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Vol. 10 (2005), Paper no. 10, pages 326-370.
Journal URL
http://www.math.washington.edu/∼ejpecp/
CONVERGENCE IN FRACTIONAL MODELS AND APPLICATIONS1
Corinne Berzin
LabSAD, BSHM, Université Pierre Mendès-France
1251 Avenue centrale, BP 47, 38040 Grenoble cedex 9, France
Corinne.Berzin@upmf-grenoble.fr
José R. León
Escuela de Matemáticas, Facultad de Ciencias
Universidad Central de Venezuela
Paseo Los Ilustres, Los Chaguaramos,
A.P. 47 197, Caracas 1 041-A, Venezuela
jleon@postgrado.ucv.ve
Abstract: We consider a fractional Brownian motion with Hurst parameter strictly

between 0 and 1. We are interested in the asymptotic behaviour of functionals of
the increments of this and related processes and we propose several probabilistic and
statistical applications.
Keywords and phrases: Level crossings, fractional Brownian motion, limit theorem, local time, rate of convergence.
AMS subject classification (2000): Primary 60F05; Secondary 60G15, 60G18,
60H10, 62F03.
Submitted to EJP on November 25, 2003. Final version accepted on February 8,
2005.

1

The research of the second author was supported in part by the project “Modelaje Estocástico
Aplicado” of the Agenda Petróleo of FONACIT Venezuela.

326

1

Introduction

Let {bα (t), t ∈ R} be the fractional Brownian motion with parameter 0 < α < 1.
Consider ϕ, a positive kernel with L1 norm equal to one, and let ϕε (·) = 1ε ϕ( ε· ) and
then define bεα (t) = ϕε ∗ bα (t), the regularized fractional Brownian motion.
Recently, there has been some interest in modeling a stock price X(t) by a fractional version of Black-Scholes model (see Black and Scholes (1973)), say:
dX(t) = X(t)(σ dbα (t) + µ dt),
with X(0) = c and α > 1/2 (see also Cutland et al. (1993)). More generally, let X(t)
be the solution of
dX(t) = σ(X(t)) dbα (t) + µ(X(t)) dt,
with X(0) = c.
First, assume that µ = 0. The model becomes
dX(t) = σ(X(t)) dbα (t),
with X(0) = c.
Lin (1995) proved that such a solution can be written as X(t) = K(bα (t)) where
K is solution of the ordinary differential equation K̇ = σ(K) with K(0) = c.
We shall consider the statisticalR problem that consists in observing, instead of
+∞
X(t), the regularization Xε (t) := 1ε −∞ ϕ((t − x)/ε)X(x) dx with ϕ as before and to
make inference about σ(·). To achieve this purpose we establish first in section 4.1.1
a convergence result for the number of crossings of Xε (·), using the following theorem
(Azaı̈s and Wschebor (1996))

Theorem 1.1 Let {bα (t), t ∈ R} be the fractional Brownian motion with parameter
0 < α < 1. Then, for every continuous function h
r 1−α Z +∞
πε
h(x)Nεbα (x) dx
2 σ2α −∞
r Z 1
Z 1
π
a.s.
ε
h(bα (u)) du
h(bα (u))|Zε (u)| du −→
=
0
Z 2 0
+∞

h(x)ℓbα (x) dx,

=

a.s.

−∞

where −→ denotes the almost-sure convergence, Nεbα (x) the number of times the
ε
(1−α) ε
regularized
ḃα (u)/σ2α with
h process biα (·) crosses level x before time 1, Zε (u) = ε
2
(1−α) ε
bα
σ2α = V ε
ḃα (u) , and process ℓ (x) is the local time in [0, 1] of bα (·) at level x.
To show the result quoted above for Xε (u), we shall use the fact that Xε (u) is
close to K(bεα (u)) and Ẋε (u) is close to K̇(bεα (u))ḃεα (u) = σ(K(bεα (u)))ḃεα (u), and this
enables us to prove that

r 1−α Z +∞
r Z 1
πε
π
h(K(bεα (u)))σ(K(bεα (u)))|Zε (u)| du,
h(x)NεX (x) dx ≃
2 σ2α −∞
2 0
327

converges almost surely to
Z 1
Z
h(K(bα (u)))σ(K(bα (u))) du =
0

1

h(X(u))σ(X(u)) du =

0

Z

+∞

h(x)σ(x)ℓX (x) dx,

−∞

where ℓX (·) is the local time for X in [0, 1].
Now suppose that 14 < α < 34 , we have the following result about the rates of
convergence in Theorem 1.1 proved in section 3.4: there exists a Brownian motion
c(·) independent of bα (·) and a constant Cα,ϕ such that,
W
r 1−α Z ∞

Z ∞
1
πε

bα
bα
√
h(x)ℓ (x) dx
h(x)Nε (x) dx −
2 σ2α −∞
ε
−∞
Z 1
D
c(u).
−→ Cα,ϕ
h(bα (u)) dW
0

Using this last result for 21 < α < 43 , we can get the same one for the number of
crossings of the process Xε (·) and we obtain in section 4.1.1
r 1−α Z ∞

Z ∞

1
πε
X
X
√
h(x)Nε (x) dx −
h(x)σ(x)ℓ (x) dx
2 σ2α −∞
ε
−∞
Z 1
D
c(u).
−→ Cα,ϕ
h(X(u))σ(X(u)) dW
0

A similar result can be obtained under contiguous alternatives for σ(·) and provides in section 4.1.2 a test of hypothesis for such a function.
We study also the rate of convergence in the following result proved by Azaı̈s and
Wschebor (1996) concerning the increments of the fractional Brownian motion given

here as Theorem 1.2.

Theorem 1.2 Let {bα (t), t ∈ R} be the fractional Brownian motion with parameter
0 < α < 1. Then, for all x ∈ R and t ≥ 0


bα (u + ε) − bα (u)
a.s.
≤ x −→ t Pr{N ∗ ≤ x},
λ 0≤u≤t:
α
ε v2α
2
where v2α
= V [bα (1)], λ is the Lebesgue measure and N ∗ is a standard Gaussian
random variable.

This result also implies that for a smooth function f , we have for all t ≥ 0

Z t 

bα (u + ε) − bα (u)
a.s.
du −→ tE [f (N ∗ )] .
f
α
ε v2α
0

(1)

It also can be shown for regularizations bεα (u) = ϕε ∗ bα (u), where ϕε (·) = 1ε ϕ( ε· ), ϕ
defined before. In the special case where ϕ = 1[−1,0] , we have εḃεα (u) = bα (u+ε)−bα (u).
328

Hence, we can write (1) in an other form say, for all t ≥ 0
!
Z t
ε(1−α) ḃεα (u)
a.s.
f
du −→ tE [f (N ∗ )] .
σ2α
0

(2)

We find the convergence rate in (2) for a function f ∈ L2 (φ(x) dx) where φ(x) dx
stands for the standard Gaussian measure. We can formulate the problem in the
following way:
Suppose that g (N ) (x) = f (x) − E [f (N ∗ )] is a function in L2 (φ(x) P
dx), whose first
(N )
non-zero coefficient in the Hermite
is aN , i.e. g (x) = ∞
n=N an Hn (x),
P∞ expansion
(N ) 2
2
(N ≥ 1) for which kg k2,φ = n=N an n!. The index N is called the Hermite’s rank.
Find an exponent a(α, N ) and a process Xg(N ) (·) such that the functional defined by
Rt
(N )
Sε (t) = ε−a(α,N ) 0 g (N ) (Zε (u)) du converges in distribution to Xg(N ) (t).
Note that similar problems have been studied by Breuer and Major (1983), Ho and
Sun (1990) and Taqqu (1977) for summations instead of integrals.
The limit depends on the value of α, and as stated in Section 3.1, α = 1 − 1/(2N )
is a breaking point. As pointed out in Section 3.3, if instead of considering the first
order increments, we take the second ones, then there is no more breaking points and
the convergence is reached for any value of α in (0, 1).
As applications of the previous results, we get in Section 4.2 the following:
Z t



D
−a(α,2)
ε
|Zε (u)|β − E |N ∗ |β du −→ Xg(2) (t),
0

and in Section 4.3 we get the rate of convergence in Theorem 1.2 and we obtain that
for all x ∈ R


Z t
D
∗
−a(α,1)
Pr(N ≤ x) du −→ Xg(1) (t),
λ{0 ≤ u ≤ t, Zε (u) ≤ x} −
ε
0

giving the form of the limit, depending also of x, and suggesting the convergence rate
in the case where 1/2 < α < 1.
We observe that all the results quoted above for the fractional Brownian motion,
have been considered in Berzin-Joseph and León (1997) for the Wiener process (corresponding to the case where α = 1/2), in Berzin et al. (2001) for the F -Brownian
motion, in Berzin et al. (1998) for a class of stationary Gaussian processes and in
Perera and Wschebor (1998) for semimartingales.
It is worth noticing that in the case of stationary Gaussian processes the results
are quite similar to those obtained in the present article for N = 2.
The paper is organized as follows. In this section we introduced the problems and
their applications. In section 2 we state some notations and the hypotheses under
which we work. Section 3 is devoted to establish the main results. The applications
are developed in section 4. Section 5 contains the proofs.
329

2

Hypotheses and notations

Let {bα (t), t ∈ R} be the fractional Brownian motion with parameter 0 < α < 1
(see for instance Mandelbrot and Van Ness (1968)), i.e. bα (·) is a centered Gaussian
process with the covariance function
i
1 2 h 2α
2α
2α
E [bα (t)bα (s)] = v2α |t| + |s| − |t − s| ,
2
2
with v2α
=

1
.
Γ(2α + 1) sin(πα)

For each t ≥ 0 and ε > 0, we define the regularized processes
Z t+ε
bα (t + ε) − bα (t)
ε
−1
bα (t) = ε
bα (u) du and Zε (t) =
.
εα v2α
t
also define, for a C 1 density ϕ with compact support included in [−1, 1] satisfying
RWe
∞
ϕ(x) dx = 1,
−∞
bεα (t)

1
=
ε

with

Z

∞

ϕ

−∞



t−x
ε



ε(1−α) ḃεα (t)
,
bα (x) dx and Zε (t) =
σ2α

Z +∞
1
=
:= V
|x|1−2α |ϕ̂(−x)|2 dx.
2π −∞
R +∞
2
(Note that for α = 1/2, σ2α
= ||ϕ||22 := −∞ ϕ2 (x) dx.)
the Hermite polynomials, which can be defined by exp(tx − t2 /2) =
P∞We shall use
n
system for the standard Gaussian mean=0 Hn (x)t /n!. They form an orthogonal P
P∞ 2
2
2
sure φ(x) dx and, if h ∈ L (φ(x) dx), h(x) = ∞
n=0 ĥn Hn (x) and ||h||2,φ =
n=0 ĥn n!.
Mehler’s formula (see Breuer and Major (1983)) gives a simple form to compute
the covariance between two
L2 functions of Gaussian random variables. Actually, if
P
k ∈ L2 (φ(x) dx), k(x) = ∞
n=0 k̂n Hn (x) and if (X, Y ) is a Gaussian random vector
such that X and Y are standard Gaussian random variables with correlation ρ then
2
σ2α

h

ε(1−α) ḃεα (t)

i

E [h(X)k(Y )] =

∞
X

ĥn k̂n n!ρn .

(3)

n=0

We will also use the following well-known property
Z z
Hk (y)φ(y)dy = −Hk−1 (z)φ(z), z ∈ R, k ≥ 1.
−∞

Let g (N ) be a function
in L2 (φ(x) dx) such that g (N ) (x) =
P∞
2
(N ) 2
with ||g ||2,φ = n=N an n! < +∞.
330

P∞

n=N

(4)
an Hn (x), N ≥ 1,

For 0 < α < 1 − 1/(2N ) or α = 1/2 and N = 1, we shall write
Z +∞
∞
X
(N ) 2
2
ρlα (x) dx,
al l!
(σa ) = 2
0

l=N

(ε)

where we define ρα (v) = E [Zε (v + u)Zε (u)] and
2 Z ∞
−v2α
(ε)
ė
ρα (x) = ρα (εx) =
ϕ̇ ∗ ϕ(y)|x
− y|2α dy
2
2σ
Z ∞ 2α −∞
1
2
=
|y|1−2α eixy |ϕ(−y)|
b
dy,
2
2πσ2α −∞

2
ė
where ϕ(y)
= ϕ̇(−y). (If α = 1/2, ρα (x) = ϕ ∗ ϕ(x)/||ϕ||
e
2 .) For ϕ = 1[−1,0] , it is easy
to show that

1
v2
ρα (x) = [|x + 1|2α − 2|x|2α + |x − 1|2α ] and 2α
= 1.
2
2
σ2α
R +∞
(1)
(2)
Note that for N = 1 and 0 < α < 1/2, since 0 ρα (x) dx = 0, (σa )2 = (σa )2 , and
(1)
(2)
for α = 1/2 with N = 1, (σa )2 = a21 /||ϕ||22 + (σa )2 .
For N ≥ 1 and 0 ≤ t ≤ 1, define
Sε(N ) (t)

=ε

−a(α,N )

Zt

g (N ) (Zε (u)) du,

0

a(α, N ) will be defined later.
Throughout the paper, C shall stand for a generic constant, whose value may
change during a proof. N ∗ will denote a standard Gaussian random variable.

3

Results

3.1
3.1.1

(N )

Convergence for Sε (t)
Case 0 < α < 1 − 1/(2N ) or α = 1/2 and N = 1

If N = 1, let us define A := {k : k ≥ 2 and ak 6= 0}. We suppose A 6= ∅ and we define
N0 = inf{k : k ∈ A}.
Theorem 3.1 a(α, N ) = 1/2 and
1)
D

c(·),
Sε(N ) (·) −→ σa(N ) W

c (·) is a Brownian motion.
where W

331

2) Furthermore,
(a) If 1/(2N ) < α < 1 − 1/(2N )
D

c(·));
(bεα (·), Sε(N ) (·)) −→ (bα (·), σa(N ) W

(b) If 1/(2N0 ) < α < 1/2 and N = 1

D

c(·));
(bεα (·), Sε(1) (·)) −→ (bα (·), σa(N0 ) W

(c) If α = 1/2 and N = 1

D

(bεα (·), Sε(1) (·)) −→ (bα (·),

a1
c (·)).
bα (·) + σa(N0 ) W
||ϕ||2

c(·) are independent. The convergence taking place in 1)
The processes bα (·) and W
and 2) is in finite-dimensional distributions.
Remark 1: If the cœfficients of the function g (N ) verify the condition
∞
X

k=N

√
3k/2 k!|ak | < ∞,
(N )

(cf. Chambers and Slud (1989), p.328), the sequence Sε (·) is tight and the convergence takes place in C[0, 1] for 1) and in C[0, 1] × C[0, 1] for 2). This will be the case
for g (N ) a polynomial.
(N )

Remark 2: In case 1), when 0 < α < 1/2, (N = 1), note that σa

(N0 )

= σa

.

Remark 3: In case 1), when 0 < α < 1/2, (N = 1) and ak ≡ 0 for k ≥ 2,
(i.e. A = ∅), the limit gives zero, so the normalization must be changed; in fact in
(1)
this case, a(α, 1) = 1 − α is the convenient normalization and Sε (·) converges in L2
towards a1 bα (·)/σ2α ; this last result is also true when α ≥ 1/2.
D
(1)
1
bα (·)).
Furthermore with this normalization, for 0 < α < 1, (bεα (·), Sε (·)) −→ (bα (·), σa2α
3.1.2

Case α = 1 − 1/(2N ) and N > 1
1
and
2
D
(N )
Sε (·)) −→

Theorem 3.2 a(α, N ) =

√
v
1
c (·)),
(bε1−1/(2N ) (·), [ln(ε−1 )]
(b1−1/(2N ) (·), 2N ![(1− 2N
)(1− N1 )]N/2 ( σ2−1/N
)N aN W
2−1/N
c (·) is a Brownian motion and the processes b1−1/(2N ) (·) and W
c(·) are indewhere W
pendent.
The convergence taking place is in finite-dimensional distributions.
− 12

332

3.1.3

Case 1 − 1/(2N ) < α < 1

Theorem 3.3 a(α, N ) = N (1 − α) and for fixed t in [0, 1]
(N )
Sε (t)

L2

−→

√

!
N Z +∞
Z +∞
N
X
i
N !aN √
λi ×
Kt
...
2πσ2α
−∞
−∞
i=1
N 

Y
1
λi |λi |− 2 −α dW (λ1 ) . . . dW (λN ),


i=1

where
Kt (λ) =

exp(itλ) − 1
.
iλ

Remark: Note that for N = 1 (α > 21 ) the limit is

3.2

(1)

Rate of convergence for Sε (t) and

Remember that for

1
2

(1)

< α < 1, Sε (t) = εα−1

P∞

a1
b (t).
σ2α α

1
2

12 and positive σ. We consider the problem of estimating
R ∞ σ when µ ≡ 0.
)X(x) dx,
Suppose we observe instead of X(t) a regularization Xε (t) = 1ε −∞ ϕ( t−x
ε
with ϕ as in section 2, where we have extended X(·) by means of X(t) = c, if
t < 0. It is easy to see that the process X(t) has a local time ℓX (x) in [0, 1] for
every level x, in fact we have ℓX (x) = ℓbα (K −1 (x))/σ(x) where K is solution of the
336

ordinary differential equation (ODE), K̇ = σ(K) with K(0) = c. Considering NεX (x)
the number of times that the process Xε (·) crosses level x before time 1 and using
Theorem 1.1 we can prove:
Proposition 4.1 Let 21 < α < 1, if h ∈ C 0 and σ ∈ C 1 then
r 1−α Z ∞
Z ∞
πε
a.s.
X
h(x)Nε (x) dx −→
h(x)σ(x)ℓX (x) dx.
2 σ2α −∞
−∞

Moreover, using Theorem 3.4 (ii) we can also obtain the following theorem.
Theorem 4.1 Let us suppose that 12 < α < 43 , h ∈ C 4 , σ ∈ C 4 , σ is bounded and
sup{|σ (4) (x)|, |h(4) (x)|} ≤ P (|x|), where P is a polynomial, then
r 1−α Z ∞
Z ∞
1
πε
X
√ [
h(x)σ(x)ℓX (x) dx],
h(x)Nε (x) dx −
2 σ2α −∞
ε
−∞
converges stably towards

Cα,ϕ

Z

0

1

c (u).
h(X(u))σ(X(u))dW

2
c(·) is still a standard Brownian motion independent of bα (·), Cα,ϕ
Here, W
given by
r
Z +∞
+∞
+∞
X
X
π
2
2l
(2)
2
Cα,ϕ = 2
ρα (v) dv and g (x) =
a2l (2l)!
|x| − 1 =
a2l H2l (x).
2
0
l=1
l=1

Remark: This type of result was obtained for a class of semimartingales, and in
particular for diffusions, in Perera and Wschebor (1998).
4.1.2

Proofs of hypothesis

Now, we observe Xε (·), solution of the stochastic differential equation, for t ≥ 0,
dXε (t) = σε (Xε (t)) dbα (t) with Xε (0) = c,
Xε (t) = c, for t < 0 and we consider testing the hypothesis
H0 : σε (·) = σ0 (·),
against the sequence of alternatives
Hε : σε (·) = σ0 (·) +

√

εd(·) +

√

εF (·,

√
ε),

where F (·, 0) = 0, σ0 , d and F are C 1 .
R +∞
Let us define the observed process Yε (·) := 1ε −∞ ϕ( .−x
)Xε (x) dx with ϕ as in section
ε
2. We are interested in observing the following functionals

r 1−α Z ∞
Z 1
1
πε
Y
h(Xε (u))σ0 (Xε (u)) du .
h(x)Nε (x) dx −
Tε (h) := √
2 σ2α −∞
ε
0
Using Theorem 3.1 2)(a) we can prove Theorem 4.2.
337

Theorem 4.2 Let us suppose that 12 < α < 43 , h ∈ C 4 , σ0 ∈ C 4 , d ∈ C 2 , F ∈ C 1 ,
(4)
σ0 is bounded and sup{|σ0 (x)|, |h(4) (x)|, |d(2) (x)|} ≤ P (|x|) where P is a polynomial
then Tε (h) converges stably towards
Cα,ϕ

Z

1

0

c (u) +
h(X(u))σ0 (X(u))dW

Z

1

h(X(u))d(X(u)) du,

0

a.s.

where X(·) = K(bα (·)) = limXε (·), and K is solution of the ODE, K̇ = σ0 (K) with
ε→0

c(·) is a standard Brownian motion independent of X(·).
K(0) = c and W

Remark 1: There is a random asymptotic bias, and the larger the bias the easier it
is to discriminate between the two hypotheses.

Remark 2: We can consider the very special case h ≡ 1 and σ0 constant. The limit
random variable is
Z 1
∗
d(σ0 bα (u) + c) du.
Cα,ϕ σ0 N +
0

Recall that the two terms in the sum are independent.

4.2

β-increments

Let
Sεβ (t)

=ε

−a(α,2)

Z tn

o
|Zε (u)|β − E |N ∗ |β du,
0

for β > 0 and 0 ≤ t ≤ 1.

As an application of Theorems 3.1 1), 3.2 (i) and 3.3, we obtain the following corollary.
Corollary 4.1 (i) If 0 < α < 43 ,
a(α, 2) =

1
2

D
(2) c
and Sεβ (·) −→ σβ W
(·),

where
(2)

(σβ )2 =

β+1

2

π


l
+∞
X
X
 (2l)!
l=1

p=0

!2 Z

l−p

(−1)
β+1
)
2p Γ(p +
l−p
(2p)!(l − p)!2
2

0

+∞


ρ2l
α (x) dx .

(ii) If α = 34 ,
a(α, 2) =

1
2

1

D

and [ln(ε−1 )]− 2 Sεβ (·) −→
338



2
v3/2
3β2β/2−1
c (·).
√
Γ((β + 1)/2) 2 W
σ3/2
4 π

(iii) If

3
4

< α < 1,

a(α, 2) = 2(1−α) and for fixed t in [0, 1],
Z
×

+∞

−∞

4.3

Z

+∞

−∞

Sεβ (t)

2(β−1)/2
−→ −β √
Γ
π
L2



β+1
2



√


1
1
Kt (λ + µ)λ|λ|−α− 2 µ|µ|−α− 2 dW (λ) dW (µ) .

2
1
2πσ2α

Lebesgue measure

Let
Sελ (t)

=ε

−a(α,1)

h

λ{0 ≤ u ≤ t, Zε (u) ≤ x}−

Z

t
∗

0

i

Pr(N ≤ x) du for x ∈ R and 0 ≤ t ≤ 1.

Thanks to Theorems 3.1 1) and 3.3 we have the following corollary.
Corollary 4.2 (i) If 0 < α < 21 , a(α, 1) =

(2)

where (σλ )2 = 2
(ii) If α =

(iii) If

1
2

1
,
2

P+∞
l=2

a(α, 1) =

1
2

and

D
(2) c
Sελ (·) −→ σλ W
(·),
i
hR
+∞ l
1
2
2
ρ
(y)
dy
.
H
(x)φ
(x)
α
l! l−1
0
1
2

and

φ(x)
D
(2) c
bα (·).
Sελ (·) −→ σλ W
(·) −
||ϕ||2

< α < 1, a(α, 1) = 1 − α and for fixed t in [0, 1]

−φ(x)
bα (t).
σ2α
P
(2)
Remark: In case (ii), if ϕ = 1[−1,0] , then (σλ )2 = 2 +∞
l=2
L2

Sελ (t) −→

1
H 2 (x)φ2 (x).
(l+1)! l−1

Thanks to Corollary 3.1.1 we can give the rate of convergence when
Indeed for 0 ≤ t ≤ 1 and x ∈ R∗ , let


φ(x)
λ
λ
−d(α,2)
Sε (t) +
Vε (t) = ε
bα (t) .
σ2α
We have the following corollary.
Corollary 4.3 (i) If

1
2

< α < 43 , d(α, 2) = α −

1
2

and

D
(2) c
Vελ (·) −→ σλ W
(·),

339

1
2

< α < 1.

(2)

where σλ is the same as previous corollary.
(ii) If α = 34 , d(α, 2) = 14 and
−1

[ln(ε )]

− 21

Vελ (·)

2
v3/2
3
c (·).
−→ − xφ(x) 2 W
8
σ3/2
D

(iii) If α > 34 , d(α, 2) = 1 − α and for fixed t in [0, 1]
2 Z +∞ Z +∞

1
1
1
1
L2
λ
Kt (λ+µ)λ|λ|−α− 2 µ|µ|−α− 2 dW (λ) dW (µ).
Vε (t) −→ √ xφ(x) √
2
2πσ2α
−∞
−∞

5
5.1

Proofs of the results
(N )

Asymptotic variance of Sε (t)

Case where 0 < α < 1 − 1/(2N ) or α = 1/2 and N = 1
h
i2
(N )
(N )
Proposition 5.1 a(α, N ) = 1/2 and E Sε (t) −→ t(σa )2 .
5.1.1

ε→0

Proof of Proposition 5.1. By Mehler’s formula (see Equation (3))
E



2
Sε(N ) (t)

+∞

2X 2
a l!
=
ε l=N l

Z

t
0

l
(t − u)(ρ(ε)
α ) (u) du.

If we let u = εx, we get
+∞
X
 (N ) 2
E Sε (t) = 2
a2l l!
l=N

Z

t
ε

0

(t − εx)ρlα (x) dx.

2
2
But |ρα (x)| is equivalent to x2α−2 α|2α − 1|v2α
/σ2α
when x tends to infinity and is
2α−2
bounded from above by C x
.
Since α < 1 − 1/(2N ) or α = 1/2 and N = 1, ||g (N ) ||22,φ < +∞ and |ρα (x)| ≤ 1, we
can use the Lebesgue’s dominated convergence theorem to get the result.
✷

Proof of Theorem 3.1. 1) We give the proof for the special case where N = 2
(0 < α < 3/4) to propose a demonstration rather different than in 2)(a). Using the
Chaos representation for the increments of the fractional Brownian motion (see Hunt
(1951)), we can write
Z ∞
1
1
bα (t) = √
[exp(iλt) − 1]|λ|−α− 2 dW (λ),
2π −∞
thus

1 ε1−α
Zε (t) = √
2π σ2α

Z

∞

1

exp(iλt)iλϕ̂(−λε)|λ|−α− 2 dW (λ),

−∞

340

making the change of variable x = ελ in the stochastic integral, we get
 
Z ∞
1
1 1
xt
Zε (t) = √
ixϕ̂(−x)|x|−α− 2 dW (x).
exp i
ε
2π σ2α −∞

We shall consider the following functional
Z t
Z t
ε
√
1
(2)
(2)
g (Zε (u)) du = ε
g (2) (Zε (εx)) dx,
Sε (t) = √
ε 0
0




where the function g (2) verifies E g (2) (N ∗ ) = 0 and E N ∗ g (2) (N ∗ ) = 0. Notice that
Z(x) := Zε (εx) is a stationary Gaussian process having spectral density
fα (x) =

x2 |ϕ̂(x)|2
.
2
2πσ2α
|x|2α+1

The function fα belongs to L2 only if 0 < α < 43 . The correlation function is
Z ∞
ρα (x) =
exp(iyx)fα (y) dy.
−∞

Now, for k ∈ N∗ and 0 = t0 < t1 < t2 < · · · < tk , let
Sε(2) (t)

=

k
X
i=1

αi [Sε(2) (ti ) − Sε(2) (ti−1 )],

where t := (t0 , . . . , tk ) and αi , i = 1, . . . , k, are defined by
ci
αi = Pk
1 ,
[ i=1 c2i (ti − ti−1 )] 2

while cj , j = 1, . . . , k, are real constants. We want to prove that
D

Sε(2) (t) −→ N (0; (σa(2) )2 ),
ε→0

where
(σa(2) )2

=2

∞
X

a2l l!

l=2

Let
(2)
Sε,M (t)

=

k
X
i=1

where
(2)
Sε,M (t)

=

√

ε

Z

0

t
ε

Z

+∞
0

(2)

ρlα (x) dx.

(2)

αi [Sε,M (ti ) − Sε,M (ti−1 )],

(2)
gM (Z(x)) dx

and

(2)
gM (y)

=

M
X
l=2

First, let us prove the following lemma
341

al Hl (y).

Lemma 5.1

D

(2)

(2)

Sε,M (t) −→ N (0; (σa,M )2 ),
ε→0

(2)

where (σa,M )2 = 2

M
X

a2l l!

l=2

Z

+∞

ρlα (x) dx.

0

Proof of Lemma 5.1. Let n be the integer part of 1ε , i.e. n := ⌊1/ε⌋. To study the
(2)
(2)
weak convergence of Sε,M (t) it is sufficient to consider that of Sn,M (t) where
(2)
Sn,M (t)

=

k
X
i=1

αi p

√
ti − ti−1

⌊nti ⌋ − ⌊nti−1 ⌋

Z

⌊nti ⌋

(2)

⌊nti−1 ⌋

gM (Z(x)) dx.

We consider the following functional
(2,m)
Sn,M (t)

=

k
X
i=1

√

ti − ti−1

αi p
⌊nti ⌋ − ⌊nti−1 ⌋

Z

⌊nti ⌋

⌊nti−1 ⌋

(2)

gM (Z (m) (x)) dx,

where Z (m) (·) is an approximation of Z(·) defined as follows, let ψ defined by

2(1 − 2|x|), − 12 ≤ x ≤ 21 ,
ψ(x) =
0,
otherwise.
R
1
Ψ(λ)/Ψ(0), then Ψ ≥ 0,
Note that ψ(x) dx = 1. Let Ψ(x) = ψ ∗ ψ(x) and ξ(λ) = 2π
R
(m)
ˆ
ˆ
ˆ
supp Ψ ⊂ [−1, 1] and ξ(λ)dλ = 1. We define ξ (λ) = mξ(mλ) and
Z +∞
1
(m)
exp(ixy)[fα ∗ ξˆ(m) ] 2 (y) dW (y).
Z (x) :=
−∞

Then (Z(x), Z (m) (x)) is a mean zero Gaussian vector verifying
E [Z(0)Z(x)] = ρα (x),
and


E Z(x)Z

(m)



(0) = r

(m)

(x) =



Ψ(x/m)
E Z (m) (0)Z (m) (x) = ρα (x)
,
Ψ(0)
Z

∞

−∞

1
1
exp(ixy)[fα ] 2 (y)[fα ∗ ξˆ(m) ] 2 (y) dy.

The covariance for Z (m) (·) has support in [−m, m] and thus Z (m) (·) is m-dependent.
h
i2
(2)
(2,m)
Lemma 5.2 gives the asymptotic value of E Sn,M (t) − Sn,M (t) .
Lemma 5.2

h
i2
(2)
(2,m)
(m)
E Sn,M (t) − Sn,M (t) ≤ k cM −→ 0,
m→+∞

342

where
(m)
cM

h Z
2

M
Z
√ X
2
= 2 2(
al l!l)

(r

(m)

0

2

(x) − ρα (x)) dx

(ρ2α (x) + (r (m) )2 (x)) dx

0

l=2

+∞

+∞

 21

+

Z

+∞

ρ2α (x)

0

 21

×

2  1 i

Ψ(x/m)
2
dx
.
1−
Ψ(0)

Proof of Lemma 5.2. Let
√

ti − ti−1

Xi = αi p
⌊nti ⌋ − ⌊nti−1 ⌋

Z

⌊nti ⌋

(2)

⌊nti−1 ⌋

(2)

[gM (Z (m) (x)) − gM (Z(x))] dx = αi

P
k

2

P
k



p
ti − ti−1 Yi .

√
to ai = αi ti − ti−1 Yi
Applying the Schwarz inequality
≤k
i=1 ai
P
(m)
and since ki αi2 (ti − ti−1 ) = 1, it is enough to prove that E [Yi ]2 ≤ cM . Notice that
i2
h
(2)
(2,m)
E [Yi ]2 = E S⌊nti ⌋−⌊nti−1 ⌋,M − S⌊nti ⌋−⌊nti−1 ⌋,M where
(2)
Sn,M

1
=√
n

Z

0

n

(2)
gM (Z(x)) dx

2
i=1 ai

(2,m)
Sn,M

and

1
=√
n

Z

0

n

(2)

gM (Z (m) (x)) dx.

Applying Lemma 4.1 of Berman (1992), which gives the required inequality not ex(2)
actly for gM but for an Hermite polynomial Hl , and Mehler’s formula (see Equation
(3)), we get Lemma 5.2.
✷
(2,m)
Now, we write Sn,M (t) as
(2,m)
Sn,M (t)

=

jn
X

bi,n ξi ,

i=1

αj

where jn = ⌊ntk ⌋, bi,n = √

√

tj −tj−1

⌊ntj ⌋−⌊ntj−1 ⌋

ξi =

Z

i

for ⌊ntj−1 ⌋ + 1 ≤ i ≤ ⌊ntj ⌋, j ∈ [1, k] and
(2)

i−1

gM (Z (m) (x)) dx.

{ξi }i∈N∗ is a strictly stationary m-dependent sequence (and then strongly mixing sequence) of real-valued random
Pjn 2variables with mean zero and strong mixing coefficients
(βn )n≥0 . Furthermore, i=1 bi,n = 1 and limn→+∞ maxi∈[1,jn ] |bi,n | = 0.
On the other hand, as in Rio (1995), defining
M2,α (Qξ1 ) =

Z

1
2

[β −1 (t/2)Qξ1 (t)]2

0

343

dt
β −1 (t/2)

,

where Qξ1 is the inverse function of t → Pr(|ξ1 | > t), β(t) = β⌊t⌋ the cadlag rate
function, β −1 the inverse function of this rate function β.
We have
Z
1

M2,α (Qξ1 ) ≤

0

β −1 (t)Q2ξ1 (t) dt.

This last integral
is finite if, and only if, E(ξ12 ) < ∞ (see Doukhan et al. (1994)).
P
2
But E(ξ12 ) ≤ M
l=2 al l! < +∞, so M2,α (Qξ1 ) < +∞.
Moreover,
Z +∞
M
X
2
al l!
ρlα (x) dx > 0,
0

l=2

for M ≥ 2, because all the terms are limits of variances hence greater or equal to
zero, and for l = 2 we have, by Plancherel’s theorem
Z ∞
Z ∞
2
fα2 (x) dx > 0.
ρα (x) dx = π
−∞

0

Then,
lim E

n→+∞

h

(2,m)
Sn,M (t)

i2

=

(m)
AM

=2

M
X
l=2

a2l l!

Z

0

+∞ 

x l
ρα (x)Ψ( m
)
dx > 0,
Ψ(0)

for M ≥ 2 and m ≥ mM and then applying Application 1 (Corollary 1, p.39 of Rio
(1995)), we finally get that
(2,m)

D

(m)

Sn,M (t) −→ N (0; AM ),
n→∞

for M ≥ 2 and m ≥ mM . Also,
(m)

D

(2)

N (0; AM ) −→ N (0; (σa,M )2 ),
m→∞

h
i2
(2)
(2,m)
and by Lemma 5.2, limm→+∞ supn E Sn,M (t) − Sn,M (t) = 0. Applying Lemma
D
(2)
(2)
1.1 of Dynkin (1988), we proved that Sn,M (t) −→ N (0; (σa,M )2 ) for M ≥ 2 and then
n→∞
Lemma 5.1 follows.
✷
Now since
h
i2
(2)
lim sup E Sε,M (t) − Sε(2) (t) = 0,
M →∞ ε>0

applying the Dynkin’s result, the proof is completed for the case where N = 2 and
0 < α < 43 . Note that this demonstration uses the crucial fact that ρα belongs to
L2 ([0, ∞[) and so can not be implemented for the other cases. For those cases, Theorem 3.1 1) can be proved using the diagram formula, going in the same way as in
Chambers and Slud (1989); indeed for this it is sufficient to adapt the following proof
344

of 2)(a).
2)(a). The following result heavily depends on the N value, known in the literature
as the Hermite’s rank.
Suppose that 1/(2N ) < α < 1 − 1/(2N ). As before, it is enough to prove that
(N )

(N )

(N )

Aε,M (t) = (bεα (t0 ) = bεα (0), . . . , bεα (tk ), Sε,M (t1 ), . . . , Sε,M (tk ) − Sε,M (tk−1 )),
converges weakly when ε → 0 to

(N )

(N )

c (t1 ), . . . , σ (W
c(tk ) − W
c (tk−1 ))),
AM (t) = (bα (t0 ) = bα (0) = 0, . . . , bα (tk ), σa,M W
a,M

where

(N )
Sε,M (t)

1
=√
ε

Z

0

t

(N )
gM (Zε (u)) du

with

=

M
X

al Hl (x),

l=N

and
(N )
(σa,M )2

(N )
gM (x)

=2

M
X

a2l l!

l=N

Z

+∞
0

ρlα (x) dx.

c(t1 ), . . . , W
c(tk )) are independent Gaussian
Furthermore (bα (t1 ), . . . , bα (tk )) and (W
vectors. We shall follow closely the arguments of Ho and Sun (1990) with necessary
modifications due to the fact that we are considering a non-ergodic situation.
Let c0 , . . . , ck , d1 , . . . , dk , be real constants, we are interested in the limit distribution of
k
k
h
i
X
X
(N )
(N )
cj bεα (tj ) +
dj Sε,M (tj ) − Sε,M (tj−1 ) .
j=0

j=1

To simplify the notation we shall write
Γε (t) =

k
X

cj bεα (tj ) and Uε (t) =

j=0

k
X
j=1

h
i
(N )
(N )
dj Sε,M (tj ) − Sε,M (tj−1 ) ,

then Γε (t) is a mean zero Gaussian random variable and
X
a2ε (t) ≡ V [Γε (t)] =
ci cj γε (ti , tj ),
i,j=0,···,k

where γε (s, t) ≡ E [bεα (s)bεα (t)] is given by Lemma 5.3 whose proof is an easy computation.
Lemma 5.3
Z

∞

−∞

Z

v2
γε (s, t) = 2α
2
∞

−∞

Z

∞

−∞

ϕ(x)|s − εx|2α dx −

ϕ(x)ϕ(y)|s − t − ε(x − y)|

2α

345

dx dy +

Z

∞

−∞

ϕ(x)|t − εx|

2α



dx .

We normalize Γε (t) defining Γ′ε (t) = Γε (t)/aε (t). The correlation between Γ′ε (t)
and Zε (s) is denoted by νε (s, t) and
νε (s, t) =

k
X

cj αε (s, tj )/aε (t),

j=0

where αε (s, t) ≡ E [bεα (t)Zε (s)] is given by the following lemma, whose proof is a
straightforward calculation.
Lemma 5.4

Z

αv 2 ε1−α h
αε (s, t) = 2α
σ2α

∞

−∞

Z

∞

Z

∞

−∞

ϕ(x)|s − εx|2α−1 sign(s − εx) dx−

ϕ(x)ϕ(y)|t − s − ε(y − x)|

−∞

2α−1

i

sign(s − t + εy − εx) dy dx .

Thus we deduce the following Lemma 5.5.
Lemma 5.5 Let β = min{α, 1 − α},
|νε (s, t)| ≤ C εβ .

The proof is a direct consequence of Lemma 5.4, according to |tj − s| > 2ε or
|tj − s| ≤ 2ε and to s > ε or s ≤ ε.
We want to study the asymptotic behaviour of
X
r
m
Dε = E [Γm
(t)U
(t)]
=
[a
(t)]
al1 al2 · · · a lr
ε
ε
ε
×

1

εr/2

Z

tj1

tj1 −1

Z

tj2

...

Z

l1 ,...,lr =N,···,M

tjr

tjr −1

tj2 −1

X

j1 ,...,jr =1,···,k

dj1 dj2 · · · djr

E [H1m (Γ′ε (t))Hl1 (Zε (s1 ))Hl2 (Zε (s2 )) · · · Hlr (Zε (sr ))] ds,

where ds = ds1 ds2 . . . dsr . We use the diagram formula. In this case:
E [H1m (Γ′ε (t))Hl1 (Zε (s1 ))Hl2 (Zε (s2 )) · · · Hlr (Zε (sr ))]
=

X

Y

ρ̂(sd1 (w) , sd2 (w) ),

G∈Γ w∈G(V ),d1 (w) 1, the levels are paired in such a way that it is
not possible for a level of the first group to link with one of the second, yielding a
factorization into two graphs, both regular, and then m and r are both even. We can
show as in Berzin et al. (1998) that the contribution of such graphs tends to
! m2
2
X
v2α
ci cj (|ti |2α + |tj |2α − |ti − tj |2α )
2 i,j=0,···,k
×(m−1)!! (r −1)!!

k
X
j=1

! r2

d2j (tj − tj−1 )

2

M
X

l! a2l

l=N

Z

0

∞

ρlα (x) dx

! 2r

.

Using the notations of Ho and Sun (1990), p. 1167, and calling Dε /Rc the contribution
of the irregular graphs in Dε :
X
r
Dε /Rc =
Aε1 × Aε2 × Aε3 × ε− 2 .
G∈Rc

Any diagram G ∈ Rc can be partitioned into three disjoint subdiagrams VG,1 , VG,2
and VG,3 which are defined as follows. VG,1 is the maximal subdiagram of G which
is regular within itself and all its edges satisfy 1 ≤ d1 (w) < d2 (w) ≤ m or m + 1 ≤
d1 (w) < d2 (w) ≤ m + r. Define
∗
VG,1
(1)
∗
VG,1
(2)

=
=

∗
{j ∈ VG,1
| 1 ≤ j ≤ m},
∗
{j ∈ VG,1
| m + 1 ≤ j ≤ m + r},

∗
where VG,1
are the levels of VG,1 .
ε
Ai is the factor of the product corresponding to the edges of VG,i , i = 1, 2, 3. The
∗
normalization for Aε1 is therefore ε−|VG,1 (2)|/2 and as shown in Berzin et al. (1998),
∗
ε−|VG,1 (2)|/2 Aε1 tends to
∗ (1)|/2
!|VG,1
2
X
v2α
∗
(|VG,1
(1)| − 1)!!(2q − 1)!!
ci cj (|ti |2α + |tj |2α − |ti − tj |2α )
2 i,j=0,...,k
!q
!q
Z ∞
k
M
X
X
d2j (tj − tj−1 )
2
ρlα (x) dx ,
a2l l!
×
j=1

l=N

347

0

∗
as ε → 0, where q = |VG,1
(2)|/2. The limit is then O(1).
ε
Consider now A2 and define VG,2 to be the maximal subdiagram of G−VG,1 , whose
∗
edges satisfy m+1 ≤ d1 (w) < d2 (w) ≤ m+r. The normalization for Aε2 is ε−|VG,2 (2)|/2 ,
∗
where VG,2
(2) are the levels of VG,2 . A graph in VG,2 is necessarily irregular, otherwise,
∗
it would have been taken into account in Aε1 . As in Berzin et al. (1998), ε−|VG,2 (2)|/2 Aε2
tends to zero as ε goes to zero. For Aε3 define

VG,3
∗
VG,3 (1)
∗
VG,3
(2)

= G − (VG,1 ∪ VG,2 ),
∗
| 1 ≤ j ≤ m},
= {j ∈ VG,3
∗
= {j ∈ VG,3 | m + 1 ≤ j ≤ m + r},
∗

∗
where VG,3
are the levels of VG,3 . The normalization for Aε3 is ε−|VG,3 (2)|/2 .
∗
We assume now that l1 , l2 ,. . . ,lr , are fixed by the graph. Let L = |VG,3
(2)|,
∗

∗

ε−|VG,3 (2)|/2 Aε3 ≤ ε−|VG,3 (2)|/2
X
×

L
Y

jξ(1) ,...,jξ(L) =1,...,k i=1

×

Y

|djξ(i) |

∗ (2)
w∈E(VG,3 ),d1 (w)∈VG,3

Z

tjξ(i)

Y

tjξ(i) −1 e∈E(V
∗
G,3 ),d1 (e)∈VG,3 (1)

|ρ(ε)
α (sd1 (w) − sd2 (w) )| dsξ(i) ,

|νε (sd2 (e) , t)|
(5)

P
where E(VG,3 ) are the edges of VG,3 and νε (s, t) = aε1(t) kj=0 cj αε (s, tj ) where αε (s, t)
∗
is given by Lemma 5.4. VG,3
(2) can be decomposed in two parts,
∗
BG = {i ∈ VG,3
(2) : g(i)(2 − 2α) ≤ 1},
∗
CG = {i ∈ VG,3 (2) : g(i)(2 − 2α) > 1},

where g(i) is the number of edges in the i-th level not connected by edges to any of
∗
the first levels. Furthermore we note BG
= {i ∈ BG : k(i)(2 − 2α) = 1} where k(i)
is the number of edges such that d1 (w) = i. As in Ho and Sun (1990), p. 1169, we
∗
can rearrange the levels in VG,3
(2) in such a way that the levels of BG are followed
by the levels of CG . Within BG and CG , the levels are also rearranged so that those
∗
with smaller g(i) come first. We have |VG,3
(2)| = |BG | + |CG |.
∗
If i ∈ VG,3 (2), we have (li − g(i)) edges coming from levels in the first group
and thanks to Lemma 5.5 their contribution to Aε3 is bounded by C εβ(li −g(i)) and
in total for these levels we get the bound C εβ i∈BG (li −g(i))+β i∈CG (li −g(i)) ; now the
(ε)
other terms are of the form: ρα (sd1 (w) − si ) which are bounded by 1, or of that one:
Z

tji

k(i)
Y

tji −1 l=1

ρ(ε)
α (si − sjl ) dsi .
348

This last integral can be bounded by



k(i)
1
(2−2α)k(i)
C ε g(i) 1i∈CG + ε
,
1i∈BG /BG∗ + ln( ) 1i∈BG∗
ε
hence, by (5)
∗

ε−|VG,3 (2)|/2 Aε3 =


O ε−(|BG |+|CG |)/2 εβ

× ε((2−2α)
and since li ≥ N , then we have

∗
ε−|VG,3 (2)|/2 Aε3 = O ε((2−2α)
×ε

(1−α−β)

i∈BG

i∈BG

i∈BG

k(i)+

k(i)+

i∈BG (li −g(i))
k(i)
i∈CG g(i)

ε

ε

ε

β(li −g(i))


) (ln( 1 ))|BG∗ | ,
ε

k(i)
i∈CG g(i) −(1−α)

g(i) (βN − 1 )|BG | β
2

i∈CG

i∈CG

i∈BG

g(i)−|CG |/2)

(li −g(i))


1 |BG∗ |
.(6)
(ln( ))
ε

We have the following bounds
(1 − α − β)

X

i∈BG

g(i) ≥ 0,

1
(βN − )|BG | ≥ 0,
2
X
β
(li − g(i)) ≥ 0,
i∈CG

(2 − 2α)

X

i∈BG

k(i) +

X
X k(i)
1
≥ (1 − α)
g(i) + |CG |.
g(i)
2
i∈B
i∈C
G

G

The last inequality is obtained by the same argument for showing (27) in Ho and Sun
(1990), p. 1170.
Three cases can occur: |BG | =
6 0, |BG | = 0 and |CG | = 0, or |BG | = 0 and
|CG | =
6 0.
6 0.
First case: |BG | =
1
∗
Since β > 1/(2N ), one has (βN −1/2)|BG | > 0 and then ε(βN − 2 )|BG | (ln( 1ε ))|BG | = o(1)
thus (6) tends to zero with ε.
∗
Second case: |BG | = 0 (then |BG
| = 0) and |CG | = 0.
In this case VG,3 = ∅ (otherwise it would have been taken in account before in VG,1 )
∗
and then VG,2 6= ∅ thus ε−|VG,2 (2)|/2 Aε2 tends to zero with ε and this gives the required
limit.

349

∗
Third case: |BG | = 0 (then |BG
| = 0) and |CG | =
6 0.
∗ (1)|
β i∈C (li −g(i))
β
|V
∗
G
In this case ε
= ε G,3
with |VG,3
(1)| > 0 (otherwise it would have
been taken in account before in VG,2 ) and (6) tends to zero with ε.
1
2

(b). Suppose that 1/(2N0 ) < α <
H1 (x) = x, then
2)(a).

(1)
Sε (t)

=ε

1
−α
2

a1
(bε (t)
σ2α α

and N = 1, since Zε (u) =

− bεα (0)) +

(N )
Sε 0 (t)

ε1−α ε
ḃ (u),
σ2α α

and

and the result follows by

(c). To conclude the proof, suppose that α = 12 and N = 1. As in (b) and since
(N )
(1)
a1
in this case σ2α = ||ϕ||2 , Sε (t) = ||ϕ||
(bεα (t) − bεα (0)) + Sε 0 (t) and we get the result
2
by using 2)(a).
Remark 3 also follows from the fact that

a1
ε1−α

✷

5.1.2

Rt
0

H1 (Zε (u)) du =

a1
(bε (t)−bεα (0)).
σ2α α

Case where α = 1 − 1/(2N ) and N > 1

Proposition 5.2

2
[ln(ε )] E Sε(N ) (t) −→ 2N !
−1

−1

ε→0



1
1−
2N



1
1−
N

N

ta2N



v2−1/N
σ2−1/N

2N

.

Proof of Proposition 5.2. We suppose t > 0. As in Proposition 5.1, we use Mehler’s
formula and we make the change of variable u = εx to get
Z t
+∞
ε
 (N ) 2
−2 X 2
−1 −1
[ln(ε )] E Sε (t) =
al l!
(t − εx)ρl1−1/(2N ) (x) dx.
ln(ε) l=N
0

2
2
Now, since |ρ1−1/(2N ) (x)| is equivalent to (1 − 1/(2N ))(1 − 1/N )x−1/N v2−1/N
/σ2−1/N
,
when x tends to infinity, and since ||g (N ) ||22,φ < +∞, we have
Z t
ε
 (N ) 2
−2 2
−1 −1
aN N !
(t − εx)ρN
[ln(ε )] E Sε (t) =
1−1/(2N ) (x) dx + O(−1/ln(ε)).
ln(ε)
0

Since

ρN
1 (x)
1− (2N
)

 
N 
N

v 2N
1
1
1
1
−1 2−1/N
1−
,
x
+ 2ε
= 1−
2N
2N
N
x
x
σ2−1/N

with x large enough, the result follows.

✷

Proof of Theorem 3.2. From Proposition 5.2 we prove that
Z t
1
aN
−1 − 21 (N )
q
[ln(ε )] Sε (t) ≃
HN (Zε (u)) du := [ln(ε−1 )]− 2 Fε(N ) (t),
ε ln( 1ε ) 0
350

h
i2
(N )
(N )
i.e. limε→0 [ln(ε−1 )]−1 E Sε (t) − Fε (t) = 0 and the result is an adaptation of
Theorem 3.1 2)(a).
✷

Case where 1 − 1/(2N ) < α < 1

5.1.3

Proposition 5.3 a(α, N ) = N (1 − α) and
h

i−1

2
E Sε(N ) (t) −→ (2α − 2)N + 1 (α − 1)N + 1
N!
ε→0

2 N
v
×a2N α(2α − 1) 2α
t(2α−2)N +2 .
2
σ2α

Proof of Proposition 5.3. We suppose t > 0 and then t ≥ 4ε. As in Propositions
5.1 and 5.2, we use Mehler’s formula and we break the integration domain into two
intervals: [0, 4ε] and [4ε, t].
For the first one, making the change of variable u = εv, we get
2ε

2N [α−(1−1/(2N ))]

+∞
X

Z

a2l l!

l=N

4
0

(t − εv)ρlα (v) dv.

Since |ρα (v)| ≤ 1 and ||g (N ) ||22,φ < +∞ this term is O(ε2N [α−(1−1/(2N ))] ) = o(1).
Now, let us have a closer look to the second interval
2
ε2N (1−α)

+∞
X
l=N

a2l l!

Z

h

t

v2
(t − u) − 2α2
2σ2α
4ε

Z

∞

−∞

Z

u
2α
il


ϕ̇(z − x)ϕ̇(z)  − x dx dz du.
ε
−∞
∞

Using a second order Taylor’s expansion of (u − εx)2α in the neighborhood of x = 0,
it becomes

Z t
+∞
2 
X
α(2α − 1)  −v2α
2
2
al l! (t − u)
2
2
σ2α
4ε
l=N
l
Z ∞Z ∞
2α−2
2
dx dz ε2(1−α)(l−N ) du,
ϕ̇(z − x)ϕ̇(z)x (u − θεx)
×
−∞

−∞

with 0 ≤ θ < 1.
Then, since 1 − 1/(2N ) < α and ||g (N ) ||22,φ < ∞, we can apply the Lebesgue’s dominated convergence theorem and the limit is given by the first term in the sum.
✷
Proof of Theorem 3.3. Define
)
G(N
ε (t)

=

aN
εN (1−α)

Z

0

351

t

HN (Zε (u)) du.

h
i2
(N )
(N )
A straightforward calculation shows that E Sε (t) − Gε (t) → 0 as ε → 0: the
proof is similar to the one of Proposition 5.3.
(N )
Thus studying the asymptotic behaviour of Gε (t) allows us to obtain the same for
(N )
Sε (t). We have seen in the proof of Theorem 3.1 1) that
Zε (u) = ε

1−α

Z

+∞

iλ exp(iλu) dWε (λ),

−∞

where the stochastic measure dWε (λ) is defined as
dWε (λ) = √

1
1
ϕ̂(−ελ)|λ|−α− 2 dW (λ),
2πσ2α

and then
)
G(N
ε (t)

aN

=

εN (1−α)

Z



t

HN ε

0

1−α

Z



+∞

iλ exp(iλu) dWε (λ) du.

−∞

Using Itô’s formula for the Wiener-Itô integral (see Dobrushin and Major (1979)), we
obtain
Z t Z +∞
Z +∞
√
(N )
...
w(λ1 , u) · · · w(λN , u) dWε (λ1 ) · · · dWε (λN ) du,
Gε (t) = N !aN
0

−∞

−∞

where
w(λ, u) = iλ exp(iλu).
As in Chambers and Slud (1989) p. 330, integrating this expression with respect to
u, we get
)
G(N
ε (t)

√ N Z
= N !i aN

+∞

−∞

···

Z

+∞

−∞

Kt (λ1 + · · · + λN )λ1 · · · λN dWε (λ1 ) · · · dWε (λN ),

where Kt (λ) = (exp(iλt) − 1)/(iλ).
So
h
i2
2N R
P

 Q 

(N )
N
N
1
2
2
2
1−2α
√
E Gε (t) = aN N ! 2πσ
|Kt
dλi .
i=1 λi |
i=1 |ϕ̂(−ελi )| |λi |
RN
2α

P

N
i=1

 Q

N
2
1−2α
λi |
when
i=1 |λi |

On one hand the inner integrand converges to |Kt
ε tends to zero.
P
 Q

N
N
2
1−2α
On the other hand, we can bound this integrand by |Kt
.
i=1 λi |
i=1 |λi |
P
 Q 

h
i2
R
(N )
N
N
2
1−2α
Thus if we prove that RN |Kt
dλi < +∞ then E Gε (t) →
i=1 λi |
i=1 |λi |
352


2
E G(N ) (t) when ε → 0 where
!
N Z +∞

Z +∞
N
X
√
i
λi ×
...
Kt
G(N ) (t) = N !aN √
2πσ2α
−∞
−∞
i=1
N 

Y
1
λi |λi |− 2 −α dW (λ1 ) . . . dW (λN ),
i=1

which is well defined.
Let us define
It :=

Z

RN

|Kt

N
X
i=1

N 
 Y
 Z
2
1−2α
λi |
|λi |
dλi =

RN

i=1

 P

N 

4 sin2 t( N
λ
)/2
Y
i
i=1
|λi |1−2α dλi .
P
2
N
i=1
i=1 λi

It is always well defined with the possible value +∞. Making the change of variables:
λ1 = 2y1 (1 − y2 − . . . − yN )/t, and λi = 2y1 yi /t, for i = 2, . . . , N , we get

 2N (1−α) Z +∞
2
2
2
(2N −3−2N α)
t
sin (y1 )|y1 |
dy1 ×
It =
t Z
−∞

1−2α
1−2α
1−2α
|1 − y2 − . . . − yN |
|y2 |
· · · |yN |
dy2 · · · dyN .
RN −1

Now, let the following change of variables y2 + y3 + . . . + yN = w2 , y3 = w2 w3 ,. . . ,
y N = w 2 wN .
(N −2)
Then y2 = w2 (1 − w3 − . . . − wN ) and the Jacobian is w2
. Thus
Z
|1 − y2 − . . . − yN |1−2α |y2 |1−2α · · · |yN |1−2α dy2 · · · dyN =
RN −1
Z ∞

|1 − w2 |1−2α |w2 |(N −1)(1−2α)+(N −2) dw2
−∞

×

Z

1−2α

RN −2

|1 − w3 − · · · − wN |

N 

Y
1−2α
|wi |
dwi .
i=3

Therefore we can apply the iteration and we have

 2 2N (1−α) Z +∞
2
2
(2N −3−2N α)
It =
t
sin (y1 )|y1 |
dy1
t
−∞

N Z ∞
Y
1−2α
(k−1)(1−2α)+(k−2)
|1 − wk |
|wk |
dwk < +∞,
×
k=2

−∞

since 1 − 1/(2N ) < α < 1.
Consider now
D

(N )

(ε) =

√

N !aN



i
√
2πσ2α

N

Kt

N
X
i=1

353

N 

Y
1
λi
ϕ̂(−ελi )λi |λi |− 2 −α .
i=1

D(N ) (ε) converges pointwise to
D

(N )

(0) =

√

N !aN



i
√
2πσ2α

N

Kt

N
X
i=1

N 

Y
1
λi
λi |λi |− 2 −α ,
i=1

and from the previous calculations and Lebesgue’s theorem D (N ) (ε) → D (N ) (0) as
ε → 0 in the L2 -norm with respect to Lebesgue’s measure and Theorem 3.3 follows.
✷
Proof of Corollary 3.1. Since Zε (u) =
Vε (t) =

ε1−α ε
ḃ (u),
σ2α α

and H1 (x) = x, if A 6= ∅,

a1 ε
[b (t) − bεα (0) − bα (t)] ε−d(α,N0 ) + Sε(N0 ) (t).
σ2α α

A straightforward calculation shows that the second order moment of the first term
above is O(ε2(α−d(α,N0 )) ) = O(ε). So 1. (i), (ii) and (iii) follows by Theorem 3.1 1),
3.2 (i) and 3.3.
bε (t) − bα (t)
√
1
and when
cα,ϕ [Bε (t) − Bε (0)] where Bε (t) = α α √
Now if A = ∅, Vε (t) = σa2α
ε cα,ϕ
D

0 < α < 1, Bε (t) −→ B(t) and this concludes the proof