Volume 04 Issue 11-2024
156
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
04
ISSUE
11
P
AGES
:
156-163
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
In the paper, we study the approximation of sub-
Gaussian random processes (r.p.’s) by Jackson
trigonometric
polynomials.
KEYWORDS
sub-Gaussian random process, modulus of continuity, trigonometric Jackson polynomial, approximation.
INTRODUCTION
A random function
℥(𝑡)
,
t
∊ 𝑇 ⊂
R
m
,
m ≥
1
is said to be pre-Gaussian [3], [5] if there exist constants k and K (0 <
k, K
<
ꝏ
) such that M
exp
{
𝑘℥(𝑡)
} ≤
K
.
Let a pre-Gaussian random function
℥(𝑡)
,
t
∊ 𝑇
be such that M
℥(𝑡)
= 0,
sup
t ∊ 𝑇
℥
2
(𝑡)
>0. Then the function
ϕ
(𝜆)
=
𝑚𝑎𝑥
|𝑥|=𝜆
𝑠𝑢𝑝
t ∊ 𝑇
𝑙𝑛𝑀𝑒𝑥𝑝{𝑥℥(𝑡)}
is defined, continuous, monotonically increasing, and convex on [0,
Λ
), for each
𝜆 ∊
[0,
Λ
),
there are left and right derivatives of the function
ϕ
(𝜆)
, where
Λ =
sup
{
𝜆
:
ϕ
(𝜆) <
ꝏ
} [5]. In [5], it was also shown
that the function
f
(
𝜆
) =
𝜑(𝜆)
𝜆
is monotonically increasing on [0,
Λ
),
lim
𝜆→
ꝏ
𝑓 (𝜆) = 𝐿 ,
0 <
L
≤
ꝏ
, the function
𝜌
(t,s) =
𝑠𝑢𝑝
𝑥≠0
|𝑥|
−1
χ
(
𝑙𝑛𝑀𝑒𝑥𝑝{𝑥[℥(𝑡) − ℥(𝑠)]})
is a semimetric on
𝑇
, where
χ
(
x
) is the inverse function to
ϕ
(𝜆)
. The metric
𝜌
is
called the natural metric of the function
℥(𝑡).
Research Article
APPROXIMATION IN A UNIFORM METRIC OF RANDOM PROCESSES BY
TRIGONOMETRIC JACKSON POLYNOMIALS
Submission Date:
November 11, 2024,
Accepted Date:
November 16, 2024,
Published Date:
November 26, 2024
Crossref doi:
https://doi.org/10.37547/ijmef/Volume04Issue11-15
Dr. Shamshiev Abdivali
Associate Professor Of The Department Of General Mathematics, Jizzakh State Pedagogical University,
Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ijmef
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 04 Issue 11-2024
157
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
04
ISSUE
11
P
AGES
:
156-163
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
Let (
𝑇, 𝜌
) be the topological space corresponding to the metric
𝜌
,
H
(
𝜀)
=
ln
(
𝜀
) is the
𝜀
-entropy oft he space (
𝑇, 𝜌
),
where
N
(
𝜀
) is the minimum possible number of points in the
𝜀
-network S(
𝜀
) of the space (
𝑇, 𝜌
).
Introduce the function
Ψ(𝜀)
=
∫ 𝐻(𝑥)[𝜒(𝐻(𝑥))]
𝜀
0
-1
dx
.
Theorem
D [5].
Let
℥(𝑡)
,
𝑡 ∊
𝑇
be a pre-Gaussian, separable with respect to some set separable on (
𝑇, 𝜌
), random
function,
L =
ꝏ
,
Ψ(𝜀)
<
ꝏ
. Then
℥(𝑡)
is bounded, continuous on (
𝑇, 𝜌
) with probability one, and for all
u
≥
𝑖𝑛𝑓
𝑝∊(0,1)
[
2
𝑝(1−𝑝)
Ψ(𝑝) +
1
1−𝑝
𝜑
′
(
𝜆(𝐻(𝑝)−0)
2(1−𝑝)
)]
, we have the estimate
P
{
𝑠𝑢𝑝
t ∊ 𝑇
℥(𝑡) ≥ 𝑢
}
≤ 𝑒𝑥𝑝
{
-
𝜑
∗
(𝑢 − Ψ
∗
(𝑢))
}
,
где
Ψ
∗
(𝑢)
=
𝑖𝑛𝑓
𝑝∊(0,1)
[
up
+
2
𝑝
Ψ(𝑝)
]
where
𝜑
∗
(𝑥)
=
𝑠𝑢𝑝
𝜆≥0
(
𝜆𝑥 −
𝜑(𝜆))
,
x
≥ 0 is the Young
-Fenchel transformation [6].
A random variable (r.v.)
℥
is said to be sub-Gaussian [10] if there is
a ≥
0
such that M
exp
{
℥𝜆
} ≤ {
𝑎
2
𝜆
2
2
} for all
𝜆 ∊
R
1
.
Denote
τ
(
℥
) =
inf
{
a ≥
0: M
exp
{
℥𝜆
} ≤ {
𝑎
2
𝜆
2
2
},
𝜆 ∊
R
1
}.
It is known [2] that a r.v.
℥
is sub-Gaussian if and only if M
℥
= 0 adn
τ
(
℥
) <
ꝏ
. It was also shown in [2] that
τ
(
℥
) =
sup
𝜆≠0
{
2𝑙𝑛M𝑒𝑥𝑝{℥𝜆}
𝜆
2
}
1
2
, and the space of all sub-
Gaussian r.v.’s
℥
with the norm
|| ℥ ||
𝑠𝑢𝑏
=
τ
(
℥
) is a Banach space.
A random function
℥(𝑡)
,
t
∊ 𝑇 ⊂
R
m
is said to be sub-Gaussian [2] if M
℥(𝑡)
= 0 and
sup
t ∊ 𝑇
τ (℥(𝑡))
<
ꝏ
.
Remark 1.
Any sub-Gaussian random function
℥(𝑡)
,
𝑡 ∊
𝑇
is pre- Gaussian, and for it,
𝜑(𝜆) =
τ
∙
𝜆
2
2
,
χ
(
x
) =
√
2𝑥
τ
,
L =
ꝏ
,
𝜑
∗
(
x
) =
𝑥
2
2τ
,
the natural metric
𝜌
(
t,s
) =
1
√τ
|| ℥(𝑡) − ℥(𝑠)||
𝑠𝑢𝑏
,
where
τ
=
sup
t ∊ 𝑇
|| ℥(𝑡)||
𝑠𝑢𝑏
.
Remark 2.
Any centered Gaussian random function
℥(𝑡)
is sub-Gaussian, and the norm
|| ℥(𝑡)||
𝑠𝑢𝑏
= {𝑀℥
2
(𝑡)}
1
2
.
Theorem D implies the following estimate, which we will use in the future.
Corollary D.
Let
℥
0
(𝑡)
,
𝑡 ∊ 𝑇
, be a sub-Gaussian, separable with respect to some separable on (
𝑇, 𝜌
0
) set, random
function, where
𝜌
0
(𝑡, 𝑠) =
1
√τ
||℥
0
(𝑡) − ℥
0
(𝑠)||
𝑠𝑢𝑏
,
𝑡, 𝑠 ∊
𝑇
,
τ
=
sup
t ∊ 𝑇
|| ℥(𝑡)||
𝑠𝑢𝑏
.
If 0 <
τ
≤ 1 и
Ψ(1)
<
ꝏ
, then, for all
u
≥ 16
Ψ(1)
,
P
{
𝑠𝑢𝑝
t ∊ 𝑇
℥
0
(𝑡) ≥ 𝑢
}
≤
exp
{
–
𝑢
2
−6𝑢
3
2
√Ψ(1)
2
}.
Proof of Corollary D.
According to Remark 1,
L =
ꝏ
, i.e., Theorem D is applicable for a sub-Gaussian random function
℥
0
(𝑡)
. Since
Volume 04 Issue 11-2024
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VOLUME
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AGES
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OCLC
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Publisher:
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𝑖𝑛𝑓
𝑝∊(0,1)
[
2
𝑝(1−𝑝)
Ψ(𝑝) +
1
1−𝑝
𝜑
′
(
𝜆(𝐻(𝑝))
2(1−𝑝)
− 0)]
=
𝑖𝑛𝑓
𝑝∊(0,1)
[
2
𝑝(1−𝑝)
Ψ(𝑝) +
√2
2
√τ𝐻(𝑝)
(1−𝑝)
2
)]
,
and
𝜑
∗
(
x
) =
𝑥
2
2τ
, then according to Theorem D,
P
{
𝑠𝑢𝑝
t ∊ 𝑇
℥
0
(𝑡) ≥ 𝑢
}
≤
exp
{-
𝑢
2
−2𝑢Ψ
∗
(𝑢) +[Ψ
∗
(𝑢)]
2
2τ
}
≤
exp
{-
𝑢
2
−2𝑢Ψ
∗
(𝑢) + [Ψ
∗
(𝑢)]
2
2
}
for all
u
≥
𝑖𝑛𝑓
𝑝∊(0,1)
[
2
𝑝(1−𝑝)
Ψ(𝑝) + +
√2
2
√τ𝐻(𝑝)
(1−𝑝)
2
)]
.
Obviously,
Ψ(𝑝) =
√2τ
2
∫ √𝐻(𝑥)
𝑝
0
dx
≥
𝑝√2τ𝐻(𝑝)
2
, i.e.
𝐻(𝑝) ≤ Ψ
𝟐
(𝑝)
𝟐
𝑝
2
τ
,
hence,
𝑖𝑛𝑓
𝑝∊(0,1)
[
2
𝑝(1−𝑝)
Ψ(𝑝) +
√2
2
√τ𝐻(𝑝)
(1−𝑝)
2
)] ≤ 𝑖𝑛𝑓
𝑝∊(0,1)
[
2Ψ(𝑝)
𝑝(1−𝑝)
+
Ψ(𝑝)
(1−𝑝)
2
)] ≤
16
Ψ (
1
2
) ≤
16
Ψ(1)
.
We obtain from here that, for all
u
≥ 16
Ψ(1)
,
P
{
𝑠𝑢𝑝
t ∊ 𝑇
℥
0
(𝑡) ≥ 𝑢
}
≤
exp
{
–
𝑢
2
−2𝑢Ψ
∗
(𝑢) +[Ψ
∗
(𝑢)]
2
2τ
},
If we take into account that
Ψ
∗
(𝑢)
≤ 6
√𝑢Ψ(1)
and
exp
{-
𝑢
2
−6𝑢
3
2
√Ψ(1)
2
}
≤ 1
as
u
≥ 16
Ψ(1)
, then we come to the
assertion of Corollary D.
Corollary D is proved.
MAIN RESULTS
Let us consider a sub-Gaussian separable, measurable separable
2𝜋
- periodic mean-square continuous real sub-
Gaussian r.p.
℥(𝑡),
𝑡
∊ 𝑅
1
. Assume that the following condition is satisfied for it
(
А
):
|| ℥(𝑡) − ℥(𝑠)||
𝑠𝑢𝑏
≤ 𝜔(|𝑡 − 𝑠|)
,
t, s
∊ 𝑅
1
,
where
𝜔(𝑧)
is the modulus of continuity, for which there exists the inverse function
𝜔
−1
(𝑥)
, and the integral
∫
𝜔(𝑧)
𝑧√|𝑙𝑛𝑧|
1
0
dz
<
ꝏ
.
It is known [11], that the r.p.
℥(𝑡)
is continuous with probability one.
We study the normalized process of deviations (n.p.d.)
𝜂
𝑛
(t) =
℥(𝑡)−𝐷
𝑛
(℥;𝑡)
С
0
𝜔(1 𝑛
⁄ )
, where
𝐷
𝑛
(℥; 𝑡)
is the Jackson operator
(trigonometric polynomial):
𝐷
𝑛
(℥; 𝑡) = 𝐷
𝑛
℥(𝑡) = ∫ ℥(𝑡 + 𝑥)
𝜋
−𝜋
𝐷
𝑛
(𝑥)𝑑𝑥 = 2𝜋 ∑
℥
𝑘
2𝑛−2
−(2𝑛−2)
𝜑
𝑘
(𝑛)
𝑒
𝑖𝑘𝑡
,
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AGES
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𝐷
𝑛
(𝑥) =
3
2𝜋(2𝑛
2
+1)𝑛
(
𝑠𝑖𝑛
𝑛𝑥
2
𝑠𝑖𝑛
𝑥
2
)
4
is the Jackson kernel,
℥
𝑘
and 𝜑
𝑘
(𝑛)
are the Fourier coefficients of
℥(𝑡)
and
𝐷
𝑛
(𝑥),
respectively,
С
0
=
𝜋√3
2
+ 1 is the Jackson constant [9, p.168]
Due to the
2𝜋
-periodicity of the n.p.d.
𝜂
𝑛
(
t
), it suffices to study it on the interval [
–
𝜋, 𝜋
].
Note that the n.p.d.
𝜂
𝑛
(
t
) was studied in [8] when
℥(𝑡)
is a stationary Gaussian r.p. and
𝜔(𝑥) = 𝑥
𝛼
,
0 <
𝛼
< 1.
Theorem 1.
If condition (A) is satisfied, then for
z
≥ 64
, the inequality
P
{
𝑚𝑎𝑥
|𝒕|≤𝜋
|
𝜂
𝑛
(𝑡)
𝛾
𝑛
| <
√𝟐
𝟐
𝑧} ≤ 2exp {−
𝑧
2
16
𝛾
𝑛
2
}
holds, where
𝛾
𝑛
= 2
√𝑙𝑛 𝑛
+
1
𝜔(1 𝑛
⁄ )
∫
𝜔(𝑥)
𝑥√|𝑙𝑛𝑥|
1
𝑛
0
dx +
√ln(𝜋 + 1)
.
Proof of Theorem 1.
We use Corollary D. To do this, we show that
𝜏
𝑛
=
|| 𝜂
𝑛
(𝑡)||
𝑠𝑢𝑏
≤ 1
for all
𝑛 ∊
N
.
Indeed, for any
𝑡 ∊
[
–
𝜋, 𝜋
] and
𝑛 ∊
N
, we have
𝜏
𝑛
=
|| 𝜂
𝑛
(𝑡)||
𝑠𝑢𝑏
=
1
С
0
𝜔(1 𝑛
⁄ )
||℥(𝑡) − 𝐷
𝑛
(℥; 𝑡) ||
𝑠𝑢𝑏
≤
≤
1
С
0
𝜔(1 𝑛
⁄ )
∫ ||℥(𝑡 + 𝑥) − ℥(𝑡)||
𝑠𝑢𝑏
𝐷
𝑛
(𝑥)
𝜋
−𝜋
𝑑𝑥 ≤
≤
1
С
0
𝜔(1 𝑛
⁄ )
∫ 𝜔(|𝑥|)𝐷
𝑛
(𝑥)
𝜋
−𝜋
𝑑𝑥 ≤
1.
(1)
The last inequality follows from the Jackson theorem ([9], p. 167).
Obviously, M
𝜂
𝑛
(𝑡) = 0
, hence, by virtue of (1), the n.p.d.
𝜂
𝑛
(𝑡)
is a sub-Gaussian r.p. for any
𝑛 ∊
N
.
Let
𝑛 ∊
N
be any fixed one. Suppose that
𝜏
𝑛
> 0
. (If
𝜏
𝑛
= 0
, then
𝜂
𝑛
(t) ≡ 0
with probability one, and for this case,
the assertion of Theorem 1 is obvious).
According to Remark 1, for
𝜂
𝑛
(t)
,
𝜑
𝑛
(𝑥)
=
𝜏
𝑛
𝑥
2
2
,
χ
(
x
) =
√
2𝑥
𝜏
𝑛
, therefore, the natural metric
𝜌
𝑛
(
t,s
) =
1
√𝜏
𝑛
||𝜂
𝑛
(𝑡) − 𝜂
𝑛
(𝑠)||
𝑠𝑢𝑏
. For
𝜌
𝑛
(
t,s
) ,
t, s
∊
[-
𝜋, 𝜋
], we have
𝜌
𝑛
(
t,s
)
=
1
С
0
√𝜏
𝑛
𝜔(1 𝑛
⁄ )
|| ∫ [℥(𝑡 + 𝑥) − ℥(𝑡) − ℥(𝑠 + 𝑥) + ℥(𝑠)]
𝜋
−𝜋
𝐷
𝑛
(𝑥)𝑑𝑥||
𝑠𝑢𝑏
≤
≤
2𝜔(|𝑡−𝑠|)
С
0
√𝜏
𝑛
𝜔(1 𝑛
⁄ )
≤
2𝜔(|𝑡−𝑠|)
√𝜏
𝑛
𝜔(1 𝑛
⁄ )
.
(2)
Using (2), we estimate the
𝜀
-entropy
𝐻
𝑛
(
𝜀
) of the space ([-
𝜋, 𝜋
],
𝜌
𝑛
).
Let
𝑁
𝑛
(
𝜀
) be the minimum possible number of points in the
𝜀
-network of the set [
–
𝜋, 𝜋
]. Then inequality (2) implies
that
𝑁
𝑛
(
𝜀
)
≤
𝑀
𝒏
(
𝜀
),
where
𝑀
𝒏
(
𝜀
) =
min
{
k
∊ 𝑁
:
2𝜔(𝜋 𝑘
⁄ )
√𝜏
𝑛
𝜔(1 𝑛
⁄ )
≤
𝜀
},
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AGES
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what implies
𝑀
𝒏
(
𝜀
)
≤
𝜋
𝜔
−𝟏
(𝜀
√𝜏𝑛
2𝜔(1 𝑛
⁄ )
)
+
1,
hence,
𝐻
𝑛
(
𝜀
) =
ln
𝑁
𝑛
(
𝜀
)
≤
ln
(
𝜋
+
𝜔
−𝟏
(
𝜀√𝜏
𝑛
2𝜔(1 𝑛
⁄ )
)
+
ln
1
𝜔
−𝟏
(
𝜀√𝜏𝑛
2
𝜔(1 𝑛
⁄ ))
,
where
𝜔
−1
(𝑥)
is the function inverse to
𝜔(𝑥)
.
Estimate
Ψ
𝑛
(1) =
∫ 𝐻
𝑛
(𝜀)[𝜒
𝑛
(𝐻
𝑛
(𝜀))]
1
0
-1
d
𝜀:
Ψ
𝑛
(1) =
√2𝜏
𝑛
2
∫ √𝐻
𝑛
(𝜀)
1
0
d
𝜀 ≤
≤
√2𝜏
𝑛
2
{
√𝑙𝑛[𝜋 + 𝜔
−1
(
√𝜏
𝑛
2
𝜔(1 𝑛
⁄ ))]
+
∫
√
|𝑙𝑛
1
𝜔
−𝟏
(
𝜀√𝜏𝑛
2
𝜔(1 𝑛
⁄ ))
|
1
0
d
𝜀
} =
=
√2𝜏
𝑛
2
{
√𝑙𝑛[𝜋 + 𝜔
−𝟏
(
√𝜏
𝑛
2
𝜔(1 𝑛
⁄ ))]
+
2
√𝜏
𝑛
∫
√|𝑙𝑛
1
𝜔
−𝟏
(𝑧𝜔(1 𝑛
⁄ )
|
√𝜏𝑛
2
0
d
z }.
Using (1), we obtain from here that
Ψ
𝑛
(1)
≤
√2
2
{
√𝑙𝑛(𝜋 + 1)
+
2 ∫ √|𝑙𝑛
1
𝜔
−𝟏
(𝑧𝜔(1 𝑛
⁄ )
|
1
0
d
z } =
=
√2
2
{
√𝑙𝑛(𝜋 + 1)
+
2√ln 𝑛 +
1
𝜔(1 𝑛
⁄ )
∫
𝜔(𝑧)
𝑧√|𝑙𝑛𝑧|
1
𝑛
0
d
z },
i.e.
Ψ
𝑛
(1)
≤
√2
2
𝛾
𝑛
<
ꝏ
for each
𝑛 ∊
N
.
(3)
Obviously, the n.p.d.
𝜂
𝑛
(t)
is continuous with probability one, therefore [4, p. 203] it is separable on ([
–
𝜋, 𝜋
],
𝜌
0
),
where
𝜌
0
=
|
t-s
|
. By virtue of (2), the metric
𝜌
𝑛
is topologically equivalent to the metric
𝜌
0
, therefore the n.p.d.
𝜂
𝑛
(𝑡)
is separable on ([
–
𝜋, 𝜋
],
𝜌
𝑛
). Hence, taking into account (1), (3) and applying Corollary D, we obtain that for all
u
≥
36
Ψ
𝑛
(1),
P
{
𝑚𝑎𝑥
|𝒕|≤𝜋
𝜂
𝑛
(𝑡) ≥ 𝑢
}
≤
exp
{
–
𝑢
2
−6𝑢
3
2
√ Ψ
𝑛
(1))
2
}.
From here, using (3), we arrive at the inequality
P
{
𝑚𝑎𝑥
|𝒕|≤𝜋
𝜂
𝑛
(t) ≥ 𝑢
}
≤
exp
{
-
𝑢
2
−6𝑢
3
2
𝛾
𝑛
√
√2
2
2
}
if
u
≥
18
𝛾
𝑛
√2
.
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Put
𝑢 =
√2
2
𝛾
𝑛
𝑣
, then for
𝑣 ≥ 36
,
P
{
𝑚𝑎𝑥
|𝒕|≤𝜋
𝜂
𝑛
(𝑡)
𝛾
𝑛
≥ 𝜔(𝑧) 𝑣
}
≤
exp
{(
√2
2
𝛾
𝑛
)
2
(
–
𝑣
2
2
+3
𝑣
3
2
)
}
.
If we assume that
𝑣 ≥ 64,
then
𝑣
2
–
6
𝑣
3
2
≥
𝑣
2
4
, therefore, for
𝑣 ≥ 64
, the following inequality holds:
P
{
𝑚𝑎𝑥
|𝒕|≤𝜋
𝜂
𝑛
(t)
𝛾
𝑛
≥
√2
2
𝑣
}
≤
exp
{
-
𝑣
2
16
𝛾
𝑛
2
)
}
.
Finally, the inequality
P
{
𝑠𝑢𝑝
|𝒕|≤𝜋
|
𝜂
𝑛
(𝑡)
𝛾
𝑛
| ≥
√2
2
𝑣
}
≤ 2
P
{
𝑠𝑢𝑝
|𝒕|≤𝜋
𝜂
𝑛
(t)
𝛾
𝑛
≥
√2
2
𝑣
}
implies the assertion of Theorem 1.
Theorem 1 is proved.
Corollary 1.
Let
Ɛ
> 0, 0 <
𝛿
< 1 and the conditions of Theorem 1 be satisfied.
If
𝜔(1 𝑛
⁄ ) 𝛾
𝑛
→
0 as
𝑛
→
ꝏ
, then, for all
𝑛 ≥
𝑛
0
+ 1,
P
{
𝑚𝑎𝑥
|𝒕|≤𝜋
|℥(𝑡) − 𝐷
𝑛
(℥; 𝑡)| < Ɛ
}
≥ 1–
𝛿
,
where
𝑛
0
=
𝑛
0
(𝑛
0
, 𝛿)
=
min
{
𝑛 ∊ 𝑁
:
𝐶
0
√2
𝜔(1 𝑛
⁄ ) (32𝛾
𝑛
+ 2√𝑙𝑛
2
𝑛
) ≤ Ɛ
}.
Proof of Corollary 1.
Put
𝑧
0
=
4
𝛾
𝑛
√𝑙𝑛
2
𝛿
.
Then, according to Theorem 1,
P
{
𝑚𝑎𝑥
|𝒕|≤𝜋
𝜂
𝑛
(𝑡)
𝛾
𝑛
≥
√2
2
(64 + 𝑧
0
)
}
≤ 2
exp
{
-
(64+𝑧
0
)
2
16
𝛾
𝑛
2
)
}
≤
2
exp
{
-
𝑧
0
2
16
𝛾
𝑛
2
)
},
i.e. P
{
𝑚𝑎𝑥
|𝒕|≤𝜋
|℥(𝑡) − 𝐷
𝑛
(℥; 𝑡)| ≥ 𝐶
0
√2 𝜔(1 𝑛
⁄ ) (32𝛾
𝑛
+ 2√𝑙𝑛
2
𝑛
)
}
≤
𝛿
,
which proves Corollary 1.
Corollary 1 is proved.
Let
℥
0
(𝑡) ∊
𝐶
Ω
2𝜋
(
𝑅
1
) be a Gaussian stationar r.p. with zero mean, unit variance and the continuous correlation
function
r
(
t
), satisfying the following condition [7], [8], [1]:
r
(
t
) = 1
–
|𝑡|
2𝛼
+ 𝑓(𝑡)
, 0 <
𝛼
≤
1,
𝑓(𝑡)
= o(
|𝑡|
2𝛼
)
, as
𝑡 → 0.
(4)
According to Remark 2,
||℥
0
(𝑡) − ℥
0
(𝑠) ||
𝑠𝑢𝑏
=
{M[℥
0
(𝑡) − ℥
0
(𝑠)]
2
}
1
2
=
{2[1 − 𝑟 (𝑡 − 𝑠)]}
1
2
,
Volume 04 Issue 11-2024
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AGES
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moreover, condition (4) implies that there exists a constant
𝐶
1
> 0
such that
{2[1 − 𝑟 (𝑡 − 𝑠)]}
1
2
≤
𝐶
1
|𝑡 − 𝑠|
𝛼
,
i.e. the r.p.
℥
0
(𝑡)
satisfies the condition of Theorem 1 with
𝜔(𝑥) = 𝐶
1
|𝑥|
𝛼
, 0 <
𝛼
≤ 1
,
0 <
𝐶
1
<
ꝏ
, and the condition of Corollary 1, hence, the following statement takes place.
Corollary 2.
There is a constant
𝐶
1
,
0 <
𝐶
1
<
ꝏ
, such that for any
𝑛 ≥
3,
0 <
𝛿
< 1, the inequality
P
{
𝑚𝑎𝑥
|𝒕|≤𝜋
|℥
0
(𝑡) − 𝐷
𝑛
(℥
0
; 𝑡)| ≥ 𝐶
0
𝐶
1
√2 [64 𝑛
−𝛼
√ln 𝑛 + 𝑛
−𝛼
(32√ln(𝜋 + 1) + 2√𝑙𝑛
2
𝛿
) +
𝟑𝟐𝑛
−𝛼
𝛼
Ɛ
𝒏
]
}
≤
𝛿
takes place, where
Ɛ
𝒏
~
𝟏
√𝒍𝒏 𝒏
.
Proof of Corollary 2.
The assertion of Corollary 2 follows from Theorem 1 if we take into account that
𝛾
𝑛
=
√𝑙𝑛(𝜋 + 1)
+
2√ln 𝑛 +
2𝑛
𝛼
𝜔(1 𝑛
⁄ )
∫
exp {−𝑢
2
}
ꝏ
√𝛼𝑙𝑛 𝑛
d
z
, когда
𝜔(𝑥) = 𝐶
1
|𝑥|
𝛼
.
For comparison, we present one result from [8]:
Let
𝑛 →
ꝏ
and
u
=
u
(
n
)
→
ꝏ
such that
n
= ]
𝜆
2𝜋𝜇
𝛼
(𝑢)
[ ,where
𝜆 ∊
(0,
ꝏ
),
𝜇
𝛼
(𝑢
) =
С
𝛼
𝑢
2−2𝛼
𝛼
𝑒
−
𝑢2
2
√2𝜋
,
С
𝛼
is a constant depending
only on
𝛼.
We denote such a coordinated change in the level of
u
and
n
by
(𝑛, 𝑢)
𝛼
→
ꝏ
.
In [8], it is proved that
lim
𝑛 →
ꝏ
𝜎
𝑛
𝑛
− 𝛼
=
𝑎
𝛼
and, moreover, if the correlation function of the r.p.
℥
0
(𝑡)
is such that
𝑟
′′
(𝑡)|𝑡|
2−𝛼
=
O(1),
t
→ 0,
then
lim
(𝑛,𝑢)
𝜆
→
ꝏ
𝑃{𝑚𝑎𝑥
|𝒕|≤𝜋
|℥
0
(𝑡) − 𝐷
𝑛
(℥
0
; 𝑡)| > 𝑢𝜎
𝑛
}
= 1
–
𝑒
−𝜆
,
where
𝜎
𝑛
2
=
{M[℥
0
(𝑡) − 𝐷
𝑛
(℥
0
; 𝑡)]
2
}
1
2
,
𝑎
𝛼
is a constant depending only on
𝛼
.
These results imply that
lim
𝑛→
ꝏ
𝑃{𝑚𝑎𝑥
|𝒕|≤𝜋
|℥
0
(𝑡) − 𝐷
𝑛
(℥
0
; 𝑡)| > 𝑛
− 𝛼
𝑏
𝛼
√𝑙𝑛 𝑛 +
1−𝛼
𝛼
+ 𝒇
𝛼,𝜆
(𝑛) }
= 1
–
𝑒
−𝜆
,
where
0 < 𝑏
𝛼
<
ꝏ
,
𝑓
𝛼,𝜆
(𝑛)
=
o (
𝑛
− 𝛼
√ln 𝑛
) ,
𝑛 →
ꝏ
.
(5)
Relation (5) and Corollary 2 show that, despite the generality of the considered class of r.p.’s, the estimate in
Theorem 1 in specific cases is close to unimprovable in the sense of order in
n.
REFERENCES
1.
Belyaev Yu. K., Simonyan A. Kh. Asymptotics of the number of deviations of a Gaussian process from an
approximating random curve. Abstracts of the II Vilnius Conference on Probability Theory and Mathematical
Statistics, 1977, Vol. I, Vilnius, p. 31-32 (in Russian)
Volume 04 Issue 11-2024
163
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
04
ISSUE
11
P
AGES
:
156-163
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
2.
Buldygin V.V., Kozachenko Yu.V. On sub-Gaussian random variables. Ukrainian Mathematical Journal, 32, 1980,
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Buldygin V.V., Kozachenko Yu.V. On local properties of realizations of some random processes and fields. Theory
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6.
Ioffe A.D., Tikhomirov V.M., Theory of Extremal Problems. Nauka, Moscow, 1974 (in Russian)
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Seleznev O.V., Approximation of periodic Gaussian processes by trigonometric polynomials, Dokl. Akad. Nauk
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Seleznev O.V., On the approximation of continuous periodic Gaussian processes by random trigonometric
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