ESAIM: Proceedings
Fractional Differential Systems:
Models, Methods and Applications
URL: http://www.emath.fr/proc/Vol.5/
Vol. 5, 1998, 43-54
DISTRIBUTION PROCESSES WITH STATIONARY
FRACTIONAL INCREMENTS
L. BEL, G. OPPENHEIM, L. ROBBIANO AND M.C. VIANO
Abstract. This paper continues an idea introducedby L. Bel, G. Oppenheim,
L. Robbiano ans M.C. Viano. We dene a linear Gaussian distribution process
with stationary k-order increments and study several properties of this process,
which depend on a specic family of distributions. For k=1, we study the cases
in which these distributions are fractional.
Resume. Ce travail continue une idee de L. Bel, G. Oppenheim, L. Robbiano
ans M.C. Viano ou nous annoncions un resultat de base demontre dans un
cas particulier. On denit un processus distribution dont les accroissements
d'ordre k sont stationnaires lineaires gaussiens. On etudie les proprietes de ces
processus qui dependentd'une famille de distributions speciquesau probleme.
Pour k=1, on s'interesse aux cas dans lesquels ces dernieres distributions sont
fractionnaires.
1.
Introduction
We continue to explore the stochastic distribution processes we began to study in
2] and 1]. These processes form a sub-family of the Gelfand-Vilenkin's family (5])
because the underlying continuity holds with relation to a stronger topology. Moreover they are dened through a lter , the impulse response, being a convolution
kernel. For this reason, we call them linear processes. We denote this family as L
They are Gaussian and stationary processes. These processes are written:
Z; h i=
s
f
:
X '
f
'
dW :
We study a little dierent family: a family of non-stationary distribution processes,
but the increments and the derivatives (with relation to the generalized functions
theory) are stationary processes and belong to L
The understanding we can gather from the study of the process called Fractional
Brownian Process, and from processes with lters whose Laplace transform is
( ) = , give a nice feeling of several results of the present paper. Of course, the
results of this paper concern more general lters.
:
F s
s
1.1. The FBM
The continuous time Fractional Brownian Motion (denoted B) (Mandelbrot-Van
Ness 7]) does not belong to Rthe family L
It can be written as ( ) = R ( ( ; ) ; (; )) s where
( ) = ;(H1+ 12 ) H ; 21 1It>0 using evident notations and with 0
1.
As we know, the increments h process
indexed
by
,
is
a
stationary
process,
R
because h ( ) = ( + ) ; ( ) = R ( ( + ; ) ; ( ; )) s h belongs
:
B t
f t
f t
s
f
s
dW
t
< H <
B B
t
B t
h
B t
t
f t
h
s
f t
s
dW :
B
c Societe de Mathematiques Appliquees et Industrielles. Typeset by LATEX.
L. Bel, G. Oppenheim, L. Robbiano, M.C. Viano: Laboratoire de Modelisation Stochastique et Statistique, B^at 425, Universite de Paris Sud, 91 405 Orsay Cedex, France. E-mail:
[email protected].
Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:1998011
44
L. BEL, G. OPPENHEIM, L. ROBBIANO AND M.C. VIANO
to L associated with the impulse response ( ) = ( + ) ; ( ) The preceeding
function , choosen by Mandelbrot, is a very good choice. It provides the property
of self-similarity in distribution to Nevertheless this property is an advantage as
well as a drawback.
An advantage, because it permits the modelisation of "fractal" situations as
for instance "electrical consumption (3]), image analysis (8]), synthesis of
articial mountains, Internet trac (11])".
A drawback because a single parameter
determines all the characteristics,
the local ones as the Holder exponents and the global ones as the speed of
convergence to 0, close to innity, of the increments covariance, which is a
mesure of long dependence property. Several authors are working to loosen
of this constraint. For instance Levy-Vehel (6]) "have dened a time varying
parameter Fractional Brownian Motion. The parameter usually constant
through time, is taken to be a smooth function ( ) of time ". We do not
go this way further, for this moment.
It is easy to associate a distribution process to the time process The derivative
process
(in the distribution sense) of belongs to the family L Its impulse
response is the Inverse Laplace transform of = ;H +1=2
As a consequence of these remarks, in order to include the FBM within the cases
to study, we had to dene, in complement to L a new family of non stationary
distribution processes whose rst order increments process belong to L and, are,
for this reason, linear stationary Gaussian distribution processes. We are able to
generalize the ideas to orders greater than 1. A member of this family is dened
using expressions similar to the following one which resembles the FBM, but in
which and are distributions:
Z
Z
; ( )
h i=
s
g t
f t
h
f t :
f
B:
H
H
H t
t
B:
@B
B
:
s
s
:
X
f
g
X '
f
'
g
' t dt
dW
1.2. Processes that can be constructed starting with Laplace
transform
s
This function belongs to a family of Laplace transforms, we have studied quite a
lot. This is the family of fractional lters. See for instance (1],10]). The fractional
basic lters Qare the Inverse Laplace Transform = L;1( ) of function such
that: ( ) = Kk=1( ; k )d
The simplest function of this set is ( ) = We know exactly the associated
when they are usual functions ( 0) or when they are distributions ( 0).
Let us have a look at the processes we are able to associate, giving greater place
to temporal, even non stationary processes or, if it is impossible, to distribution
processes. Let vary in R, partitionned as follows:
let = ; ; 21 . Then
8 ( )
if 2
>
>
1 vp( H ; 21 ) if 2 R+ n
<
L;1( ) =
;( + 12 )
>
>
if 2 R;
: ;( 1+ 1 ) H; 12
2
3
1
We know that for ;
; there exists the FBM, a time non stationary
2
2
process, characterized by the parameter , 0
1
3
1
Let us examine the left part of the interval ; ; ]. When
2 2
2
+
1
2
;1
;
;
then ; 1
, there exists
2
2 for = 1 2
f
f
F s
s
F
F s
s :
f
<
NN
H
s
t
H
t
H
< <
H
k
< <
ESAIM: Proc., Vol. 5, 1998, 43-54
F
k:
a
k
k
::: < H <
k
:
< H < k
DISTRIBUTION PROCESSES WITH STATIONARY FRACTIONAL INCREMENTS
45
a time process whose k-th order increments are stationary. This process is
dened by
Z X 1 j (j ) ()=
( ; );
(; ) s
R
0j k;1 !
X t
f t
t f
j
s
dW
where (j ) is the -derivative of .
0. The
Let us analyse the right of the interval ; 32 ; 12 ]. In this case
Inverse Laplace Transform is, either a function that does not belong to 2
close to 0, or is a distribution. Nevertheless, in all the cases, it is possible to
dene a stationary linear distribution Gaussian process. For these processes
we know several properties (1]).
f
s
j
f
H >
L
1.3. Further extensions
Leaning on the intuitions of the FBM, we extend the ideas in several directions:
from time processes to distribution processes,
from order 1 to order increments,
from the function (associated with the FBM) to couples of usual functions
or distributions .
These extensions constitute the center of this paper.
k
f
f g
1.4. Plan
After this introduction, we dene and study the distribution processes whose
increments h are stationary linear processes. We present the properties of
and particularly when h is a fractional process.P The last part concerns the
processes whose th-order increments, (k) h ( ) = kj=0 kj (;1)k;j ( + ) are
stationary processes.
X
X
X
X
k
2.
X
t
C
X t
jh Linear Generalized Processes with stationary increments
The time processes that can be written as
Xt
=
Z
( (
f t
; s) ; g(;s)) dWs
where and are functions belonging to 2 (Rn) are processes with stationary
increments. Indeed
Z
h( ) = ( + ) ; ( ) = h( ; ) s
f
g
L
X
t
X t
h
X t
f
t
s dW
has the same distribution as h ( + ) for every .
Let be the distribution process associated with t We get
X
X
hX 'i
=
=
t
Z Z
Z
((
f t
f
X :
'
; s) ; g(;s)) '(t)dt
; g
Z
()
' t dt
dWs
(2.1)
dWs
R
The processes dened by (2.1) for distributions and such that ; belongs to 2 (Rn) for every 2 C01 (Rn) have stationary increments. This means
that the process h = h ; is a stationary distribution process:
h h i = h ;h i = ;h i
Z
( ;h ; ) s
=
(2.2)
f
L
g
f
'
g
'dt
'
X
X
X
X
'
X f
'
X '
'
' dW
ESAIM: Proc., Vol. 5, 1998, 43-54
46
L. BEL, G. OPPENHEIM, L. ROBBIANO AND M.C. VIANO
has the same distribution as h h i.
The next theorem gives a necessary and sucient condition on et for the process
written in (2.1) to be well dened.
Let s (Rn) denote the -index Sobolev space 9], h is = (1 + j j2)s=2 .
We shall note t = j .
R
Theorem 2.1. Let and 2 D0 (Rn ), ; ( ) 2 2 (Rn) for every
2 C01 (Rn ) if and only if
) j 2 ;1(Rn ) 8 = 1
) ; 2 ;1 (Rn)
Remark 2.2. If 2 2 (Rn ), the hypotheses of the theorem are satised for every
2 ;1 (Rn ) and the result is a direct consequence of Theorem 6 of 2] (i.e. 2
2 (R) for all 2 C01 if and only if 2 ;1 ). The hypothesis j 2 ;1 (Rn) is
equivalent to
h i;1 j j b( )j 2 2 (Rn)
=1
It is less stringent than the hypothesis 2 2(Rn ) because it implies that b( )
belongs to 2 close to innity and j b belongs to 2 close to 0.
Proof. We shall need the following lemma:
Lemma 2.3. Let 2 D0(Rn ). If 8 = 1
j 2 ;1 (Rn) then 8 2 C01 (Rn)
X
'
f
g
X
H
s
@ jg
@ g
f
g
f
'
g
' t dt
L
'
g
f
i
@ g
ii
f
H
g
j
H
::: n
:
L
H
f
L
'
f
g H
@ g
L
j
g
L
g
j
g
'
; g
g L
::: n
Z
f
'
R
; g
'
H
Z
()
' t dt
( ) 2 2 (Rn ) Then
( j ) = j 2 2 (Rn )
and from the Theorem 6 of 2], there exists such that j 2
Moreover, if ( ) 2 2 (Rn) then j ( ( )) 2 ;1(Rn ). As
Z
()
j ( ( )) = j ( ) ; j ()
Suppose that for every
R
@ g
2 L2 ( n )
()
' t dt
Suppose the lemma has been proved.
Let
( ) = H '
::: n
L
g
we have
2 C01 ( n ),
'
H '
H @ '
L
@ f
'
:
L
H '
L
@
H '
@
g
'
' t dt
@ f
@
H '
H
@
f
@
R
we have j () Z( ) 2 ;1 (Rn).
Let such that ( ) = 1 we get
'
H
'
g
H
R
( n).
' t dt
H
' t dt
() 2 ;1(Rn )
@j g
Furthermore
H
( ) = ( ; ) H '
f
g
'
8j
=1
: : : n:
; (
g '; g
Z
() )
' t dt
the second term belongs to 2 (Rn ), thanks to the lemma we get
8 2 C01 (Rn)
( ; ) 2 2(Rn )
and ; 2 ;1 (Rn ) from Theorem 6 of 2].
Conversely suppose that the hypotheses i) and ii) of the Theorem are satised,
Z
( ) = ( ; ) + ( ; ( ) )
L
'
f
g
f
g
'
L
H
H '
ESAIM: Proc., Vol. 5, 1998, 43-54
f
g
'
g
'
g
' t dt
DISTRIBUTION PROCESSES WITH STATIONARY FRACTIONAL INCREMENTS
47
The rst term belongs to 2(Rn ) from the Theorem 6 of 2] and the second term
belongs to 2 (Rn) thanks to the lemma. In consequence ( ) 2 2 (Rn ) for every
2 C01 (Rn ).
L
L
H '
L
'
Proof of lemma 2.3. If 8j = 1 : : : n
kg ' ; g
Z
@j g
( ) k2L2 (R ) =
' t dt
so g
'
; g
()
' t dt
Z
n
C
C
C
b
g
R and
2 S 0 ( n)
b
b
j g(;
) j2j '(
) ; '(0) j2
Z
Z
R then
2 H ;1( n ),
b
d
\
h
i;2 j j2j g(;
) j2 d
n Z
X
h
i;2 j @j g(
) j2d
j =1
n
X
j =1
k@j gk2H ;1 (Rn )
<
+1
R
2 L2 ( n).
2.1. Properties of the derivative
Let be a process dened by (2.1), the process j is a stationary linear distribution process and the results of 2] permits to settle easily its properties.
We recall that, for the distribution process, the regularity spaces
s(Rn 2( )), 2 R are dened by (2]):
s(Rn 2( )) =
f 2 S 0 (Rn 2 ( )) 8 ;1 kk kL2 () 1
2;ksg
L (R )
X
C
@ X
L
C
s
L
X
L
k
X
n
C
:
the k being the operators entering in the Littlewood-Paley decomposition.
Proposition 2.4. j is a linear distribution process associated with the lter
j
@ X
,
- the spectral density is j j fb(
) j2,
- the covariance function is = F ;1 (j j fb(
) j2),
- @j X belongs to C s(Rn L2 ( )) if and only if @j f belongs to B2s1 (Rn).
@ f
Proof. Let X be a distribution process dened by (2.1) then
h@j X 'i = ;hX @j 'i =
Z
@j f
'dWs :
\
If and satisfy the hypotheses of Theorem 2.1, j belongs to ;1 (Rn ) and
2
j is a linear stationary distribution process with a spectral density j j ( ) j =
j j b( ) j2 its covariance is given by
= F ;1(j j b( ) j2) and it belongs to
s(Rn 2( )) if and only if
s
n
j belongs to 21 (R ) (2]).
f
g
@ f
@ X
H
@ f f C
L
@ f
Proposition 2.5. If
longs to
R
s
n).
C
loc (
@j f
f B
belongs to B2s;+11 (Rn ), the process dened by (2.1) be-
Proof. If for every j , @j f belongs to B2s;+11 (Rn) the process @j X belongs to C s;1(Rn )
and
X
belongs to
s
Cloc
(Rn 2( )).
L
ESAIM: Proc., Vol. 5, 1998, 43-54
48
L. BEL, G. OPPENHEIM, L. ROBBIANO AND M.C. VIANO
2.2. Properties of the increments
Let be a process (2.1), the process h is a linear stationary distribution process
and in this case we can also settle easily its properties.
Proposition 2.6. h is a linear distribution process associated with the lter
h:
- the spectral density is 4j b( )j2 sin2( 2 ),
X
X
X
f
h
f - the covariance function is h = F ;1 4jfb(
)j2 sin2( h
) ,
2
- X belongs to C s+1(Rn L2( )) if @j f belongs to B2s1 (Rn ) for every j .
Proof.
hXh 'i
Z
=
Z
=
( ;h )
f
' dWs
( h ; ) f
f
'dWs
So h is the lter of the linear distribution process h . In consequence its
density is:
jF ( h )j2 = j b( )j2 ( ih ; 1)( ;ih ; 1)
= 4j b( )j2 sin2( 2 )
and its covariance is: h = F ;1 4j b( )j2 sin2 ( 2 ) 5].
Suppose now that for every , j 2 2s1 (Rn), is the truncature function entering in the Littlewood-Paley decomposition of 9] for 0
dh k2L2(R )
kk h k2L2 (R ) = k (2;k )
f
X
f
f e
e
h
f h
f j
@ f
B
k
f
= 4
for = ;1
Z
n
f
n
2k+1 <jj<2k+2
C
2;k
Z
2 (2;k )jfb(
)j2 sin( h
)d
=
j2
f j
d
d
k(
)fh k2L2 (Rn )
= 4
Z
C
C
( )2 j b( )j2 sin( 2 )
jZj<2
( )j j2 b( )j2
jj<2
In consequence h belongs to 2s+1
1 (Rn ) and s+1(Rn 2( )).
f
(2;k ) 2
j b( )j2
;k
k
C
2
2k+1 <jj<2k+2
;
k
(
s
+1)
C2
k;1fh k2L2(Rn )
:
B
f Xh
f h
d
d
belongs to
L
2.3. Processes with stationary fractional increments
We have dened in 1] the distribution fractional ARMA processes by
Z
h i=
s
X '
ESAIM: Proc., Vol. 5, 1998, 43-54
f
'dW
DISTRIBUTION PROCESSES WITH STATIONARY FRACTIONAL INCREMENTS
with
C
f
C
= L;1 ( )
F
F
K
Y
=
k=1
(
s
49
; ak )dk
the parameters k 2 and k 2 satisfying the following conditions:
- < ( k ) 0 if k singular
- if < ( k ) = 0 and k singular, then < ( k ) ; 12 .
We introduce now the distribution processes with fractional increments in the following way: they are processes that satisfy (2.1) with
= L; 1 ( + )
= L; 1 ( )
(2.3)
with
a
e a
a
d
e a
a
e d
f
( )=
F s
C
Proposition 2.7. If
- =d+
J
X
j =1
j <
F
K
Y
(
k=1
;
s
G
; ak
1,
2
>
g
) dk
( )=
G s
G
d
s
J
Y
(
j =1
s
; bj )j
1
( j ) ; 12 if < ( j ) = 0 and
(2.4)
- <e(ak ) < 0 if ak singular or <e(dk ) > ; 2 if <e(ak ) = 0 and ak singular
- <e(bj ) < 0 if bj singular or <e e b
>
bj
singular,
- <e(d) > ; 3
2
then the process X dened by (2.1) with (2.3) and (2.4) is a distribution process
with stationary increments.
Proof. With these conditions on ak and dk , from 1], we have L;1(F ) 2 H ;1 (R)
;1 (R). The condition = d + X j
j =1
J
1 implies that ( )
2
1
3
; if < ( j ) = 0
is 2 close to innity, while the conditions
; and j
2
2
assure that h i;1 F ( ) = h i;1 ( ) could be integrated at 0, which implies that
2 ;1 (R).
Property 2.8. The process with fractional increments dened by (2.1) with (2.3)
; sup(D); 21 (R 2( )).
and (2.4) belongs to loc
thus
f
;g 2
H
L
@g
;
d >
@g
<
G s
>
e b
G i
H
C
Proof. It was proved in 1] that
L
@f
2
B
;D ; 3
21 2 (R),
@g
2
;D ; 3
21 2 (R) so the sum
B
1
belongs to 2;1sup(D); 2 (R) and, applying proposition 2.5, we get the result.
Proposition 2.9. The increments h of the fractional process dened by (2.1)
B
X
with (2.3) and (2.4) posses the following properties:
- their spectral density is 4(j(F + G)(i
)j)2 sin2( h
2 ),
- their covariance function is h = F ;1 4j(F + G)(
)j2 sin2 ( h
2) ,
3
- they belong to C ; sup(D); 2 (R L2( )),
- they are mixing if and only if for singular ak , <e(ak ) < 0.
Proof. The three rst properties are direct consequences of proposition (2.6). Let
us prove the fourth.
ESAIM: Proc., Vol. 5, 1998, 43-54
50
L. BEL, G. OPPENHEIM, L. ROBBIANO AND M.C. VIANO
Let (f ) be the stationary process associated with the lter .
h f i = h (f ) ;hi
Let 2 C01 (] ; 1 0]) and 2 C01 ( +1).
( (f ) ( ;h ) (f ) ( ;h )) = ( (f ) ( h ;h ) (f ) ( h ;h ))
supp( h h ) ] ; 1 0] and supp( h h ) ; 1 thus if < ( k ) 0 for all
singular k , from the Theorem 9 of 1]
cor( h ( ) h ( )) ;b(T ;h) h ;T
and h is a mixing process.
If < ( k ) = 0, for some singular k then h is not a mixing process, from Theorem
11 of 1].
Let us examine the case where = 0, that is when = = L;1( ). With the
hypothesis on , ( ; ) ; (; ) belongs to 2 and the process is a continuous time
usual process. If one writes
X
f
X
'
'
E X
'
X
'
T
:X
E X
'
T
'
:X
h
e a
<
a
X
X
Ce
C e
X
e a
a
X
F
g
g t
s
g
f
s
( )=
G s
g
G
L
s
J
Y
(
; bj )j
s
j =1
with
bj
6= 0
we can see that the dened processes are a simple extension of the Brownian
Fractional Processes, which correspond to the case where ( ) = ;H ; 21 and
1 we have ; ; 12 ; 32 . The only
( ) = ;(H1+ 12 ) H ; 12 1It>0. If 0
processes of the family that are self-similar are the FBM. Nevertheless, in this family we have a greater variety of sample regularity, since, deduced from property 2.8,
the regularity index is ; sup( ) ; 12 .
The derivative of the FBM is a stationary distribution process, the fractional Gaussian white noise, which density is given by (4]) :
1
1
j ;( + ) b( ) j2=j ( );H ; 2 j2=j j1;2H
2
G s
< H
t
g t
<
s
H
>
:
D
H
3.
g i
i
Distribution Processes with stationary kth-order
increments
The time processes that may be written as
()=
X t
Z
(
f t
; s) ;
X
jjk;1
t g
(; )
s
dWs
with , in 2 (Rn ), have stationary th-order increments. Indeed
f
g
L
k
(k)Xh (t) =
=
k
X
j =0
Z
j
Ck
(;1)k;j ( + )
X t
(hk) (
f t
jh
; s)dWs
are stationary processes. We may dene, similarly, distribution processes with
stationary th-order increments by:
k
hX
Z0
X
i = @ ( ) ;
'
f
' s
jjk;1
g
(; )
s
Then we have a result that generalises Theorem 2.1
ESAIM: Proc., Vol. 5, 1998, 43-54
Z
()
' t t
dt
1
A
dWs
DISTRIBUTION PROCESSES WITH STATIONARY FRACTIONAL INCREMENTS
Theorem 3.1. Let f andZg, j j k ; 1 2 D0 (Rn),
X
2 n
f '(s) ;
g (;s)
'(t)t dt 2 L (R ) for every
jjk;1
if
N N
'
R
2 C01 ( n)
51
if and only
i) for every 2 n, 1 j j k, j j + j j= k @ g 2 H ;jj (Rn ),
ii) 2 n, 1 j j k ; 1, @ g0 ; !g 2 H ;jj (Rn),
iii) f ; g0 2 H ;1 (Rn ).
Proof. In order to simplify, we prove the result in R. We note f = f and gj = gj .
We prove the theorem by induction. It has been proved for = 1. Suppose it is
true for ; 1. Let
k
k
fg
k (')
H
=
f
';
X Z
k;1
j =0
gj
()
j
t ' t dt
By induction hypothesis, kfg;1( ) 2 2 (R) for every 2 C01 (R) if and only if
) j k;1;j 2 ;j (R)
=1
;1
j
j
;
j
)
(R)
=1
;2
0 ; (;1) ! j 2
;1
) ; 02
(R )
Suppose that kfg ( ) 2 2(R) for every 2 C01 (R).
H
i
@ g
ii
@ g
iii
L
'
H
j
j g
f
H
'
g
'
H
fg
k (@')
j
::: k
H
L
H
::: k
'
=
@f
'+
=
@f
'+
eg
fe
=
Hk
X Z
k;1
j =1
k;2
X
j =0
gj
gj
jt
Z
+1
j ;1 '(t)dt
( + 1) j ( )
j
t ' t dt
;1(')
eg
2
with e = and ej = ( + 1) j +1. kfe
;1( ) 2 (R) for every
because of hypothesis i) of the induction.
je
j
; j (R )
for = 1
k;1;j = ( ; )
k;j 2
Moreover
f
@f
g
j
@ g
k
k
@ gj
=
H
j @ g
fg
k
@ (H
k ('))
We have
g
j k;j g 2
j
H
L
H
= (@ k f ) ' ;
@ @
'
j
X
k;1
j =0
(@ k gj )
Z
j
'
::: k
()
t ' t dt
R), thus,
2 C01 (
; 1:
2 H ;k (
(3.1)
R)
;k (R) for j = 1 : : : k ; 1 by (3.1) and
)) = ( k ) 2 2(R) ;k (R)
thus k 0 k ( ) 2 ;k (R) and together with (3.1)
j
;j (R)
for = 1
k ;j 2
Thus we obtain condition i).
Let 1 ,
@ g
R
fg
Hk
t ' t dt
(
k
@ '
=
L
H
k
fg l
k (@ ')
H
'
H
H
@ g
l
@ f
l ' ; (;1)l
@ f
2 L2 (R)
j
X Z
k ;1
j =l
gj
(
j j
: : : k:
; 1) : : : (j ; l + 1)tj ;l '(t)dt
(3.2)
ESAIM: Proc., Vol. 5, 1998, 43-54
52
L. BEL, G. OPPENHEIM, L. ROBBIANO AND M.C. VIANO
Moreover
fg
l
@ (H
k ('))
=
l
@ f ';
=
l
@ f ';
2 H ;l (
For
l
k ; j , @ l gj
X
k;1
j =0
k;l;1
X
@
l
@ f ' ;
but
X
k;l;1
l ' ;
@ f
X
j =0
@ gj
j =0
gj
k;l;1
Z
l
j =0
R)
= l;(k;j ) k;j
@
(@ l gj )
2 H ;l (
;
j =0
Z
j
j
t '(t)dt ;
k ;1
X
j =k;l
l
@ g
Z
j
()
j
t ' t dt
R), by condition i), and in consequence
R)
2 H ;l (
j
()
(
; 1):::(j ; l + 1)tj ;l '(t)dt
t ' t dt
XZ
k;1
gj
j =l
X Z
l
X
()
() =
j
l ' ; (;1)l
k ;1
k;l;1
Z
l
@ gj
l
@ g
j
t ' t dt
t ' t dt
@ f
+(;1)
Z
gj
j =l
l
@ g
Z
j
(
j j
j j
; 1):::(j ; l + 1)tj ;l '(t)dt
()
j
t ' t dt
The rst term belongs to 2 (R) from (3.2) thus to ;l (R) and
L
(;1)l
X Z
k;1
j =l
gj
(
j j
; 1):::(j ; l + 1)tj ;l '(t)dt ;
X l
(;1) ( + )( +
k;l;1
j
j =0
l
H
j
j
l
j
l
j =0
l
l ; 1):::(j + 1)gj +l ; @ gj
This relation being true for every
(;1)l ( + )( +
X
k;l;1
'
Z
Z
l
@ gj
j
()
t ' t dt
@ g
@ g
H
Z
g
ESAIM: Proc., Vol. 5, 1998, 43-54
H
j
()
' t dt
;
R)
2 H ;l (
R), we obtain
R) for = 0
j
For = 0, we get (;1)l ! l ; l 0 2 ;l (R) for = 1
condition ii).
Lemma 3.2. If j 2 ;j (R) for = 1
, then for every
l g
() =
2 C01 (
; 1):::(j + 1)gj +l ; @ l gj 2 H ;l (
j
j
t ' t dt
l
::: k
::: k
j ;1
X
1
!
h=0
h
@
R)
h g th ' 2 L2 (
::: k
;1
'
;1;l
and we obtain
2 C01 (
R),
DISTRIBUTION PROCESSES WITH STATIONARY FRACTIONAL INCREMENTS
Let us suppose the lemma has been proved.
fg
H
k (')
=
X( Z
k;1
';
f
kX
;1 ( X
;
= (
l=0
h=0
kX
;1 ( X
l
l=1
l;h
tl '
!
h
; g0 ) ' ;
f
;
gj
j =0
l
h
@ g
j
t '(t)dt ;
X( Z
k;1
@
h=0
gj
j =0
hg
l;h
)
!
h=0
h
)
tj +h '
X
k;1;j h
@ g
j
t '(t)dt ;
tl '
!
h
X
k;1;j h
@ gj
h=0
)
53
h
!
j
tj +h '
)
(3.3)
By the lemma, the rst sum belongs to 2 (R). We can write
l
X
@
h=0
hg
(Xl
= 1!
l;h
!
h
h=0
l
+
Xl
h=0
L
h (;@ l g
Cl
0 + (;1)l;h (l ; h)!@ h gl;h )
(;1)l;h Clh @ l g0
)
The second term is equal to 0 and by condition ii)
; l 0 + (;1)l;h ( ; )! h l;h = h (; l;h 0 + (
@ g
l
f
@ g
@
@
g
l
R)
; h)!gl;h ) 2 H ;l (
h
@ gj
t '
h
;g
L
f
g
'
b(0) = b( ) ; b( ) ;
'
thus
' @ ' :::
b b
b
If j 2 ;j (R), j jj bh i;j 2 2(R) and
b( ) ; b(0) ; b( ) ; b( ) ;
now h b( ) h b( ) = F ( h h ), thus
Z
b
; j ;1@ j ;1 '(
) + O(
j )
b
j'(0) ; '(
) ; @ '(
) ; : : : ; j ;1@ j ;1'(
)j C
H
g g @
g
L
H
Proof of lemma 3.2. Let ' 2 C01 (R),
@ g
l belong to 2 (R) and ( ; 0 ) 2 2 (R) in conse!
h=0
0 belongs to ;1 (R) and we obtain the condition iii).
thus the terms
quence
Xl
h
g '
' ()
' t dt
b L
' @
g
@ ' :::
R)
; j ;1@ j ;1 '(
) 2 L2 (
R)
; g ' ; @g t' ; : : : ; @ j ;1g tj ;1' 2 L2 (
i
@ g
ii
@ g
f
H
j
j g
g
t '
Conversely, we suppose
) j k;j 2 ;j (R)
=1
j
j
;
j
)
(R)
0 ; (;1) ! j 2
;1
) ; 02
(R )
iii
j
jj
h
ij
H
::: k
j
=1
::: k
:
;1
H
ESAIM: Proc., Vol. 5, 1998, 43-54
54
Let
L. BEL, G. OPPENHEIM, L. ROBBIANO AND M.C. VIANO
R),
2 C01 (
'
fg
H
k (')
= (
f
;
; g0 ) ' ;
X (Xl
k ;1
X( Z
k;1
gj
j =0
(;1)h @
hg
()
j
;
t ' t dt
l;h
tl '
X
k;1;j
)
h
@ gj
(;1)h !
h=0
h
tj +h '
)
!
Thanks to condition iii) the rst term belongs to 2 (R), thanks to condition i) and
the lemma, the second term belongs to 2 (R) and thanks to condition ii) the third
term belongs to 2 (R). Thus for every 2 C01 (R), kfg ( ) 2 2 (R).
l=1
h=0
h
L
L
L
Let
that,
k
;1
'
< H < k
H ; 21 1I
, ()=
f t
t
t>0
H
'
L
and j ( ) = 1! (j ) ( ). Then, it is obvious
g
t
j
f
t
= 0 2 ;j (R)
=1
;1
; 0 = 0 2 ;1 (R)
j
k (k) and h i;1 j k bj = h i;1j j;H ; 21 +k 2 2(R), so k 2 ;1(R) k ;j =
;j (R) for = 1
. Thus the process
j ; j !
gj
0
@ g
f
@ g
H
H
g
@ f
j
: : :k
j
::: k
H
@ f
Z
C L
X
k ;1 j
t
@ f
H
(j ) (; ) s
!
j =0
is well dened its order increments form a stationary process and its ; 1thorder derivative is a Fractional Brownian process with a parameter value equal to
; + 1.
Xt
k
H
=
R
(
f t
; s) ;
j
f
k
s dW
k
k
References
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ARMA type, mixing properties, Probability and Mathematical Statistics, 16(2), 1996, 311{
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Applied Mathematics and Stochastic Analysis, 11(1), 1998, 43{58.
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courbe de charge electrique par un processus brownien fractionnaire, Revue de Statistique
Appliquee, 92-59,1992,1-21.
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1964.
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Time Fractional Processes, Journal of Times Series Analysis, 16(3), 1995, 323 - 338.
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ESAIM: Proc., Vol. 5, 1998, 43-54