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Nom original: Article3_Sghir_Aissa.pdfTitre: Strong approximation of some additive functionals of symmetric stable processAuteur: M. Ait Ouahra, A. Kissami and A. Sghir

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M athematical
I nequalities
& A pplications
Volume 17, Number 4 (2014), 1327–1336

doi:10.7153/mia-17-96

STRONG APPROXIMATION OF SOME ADDITIVE
FUNCTIONALS OF SYMMETRIC STABLE PROCESS
M. A IT O UAHRA , A. K ISSAMI AND A. S GHIR
(Communicated by N. Elezovi´c)
Abstract. This paper deals with some additive functionals based on the local time of symmetric
stable process. In concrete, we obtain some L p -inequalities of the local time and the fractional
derivative of the local time of symmetric stable process of index 1 < α 2 . As an application, we generalize the well known Barlow-Yor [4] inequality, which we use to give a strong
approximation version, (almost surely estimate), of occupation times problem of this process.
Our results generalize those obtained by Csaki et al. [7] for Brownian motion, and Ait Ouahra
and Ouali [2] for symmetric stable process of index 1 < α 2 in L p -norm.

1. Introduction
The strong approximations of Brownian additive functionals has been studied by
Csaki et al. [7] as counterparts of limit theorem for additive functionals which feature
the fractional derivative of Brownian local time. We first recall their result.


T HEOREM 1. Let f be a Borel function on R such that R |x|k | f (x)|dx < ∞, for
some k > 0 . Then for any 0 < γ < 32 (with γ = 1 ) and all sufficiently small ε > 0 ,
when t goes to infinity, we have
t
0

Dγ −1 f (Bs )ds =



γ
I( f )
Dγ −1 lt. (0) + o(t 1− 2 −ε ),
Γ(1 − γ )

a.s.

where I( f ) = R f (x)dx , ltx is the local time of the Brownian motion B and Dγ −1 is a
fractional derivative of order γ − 1 , (see definition below).
This theorem and the law of the iterated logarithm, (LIL for brevity), of Csaki et
al. [8], proved for Dγ −1 Lt. (0), together imply that there exists a constant 0 < c(γ ) < ∞,
depending only on γ , such that
t

γ −1 f (B )ds
s
0D
γ
γ
1−
t→∞ t
2 (log logt) 2

lim sup

= c(γ )

a.s.

On the other hand, Ait Ouahra and Ouali [2] have established the following result, in
L p -norm, for some self similar process, namely symmetric stable process X of index
1 < α 2 and fractional Brownian motion with Hurst parameter 0 < H < 1 .
Mathematics subject classification (2010): 60J55.
Keywords and phrases: Strong approximation, additive functional, stable process, fractional derivative,
local time, Barlow-Yor inequality.
c

, Zagreb
Paper MIA-17-96

1327

1328

M. A IT O UAHRA , A. K ISSAMI AND A. S GHIR



T HEOREM 2. Let f be a Borel function on R such that R |x|k | f (x)|dx < ∞, for
some k > 0 . Then for any 0 γ < α −1
2 and all sufficiently small ε > 0 and p 1 ,
when t goes to infinity, we have


t
0

Dγ f (Xs )ds 2p =

α −1 γ
I( f )
Dγ Lt. (0) 2p + o(t α − α −ε ),
Γ(1 − γ )

1

where . 2p = (E0 |.|2p ) 2p and Ltx the local time of symmetric stable process X .
R EMARK 1. The same estimation in Theorem 2 can be obtained for the self similar process called Sub-fractional Brownian motion, (sfBm for brevity), of Hurst paramter
0 < H < 1 , (see definition of sfBm in Bojdecky et al. [5]).
Our purpose in this paper is to extend the result of Csaki et al. [7], to symmetric
stable process of index 1 < α 2 . We will prove the following theorem.


T HEOREM 3. Let f be a Borel function on R such that R |x|k | f (x)|dx < ∞, for
some k > 0 . Then for any 0 < γ < α −1
2 and all sufficiently small ε > 0 , when t goes
to infinity, we have
t
0

Dγ f (Xs )ds =

α −1 γ
I( f ) γ .
D Lt (0) + o(t α − α −ε ),
Γ(−γ )

a.s.

The remainder of this paper is organized as follows: In the next section, we establish some L p -inequalities for some additive functionals of symmetric stable process.
Finally, in the last section, we give the prove of Theorem 3.
Most of the estimates in this paper contain unspecified positive constants. We use
the same symbol C for these constants, even when they vary from one line to the next.
2. Some inequalities for some additive functionals of symmetric stable process
Throughout this paper, X = (Ω, F , Ft , θt , Xt , Px ) will denote the canonical realization of a real valued symmetric stable process of index 1 < α 2 , with X0 = 0 . The
sample paths of Xt are right-continuous with left limits a.s. (c`adl`ag for brevity), and
has stationary independent increments with characteristic function
E0 exp(iλ Xt ) = exp(−t|λ |α ),

∀t 0, λ ∈ R.

E0 denotes the expectation with respect to the distribution P0 of the process starting
from 0 . And (θt ) : Ω → Ω are the translation operators defined by (θt (ω ))(s) = ω (t +
s).
Notice that for α = 2 , X is a Brownian motion.
It is known from Barlow [3] and Boylan [6] that the local time {Ltx ; t 0, x ∈ R}
of X exists and is jointly continuous in t and x with compact support and satisfies the
scaling property


1
1
α −1
(1)
∀λ > 0,
(Ltx ,t 0, x ∈ R, Py ) d λ − α Lxλλt α ,t 0, x ∈ R, Pyλ α

A PPROXIMATION OF ADDITIVE FUNCTIONALS OF SYMMETRIC STABLE PROCESS

1329

where d means the equality in the sense of the finite-dimensional distributions.
In addition, Ltx is an additive functional
x
Ltx (ω ) = Lxs (ω ) + Lt−s
(θs (ω )),

(2)

and satisfies the occupation density formula
t
0

f (Xs (ω ))ds =


R

f (x)Ltx (ω )dx,

(3)

for any bounded or nonnegative Borel function f .
Moreover, by Lemma 3.3 in Marcus and Rosen [10] and Theorem 1 in Ait Ouahra
and Eddahbi [1] and the Kolmogorov criterion, for all T > 0 fixed, we have almost
surely: ∀ 0 < β < αα−1 , ∃ 0 < C < ∞ such that ∀ 0 t, s T , |x| M , where M is
a constant
|Ltx − Lxs | C|t − s|β .
∀ 0 < β1 < α2−1
α , ∀ 0 < β2 <
where M is a constant

α −1
2 ,

∃ 0 < C < ∞ such that ∀ 0 t, s T , |x|, |y| M ,

y

|Ltx − Lt − Lxs + Lys | C|t − s|β1 |x − y|β2 .
For 0 < γ <

α −1
2 ,

(4)

(5)

we define the fractional derivative of Ltx as follows:
Htx := Dγ Lt. (x) =

1
Γ(−γ )

t
0

ds
,
(Xs − x)1+γ

where yγ := |y|γ sgn(y).
According to Fitzsimmons and Getoor [9], Htx is an additive functional satisfying
the scaling property
(Htx ,t 0, x ∈ R, Py ) d



1
1
α −1 γ
λ − α − α Hλxλt α ,t 0, x ∈ R, Pyλ α ∀λ > 0.

(6)

On the other hand, Ait Ouahra and Eddahbi [1] showed in Lemma 1 that, for all T > 0
fixed, we have almost surely ∀ 0 < β < αα−1 − αγ , ∃ 0 < C < ∞ such that ∀ 0 t, s
T , |x| M , where M is a constant
|Htx − Hsx | C|t − s|β .

(7)

We refer the reader for a complete survey on the fractional derivative to Samko et al.
[11] and the references therein.
The next lemma follows easily from the scaling properties (1) and (6).

1330

M. A IT O UAHRA , A. K ISSAMI AND A. S GHIR

L EMMA 1. For all 1 < α 2 , 0 < ν <

α −1
2

and p 1 , we have





α −1
x
0
x
0
α
sup L1 ,t 0, P
d t
sup Lt ,t 0, P
x∈R
x∈R



.
α −1 + 1
0
d t α pα L.1 p,R ,t 0, P0
Lt p,R ,t 0, P




α −1 γ
sup Htx ,t 0, P0 d t α − α sup H1x ,t 0, P0
x∈R
x∈R


α −1 γ 1

.
0
sup Hs p,R ,t 0, P
d t α − α + pα H1. p,R ,t 0, P0
0 s t




sup sup

0 s t x =y

|Lxs − Lys |
,t 0, P0
|x − y|ν


where . p,R = (

R |.|

1

p) p





d

t

α −1 − ν
α
α

sup
x =y

|Lx1 − Ly1 |
,t 0, P0
|x − y|ν

(8)
(9)
(10)
(11)

(12)

.

In order to establish our results, we need the following lemma, (see Fitzsimmons
and Getoor [9]).
L EMMA 2. Let (At )t 0 be a continuous increasing (Ft )-adapted real valued
process with A0 = 0 . Assume that:
(i) At As + K.At−s ◦ θs , ∀ t, s 0 , for some constant K > 0 .
(ii) There exists a constant q > 0 such that
lim

1

sup Py (Aλ > λ q z) = 0.

z→∞ λ >0,y∈R

Then for each p > 0 , there exists a constant 0 < C < ∞ such that
1

At p Ct q ,

∀ t 0,

1

where . p = (E0 |.| p ) p .
The first application of Lemma 2 is the following result.
L EMMA 3. For each p, p 1 , there is a constant 0 < C < ∞ such that
sup Ltx p Ct

α −1
α

(13)

x∈R

sup |Ltx − Lxs | p C|t − s|

α −1
α

x∈R

Lt. p ,R p Ct

α −1 + 1
α
p α

sup Hs. p ,R p C|t − s|
0 s t

α −1 − γ + 1
α
α
p α

(14)
(15)
(16)

A PPROXIMATION OF ADDITIVE FUNCTIONALS OF SYMMETRIC STABLE PROCESS

1331

Proof. (13): We apply Lemma 2 with
At = sup Ltx ,

q=

x∈R

α − 1 −1

α

.

Clearly, At is increasing and satisfies (i) in Lemma 2 by (2).
Using (3), we get Ltx (τy (ω )) = Ltx−y (ω ) where τy : Ω → Ω is the translation ω →
ω (.) + y. Since X is spatially homogeneous i.e. ( τy (P0 ) = Py ), it follows that the
Py distribution of (At ) does not depend on y. Thus, by applying (8) and the fact that
0 < A1 < ∞ by (4), we get
lim

1

1

sup Py (Aλ > λ q z) = lim sup P0 (Aλ > λ q z) = lim P0 (A1 > z) = 0.

z→∞ λ >0,y∈R

z→∞ λ >0

z→∞

Which completes the proof of (13).
Now, we verify (14). From the Markov property of X in s and (13), we have
x
E0 (sup |Ltx − Lxs | p ) = E0 (sup |Lt−s
◦ θs | p )
x∈R

x∈R

x
= E0 (sup |Lt−s
| p ◦ θs )
x∈R

x
= E0 (E0 (sup |Lt−s
| p ◦ θs /Fs )

=
=




x∈R

x−y

0

P (Xs ∈ dy)E0 (sup |Lt−s | p )
x∈R

x
P0 (Xs ∈ dy)E0 (sup |Lt−s
| p)
x∈R

C|t − s|

α −1
α

.

This gives the desired estimate.
Next, we apply Lemma 2 with
At = Lt. p ,R ,

q=

α −1

α

+

1 −1
.
p α

Clearly, At is increasing and satisfies (i) in Lemma 2.
The Py distribution of (At ) does not depend on y by the translation invariance
of the norm . p ,R . The finiteness of A1 follows from (4) and the fact that Ltx has a
compact support. Thus by a scaling property (9), we get
lim

1

sup Py (Aλ > λ q z) = lim P0 (A1 > z) = 0.

z→∞ λ >0,y∈R

z→∞

Finally, using the same method as above, we obtain (16).



A more interesting application of Lemma 2 is the following inequality which generalize the well known Barlow-Yor inequality to symmetric stable process of index
1 < α 2 . (See Barlow and Yor [4] for the Brownian motion case).

1332

M. A IT O UAHRA , A. K ISSAMI AND A. S GHIR

L EMMA 4. For all t > 0 , 0 < ν < α −1
2 and p 1 , there exists a constant 0 <
C < ∞, such that



α −1 ν
|Lxs − Lys |


Ct α − α .
sup sup
0 s t x =y |x − y|ν
p

Proof. We apply Lemma 2 with
At = sup sup
0 s t x =y

|Lxs − Lys |
,
|x − y|ν

q=

α −1

α



ν −1
.
α

Clearly, (At ) is increasing and satisfies (i) in Lemma 2.
In fact, for all 0 s t , we have
At = sup sup
0 r t x =y



|Lxr − Lyr |
|x − y|ν

|Lx − Lyr |
|Lxr − Lyr |
max sup sup r
;
sup
sup
ν
ν
s r t x =y |x − y|
0 r s x =y |x − y|



= max {As ; At−s ◦ θs }
As + At−s ◦ θs .
Moreover,
At−s ◦ θs =
=
=

sup sup

|Lxr ◦ θs − Lyr ◦ θs |
|x − y|ν

sup sup

|Lxr ◦ θs − Lyr ◦ θs |
|x − y|ν

sup sup

|Lxr+s − Lxs − Lyr+s + Lys |
|x − y|ν

0 r t−s x =y

s r+s t x =y

s r+s t x =y

= sup sup
s r t x =y

|Lxr − Lxs − Lyr + Lys |
.
|x − y|ν

It follows from (5) that
At − As At−s ◦ θs C sup |r − s|β1 C|t − s|β1 ,
s r t

for any 0 < β1 < α2−1
α . Which implies that At is continuous.
x
Since Lt (τy (ω )) = Ltx−y (ω ) and X is spatially homogenous, it follows that the Py
distribution of At does not depend on y. Thus, by (12) we have



x − Ly |
1
1
|L
y
0
0
1
1
> z = 0.
lim sup P (Aλ > λ q z) = lim sup P (Aλ > λ q z) = lim P sup
ν
z→∞ λ >0,y∈R
z→∞ λ >0
z→∞
x =y |x − y|

A PPROXIMATION OF ADDITIVE FUNCTIONALS OF SYMMETRIC STABLE PROCESS

1333

Finally, by (5) and the fact that Lx0 = 0 , we get
sup
x =y

|Lxs − Lys |
< ∞,
|x − y|ν

a.s.



This completes the proof of Lemma (4).

The proof of Theorem 3 is based on the following
C OROLLARY 1. For any ν ∈]0, α −1
2 [ and ε > 0 , when t → ∞,
sup
x =y

α −1 ν
|Lxs − Lys |
= o(t α − α +ε ),
|x − y|ν

a.s.

(17)

Proof. Using Tchebychev’s inequality and Lemma 4 with p = ε2 , for any n 1 ,



α −1 − ν +ε
|Lxs − Lys |
0
>n α α
sup sup
C(ν , ε )n−2 .
P
ν
0 s n x =y |x − y|
Then, by the Borel-Cantelli lemma, we get as n → +∞, almost surely,
y

sup sup

0 s n x =y

α −1 ν
|Lxs − Ls |
= O(n α − α +ε ).
ν
|x − y|

Since At is increasing and since ε can be arbitrarily small, we have proved the corollary.
3. Proof of Theorem 3
The idea of the proof is inspired from that used in Csaki et al. [7] for the Brownian
motion case. For this, we need the following lemmas.
L EMMA 5. For any 0 < γ < δ < α −1
2 and ε > 0 , when t → ∞,


1 Lx+y − Lx−y
α −1 δ


t
t
sup
dy = o(t α − α +ε ),
a.s.
1+γ


y
x∈R 0


∞ Lx+y − Ly
α −1 δ


t
t
sup
dy = o(t α − α +aδ +ε ),
1+γ


y
a
1
|x| t

a.s.

(18)

(19)

for some a > 0 .
Proof. We have almost surely,



1 Lx+y − Lx−y
α −1 δ
|Ltx+y − Ltx−y | 1 dy


t
t
sup
dy
sup
= o(t α − α +ε ),

sup

1+γ
δ
1+γ −δ


y
y
y
0
0
x∈R
x∈R 0<y 1

1334

M. A IT O UAHRA , A. K ISSAMI AND A. S GHIR

where we have used in the last equality (17) and the fact that δ > γ .
Next, by (17), almost surely,


x+y
y
∞ Lx+y − Ly
α −1 δ
|Lt − Lt | ∞ dy


t
t
δ

sup
dy
|x|
sup
= o(t α − α +aδ +ε ).
sup

1+γ
1+γ
δ


y
y
a
a
|x|
1
1
y∈R
|x| t
|x| t
α −1
2

L EMMA 6. For any 0 < γ <



and ε > 0 , when t → ∞,

t



γ
ds

= o(t αα−1 − α +ε ),
sup

1+
γ
x∈R 0 (Xs − x)

a.s.

(20)

Proof. This lemma follows immediately by the same arguments used in the proof
of Corollary 1. More precisely, we apply Lemma 2 with
At = sup sup |Htx |,
0 s t x∈R

q=

α − 1

α



γ −1
.
α



We are now able to prove Theorem 3.
Proof of Theorem 3. By Fubini’s theorem, we have
t

I( f ) γ .
D Lt (0)
Dγ f (Xs )ds −
Γ(−γ )
0



t
t
ds
ds
1
=

f (x)dx
1+γ
Γ(−γ ) R 0 (Xs − x)1+γ
0 Xs
1
(I1 (t) + I1(t)),
=
Γ(−γ )

I(t) =

where
I1 (t) =
I2 (t) =




|x|>t a

t
0




|x| t a

t
0

ds

(Xs − x)1+γ
ds

(Xs − x)1+γ


t
ds
0

1+γ

Xs


t
ds
0

1+γ

Xs

f (x)dx,
f (x)dx,

for some 0 < a α1 .
Let us deal with the first term I1 (t). By (20), we get
t

t

ds
ds

I1 (t) sup

|x|−k |x|k | f (x)|dx
1+γ
1+γ
|x|>t a
0 (Xs )
|x|>t a 0 (Xs − x)


o(t

α −1 − γ −ak+ε
α
α

)

= o(t

α −1 − γ −ak+ε
α
α

).

|x|>t a

|x|k | f (x)|dx
a.s.

1335

A PPROXIMATION OF ADDITIVE FUNCTIONALS OF SYMMETRIC STABLE PROCESS

Now, we deal with I2 (t). Using (3) and the fact that f is integrable, we obtain
t

t

ds
ds


| f (x)|dx
I2 (t) sup
1+γ
1+γ
|x| t a
0 (Xs )
|x| t a 0 (Xs − x)


∞ Lx+y − Lx−y − Ly − L−y

t
t
t
t
C sup


y1+γ
|x| t a 0




1 Lx+y − Lx−y − Ly − L−y
∞ Lx+y − Lx−y − Ly − L−y


t
t
t
t
t
t
t
t
C sup
+ C sup
,


y1+γ
y1+γ
|x| t a 0
|x| t a 1
which, in view of (18) and (19), implies
I2 (t) = o(t

α −1 − δ +ε
α
α

Then
I(t) = o(t

) + o(t

α −1 − δ +aδ +ε
α
α

α −1 − δ +aδ +ε
α
α

) + o(t

) = o(t

α −1 − δ +aδ +ε
α
α

α −1 − γ −ak+ε
α
α

).

).

a.s.

a.s.

Choosing
a=
it is clear that 0 < a

1
α

δ −γ
,
α (δ + k)

. It follows that
I(t) = o(t b+ε ),

with
b=

a.s.

α −1 γ
δ −ν
α −1 γ
<
− −k
− .
α
α
α (δ + k)
α
α

Then for all sufficiently small ε > 0 , when t → ∞,
I(t) = o(t
The theorem is proved.

α −1 − γ −ε
α
α

),

a.s.



R EMARK 2.
1. It would be interesting to prove the LIL for Dγ Lt. (0) in case of symmetric stable
process of index 1 < α 2 . This allows to deduce the LIL of the functional
t γ
0 D f (Xs )ds.
2. The L p - estimate of Ait Ouahra and Ouali [2] is proved for fractional Brownian
motion of Hurst parameter 0 < H < 1 which is a non Markovian process. The
question which arises is if we can also extend previous results to this process.

1336

M. A IT O UAHRA , A. K ISSAMI AND A. S GHIR
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[2] M. A IT O UAHRA AND M. O UALI , Occupation time problems for fractional Brownian motion and
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(Received June 26, 2011)

M. Ait Ouahra
Facult´e des Sciences Oujda
Laboratoire de Mod´elisation Stochastique
et D´eterministe et URAC 04. B.P. 717. Maroc
e-mail: ouahra@gmail.com
A. Kissami
Facult´e des Sciences Oujda
Laboratoire de Mod´elisation Stochastique
et D´eterministe et URAC 04. B.P. 717. Maroc
e-mail: Kissami@fso.ump.ma
A. Sghir
Facult´e des Sciences Oujda
Laboratoire de Mod´elisation Stochastique
et D´eterministe et URAC 04. B.P. 717. Maroc
e-mail: semastai@hotmail.fr

Mathematical Inequalities & Applications

www.ele-math.com
mia@ele-math.com


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