On accuracy of simulation of gaussian stationary processes in L2([0, T])
A theorem about simulation of a Gaussian stochastic process with given accuracy and reliability in L2([0, T ]) using wavelets has been proved.
Збережено в:
| Дата: | 2006 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2006
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/4469 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On accuracy of simulation of gaussian stationary processes in L2([0, T]) / Y. Turchyn // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 255–260. — Бібліогр.: 5 назв.— англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-4469 |
|---|---|
| record_format |
dspace |
| spelling |
Turchyn, Y. 2009-11-11T15:29:55Z 2009-11-11T15:29:55Z 2006 On accuracy of simulation of gaussian stationary processes in L2([0, T]) / Y. Turchyn // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 255–260. — Бібліогр.: 5 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4469 A theorem about simulation of a Gaussian stochastic process with given accuracy and reliability in L2([0, T ]) using wavelets has been proved. en Інститут математики НАН України On accuracy of simulation of gaussian stationary processes in L2([0, T]) Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On accuracy of simulation of gaussian stationary processes in L2([0, T]) |
| spellingShingle |
On accuracy of simulation of gaussian stationary processes in L2([0, T]) Turchyn, Y. |
| title_short |
On accuracy of simulation of gaussian stationary processes in L2([0, T]) |
| title_full |
On accuracy of simulation of gaussian stationary processes in L2([0, T]) |
| title_fullStr |
On accuracy of simulation of gaussian stationary processes in L2([0, T]) |
| title_full_unstemmed |
On accuracy of simulation of gaussian stationary processes in L2([0, T]) |
| title_sort |
on accuracy of simulation of gaussian stationary processes in l2([0, t]) |
| author |
Turchyn, Y. |
| author_facet |
Turchyn, Y. |
| publishDate |
2006 |
| language |
English |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
A theorem about simulation of a Gaussian stochastic process with given accuracy and reliability in L2([0, T ]) using wavelets has been proved.
|
| issn |
0321-3900 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/4469 |
| citation_txt |
On accuracy of simulation of gaussian stationary processes in L2([0, T]) / Y. Turchyn // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 255–260. — Бібліогр.: 5 назв.— англ. |
| work_keys_str_mv |
AT turchyny onaccuracyofsimulationofgaussianstationaryprocessesinl20t |
| first_indexed |
2025-11-25T13:48:46Z |
| last_indexed |
2025-11-25T13:48:46Z |
| _version_ |
1850513606900187136 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 255–260
YEVGEN TURCHYN
ON ACCURACY OF SIMULATION OF GAUSSIAN
STATIONARY PROCESSES IN L2([0, T ])
A theorem about simulation of a Gaussian stochastic process with given
accuracy and reliability in L2([0, T ]) using wavelets has been proved.
1. Introduction
Problems of expansion of random processes in series over uncorrelated
random variables using wavelets were considered in [3], [4], [5] and other
papers. There has been proved in [3] a theorem about accuracy and reli-
ability of such expansions in Lp([0, T ]) for Gaussian wide-sense stationary
processes. We’ll give a refinement of this theorem for L2([0, T ]). Since
Gaussian processes are widely used in financial and actuarial mathematics
results of the article may be used in these areas.
First of all we need to list some necessary facts.
Let ϕ ∈ L2(R) be such a function that the following assumptions hold:
i)
∑
k∈Z |ϕ̂(y+2πk)|2 = 1 almost everywhere, where ϕ̂(y) is the Fourier
transform of ϕ,
ϕ̂(y) =
∫
R
exp{−iyx}ϕ(x)dx;
ii) There exists a function m0 ∈ L2([0, 2π]) such that m0(x) has pe-
riod 2π and almost everywhere
ϕ̂(y) = m0
(y
2
)
ϕ̂
(y
2
)
;
iii) ϕ̂(0) �= 0 and the function ϕ̂(y) is continuos at 0.
Function ϕ(x) is called f-wavelet. Let ψ(x) be the inverse Fourier trans-
form of the function
ψ̂(y) = m0
(y
2
+ π
)
exp
{
− i
y
2
}
ϕ̂
(y
2
)
.
Function ψ(x) is called m-wavelet. Let ϕjk(x) = 2
j
2 ϕ(2jx − k), ψjk(x) =
2
j
2 ψ(2jx−k), k ∈ Z, j = 0, 1, 2, . . . , It is known that the family of functions
{ϕ0k, ψjk, j = 0, 1, 2, . . . , k ∈ Z} is an orthonormal basis in L2(R) (see, for
2000 Mathematics Subject Classification. Primary 60G10, 60G15, 42C40.
Key words and phrases. Stochastic processes, simulation, wavelets.
255
256 YEVGEN TURCHYN
example, the book [1]). Such a basis is called wavelet basis. Any function
f ∈ L2(R) can be represented in the form
(1) f(x) =
∑
k∈Z
α0kϕ0k(x) +
∞∑
j=0
∑
k∈Z
βjkψjk(x),
where α0k =
∫
R
f(x)ϕ0k(x)dx, βjk =
∫
R
f(x)ψjk(x)dx,
∑
k∈Z
|α0k|2 +
∞∑
j=0
∑
k∈Z
|βjk|2 < ∞.
That is, series (1) converges in the norm of the space L2(R). Representation
(1) is called wavelet representation.
Let X = {X(t), t ∈ R} be a centered wide-sense stationary random
process (from this moment we will refer to wide-sense stationary processes
simply as stationary processes), X(t, ω) ∈ L2(Ω, F, P ) for all t ∈ R (where
(Ω, F, P ) is the standard probability space to which belong all random vari-
ables X(t, ω) ), R(τ) = EX(t + τ)X(t).
There has been proved the following theorem in [3].
Theorem 1. Let X = {X(t), t ∈ R} be centered stationary random
process, R(τ) = EX(t+τ)X(t). Suppose that R(τ) is a continuous function
and process X(t) has spectral density, i.e. R(τ) =
∫
R
exp{−iτλ}f(λ)dλ,
where f is real-valued, f(λ) ≥ 0,
∫∞
−∞ f(λ)dλ = R(0) < ∞. Let {ϕ0k(x),
ψjk(x), k ∈ Z, j = 0, 1, 2, . . .} be a wavelet basis. Then
(2) X(t) =
∑
k∈Z
ξ0kα0k(t) +
∞∑
j=0
∑
k∈Z
ηjkβjk(t),
where
(3) α0k(t) =
1√
2π
∫
R
(f(y))1/2 exp{−iy(t − k)}ϕ̂(y)dy,
(4) βjk(t) =
1√
2π2j/2
∫
R
(f(y))1/2 exp
{
−iy
(
t − k
2j
)}
ψ̂
( y
2j
)
dy,
ξ0k and ηjk are centered random variables such that Eξ0kξ0l = δkl, Eηmkηnl =
δmnδkl, Eξ0kηnl = 0, δkl is Kronecker delta and series (2) converges in mean
square.
The expansion (2) has been used in [3] for modeling of Gaussian stochas-
tic processes and obtaining inequality for given accuracy and reliability of
appropriate model in Lp([0, T ]). We’ll give refinement of these theorems for
L2([0, T ]).
ON ACCURACY OF SIMULATION OF GAUSSIAN PROCESSES 257
2. Simulation of stationary Gaussian processes.
If we have a Gaussian process X(t) which satisfies conditions of theorem
1, then we can consider as a model of X(t) a process
X̂(t) =
N0−1∑
k=−(N0−1)
ξ0kα0k(t) +
N−1∑
j=0
Mj−1∑
k=−(Mj−1)
ηjkβjk(t),
where ξ0k, ηjk are independent random variables with distribution N(0, 1),
α0k(t) and βjk(t) are calculated using formulae (3) and (4), N0 > 1, N > 1,
Mj > 1 (j = 0, . . . , N − 1).
Definition. Model X̂(t) approximates process X(t) with given reliability
1 − δ, 0 < δ < 1, and accuracy ε > 0 in Lp([0, T ]) if
P
{(∫ T
0
|X(t) − X̂(t)|pdt
)1/p
> ε
}
≤ δ.
There has been proved the following theorem in [3].
Theorem 2. Stochastic process X̂(t) approximates process X(t) with reli-
ability 1 − δ and accuracy ε in Lp([0, T ]) (p ≥ 1, 0 < δ < min{1, 2e−p/2}),
(5) P
{(∫ T
0
|X(t) − X̂(t)|pdt
)1/p
> ε
}
≤ δ
if
sup
t∈[0,T ]
E|X(t) − X̂(t)|2 = sup
t∈[0,T ]
( ∑
k:|k|≥N0
|α0k(t)|2 +
N−1∑
j=0
∑
k:|k|≥Mj
|βjk(t)|2+
∞∑
j=N
∑
k∈Z
|βjk(t)|2
)
≤ ε2
2T 2/p ln 2
δ
.
This result can be made more exact for p = 2. We’ll give first some
auxiliary facts.
There has been proved the following assertion in [2], which we give here
in simplified form.
Theorem 3. Let {T,U, μ} be a measurable space. Consider a random
series
(6) S(t) =
∞∑
k=1
ξkfk(t), t ∈ T,
where ξ = {ξk, k = 1, 2, . . .} is a family of Gaussian random variables and
f = {fk(t), k = 1, 2, . . .} is a family of real-valued L2(T) functions. Let the
258 YEVGEN TURCHYN
random variables in (6) be either uncorrelated with Eξ2
k = σ2
k or the system
of functions fk(t) be orthogonal, that is,∫
T
fk(t)fl(t)dμ(t) = δkla
2
k,
where δkl is the Kronecker symbol. If the series
∞∑
k=1
σ2
ka
2
k < ∞
converges, then the series (6) is mean square convergent in L2(T) and for
all x >
√
An and n=1,2, . . . we have
(7) P
⎧⎨
⎩
∥∥∥∥∥
∞∑
k=n
ξkfk(t)
∥∥∥∥∥
L2(T)
> x
⎫⎬
⎭ ≤ e1/2 x√
An
exp
{
− x2
2An
}
,
where An =
∑∞
k=n σ2
ka
2
k.
The following statement about bounds for coefficients α0k(t) and βjk(t)
was proved in [3].
Lemma 1. Let R(τ) be a covariance function which satisfies conditions of
theorem 1, R(τ) =
∫
R
exp{−iτλ}f(λ)dλ. Let ϕ̂(y) be the Fourier transform
of a f-wavelet ϕ(x). Let ϕ̂(y) be a continuous function and assertions i) –
iii) hold true for all y ∈ R. Let ψ̂(y) be the Fourier transform of m-wavelet
ψ(x) corresponding to ϕ(x). Let g(y) =
√
f(y) and there exist g′(y), ψ̂′(y),
ϕ̂′(y); |ψ̂(y)| < C1, |ψ̂′(y)| < C2, |ϕ̂(y)| is bounded,∫
R
g(y)dy < ∞,
∫
R
|g′(y)||y|dy < ∞,
∫
R
g(y)|y|dy < ∞,
∫
R
|g′(y)||ϕ̂(y)|dy < ∞,
∫
R
g(y)|ϕ̂′(y)|dy < ∞,
α0k(t) and βjk(t) are given in (3) and (4). If k �= 0 then for all t ∈ R
|βjk(t)| ≤ A + B|t|
|k|2j/2
,
where
A =
C2√
2π
∫
R
(|g′(y)||y|+ g(y))dy, B =
C2√
2π
∫
R
g(y)|y|dy;
|α0k(t)| ≤ A1 + B1|t|
|k| ,
where
A1 =
1√
2π
∫
R
(|g′(y)||ϕ̂(y)|+ g(y)|ϕ̂′(y)|)dy,
B1 =
1√
2π
∫
R
g(y)|y||ϕ̂(y)|dy.
ON ACCURACY OF SIMULATION OF GAUSSIAN PROCESSES 259
For all t ∈ R and j = 0, 1, 2, . . .
|βj0(t)| ≤ C2√
2π23j/2
∫
R
g(y)|y|dy, |α00(t)| ≤ 1√
2π
∫
R
g(y)|ϕ̂(y)|dy.
Now we can formulate the main result of our paper.
Theorem 4. Let Gaussian process X(t) satisfy restrictions of theorem 1,
f-wavelet ϕ and m-wavelet ψ satisfy conditions of lemma 1. Model X̂(t)
approximates process X(t) with accuracy ε and reliability 1− δ (0 ≤ δ ≤ 1)
in L2([0, T ]),
(8) P
{(∫ T
0
|X(t) − X̂(t)|2dt
)1/2
> ε
}
≤ δ,
if An < ε2
x2
δ
, where xδ (xδ ≥ 1) is a root of equation e1/2xe−x2/2 = δ,
An =
∑
k:|k|≥N0
∫ T
0
|α0k(t)|2dt+
+
N−1∑
j=0
∑
k:|k|≥Mj
∫ T
0
|βjk(t)|2dt +
∞∑
j=N
∑
k∈Z
∫ T
0
|βjk(t)|2dt,
α0k(t) and βjk(t) are defined by formulae (3) and (4).
Proof. Let’s consider stochastic process
(9) X̃(t) =
∑
k∈Z
ξ∗0kα0k(t) +
∞∑
j=0
∑
k∈Z
η∗
jkβjk(t),
where ξ∗0k, η
∗
jl (j = 0, 1, 2, . . . ; k, l ∈ Z) are i.i.d. random variables with
distribution N(0, 1). It’s easy to see that correlation functions of processes
X(t) and X̃(t) are the same. Therefore we may consider process X̃(t)
instead of process X(t). If we apply theorem 3 to series (9) we obtain
inequality (8).
Corollary. Model X̂(t) approximates process X(t) with accuracy ε and
reliability 1 − δ in L2([0, T ]) if the following inequalities are satisfied:
N0 > 1 +
6D2
ε1
, N > log2
(
3
ε1
(
8D1 +
8D2T
7
))
,
Mj >
12D1
ε1
(
1 − 1
2N
)
+ 1,
(j = 0, 1, . . . , N − 1), where ε1 = ε2
x2
δ
,
D =
C2√
2π
∫
R
g(y)|y| dy,
D1 = A2T + ABT 2 + 1
3
B2T 3, D2 = A2
1T + A1B1T
2 + 1
3
B2
1T
3; g(y), A, B,
A1, B1 C2 are defined in lemma 1, xδ is defined in theorem 4.
260 YEVGEN TURCHYN
Proof. If we apply lemma 1 it’s easy to see that under conditions of the
corollary the following inequalities are true:
An <
2D2
N0 − 1
+
4D1
2N−1
+ D2T
1
7 · 8N−1
+ D1
N−1∑
j=0
1
2j−1(Mj − 1)
,
2D2
N0 − 1
<
ε1
3
,
4D1
2N−1
+ D2T
1
7 · 8N−1
<
ε1
3
,
D1
N−1∑
j=0
1
2j−1(Mj − 1)
<
ε1
3
.
Statement of the corollary immediately follows from these inequalities and
theorem 4. �
3. Conclusions
There has been obtained a theorem about accuracy and reliability of
simulation in L2([0, T ]) for a certain class of stationary Gaussian processes.
This theorem is more exact than previous result for Lp([0, T ]) when p = 2.
Author expresses his thanks to professor Yuriy V. Kozachenko for valu-
able discussions.
References
1. W.Härdle, G. Kerkacharian, D. Picard, A. Tsybakov, Wavelets, Approximation
and Statistical Applications, Springer, New York, (1998).
2. Yu.V. Kozachenko, A.O. Pashko, Accuracy of simulation of stochastic processes
in norms of Orlicz spaces. I, Theory Probab. Math. Statist., 58, (1999), 51–92.
3. Yu. Kozachenko, E. Turchyn, On a generalization of Walter-Zhang’s wavelet-
based KL-like expansion for stationary random processes, submitted to Applied
and Computational Harmonic Analysis.
4. G. Walter, J. Zhang, A wavelet-based KL-like expansion for wide-sense stationary
random processes, IEEE Trans. Signal Process., 42, (1994), no.7, 1737–1745.
5. G. Walter, J. Zhang, Wavelets based on band-limited processes with a KL-type
property, Proc. SPIE Conf. Visual Inform. Processing II (Orlando, FL), Apr.
1993, 336–343.
Department of Higher Mathematics, Dnipropetrovsk State Agricultural
University, Voroshylova str., 25, Dnipropetrovsk, Ukraine
E-mail address: turchyn@a-teleport.com
|