Stochastic processes in some Besov spaces
The norm of increments of stochastic process in space Lq[a, b] is estimated and conditions under which trajectories of process belong to some Besov spaces are found.
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irk-123456789-44992009-11-20T12:00:41Z Stochastic processes in some Besov spaces Yakovenko, T. The norm of increments of stochastic process in space Lq[a, b] is estimated and conditions under which trajectories of process belong to some Besov spaces are found. 2007 Article Stochastic processes in some Besov spaces / T. Yakovenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 308-315. — Бібліогр.: 3 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4499 en Інститут математики НАН України |
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The norm of increments of stochastic process in space Lq[a, b] is estimated and conditions under which trajectories of process belong to some Besov spaces are found. |
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Yakovenko, T. |
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Yakovenko, T. Stochastic processes in some Besov spaces |
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Yakovenko, T. |
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Yakovenko, T. |
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Stochastic processes in some Besov spaces |
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Stochastic processes in some Besov spaces |
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Stochastic processes in some Besov spaces |
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Stochastic processes in some Besov spaces |
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Stochastic processes in some Besov spaces |
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stochastic processes in some besov spaces |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4499 |
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Stochastic processes in some Besov spaces / T. Yakovenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 308-315. — Бібліогр.: 3 назв.— англ. |
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AT yakovenkot stochasticprocessesinsomebesovspaces |
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2025-07-02T07:43:46Z |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.308-315
TETYANA YAKOVENKO
STOCHASTIC PROCESSES IN SOME BESOV
SPACES
The norm of increments of stochastic process in space Lq[a, b] is esti-
mated and conditions under which trajectories of process belong to
some Besov spaces are found.
1. Introduction
In this paper the stochastic process from Lp(Ω) is considered. We find
conditions under which trajectories of this process belong with probability
one to the Besov space Bαp
q [a, b] when 0 < α < 1 and p < q.
The paper consists of 4 sections. In Section 2 the norm of increments of
the stochastic process is estimated. In Section 3 the definition of the Besov
space is given. Then, it was obtained the estimation for the modulus of
continuity of stochastic process. It gives the possibility to find the condition
under which the trajectories of stochastic process belong to some Besov
space with probability one. Section 4 is the conclusion of this paper.
2. Estimates for increments of the stochastic processes
Let’s consider stochastic process X = {X(t), t ∈ [a− δ, b + δ]}, a < b, δ > 0
and denote the increments of this process as ΔhX(t) = X(t + h) − X(t),
t ∈ [a, b], |h| < δ.
Theorem 1. Assume that 1 < p < q < ∞. Let X = {X(t), t ∈ [a−δ, b+δ]},
a < b, δ > 0 to be separable measurable stochastic process from space Lp(Ω)
for which the following condition on its increments holds true:
sup
t ∈ [a, b]
|h| ≤ δ,
‖ΔhX(t)‖Lp(Ω) = sup
t ∈ [a, b]
|h| ≤ δ,
(E|X(t + h) − X(t)|p)1/p ≤ σ(δ), (1)
where σ(δ), δ > 0 is a continuous nondecreasing function such that σ(δ) → 0
as δ → 0.
Then
2000 Mathematics Subject Classifications. Primary 60G17; Secondary 60G07.
Key words and phrases. Stochastic processes, Besov space, moduli of continuity.
308
STOCHASTIC PROCESSES IN SOME BESOV SPACES 309
1) there exists m = m(h) ∈ {1, 2, ...} :
∥∥∥‖ΔhX(·)‖Lq[a,b]
∥∥∥
Lp(Ω)
=
⎛
⎝E
(∫ b
a
|X(t + h) − X(t)|qdt
) p
q
⎞
⎠
1
p
≤
≤ 21+ 1
q
− 1
p (b − a)
1
p
∞∑
k=m−1
(εk+1)
1
q
− 1
p [σ(6εk+1) + σ(6εk)] =: Bm,
where sequence {εk > 0}k≥0 is such that for all k ≥ 0 εk > εk+1,
εk → 0, k → ∞ and there exists the sequence {0 < αk < 1}k≥0, for
which the fraction σ(εk)
αk
→ 0, k → ∞ and the series
∑∞
k=0 αk < ∞;
2) if Bm < ∞ then ‖ΔX(·)‖Lq [a,b] belongs to Lp(Ω) and for all x > 0
P
{
‖ΔX(·)‖Lq[a,b] > x
}
≤
(
Bm
x
)p
.
Proof. Let’s divide the interval [a − δ, b + δ] into measurable sets {Br
k, r =
1, 2, ..., N(εk)}k≥0 with respect to the sequence {εk > 0}k≥0 according to
partition procedure placed in [1]. This partition has the following properties:
1) B1
0 = [a − δ, b + δ], (ε0 = b−a
2
+ δ);
2) ∀k ≥ 1 Bu
k
⋂
Br
k = ∅ if u
= r and
⋃N(εk)
r=1 Br
k = [a − δ, b + δ];
3) ∀k ≥ 1 ∀Br
k ∃Bl
k−1: Br
k ⊂ Bl
k−1 and Br
k
⋂
Bl
k−1 = Br
k;
4) ∀k ≥ 1 ∀Br
k ∃trk : ∀t ∈ Br
k 2εk ≤ |t − trk| < 6εk.
Now we can define the process:
Xk(t) =
N(εk)∑
r=1
X(trk)χBr
k
(t), k ≥ 0, t ∈ [a − δ, b + δ], χBR
k
=
{
1, t ∈ Br
k;
0, t /∈ Br
k.
As long as the proof of this theorem is similar to the proof of the
lemma 3.5 [1] it was reduced to the statements that are essentially different.
So, if the points t and t+h belong to the same set Br
k then Xk(t) = Xk(t+
h) with probability one and ΔhXk(t) = 0 with probability one. Further we
will assume that process X is not degenerate. The properties of the sequence
εk imply that there exists the number m = m(h) ∈ {1, 2, ...} starting from
which the points t and t+h will be in different sets of the partition {Br
m}εm
r=1.
Then, taking into consideration that Xm−1(t) = Xm−1(t + h) we have for
some n > m:
|ΔhX(t)| = |X(t+h)−X(t)| = |X(t+h)−Xm−1(t+h)+Xm−1(t)−X(t)| ≤
310 TETYANA YAKOVENKO
≤ |X(t + h) − Xn(t + h)| +
n−1∑
k=m−1
|Xk+1(t + h) − Xk(t + h)|+
+|X(t) − Xn(t)| +
n−1∑
k=m−1
|Xk+1(t) − Xk(t)|.
So, for all t ∈ [a, b] as n → ∞:
|ΔhX(t)| ≤
∞∑
k=m−1
[
|Xk+1(t + h) − Xk(t + h)| + |Xk+1(t) − Xk(t)|
]
.
The condition (1) yields:
‖Xk+1(t) − Xk(t)‖Lp(Ω) ≤ ‖Xk+1(t) − X(t)‖Lp(Ω) + ‖X(t) − Xk(t)‖Lp(Ω) ≤
σ(6εk+1) + σ(6εk).
Then
‖ΔhX(·)‖Lq[a,b] =
(∫ b
a
|X(t + h) − X(t)|q
)1/q
≤
≤
∥∥∥∥∥∥
∞∑
k=m−1
{ |Xk+1(t) − Xk(t)|
‖Xk+1(t) − Xk(t)‖Lp(Ω)
‖Xk+1(t) − Xk(t)‖Lp(Ω) +
|Xk+1(t + h) − Xk(t + h)|
‖Xk+1(t + h) − Xk(t + h)‖Lp(Ω)
‖Xk+1(t + h) − Xk(t + h)‖Lp(Ω)
}∥∥∥∥∥
Lq[a,b]
≤
≤
∞∑
k=m−1
⎧⎨
⎩
∥∥∥∥∥ |Xk+1(t + h) − Xk(t + h)|
‖Xk+1(t + h) − Xk(t + h)‖Lp(Ω)
∥∥∥∥∥
Lq[a,b]
+
+
∥∥∥∥∥ |Xk+1(t) − Xk(t)|
‖Xk+1(t) − Xk(t)‖Lp(Ω)
∥∥∥∥∥
Lq[a,b]
⎫⎬
⎭
[
σ(6εk+1) + σ(6εk)
]
≤
≤
∞∑
k=m−1
⎧⎨
⎩
∥∥∥∥∥ |Xk+1(t + h) − Xk(t + h)|
‖Xk+1(t + h) − Xk(t + h)‖Lp(Ω)
∥∥∥∥∥
Lp[a,b]
+
+
∥∥∥∥∥ |Xk+1(t) − Xk(t)|
‖Xk+1(t) − Xk(t)‖Lp(Ω)
∥∥∥∥∥
Lp[a,b]
⎫⎬
⎭ (2εk+1)
1
q
− 1
p
[
σ(6εk+1) + σ(6εk)
]
= Bm.
The expression for Bm we will get from the fact that Condition M (see
[1]) is fulfilled for the space Lp with constant (b − a)1/p.
The second statement follows from lemma 3.1 [2,p.66].
Corollary 1. If the stochastic process X = {X(t), t ∈ [a−δ, b+δ]} fatisfies
the conditions of theorem 1, then for all 0 < θ1 < 1 the following inequality
takes place:
∥∥∥‖ΔhX(·)‖Lq[a,b]
∥∥∥
Lp(Ω)
=
⎛
⎝E
(∫ b
a
|X(t + h) − X(t)|qdt
)p
q
⎞
⎠
1
p
≤
STOCHASTIC PROCESSES IN SOME BESOV SPACES 311
≤ 2(b − a)
1
p
3
1
q
− 1
p
· 1 + θ1
θ(1 − θ)
∫ θ1σ( b−a
2
+δ)
0
(σ(−1)(u))
1
q
− 1
p du =: B̃m (2)
Proof. Lets choose the sequence {εk}k≥0 in such way
ε0 =
b − a
2
+ δ, γ0 = σ(ε0), εk =
σ(−1)(θk
1γ0)
6
.
This sequence has appropriable for theorem 1 properties. So,
Bm = 2(b − a)
1
p
∞∑
k=m−1
(
σ(−1)(θk+1
1 γ0)
3
) 1
q
− 1
p
[θk+1
1 γ0 + θk
1γ0] ≤
≤ 2(b − a)
1
p
∞∑
k=m−1
θk+1
1 γ0 + θk
1γ0
θk+1
1 γ0 − θk+2
1 γ0
∫ θk+1
1 γ0
θk+2
1 γ0
(
σ(−1)(u)
3
) 1
q
− 1
p
du ≤
≤ 2(b − a)
1
p
3
1
q
− 1
p
· 1 + θ1
θ1(1 − θ1)
∫ θ1γ0
0
(σ(−1)(u))
1
q
− 1
p du = B̃m.
Corollary 2. Assume that in the theorem 1 function σ(δ) = Cδτ , where
C > 0 is some constant, τ > 1
p
− 1
q
. Then the increments of the process
X = {X(t), t ∈ [a − δ, b + δ]} belong to the space Lq[a, b] with probability
one and for all 0 < θ1 < 1 the inequality follows:
∥∥∥‖ΔhX(·)‖Lq[a,b]
∥∥∥
Lp(Ω)
=
⎛
⎝E
(∫ b
a
|X(t + h) − X(t)|qdt
)p
q
⎞
⎠
1
p
≤
≤ 2 · 3 1
p
− 1
q C(b − a + 2δ)
1
q
+τ
1
τ
(
1
q
− 1
p
)
+ 1
· 1 + θ1
1 − θ1
· θ
m
τ ( 1
q
− 1
p)+m−1
1 =: B∗
m.
Proof. Indeed, in this case
B̃m =
2(b − a)
1
p
3
1
q
− 1
p
· 1 + θ1
θ1(1 − θ1)
∫ θm
1 C( b−a
2
+δ)
τ
0
(
u
C
) 1
τ ( 1
q
− 1
p)
du =
=
2 · 3 1
p
− 1
q (b − a)
1
p
C
1
τ ( 1
q
− 1
p)
· 1 + θ1
θ1(1 − θ1)
·
(
θm
1 C
(
b−a
2
+ δ
)τ) 1
τ (
1
q
− 1
p)+1
1
τ
(
1
q
− 1
p
)
+ 1
≤ B∗
m.
Remark 1. Similar results hold for stochastic processes in more general
case when stochastic process belongs to the Orlicz space of random variables.
The estimation for the norm of increments in various functional Orlicz space
could be obtained by the same way. More information can be found in [1,2].
312 TETYANA YAKOVENKO
3. Conditions under which trajectories of the process
belong to some Besov spaces
Let’s remind the definition of the Besov space at first. We start from intro-
ducing the moduli of continuity of the first and the second order and some
of their properties.
Definition 2. Let f be a function in Lq(T ), 1 ≤ q ≤ ∞, T ⊆ R. Let’s
denote Δhf = f(t + h) − f(t) and Δ2
hf = ΔhΔhf . For δ > 0 the moduli of
continuity are determined as
ω1
q (f, δ) = sup
|h|≤δ
‖Δhf‖Lq(T ), ω2
q (f, δ) = sup
|h|≤δ
‖Δ2
hf‖Lq(T ).
Remark 2. For any function f from the space Lq(R) ω1
q(f, δ) and ω2
q (f, δ)
are non-decreasing functions of δ and
ω2
q (f, δ) ≤ 2ω1
q (f, δ) ≤ 4‖f‖Lq(T ).
Let 1 ≤ p ≤ ∞ be given, and let the function y(δ) on [0,∞) be such
that ‖y(δ)‖∗p < ∞, where
‖y(δ)‖∗p =
⎧⎨
⎩
(∫∞
0 |y(δ)|p dδ
δ
) 1
p , if 1 ≤ p < ∞;
ess supδ |y(δ)|, if p = ∞.
Clearly, ‖ · ‖∗p is a norm in the weighted Lp-space Lp
(
[0,∞), dδ
δ
)
, if p < ∞.
Definition 2. Let 1 ≤ q, p ≤ ∞ and s = n + α, with n ∈ {0, 1, ...} and
0 < α ≤ 1. The Besov space Bsp
q (T ) is the space of all functions f such that
f ∈ W n
q (T ) and ω2
q (f
(n), δ) = y(δ)δα,
where W n
q (T ) is the Sobolev space and ‖y(δ)‖∗p < ∞.
The space Bsp
q (T ) is equipped with the norm
‖f‖Bsp
q (T ) = ‖f‖W n
q (T ) +
∥∥∥∥∥ω
2
q (f
(n), δ)
δα
∥∥∥∥∥
∗
p
.
Remark 3. If 0 < α < 1 we can use ω1
q instead of ω2
q in the definition of
Besov spaces. But this is not true in the case if α = 1 (See [3]).
Definition 3. The stochastic process X = {X(t), t ∈ T} belongs to Besov
space Bsp
q (T ) with probability one if all its trajectories belong to this func-
tional space with probability one.
Remark 4. Further, when convergence of the integral
∫ c
0 f(t)dt doesn’t
depend on value c we will use the sign
∫
0+.
Theorem 2.Assume that 1 < p < q < ∞ and separable measurable stochas-
tic process X = {X(t), t ∈ [a − δ, b + δ]}, a < b, 0 < δ < ∞ belongs to the
space Lp(Ω). Besides, let its increments satisfy two conditions:
STOCHASTIC PROCESSES IN SOME BESOV SPACES 313
a) sup
t ∈ [a, b]
|h| ≤ δ,
‖ΔhX(t)‖Lp(Ω) = sup
t ∈ [a, b]
|h| ≤ δ,
(E|X(t + h) − X(t)|p)1/p ≤ Cpδ
τp,
b) sup
t ∈ [a, b]
|h| ≤ δ,
‖ΔhX(t)‖Lq(Ω) = sup
t ∈ [a, b]
|h| ≤ δ,
(E|X(t + h) − X(t)|q)1/q ≤ Cqδ
τq ,
where Cp, Cq > 0 are some constants, τp > 1
p
− 1
q
and τq > 0.
Then
1) ω1
q (X, δ) = sup|h|≤δ
(∫ b
a |X(t + h) − X(t)|qdt
)1/q ∈ Lp(Ω)
2) ‖ω1
q (X, δ)‖Lp(Ω) =
(
E
(
sup|h|≤δ
(∫ b
a |X(t + h) − X(t)|qdt
)1/q
)p)1/p
≤
21/p+2τq C2
q (b−a)2/qδ2τq
2δ
p−1
+2τqCq(b−a)1/qδτq−2
√
δ2
(p−1)2
+
2τq Cq(b−a)1/qδ1+τq
p−1
.
Proof. Let’s determine the stochastic process ξ(h) = ‖ΔhX(·)‖Lq[a,b], |h| < δ
(so, ω1
q (X, δ) = sup|h|≤δ ξ(h)). We can conclude from the corollary 2 that
ξ ∈ Lp(Ω). Then, Lyapunov inequality, Fubini theorem and condition b)
yield
β := sup
h1,h2∈[−δ,δ]
‖ξ(h1) − ξ(h2)‖Lp(Ω) =
= sup
h1,h2∈[−δ,δ]
∥∥∥‖Δh1X(·)‖Lq[a,b] − ‖Δh2X(·)‖Lq[a,b]
∥∥∥
Lp(Ω)
≤
≤ sup
h1,h2∈[−δ,δ]
∥∥∥‖Δh1X(·) − Δh2X(·)‖Lq[a,b]
∥∥∥
Lp(Ω)
=
= sup
h1,h2∈[−δ,δ]
⎛
⎝E
(∫ b
a
|X(t + h1) − X(t + h2)|qdt
) p
q
⎞
⎠
1
p
≤
≤ sup
h1,h2∈[−δ,δ]
(
E
∫ b
a
|X(t + h1) − X(t + h2)|qdt
) 1
q
=
= sup
h1,h2∈[−δ,δ]
(∫ b
a
E|X(t + h1) − X(t + h2)|qdt
) 1
q
≤
≤
(∫ b
a
(Cq(2δ)
τq)q dt
) 1
q
≤ Cq(2δ)
τq(b − a)1/q < ∞.
Since, the entropy integral
∫
0+
(
δ
ε
+ 1
)1/p
dε ∼
∫
0+
dε
ε1/p
< ∞, as p > 1,
then the theorem 3.3 [2, p.120] implies that
314 TETYANA YAKOVENKO
1) sup|h|≤δ ξ(h) = ω1
q (X, δ) ∈ Lp(Ω)
2)
∥∥∥sup|h|≤δ ξ(h)
∥∥∥
Lp(Ω)
=
∥∥∥ω1
q(X, δ)
∥∥∥
Lp(Ω)
≤
≤ inf |h|≤δ ‖ξ(h)‖Lp(Ω) + inf0<θ2<1
1
θ2(1−θ2)
θ2β∫
0
(
δ
ε
+ 1
)1/p
dε.
Let’s calculate each term of latter inequality.
inf
|h|≤δ
‖ξ(h)‖Lp(Ω) = inf
|h|≤δ
∥∥∥‖X(· + h) − X(·)‖Lq[a,b]
∥∥∥
Lp(Ω)
= 0, when h = 0.
∫ θ2β
0
(
δ
ε
+ 1
)1/p
dε =
∫ δ
0
(
2δ
ε
)1/p
dε +
∫ θ2Cq(2δ)τq (b−a)1/q
δ
21/pdε =
= (2δ)1/p p
p − 1
δ1−1/p + 21/p
(
θ2Cq(2δ)
τq(b − a)1/q − δ
)
=
=
21/p
p − 1
δ + 21/p+τqCq(b − a)1/qδτqθ2 = A + Bθ2.
Function A+Bθ2
θ2(1−θ2)
possesses its minimum value B2
(2A+B)−2
√
A(A+B)
at point
θ2 =
√
A(A+B)−A
B
. So,
∥∥∥ω1
q (X, δ)
∥∥∥
Lp(Ω)
≤ 21/p+2τqC2
q (b − a)2/qδ2τq
2δ
p−1
+ 2τqCq(b − a)1/qδτq − 2
√
δ2
(p−1)2
+ 2τq Cq(b−a)1/qδ1+τq
p−1
.
And now let’s find the conditions under which stochastic process X =
{X(t), t ∈ [a − δ, b + δ]} belongs to the Besov space when 1 < p < q < ∞,
n = 0, 0 < α < 1. Under this assumptions we can use ω1
q instead of ω2
q
in definition of the Besov space and norm in this particular Besov space
Bαp
q [a, b] is following:
‖X‖Bαp
q [a,b] = ‖X‖Lq[a,b] +
∥∥∥∥∥ω
1
q(X, δ)
δα
∥∥∥∥∥
∗
p
=
=
(∫ b
a
|X(t)|qdt
)1/q
+
⎛
⎜⎝∫ ∞
0
∣∣∣∣∣∣∣
sup|h|≤δ
(∫ b
a |X(t + h) − X(t)|qdt
)1/q
δα
∣∣∣∣∣∣∣
p
dδ
δ
⎞
⎟⎠
1/p
.
Theorem 3. If for stochastic process X supt∈[a,b] (E|X(t)|p)1/p < ∞ and
all the assumptions of the theorem 2 hold true, then:
1) for 0 < τq ≤ 1 the stochastic process X belongs to the Besov space
Bαp
q [a, b], 0 < α < τq with probability one;
STOCHASTIC PROCESSES IN SOME BESOV SPACES 315
2) for τq > 1 the stochastic process X belongs to the Besov space Bαp
q [a, b]
with probability one for all 0 < α < 1.
Proof. Applying the theorem 4.1 [1] to the process X we will get that it
belongs to the space Lq[a, b] with probability one. Then:
for 0 < τq ≤ 1 the estimation for
∥∥∥ω1
q(X, δ)
∥∥∥
Lp(Ω)
from the theorem 2
21/p+2τqC2
q (b − a)2/qδ2τq
2δ
p−1
+ 2τqCq(b − a)1/qδτq − 2
√
δ2
(p−1)2
+ 2τq Cq(b−a)1/qδ1+τq
p−1
∼ δτq as δ → 0
and
E
∫
0+
∣∣∣∣∣ω
1
q (X, δ)
δα
∣∣∣∣∣
p
dδ
δ
=
∫
0+
E
∣∣∣∣∣ω
1
q (X, δ)
δα
∣∣∣∣∣
p
dδ
δ
=
∫
0+
E
∣∣∣ω1
q(X, δ)
∣∣∣p dδ
δαp+1
=
=
∫
0+
(
‖ω1
q (X, δ)‖Lp(Ω)
)p dδ
δαp+1
∼
∫
0+
δτqp
δαp+1
dδ < ∞, as 0 < α < τq;
for τq > 1
21/p+2τqC2
q (b − a)2/qδ2τq
2δ
p−1
+ 2τqCq(b − a)1/qδτq − 2
√
δ2
(p−1)2
+ 2τq Cq(b−a)1/qδ1+τq
p−1
∼ δ2τq−1 as δ → 0
and
E
∫
0+
∣∣∣∣∣ω
1
q(X, δ)
δα
∣∣∣∣∣
p
dδ
δ
∼
∫
0+
(
δ2τq−1
δα
)p
dδ
δ
< ∞, as 0 < α < 2τq − 1.
Since, 2τq − 1 ≥ 1 the last statement is true for all 0 < α < 1.
4. Conclusion
In this paper the processes in the Besov space are investigated. It was found
the conditions under which the trajectories of the stochastic process from
space Lp(Ω) belong to certain Besov space Bsp
q [a, b] when s = α and p < q.
Bibliography
1. Kozachenko, Yu.V. and Yakovenko, T.O., Conditions under which stochas-
tic processes belong to some function Orlicz spaces. Bulletin of Kiyv Uni-
versity 5 , Kiev, (2002), 64–74.
2. Buldygin, V.V. and Kozachenko, Yu.V., Metric characterization of random
variables and random processes, Amer. Math. Soc., Providence, RI (2000).
3. Härdle, W., Kerkyacharian, G., Picard, D., Tsybakov, A., Wavelets, Appro-
ximation, and Statistical Applications, Springer-Varlag, New York, (1998).
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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