xm(n)∈Bsuch that inB-norm,

xmj(n)→xm(n), j → ∞; xm<+∞. (1.117)

Taking the limit injin both sides of equality (1.115), we conclude thatxmis a solution of (1.113). By (1.112) and (1.113), it follows that

x−xm≤Lδmxm, xxm

∞≤ Lδm 1−Lδm

y∞, m≥1.

(1.118)

Theorem1.22(see [12]). Let operators from(B)

A(n), Am(n), n∈Z, m1, (1.119)

satisfy the following conditions for fixedp∈N:

(i) for alln∈Z,A(n+p)=A(n);

(ii) σ (A(p−1)A(p2)···A(1)A(0))∩S= ∅;

(iii) sup(Am(n)−A(n) |n∈Z)→0,m→ ∞.

Let{y(n)}be ap-periodic sequence inB. Then the equation

x(n+1)=A(n)x(n)+y(n), n∈Z (1.120)

and, for everymgreater than somem0∈N, the equation

xm(n+1)=Am(n)xm(n)+y(n), n∈Z (1.121)

have a unique p-periodic solution {x(n)}and a unique bounded solution {xm(n)}, respectively; and

xxm

0, m → ∞. (1.122)

(17)

1.9. Boundary value problem for difference equation. We return to (1.5) and con-sider the following boundary value problem. Let the integer numbers{n1, n2} ⊂Z, n1< n2be given. The boundary value problem to (1.5) is the following. Leta1,a2, and

{y(n1), y(n1+1), . . . , y(n21)}be arbitrary elements fromB. It is possible to find elements{x(n1), x(n1+1), . . . , x(n2)} ⊂Bsuch that

x(n+1)=Ax(n)+y(n), n1≤n≤n2−1;

xn1=a1, xn2=a2. (1.123)

However, it is easy to see that this problem does not have a solution in general. We consider the correct boundary value problem to (1.5) with an operatorA satisfying condition (1.6). LetB−=P−B,B+=P+B, bothB−andB+are invariant underA.

Theorem1.23. LetAbe an operator satisfying condition (1.6). For arbitrarya1 B,a2∈B+, and{y(n1), y(n1+1), . . . , y(n21)} ⊂B, the following problem:

x(n+1)=Ax(n)+y(n), n1≤n≤n2−1;

P−xn1=a1, P+xn2=a2 (1.124)

has a unique solution{x(n1), x(n1+1), . . . , x(n2)}.

Proof. Ignoring a trivial case, we suppose thatσandσ+. LetA:=AP, A+:=AP+, andx−:=P−x, x+:=P+xforx∈B. If{x(n1), x(n1+1), . . . , x(n2)}is a solution of (1.124) in B, then {x−(n1), x−(n1+1), . . . , x−(n2)} is a solution of the problem

x−(n+1)=A−x−(n)+y−(n), n1≤n≤n2−1;

x−n1=a1, (1.125)

inB−; and{x+(n1), x+(n1+1), . . . , x+(n2)}is a solution of the problem

x+(n+1)=A+x+(n)+y+(n), n1≤n≤n2−1;

x+n2=a2, (1.126)

inB+. Rewrite (1.126) as

x+(n)=A−1+ x+(n+1)−A−1+ y+(n), n1≤n≤n2−1; x+n2=a2.

(1.127)

Problem (1.125) is an value problem and problem (1.127) is a reverse initial-value problem. It is trivial that

x(n)= n−n11

j=0

Ajy(n−j−1)+An−n1

a1, n > n1,

x+(n)= − 1

j=n−n2

Aj+y+(n−j−1), n < n2.

(1.128)

(18)

Writex(n;n1, n2, a1, a2):=x(n)for a solution of problem (1.124) withn1≤n≤n2.

Theorem1.24. LetAbe an operator satisfying condition (1.6) and let{y(n)}be a bounded sequence inB. For anyn∈Zandc >0,

lim n1→−∞,n2→+∞

sup a1≤c,a2≤c

x

n;n1, n2, a1, a2−z(n)=0, (1.129)

wherezis a unique bounded solution to (1.5).

Proof. This follows from the proofs of Theorems1.3and1.23.

2. 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 Sections1and2are the basis to study the stochastic difference equations with periodic disturbances or periodic structure. The definition of the second-order periodicC-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 pro-cesses 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 ran-dom sequences in a Banach space can be found in [3,31,34].

2.1. Periodic in distribution random sequences. Let(B, · )be a complex sepa-rable Banach space with the zero element ¯0 and letB∗be its dual space. LetᏮ(B)be the Borelσ-algebra onB. We refer to [27] for the basicB-valued random variables or random elements theory. All random elements which arise in the next sections are con-sidered on a complete probability space(,,P). Further, all equalities, inequalities, and so on, with the random elements or random variables mean those almost surely.

Definition2.1. The mapping

x:Z×→B (2.1)

is called aB-valued random process with discrete timeorrandom sequence inBif, for eachn∈Z, the mapping

(19)

is aB-valued random element. For each ω∈Ω, the sequence {x(n, ω):n∈Z}is calledtrajectoryof the random process:

x(n):n∈Z or x(n, ω):n∈Z. (2.3)

Let

BZ:=u(n):n∈Z| ∀n∈Z:u(n)∈B (2.4)

be the set of all sequences with elements fromB; and, for eachj∈Z, let

πju(n):n∈Z:=u(j). (2.5)

Theσ-algebra in the spaceBZis defined as

ᏮZ:=σ a

j∈Z

πj−1

(B)

. (2.6)

For every random process{x(n):n∈Z}there is a correspondent probability mea-sureµxonᏮZ, which is defined by the equality

µx(A):=P

ω|x(n, ω):n∈Z∈A, A∈ᏮZ. (2.7)

By the well-known Kolmogorov theorem, see, for instance, [17], the measureµx is uniquely restored by collection of the finite-dimensional distributions

µxI :I⊂Z, Iis finite set

, (2.8)

where for every givenI, the measureµI

xis a unique extension on

σ a

j∈I πj−1

(B)

, (2.9)

of the function

µxI

j∈I Aj

 :=P

j∈I

ω|x(j, ω)∈Aj

, Aj∈(B), j∈I. (2.10)

Define, for everyj∈Z, the mapθ(j):BZ→BZas

θ(j)x(n):n∈Z:=x(n+j):n∈Z. (2.11)

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

µθ(j)x (2.12)

be the measure which corresponds to the process

(20)

We have

µθ(j)x=µxθ(j)−1. (2.14) Let a numberp∈Nbe fixed.

Definition2.2. AB-valued random process{x(n):nZ}is calledperiodic of the periodpin distributionorp-periodic in distribution, if, for every

n∈N, m1, m2, . . . , mn

Z, A1, A2, . . . , An

(B), (2.15)

the function

Zn→P  n

j=1

ω|xmj+n, ω

∈Aj

(2.16)

is periodic onZof periodp.

Remarks. (1) A periodic of periodp=1 in distribution random processes is sta-tionary.

(2) A stationary random process is periodic of periodpin distribution for every p∈N.

Definition2.3. TheB-valued random processes

xj(n):n∈Z

, 1≤j≤m, (2.17)

are calledperiodically with period p connected orperiodically connected, if theBm -valued random process

x1(n), x2(n), . . . , xm(n)

:n∈Z (2.18)

is periodic of periodpin distribution.

We refer to [9, Chapter 2] for basic properties of periodic in distribution processes and list only several that we need later.

Lemma2.4. A process{x(n):n∈Z}is periodic in distribution of periodpif and only if

µxθ(p)−1=µx. (2.19)

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). SeeDefinition 2.2.

Lemma2.5. AB-valued process{x(n):n∈Z}is periodic of periodpin distribution if and only if theBp-valued process

x(n):=x(np), x(np+1), . . . , x(np+p−1):n∈Z (2.20)

is stationary.

Proof. We first show that if {x(n):n∈Z}is p-periodic in distribution, then process (2.20) is stationary. Let

n∈N, m1, . . . , mn

(21)

where

Aj=Aj0×···×Aj(p−1), Ajk∈(B), 1≤j≤n,0≤k≤p−1 (2.22) is given. By uniqueness of an extension of measure from semi-algebra to theσ-algebra generated by semi-algebra, (see, e.g., [22] andDefinition 2.2) we have

P  n

j=1

ω|xmj+1∈Aj  =P

 n

j=1 p−1

k=0

ω|xmj+1p+k∈Ajk  

=P  n

j=1 p1 k=0

ω|xmjp+k

∈Ajk

=P  n

j=1

ω|xmj∈Aj  .

(2.23)

Now, assume that (2.20) is stationary. Given any

n∈N, m1, . . . , mn⊂Z, A1, . . . , An⊂(B), (2.24)

putmj=(pj+1)p+kj, where{p1, . . . , pn} ⊂Zand 0≤kj≤p−1, 1≤j≤n. For the sets

Cjl:=   

Aj, l=kj; B, lkj;

(2.25)

where 1≤j≤n, 0≤l≤p−1, and

Cj:=Cj0×···×Cj(p−1), 0≤j≤n, (2.26) fromDefinition 2.2for (2.20), we have

P  n

j=1

ω|xmj+p∈Aj  =P

j=1

ω|xpj+1p+kj∈Aj  

=P  n

j=1

ω|xpj+1

∈Cj =P   j=1

ω|xpj∈Cj  

=P  n

j=1

ω|xmj

∈AJ

.

(2.27)

Theorem2.6. Let{x(n):n∈Z}be ap-periodic in distribution random process in B,B1a complex separable Banach space, andm∈N. Suppose that a map

(22)

satisfies the conditions

∀n∈Z ∀x∈Bm:g(n+p, x)=g(n, x); g(n,·)∈CBm, B1. (2.29)

Then, for everyn1, . . . , nmfromZ, theB1-valued random process

gn, xn+n1, xn+n2, . . . , xn+nm

:n∈Z (2.30)

isp-periodic in distribution.

Proof. Let

k∈N, t1, . . . , tk

Z, A1, . . . , Ak

B1 (2.31)

be given. We have, clearly,

Cj:=

x∈Bm|gt j+p, x

∈Aj

=x∈Bm|gt j, x

∈Aj

, 1≤j≤k. (2.32)

Define forj, 1≤j≤k,

xtj:=xtj+n1, xtj+n2, . . . , xtj+nm. (2.33)

To prove the theorem, we useDefinition 2.2to show that

P  k

j=1

ω|xtj+p

∈Cj

=P  k

j=1

ω|xtj

∈Cj

. (2.34)

By periodicity of the process{x(n):n∈Z}, equality (2.34) holds for the sets

Cj=Cj1×···×Cjm, Cjl∈(B),1≤j≤k, 1≤l≤m. (2.35) Therefore, (2.34) holds by the uniqueness of extension of measure.

Corollary2.7. Let{x(n):n∈Z}be ap-periodic in distribution random process inB.

(i) The process

x(n):n∈Z (2.36)

isp-periodic in distribution inR.

(ii) For anyg∈B∗, the process

g, x(n) :n∈Z (2.37)

isp-periodic in distribution inC.

(iii) If a functiona:Z→Bisp-periodic, then the process

a(n)+x(n):n∈Z (2.38)

(23)

(iv) If a functionA:Z(B, B1)isp-periodic, then the process

A(n)x(n):n∈Z (2.39)

isp-periodic in distribution inB1.

(v) If the processes{xj(n):n∈Z},j=1,2are periodically connected with periodp, then the process

x1(n)+x2(n):n∈Z (2.40)

isp-periodic in distribution inB.

Theorem2.8. Letm∈Nand let{x(n):n∈Z}bep-periodic in distribution inB. Suppose thatF∈C(Bm,C)and numbers{n1, . . . , n

m} ⊂Zsuch that

EFxn1, . . . , xnm<+∞. (2.41)

Then

∀j∈Z:EFxn1+jp, . . . , xnm+jp

=EFxn1, . . . , xnm

. (2.42)

Proof. We establish the result forj=1. For{n1, . . . , nm} ⊂Z,m∈N, letµ(n1, . . . , nm;·)be the distribution onᏮ(Bm)of the random element(x(n1), . . . , x(nm)). Note that

µn1, . . . , nm;A1×···×Am

=P  m

k=1

ω:xnk

∈Ak

, (2.43)

for every measurable parallelepiped:

A1×A2×···×Am, A1, . . . , Am⊂(B). (2.44)

Then, according toDefinition 2.2, we have

µn1, . . . , nm;A1×···×Am=µn1+p, . . . , nm+p;A1×···×Am. (2.45)

Therefore, the measuresµ(n1, . . . , nm;·)andµ(n1+p, . . . , nm+p)coincide, since the σ-algebraᏮ(Bm)is generated by measurable parallelepipeds. Then

EFxn1, . . . , xnm

= !

BmF

u1, . . . , um

dµn1, . . . , nm;·

= !

BmF

u1, . . . , um

dµn1+p, . . . , nm+p;·

=EFxn1+p, . . . , xnm+p

.

(2.46)

Corollary2.9. Let{x(n):nZ}bep-periodic in distribution inBand

Ex(k)α<+∞, 1≤k≤p, (2.47)

whereα >0. Then the function

Zn→Ex(n)α (2.48)

(24)

Theorem2.10. The following assertions are equivalent:

(i) a process{x(n):n∈Z}isp-periodic in distribution inB;

(ii) for allm∈N, for all{n1, . . . , nm} ⊂Z, for allF∈C(Bm),supBm|F|<+∞, the

function

Zn→EFxn1+n, . . . , xnm+n (2.49)

isp-periodic.

Proof. (i) implies (ii). This follows directly fromTheorem 2.8.

(ii) implies (i). It is enough to show that the measuresµ(n1, . . . , nm;·)andµ(n1+ p, . . . , nm+p)are equal for every m∈Nand {n1, . . . , nm} ⊂Z. Therefore, we may equivalently prove the equality of the corresponding characteristic functions, see [2]. However, byTheorem 2.8, it follows that for everyg∈(Bm),

ˆ

µn1, . . . , nm

(g):=EexpiReg,xn1, . . . , xnm =expiReg,xn1+p, . . . , xnm+p =: ˆµn1+p, . . . , nm+p(g).

(2.50)

Corollary2.11. A random process{x(n):nZ}inBisp-periodic in distribution, if and only if the functions

Zn→Ecos

 m

j=1

Regj, xnj+n  ,

Zn→Esin

 m

j=1

Regj, x

nj+n  

(2.51)

arep-periodic onZfor every

m∈N, n1, . . . , nm

Z, g1, . . . , gm

⊂B∗. (2.52)

Theorem2.12. Let{x(n):n∈Z}be ap-periodic in distribution process inBand let a functionf:BZ→B1beᏮZ(B1)be measurable.

Then the process

fx(n+k):k∈Z:n∈Z (2.53)

isp-periodic in distribution inB1.

Proof. Given

m∈N, n1, . . . , nm

Z, A1, . . . , Am

B1, (2.54)

define

(25)

It is obvious that

P  m

k=1

ω|fxnk+j:j∈Z∈Ak  

=P  m

k=1

ω|xnk+j

:j∈Z∈Ck

=P  m

k=1

ω|θnk

x(j):j∈Z∈Ck

=P  m

k=1

ω|x(j):j∈Z∈θ−nk

Ck

=P

ω|x(j):j∈Z m

k=1 θ−nk

Ck

=:µx  m

k=1

θ−nkCk  .

(2.56)

ByLemma 2.4, we have

µx  m

k=1

θ−nkCk  =µx

 m

k=1

θ−nk−pCk

; (2.57)

and, it follows from (2.56), that

P  m

k=1

ω|fxnk+j

:j∈Z∈Ak

=P  m

k=1

ω|fxnk+p+j

:j∈Z∈Ak

.

(2.58)

Example2.13. Let{x(n):n∈Z}be ap-periodic in distribution process such that

Ex(k)<+∞, 1≤k≤p. (2.59)

Assume that

A(k):k∈Z(B), k∈Z

A(k)<+∞. (2.60)

Then, for everyn∈Z, the series

y(n):= k∈Z

A(k)x(n−k) (2.61)

converges almost surely inB-norm and{y(n):n∈Z}isp-periodic in distribution.

Theorem2.14. Let{xt(n):n∈Z},t∈N, be a sequence ofp-periodic in distribution processes and let{x(n):n∈Z}be a process such that, for everyn∈Z,

∀ε >0 :Pxt(n)−x(n)≥ε

(26)

Then{x(n):n∈Z}isp-periodic in distribution inB.

Proof. Let

µtn1, . . . , nm;·, µn1, . . . , nm;· (2.63)

be the distributions of the random elements

xtn1, . . . , xtnm, xn1, . . . , xnm, (2.64)

respectively, form∈N,{n1, . . . , nm} ⊂Z. Then

µtn1, . . . , nm;· →µn1, . . . , nm;·, t → ∞ (2.65)

in distribution. Hence, for every

F∈CBm,C, sup

Bm |F|<+∞, (2.66)

we have

EFxt

n1, . . . , xt

nm

EFxn1, . . . , xnm

, t → ∞. (2.67)

By (2.67) and byTheorem 2.8,

EFxn1+p, . . . , xnm+p

=EFxn1, . . . , xnm

. (2.68)

Therefore, byTheorem 2.10{x(n):n∈Z}isp-periodic in distribution.

Example2.15. Let{x(n):n∈Z}be ap-periodic in distribution process inBsuch that

Ex(k)<+∞, 1≤k≤p. (2.69)

Let{ξk:k∈Z}be a stationary process inCsuch thatE|ξ0|<+∞. If the processes {x(n)}and{ξk}are independent, then the processes

y(n):= n

k=−∞

ξn−k

1+(n−k)2x(k), z(n):=

k∈Z

ξk

1+k2x(n−k); n∈Z, (2.70) arep-periodic in distribution.

2.2. 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(,,P)be a complete probability space, and let(H, (·,·)) be a complex separable Hilbert space with inner product (·,·) and corresponding norm·.

Letx(k)andx(n)beH-valued random variables such that

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Then the Bochner integrals

Ex(k)= !

Hx(k)dP, Ex(n)= !

Hx(n)dP (2.72)

are well defined. Define

r (k, n):=Ex(k)−Ex(k), x(n)−Ex(n). (2.73)

The function

bk,n(u, v):=Ex(k)−Ex(k), ux(n)−Ex(n), v, {u, v} ⊂H (2.74)

is a continuous bilinear mapping. By the Riesz theorem, there is a unique linear bound-ed operatorSk,nsuch that

bk,n(u, v)=u, Sk,nv, {u, v} ⊂H. (2.75)

The operatorSk,nis called jointcovarianceofx(k)andx(n). It is known (see, e.g., [28,39]) that

(i) Sk,n =Sn,k;

(ii) the operatorSn,nis selfadjoint nonnegative and nuclear; (iii) the following relations hold:

trSn,n=Ex(n)−Ex(n)2, trSk,n∗ Sk,n1/2<+∞, u, Sk,nv2

Sk,ku, u

Sn,nv, v

; {u, v} ⊂H;

(2.76)

(iv) for every orthonormal basis{ej:j≥1}, the equality

r (k, n)=

j=1

ej, Sk,nej

(2.77)

holds.

Definition2.16. Let{x(n):nZ}be anH-valued random process such that

∀n∈Z:Ex(n)2<+∞. (2.78)

The process{x(n):n∈Z}is calledweakly second-order (WSO)p-periodic, if (i) for alln∈Z,Ex(n+p)=Ex(n);

(ii) for all{k, n} ⊂Z,r (k+p, n+p)=r (k, n). A WSO 1-periodic process is calledWSO stationary.

Lemma2.17. If{x(n):n∈Z}is WSOp-periodic inH, then theHp-valued process

x(n):=x(np), x(np+1), . . . , x(np+p−1), n∈Z (2.79)

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Proof. Let {x(n): n∈Z} be a WSO p-periodic process in H. It can be easily checked thatEx(n) =Ex(0), n∈Zand

Ex(n) Ex(n), x(m)−Ex(m) = p−1

j=0

r (np+j, mp+j)

= p1 j=0

r(n−m)p+j, j.

(2.80)

Remark2.18. If p2 then the converse toLemma 2.17is not true. Indeed, let H=R2and {ξ

n :n∈Z}be a sequence of independent identically distributed real random variables withEξ0=0,Eξ2

0=1. Define

x(n):=x(2n), x(2n+1), n∈Z, (2.81)

where

x(2n):=ξn; x(2n+1):=ξn, n∈Z\{1,2}, x(3):=ξ2, x(5):=ξ1. (2.82)

Then the process{x(n) :n∈Z}is WSO stationary sinceEx(n) =(0,0)for alln∈Z, and

Ex(n), x(m)=Ex(2n)x(2m)+Ex(2n+1)x(2m+1)

=2δ(n−m), {n, m} ⊂Z. (2.83)

However, the process{x(n):n∈Z}is not WSO 2-periodic since

Ex(0)x(1)=1, Ex(4)x(5)=0. (2.84)

Definition2.19. Let{x(n):nZ}be a random process inHsuch that

Ex(n)2<+∞, n∈Z. (2.85) The process{x(n):n∈Z}is calledsecond-order (SO)p-periodic if

(i) for alln∈Z,Ex(n+p)=Ex(n); (ii) for all{k, n} ⊂Z,Sk+p,n+p=Sk,n.

An SO 1-periodic process is second-order stationary.

Remarks. (1) By (2.77), it follows that SOp-periodic process is WSOp-periodic. (2) ByTheorem 2.8, it follows that ap-periodic processxsuch thatEx(k)2<+∞, 1≤k≤p, is SOp-periodic.

Lemma2.20. Let{x(n):n∈Z}be a process inHsuch that

∀n∈Z:Ex(n)2<+∞. (2.86)

The process{x(n):n∈Z}is SOp-periodic if and only if theHp-valued process

x(n):=x(np), x(np+1), . . . , x(np+p−1), n∈Z (2.87)

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Proof. We first assume that{x(n):n∈Z}is SOp-periodic. Then from condition (i) ofDefinition 2.19, we haveEx(n) =Ex(0), n∈Z. Let ˜Sn,mbe the joint covariance of elementsx(n) andx(m). It can be easily checked that

˜ Sn,m=

Snp+j,mp+l p−1

j,l=0, {n, m} ⊂Z. (2.88) By condition (ii) ofDefinition 2.19, it follows that function ˜Sn,mis a function ofn− monly.

Now assume that{x(n) :n∈Z}is SO stationary inHp. Then the equality

Ex(n +1)=Ex(n), n∈Z (2.89)

implies (i) fromDefinition 2.19. Since ˜Sn,mis a function only ofn−m, forSnp+j,mp+l, we have

u, Snp+j,mp+lv=Ex(np+j), ux(mp+l), v=u, S˜n−m,0v, (2.90)

whereu=(uδj,k) k=p

k=0andv=(vδl,k) k=p k=0.

Theorem2.21. AnH-valued process{x(n):n∈Z}satisfying

Ex(n)2<+∞, n∈Z, (2.91)

is SOp-periodic, if and only if theC2-valued process

x(n), u,x(n), v:n∈Z (2.92)

is SOp-periodic for every{u, v} ⊂H.

Proof. First, suppose that{x(n):nZ}is SOp-periodic. It is obvious that

Ex(n+p), u,x(n+p), v=Ex(n), u,x(n), v, n∈Z. (2.93)

It can be verified that the joint covariance of vectors

x(k), u,x(k), v, x(n), u,x(n), v (2.94)

is the matrix

σp,q(k, n) 2

p,q=1, (2.95)

where

σ1,1=

u, Sk,nu

, σ1,2=

u, Sk,nv

=σ2¯,1, σ2,2=

v, Sk,nv

;

σp,q(k+p, n+p)=σp,q(k, n), p, q=1,2,{k, n} ⊂Z.

(2.96)

Now let process (2.92) be SOp-periodic. Then

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so that

∀n∈Z,∀u∈H:Ex(n+p)−Ex(n), u=0. (2.98)

Therefore,

∀n∈Z:Ex(n+p)=Ex(n). (2.99)

Then, using equality

σ1,2(k+p, n+p)=σ1,2(k, n), {u, v} ⊂H, (2.100) we see that

Sk+p,n+p=Sk,n, {k, n} ⊂Z. (2.101)

Corollary2.22. Let{x(n):nZ}be SOp-periodic process inHand

m∈N, A1, A2, . . . , Am

(H), n1, n2, . . . , nm

Z. (2.102)

Then the process inH

"

y(n):= m

j=1 Ajx

n+nj

:n∈Z

#

(2.103)

is SOp-periodic.

Proof. It is simple. The joint covariance of the elementsy(k)and y(n)is the operator

m

p,q=1

ApSk+np,n+nqA∗q. (2.104)

Theorem2.23. Let{xt(n):n∈Z},t≥1, be a sequence ofH-valued SOp-periodic processes and{x(n):n∈Z}be a process inHsuch that

∀n∈Z:Ext(n)−x(n) 2

0, t → ∞. (2.105)

Then the process{x(n):n∈Z}is SOp-periodic.

Proof. It is enough to prove that, for everynZ, Ext(n) Ex(n), t → ∞,

Ext(k)−Ext(k), u

,xt(n)−Ext(n), v

Ex(k)−Ex(k), ux(n)−Ex(n), v, t → ∞; {u, v} ⊂H.

(2.106)

From the Cauchy inequality, this follows in an obvious way.

Remark2.24. About the harmonic analysis of WSO stationary and WSO periodic

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3. Stationary and periodic solutions of difference equations. This section con-tains 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 pertur-bation of operator coefficients. The equations with unbounded operator coefficient and nonlinear equations are also considered.

3.1. Linear difference equations. A stochastic process with discrete time in a com-plex separable Banach spaceB by definition is a collection ofB-valued random ele-ments{x(n):n∈Z}defined on a complete probability space(,,P).

Two processes{x(n):n∈Z}and{y(n):n∈Z}are said to bestochastic equiva-lentif

∀n∈Z:Px(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.

Theorem3.1(see [7,9]). LetA∈(B). The following statements are equivalent: (i) the equation

x(n+1)=Ax(n)+y(n), n∈Z (3.2)

has a unique stationary solution{x(n):n∈Z}inB withEx(0)<+∞for every stationary process{y(n):n∈Z}inBwithEy(0)<+∞;

(ii) σ (A)∩S= ∅.

Proof. (i) implies (ii). LetzB,z0, andtS be fixed. Let θbe the random variable with the uniform distribution over[0,2π ]. The process

y(n):= −zeiθtn:n∈Z (3.3)

is stationary inB. Let{x(n):n∈Z}be a unique solution for (3.2) corresponds to (3.3). By Hahn-Banach theorem, there is an elementf∈B∗ such thatf , z =1. Then, by (3.2),

tne=f , Ax(n)x(n+1) , nZ, (3.4)

therefore theC-valued process

x(n), eiθtn:nZ (3.5)

is stationary. Hence the process

x(n)t−ne−iθ:n∈Z (3.6)

is also stationary, moreover,

References

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