Stationary processes in functional spaces Lq( R )
The paper is devoted to the problem of establishing the conditions on the stochastic process to belong it to the functional space Lq(R) with probability one. The corresponding results were obtained for the strictly Orlicz, stationary in wide sense processes.
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irk-123456789-45252009-11-25T12:00:39Z Stationary processes in functional spaces Lq( R ) Yakovenko, T. The paper is devoted to the problem of establishing the conditions on the stochastic process to belong it to the functional space Lq(R) with probability one. The corresponding results were obtained for the strictly Orlicz, stationary in wide sense processes. 2007 Article Stationary processes in functional spaces Lq( R ) / T. Yakovenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 210–218. — Бібліогр.: 7 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4525 en Інститут математики НАН України |
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The paper is devoted to the problem of establishing the conditions on the stochastic process to belong it to the functional space Lq(R) with probability one. The corresponding results were obtained for the strictly Orlicz, stationary in wide sense processes. |
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Yakovenko, T. Stationary processes in functional spaces Lq( R ) |
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Yakovenko, T. |
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Yakovenko, T. |
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Stationary processes in functional spaces Lq( R ) |
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Stationary processes in functional spaces Lq( R ) |
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Stationary processes in functional spaces Lq( R ) |
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Stationary processes in functional spaces Lq( R ) |
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Stationary processes in functional spaces Lq( R ) |
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stationary processes in functional spaces lq( r ) |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4525 |
| citation_txt |
Stationary processes in functional spaces Lq( R ) / T. Yakovenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 210–218. — Бібліогр.: 7 назв.— англ. |
| work_keys_str_mv |
AT yakovenkot stationaryprocessesinfunctionalspaceslqr |
| first_indexed |
2025-07-02T07:44:58Z |
| last_indexed |
2025-07-02T07:44:58Z |
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1836520359488126976 |
| fulltext |
Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.210–218
TETYANA YAKOVENKO
STATIONARY PROCESSES IN FUNCTIONAL
SPACES Lq(R)
The paper is devoted to the problem of establishing the conditions
on the stochastic process to belong it to the functional space Lq(R)
with probability one. The corresponding results were obtained for
the strictly Orlicz, stationary in wide sense processes.
1. Introduction
This paper deals with the problem of establishing the condition on the
stochastic process from the space of random variables Lp(Ω) under which it
belongs to the functional space Lq(R) with probability one. This two spaces
are actually Orlicz spaces generated by the functions V (x) = |x|p, p ≥ 1 and
U(x) = |x|q, q ≥ 1. The general theory on Orlicz functions and spaces is
contained in [1]. Random variables and stochastic processes in Orlicz spaces
were investigated in book [2]. For the stochastic process defined in compact
set the corresponding conditions of belonging were established for two cases:
for 1 ≤ p < q in [3] and for 1 ≤ q ≤ p in [4]. The stochastic processes
with noncompact parametric set were examined in [5,6] for 1 ≤ p < q. In
this paper we have completed that results with treatment the case when
1 ≤ q ≤ p. The results obtained were applied to strictly Orlicz, stationary
in wide sense stochastic processes in both this cases.
2. Some facts from Orlicz space theory.
Definition 2.1 [1] A function U = U(x), x ∈ R is called an Orlicz S-
function if it is continuous, even, convex and U(0) = 0, U(x) > 0 as x �= 0.
Example 2.2 The functions U(x) = A|x|α, A > 0, α ≥ 1 and V (x) =
exp{B|x|β} − 1, B > 0, β ≥ 1 are the S-functions.
2000 Mathematics Subject Classifications. Primary 60G17, secondary 60G07
Key words and phrases. Stationary process, Orlicz space, noncompact parametric
set, strictly Orlicz random variables.
210
STATIONARY PROCESSES IN SPACES Lq(R) 211
Let {T,B, μ} be a measurable space.
Definition 2.3 [2] The space of measurable functions f = {f(t), t ∈ T}
such that for all f there exists a constant rf > 0 for which∫
T
U
(
f(t)
rf
)
dμ(t) < ∞
is called the Orlicz space LU(T, μ) generated by the S-function U .
Remark 2.4 [2] The space LU (T, μ) is a Banach space with respect to the
Luxemburg norm
‖f(t)‖LU (T) = inf
{
r > 0 :
∫
T
U
(
f(t)
r
)
dμ(t) ≤ 1
}
.
Definition 2.5 [2] Let {Ω,B, P} be a standard probability space. Then
LU (Ω, P ) is the Orlicz space of random variables generated by the S-function
U . The Luxemburg norm in this space is determined as
‖ξ‖LU (Ω) = inf
{
r > 0 : EU
(
ξ
r
)
≤ 1
}
.
Definition 2.6 [2] A stochastic process X = {X(t), t ∈ T} belongs to the
Orlicz space LU (Ω, P ) if for all t ∈ T random variables X(t) belong to the
space LU(Ω).
Definition 2.7 [2] A stochastic process X = {X(t), t ∈ T} belongs to the
Orlicz space LU(T, μ) with probability one if the trajectories of the process
X are measurable and X(t) ∈ LU(T, μ) with probability one.
Condition M. [3] It is said that condition M is satisfied for S-function V
in a set G if for all measurable stochastic processes Z = {Z(t), t ∈ G} such
that Z(t) ∈ LV (Ω) and ‖Z(t)‖LV (Ω) = 1, random variables ‖Z(t)‖LV (G) also
belong to LV (Ω) and there exists an absolute constant aV such that∥∥‖Z(t)‖LV (G)
∥∥
LV (Ω)
≤ aV .
Example 2.8 [3] The functions U(x) = A|x|α, where A > 0, α ≥ 1 and
V (x) = exp{B|x|β} − 1, where B > 0, β ≥ 1 satisfy the condition M in
every compact set G.
Consider the measurable parametric space (T,B(T), μ) with σ− finite
measure μ. That is, there exists such partition {Tl ∈ B(T)}l≥1 of space T
212 TETYANA YAKOVENKO
that T =
⋃∞
l=1 Tl and μ(Tl) < ∞. Furthermore, let for all l
′
, l
′′
the following
condition holds : Tl′
⋂
Tl′′ = ∅ as l
′ �= l
′′
.
Assume that U and V are such Orlicz S− functions that W = W (x) =
V −1(U(x)), x ∈ R is convex function too. In this case the following theorem
takes place.
Theorem 2.9 [5] Assume that the stochastic process X = {X(t), t ∈ T}
(μ(T) = ∞), which belongs to the Orlicz space of random variables LV (Ω),
is separable, measurable and
∀B ∈ B(T ) sup
ρ(t, s) ≤ h
t, s ∈ B
‖X(t) − X(s)‖LV (Ω) ≤ σB(h)
where σB = σB(h), h > 0 is the continuous, monotone nondecreasing func-
tion and σB(h) → 0 as h → 0.
Then, if the condition M is satisfied for the Orlicz S-function V in every
set Tl from the partition T and there exists such sequence {0 < δl < 1}l≥1
that
∑
l≥1
δl < ∞ and
∑
l≥1
(
V
(
δl∥∥‖X(t)‖LU (Tl)
∥∥
LV (Ω)
))−1
< ∞,
then the stochastic process X = {X(t), t ∈ T} belongs to the space LU (T)
with probability one.
3. Conditions for stochastic process belonging to the
functional space Lq(R).
Consider stochastic processes from the Orlicz space of random variables
Lp(Ω). This space is generated by the S-function V (x) = |x|p, p ≥ 1. The
functional space Lq(R) is generated by the Orlicz S-function U(x) = |x|q,
q ≥ 1. In this section we will present sufficient conditions for belonging of
the stochastic processes defined on set of all real numbers R to functional
space Lq(R) with probability one in two cases, according to the type of
subordination between p and q : 1 ≤ q ≤ p and 1 ≤ p < q.
Consider measurable parametric space (R,B(R), μ), where B(R) is a σ-
algebra of Borelian subsets on R, μ is the Lebegue measure on R.
Let partition the set R into half-open intervals [xl−1, xl) with properties
∀l ≥ 1 Δxl = |xl − xl−1| < ∞;
⋃
l∈
[xl−1, xl) = R; (1)
STATIONARY PROCESSES IN SPACES Lq(R) 213
[xl′−1, xl′)
⋂
[xl′′−1, xl′′) = ∅ , as l′ �= l′′.
The partition determined by the sequence⎧⎨
⎩xl = sign(l)
|l|∑
i=1
1
iγ
, 0 < γ < 1
⎫⎬
⎭
l∈
(2)
has property (1) and for all l ≥ 1
xl = |x−l| =
l∑
i=1
1
iγ
> (l + 1)1−γ − 1 ∼ l1−γ as l → ∞
Really, since for all 0 < γ < 1, u ≥ 1 function 1/uγ is monotonely decreas-
ing. Then for all i ≥ 1
1
iγ
=
∫ i+1
i
1
iγ
du ≥
∫ i+1
i
1
uγ
du =
u1−γ
1 − γ
∣∣∣∣
i+1
i
=
1
1 − γ
(
(i + 1)1−γ − i1−γ
)
,
and thus
xl = |x−l| =
l∑
i=1
1
iγ
≥ 1
1 − γ
l∑
i=1
(
(i + 1)1−γ − i1−γ
)
=
=
1
1 − γ
(
(l + 1)1−γ − 1
)
> (l + 1)1−γ − 1 ∼ l1−γ as l → ∞.
Theorem 3.1 Let 1 ≤ q ≤ p. Suppose that the measurable stochastic
process X = {X(t), t ∈ R} belongs to the space of random variables Lp(Ω)
and for all −∞ < a < b < +∞ such that b − a ≤ 1
sup
a≤t<b
(E|X(t)|p)1/p ≤ A
(max{|a|, |b|})τ ,
where A > 0, τ > 1
1−γ
+ 1
p
and 0 < γ < 1. Then trajectories of the process
X belong to the functional space Lq(R) with probability one.
Proof. Let partition the set R using sequence {xl}l∈ (2). Then
Δxl = |xl − xl−1| =
{ 1
lγ
, l ≥ 1;
1
(−l+1)γ , l ≤ 0.
Notice, that for all l ∈ Z xl = −x−l and Δxl = Δx−l+1.
Theorem 4.1 [4] and properties of the sequence (2) imply that for every
l ≥ 1
∥∥‖X(t)‖Lq[xl−1,xl)
∥∥
Lp(Ω)
≤ (1 + Δxl)Δx
1/q
l
xτ
l
≤
(
1 + 1
lγ
)
1
lγ/q
((l + 1)1−γ − 1)τ ≤
≤ 2
lγ/q ((l + 1)1−γ − 1)τ ∼ 2
lγ/q+τ(1−γ)
=: B+
l , as l → +∞,
214 TETYANA YAKOVENKO
and for l ≤ 0,
∥∥∥‖X(t)‖Lq [xl−1,xl)
∥∥∥
Lp(Ω)
≤ (1 + Δxl)Δx
1/q
l
|xl−1|τ =
=
(1 + Δxl)Δx
1/q
l
xτ
−l+1
≤
(
1 + 1
(1−l)γ
)
1
(1−l)γ/q
((2 − l)1−γ − 1)τ ≤
≤ 2
(1 − l)γ/q ((2 − l)1−γ − 1)τ ∼ 2
(1 − l)γ/q+τ(1−γ)
=: B−
l , as l → −∞.
It is obvious that for all l ≥ 1 B+
l = B−
−l+1. Consider the following sequence
δl =
{ 1
lε
, l ≥ 1;
1
(−l+1)ε , l ≤ 0,
ε > 1.
Then
+∞∑
l=−∞
δl =
0∑
l=−∞
1
(−l + 1)ε +
+∞∑
l=1
1
lε
= 2
+∞∑
l=1
1
lε
< ∞,
and
+∞∑
l=−∞
(
Bl
δl
)p
=
0∑
l=−∞
(
B−
l
(−l + 1)ε
)p
+
+∞∑
l=1
(
B+
l
lε
)p
=
= 2
+∞∑
l=1
(
B+
l
lε
)p
∼
∑
l≥1
1
lγ+τ(1−γ)p−εp
.
The last series converges as γ + τ(1 − γ)p − εp > 1, that is
1 < ε < (1 − γ)
(
τ − 1
p
)
. (3)
According to the conditions of the theorem
(1 − γ)
(
τ − 1
p
)
> (1 − γ)
(
1
1 − γ
+
1
p
− 1
p
)
= 1.
and such ε exists. So, the statement of the theorem follows from the Theo-
rem 2.9. �
Theorem 3.2 [6] Let 1 ≤ p < q. We consider separable measurable stochas-
tic process X = {X(t), t ∈ R}, which belongs to the space of random vari-
ables Lp(Ω). Suppose that for some −∞ < a < b < +∞, b − a ≤ 1:
i) supa≤t<b (E|X(t)|p)1/p ≤ A
(max{|a|,|b|})τ ,
where A > 0, τ > 1
1−γ
(
1 + 1
p
− γ
q
)
, 0 < γ < 1;
STATIONARY PROCESSES IN SPACES Lq(R) 215
ii) sup |t − s| ≤ h
t, s ∈ [a, b)
(E|X(t) − X(s)|p)1/p ≤ Ca,bh
α, where h> 0, α > 1
p
− 1
q
;
iii) ∃ 0 < c < ∞ : Ca,b ≤ c
(b−a)α supa≤t<b (E|X(t)|p)1/p .
In this case trajectories of the stochastic process X belong to the functional
space Lq(R) with probability one.
Remark 3.3 Conditions in the Theorem 3.1 and Theorem 3.2 on stochastic
process will be weaker if γ is closer to 0.
4. Stationary strictly Orlicz stochastic processes.
In this section we consider special kind of stochastic processes, namely
stationary strictly Orlicz processes. Let us recall some definitions first.
Definition 4.1 [7] Let V = V (x), x ∈ R be Orlicz S-function, such that,
there exist x0 > 0 and k > 0 that for all x > x0, x2 ≤ V (kx). Family Δ of
random variables ξi (Eξi = 0) from space LV (Ω) is called strictly Orlicz if
there exists constant CΔ, such that for every finite set of random variables
ξi ∈ Δ, i ∈ I and for all λi ∈ R the following inequality holds true
∥∥∥∥∥
∑
i∈I
λiξi
∥∥∥∥∥
LV (Ω)
≤ CΔ
⎛
⎝E
(∑
i∈I
λiξi
)2
⎞
⎠
1/2
.
CΔ is determining constant for the family of strictly Orlicz random variables
Δ.
Properties of the Orlicz S-function V imply that there exists constant
B > 0 such that⎛
⎝E
(∑
i∈I
λiξi
)2
⎞
⎠
1/2
≤ B
∥∥∥∥∥
∑
i∈I
λiξi
∥∥∥∥∥
LV (Ω)
,
i.e. norms ‖ · ‖LV (Ω) and (E(·)2)1/2 are equivalent for family Δ.
Example 4.2 Family of Gaussian centered random variables is strictly Orlicz
in the exponential Orlicz space Exp2(Ω)(See book [2]).
Definition 4.3 Stochastic process X = {X(t), t ∈ T} belonging to space
LV (Ω) is called strictly Orlicz if the family of random variables {X(t)}t∈T
is strictly Orlicz.
Definition 4.4 Stochastic process X = {X(t), t ∈ R} is called station-
ary in wide sense if EX(t) = m = const and its correlation function
216 TETYANA YAKOVENKO
R(t, s) = r(t − s), t, s ∈ R.
In the space Lp(Ω), p ≥ 2 the family of the strictly Orlicz random vari-
ables can be easily determined. For more details see [7].
Theorem 4.5 Let 1 ≤ q ≤ p, Y = {Y (t), t ∈ R} be centered, measurable,
stationary in wide sense, strictly Orlicz in Lp(Ω) stochastic process with
determining constant CΔp and correlation function R(t, s) = r(t− s), t, s ∈
R.
Consider function c(t) = A
(|t|+1)τ , t ∈ R, A = CΔp
√
r(0), τ > 1
1−γ
+ 1
p
,
0 < γ < 1. Then stochastic process X(t) = {c(t)Y (t), t ∈ R} belongs to the
functional space Lq(R) with probability one.
Proof. The statement of the theorem follows from the Theorem 3.1. Indeed,
for all −∞ < a < b < +∞, b−a ≤ 1 (Note: 0 must not be inside the interval
(a, b))
sup
a ≤ t < b
‖X(t)‖Lp(Ω) = sup
a ≤ t < b
(E|X(t)|p)1/p = (4)
= sup
a ≤ t < b
(E|c(t)Y (t)|p)1/p ≤ CΔp sup
a ≤ t < b
|c(t)| (E|Y (t)|2)1/2
=
= CΔp
√
r(0) sup
a ≤ t < b
|c(t)| =
CΔp
√
r(0)
(min{|a|, |b|} + 1)τ
≤ A
(max{|a|, |b|})τ ,
since b − a = max{|a|, |b|} − min{|a|, |b|} ≤ 1. �
Theorem 4.6 Let 2 ≤ p ≤ q and Y = {Y (t), t ∈ R} be centered, measur-
able, separable, stationary in wide sense, strictly Orlicz in Lp(Ω) stochastic
process with determining constant CΔp and correlation function R(t, s) =
r(t − s), t, s ∈ R (r(0) �= 0), such that for all 0 < h ≤ 1 r(0) − r(h) ≤
C0h
2α, C0 > 0, 1/p − 1/q < α ≤ 1.
Consider function c(t) = A
(|t|+1)τ , t ∈ R, τ > 1
1−γ
(
1 + 1
p
− γ
q
)
, 0<γ <1,
A = CΔp
√
r(0). Then stochastic process X(t) = {c(t)Y (t), t ∈ R} belongs
to the functional Orlicz space Lq(R) with probability one.
Proof. For stochastic process X(t) = {c(t)Y (t), t ∈ R} the inequality (4)
takes place, i.e the first condition of Theorem 3.2 holds.
Now, let’s check the other two conditions of the Theorem 3.2.
For −∞ < a < b < +∞, b − a ≤ 1
sup
|t − s| ≤ h
t, s ∈ [a, b)
‖X(t) − X(s)‖Lp(Ω) = sup
|t − s| ≤ h
t, s ∈ [a, b)
(E|X(t) − X(s)|p)1/p ≤
≤ CΔp sup
|t − s| ≤ h
t, s ∈ [a, b)
(
E|X(t) − X(s)|2)1/2 =
STATIONARY PROCESSES IN SPACES Lq(R) 217
= CΔp sup
|t − s| ≤ h
t, s ∈ [a, b)
(
E|c(t)Y (t) − c(s)Y (s)|2)1/2 =
= CΔp sup
|t − s| ≤ h
t, s ∈ [a, b)
(
E|c(t)Y (t) − c(s)Y (t) + c(s)Y (t) − c(s)Y (s)|2)1/2 ≤
≤ CΔp sup
|t − s| ≤ h
t, s ∈ [a, b)
[
|c(t) − c(s)| (EY 2(t)
)1/2 + c(s)
(
E(Y (t) − Y (s))2
)1/2
]
=
= CΔp sup
|t − s| ≤ h
t, s ∈ [a, b)
[
|c(t) − c(s)|
√
r(0) + c(s)
√
2(r(0) − r(t − s))
]
=
= CΔp
[
(c(min{|a|, |b|}) − c(min{|a|, |b|} + h))
√
r(0)+
+ c(min{|a|, |b|})
√
2(r(0) − r(h))
]
.
As long as function c(t) is continuous then there exists
θ ∈ [min{|a|, |b|}, min{|a|, |b|} + h] such that
c(min{|a|, |b|}) − c(min{|a|, |b|} + h) =
=
A
(min{|a|, |b|} + 1)τ
− A
(min{|a|, |b|} + h + 1)τ
=
=
−τA
(|θ| + 1)τ+1
(min{|a|, |b|} − min{|a|, |b|} − h) =
=
τA
(|θ| + 1)τ+1
h ≤ τA
(min{|a|, |b|} + 1)τ+1
h
then
sup
|t − s| ≤ h
t, s ∈ [a, b)
‖X(t) − X(s)‖Lp(Ω) ≤
≤ CΔp
[
Aτ
√
r(0) · h
(min{|a|, |b|} + 1)τ+1
+
A
√
2C0 · hα
(min{|a|, |b|} + 1)τ
]
≤
≤ ACΔp
√
r(0)hα
(min{|a|, |b|} + 1)τ
[
τ
(min{|a|, |b|}+ 1)
+
√
2C0
r(0)
]
≤
≤
ACΔp
√
r(0)
(
τ +
√
2C0
r(0)
)
(min{|a|, |b|} + 1)τ
hα = Ca,bh
α.
So, condition ii) holds true. Let’s check condition iii). Since p ≥ 2 then
there exists constant C̃ such that ‖ · ‖L2(Ω) ≤ C̃‖ · ‖Lp(Ω). So,
Ca,b =
ACΔp
(
τ +
√
2C0
r(0)
)
(min{|a|, |b|} + 1)τ
√
r(0) ≤
218 TETYANA YAKOVENKO
≤
ACΔp
(
τ +
√
2C0
r(0)
)
(b − a)α
· (b − a)α · sup
a ≤ t < b
|b − a| ≤ 1
(
E|X(t)|2)1/2 ≤
≤
AV CΔp
(
τ +
√
2C0
r(0)
)
C̃
(b − a)α
sup
a ≤ t < b
|b − a| ≤ 1
(E|X(t)|p)1/p .
This concludes that all conditions of Theorem 3.2 hold for stochastic process
X(t) = {c(t)Y (t), t ∈ R}. The theorem has been proved. �
References
1. Krasnosel’skĭi, M. A. and Rutickĭi, Ya. B., Convex functions and Orlicz
spaces. Fizmatgiz, Moscow, (1958). English transl., Noordhof, Gröningen
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine.
E-mail address: yata452@univ.kiev.ua
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