Columbia International Publishing
Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 137-148
doi:10.7726/jams.2016.1011
Research Article
A New Wavelet-based Expansion of a Random Process
Ievgen Turchyn 1*
Received: 28 July 2016; Published online: 1 October 2016
© The author 2016. Published with open access at www.uscip.us
Abstract
There has been obtained an expansion of a second-order stochastic process into a system of wavelet-based
functions. This expansion is used for simulation of a sub-Gaussian stochastic process with given accuracy and
reliability.
Keywords: Wavelets; Sub-Gaussian stochastic processes; Expansion
1. Introduction
One noteworthy class of expansions of random processes is formed by wavelet-based expansions.
Wavelet-based expansions with uncorrelated terms are especially important since they are
convenient for simulation and approximation of random processes.
Different wavelet-based expansions of stochastic processes were studied by Walter and Zhang
(1994), Meyer et al. (1999), Pipiras (2004), Zhao et al. (2004), Didier and Pipiras (2008).
Kozachenko and Turchyn (2008) obtained a theorem about a wavelet-based expansion for a wide
class of stochastic process. Namely, a centered second-order process 𝑋(𝑡) with the correlation
function
𝑅(𝑡, 𝑠) = ∫ ‍𝑢(𝑡, 𝑦)𝑢(𝑠, 𝑦)𝑑𝑦
ℝ
(𝑢(𝑡,⋅) ∈ 𝐿2 (ℝ)) can be represented as the series
𝑋(𝑡) = ∑𝑘∈ℤ ‍𝜉0𝑘 𝑎0𝑘 (𝑡) + ∑∞
𝑗=0 ‍∑𝑘∈ℤ ‍𝜂𝑗𝑘 𝑏𝑗𝑘 (𝑡)
(1)
which converges in 𝐿2 (Ω) for any fixed 𝑡, where
______________________________________________________________________________________________________________________________
*Corresponding e-mail: [email protected]
1 Oles Honchar Dnipropetrovsk National University
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Ievgen Turchyn / Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 137-148
1
√2𝜋
∫ ‍𝑢(𝑡, 𝑦)𝜙̂0𝑘 (𝑦)𝑑𝑦,‍‍‍‍𝑏𝑗𝑘 (𝑡) =
1
∫ ‍𝑢(𝑡, 𝑦)𝜓̂𝑗𝑘 (𝑦)𝑑𝑦,
√2𝜋
ℝ
ℝ
̂
𝜙(𝑦) is an 𝑓-wavelet, 𝜓(𝑦) is the corresponding 𝑚-wavelet, 𝜙0𝑘 (𝑦) and 𝜓̂𝑗𝑘 (𝑦) are the Fourier
transforms of 𝜙0𝑘 (𝑦) = 𝜙(𝑦 − 𝑘) and 𝜓𝑗𝑘 (𝑦) = 2𝑗/2 𝜓(2𝑗 𝑦 − 𝑘) correspondingly, 𝜉0𝑘 and 𝜂𝑗𝑘 are
uncorrelated random variables.
𝑎0𝑘 (𝑡) =
The rate of convergence of this expansion in different functional spaces was studied by Turchyn
(2006), Kozachenko and Turchyn (2008), Turchyn (2011a), Turchyn (2011b). Conditions for
uniform convergence with probability 1 and in probability were obtained by Kozachenko and
Turchin (2009). A more general expansion of a random process (based on an arbitrary orthonormal
basis in 𝐿2 (𝐓)) was considered by Kozachenko et al. (2011).
We propose a new wavelet-type expansion of a second-order stochastic process based on a basis in
𝐿2 (ℝ2 ) which is built using wavelets. This expansion is similar to (1) but is more flexible since we
use two wavelets for this expansion instead of one wavelet in (1). We find the rate of convergence
of our expansion in 𝐿𝑝 ([0, 𝑇]) for a sub-Gaussian process in terms of simulation with given accuracy
and reliability. A model based on our expansion can be used for simulation of a Gaussian process.
2. Auxiliary Facts
Let 𝜙 ∈ 𝐿2 (ℝ) be such a function that the following assumptions hold:
i)
∑ ‍|𝜙̂(𝑦 + 2𝜋𝑘)|2 = 1‍‍‍‍a. e.,
𝑘∈ℤ
where 𝜙̂(𝑦) is the Fourier transform of 𝜙,
𝜙̂(𝑦) = ∫ ‍exp{−𝑖𝑦𝑥}𝜙(𝑥)𝑑𝑥;
ℝ
ii) there exists a function 𝑚0 ∈ 𝐿2 ([0,2𝜋]) such that 𝑚0 (𝑥) has period 2𝜋 and almost everywhere
𝑦
𝑦
𝜙̂(𝑦) = 𝑚0 ( ) 𝜙̂ ( ) ;
2
2
iii) 𝜙̂(0) ≠ 0 and the function 𝜙̂(𝑦) is continuos at 0.
The function 𝜙(𝑥) is called a 𝑓-wavelet. Let 𝜓(𝑥) be the inverse Fourier transform of the function
𝑦
𝑦
𝑦
𝜓̂(𝑦) = 𝑚0 ( + 𝜋) exp{−𝑖 }‍𝜙̂ ( ).
2
2
2
The function 𝜓(𝑥) is called a 𝑚-wavelet. Let
𝜙𝑗𝑘 (𝑥) = 2𝑗/2 𝜙(2𝑗 𝑥 − 𝑘), 𝜓𝑗𝑘 (𝑥) = 2𝑗/2 𝜓(2𝑗 𝑥 − 𝑘),‍‍‍‍𝑘 ∈ ℤ, 𝑗 = 0,1, …
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Ievgen Turchyn / Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 137-148
It is known that the family of functions {𝜙0𝑘 , 𝜓𝑗𝑘 , 𝑗 = 0,1, … ; 𝑘 ∈ ℤ} is an orthonormal basis in
𝐿2 (ℝ). Obviously the system {𝜙̂0𝑘 /√2𝜋, 𝜓̂𝑗𝑘 /√2𝜋, 𝑗 = 0,1, … ; 𝑘 ∈ ℤ} also is an orthonormal basis in
𝐿2 (ℝ) and therefore any function 𝑓 ∈ 𝐿2 (ℝ) can be represented as
𝑓(𝑥) =
1
√2𝜋
∑𝑘∈ℤ ‍𝑎0𝑘 𝜙̂0𝑘 (𝑥) +
1
√2𝜋
̂
∑∞
𝑗=0 ‍∑𝑘∈ℤ ‍𝑏𝑗𝑘 𝜓𝑗𝑘 (𝑥),
(2)
where
𝑎0𝑘 =
1
∫ ‍𝑓(𝑥)𝜙̂0𝑘 (𝑥)𝑑𝑥,
√2𝜋 ℝ
𝑏𝑗𝑘 =
1
∫ ‍𝑓(𝑥)𝜓̂𝑗𝑘 (𝑥)𝑑𝑥,
√2𝜋 ℝ
series (2) converges in the norm of the space 𝐿2 (ℝ).
More detailed information about wavelets is contained, for instance, in Härdle et al. (1998) or
Walter and Shen (2000).
Definition 2.1. A centered random variable 𝜉 is called sub-Gaussian if there exists such a constant
𝑎 ≥ 0 that
Eexp{𝜆𝜉} ≤ exp{𝜆2 𝑎2 /2},‍‍‍‍𝜆 ∈ ℝ.
The class of all sub-Gaussian random variables on a standard probability space {Ω, ℬ, 𝑃} is a Banach
space with respect to the norm
𝜏(𝜉) = inf{𝑎 ≥ 0: Eexp{𝜆𝜉} ≤ exp{𝜆2 𝑎2 /2}, 𝜆 ∈ ℝ}.
Definition 2.2 A sub-Gaussian random variable 𝜉 is called strictly sub-Gaussian if
𝜏(𝜉) = (E𝜉 2 )1/2 .
Example. A centered Gaussian random variable is strictly sub-Gaussian.
Definition 2.3. A stochastic process 𝑋 = {𝑋(𝑡), 𝑡 ∈ 𝑻} is called sub-Gaussian if all the random variables 𝑋(𝑡), 𝑡 ∈ 𝑻, are sub-Gaussian.
Example. Any centered Gaussian process is sub-Gaussian.
Details about sub-Gaussian random variables and processes can be found in Buldygin and
Kozachenko (2000).
3. A wavelet-based Expansion
Theorem 3.1. (Kozachenko et al. (2011)) Suppose that {𝑋(𝑡), 𝑡 ∈ 𝑻} is a centered second-order
random process with the correlation function 𝑅(𝑡, 𝑠) = E𝑋(𝑡)𝑋(𝑠), ‍(𝛬, ℬ𝛬 , 𝜇) is a measure space,
{𝑔𝑘 (𝜆), 𝑘 ∈ ℤ} is an orthonormal basis in 𝐿2 (𝛬, 𝜇) and 𝑢(𝑡,⋅) ∈ 𝐿2 (𝛬, 𝜇), 𝑡 ∈ 𝑻.
The correlation function 𝑅(𝑡, 𝑠) of 𝑋(𝑡) can be represented as
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Ievgen Turchyn / Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 137-148
𝑅(𝑡, 𝑠) = ∫𝛬 ‍𝑢(𝑡, 𝜆)𝑢(𝑠, 𝜆)𝑑𝜇(𝜆),
(3)
if and only if the process 𝑋(𝑡) can be represented as
𝑋(𝑡) = ∑𝑘∈ℤ ‍𝑎𝑘 (𝑡)𝜉𝑘 ,‍‍‍𝑡 ∈ 𝑻,
(4)
where (4) converges in 𝐿2 (𝛺), 𝑡 ∈ 𝑻,
𝑎𝑘 (𝑡) = ∫𝛬 ‍𝑢(𝑡, 𝜆)𝑔𝑘 (𝜆)𝑑𝜇(𝜆),
𝜉𝑘 are centered random variables such that E𝜉𝑘 ‍𝜉𝑙 = 𝛿𝑘𝑙 .
The following result enables us to expand a second-order process in a random series with
uncorrelated terms which is built using two wavelets.
Theorem 3.2. Let 𝑋 = {𝑋(𝑡), 𝑡 ∈ [0, 𝑇′]} be a centered random process such that E|𝑋(𝑡)|2 < ∞ for all
𝑡 ∈ [0, 𝑇′]‍. Let 𝑅(𝑡, 𝑠) = E𝑋(𝑡)𝑋(𝑠) and there exists such a Borel function 𝑢(𝑡, 𝑦1 , 𝑦2 ), 𝑦𝑖 ∈ ℝ,‍
𝑡 ∈ [0, 𝑇′], that ∫ℝ2 ‍|𝑢(𝑡, 𝑦1 , 𝑦2 )|2 𝑑𝑦1 𝑑𝑦2 < ∞ (𝑖 = 1,2)‍for all 𝑡 ∈ [0, 𝑇′] and
𝑅(𝑡, 𝑠) = ∫ ‍𝑢(𝑡, 𝑦1 , 𝑦2 )𝑢(𝑠, 𝑦1 , 𝑦2 )𝑑𝑦1 𝑑𝑦2 ‍.‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(5)
ℝ2
Let 𝜙 (1) (𝑥), 𝜓 (1) (𝑥) and 𝜙 (2) (𝑥), 𝜓 (2) (𝑥) be two pairs of a 𝑓-wavelet and the corresponding 𝑚-wavelet. Then the process 𝑋(𝑡) can be represented as the following series which converges for any
𝑡 ∈ [0, 𝑇′] in 𝐿2 (𝛺):
∞
𝑋(𝑡) = ∑ ‍ ∑ ‍𝑎0,𝑘1 ,𝑘2 (𝑡)𝜉0,𝑘1 ,𝑘2 + ∑ ‍∑ ‍∑ ‍𝑏𝑘;𝑗,𝑙 (𝑡)𝜂𝑘;𝑗,𝑙
𝑘1 ∈ℤ 𝑘2 ∈ℤ
∞
𝑘∈ℤ 𝑗=0 𝑙∈ℤ
∞
∞
+ ∑ ‍∑ ‍∑ ‍𝑑𝑘;𝑗,𝑙 (𝑡)𝜁𝑘;𝑗,𝑙 + ∑ ‍ ∑ ‍ ∑ ‍ ∑ ‍𝑒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 (𝑡)𝜒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 ,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(6)
𝑘∈ℤ 𝑗=0 𝑙∈ℤ
𝑗1 =0 𝑙1 ∈ℤ 𝑗2 =0 𝑙2 ∈ℤ
where
𝑎0,𝑘1 ,𝑘2 (𝑡) =
1
(1)
(2)
∫ ‍𝑢(𝑡, 𝑦1 , 𝑦2 )𝜙̂0,𝑘 (𝑦1 )‍𝜙̂0,𝑘 (𝑦2 )𝑑𝑦1 𝑑𝑦2 ,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(7)
1
2
2𝜋 ℝ2
𝑏𝑘;𝑗,𝑙 (𝑡) =
1
(1)
(2)
∫ ‍𝑢(𝑡, 𝑦1 , 𝑦2 )𝜙̂0,𝑘 (𝑦1 )‍𝜓̂𝑗,𝑙 (𝑦2 )𝑑𝑦1 𝑑𝑦2 ,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(8)
2𝜋 ℝ2
𝑑𝑘;𝑗,𝑙 (𝑡) =
1
(1)
(2)
∫ ‍𝑢(𝑡, 𝑦1 , 𝑦2 )𝜓̂𝑗,𝑙 (𝑦1 )‍𝜙̂0,𝑘 (𝑦2 )𝑑𝑦1 𝑑𝑦2 ,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(9)
2𝜋 ℝ2
𝑒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 (𝑡) =
1
(1)
(2)
∫ ‍𝑢(𝑡, 𝑦1 , 𝑦2 )𝜓̂𝑗 ,𝑙 (𝑦1 )‍𝜓̂𝑗 ,𝑙 (𝑦2 )𝑑𝑦1 𝑑𝑦2 ,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(10)
1
1
2 2
2𝜋 ℝ2
all the random variables 𝜉0,𝑘1 ,𝑘2 , 𝜂𝑘;𝑗,𝑙 , 𝜁𝑘;𝑗,𝑙 , 𝜒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 are centered and uncorrelated,
E|𝜉0,𝑘1 ,𝑘2 |2 = E|𝜂𝑘;𝑗,𝑙 |2 = E|𝜁𝑘;𝑗,𝑙 |2 = E|𝜒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 |2 = 1.
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Ievgen Turchyn / Journal of Applied Mathematics and Statistics
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Proof. It is enough to apply Theorem 3.1 to the process 𝑋(𝑡) and the orthonormal basis ℱ defined as
ℱ = ℱ1 ∪ ℱ2 ∪ ℱ3 ∪ ℱ4 , where
(1)
(2)
ℱ1 = 𝜙̂0,𝑘1 (𝑦1 )𝜙̂0,𝑘2 (𝑦2 )/(2𝜋), 𝑘1 , 𝑘2 ∈ ℤ,
(1)
(2)
ℱ2 = 𝜙̂0,𝑘1 (𝑦1 )𝜓̂𝑗2 ,𝑘2 (𝑦2 )/(2𝜋), 𝑘1 , 𝑘2 ∈ ℤ, 𝑗2 = 0,1, …,
(1)
(2)
ℱ3 = 𝜓̂𝑗1 ,𝑘1 (𝑦1 )𝜙̂0,𝑘2 (𝑦2 )/(2𝜋), 𝑘1 , 𝑘2 ∈ ℤ, 𝑗1 = 0,1, …,
(1)
(2)
ℱ4 = 𝜓̂𝑗1 ,𝑘1 (𝑦1 )𝜓̂𝑗2 ,𝑘2 (𝑦2 )/(2𝜋), 𝑘1 , 𝑘2 ∈ ℤ, 𝑗1 , 𝑗2 = 0,1, …
Remark. We can generalize expansion (6) if we consider a centered random field 𝑋 = {𝑋(𝑡), 𝑡 ∈ 𝐓}
(where 𝐓 ⊂ ℝ𝑑 ) which correlation function can be represented as
𝑅(𝑡, 𝑠) = ∫ ‍𝑢(𝑡, 𝑦1 , 𝑦2 , … , 𝑦𝑛 )𝑢(𝑠, 𝑦1 , 𝑦2 , … , 𝑦𝑛 )𝑑𝑦1 𝑑𝑦2 … 𝑑𝑦𝑛 ,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(11)
ℝ𝑛
𝑢(𝑡,⋅,⋅, … ,⋅) ∈ 𝐿2 (ℝ𝑛 ), and apply Theorem 3.1 to the orthonormal basis in 𝐿2 (ℝ𝑛 ) which is the
(𝑖)
(𝑖)
tensor product of orthonormal bases {𝜙̂0,𝑘 (𝑥)/√2𝜋, 𝑘 ∈ ℤ; 𝜓̂𝑗,𝑙 (𝑥)/√2𝜋, 𝑗 = 0,1, … ; 𝑘, 𝑙 ∈ ℤ} ,
𝑖 = 1,2, … , 𝑛 (𝜙 (𝑖) (𝑥) and 𝜓 (𝑖) (𝑥) are pairs of a 𝑓-wavelet and the corresponding 𝑚-wavelet).
Let us consider the case 𝑑 = 𝑛. If 𝑋 = {𝑋(𝑡), 𝑡 ∈ 𝐓 ⊂ ℝ𝑑 } is a centered stationary random field
which has the spectral density 𝑓(𝑦), 𝑦 ∈ ℝ𝑑 , then its correlation fucntion 𝑅(𝑡, 𝑠) can be represented
as (11) if we set
𝑢(𝑡, 𝑦) = √𝑓(𝑦)exp{−𝑖(𝑡, 𝑦)},
where 𝑡 = (𝑡1 , 𝑡2 , … , 𝑡𝑑 ), 𝑦 = (𝑦1 , 𝑦2 , … , 𝑦𝑑 ) . So 𝑋(𝑡) can be expanded into a series of types
(4)‍and‍(6).
4. Inequalities for the Coefficients
Lemma 4.1. Let 𝑋(𝑡) be a stochastic process which satisfies the conditions of Theorem 3.2 together
with the f-wavelets 𝜙 (1) (𝑥), 𝜙 (2) (𝑥) and the corresponding m-wavelets 𝜓 (1) (𝑥), 𝜓 (2) (𝑥), the function
𝑢(𝑡, 𝑦1 , 𝑦2 ) from (5) is such that
|𝑢(𝑡, 𝑦1 , 𝑦2 )| ≤ 𝑣1 (𝑡, 𝑦1 )‍𝑣2 (𝑡, 𝑦2 ),
(12)
(𝑖)
where 𝑣𝑖 (𝑡,⋅) ∈ 𝐿2 (ℝ)‍(𝑖 = 1,2). Assume that 𝜙̂ (𝑦) are absolutely continuous, 𝑣𝑖 (𝑡, 𝑦) are absolutely
continuous with respect to 𝑦 for any fixed 𝑡, there exist 𝜕𝑣𝑖 (𝑡, 𝑦)/𝜕𝑦, 𝑑𝜙̂ (𝑖) (𝑦)/𝑑𝑦, 𝑑𝜓̂ (𝑖) (𝑦)/𝑑𝑦 and
𝑑𝜓̂ (𝑖) (𝑦)
|
| ≤ 𝐶𝑖 ,
𝑑𝑦
|𝑣𝑖 (𝑡, 𝑦)| ≤ 𝐻𝑖 (𝑡)‍𝑣𝑖∗ (𝑦),
∂𝑣𝑖 (𝑡, 𝑦)
|
| ≤ 𝐺𝑖 (𝑡)‍𝑣𝑖∗∗ (𝑦),
∂𝑦
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sup 𝐻𝑖 (𝑡) < ∞,
𝑡∈[0,𝑇]
sup 𝐺𝑖 (𝑡) < ∞,
𝑡∈[0,𝑇]
∫ ‍𝑣𝑖∗ (𝑦)|𝑦|𝑑𝑦 < ∞,
ℝ
∫ ‍𝑣𝑖∗ (𝑦)𝑑𝑦 < ∞,
ℝ
∫ ‍𝑣𝑖∗ (𝑦) |
ℝ
𝑑𝜙̂ (𝑖) (𝑦)
| 𝑑𝑦 < ∞,
𝑑𝑦
∫ ‍𝑣𝑖∗ (𝑦)|𝜙̂ (𝑖) (𝑦)|𝑑𝑦 < ∞,
ℝ
∫ ‍𝑣𝑖∗∗ (𝑦)|𝑦|𝑑𝑦 < ∞,
ℝ
∫ ‍𝑣𝑖∗∗ (𝑦)|𝜙̂ (𝑖) (𝑦)|𝑑𝑦 < ∞,
ℝ
lim 𝑣𝑖 (𝑡, 𝑦)|𝜙̂ (𝑖) (𝑦)| = 0,‍‍‍‍𝑡 ∈ ℝ,
|𝑦|→∞
lim 𝑣𝑖 (𝑡, 𝑦)|𝜓̂ (𝑖) (𝑦/2𝑗 )| = 0,‍‍‍‍𝑡 ∈ ℝ,
|𝑦|→∞
𝑗 = 0,1, …,
𝑖 = 1,2. Define
(𝑖)
𝑆1
(𝑖)
=
𝑑𝜙̂ (𝑖) (𝑦)
∗
∫ ‍𝑣𝑖 (𝑦) |
| 𝑑𝑦,
𝑑𝑦
√2𝜋 ℝ
𝑆2 =
1
1
√2𝜋
(𝑖)
𝑄1 =
∫ ‍𝑣𝑖∗∗ (𝑦)|𝜙̂ (𝑖) (𝑦)|𝑑𝑦,
ℝ
𝐶𝑖
∫ ‍𝑣𝑖∗ (𝑦)𝑑𝑦,
√2𝜋 ℝ
𝐶𝑖
(𝑖)
∫ ‍𝑣𝑖∗∗ (𝑦)|𝑦|𝑑𝑦,
𝑄2 =
√2𝜋 ℝ
1
∫ ‍𝑣𝑖∗ (𝑦)|𝜙̂ (𝑖) (𝑦)|𝑑𝑦,
𝐿(𝑖) =
√2𝜋 ℝ
𝐶𝑖
∫ ‍𝑣𝑖∗ (𝑦)|𝑦|𝑑𝑦,
𝑊 (𝑖) =
√2𝜋 ℝ
𝑖 = 1,2. Then the following inequalities hold:
|𝑎0,𝑘1 ,𝑘2 (𝑡)| ≤
𝐴𝑎,𝑎 (𝑡)
,‍‍‍‍𝑘1 ≠ 0, 𝑘2 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(13)
|𝑘1 ||𝑘2 |
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(2016) Vol. 3 No. 3 pp. 137-148
𝐴𝑎,0 (𝑡)
,‍‍‍‍𝑘1 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(14)
|𝑘1 |
𝐴0,𝑎 (𝑡)
|𝑎0,0,𝑘2 (𝑡)| ≤
,‍‍‍‍𝑘2 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(15)
|𝑘2 |
‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍|𝑎0,0,0 (𝑡)| ≤ 𝐴0,0 (𝑡),‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(16)
|𝑎0,𝑘1 ,0 (𝑡)| ≤
𝐵𝑎,𝑏 (𝑡)
,‍‍‍‍𝑘 ≠ 0, 𝑙 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(17)
2𝑗/2 |𝑘||𝑙|
|𝑏𝑘;𝑗,𝑙 (𝑡)| ≤
|𝑏0;𝑗,𝑙 (𝑡)| ≤
𝐵0,𝑏 (𝑡)
,‍‍‍‍𝑙 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(18)
2𝑗/2 |𝑙|
|𝑏𝑘;𝑗,0 (𝑡)| ≤
𝐵𝑎,0 (𝑡)
,‍‍‍‍𝑘 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(19)
|𝑘|23𝑗/2
𝐵0,0 (𝑡)
,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(20)
23𝑗/2
𝐷𝑎,𝑏 (𝑡)
|𝑑𝑘;𝑗,𝑙 (𝑡)| ≤ 𝑗/2
,‍‍‍‍𝑘 ≠ 0, 𝑙 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(21)
2 |𝑘||𝑙|
|𝑏0;𝑗,0 (𝑡)| ≤
𝐷0,𝑏 (𝑡)
,‍‍‍‍𝑙 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(22)
2𝑗/2 |𝑙|
𝐷𝑎,0 (𝑡)
|𝑑𝑘;𝑗,0 (𝑡)| ≤
,‍‍‍‍𝑘 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(23)
|𝑘|23𝑗/2
|𝑑0;𝑗,𝑙 (𝑡)| ≤
|𝑑0;𝑗,0 (𝑡)| ≤
|𝑒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 (𝑡)| ≤
𝐷0,0 (𝑡)
,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(24)
23𝑗/2
𝐸𝑏,𝑏 (𝑡)
,‍‍‍‍𝑙1
𝑗
/2
2 1 ‍2𝑗2 /2 ‍|𝑙1 ||𝑙2 |
≠ 0, 𝑙2 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(25)
|𝑒𝑗1 ,0;𝑗2 ,𝑙2 (𝑡)| ≤
𝐸0,𝑏 (𝑡)
,‍‍‍‍𝑙2 ≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(26)
23𝑗1 /2 ‍2𝑗2 /2 ‍|𝑙2 |
|𝑒𝑗1 ,𝑙1 ;𝑗2 ,0 (𝑡)| ≤
𝐸𝑏,0 (𝑡)
,‍‍‍‍𝑙1
𝑗
/2
1
2
‍23𝑗2 /2 ‍|𝑙1 |
|𝑒𝑗1 ,0;𝑗2 ,0 (𝑡)| ≤
≠ 0,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(27)
𝐸0,0 (𝑡)
,‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍(28)
3𝑗
2 1 /2 ‍23𝑗2 /2
where
(1)
(1)
(2)
(2)
𝐴𝑎,𝑎 (𝑡) = (𝑆1 𝐻1 (𝑡) + 𝑆2 𝐺1 (𝑡))(𝑆1 𝐻2 (𝑡) + 𝑆2 𝐺2 (𝑡)),
(2)
(2)
(1)
(1)
𝐴0,𝑎 (𝑡) = (𝑆1 𝐻2 (𝑡) + 𝑆2 𝐺2 (𝑡))𝐿(1) 𝐻1 (𝑡),
𝐴𝑎,0 (𝑡) = (𝑆1 𝐻1 (𝑡) + 𝑆2 𝐺1 (𝑡))𝐿(2) 𝐻2 (𝑡),
𝐴0,0 (𝑡) = 𝐿(1) 𝐿(2) 𝐻1 (𝑡)𝐻2 (𝑡),
(1)
(1)
(2)
(2)
𝐵𝑎,𝑏 (𝑡) = (𝑆1 𝐻1 (𝑡) + 𝑆2 𝐺1 (𝑡))(𝑄1 𝐻2 (𝑡) + 𝑄2 𝐺2 (𝑡)),
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(2016) Vol. 3 No. 3 pp. 137-148
(2)
(2)
𝐵0,𝑏 (𝑡) = 𝐿(1) 𝐻1 (𝑡)(𝑄1 𝐻2 (𝑡) + 𝑄2 𝐺2 (𝑡)),
(1)
(1)
𝐵𝑎,0 (𝑡) = 𝑊 (2) 𝐻2 (𝑡)(𝑆1 𝐻1 (𝑡) + 𝑆2 𝐺1 (𝑡)),
𝐵0,0 (𝑡) = 𝐿(1) 𝑊 (2) 𝐻1 (𝑡)𝐻2 (𝑡),
(2)
(2)
(1)
(1)
𝐷𝑎,𝑏 (𝑡) = (𝑆1 𝐻2 (𝑡) + 𝑆2 𝐺2 (𝑡))(𝑄1 𝐻1 (𝑡) + 𝑄2 𝐺1 (𝑡)),
(1)
(1)
𝐷0,𝑏 (𝑡) = 𝐿(2) 𝐻2 (𝑡)(𝑄1 𝐻1 (𝑡) + 𝑄2 𝐺1 (𝑡)),
(2)
(2)
𝐷𝑎,0 (𝑡) = 𝑊 (1) 𝐻1 (𝑡)(𝑆1 𝐻2 (𝑡) + 𝑆2 𝐺2 (𝑡)),
𝐷0,0 (𝑡) = 𝐿(2) 𝑊 (1) 𝐻1 (𝑡)𝐻2 (𝑡),
(1)
(1)
(2)
(2)
𝐸𝑏,𝑏 (𝑡) = (𝑄1 𝐻1 (𝑡) + 𝑄2 𝐺1 (𝑡))(𝑄1 𝐻2 (𝑡) + 𝑄2 𝐺2 (𝑡)),
(2)
(2)
(1)
(1)
𝐸0,𝑏 (𝑡) = 𝑊 (1) 𝐻1 (𝑡)(𝑄1 𝐻2 (𝑡) + 𝑄2 𝐺2 (𝑡)),
𝐸𝑏,0 (𝑡) = 𝑊 (2) 𝐻2 (𝑡)(𝑄1 𝐻1 (𝑡) + 𝑄2 𝐺1 (𝑡)),
𝐸0,0 (𝑡) = 𝑊 (1) 𝑊 (2) 𝐻1 (𝑡)𝐻2 (𝑡).
Proof. Let us prove, for instance, inequality (13). We have
|𝑎0,𝑘1 ,𝑘2 (𝑡)| ≤
1
(1)
(2)
|∫ ‍𝑣 (𝑡, 𝑦1 )𝜙̂0,𝑘 (𝑦1 )𝑑𝑦1 |‍‍|∫ ‍𝑣2 (𝑡, 𝑦2 )𝜙̂0,𝑘 (𝑦2 )𝑑𝑦2 |.
1
2
2𝜋 ℝ 1
ℝ
Estimating the integrals in the right-hand side by means of Lemma 1 from Turchyn (2011a), we
obtain (13).
Inequalities (14)–(28) are proved in a similar way.
5. Simulation
Expansion (6) may be used for simulation of stochastic processes.
If a process 𝑋(𝑡) satisfies the conditions of Theorem 3.2, then we can consider as a model of 𝑋(𝑡)
the process
𝑎
𝑎
𝑏
𝑏
𝑏
𝑘 −1
𝑘 −1
𝑗 −1
−1
𝑙 −1
𝑋̂(𝑡) = ∑𝑘1 =−(𝑘 𝑎 −1) ‍‍‍∑𝑘2 =−(𝑘 𝑎 −1) ‍𝑎0,𝑘1 ,𝑘2 (𝑡)𝜉0,𝑘1 ,𝑘2 + ∑𝑘𝑘=−(𝑘
𝑏 −1) ‍‍‍∑𝑗=0 ‍‍‍∑𝑙=−(𝑙 𝑏 −1) ‍𝑏𝑘;𝑗,𝑙 (𝑡)𝜂𝑘;𝑗,𝑙
1
1
𝑗 𝑑 −1
𝑑
2
2
𝑑
−1
𝑙 −1
+ ∑𝑘𝑘=−(𝑘
𝑑 −1) ‍‍‍∑𝑗=0 ‍‍‍∑𝑙=−(𝑙 𝑑 −1) ‍𝑑𝑘;𝑗,𝑙 (𝑡)𝜁𝑘;𝑗,𝑙
𝑗 𝑒 −1
𝑙 𝑒 −1
𝑗 𝑒 −1
𝑙 𝑒 −1
+ ∑𝑗11 =0 ‍‍‍∑𝑙1 =−(𝑙𝑒 −1) ‍‍‍∑𝑗22 =0 ‍‍‍∑𝑙2 =−(𝑙𝑒 −1) ‍𝑒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 (𝑡)𝜒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 ‍,
1
1
2
2
(30)
where 𝜉0,𝑘1 ,𝑘2 , 𝜂𝑘;𝑗,𝑙 , 𝜁𝑘;𝑗,𝑙 , 𝜒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 are the random variables from expansion (6), the functions
𝑎0,𝑘1 ,𝑘2 (𝑡), 𝑏𝑘;𝑗,𝑙 (𝑡), 𝑑𝑘;𝑗,𝑙 (𝑡), 𝑒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 (𝑡) are calculated using (7)–(10), the parameters
𝑘1𝑎 , 𝑘2𝑎 , 𝑘𝑏 , 𝑗𝑏 , 𝑙 𝑏 , 𝑘 𝑑 , 𝑗 𝑑 , 𝑙 𝑑 , 𝑗1𝑒 , 𝑙1𝑒 , 𝑗2𝑒 , 𝑙2𝑒 are strictly greater than 1.
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Ievgen Turchyn / Journal of Applied Mathematics and Statistics
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Remark. If 𝑋(𝑡) is a Gaussian process then 𝜉0,𝑘1 ,𝑘2 , 𝜂𝑘;𝑗,𝑙 , 𝜁𝑘;𝑗,𝑙 , 𝜒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 in (30) are independent
random variables with distribution 𝑁(0,1) and the model 𝑋̂(𝑡) can be used for computer simulation
of 𝑋(𝑡).
Definition 5.1. We say that the model 𝑋̂(𝑡) approximates a process 𝑋(𝑡) with given reliability 1 − 𝛿
(0 < 𝛿 < 1) and accuracy 𝜀 > 0 in 𝐿𝑝 ([0, 𝑇]) if
1/𝑝
𝑇
𝑃 {(∫ ‍|𝑋(𝑡) − 𝑋̂(𝑡)|𝑝 𝑑𝑡)
> 𝜀} ≤ 𝛿.
0
Remark. We will consider only real-valued 𝑓-wavelets and 𝑚-wavelets below.
Definition 5.2. We say that the condition R1 holds for a stochastic process 𝑋(𝑡) if it satisfies the
conditions of Theorem 3.2, 𝑢(𝑡,⋅,⋅) ∈ 𝐿1 (ℝ2 ) ∩ 𝐿2 (ℝ2 ) and inverse Fourier transform 𝑢̃(𝑡, 𝑦1 , 𝑦2 ) of the
function 𝑢(𝑡, 𝑦1 , 𝑦2 ) with respect to (𝑦1 , 𝑦2 ) is a real-valued function.
Theorem 5.1. (Turchyn (2011a)) Suppose that 𝑋(𝑡) is a sub-Gaussian random process that satisfies
condition R1, 𝑋̂(𝑡) is the model of the process defined by (30), random variables 𝜉0,𝑘1 ,𝑘2 , 𝜂𝑘;𝑗,𝑙 , 𝜁𝑘;𝑗,𝑙 ,
𝜒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 in expansion (6) of 𝑋(𝑡) are independent and strictly sub-Gaussian, 𝑝 ≥ 1, 𝛿 ∈ (0; 1), 𝜀 > 0,
𝑇 ∈ (0, 𝑇′).
If
sup E|𝑋(𝑡) − 𝑋̂(𝑡)|2 ≤ min {
𝑡∈[0,𝑇]
𝜀2
𝜀2
,‍‍‍
},
2𝑇 2/𝑝 ln(2/𝛿) 𝑝𝑇 2/𝑝
then the model 𝑋̂(𝑡) approximates the process 𝑋(𝑡) with reliability 1 − 𝛿 and accuracy 𝜀 in 𝐿𝑝 ([0, 𝑇]).
The following theorem is the main result of the article.
Theorem 5.2 Suppose that a sub-Gaussian random process 𝑋 = {𝑋(𝑡), 𝑡 ∈ [0, 𝑇′]} satisfies the
condition R1 and the conditions of Lemma 4.1 together with 𝑓-wavelets 𝜙 (1) (𝑥), 𝜙 (2) (𝑥) and the
corresponding 𝑚 -wavelets 𝜓 (1) (𝑥), 𝜓 (2) (𝑥) , the random variables 𝜉0,𝑘1 ,𝑘2 , 𝜂𝑘;𝑗,𝑙 , 𝜁𝑘;𝑗,𝑙 , 𝜒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 in
expansion (6) of 𝑋(𝑡) are independent and strictly sub-Gaussian, 𝑝 ≥ 1, 𝑇 ∈ (0, 𝑇′), 𝛿 ∈ (0; 1), 𝜀 > 0.
Denote by 𝐴̃𝑎,0 (𝑇), … , 𝐸̃0,𝑏 (𝑇) the suprema of the functions 𝐴𝑎,0 (𝑡), … , 𝐸0,𝑏 (𝑡) correspondingly on
[0, 𝑇] (where 𝐴𝑎,0 (𝑡), … , 𝐸0,𝑏 (𝑡) are defined in Lemma 4.1),
𝜀2
𝜀2
𝜀1 = min { 2/𝑝
, 2/𝑝 }.
2𝑇 ln(2/𝛿) 𝑝𝑇
If
𝑘1𝑎 ≥ 1 + (16(𝐴̃𝑎,0 (𝑇))2 + 64(𝐴̃𝑎,𝑎 (𝑇))2 )/𝜀1 ,
𝑘2𝑎 ≥ 1 + (16(𝐴̃0,𝑎 (𝑇))2 + 64(𝐴̃𝑎,𝑎 (𝑇))2 )/𝜀1 ,
𝑘𝑏 ≥ 1 + ((256/7)(𝐵̃𝑎,0 (𝑇))2 + 256(𝐵̃𝑎,𝑏 (𝑇))2 )/𝜀1 ,
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𝑙 𝑏 ≥ 1 + (256(𝐵̃𝑎,𝑏 (𝑇))2 + 64(𝐵̃0,𝑏 (𝑇))2 )/𝜀1 ,
𝑗𝑏 ≥ max {1 + log 2
64(𝐵̃0,𝑏 (𝑇))2 + 256(𝐵̃𝑎,𝑏 (𝑇))2
64(𝐵̃𝑎,0 (𝑇))2 + 16(𝐵̃0,0 (𝑇))2
, 1 + log 8
},
𝜀1
7𝜀1
̃𝑎,0 (𝑇))2 + 256(𝐷
̃𝑎,𝑏 (𝑇))2 )/𝜀1 ,
𝑘 𝑑 ≥ 1 + ((256/7)(𝐷
̃𝑎,𝑏 (𝑇))2 + 64(𝐷
̃0,𝑏 (𝑇))2 )/𝜀1 ,
𝑙 𝑑 ≥ 1 + (256(𝐷
𝑗 𝑑 ≥ max {1 + log 2
̃0,𝑏 (𝑇))2 + 256(𝐷
̃𝑎,𝑏 (𝑇))2
̃𝑎,0 (𝑇))2 + 16(𝐷
̃0,0 (𝑇))2
64(𝐷
64(𝐷
, 1 + log 8
},
𝜀1
7𝜀1
𝑗1𝑒 ≥ max{1 + log 2
768(𝐸̃𝑏,𝑏 (𝑇))2 + 768(𝐸̃𝑏,0 (𝑇))2 /7
24(𝐸̃0,𝑏 (𝑇))2 /7 + 24(𝐸̃0,0 (𝑇))2 /49
, 2 + log 8
},
𝜀1
𝜀1
𝑗2𝑒 ≥ max{1 + log 2
768(𝐸̃𝑏,𝑏 (𝑇))2 + 768(𝐸̃0,𝑏 (𝑇))2 /7
24(𝐸̃𝑏,0 (𝑇))2 /7 + 24(𝐸̃0,0 (𝑇))2 /49
, 2 + log 8
},
𝜀1
𝜀1
𝑙1𝑒 ≥ 1 +
768(𝐸̃𝑏,𝑏 (𝑇))2 + 768(𝐸̃𝑏,0 (𝑇))2 /7
,
𝜀1
𝑙2𝑒 ≥ 1 +
768(𝐸̃𝑏,𝑏 (𝑇))2 + 768(𝐸̃0,𝑏 (𝑇))2 /7
,
𝜀1
then the model 𝑋̂(𝑡) defined by (30) approximates the process 𝑋(𝑡) with reliability 1 − 𝛿 and
accuracy 𝜀 in 𝐿𝑝 ([0, 𝑇]).
Proof. It is easy to see using Lemma 4.1 that under the assumptions of the theorem
𝑎
𝑘 −1
E|𝑋(𝑡) − 𝑋̂(𝑡)|2 = ∑𝑘1 :|𝑘1 |≥𝑘1𝑎 ‍‍‍∑𝑘2 ∈ℤ ‍|𝑎0,𝑘1 ,𝑘2 (𝑡)|2 + ∑𝑘1 =−(𝑘 𝑎 −1) ‍‍‍∑𝑘2 :|𝑘2 |≥𝑘2𝑎 ‍|𝑎0,𝑘1 ,𝑘2 (𝑡)|2
1
2
+ ∑𝑘:|𝑘|≥𝑘 𝑏 ‍‍‍∑∞
𝑗=0 ‍‍‍∑𝑙∈ℤ ‍|𝑏𝑘;𝑗,𝑙 (𝑡)|
+
1
𝑏 −1
∞
2
∑𝑘𝑘=−(𝑘
𝑏 −1) ‍‍‍∑𝑗=𝑗 𝑏 ‍‍‍∑𝑙∈ℤ ‍|𝑏𝑘;𝑗,𝑙 (𝑡)|
𝑗 𝑏 −1
𝑏
−1
∞
2
2
+ ∑𝑘𝑘=−(𝑘
𝑏 −1) ‍‍‍∑𝑗=0 ‍‍‍∑𝑙:|𝑙|≥𝑙 𝑏 ‍|𝑏𝑘;𝑗,𝑙 (𝑡)| + ∑𝑘:|𝑘|≥𝑘 𝑑 ‍‍‍∑𝑗=0 ‍‍‍∑𝑙∈ℤ ‍|𝑑𝑘;𝑗,𝑙 (𝑡)|
𝑑
𝑗 𝑑 −1
𝑑
−1
∞
𝑘 −1
2
2
+ ∑𝑘𝑘=−(𝑘
𝑑 −1) ‍‍‍∑𝑗=𝑗 𝑑 ‍‍‍∑𝑙∈ℤ ‍|𝑑𝑘;𝑗,𝑙 (𝑡)| ‍ + ‍ ∑𝑘=−(𝑘 𝑑 −1) ‍‍‍∑𝑗=0 ‍‍‍∑𝑙:|𝑙|≥𝑙 𝑑 ‍|𝑑𝑘;𝑗,𝑙 (𝑡)|
𝑗 𝑒 −1
∞
∞
1
2
2
+ ∑∞
𝑗1 =𝑗1𝑒 ‍∑𝑗2 =0 ‍‍‍∑𝑙1 ∈ℤ ‍‍‍∑𝑙2 ∈ℤ ‍|𝑒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 (𝑡)| + ∑𝑗1 =0 ‍∑𝑗2 =𝑗2𝑒 ‍‍‍∑𝑙1 ∈ℤ ‍‍‍∑𝑙2 ∈ℤ ‍|𝑒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 (𝑡)|
𝑗 𝑒 −1
𝑗 𝑒 −1
𝑗 𝑒 −1
𝑗 𝑒 −1
𝑙 𝑒 −1
+ ∑𝑗11 =0 ‍∑𝑗22 =0 ‍‍‍∑𝑙1 :|𝑙1 |≥𝑙1𝑒 ‍‍‍∑𝑙2 ∈ℤ ‍|𝑒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 (𝑡)|2 + ∑𝑗11 =0 ‍∑𝑗22 =0 ‍‍‍∑𝑙1 =−(𝑙𝑒 −1) ‍‍‍∑𝑙2 :|𝑙2 |≥𝑙2𝑒 ‍|𝑒𝑗1 ,𝑙1 ;𝑗2 ,𝑙2 (𝑡)|2
1
1
≤ 𝜀1
for all 𝑡 ∈ [0, 𝑇]. Therefore
sup E|𝑋(𝑡) − 𝑋̂(𝑡)|2 ≤ 𝜀1
𝑡∈[0,𝑇]
and it remains to apply Theorem 5.1.
Example. It is easy to see that a centered Gaussian process 𝑋 = {𝑋(𝑡), 𝑡 ∈ [0, 𝑇′]} with the
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correlation function
𝑅(𝑡, 𝑠) = ∫ℝ ‍𝑢(𝑡, 𝑦1 , 𝑦2 )𝑢(𝑠, 𝑦1 , 𝑦2 )𝑑𝑦1 𝑑𝑦2 ,
where
𝑢(𝑡, 𝑦1 , 𝑦2 ) =
𝑡 4𝑙
2𝑚1
(1 + 𝑡 2𝑙 )2 + 𝐵(1 + 𝑡 2𝑙 )(𝑦1
2𝑚2
+ 𝑦2
2𝑚1 2𝑚2 ,
𝑦2
) + 𝐴𝑦1
𝑙, 𝑚1 , 𝑚2 ∈ ℕ, 𝑚1 ≥ 2, 𝑚2 ≥ 2, 𝐴 ≥ 1, 𝐵 ≥ 1, together with two Daubechies wavelets of arbitrary
order satisfies the conditions of Theorem 5.2.
6. Conclusion
We consider an expansion of a second-order stochastic process based on two wavelets. This
expansion may be regarded as a generalization of an expansion from Kozachenko and Turchyn
(2008). We use our expansion to build a model of a stochastic process and obtain a theorem about
simulation of a sub-Gaussian stochastic process with given accuracy and reliability in 𝐿𝑝 ([0, 𝑇]) by
this model.
Acknowledgements
The author would like to thank professor Yuriy V. Kozachenko for valuable discussions which
helped to substantially improve the quality of the paper and an anonymous referee for the
constructive comments .
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