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Serials Publications

Communications on Stochastic Analysis
Vol. 11, No. 4 (2017) 383-397

www.serialspublications.com

OCCUPATION TIME PROBLEM OF CERTAIN
SELF-SIMILAR PROCESSES RELATED TO THE
FRACTIONAL BROWNIAN MOTION
AISSA SGHIR, MOHAMED AIT OUAHRA, AND SOUFIANE MOUSSATEN

Abstract. In this paper, we prove some limit theorem for occupation time
problem of certain self-similar processes related to the fractional Brownian
motion, namely the bifractional Brownian motion, the subfractional Brownian motion and the weighted fractional Brownian motion. The key ingredients
to prove our results is the well known Potter’s Theorem involving slowly varying functions. We give also the Lp -estimate version of strong approximation
of our limit theorem.

1. Introduction
Throughout this paper, we use the same symbol Y τ := (Ytτ , t ≥ 0) to denote
each of the Gaussian τ -self-similar processes: the fractional Brownian motion (τ =
H, fBm for short), the bifractional Brownian motion (τ = HK, bfBm for short),
the subfractional Brownian motion (τ = α2 , sfBm for short) and the weighted
τ
fractional Brownian motion (τ = 1+b
2 , wfBm for short) and we denote L :=
τ
(L (t, x) , t ≥ 0 , x ∈ R) its local time, (see the definitions below). For any fixed
t0 ≥ 0 and any ρ > 0, we define the tangent process related to Y τ as follows:
(
)
Ytτ0 +ρt − Ytτ0
ρ,τ
ρ,τ
Y
:= Yt =
, t≥0 ,
ρτ
and we denote Lρ,τ := (Lρ,τ (t, x) , t ≥ 0 , x ∈ R) its local time.
It is proved recently in [1], and in [18] that when ρ goes to zero, the process
(Ytρ,τ , t ≥ 0) converges, in the sense of the finite dimensional distributions, to the
fBm of Hurst parameter τ , (up to a multiplicative constant). Notice that in the
fBm case we have: (Y ρ,H , t ≥ 0) d (Y H , t ≥ 0), where d denotes the equalities
of the finite dimensional distributions.
The aim of the present paper is to obtain a limit theorem for normalized occupation time integrals of the form:
∫ nt
1
f (Ysρ,τ )ds,
(1.1)
n1−τ (1+γ) 0
Received 2017-3-25; Communicated by the editors.
2010 Mathematics Subject Classification. 60G18; 60J55; 60F17.
Key words and phrases. Limit theorems, strong approximation, self-similar process, bifractional Brownian motion, subfractional Brownian motion, weighted fractional Brownian motion,
local time, fractional derivative; slowly varying function.
383

384

AISSA SGHIR, MOHAMED AIT OUAHRA, AND SOUFIANE MOUSSATEN

l,γ
l,γ
where f = K±
g, and K±
is the generalized fractional derivative of order γ > 0
generated by a slowly varying function l, (see the definitions below), and g ∈
C β ∩ L1 (R) with compact support. C β is the space of functions satisfying a H¨older
condition of order some β > 0. The process in (1.1) was studied in the case of the
γ
classical fractional derivative D±
where l ≡ 1, and we refer to Yamada [24],[25]
1
for Brownian motion case (τ = 2 ) and to Shieh [21] for fBm case (τ = H). Notice
that even if f is not a fractional derivative of some function g, the limiting process
in (1.1) is a fractional derivative of local time.
We end this section by the definitions of the Gaussian τ -self-similar processes
studied in this paper. The first process is the bfBm with parameters H ∈ (0, 1)
and K ∈ (0, 1] introduced in [14]. It is a (τ = HK)-self-similar Gaussian process,
centered, starting from zero, with covariance function:
]
K
1 [
RH,K (t, s) = K (t2H + s2H ) − |t − s|2HK .
2
The case K = 1 corresponds to the fBm [16] of Hurst parameter τ = H ∈ (0, 1).
The second process is the sfBm with parameter α ∈ (0, 2). It is an extension of
Brownian motion (H = 21 ) or (α = 1), which preserves many properties of fBm
but not the stationarity of the increments. It was introduced by Bojdecki et al.
[7]. It is a (τ = α2 )-self-similar Gaussian process, centered, starting from zero,
with covariance function:
1
RH (t, s) = tα + sα − [(t + s)α + |t − s|α ].
2
The third process is the wfBm with parameters a and b introduced in [8]. It
is a (τ = 1+b
2 )-self-similar Gaussian process, centered, starting from zero, with
covariance function:
∫ s∧t [
]
b
b
Ra,b (t, s) =
ua (t − u) + (s − u) du,
0

where a > −1, −1 < b < 1, and |b| < 1 + a. Clearly, if a = 0, the process coincides
with the fBm with Hurst parameter 12 (1 + b), (up to a multiplicative constant).
The remainder of this paper is organized as follows: In the next section, we
present some basic facts about local time and the generalized fractional derivative.
In section 3, we give the proof of our limit theorem. Finally, in the last section, we
state and prove strong approximation version of our limit theorem, more precisely,
we show the Lp -estimate version.
Notice that most of the estimates in this paper contain unspecified finite positive
constants. We use the same symbol C to denote these constants, even when they
vary from one line to the next.
2. Local Time and the Generalized Fractional Derivatives
We begin this section by a briefly survey on local time and we refer to [12].
Let X := (Xt , t ≥ 0) be a real-valued separable random process with Borel
sample functions. For any Borel set B ⊂ R+ , the occupation measure of X on B
is defined as:
µB (A) = λ{s ∈ B ; Xs ∈ A},
∀A ∈ B(R),

OCCUPATION TIME PROBLEM-FRACTIONAL BROWNIAN MOTION

385

where λ is the one-dimensional Lebesgue measure on R+ . If µB is absolutely
continuous with respect to λ, we say that X has a local time on B denoted by
L(B, .). Moreover, the local time satisfies the occupation density formula: for
every Borel set B ⊂ R+ and every measurable function f : R → R+ , we have


f (Xt )dt =
f (x)L(B, x)dx,
B

R

and we have the following representation of local time:
)
∫ (∫ t
1
iu(Xs −x)
e
ds du.
L(t, x) := L([0, t], x) =
2π R
0

(2.1)

This representation due to the Fourier analysis for local time, have played a central
role to study the regularities properties of local time of our processes. Tudor
and Xiao [22] have proved, by using Lamperti’s transform and the concept of
strong local nondeterminism introduced by Berman [5], the existence and the
joint continuities of local time of bfBm. The case of fBm was given by Xiao [23].
Mendy [17] have studied the local time of sfBm for any α ∈ (0, 1), by using a
decomposition in law of sfBm given in [4]. Notice that the same arguments used
in [17] with a decomposition in law of bfBm given in [15] give easily the H¨older
regularities of bfBm. The case of wfBm for any a ≥ 0 and −1 < b < 1 was given
in [18]. Finally and more precisely, we have the following H¨older regularities of
the local time Lτ where τ = HK ∈ (0, 1) for the bfBm, τ = H ∈ (0, 1) for the
fBm, τ = α2 ∈ (0, 12 ) for the sfBm and τ = 1+b
2 ∈ (0, 1) for the wfBm.
Theorem 2.1. For any integer p ≥ 1, there exists a constant δ > 0 and C > 0
such that for any t ≥ 0, any h ∈ (0, δ), any x, y ∈ R and any 0 < ξ < 1−τ
2τ , there
hold:
∥Lτ (t + h, x) − Lτ (t, x)∥2p ≤ Ch1−τ ,
(2.2)
∥Lτ (t + h, y) − Lτ (t, y) − Lτ (t + h, x) + Lτ (t, x)∥2p ≤ C|y − x|ξ h1−τ (1+ξ) , (2.3)
1

where ∥.∥2p = (E|.|2p ) 2p .
Remark 2.2. Following the same arguments used in [1] to prove Theorems 3.1 and
3.2, and the motivation in [9]: page 862, it is easy to see that the tangent process
Y ρ,τ has the local time:
Lτ (t0 + ρt, ρτ x + Ytτ0 ) − Lτ (t0 , ρτ x + Ytτ0 )
Lρ,τ (t, x) =
.
ρ1−τ
In fact, by virtue of (2.1) and a changes of variables: v = ρuτ and z = t0 + ρs, we
have
Lτ (t0 + ρt, ρτ x + Ytτ0 ) − Lτ (t0 , ρτ x + Ytτ0 )
ρ1−τ
(∫ ∫ t0 +ρt
)
∫ ∫ t0
1
iv (Yzτ −(ρτ x+Ytτ0 ))
iv (Yzτ −(ρτ x+Ytτ0 ))
e
=
dzdv −
e
dzdv
2πρ1−τ
R 0
R 0
∫ ∫ t0 +ρt
τ
τ
τ
1
eiv(Yz −(ρ x+Yt0 )) dzdv
=
1−τ
2πρ
R t0

386

AISSA SGHIR, MOHAMED AIT OUAHRA, AND SOUFIANE MOUSSATEN

=

1


∫ (∫

t

ρ,τ

eiu(Ys
R

−x)

)
ds du = Lρ,τ (t, x).

0

An immediate consequence of Theorem 2.1 is the following result concerning
the regularities properties of the local time Lρ,τ .
Theorem 2.3. For any integer p ≥ 1, there exists a constant C > 0 such that
for any s, t ≥ 0, any x, y ∈ R, any 0 < ξ < 1−τ
2τ and any ρ > 0 sufficiently small,
there hold:
∥Lρ,τ (t, x) − Lρ,τ (s, x)∥2p ≤ C | t − s |1−τ
(2.4)
∥Lρ,τ (t, y)−Lρ,τ (s, y)−Lρ,τ (t, x)+Lρ,τ (s, x)∥2p ≤ C|y −x|ξ | t−s |1−τ (1+ξ) (2.5)
Proof. The proofs of (2.4) and (2.5) are similar. Let us deal for exemple with
(2.5). By virtue of (2.3), we have
∥Lρ,τ (t, y) − Lρ,τ (s, y) − Lρ,τ (t, x) + Lρ,τ (s, x)∥2p
=

∥Lτ (t0 + ρt, ρτ y + Ytτ0 ) − Lτ (t0 + ρs, ρτ y + Ytτ0 )
ρ1−τ
τ
τ
−L (t0 + ρt, ρ x + Ytτ0 ) + Lτ (t0 + ρs, ρτ x + Ytτ0 )∥2p
+
ρ1−τ

≤ C|y − x|ξ | t − s |1−τ (1+ξ) .


This gives the desired estimate.

Remark 2.4. The passage through the tangent process allowed us to obtain regularities without the condition | t − s |< δ, it was the motivation for which we chose
the tangent process Y ρ,τ instead of choosing the process Y τ in (1.1).
Now, we give the definition of the generalized fractional derivatives and we
refer to [11] and the references therein. For this, we collects some basic facts
about slowly varying function and we refer for example to Bingham et al. [6] and
Seneta [20].
Definition 2.5. A measurable function l : R+ → R+ is slowly varying at infinity
(in Karamata’s sense), if for all t positive, we have
lim

x→+∞

U (tx)
= 1.
U (x)

We are interested in the behavior of l at +∞, then in what follows, we assume
that l is bounded on each interval of the form [0, a], (a > 0). This assumption is
provided by Lemma 1.3.2 in [6]. For γ > 0, let kγ the function defined by:
{
l(y)
if y > 0,
y 1+γ ,
kγ (y) :=
0,
if y ≤ 0,
where l is slowly varying function at +∞, continuously differentiable on [a, +∞[,
(a > 0), (this property is given by Theorem 1.3.3 in [6]), and l(x) > 0 for all x > 0
and l(0+ ) = 1.

OCCUPATION TIME PROBLEM-FRACTIONAL BROWNIAN MOTION

387

For any γ ∈]0, β[ and g ∈ C β ∩ L1 (R), we define:
∫ +∞
1
l,γ
g(x) :=

kγ (y) [g(x ± y) − g(x)] dy,
Γ(−γ) 0
and we put:
l,γ
l,γ
.
K l,γ := K+
− K−

The following theorem called Potter’s Theorem, (see Theorem 1.5.6 in [6]), has
played a central role in the proof of our main results.
Theorem 2.6. 1) If l is slowly varying function, then for any chosen constants
A > 1 and δ > 0, there exists X = X(A, δ) such that:
{( ) ( ) }
y δ y −δ
l(y)
≤ A max
,
(x ≥ X, y ≥ X).
l(x)
x
x
2) If further, l is bounded away from 0 and ∞ on every compact subset of [0, +∞[,
then for every δ > 0, there exists A = A(δ) > 1 such that:
{( ) ( ) }
l(y)
y δ y −δ
≤ A max
,
(x > 0, y > 0).
l(x)
x
x
l,γ
l,γ
satisfy the switching identity:
and K−
Remark 2.7. 1) K+


l,γ
l,γ
f (x)dx,
g(x)K+
f (x)K− g(x)dx =

(2.6)

R

R

for any f, g ∈ C β ∩ L1 (R) and γ ∈]0, β[.
2) For h : R → R and a > 0, we denote by ha the function x → h(ax). Then,
l( . ),γ

l,γ

(ha ) = aγ (K± a

)a ,

∀ γ > 0,

∀ a > 0,

(2.7)

where l( a. ) : x 7−→ l( xa ).
3) If we take l ≡ 1, we recover the definition of the classical fractional derivative
denoted by: Dγ , (see [19], [24] and the references therein), where
{ 1
if y > 0,
y 1+γ ,
kγ (y) :=
0,
if y ≤ 0,
γ

g(x) :=

1
Γ(−γ)


0

+∞

g(x ± y) − g(x)
dy,
y 1+γ

and
γ
γ
Dγ := D+
− D−
.

Now we are ready to state and prove the main results of this section.
l,γ
Theorem 2.8. Let 0 < γ < ξ and K ∈ {K±
, K l,γ }. For any integer p ≥ 1, there
exists a constant C > 0, such that for any t, s ≥ 0 and any x ∈ R and any ρ > 0
sufficiently small, there hold:

∥KLρ,τ (t, .)(x) − KLρ,τ (s, .)(x)∥2p ≤ C | t − s |1−τ (1+γ) .

388

AISSA SGHIR, MOHAMED AIT OUAHRA, AND SOUFIANE MOUSSATEN

l,γ
Proof. We treat only the case K = K+
, the other cases are similar. Let b =
l,γ
|t − s|τ . By the definition of K+ , we have
l,γ ρ,τ
l,γ ρ,τ
∥K+
L (t, .)(x) − K+
L (s, .)(x)∥2p
∫ +∞
1
∥Lρ,τ (t, x + u) − Lρ,τ (t, x) − Lρ,τ (s, x + u) + Lρ,τ (s, x)∥2p

l(u)
du
|Γ(−γ)| 0
u1+γ
≤ I1 + I2 .

where
I1 :=

1
|Γ(−γ)|
×



b

l(u)
0

∥Lρ,τ (t, x + u) − Lρ,τ (t, x) − Lρ,τ (s, x + u) + Lρ,τ (s, x)∥2p
du
u1+γ

and
1
I2 :=
|Γ(−γ)|



+∞

l(u)
b

∥Lρ,τ (t, x + u) − Lρ,τ (t, x) − Lρ,τ (s, x + u) + Lρ,τ (s, x)∥2p
du.
u1+γ
We estimate I1 and I2 separately. Since l is bounded on each compact in R+ ,
it follows from (2.5) that:
×

I1 ≤ C | t − s |1−τ (1+ξ) bξ−γ
≤ C | t − s |1−τ (1+γ) .
Potter’s Theorem with 0 < ξ < γ implies the existence of A(ξ) > 1 such that:
( u )ξ
l(u) ≤ A(ξ)l(b)
.
b
Combining this fact with (2.4), we obtain:
I2 ≤ C | t − s |1−τ (1+γ) .


The proof of Theorem 2.8 is done.

We end this section by the following result. It will be useful in the sequel to
prove the tightness in our limit theorem.
l,γ
Corollary 2.9. Let 0 < γ < ξ and K ∈ {K±
, K l,γ }. For any integer p ≥ 1,
there exists a constant C > 0, such that for any t, s ≥ 0, any x ∈ R, any ρ > 0
sufficiently small and any n sufficiently large, there holds:
)
(
)
(

l . ,γ
(x)
( x )
.
l



−τ
−τ
Lρ,τ (t, .) τ − K n
Lρ,τ (s, .) τ
[l(nτ )]−1 K n

n
n
2p

≤C |t−s|
,
)
(
)
x
.
: x 7−→ l n−τ
.
where l n−τ
1−τ (1+γ)

(

OCCUPATION TIME PROBLEM-FRACTIONAL BROWNIAN MOTION

)

(
l

.
n−τ

Proof. We treat only the case K
(+ )
l

.
n−τ



, the other cases are similar. Let b =



|t − s|τ . By the
of K+
,(we have
( definition
)
)
.
.
l −τ
(
)
( )

l

n
n−τ


[l(nτ )]−1 K+
Lρ,τ (t, .) nxτ − K+
Lρ,τ (s, .) nxτ


389

2p

b

1
l(nτ u)
|Γ(−γ)| 0 l(nτ )


ρ,τ

L (t, nxτ + u) − Lρ,τ (s, nxτ + u) − Lρ,τ (t, nxτ ) + Lρ,τ (s, nxτ )
2p
×
du
u1+γ
∫ +∞
1
l(nτ u)
+
|Γ(−γ)| b
l(nτ )


ρ,τ

L (t, nxτ + u) − Lρ,τ (s, nxτ + u) − Lρ,τ (t, nxτ ) + Lρ,τ (s, nxτ )
2p
×
du
1+γ
u
:= J1 + J2 .
We estimate J1 and J2 separately. It follows from (2.5) that:


J1 ≤ C sup

u∈R+

l(nτ u)
| t − s |1−τ (1+δ) bδ−γ
l(nτ )

l(nτ u)
| t − s |1−τ (1+γ) .
τ
u∈R+ l(n )
Potter’s Theorem with 0 < ξ < γ implies the existence of A(ξ) > 1 such that:
u
l(nτ u) ≤ A(ξ)l(nτ b)( )ξ .
b
Combining this fact with (2.4), we obtain:
≤ C sup

J2 ≤ C

l(nτ b)
| t − s |1−τ (1+γ) .
l(nτ )

Finally, by using the fact that:
l(nτ u)
= 1,
n→+∞ l(nτ )
lim



we complete the proof of Corollary 2.9.
3. Limit Theorems
The main result of this section is the following result.

l,γ
Theorem 3.1. Let 0 < γ < ξ < 1−τ
2τ . Suppose that f = K± g where g ∈
β
1
C ∩ L (R) with compact support for some γ < β. Then when n → +∞ and
ρ → 0, the sequence of processes:
)
(
∫ nt
ρ,τ
1−τ (1+γ)
τ −1
f (Ys )ds
,
[n
l(n )]
0

t≥0

390

AISSA SGHIR, MOHAMED AIT OUAHRA, AND SOUFIANE MOUSSATEN

converges in law to the process:
)
(
γ τ
L (t, .)(0)
I(g)c−1−γ D∓

,
t≥0


where I(g) = R g(x)dx, Lτ is the local time of the fBm B τ of Hurst parameter τ
and the constant c is given by:
√ a2
1−K
2t
c = 1 (for sfBm), c = 2 2 (for bfBm) and c = √ 0 (for wfBm).
1+b
Remark 3.2.
1) Notice that even if f is not a fractional derivative of some function g, the
limiting process is a fractional derivative of local time.
2) We recall that τ = HK ∈ (0, 1) for bfBm, τ = H ∈ (0, 1) for fBm, τ = α2 ∈ (0, 21 )
for sfBm and τ = 1+b
2 ∈ (0, 1) for wfBm.
Proof of Theorem 3.1. The convergence of the finite dimensional distributions
follows easily by using the same arguments used in [9] to prove Proposition 5.2 in
case of the well known multifractional Brownian motion, and Remark 3.18 in [11].
In fact, according to [1], section 2.3, the process: (Ytρ,τ , t ≥ 0) converges, in the
sense of finite dimensional distributions, when ρ → 0, to the process: (c.Btτ , t ≥ 0),
where c is the constant appeared in Theorem 3.1. Therefore by combining the fact
that f is locally Riemann integrable and Theorem VI.4.2 in [13], we obtain
∫ nt
∫ nt
f (Ysρ,τ )ds −→
f (c.Bsτ )ds
as
ρ → 0+ .
0

0

Using the occupation density formula, the scaling property of local time, (2.6) and
(2.7), one can write:
∫ nt
c1+γ [n1−τ (1+γ) l(nτ )]−1
f (c.Bsτ )ds
∫0
(
x)
γ 1−τ (1+γ)
τ −1
= c [n
l(n )]
f (x)Lτ nt,
dx
c
R

( x )
γ
= [l(nτ )]−1 (cnτ )
f (x)Lτ t, τ dx
cn
∫R
( (
. ))
γ
τ −1
τ γ
= [l(n )] (cn )
g(x)K∓ Lτ t, τ
(x)dx
cn
R

( (
l(cnτ )
. ))
γ
τ −1
τ γ
=
[l(cn
)]
(cn
)
g(x)K∓
Lτ t, τ
(x)dx.
τ
l(n )
cn
R
According to [11], Remark 3.18, as n → ∞, we have
( (
. ))
−1
l,γ
γ τ
[l(cnτ )] (cnτ )γ K∓
Lτ t, τ
(x) −→ D∓
L (t, .)(0).
cn
By the definition of the slowly varying function l, we have
l(cnτ )
= 1.
n→+∞ l(nτ )
lim

OCCUPATION TIME PROBLEM-FRACTIONAL BROWNIAN MOTION

391

Finally, we obtain the convergence in the sense of finite dimensional distributions.
To end the proof of Theorem 3.1, we need only to show the tightness of the
sequence:

nt

Ant := [n1−τ (1+γ) l(nτ )]−1

f (Ysρ,τ )ds.
0

By the occupation density formula, the scaling property of local time and (2.6),
we have

)
(∫ nt
∫ ns


1
n
n
ρ,τ
ρ,τ

f (Yu )du
∥At −As ∥2p = τ 1−τ (1+γ)
f (Yu )du −

l(n )n
0
0
2p


( x)
( x)


ρ,τ
t, τ dx − f (x)Lρ,τ s, τ dx
= nτ γ [l(nτ )]−1
f (x)L

n
n
R
R
2p


[
( x)
( x )]

l,γ
ρ,τ
t, τ − Lρ,τ s, τ dx
= nτ γ [l(nτ )]−1
K± g(x) L

n
n
R
2p

[
(
)
(
) ]


.
.
l,γ ρ,τ
l,γ ρ,τ
τγ
τ −1
t, τ (x) − K∓ L
= n [l(n )] g(x) K∓ L
s, τ (x) dx
.
n
n
R
2p
Therefore, it follows from (2.7), that:
∥Ant − Ans ∥2p ≤ C[l(nτ )]−1
(
)
(
)

.
.
( l n−τ
(x)
( x ))

l

−τ
n


×
Lρ,τ (t, .) τ − K∓
Lρ,τ (s, .) τ dx
g(x) K∓
n
n
2p
S

τ −1
≤C
∥g∥∞ [l(n )]
S
(
)
)
(
.
.
( l −τ
(x)
( x ))
l


−τ
n
n


Lρ,τ (t, .) τ − K∓
Lρ,τ (s, .) τ dx,
× K∓
n
n
2p
where S = supp(g).
Thanks to Corollary 2.9, for n sufficiently large, we have
∥Ant − Ans ∥2p ≤ C | t − s |1−τ (1+γ) .
Finally, we can take p(1 − τ (1 + γ)) > 1 and the tightness is proved.
4. Strong Approximation
In this section, we give a strong approximation of Theorem 3.1, more precisely
the Lp -estimate. Our main result in this paragraph reads:
Theorem 4.1. Let f be a Borel function on R satisfying:

|x|k |f (x)|dx < ∞,

(4.1)

R

for some k > 0. Then, for any sufficiently small ε > 0 and ρ > 0, and any integer
p ≥ 1, when t goes to infinity, we have
∫ t

I(f )


∥Dγ Lτ (t, .)(0)∥2p + o(t1−τ (1+γ)−ε ).
K l,γ f (Ysρ,τ )ds =

Γ(−γ)
2p
0
0<γ<ξ<

1−τ


and Lτ is the local time of the fBm B τ of Hurst parameter τ .

392

AISSA SGHIR, MOHAMED AIT OUAHRA, AND SOUFIANE MOUSSATEN

In order to prove Theorem 4.1, we shall first state and prove some technical
lemmas. The proofs are similar to that given by Ait Ouahra and Ouali [2] in case
of fractional derivative and fBm.
Lemma 4.2. Let 0 < γ < ξ < 1−τ
2τ . For any sufficiently small ε > 0 and ρ > 0,
and any integer p ≥ 1, when t goes to infinity, we have

2p
sup K l,γ Lρ,τ (t, .)(x) 2p = o(t2p(1−τ (1+γ))+ε ).
x∈R

Proof. Using Theorem 2.8 for s = 0 and the fact that K l,γ Lρ,τ (0, .)(x) = 0 a.s.,
we get:

2p
sup K l,γ Lρ,τ (t, .)(x) 2p ≤ Ct2p(1−τ (1+γ)) .
x∈R

The conclusion follows immediately.



In the same way, using (2.5) for s = 0 and the fact that Lρ,τ (0, x) = 0 a.s., we
get the following result.
Lemma 4.3. Let 0 < ξ < 1−τ
2τ . For any sufficiently small ε > 0 and ρ > 0, and
any integer p ≥ 1, when t goes to infinity, we have
sup
x̸=y

∥Lρ,τ (t, x) − Lρ,τ (t, y)∥2p
2p
= o(t2p(1−τ (1+ξ))+ε ).
|x − y|2pξ

Lemma 4.4. Let 0 < γ < ξ < 1−τ
2τ . For any sufficiently small ε > 0 and ρ > 0,
and any integer p ≥ 1, when t goes to infinity, we have
∫ 1
2p

Lρ,τ (t, x + y) − Lρ,τ (t, x − y)
2p(1−τ (1+ξ))+ε

sup
l(y)
dy
).
= o(t
y 1+γ
x∈R
0
2p
Proof. We have


sup

x∈R

2p
Lρ,τ (t, x + y) − Lρ,τ (t, x − y)

dy

y 1+γ
0
2p
2p
2p ∫ 1
ρ,τ
ρ,τ

∥L (t, x + y) − L (t, x − y)∥2p
l(y)

≤ sup sup
dy

2pξ
1+γ−ξ
y
x∈R 0<y≤1
0 y
1

l(y)

By virtue of Lemma 4.3 and the fact that l is bounded on [0, 1], we deduce the
lemma.

Lemma 4.5. Let 0 < γ < ξ < 1−τ
2τ . For any sufficiently small ε > 0 and ρ > 0,
and any integer p ≥ 1, when t goes to infinity, we have
2p
∫ ∞

Lρ,τ (t, x + y) − Lρ,τ (t, y)
= o(t2p(1−τ (1+ξ))+2paξ+ε ),
sup
l(y)
dy


y 1+γ
|x|≤ta
1
2p
for some a > 0.
Proof. We have


sup

|x|≤ta


1

2p
Lρ,τ (t, x + y) − Lρ,τ (t, y)
dy
l(y)

y 1+γ
2p

OCCUPATION TIME PROBLEM-FRACTIONAL BROWNIAN MOTION

2p
l(y)
≤ sup sup ∥L (t, x + y) − L
dy
y 1+γ
|x|≤ta y∈R
1
∫ ∞
2p
∥Lρ,τ (t, x + y) − Lρ,τ (t, y)∥2p
l(y)
2p
2pξ
≤ sup |x| sup
dy .

|x|2pξ
y 1+γ
y∈R
|x|≤ta
1
Using Potter’s Theorem for x = 1, y ≥ 1 and 0 < δ < γ, we obtain:
∫ +∞
l(y)
dy < ∞.
1+γ
y
1
Finally, by virtue of Lemma 4.3, we deduce the desired estimate.
ρ,τ

ρ,τ

(t, y)∥2p
2p






393



(4.2)


Lemma 4.6. Let f be a Borel function on R satisfying (4.1) for some k > 0.
Then, for any sufficiently small ε > 0 and ρ > 0, and any integer p ≥ 1, when t
goes to infinity, we have:
∫ t

I(f )


K l,γ f (Ysρ,τ )ds =
∥K l,γ Lρ,τ (t, .)(0)∥2p + o(t1−τ (1+γ)−ε ),

Γ(−γ)
2p
0
where 0 < γ < ξ <

1−τ
2τ .

Proof. By the occupation density, we have
∫ t
2p


I(f ) l,γ ρ,τ
l,γ
ρ,τ

I(t) :=
K
L
(t,
.)(0)
K
f
(Y
)ds

s


Γ(−γ)
0
2p

2p


l,γ ρ,τ
l,γ ρ,τ

= C (K L (t, .)(x) − K L (t, .)(0))f (x)dx

R

2p

≤ C(I1 (t) + I2 (t)),
where


2p




l,γ ρ,τ
l,γ ρ,τ
I1 (t) :=
(K L (t, .)(x) − K L (t, .)(0))f (x)dx ,
|x|>ta

2p

and


2p




I2 (t) :=
(K l,γ Lρ,τ (t, .)(x) − K l,γ Lρ,τ (t, .)(0))f (x)dx ,
|x|≤ta

2p

for some 0 < a ≤ τ .
Let us deal with the first term I1 (t). Lemma 4.3 and (4.1) imply that:

2p




2p
I1 (t) ≤ sup ∥K l,γ Lρ,τ (t, .)(x) − K l,γ Lρ,τ (t, .)(0)∥2p
|x|−k |x|k |f (x)|dx


a
a
|x|>t
|x|>t

2p



2p
−2pak
l,γ ρ,τ
l,γ ρ,τ
k
≤t
sup ∥K L (t, .)(x) − K L (t, .)(0)∥2p
|x| |f (x)|dx
|x|>ta

|x|>ta
= o(t2p(1−τ (1+γ))−2pak+ε ).
Now, we deal with I2 (t). By the definition of K l,γ and the fact that f is integrable,
we have:
I2 (t) ≤

394

AISSA SGHIR, MOHAMED AIT OUAHRA, AND SOUFIANE MOUSSATEN



sup

a

|x|≤t


0

2p
Lρ,τ (t, x + y) − Lρ,τ (t, x − y) − Lρ,τ (t, y) + Lρ,τ (t, −y)
l(y)
dy

y 1+γ
2p

2p




×
|f (x)|dx
|x|≤ta




≤ C sup

a
|x|≤t

+


C sup

a
|x|≤t


1

1
0

2p
Lρ,τ (t, x + y) − Lρ,τ (t, x − y) − Lρ,τ (t, y) + Lρ,τ (t, −y)
dy
l(y)

y 1+γ
2p

2p
[Lρ,τ (t, x + y) − Lρ,τ (t, y)] − [Lρ,τ (t, x − y) − Lρ,τ (t, −y)]
l(y)
dy

y 1+γ
2p

which, in view of Lemmas 4.4 and 4.5, implies:
I2 (t) = o(t2p(1−τ (1+ξ))+ε ) + o(t2p(1−τ (1+ξ))+2paξ+ε )
= o(t2p(1−τ (1+ξ))+2paξ+ε ).
Then
I(t) = o(t2p(1−τ (1+γ))−2pka+ε ) + o(t2p(1−τ (1+ξ))+2paξ+ε ).
Choosing:
τ (ξ − γ)
.
ξ+k
It is clear that 0 < a ≤ τ . We finally get:
a=

I(t) = o(t2pb+ε ),
with
b=

ξ(1 − τ (1 + γ)) + k(1 − τ (1 + ξ))
.
k+ξ

Clearly b < 1 − τ (1 + γ), because γ < ξ. Then for all sufficiently small ε > 0, when
t goes to infinity, we have
I(t) = o(t2p(1−τ (1+γ))−ε ),
which gives the desired estimate.



Now, to end the proof of Theorem 4.1, it suffices to establish the following
result.
Lemma 4.7. Let f be a Borel function on R satisfying (4.1) for some k > 0.
Then, for any sufficiently small ε > 0 and ρ > 0, and any integer p ≥ 1, when t
goes to infinity, we have:

I(f ) l,γ ρ,τ
∥K L (t, .)(0)∥2p − ∥Dγ Lτ (t, .)(0)∥2p = o(t1−τ (1+γ)−ε ),
Γ(−γ)
where Lτ is the local time of the fBm of Hurst parameter τ .

OCCUPATION TIME PROBLEM-FRACTIONAL BROWNIAN MOTION

395

Proof. Using (4.1), we get:
J(t) :=

I(f )
∥K l,γ Lρ,τ (t, .)(0)∥2p ≤ C(J1 (t) + J2 (t)),
Γ(−γ)


where
J1 (t) := sup ∥K l,γ Lρ,τ (t, .)(0)∥2p
|x|>ta

|x|>ta

|x|−k |x|k |f (x)|dx,

and
J2 (t) := sup ∥K l,γ Lρ,τ (t, .)(0)∥2p .
|x|≤ta

The same arguments used in the proof of Lemma 4.6 implies that:
J1 (t) = o(t1−τ (1+γ)−ka+ε ).
For J2 (t), we have by Lemma 4.3:
J2 (t) = o(t1−τ (1+ξ)+ε ),
therefore
J2 (t) = o(t1−τ (1+ξ)+aξ+ε ),
Consequently
J(t) = o(t1−τ (1+γ)−ε ).
On the other hand, using Remark 2.2 and the fact that the fBm B τ satisfies:
(B ρ,τ , t ≥ 0) d (B τ , t ≥ 0),
we get:

(

) (
)
K l,γ Lτ (t, .)(0) , t ≥ 0 d K l,γ Lρ,τ (t, .)(0) , t ≥ 0 .

Therefore
I(f )
I(f )
∥K l,γ Lτ (t, .)(0)∥2p =
∥K l,γ Lρ,τ (t, .)(0)∥2p = o(t1−τ (1+γ)−ε ).
Γ(−γ)
Γ(−γ)
In particular, if we take l ≡ 1, we get:
I(f )
∥Dγ Lτ (t, .)(0)∥2p = o(t−τ (1+γ)−ε ).
Γ(−γ)
The proof of Lemma 4.7 is done.

Finally, combining Lemma 4.6 and Lemma 4.7, the proof of Theorem 4.1 is
completed.
Remark 4.8. We should point out that in this paper we only study the Lp -estimate
of our limit theorem. This is enough for the purpose of this study. We will study
the a.s., estimate in future work and apply this
∫ t idea to study the law of the iterated
logarithm of stochastic process of the form 0 K l,γ f (Ysρ,τ )ds. For the a.s. estimate
in case of Brownian motion and the symmetric stable process of index α ∈ (1, 2],
we refer respectively to [10] and [3].
Acknowledgment. The authors are very grateful to the associate editor and
anonymous referees whose remarks and suggestions greatly improved the presentation of this paper.

396

AISSA SGHIR, MOHAMED AIT OUAHRA, AND SOUFIANE MOUSSATEN

References
1. Ait Ouahra, M., Moussaten, S. and Sghir, A.: On limit theorems of some extensions of
fractional Brownian motion and their additive functionals, Stochastics and Dynamics 17
(2016), no. 3, 1–14.
2. Ait Ouahra, M. and Ouali M.: Occupation time problems for fractional Brownian motion
and some other self-similar processes. Random Oper. Stochastic Equations, 17 (2009), no. 1,
69–89.
3. Ait Ouahra, M., Kissami, A. and Sghir, A.: Strong approximation of some additive functionals of symmetric stable, Math. Inequa. Appl. 17 (2014), no. 4, 1327–1336.
4. Bardina, X. and Bascompte, D.: Weak convergence towards two independent Gaussian processes from a unique Poisson process, Collect. Math. 61 (2010), no. 2, 191–204.
5. Berman, S. M.: Local nondeterminism and local times of Gaussian processes, Indiana Univ.
Math. J. 23 (1973), 69–94.
6. Bingham, N. H., Goldie, C. M. and Teugels, J. L.:Regular Variation, Cambridge University
Press, Cambridge, 1987.
7. Bojdecki, T., Gorostiza, L. and Talarczyk, A.: Sub-fractional Brownian motion and its
relation to occupation times, Statist. Probab. Lett. 69 (2004), 405–419.
8. Bojdecki, T., Gorostiza, L. and Talarczyk, A.: Some Extensions of fractional Brownian
motion and Sub-Fractional Brownian Motion Related to Particle Systems, Elect. Comm. in
Probab. 12 (2007), 161–172.
9. Boufoussi, B., Dozzi, M. and Guerbaz, R.: Sample path properties of the local time of
multifractional Brownian motion, Bernoulli 13 (2007), no. 3, 849–867.
10. Csaki, E., Shi, Z. and Yor, M.: Fractional Brownian motions as ”higher-order” fractional
derivatives of Brownian local times, in: Limit Theorems in Probability and Statistics. I
(2000), 365–387, Balatonlelle, Janos Bolyai Math. Soc., Budapest.
11. Fitzsimmons, P. J. and Getoor, R. K.: Limit theorems and variation properties for fractional
derivatives of the local time of a stable process, Ann. Inst. H. Poincar´
e. 28 (1992), no. 2,
311–333.
12. Geman, D. and Horowitz, J.: Occupation densities, Ann. Probab. 8 (1980), no. 1, 1–67.
13. Gihman, I.I. and Skorohod, A.V.: The Theory of Stohastic Processes, I. New York, Springer,
(1974).
14. Houdr´
e, C. and Villa, J.: An example of infinite dimensional quasi-helix,Contemp. Math.
(Amer. Math. Soc.) 336 (2003), 195–201.
15. Lei, P. and Nualart, D.: A decomposition of the bifractional Brownian motion and some
applications,Statist. Probab. Lett. 79 (2009), 619–624.
16. Mandelbrot, B. and Ness, J. W. V.: Fractional Brownian motion, fractional Gaussien noises
and applications,SIAM Reviews. 10 (1968), no. 4, 422–437.
17. Mendy, I.: On the local time of sub-fractional Brownian motion, Ann. math. Blaise. Pascal.
17 (2010), no. 2, 357–374.
18. Mendy, I.: Local time of Weight fractional Brownian motion, J. Num. Math. Stoch. 5 (2013),
no. 1, 35–45.
19. Samko, S. G., Kilbas, A. A. and Marichev, O. I.: Fractional Integrals and Derivatives,
Theory and Applications. Gordon and Breach Science Publishers, Yverdon, 1993.
20. Seneta, E.: Regularly Varying Functions, Springer-Verlag. New York and Berlin, 1976.
21. Shieh, N. R.: Limits theorems for local times of fractional Brownian motions and some other
self-similar processes, J. Math. Kyoto Univ. 36 (1996), no. 4, 641–652.
22. Tudor, C. A. and Xiao, Y.: Sample path properties of bifractional Brownian motion,
Bernoulli 13 (2007), no. 4, 1023–1052.
23. Xiao, Y.: H¨
older conditions for the local times and the Hausdorff measure of the level sets
of Gaussian random fields,Probab. Theorey Related Fields. 109 (1997), no. 1, 129–157.
24. Yamada, T.: On the fractional derivative of Brownian local time, J. Math. Kyoto Univ. 25
(1985), no. 1, 49–58.

OCCUPATION TIME PROBLEM-FRACTIONAL BROWNIAN MOTION

397

25. Yamada, T.: On some limit theorems for occupation times of one-dimensional Brownian
motion and its continuous additive functionals locally of zero energy,J. Math. Kyoto Univ.
26 (1986), no. 2, 309–322.
´partement de Mathe
´matiques et Informatique, Faculte
´ des Sciences
Aissa Sghir: De
´ Mohammed Premier, Laboratoire de Mode
´lisation Stochastique et
Oujda, Universite
´terministe et URAC 04, BP, 717, Maroc
De
E-mail address: sghir.aissa@gmail.com
´partement de Mathe
´matiques et Informatique, Faculte
´ des
Mohamed Ait Ouahra: De
´ Mohammed Premier, Laboratoire de Mode
´lisation StochasSciences Oujda, Universite
´terministe et URAC 04, BP, 717, Maroc
tique et De
E-mail address: ouahra@gmail.com
´partement de Mathe
´matiques et Informatique, Faculte
´ des
Soufiane Moussaten: De
´ Mohammed Premier, Laboratoire de Mode
´lisation StochasSciences Oujda, Universite
´terministe et URAC 04, BP, 717, Maroc
tique et De
E-mail address: s.moussaten@ump.ca.ma


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