Publicacions
Ma emá iques,
Vol 36
(1992),
981-987
.
A
bs ac
ON
THE
DENSITYOF
SOME
WIENER
FUNCTIONALS
:
AN
APPLICATION
OF
MALLIAVIN
CALCULUS
ANTON1
SINTEs
BLANC
Dedica ed
o
P o esso
Pe e
Menal
i
B u al, in
memo iam
Using
a
ep esen a ion
as
an
in ini e
linea
combina ion
o
chi-
squa e
independen
andom
a iables,
i
is
shown
ha
some
Wiene
unc ionals,
appea ing
in
empi ical
cha ac e is ic
p ocess
asymp o ic
heo y,
ha e
densi ies
which
a e
empe ed
in
he
p op-
e ly
in ini e
case
and
exponen ially
decaying
in
he
ini e
case
.
In oduc ion
Le
F
be
a
p obabili y dis ibu ion
unc ion
on
R,
wi h
cha ac e is-
ic
unc ion
c( )
:=
E(exp{i X}),
X
being
a
eal
andom
a iable
wi h
dis ibu ion
unc ion
F
.
Le
X
1
,
X2,
...
be
a
sequence
o
independen
copies
o
X
de ined
on
some
p obabili y
space
(S2,
.P,
P),
and
F
n
he
empi ical
dis ibu ion
unc-
ion o
he
i s
n
a iables
.
The
empi ical
cha ac e is ic
unc ion
c
,( ) is
he
( andom)
cha ac e -
is ic
unc ion
o
F,
and
has
been
used
in
se e al
s a is ical
applica ions
since
a
leas
C ame 's
amous
book
.
In
he
la e
70's a
sys ema ic
s udy
o
i s
p ope ies
and
s a is ical
ap-
plica ions
was
ini ia ed
by
Feue e ge
and
Mu eika
.
In
pa icula
hey
p o ed
he
i s
limi
heo ems
o
he empi ical
cha ac e is ic
p ocess
Y,~( )
:=
~,l,-n,(c

( )-c( )),
unde
s ong
mo ien
condi ions
on F, see
[2]
.
In
1981
M
.
Ma cus,
[3],
ound
necessa y
and
su icien
condi ions
o
he p ocesses
{X,,( )},°,°__i,
E
o
weakly
con e ge
o
a
limi
p ocess
o
he
ollowing
o m
+
Y( ) :=

+00
exp{i x}d
00
I
:
982

A
.
SINTES
BLANC
whe e
B
is
B ownian
b idge
on
[0,1],
as
C[T1,T2]
alued
andom
ele-
men s
.
The
se me
yea
S
.
Csó gó,
[1,
Theo em
3],
.ob ained
s ong
app oxima-
ions
o
Yn,( ),
unde
he
assump ion
h(x)[1
-
F(x)
+F(-
.x)]
=
O(1)
,

as
x
goes
o
+oo
whe e
h
is
a
con inuous
unc ion
on
(0,
+oo) and
h(x)x
-
"
inc eases
o
+oo
as
x
inc eases
o
+oo,
o
some
posi i o
u
.
F om
his
s ong
app oxima ions
S
.
Csó gó
de i ed
a es
.
,o
con e -
gence
o
se e al
unc iónals
o
he
pa hs
o
he empi ical
cha ac e is ic
p ocese,
unde
he
u he
assump ion
o
exis en e
and
boundedness
o
a
densi y
o
he
limi
unc ional
.
He e
we
wan
o
s udy
he
ollowing
one
which
is
use ul in
es ing
o
symme y
o
F
.,
namely
:=

¡'Tz
.
[I~ n(X( ))]
2dH( )
whe e
0-V
is
a
gi en
dis ibu ion
unc ion
wi h
suppo
in
[T
1
,T2],
and
-oo
<
T
I
<
T2
<
+oo
.
The
main
esul
in
he
p esen
wo k
is
ha
wi h
g ea
gene ali y,
T
has
a
densi y
which
in ac
sa is ies
much
s onge
boundedness
condi ions
han
hose
needed
o
Csó gó's
a es o
con e gen e
o
hold
ue
.
I
is
he
ollowing
Theo em
.
Assume
ha
he
andom
a iable
T, as
de ined
abone,
is
non
degene a e
( his
is
a
condi ion
on
bo h,
F
and
Hl)
.
Then
T
has
a
smoo h
densi y,
which
is
ei he
a
empe ed
unc ion o
an
exponen ially
decaying
one
(a
in ini y)
.
The
main
ing edien e
o
he
p oo
a e
Mallia in
calculus,
Cauchy's
o mula
and
Fou ie
ans o m
.
In
he
nex
pa ag aph
we
ecall,
e y
b ie ly,
he
p incipal
de ini ions
and
esul s
we
a e
going
o
use,
and
gi e
some
e e en es
whe e
comple e
and
de ailed
exposi ions
can
be
ound
.
Finally, in
pa ag aph,
3
we
gi e
he
p oo
o
he
heo em
.
The
ools
Le
L2
[0,
1]
be
he
Hilbe
space
o
squa e
in eg able
unc ions
wi h
espec
o
he
Lebesgue
measu e on
he
Bo el
u- ield,
C3,
i 1
he
uni
in e al
[0,1]
.
Fo
hE
L2
[0,1]
;
deno e
VY(h)
he
Wiene
in eg al o
h
wi h
espec
o
he
Wiene
p ocess,
W,
on
[0,1]
.
We
hink
o
W
as
a
Gaussian
o hogonal
mensu e on
he
space
([0,
1],
L3,
A),
i is,
a
ze o
mean
Gaussian
p ocess
{W
(B)
:
B
E
13}
de ined
on
some
p obabili y
space
(S2,
F
;
P),
wi h
co-
a iance unc ion
gi en
by
E(W(BI^B2))
=
A(B
1
n
B2),
whe e
A
is
Lebesgue
measu e on
[0,1]
.
A ound
1950
K
.
I o
showed
ha
each
squa e
in eg able unc ional
F
E

P)
can be
de eloped
in
he
o m
whe e
I,,(

j
a e
mul iple
I o-Wiene
in eg als
o
he
(de e minis ic)
unc ions

,
E
L2([0,
1]m,
13'
n
,
A'
n
)
.
In
pa icula
II
(
I)
=
W( 1)
.
This
expansion
is
some imes
known
as
he
Wiene
chaos
decomposi ion
o
F
.
S oock's
o mula
iden i ies
he
ke nels,
,
in
e ms
o
i e a ed
Malli-
a in
de i a i es
o
F
:
Fo
F
E
L2
(52,
.F,
P)
and
hE
L2
[0,1],
he
Mallia in
de i a i e
o
F
in
he
h
di ec ion,
Dj
L
F,
can
be
de ined
as
i
he
se ies
con e ges
in
L2
(S2,
.F,
P),
whe e
(,
) is
inne
p oduc
.

All
his
(and
much
mo e) can
be
ound
in
[4]
and
[5]
.
To
apply
his
heo y
o
ou
unc ional
T,
we
w i e
i
down
in
e ms
o
as
ollows
:
AN
APPLICATION
OF
MALLIAVIN
CALCULUS

983
D
h
F
:=
7n=1
00
F=
E
(F)
+
Y~
I
(
m
)
=1
,,,,
=
(m!)-1E(D-F)
.
(h

I a-1( a( l
.
.
.
, a-1,*))i
T=
TZ
[W(h )]
2
dH( )
T
l
whe e
h
(y)
:=
sin( F -1
(y))
-
Im(c( )),
o
y
E
[0,1],
c( )
being
he
cha ac e is ic
unc ion
o F,
Le
.
c( )
:=
E(exp{iA}),
whe e
k
is
a
F-
dis ibu ed
andom
a iable
.
Le
us
calcula e
he
Mallia in
de i a i es
D[W
(h )]
2
=
2W(h )h
D
2
[W(h )]
2
=
2h
®
h
and
he
de i a i es
o
o de
g ea e
o
equal
han
3
a e
all
ze o
.
984

A
.
SINTEs
BLANC
Hence
;
by
linea i y
and om
S oock's
o mula
we
ge
.
1
T'
D
2
T
_

2h
®h
d~l
( )
T
1
7T
=
E(T)
+
12
(
.l

2
T, h
®
h dH( )
Now
we
ecall
some
mo e
de ini ions
and
esul s
ha
will
also
be
needed
in
he
nex
pa ag aph
.
A
unc ion
on
R
is
called
a empe ed
unc ion
i i
is
a
smoo h
unc ion
and
o
each
N
E
Nl
( he se o
na u al
numbe s)
sup
sup(1+x
2
)
N
j
(p
)(x)1
< +
oo
p<N
xER
whe e
(p)
is
he
p- h
de i a i e
o
.
Le
S
deno e
he
space
o
empe ed
unc ions
on R
.
The
Fou ie
ans o m
o
a
unc ion
E S
is
de ined
as
( )
:_
(27 )
-
z
~

exp(-i x) (x)dx
00
I
u ns
ou
ha
his
ans o ma ion
de ines
a
con inuous
one- o-one
mapping
o
S
on o
S,
whose
in e se
is
also
con inuous
.
Mo eo e ,
i
has
pe iod
4,
due
o
he
in e sion
o mula
which
is
undamen al
+0
(
.x)
=
(27 )
-
z

exp(i x)
( )d
.
_
00
A
e y
good
e e en e
o
his
is
[6]
.
P oo
o
he
heo em
The
idea
o
he
p oo
is
qui e
simple
:
o
show
ha
he
cha ac e is ic
unc ion
o
T
is
a
empe ed
unc ion
in
he
p ope ly
in ini e
case
;
and
di ec
calcula ion
in
he
ini e
case
(dis ine ion
o
he
wo
cases
will
be
clea in
a
momen )
.
Le
us
go back
o
S oock's ep esen a ion
o
T
;
and
ecall
he
explici
o m
o
he
ke nel
in
he'
double
I o-Wiene
in eg al
he e
.
AN
APPLICATION
O
1VIALLIAVIN
CALCULUS

985
I
is
known
ha
such
a
ke nel
can be
de eloped
in
a
L
2
-con e gen
se ies
.
K( ,s)
=
L
Aiei( )ei(s)
whe e
A
;,,
( espec i ely
e?(-)),
a e he
eigen alues, ( espec i ely
eigen ec-
o s)
o
he
ollowing
sel -adjoin
non-nega i e
in eg al
ope a o
I
o(
.)-
K(-,
s)O(s)ds
0
en
L
2
[0
;
1]
;
he
e&)'s can
be
assumed
o
be
a
comple e
o hono mal
sys em
in
L
2
[0,
1],
and j
:°
°I
A
;,
<
+oo
.
I
is
also well
known
ha
o
hE
L
2
[0,
1],
wi h
1111112
=
1
I2(h
(D
h)
=
W(h)
2
-
1
and
combina ion
o
his
ac s
leads
us
o
he
ep esen a ion
we
had
in
mind,
namely
_
E(T)
+

~i(W(ei)2
-
1)
_

AáW(ei)2
wi h
F-00,
' i
<+
00
.
We
ema k
he e
ha
om
his
ep esen a ion
i is
clea
ha
T
is
non-degene a e
i
S oock's
ke nel,
D2
T,
is
non-degene a e
.
To
p oceed
we
dis inguish
wo
cases
:
he
one whe e
only
a
ini e nu i-
be
o
he
Ai
a e
non
ze o,
and
he one
whe e
he e
a e
in ini ely
many
non
ze o
.
We
ea
only
he
second
case, as
he
i s
one
is
e y
elemen-
a y
.
So, o
inish
he
p oo
o
he
heo em
i is
enough
o
p o e
nex
lemma,
Lemma
.
Le
{%i}.°°,
be
independen
iden ically
dis ibu ed
andom
a iables
wi h a
common
chi-squa e
dis ibu ion,
¡
.e
.
a
dis ibu ion
wi h
densi yx
-
z
-exp(-2)110,-1( ,)
.
Le
{Ai }°°
I
be
a
noninc easing
sequence
o
posi i e
eal
numbe s
such
ha

ñ
i
<
+oo
.
Then
he
andom
a iable
,1

AiXi has
a
empe ed
p obabili y
densi y
anc ion
.
P oo -
I
is
well
known
ha
he
cha ac e is ic
unc ion
o
he
andom
a iable
.D
is
00
.
( )
_
l
(1
-
2A,,i )

2
k-I

986

A
.
SINTEs
BLANC
I
is
easily
seen
ha
his
unc ion
possesses
an
analy ic
con inua ion
on
he
s ip
-
(
4,
1) -1
<
Im(z)
<
(
4,
1)
-1
,
which
is
o
he
same
o m,
Le
.
(z)
=
l
(1
-
2Akiz)
-12
k=1
whe e
i
:=
V
/-
--
1
.
Obse e
ha
o
zin his s ip
Z
<
Re(1
-
2¡, z)
<
2
,
so ha
we
can
use
he
b anch
o
y
/'z
-
which
coincides
wi h
he
usual
squa e
oo
unc ion
on
he
posi i e
eal
numbe s
.,
and
we
can
apply
Theo em
(15
.6)
om
[7]
;
(he e
we
use
he
hypo hesis
-°°1
~i
<+
oo)
.
Fo
>
0
apply
Cauchy's
o mula
o
he ec angle
F
de ined
by
he
poin s ( /2)
T-
i(4A1)
-1
,
2
:F
i(4A1)
-
'
;
o
ge
(P)( )
_
(P!)
1
:
1

(z
-
(
z
)
)
P+1
dz
j=1
j
whe e
{Fj}~_1
a e
he
ou
sides
o
he ec angle
F
.
To
bound
his
in eg als
we
use he
ac
ha
o
each
na u al
numbe
MEN
Fo
ins an e
he
in eg al
on
he
le
side,
le
us
say
F
I
,
o
he ec angle
F,
can be
bounded
as
ollows
(4A1)
-1
(( /2)
+
is)j
j(( /2)
-
is)j
-1-
Pds
<
(4ñ1)
-1
M

1
<
2PAi
1
P
+1

((
1 /4)
+
A2 2)l
-`
k=1
whe e
we
used
(1)
;
and
he
same
ype
o
bound
is
ob ained
o
he
in eg al
on
he
igh
side,
1'3,
o
F
.
In a
simila
way
he
bound
in
(2)
is
used
o ea
he
in eg als
on
he
ho i zon al
sides
o
he
ec angle,
F2
and
I'
4
.
And
his
p o es
he
lemma
in
he
case
>
0
.
(1)
m
_,
a
SUP
j
« /2)
+
is)~
<
((
1
/
4
)
+
isi<(4a1)-1
(~
~kk 2))
k=1
(
2
)
n~
1
_
SUP
l (s
+
i(4a1)
-1
)I
~
(~
(1
+
>,2 2»
( /2)<isi<2
k=1
AN
APPLICATION
01
"
MALLIAVINCALCULUS

987
Fo
<
0,
wi h
small
changos,
all
wo ks
he
same
way,
and
o
=
0
he e
is
no p oblem
a
all
.
Thus
he
lemma
is
p o ed
.
Rema k
.
In
he
abo e
p oo
we
could
ha e
used
he
asymp o ic
ep-
esen a ion
ob ained
in
[8],
ins ead
o
ou
lemma
.
I
is
clea
om
[8]
ha
he
densi y
(x)
is
expo ien ially
decaying
a
in ini y,
e en
in
he
p ope ly
in ini o
case
.
Howe e ,
i is
no
clea
how
can
we
ge he
em-
pe a e
cha ac e
o
om
Zolo a e 's
ep esen a ion,
wi hou
u he
wo k
.
In
ha
poin
ou
p oo
seems
o
be
sho e
and
clea e
.
Acknowledgmen s
.
1
wan
o
hank
P o esso s
Paul
Mallia in
and
Da id
Nuala
o
hei
kind
a en ion
and
help
on
some
p oblems
ela ed
o
he
subjec
o his
pape
.
Thanks
also
o
P o esso
E a is
Giné,
o
poin
me
ou
e e en e
[8]
.
Re e en es
1
.

S
.
Csói3
.Gó,
Limi
beha io
o
he empi ical
cha ac e is ic
unc ion,
The
Annals
o
P obabili y
9,
no
.
1
(1981),
130-144
.
2
.

A
.
FEUERVERGER
AND
R
.
A
.
MUREIKA,
The
empi ical cha ac e -
is ic
unc ion
and
i s
applica ions,
Ann
.
S a is
.
5
(1977),
88-97
.
3
.

M
.
B
.
MARCUS,
Weak
con e gen e
o
he
empi ical
cha ac e is ic
unc ion,
Au a
.
P obabili y
9
(1981),
194-201
.
4
.
D
.
NUALART
AND
M
.
ZAKAI,
Gene alized
s ochas ic
in eg als
and
he
Mallia in
calculus,
P obab
.
Theo y Rel
.
Fields
73
(1986),
255-280
.
5
.

D
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