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Nom original: Article12_Sghir_Aissa.pdfTitre: On limit theorems of some extensions of fractional Brownian motion and their additive functionalsAuteur: M. Ait Ouahra, S. Moussaten, and A. Sghir

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Stochastics and Dynamics
Vol. 17, No. 3 (2017) 1750022 (14 pages)
c World Scientific Publishing Company

DOI: 10.1142/S0219493717500228

On limit theorems of some extensions of fractional
Brownian motion and their additive functionals

M. Ait Ouahra∗ and S. Moussaten†
Facult´
e des Sciences, Laboratoire de Mod´
elisation Stochastique
et D´
eterministe et URAC 04, Oujda, BP, 717, Maroc
∗m.aitouahra@ump.ma
†s.moussaten@ump.ac.ma
A. Sghir
Facult´
e des Sciences Mekn`
es,

epartement de Math´
ematiques et Informatique,
BP, 11201, Zitoune, Maroc
sghir.aissa@gmail.com
Received 29 January 2016
Revised 6 April 2016
Accepted 19 April 2016
Published 16 June 2016
This paper is divided into two parts. The first deals with some limit theorems to certain
extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part
we give the similar results of their continuous additive functionals; more precisely, local
time and its fractional derivatives involving slowly varying function.
Keywords: Bifractional Brownian motion; subfractional Brownian motion; weighted
fractional Brownian motion; increment process; tangent process; local time; fractional
derivative.
AMS Subject Classification: 60G18, 60J55, 60F05

1. Introduction
It is well known that the stationarity of increments is a very powerful ingredient
for studying the sample paths properties of local times of a large classes of locally
nondeterministic Gaussian processes, (see Berman [3, 4] and Xiao [21]). A nice
solution in case of Gaussian processes that do not have stationary increments is
the Lamperti transformation which provides a powerful connection between selfsimilar processes and stationary processes, (see Tudor and Xiao [20] and Yan et al.
[23]). It is natural to expect the same when dealing with asymptotically stationary
increments. Thus it will be of some interest to know if the local times satisfy a kind
of this property.
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M. A. Ouahra, S. Moussaten & A. Sghir

In this paper, firstly, we will study some extensions of fractional Brownian
motion that do not have stationary increments.
Let B H,K := (BtH,K , t ≥ 0) be the bifractional Brownian motion, (bfBm for
short), with parameters H ∈ (0, 1) and K ∈ (0, 1] introduced in [11]. The case
K = 1 corresponds to the fractional Brownian motion [15] of Hurst index H ∈
(0, 1) denoted by B H , (fBm for short). The increments of the bfBm B H,K are not
stationary for any K ∈ (0, 1) except the case of fBm, however, Maejima and Tudor
[14] have proved that, when h → +∞, the increment process
H,K
− BhH,K , t ≥ 0)
(Bt+h

converges modulo a constant, in the sense of the finite dimensional distributions,
to the fBm (BtHK , t ≥ 0). We can interpret this result like the bfBm has stationary
increments for h large enough. Therefore the dependence of the increment process
depending on h decreases. The key ingredient in the proof of the previous result is a
decomposition in law of the bfBm presented in [13] as follows: Let W := (Wθ , θ ≥ 0)
be a standard Brownian motion independent of B H,K . For any K ∈ (0, 1), let
X K := (XtK , t ≥ 0) be the centered Gaussian process defined by
+∞
(1+K)
(1 − e−θt )θ− 2 dWθ .
XtK :=
0

The authors in [13] showed, by setting
XtH,K := XtK2H ,

(1.1)

that
(1.2)
(C1 (K)XtH,K + BtH,K ; t ≥ 0) d (C2 (K)BtHK ; t ≥ 0),

(1−K)
2−K K
2
where C1 (K) =
and d means equality of all finite
Γ(1−K) , C2 (K) = 2
dimensional distributions.
In the same spirit, we prove a similar result in the case of subfractional Brownian
motion (see the definition in Sec. 2).
In the case of weighted fractional Brownian motion (see the definition in Sec. 2)
the similar result with some normalization is given in [7].
We also give some limit theorems to these two extensions. These results are
analogous to those given in [14] in the bifractional Brownian motion case.
We end this first part by similar results for tangent process of bifractional and
subfractional Brownian motions.
The second part of this paper is concerned with similar results for some additive
functionals of these extensions namely, local time and its fractional derivatives
involving slowly varying function. The proofs are based on integral representation
of local time initiated by Berman [4].
This paper is structured as follows. In Sec. 2 we give some result on the increment
process of subfractional Brownian motion (sfBm for short). The proof is based on
the decomposition in law of sfBm (see (2.2) and (2.3) below). Our results are similar
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to those of Maejima and Tudor [14]. In Sec. 3 we prove the similar limit theorems
in the weighted fractional Brownian motion (wfBm for short) case. In Sec. 4 we
treat the case of tangent process of these extensions. The last section is devoted to
some results of associated continuous additive functionals like: local time and its
generalized fractional derivatives.
2. On Some Processes Extensions of fBm
2.1. On the subfractional Brownian motion
The sfBm is an extension of standard Brownian motion which preserves many
properties of fBm but not the stationarity of the increments. It was introduced by
Bojdecki et al. [6]. It is a continuous centered Gaussian process, starting from zero,
with covariance function
1
E(Stλ Ssλ ) = tλ + sλ − [(t + s)λ + |t − s|λ ],
2
where λ ∈ (0, 2).
The increments of sfBm are not stationary and satisfy for all s ≤ t,
[(2 − 2λ−1 ) ∨ 1](t − s)λ ≤ E[Stλ − Ssλ ]2 ≤ [(2 − 2λ−1 ) ∧ 1](t − s)λ .
Let X λ be the Gaussian process appeared in Sec. 1. The authors in [1] have defined
X λ for λ ∈ (1, 2) and have proved that

Γ(1 − λ) λ


[t + sλ − (t + s)λ ], if λ ∈ (0, 1),


λ
E(Xtλ Xsλ ) =
(2.1)
 Γ(2 − λ)

λ
λ
λ

[(t + s) − t − s ], if λ ∈ (1, 2).

λ(λ − 1)
Moreover, the sfBm S λ has the following decompositions in law:
(2.2)
(Stλ , t ≥ 0) d (Btλ + C3 (λ)Xtλ , t ≥ 0), if λ ∈ (0, 1),

λ
where C3 (λ) = 2Γ(1−λ)
and W (the standard Brownian motion appeared in the
definition of X λ ) and B λ are independent, (B λ is fBm of Hurst parameter H = λ2 ).
(Btλ , t ≥ 0) d (Stλ + C4 (λ)Xtλ , t ≥ 0), if λ ∈ (1, 2),

λ(λ−1)
where C4 (λ) = 2Γ(2−λ)
and W and S λ are independent.
1

Remark 2.1. If we take in (1.1): K = λ and H = 12 , we get Xt2
will be useful in the sequel.



(2.3)

= Xtλ . This fact

The first application of Remark 2.1, (2.2) and (2.3) is the following result. The
proof is similar to that given in [14] in case of the bfBm.
Theorem 2.1. Let λ ∈ (0, 2). Then, the increment process
λ
− Shλ , t ≥ 0)
(Sh+t

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converges, in the sense of the finite dimensional distributions, when h → ∞, to the
fBm (Btλ , t ≥ 0).
We can understand this last result by considering the sfBm noise, defined for
every integer n ≥ 0 as follows:

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λ
− Snλ .
Yn = Sn+1

For every a ∈ N and every n ≥ 0, we put
λ
λ
λ
Rλ (a, a + n) = E(Ya Ya+n ) = E[(Sa+1
− Saλ )(Sa+n+1
− Sa+n
)].

Then, using (2.2) and (2.3), we get
Rλ (a, a + n) = g(n) + fa (n),
where
λ
g(n) = E[B1λ (Bn+1
− Bnλ )],

and
λ
λ
λ
fa (n) = C32 (λ)E[(Xa+1
− Xaλ )(Xa+n+1
− Xa+n
)] if λ ∈ (0, 1),
λ
λ
λ
fa (n) = −C42 (λ)E[(Xa+1
− Xaλ )(Xa+n+1
− Xa+n
)]

if λ ∈ (1, 2).

Theorem 2.2. For each n, we have
lim fa (n) = 0.

a→∞

Therefore the sfBm noise converges to a stationary sequence.
Another interesting study of the function fa (n) for large n, is that the longrange dependence decays at a higher rate for sfBm than for fBm. This justifies the
name subfractional Brownian motion for S λ . More precisely, we have the following
result.
Theorem 2.3. For large n, we have
Rλ (a, a + n) =

λ
(λ − 1)(2 − λ)(2a + 1)nλ−3
2
λ
− (λ − 1)(λ − 2)(λ − 3)(2a + 1)2 nλ−4 + · · ·
4

if λ = 1.

Consequently, for every integer a, we have

Rλ (a, a + n) < ∞.
n≥0

We end this section with an approximation in law of the sfBm S λ for λ ∈ (1, 2).
The proof is similar to that given in [14] in case of the bfBm B H,K where 2HK > 1,
(see Remark 6.3 in [14]). For this, let (φj , j = 1, 2, . . .) be a sequence of standard
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normal random variables and (ηj = f (φi ), j = 1, 2, . . .), where f is the function
appeared in [14] defined as follows:
f (x) =

+∞


ck Hk (x),

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k=1

ck = E[f (φt )Hk (φt )] and Hk (x) is the kth Hermite polynomial with highest coefficient 1.
For x ≥ 0 and y ≥ 0, define a function g(x, y) by
1
1
g(x, y) = − λ(λ − 1)(x + y)λ−2 + λ(λ − 1)|x − y|λ−2 .
2
2
Theorem 2.4. Assume that E(φi φj ) = g(i, j). Then
(1) The sequence of processes,


n− λ2

[nt]



φj , t ≥ 0,

j=1

converges, in the sense of the finite dimensional distributions, when n → ∞, to
the sfBm (Stλ , t ≥ 0).
(2) The sequence of processes,


[nt]

λ
 n− 2
ηj , t ≥ 0,
j=1

converges, in the sense of the finite dimensional distributions, when n → ∞, to
the sfBm (c1 Stλ , t ≥ 0) where c1 is the constant appeared in [14].
2.2. On the weighted fractional Brownian motion
The weighted fractional Brownian motion, (wfBm for short), with parameters a and
b introduced in [7] is a centered Gaussian process ξ := {ξt , t ≥ 0} with covariance
function:
s∧t
ua [(t − u)b + (s − u)b ]du,
(2.4)
Q(s, t) = E(ξt ξs ) =
0

where a > −1, −1 < b < 1 and |b| < 1 + a. Clearly, if a = 0 in (2.4), the
process coincides with the fBm with Hurst parameter H = 12 (b + 1), (up to a
multiplicative constant) and it admits the explicit significance. Thus, wfBm’s are
family of processes which extend fBm’s and preserve many properties like: selfsimilarity, path continuity, long-range dependence, non-semimartingale, and others.
The increments of wfBm are not stationary, but the authors in [7] proved that the
family of process
a

(h− 2 (ξt+h − ξh ), t ≥ 0)
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converges, as h → ∞, in the sense of the finite
dimensional distributions to the fBm
1+b
2
.
with Hurst parameter 2 , multiplied by 1+b
a

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Remark 2.2. It is necessary to normalize by h 2 , because the term E((ξt+h −
ξh )(ξs+h − ξh )) explodes as h → +∞, which is not the case for the sfBm and bfBm.
The following result give an approximation in law of the wfBm in the sense of
Theorem 2.4, by considering the same assumptions, and replacing the expression
of the function g by:
g(x, y) = b(x ∧ y)a |x − y|b−1 ,
where a > −1, 0 < b < 1 and b < 1 + a.
Theorem 2.5. (1) The sequence of processes,


[nt]
a+b+1
 n− 2
φj , t ≥ 0,
j=1

converges, in the sense of the finite dimensional distributions, when n → ∞, to the
wfBm (ξt , t ≥ 0) with parameters a and b.
(2) The sequence of processes,


[nt]
a+b+1
 n− 2
ηj , t ≥ 0,
j=1

converges, in the sense of the finite dimensional distributions, when n → ∞, to the
wfBm (c1 ξt , t ≥ 0) with parameters a and b.
2.3. The behavior of the tangent process of bfBm, sfBm and wfBm
The first main result of this section is an approximation in law of the fBm via the
tangent process generated by the bfBm.
Theorem 2.6. Let H ∈ (0, 1) and K ∈ (0, 1]. For every t0 > 0, the tangent process
H,K

Bt0 +εu − BtH,K
0
,u ≥ 0
εHK
converges, in the sense of the finite dimensional distributions, when ε → 0, to
(2

(1−K)
2

BuHK , u ≥ 0).

Proof. Clearly the case K = 1 is trivial. Indeed we use the fact that the fBm is
self-similar with stationary increments. Now, let K ∈ (0, 1). By virtue of (1.2), it
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suffices to prove that the tangent process generated by X H,K converges to zero.
Using (1.1) and (2.1), there exists a constant C(H, K) > 0 such that


H,K 2

X
XtH,K
t0
0 +εu
 = C(H, K)t2(HK−1)
u2 ε2(1−HK) (1 + o(1)), (2.5)
E
0
εHK
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which tends to zero, when ε → 0, since 1 − HK > 0.
A similar result is obtained for the sfBm.
Theorem 2.7. Let λ ∈ (0, 2). For every t0 > 0, the tangent process


Stλ0 +εu − Stλ0
,u ≥ 0
λ
ε2
converges, in the sense of the finite dimensional distributions, when ε → 0, to the
fBm (Buλ , u ≥ 0).
Proof. This result is an easy consequence of (2.1) and (2.5) and Remark 2.1.
Remark 2.3. Concerning the wfBm, Mendy [17], in Lemma 3.3, proved that for
every a ≥ 0, −1 < b < 1 and every t0 ∈ [α, β] ⊂ (0, ∞),
√ a



b+1
ξt0 +εu − ξt0
2t02
2
lim law
= law √
.
Bu
b+1
ε→0
b+1
ε 2
u≥0
u≥0
3. On Some Additive Functionals of bfBm, sfBm and wfBm
3.1. The case of local time
Before the statement and proof of our main results in this section, we give a brief
survey on local time and we refer to [10]. 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),

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+ ,


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

R

and we have the following representation:

t
1
iu(Xs −x)
e
ds du.
L(t, x) := L([0, t], x) =
2π R
0
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We denote LH,K , (respectively Lλ and La,b ), the local time of the bfBm B H,K ,
(respectively of the sfBm S λ and of the wfBm ξ). Tudor and Xiao [20] have proved,
by using Lamperti’s transform and the concept of strong local nondeterminism, the
existence and the joint continuities of LH,K for any H ∈ (0, 1) and K ∈ (0, 1).
Mendy proved by using the concept of local nondeterminism for Gaussian process
introduced by Berman [4], the existence and the joint continuity of Lλ for every
λ ∈ (0, 1) in [16], and the existence and the joint continuity of La,b for a ≥ 0,
−1 < b < 1 in [17].
Now, we are ready to state and prove our main results in this section.
Theorem 3.1. Let H ∈ (0, 1) and K ∈ (0, 1). Then, the processes
(2

1−K
2

(LH,K (t + h, x2

1−K
2

+ BhH,K ) − LH,K (h, x2

1−K
2

+ BhH,K )), t ≥ 0, x ∈ R)

converges, in the sense of the finite dimensional distributions, when h → ∞, to
(L(t, x), t ≥ 0, x ∈ R),
the local time of the fBm B HK .
Proof. By virtue of (3.1), we have,
2

1−K
2

1−K

1−K

(LH,K (t + h, x2 2 + BhH,K ) − LH,K (h, x2 2 + BhH,K ))



t+h
1−K
H,K
1−K 1
H,K
2 +B
−iu(x2
)
iuB
h
e
e r dr du
=2 2
2π R
h

t

K−1
K−1
H,K
H,K
1
2 B
h+s ds
=
e−iu(x+2 2 Bh )
eiu2
du
2π R
0

«


t iu 2 K−1
H,K
H,K
2 (B
−B
)
1
h+s
h
e−iux
e
ds du.
=
2π R
0
K−1

H,K
−BhH,K , t ≥
The term in the last equality is the local time of the process 2 2 (Bt+h
0) which converges to (BtHK , t ≥ 0). Therefore, our result is an immediate consequence of propositions (4.1) and (4.2) in Jolis and Viles [12] which allows us to
identify the limits law as local times.

In the same way, we obtain the following series of results.
Theorem 3.2. Let H ∈ (0, 1) and K ∈ (0, 1). Then, the process

1−K
1−K
H,K
(t0 + εu, εHK x2 2 + BtH,K
) − LH,K (t0 , εHK x2 2 + BtH,K
)
1−K (L
0
0
,
2 2
1−HK
ε

u ≥ 0, x ∈ R

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converges, in the sense of the finite dimensional distributions, when ε → 0, to
(L(u, x), u ≥ 0, x ∈ R).
Theorem 3.3. Let λ ∈ (0, 1). Then, the process

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(Lλ (t + h, x + Shλ ) − Lλ (h, x + Shλ ), t ≥ 0, x ∈ R)
converges, in the sense of the finite dimensional distributions, when h → ∞, to
(L (t, x), t ≥ 0, x ∈ R),
the local time of the fBm B λ .
Theorem 3.4. Let λ ∈ (0, 1). Then the process

λ
λ
Lλ (t0 + εu, ε 2 x + Stλ0 ) − Lλ (t0 , ε 2 x + Stλ0 )
λ

ε1− 2


, u ≥ 0, x ∈ R

converges in law, when ε → 0, to
(L (u, x), u ≥ 0, x ∈ R).

2
Theorem 3.5. Let a ≥ 0, −1 < b < 1 and Ca,b = 1+b
. Then, the process
a

a

a

(Ca,b h 2 (La,b (t + h, h 2 xCa,b + ξh ) − La,b (h, h 2 xCa,b + ξh )), t ≥ 0, x ∈ R)
converges, in the sense of the finite dimensional distributions, when h → ∞, to
(L (t, x), t ≥ 0, x ∈ R),
the local time of the fBm B

1+b
2

.
√ a
2t 2

0
Theorem 3.6. Let a ≥ 0, −1 < b < 1 and Da,b = √1+b
. Then the process


1+b
1+b
La,b (t0 + εu, ε 2 xDa,b + ξt0 ) − La,b (t0 , ε 2 xDa,b + ξt0 )
, u ≥ 0, x ∈ R
Da,b
1−b
ε 2

converges in the sense of the finite dimensional distributions, when ε → 0, to
(L (u, x), u ≥ 0, x ∈ R).
3.2. The case of generalized fractional derivative
We begin this section with the definition of the generalized fractional derivatives
with kernel depending on slowly varying function. For more details about slowly
varying functions, we refer the reader to Bingham et al. [5].
Definition 3.1. We say that a measurable function l : R+ → R+ is slowly varying
at infinity, if for all positive t, we have
l(tx)
= 1.
x→+∞ l(x)
lim

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We are interested in the behavior of l at +∞, then we can assume for example
that l is bounded on each interval of the form [0, a], where a > 0.
For any γ ∈ ]0, β[ and g ∈ C β ∩ L1 (R), we define
+∞
1
g(x ± y) − g(x)
l,γ
g(x) :=
l(y)
dy,

Γ(−γ) 0
y 1+γ
where C β is the H¨
older space of order β > 0.
> 0, (see Bingham et al. [5],
Note that l(x) = o(xβ ) as x → +∞ for any
+∞β l(y)
< +∞. Consequently, if
Proposition 1.3.6), so when γ > 0, we get 1
1+γ
y

l,γ
g(x) defined bounded continuous
g ∈ C β ∩ L1 (R) for some γ ∈ ]0, β[, then K±
function. We put
l,γ
l,γ
− K−
,
K l,γ := K+

and is called the generalized fractional derivative.
Remark 3.1. The classical fractional derivative Dγ corresponds to l ≡ 1. These
additive functionals appeared in some limit theorems discussed by Yamada [22]
for Brownian motion, in Shieh [19] for fBm and Fitzsimmons and Getoor [9] for
symmetric stable process. Moreover, according to Samko et al. [18], if f ∈ C β ∩L1 (R)
γ
, Dγ }.
then Df ∈ C β−γ , where D ∈ {D±
We also need the following lemma concerning the continuity of the generalized
fractional derivative. It is a generalization of the result of Eddahbi and Vives [8] in
case of fractional derivative.
Let C(R, E) be the space of continuous functions on R endowed with the sup
norm . ∞,E and C β (R, E) the space of β-H¨older continuous E-valued functions
endowed with the norm
f ∞,β,E := f ∞,E + f β,E
:= sup f (x) E + sup
x

x =y

f (x) − f (y) E
.
|x − y|β

l,γ
Lemma 3.1. Let K γ ∈ {K±
, K l,γ }. Then K γ is a bounded linear operator from
β
C (R, E) to C(R, E).
l,γ
Proof. Let f be a function in the space C β (R, E). Assume that K γ = K+
. We
have
+∞

1
f (x + y) − f (x)
l,γ

l(y)
dy
K+ f (x) E =

Γ(−γ)
y 1+γ
0





E

1

1
f (x + y) − f (x) E
l(y)
dy
Γ(−γ)
y 1+γ
0

+∞
f (x + y) − f (x) E
+
l(y)
dy .
y 1+γ
1
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Limit theorems of some extensions of fractional Brownian motion

+∞ l(y)
Using the fact that l is bounded on [0, 1], 1
y 1+γ < +∞ and 0 < γ < β, we get
1
+∞
l(y)
1
1
l,γ
δ−γ−1
f β,E
f ∞,E
K+ f (x) E ≤
y
dy +
dy
1+γ
Γ(−γ)
Γ(−γ)
y
0
1
≤ C(γ)( f β,E + f ∞,E )
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:= C(γ) f ∞,β,E ,
hence the proof is complete.
Now we are ready to state and prove the main results of this section.
l,γ
, K l,γ }. Then, the process
Theorem 3.7. Let H ∈ (0, 1), K ∈ (0, 1) and K γ ∈ {K±



`

K

l

2


1−K
2

´



`

LH,K (t + h, .)(x2

1−K
2

+ BhH,K ) − K
2

l

2


1−K
2

´



LH,K (h, .)(x2

K−1
(1+γ)
2

1−K
2

+ BhH,K )

,
«

t ≥ 0, x ∈ R

converges, in the sense of the finite dimensional distributions, when h → ∞, to
(K l,γ L(t, .)(x), t ≥ 0, x ∈ R),
where K l,γ L(t, .)(x) is the generalized fractional derivative of the local time of the
fBm B HK .
Proof. Using (3.1), we get


2

1−K
2

(1+γ)

−K

K
l



l

2


1−K
2


1−K
2 2



LH,K (t + h, .)(x2

1−K
2







+ BhH,K )

L

H,K

(h, .)(x2

1−K
2

+

BhH,K )


t+h
1−K
H,K
2 2 (1+γ) +∞
eiuBr
2πΓ(−γ) 0
R h
y

l 1−K
1−K
1−K
H,K
H,K
2 2
× e−iu(x2 2 +Bh +y) − e−iu(x2 2 +Bh −y)
drdudy
y 1+γ
t

1−K
K−1
H,K
2 2 (1+γ) +∞
2 B
h+s
=
eiu2
2πΓ(−γ) 0
R 0
y

l 1−K
K−1
K−1
H,K
H,K
2 2
× e−iu(x+2 2 (Bh +y)) − e−iu(x+2 2 (Bh −y))
dsdudy
y 1+γ
+∞ t
K−1
H,K
H,K
1
2 (B
)
h+s −Bh
eiu2
=
2πΓ(−γ) 0
R 0

=

×(e−iu(x+y) − e−iu(x−y) )

l(y)
dsdudy.
y 1+γ
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M. A. Ouahra, S. Moussaten & A. Sghir

The term in the last equality is the generalized fractional derivative of the local
K−1
H,K
− BhH,K , t ≥ 0) which converges to the generalized
time of the process 2 2 (Bt+h
fractional derivative of the local time of the process (BtHK , t ≥ 0) when h −→ ∞
by using Theorem 3.1 and Lemma 3.1.

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In the same way we obtain the following results.
Theorem 3.8. Let H ∈ (0, 1) and K ∈ (0, 1). Then, the process

2

1−K
2

(1+γ) K

l

`



´

1−K
εHK 2 2



l

LH,K (t0 + εu, .)(Xεx ) − K
ε1−HK(1+γ)

`



´

1−K
εHK 2 2



LH,K (t0 , .)(Xεx )

,
«

u ≥ 0, x ∈ R

converges, in the sense of the finite dimensional distributions, when ε → 0, to
(K l,γ L(u, .)(x), u ≥ 0, x ∈ R),
with Xεx = εHK x2

1−K
2

+ BtH,K
.
0

Theorem 3.9. Let λ ∈ (0, 1). Then, the process
(K l,γ Lλ (t + h, .)(x + Shλ ) − K l,γ Lλ (h, .)(x + Shλ ), t ≥ 0, x ∈ R)
converges, in the sense of the finite dimensional distributions, when h → ∞, to
(K l,γ L (t, .)(x), t ≥ 0, x ∈ R),
where K l,γ L (t, .)(x) is the generalized fractional derivative of the local time of the
fBm B λ .
Theorem 3.10. Let λ ∈ (0, 1). Then, the process


l •λ ,γ
l •λ ,γ λ
λ
λ
λ
K ε2
L (t0 + εu, .)(ε 2 x + St0 ) − K ε 2
Lλ (t0 , .)(ε 2 x + Stλ0 )

,


λ

ε1− 2 (1+γ)
u ≥ 0, x ∈ R

converges, in the sense of the finite dimensional distributions, when ε → 0, to
(K l,γ L (u, .)(x), u ≥ 0, x ∈ R).

2
Theorem 3.11. Let a ≥ 0, −1 < b < 1 and Ca,b = 1+b
. Then, the process


K

l

`


a
h 2 Ca,b

´



a,b

L

a
2

(t + h, .)(h xCa,b + ξh ) − K
a

l

`


a
h 2 Ca,b

´



a

La,b (h, .)(h 2 xCa,b + ξh )

,
«

(h 2 Ca,b )−(1+γ)
t ≥ 0, x ∈ R

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Limit theorems of some extensions of fractional Brownian motion

converges, in the sense of the finite dimensional distributions, when h → ∞, to


(K l,γ L (t, .)(x), t ≥ 0, x ∈ R.

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Theorem 3.12. Let a ≥ 0, −1 < b < 1 and Da,b =
l

K
1+γ
Da,b




b+1
ε 2 Da,b





La,b (t0 + εu, .)(Yεx ) − K
ε

l



√ a
2t 2
√ 0 .
b+1

Then, the process


b+1
ε 2 Da,b

1−( b+1
2 )(1+γ)





La,b (t0 , .)(Yεx )

,


u ≥ 0, x ∈ R
converges, in the sense of the finite dimensional distributions, when ε → 0, to


(K l,γ L (u, .)(x), u ≥ 0, x ∈ R),
with Yεx = ε

b+1
2

xDa,b + ξt0 .

Remark 3.2. Bardina and Es-Sebaiy [2] proved that the bfBm B H,K with H ∈
(0, 1) and K ∈ (0, 1) can be extended for K ∈ (1, 2) such that HK ∈ (0, 1). The
existence and the continuity of the local time of this process can be showed easily
by using the same arguments in [16] in case of the sfBm of parameter H ∈ (0, 1).
Finally, we believe that all results proved in this paper are valid for this extension.
Acknowledgment
The authors are very grateful to the associate editor and anonymous referees whose
remarks and suggestions greatly improved the presentation of this paper.
References
1. X. Bardina and D. Bascompte, A decomposition and weak approximation of the subfractional Brownian motion, arXiv:0905-4360.
2. X. Bardina and K. Es-Sebaiy, An extension of bifractional Brownian motion, Comm.
Stoch. Anal. 5 (2011) 333–340.
3. S. M. Berman, Local times and sample function properties of stationary Gaussian
processes, Trans. Amer. Math. Soc. 137 (1969) 277–299.
4. S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana
Univ. Math. J. 23 (1973) 69–94.
5. N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation (Cambridge Univ.
Press, 1987).
6. T. Bojdecki, L. Gorostiza and A. Talarczyk, Sub-fractional Brownian motion and its
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the fractional Brownian motion, J. Math. Kyoto Univ. (JMKYAZ) 43 (2003) 349–368.
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2nd Reading
June 14, 2016 10:41 WSPC/S0219-4937 168-SD

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Stoch. Dyn. Downloaded from www.worldscientific.com
by UNIVERSITY OF SOUTHERN CALIFORNIA @ LOS ANGELES on 06/19/16. For personal use only.

M. A. Ouahra, S. Moussaten & A. Sghir

9. P. J. Fitzsimmons and R. K. Getoor, Limit theorems and variation properties for
fractional derivatives of the local time of a stable process, Ann. Inst. H. Poincare 28
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local time of the fractional Brownian motion, J. Theor. Probab. 20 (2007) 133–152.
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some applications, Statist. Probab. Lett. 79 (2009) 619–624.
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Anal. 2 (2008) 369–383.
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noises and applications, SIAM Rev. 10 (1968) 422–437.
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Pascal. 17 (2010) 357–374.
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Theory and Applications (Gordon and Breach, 1993).
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other self-similar processes, J. Math. Kyoto Univ. 36 (1996) 641–652.
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older conditions for the local times and the Hausdorff measure of the level
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motion, Stochastics 86 (2014) 721–758.

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