Modeling of Stochastic Processes in Lp(T) Using Orthogonal Polynomials

Modeling of Stochastic Processes in

L

p

(

T

)

Using

Orthogonal Polynomials

Oleksandr Mokliachuk

National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kiev, Ukraine ∗_{Corresponding Author: [email protected]}

Copyright c⃝2014 Horizon Research Publishing All rights reserved.

Abstract

In this paper, models that approximate stochastic processes from the spaceSubφ(Ω) with given

reliability and accuracy in Lp(T) are considered for

some specific functions φ(t). For processes that are decomposited in series using orthonormal bases, such models are constructed in the case where elements of such decomposition cannot be found explicitly.

Keywords

Models of stochastic processes, φ -sub-Gaussian processes, orthogonal polynomials

1

Introduction

In different applications of the theory of stochastic processes it is vital to construct the model of the studied process. One way to construct the model of a stochastic process is to represent that process as the infinite series with respect to orthonormal polynomial basis and select the sum of firstNelements of this series to be the model. Consider stochastic process of the second orderX =

{X(t), t ∈ T}, EX(t) = 0 ∀t ∈ T, and let B(t, s) =

EX(t)X(s) be the correlation function of this stochastic processX. Then the next statement holds true.

Theorem 1 [1] (On decomposition of the stochastic process using an orthonormal basis) Let X(t), t ∈ T

be stochastic process of the second order, EX(t) = 0

∀t∈T, letB(t, s) =EX(t)X(s)be the correlation func-tion of X, let f(t, λ) be some function from L2(Λ, µ)

space, and let {gk(λ), k ∈ Z} be the orthonormal basis

in L2(Λ, µ) space. Then, correlation function B(t, s) is

represented in the form

B(t, s) = ∫

Λ

f(t, λ)f(s, λ)dµ(λ)

if and only if the process can be represented in the form

X(t) =

∞

∑

k=1

ak(t)ξk, (1)

where

ak(t) =

∫

Λ

f(t, λ)gk(λ)dµ(λ), (2)

ξk are centered uncorrelated random variables that

sat-isfy the conditions: Eξk = 0,Eξkξl=δkl,Eξk2= 1.

When constructing models of the processes, it is diffi-cult or impossible to find ak(t) explicitly. In that case,

we have to use approximations of these elements. Let us now introduce the model of such process.

Definition 1 Let stochastic process X ={X(t), t∈T}

allow decomposition (1). We will call stochastic process

XN ={XN(t), t∈T} model of the process X, if

XN(t) = N

∑

k=1

ξkˆak(t), (3)

whereˆak(t)are approximations of functionsak(t)in the

form (2),ξk are centered uncorrelated random variables,

Eξk= 0,Eξkξl=δkl,Eξ2k= 1.

We will consider in details the case where ξk are

in-dependent φ-sub-Gaussian random variables.

Since ˆak(t) are approximations of functionsak(t), they

will introduce some error in the model of stochastic pro-cess. The next theorem deals with that case.

Theorem 2 [2]

LetX ∈Subφ(Ω),X ={X(t), t∈[0, T]}be a

stochas-tic process,

φ(t) = {

t2 γ, t <1 tγ

γ, t≥1

,

where γ >2. Let

cN =

∫ T

0

(_N ∑

k=1

τ_φ2(ξk)δ2k(t)+

+

∞

∑

k=N+1

τ_φ2(ξk)a2k(t)

)p/2

dt <∞.

ModelXN approximates stochastic processX with given

reliability1−αand accuracyδ in the spaceLp(0, T), if

{

cN ≤δ/(βlnα2) p/β

cN < δ/pp(1−1/γ)

,

A system of orthonormal functions can be considered as basis{gk(λ)}. It is interesting to consider bases that

consist of sets of orthogonal polynomials. The classical examples of such polynomial sets are Chebyshev polyno-mials, Legendre polynopolyno-mials, Hermite polynopolyno-mials, La-guerre polynomials, Jacobi polynomials.

For a system of polynomials{Pn(x)}that are

orthogo-nal with the weight functionh(x) on some interval (a, b) and for some fixedx∈(a, b) the next series can be con-sidered:

GF(x, ω) =

∞

∑

n=0

Pn(x)

n! ω

n_.

Under some minimal conditions, this series has pos-itive convergence radius. In such a case, the function

GF(x, ω) is called generating function of the polynomial set{Pn(x)} [3].

If for some functional basis{gk(λ)}a generating

func-tion exists, namely, if for someω

GFg(x, ω) = ∞

∑

k=0

gk(λ)ωk,

then, under additional conditionτφ(ξk) =τ ωk, the next

equality holds true for the processX(t):

τφ(X(t)) =τφ

(_∞ ∑

k=0

ξk

∫ b

a

f(t, λ)gk(λ)dλ

) =

=τφ

(∫ b

a

f(t, λ) (_∞

∑

k=0

ξkgk(λ)

) dλ

)

≤

≤τ ∫ b

a

f(t, λ) (_∞

∑

k=0

ωkgk(λ)

) dλ=

=τ ∫ b

a

f(t, λ)GFg(λ, ω)dλ.

2

Modeling of stochastic

pro-cesses in

L

p

(0

, T

)

using the

Her-mite polynomials

Let X = {X(t), t ∈ [0, T]} ∈ Subφ(Ω) be stochastic

process of the second order,EX(t) = 0. Let the corre-lation function of the processX,B(t, s) =EX(t)X(s), be represented as

B(t, s) = ∫ _∞

−∞

f(t, λ)f(s, λ)dλ,

where f(t, λ), t ∈[0, T], λ∈R is a family of functions from L2(R). Since Hermite functions [4] form an

or-thonormal basis, the stochastic processX, according to the theorem 1, can be represented as

X(t) =

∞

∑

k=0

ξk

∫ _∞

−∞

f(t, λ) ˆHk(λ)dλ,

where ξk are centered uncorrelated random variables,

Eξk = 0, Eξkξl = δkl, Eξk2 = 1; ˆHk(λ) are Hermite

functions:

ˆ

Hk(λ) =

H_√k(λ)

k! 1

4 √

2πexp{− λ2

2 }, (4)

whereHk(λ) are the Hermite polynomials:

Hk(λ) = (−1)keλ

2_/2 dn

dλne −λ2/2_.

Theorem 3 Let a stochastic process X = {X(t), t ∈

[0, T]} belong to the space Subφ(Ω)with

φ(t) = {

t2 γ, t <1 tγ

γ, t≥1

forγ >2, let processX(t)can be represented in the form (1), and let the series {Hˆk(t)} of Hermite functions be

the basis. Let

cN =

∫ T

0

(∫ ∞

−∞

Zf2(t, λ)dλ ∞

∑

k=N+1

τ2 φ(ξk)

k2_{+ 3k+ 2}+

+

N

∑

k=1

τ_φ2(ξk)δk2(t)

)p/2

dt <∞,

Zf(t, λ) =

∂2f_∂λ(t, λ2 )−λ

∂f(t, λ)

∂λ +

λ2₋₂

4 f(t, λ) ,

where functionf(t, s)is twice differentiable and bounden with respect to the variable s, Zf(λ) is integrable on

R. The model XN(t), defined in (3), approximates the

stochastic process X(t) with given reliability 1−α and accuracyδ inLp(0, T)spaces, if

{

cN ≤δ/(βln2_α)p/β

cN < δ/pp(1−1/γ)

,

where1/γ+ 1/β= 1.

Proof. According to the theorem conditions,

ak(t) =

∫ _∞

−∞

f(t, λ) ˆHk(λ)dλ=

= 1

4 √

π ∫ _∞

−∞

f(t, λ)Hk(λ)e

−λ2/4

√

k! dλ.

Using properties of Hermite polynomials[4], we can show that

∂Hk(t)

∂t =kHk−1(t).

Using integration by parts, we get:

ak(t) =

1

4 √

π ∫ _∞

−∞

f(t, λ)Hk(λ)e

−λ2_/4 √

k! dλ=

= 1

4 √

π ∫ _∞

−∞

f(t, λ) e

−λ2/4

√

k+ 1√(k+ 1)!

∂Hk+1(λ)

∂λ dλ=

= 1

4 √

πf(t, λ)

e−λ2/4

√

k+ 1√(k+ 1)!Hk+1(λ)

λ=∞

λ=−∞

−

−√41_π

∫ _∞

−∞

∂(f(t, λ)e−λ2/4)

∂λ

Hk+1(λ)

√

Since Hk(λ) exp{−λ2/4} tends to zero asλ → ±∞,

andf(t, λ) is bounded, we get

ak(t) =−

1

4 √

π ∫ _∞

−∞

∂(f(t, λ)e−λ2/4₎

∂λ

Hk+1(λ)

√

k+ 1√(k+ 1)!dλ. Integration by parts one more time gives

ak(t) =−

1

4 √

π ∫ _∞

−∞

∂(f(t, λ)e−λ2/4₎

∂λ

Hk+1(λ)

√

k+ 1√(k+ 1)!dλ=

=−√₄1

π ∫ _∞

−∞

∂(f(t, λ)e−λ2/4)

∂λ ×

×√ 1

(k+ 1)(k+ 2)√(k+ 2)!

∂Hk+2(λ)

∂λ dλ=

=− √₄1

π

∂(f(t, λ)e−λ2/4)

∂λ ×

×√ 1

(k+ 1)(k+ 2)√(k+ 2)!Hk+1(λ)

λ=∞

λ=−∞

+

+√₄1

π ∫ _∞

−∞

∂2(f(t, λ)e−λ2/4)

∂λ2 ×

×_√ Hk+2(λ)

(k+ 1)(k+ 2)√(k+ 2)!dλ. It is certain that Hk+1(λ)∂(f(t, λ)e−λ

2_/4

)/∂λ tends to zero as λ → ±∞, since ∂(f(t, λ)e−λ2/4)/∂λ =

e−λ2/4(∂f(t, λ)/∂λ−λf(t, λ)),Hk+1(λ)e−λ

2_/4

→0, and

∂f(t, λ)/∂λ−λf(t, λ) is bounded, because f(t, λ) and

λf(t, λ) are bounded due to the conditions of the theo-rem. Then,

ak(t) =

1

4 √

π ∫ _∞

−∞

∂2(f(t, λ)e−λ2/4)

∂λ2 ×

×_√ Hk+2(λ)

(k+ 1)(k+ 2)√(k+ 2)!dλ. Because ˆHk is an orthonormal basis,

∫_∞

−∞Hˆk2(t)dt= 1.

That’s why 1

4 √

π ∫ _∞

−∞

∂2_{(f(t, λ)e}−λ2_/4

)

∂λ2 ×

×_√ Hk+2(λ)

(k+ 1)(k+ 2)√(k+ 2)!dλ=

= √ 1

(k+ 1)(k+ 2) ∫ ∞

−∞

∂2(f(t, λ)e−λ2/4)

∂λ2 ×

×Hk+2(λ)e−λ

2_/4

√

(k+ 2)!√4_2πe

λ2/4_dλ=

= √ 1

(k+ 1)(k+ 2) ∫ _∞

−∞

∂2_{(f(t, λ)e}−λ2/4₎

∂λ2 ×

×gk+2(λ)eλ

2_/4

dλ≤

≤√ 1

(k+ 1)(k+ 2) (∫ ∞

−∞

(

∂2(f(t, λ)e−λ2/4)

∂λ2 ×

×eλ2/4 )2

dλ )1

2(∫ ∞

−∞

g_k+22 (λ)dλ )1

2

=

=√ 1

(k+ 1)(k+ 2) (∫ _∞

−∞

(

∂2(f(t, λ)e−λ2/4)

∂λ2 ×

×eλ2/4 )2

dλ )1

2

Besides,

∂2(f(t, λ)e−λ2/4)

∂λ2 e

λ2/4₌ ∂

∂λ (

∂f(t, λ)

∂λ e

−λ2/4₋

−1

2e

−λ2/4_{λf(t, λ)}

)

eλ2/4= (

∂ ∂λ

(

∂f(t, λ)

∂λ e

−λ2/4

)

−

− ∂

∂λ (

1 2e

−λ2/4_{λf(t, λ)}

))

eλ2/4= (

∂2f(t, λ)

∂λ2 e −λ2/4₋

−1

2e

−λ2/4_λ∂f(t, λ)

∂λ −

1 2e

−λ2/4_λ∂f(t, λ)

∂λ −

−f(t, λ) ∂

∂λ (

1 2e

−λ2_/4

λ ))

eλ2/4= (

∂2_{f(t, λ)}

∂λ2 e −λ2_/4

−

−e−λ2/4λ∂f(t, λ)

∂λ −

−f(t, λ) (

1 2e

−λ2_/4 −1

4e

−λ2_/4

λ2 ))

eλ2/4=

= ∂

2_{f(t, λ)}

∂λ2 −λ

∂f(t, λ)

∂λ +

λ2₋₂

4 f(t, λ). Finally, we obtain

ak(t)≤

(

1 (k+ 1)(k+ 2)

∫ _∞

−∞

Z_f2(t, λ)dλ )1

2

,

where

Zf(t, λ) =

∂2f_∂λ(t, λ2 )−λ

∂f(t, λ)

∂λ +

λ2₋₂

4 f(t, λ) .

Therefore,

cN =

∫ T

0

(_N ∑

k=1

τ_φ2(ξk)δ2k(t)+

+

∞

∑

k=N+1

τ_φ2(ξk)a2k(t)

)p/2

≤

∫ T

0

(∫ ∞

−∞

Z_f2(t, λ)dλ

∞

∑

k=N+1

τ2 φ(ξk)

k2_{+ 3k+ 2} +

+

N

∑

k=1

τ_φ2(ξk)δk2(t)

)p/2

dt.

Finally, the statement of this theorem is derived from the theorem 2.

Theorem 4 Let stochastic process X = {X(t), t ∈

[0, T]} belong to the space Subφ(Ω),

φ(t) = {

t2

γ, t <1 tγ

γ, t≥1

for γ > 2, let the process X(t) allow representation in the form (1), and the seriesHˆk(t)of Hermite functions

is the basis. Letτφ(ξk) =τ ωk,|ω|<1, and

cN =

∫ T

0

( τ √

(1−ω2₎

(∫ _∞

−∞

f2(t, λ)dλ )1/2 − − N ∑ k=0

τ ωkˆak(t)

)p

dt <∞.

Model XN(t), provided in (3), approximates X(t)

with given reliability 1−αand accuracy δ in the space

Lp(0, T), if

{

cN ≤δ/(βlnα2) p/β

cN < δ/pp(1−1/γ)

,

where1/γ+ 1/β= 1.

Proof. According to the statement of the theorem,

ak(t) =

∫ _∞

−∞

f(t, λ) ˆHk(λ)dλ=

= √₄1

π ∫ _∞

−∞

f(t, λ)Hk(λ)e

−λ2_/2 √

k! dλ.

Using the properties of Hermite polynomials{Hk(λ)},

we get

GFH2(λ, ω) =

∞

∑

k=1

H2 k(λ)

k!2k ω

k _=√ 1

1−ω2exp

{ 2λ2_ω

1 +ω }

.

Moreover, such series converges for|ω|<1. Under the conditions of the theorem, τφ(ξk) = τ ωk. That’s why,

for the processX(t) the next condition holds true:

τφ(X(t)) =τφ

(_∞ ∑ k=1 ξk ∫ _∞ −∞

f(t, λ) ˆHk(λ)dλ

)

≤

≤τ ∫ _∞

−∞

f(t, λ)

∞

∑

k=0

ωkHˆk(λ)dλ≤

≤τ (∫ _∞

−∞

f2(t, λ)dλ )1/2 × ×  ∫ ∞ −∞ (_∞ ∑ k=0

Hk(λ) exp{−λ2/2}

√

k!2k√_π ω k )2 dλ   1/2 ≤ ≤τ (∫ _∞ −∞

f2(t, λ)dλ )1/2 × ×(∫ ∞ −∞ ∞ ∑ k=0 H2

k(λ) exp{−λ2}

k!2k√_π ω 2k_dλ )1/2 = =τ (∫ ∞ −∞

f2(t, λ)dλ )1/2 × × ( 1 √ π ∫ _∞ −∞

GFH2(λ, ω2) exp{−λ2}dλ

)1/2

=

=τ (∫ _∞

−∞

f2(t, λ)dλ )1/2 × × ( 1 √

π(1−ω4₎

∫ _∞

−∞

exp {

2λ2_ω

1 +ω }

exp{−λ2}dλ )1/2

=

=τ (∫ _∞

−∞

f2(t, λ)dλ )1/2 × × ( 1 √

π(1−ω4₎

∫ _∞

−∞

exp {

λ2ω

2₋₁

ω2_{+ 1}

} dλ )1/2 = =τ (∫ _∞ −∞

f2(t, λ)dλ )1/2 × × ( 1 √

π(1−ω4₎

√ π1 +ω

2

1−ω2

)1/2

=

=√ τ

(1−ω2₎

(∫ _∞

−∞

f2(t, λ)dλ )1/2

.

Given these considerations, the estimator of the model of the process will take the next form:

τφ(X(t)−XN(t)) =τφ

(_∞ ∑

k=0

ξkak(t)− N

∑

k=0

ξkˆak(t)

) =

=τφ

(_N ∑

k=0

ξkδk(t) + ∞

∑

k=N+1

ξkak(t)

) ≤ ≤ N ∑ k=0

τφ(ξk)δk(t) + ∞

∑

k=N+1

τφ(ξk)ak(t) =

=

∞

∑

k=0

τ ωkak(t)− N

∑

k=0

τ ωkaˆk(t)≤

≤ √ τ

(1−ω2₎

(∫ ∞

−∞

f2(t, λ)dλ )1/2

−

N

∑

k=0

τ ωkˆak(t).

3

Modeling

of

stochastic

pro-cesses

in

L

p

(0

, T

)

using

the

Chebyshev polynomials

Let the process X(t) has the same properties as the process from the previous section. Let orthonotmal Chebyshev polynomials be used as the basis:

ˆ

Tn(λ) =

√ 2

πTn(λ),

where

Tn(λ) = cos(narccosλ).

In such a case we can proof the next theorem.

Theorem 5 Let a stochastic process X = {X(t), t ∈

[0, T]} belong to the space Subφ(Ω) with

φ(t) = {

t2

γ, t <1 tγ

γ, t≥1

for γ > 2, let X(t) is representated in the form (1), and let the system of orthonormal Chebyshev polyno-mials {Tˆk(t)} be used as the basis. Let τφ(ξk) = τ ωk,

0< ω <1, and

cN =

∫ T

0

(√ 2

πτ (∫ 1

−1

f2(t, λ)dλ )1/2_√

DT(ω)−

−

N

∑

k=0

τ ωkaˆk(t)

)p

dt <∞,

DT(ω) = 2

1

ω(4 + 3ω2_+ω4₎

(

ω(5 + 5ω2+ 2ω2)+

+ (4 + 7ω2+ 4ω4+ω6) ln{(ω2−ω+ 2)/(ω2+ω+ 2)}).

Model XN(t), determined in (3), approximates the

process X(t) with given reliability 1−α and accuracy

δ in the spaceLp(0, T), if

{

cN ≤δ/(βln_α2)p/β

cN < δ/pp(1−1/γ)

,

where1/γ+ 1/β= 1.

Proof. Under the conditions of the theorem,

ak(t) =

∫ 1

−1

f(t, λ) ˆTn(λ)dλ=

∫ 1

−1

f(t, λ) √

2

πTn(λ)dλ.

The generating function of orthogonal Chebyshev polynomials of the first kind{Tk(λ)}has the next form:

GFT(λ, ω) = ∞

∑

k=0

Tk(λ)ωk =

1−ωλ

2−ωλ+ω2

for 0 < ω < 1. Under the conditions of the theorem,

τφ(ξk) =τ ωk. Thats why for the processX(t) the next

condition is true:

τφ(X(t)) =τφ

(_∞ ∑

k=1

ξk

∫ 1

−1

f(t, λ) ˆTk(λ)dλ

)

≤

≤τ ∫ 1

−1

f(t, λ)

∞

∑

k=0

ωkTˆk(λ)dλ≤

≤τ (∫ 1

−1

f2(t, λ)dλ )1/2

×

×

 ∫ 1

−1

(_∞ ∑

k=0

√ 2

πTk(λ) )2

dλ  

1/2

=

= √

2

πτ (∫ 1

−1

f2(t, λ)dλ )1/2

×

×(∫

1

−1

(

1−ωλ

2−ωλ+ω2

)2

dλ )1/2

.

Lets calculate the second integral of the last expres-sion separetely.

∫ 1

−1

(

1−ωλ

2−ωλ+ω2

)2

dλ=

= 2 1

ω(4 + 3ω2_+ω4₎

(

ω(5 + 5ω2+ 2ω2)+

+ (4 + 7ω2+ 4ω4+ω6) ln{(ω2−ω+ 2)/(ω2+ +ω+ 2)}) :=DT(ω).

Then the estimator of τφ(X(t)) will take the form:

τφ(X(t))≤

√ 2

πτ (∫ 1

−1

f2(t, λ)dλ )1/2_√

DT(ω).

Given these considerations, the estimator of the model of the process will take the following form:

τφ(X(t)−XN(t)) =τφ

(_∞ ∑

k=0

ξkak(t)− N

∑

k=0

ξkˆak(t)

) =

=τφ

(_N ∑

k=0

ξkδk(t) + ∞

∑

k=N+1

ξkak(t)

)

≤

≤

N

∑

k=0

τφ(ξk)δk(t) + ∞

∑

k=N+1

τφ(ξk)ak(t) = ∞

∑

k=0

τ ωkak(t)− N

∑

k=0

τ ωkˆak(t)≤

≤

√ 2

πτ (∫ 1

−1

f2(t, λ)dλ )1/2_√

DT(ω)− N

∑

k=0

τ ωkˆak(t).

We can also proof the similar theorem in the case of Chebyshev polynomials of the second kind:

Un(λ) =

sin((n+ 1) arccosλ)

√

1−λ2

ˆ

Un(λ) =

√ 2

πUn(λ).

Generating function of the series{Un(λ)}has the form

GFU(λ, ω) = ∞

∑

k=0

ωkUn(λ).

Let the processX(t) has the same properties as in the previous theorem Then, the next proposition holds true.

Theorem 6 Let stochastic process X = {X(t), t ∈

[0, T]} belong to the space Subφ(Ω),

φ(t) = {

t2 γ, t <1 tγ

γ, t≥1

forγ >2, let processX(t)can be represented in the form (1), and let series {Uˆk(t)} of orthonormal Chebyshev

polynomials be the basis. Let also τφ(ξk) = τ ωk, 0 <

ω <1, and

cN =

∫ T

0

( 2τ

√

π(ω2−₁₎

(∫ 1

−1

f2(t, λ)dλ )1/2

−

−

N

∑

k=0

τ ωkˆak(t)

)p

dt <∞.

Model XN(t), determined in (3), approximates the

process X(t) with given reliability 1−α and accuracy

δin the spaceLp(0, T), if

{

cN ≤δ/(βln_α2)p/β

cN < δ/pp(1−1/γ)

,

where1/γ+ 1/β= 1.

Proof. Under the conditions of the theorem, we have:

ak(t) =

∫ 1

−1

f(t, λ) ˆUn(λ)dλ=

∫ 1

−1

f(t, λ) √

2

πUn(λ)dλ.

Generation function of the series of orthogonal Cheby-shev polynomials of the second kind {Uk(λ)} has the

following form:

GFT(λ, ω) = ∞

∑

k=0

Uk(λ)ωk =

1 1−2ωλ+ω2

0 < ω < 1. As stated in the theorem’s conditions,

τφ(ξk) = τ ωk. That’s why for stochastic process X(t)

the next statement holds true:

τφ(X(t)) =τφ

(_∞ ∑

k=1

ξk

∫ 1

−1

f(t, λ) ˆUk(λ)dλ

)

≤

≤τ ∫ 1

−1

f(t, λ)

∞

∑

k=0

ωkUˆk(λ)dλ≤

≤τ (∫ 1

−1

f2(t, λ)dλ )1/2

×

×

 ∫ 1

−1

(_∞ ∑

k=0

√ 2

πUk(λ) )2

dλ  

1/2

=

= √

2

πτ (∫ 1

−1

f2(t, λ)dλ )1/2

×

×(∫ 1

−1

( 1 1−2ωλ+ω2

)2

dλ )1/2

=

= √

2

πτ (∫ 1

−1

f2(t, λ)dλ

)1/2 √

2 (ω2−₁₎ =

=√ 2τ

π(ω2−₁₎

(∫ 1

−1

f2(t, λ)dλ )1/2

.

Taking into account these considerations, the estima-tor of the model of the stochastic process will fit the next condition:

τφ(X(t)−XN(t)) =τφ

(_∞ ∑

k=0

ξkak(t)− N

∑

k=0

ξkˆak(t)

) =

=τφ

(_N ∑

k=0

ξkδk(t) + ∞

∑

k=N+1

ξkak(t)

)

≤

≤

N

∑

k=0

τφ(ξk)δk(t) + ∞

∑

k=N+1

τφ(ξk)ak(t) =

=

∞

∑

k=0

τ ωkak(t)− N

∑

k=0

τ ωkaˆk(t)≤

≤√ 2τ

π(ω2−₁₎

(∫ 1

−1

f2(t, λ)dλ )1/2

−

N

∑

k=0

τ ωkˆak(t).

Statement of this theorem follows from the theorem 2 and the last inequality.

4

Conclusions

Theorems are proved that allow to construct models of stochastic processes from Subφ(Ω) in the case where

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