Approximation of random processes in the space L2([0, T])
The estimation for distribution of the norms of strictly sub-Gaussian random processes in the space L2(T) is obtained. The approximation of some classes of strictly sub-Gaussian random processes with given accuracy and reliability is considered.
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irk-123456789-45132009-11-25T12:00:43Z Approximation of random processes in the space L2([0, T]) Kamenschykova, O. The estimation for distribution of the norms of strictly sub-Gaussian random processes in the space L2(T) is obtained. The approximation of some classes of strictly sub-Gaussian random processes with given accuracy and reliability is considered. 2007 Article Approximation of random processes in the space L2([0, T]) / O. Kamenschykova // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 64–68. — Бібліогр.: 3 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4513 en Інститут математики НАН України |
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The estimation for distribution of the norms of strictly sub-Gaussian random processes in the space L2(T) is obtained. The approximation of some classes of strictly sub-Gaussian random processes with given accuracy and reliability is considered. |
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Article |
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Kamenschykova, O. |
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Kamenschykova, O. Approximation of random processes in the space L2([0, T]) |
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Kamenschykova, O. |
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Kamenschykova, O. |
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Approximation of random processes in the space L2([0, T]) |
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Approximation of random processes in the space L2([0, T]) |
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Approximation of random processes in the space L2([0, T]) |
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Approximation of random processes in the space L2([0, T]) |
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Approximation of random processes in the space L2([0, T]) |
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approximation of random processes in the space l2([0, t]) |
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Інститут математики НАН України |
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2007 |
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Approximation of random processes in the space L2([0, T]) / O. Kamenschykova // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 64–68. — Бібліогр.: 3 назв.— англ. |
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2025-07-02T07:44:25Z |
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Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.64–68
OLEXANDRA KAMENSCHYKOVA
APPROXIMATION OF RANDOM PROCESSES IN
THE SPACE L2(T )
The estimation for distribution of the norms of strictly sub-Gaussian
random processes in the space L2(T ) is obtained. The approximation
of some classes of strictly sub-Gaussian random processes with given
accuracy and reliability is considered.
1. Introduction
In the paper [3] we constructed the approximations of strictly ϕ-sub-
Gaussian random processes by broken lines such that this broken line ap-
proximates the process with given accuracy and reliability in the norm of
C[0, 1].
In this paper we consider the approximation of strictly sub-Gaussian
random processes by broken lines in the space L2(T ). We obtain the in-
equality for the norm of strictly sub-Gaussian random process and use it to
construct the approximation of the initial process.
We recall some basic facts about strictly sub-Gaussian random processes.
Let (Ω,B, P ) be a standard probability space.
Definition 1. [1] A random variable ξ is called sub-Gaussian (ξ ∈ Sub(Ω)),
if Eξ = 0 and ∃a > 0 such that E exp{λξ} ≤ exp
{
λ2a2
2
}
for all λ ∈ R1.
Proposition 1. [1] The space Sub(Ω) is a Banach space with respect to
the norm τϕ(ξ) = inf{a ≥ 0 : E exp (λξ) ≤ exp(ϕ(aλ)), λ ∈ R}.
Definition 2. [2] A random variable ξ is called strictly sub-Gaussian if
Eξ = 0 and Eξ2 = τ 2(ξ).
Definition 3. [2] A family Δ of sub-Gaussian random variables is called
strictly sub-Gaussian if for any finite or countable set Δ of random variables
2000 Mathematics Subject Classifications. 60E15
Key words and phrases. Approximation, SSub(Ω) processes,broken lines
64
APPROXIMATION OF RANDOM PROCESSES IN L2(T ) 65
{ξi, i ∈ I} and for all λi ∈ R : τ 2
(∑
i∈I
λiξi
)
= E
(∑
i∈I
λiξi
)2
.
Definition 4. [2] A vector
−→
ξ
T
= (ξ1, ..., ξn), where ξk are random vari-
ables from the family of strictly sub-Gaussian random variables, is called a
strictly sub-Gaussian random vector.
Definition 5. [2] A random process X = {X(t), t ∈ T} is called a
strictly sub-Gaussian (X(t) ∈ SSub(Ω)) if a family of random variables
{X(t), t ∈ T} is strictly sub-Gaussian.
Let X = {X(t), t ∈ T}, T = [0, 1], be a strictly sub-Gaussian process.
Denote by S := {tk}k=N
k=0 = { k
N
, k = 0, N} the uniform partition of the
segment [0, 1] into N parts. We approximate the random process {X(t), t ∈
T} by an interpolation broken line XN(t) for given values {X(tk)}, k = 0, N,
i.e.
XN (t) = α1X(tk) + α2X(tk+1), t ∈ [tk, tk+1], k = 0, N − 1,
where α1 = 1 − (t − tk)N, α2 = (t − tk)N.
The problem is to restore the process {X(t), t ∈ T} by the broken
line {XN(t), t ∈ T} with given accuracy ε and reliability 1 − δ in the
norm of L2(T ) knowing the values of given process in corresponding points
{k/N, k = 0, N}.
Denote by YN(t) := X(t)−XN(t), t ∈ T, the deviation random process.
We assume that for given process {X(t), t ∈ T} the next inequality is
satisfied:
sup
t∈T
E|X(t + h) − X(t)|2 ≤ b2(h), (1)
where b(h), h > 0 is a known monotonically increasing continuous function
and b(h) ↓ 0 as h ↓ 0.
As an example we consider power an logarithmic deviation functions
b(h).
2. Accuracy of approximation of strictly sub-Gaussian
processes in L2(T )
Definition 6. The broken line XN(t) approximates the process X(t) with
given accuracy ε > 0 and reliability 1 − δ, 0 < δ < 1 in L2(T ) if the next
inequality is satisfied:
P
⎧⎪⎨
⎪⎩
⎛
⎝∫
T
|X(t) − XN(t)|2 dt
⎞
⎠
1/2
> ε
⎫⎪⎬
⎪⎭ ≤ δ.
66 OLEXANDRA KAMENSCHYKOVA
Theorem. Let X = {X(t), t ∈ T} be a strictly sub-Gaussian random pro-
cess, (T,L, μ) be a measurable space. Assume
∫
T
(EX2(t))dμ(t) < ∞, then
with probability one there exists
∫
T
X2(t)dμ(t) and for any ε >
∫
T
(EX2(t))dμ(t)
the inequality holds
P
⎧⎨
⎩
∫
T
X2(t)dμ(t) > ε
⎫⎬
⎭ ≤
≤ e
1
2
⎛
⎝ ε∫
T
(EX2(t))dμ(t)
⎞
⎠
1
2
· exp
⎧⎨
⎩ −ε
2
∫
T
(EX2(t))dμ(t)
⎫⎬
⎭ . (2)
Proof. The existence of
∫
T
X2(t)dμ(t) follows from the Fubini‘s theorem.
Assume
−→
ξ
T
= (ξ1, ..., ξn) is a strictly sub-Gaussian random vector, A –
a symmetrical non-negatively defined matrix, η =
−→
ξ
T
A
−→
ξ , then for ε > Z1
the next inequality is satisfied (ex. 1.2.2, [2]):
P{η > ε} ≤ e
1
2
(
ε
Z1
) 1
2
exp
{
− ε
2Z1
}
, (3)
where Z1 = E
−→
ξ
T
A
−→
ξ . Let Λ = {ti}i=n
i=0 = {0 = t0 < ... < tn = 1} be a
partition of the segment T. Let ξi = X(ti), i = 1, n and let
A =
⎛
⎜⎜⎜⎝
√
Δt1 0 . . . 0
0
√
Δt2 . . . 0
...
...
. . .
...
0 0 . . .
√
Δtn
⎞
⎟⎟⎟⎠ .
Then the inequality (3) becomes
P
{
n∑
i=1
X2(ti)Δti > ε
}
≤
≤ e
1
2
⎛
⎜⎜⎝ ε
E
n∑
i=1
X2(ti)Δti
⎞
⎟⎟⎠
1
2
exp
⎧⎪⎪⎨
⎪⎪⎩− ε
2E
n∑
i=1
X2(ti)Δti
⎫⎪⎪⎬
⎪⎪⎭ ,
where ε > E
n∑
i=1
X2(ti)Δti.
APPROXIMATION OF RANDOM PROCESSES IN L2(T ) 67
In the last inequality we proceed to the limit in the mean square when
max1≤i≤n Δti → 0. As
∫
T
X2(t)dt = l.i.m.
n∑
i=1
X2(ti)Δti, we obtain (2). �
3. Some examples of approximation in L2(T )
As the process X = {X(t), t ∈ T} is a strictly sub-Gaussian, the pro-
cesses {XN(t), t ∈ T} and {YN(t), t ∈ T} are also strictly sub-Gaussian
([3]).
Let‘s apply the theorem above to the deviation process YN(t).
Assume the process {X(t), t ∈ T} is a stationary. The right side of
the expression in (2) increases on
∫
T
(EX2(t))dμ(t) (if
∫
T
(EX2(t))dμ(t) > ε)
so using the inequality supt∈T EY 2
N(t) ≤ b2( 1
N
) ([3]), we obtain the next
estimation:
P {‖YN(t)‖L2 > ε} ≤ e
1
2 ε
b( 1
N
)
· exp
{ −ε2
2b2( 1
N
)
}
,
where ε > b( 1
N
).
So the desired rate of interpolation N for approximation of stationary
strictly sub-Gaussian random process by the broken line with given accuracy
ε > 0 and reliability 1 − δ, 0 < δ < 1 in L2([0, 1]) can be found from the
inequalities {
e
1
2 ε
b( 1
N
)
· exp
{
−ε2
2b2( 1
N
)
}
≤ δ,
ε > b( 1
N
),
(4)
where b(h) is a deviation function of the process X(t).
Example 1. Power function b(h).
Assume in (1) b(h) = chα, 0 ≤ α ≤ 1, c is a positive constant.
Let ε = 0.01, δ = 0.01, c = 1, α = 1. Then the condition (4) is satisfied
for N ≥ 358.
Example 2. Logarithmic function b(h).
Assume in (1) b(h) = c
(ln(1+ 1
h
))μ , μ > 1
2
, c is a positive constant.
Let ε = 0.01, δ = 0.01, c = 1, μ = 4. Then we obtain that the condition
(4) is satisfied for N ≥ 1204.
References
1. Buldygin, V. V., Kozachenko, Y. V., Metric characterization of random
variables and random processes, American Mathematical Society, Provi-
dence, R I, (2000).
68 OLEXANDRA KAMENSCHYKOVA
2. Kozachenko, Y. V., Pashko, A. O., Simulation of random processes, Kyiv
University, Kyiv, (1999).
3. Kamenshykova, O., Linear interpolation of SSubϕ(Ω) processes, Bulletin
of the University of Uzhgorod: Mathematics and Informatics, 14, (2007),
39–49.
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: –@mechmat.univ.kiev.ua
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