X
X
R
X
2 3
0
0
0
0
1
0 2
2
2
2
2 1
2
0
2
empiricalprocess
12 3
1
2
3
( +1)2
+ + 2
2
2
0
(1+ ) 2
[ ]
=1
;
t
t t
t
t
t
t t t
t
s t t s s
t
D =
t
s t s k t k
D
D=
N
Nt
s
s
t
t X ;t
X x F x f xX :::
F x P X x f x F x
X
X ;t
X G Y Y;t
: X b ; t ;
: b L t t ;
D ; L
;t
: r t EX X b b L t t
L t dL t d u u du
: F x;t
N
X x F x ;
X ;t
: F x P X x
Abstract.
Z
1
1. Intro duction.
Z
Z
Z
Z
1
Z
LiudasGiraitis andDonatasSurgailis
Heidelb erg University,ImNeuenheimerFeld294,69120Heidelb erg,Germany
Research supp ortedbytheAlexander vonHumb oldt-Foundation
Institute ofMathematicsandInformatics,2600Vilnius,Lithuania
Acentrallimittheoremforthenormalizedempiricalpro cess,basedona(non-Gaussian)moving
averagesequence withlongmemory,isestablished,generalizing theresultsofDehlingandTaqqu
(1989). Thepro ofisbased onthe(App ell)expansion
( )= ( )+ ( ) +
oftheindicatorfunction,where ( )= [ ]isthemarginaldistributionfunction, ( )= ( ),and
thecovarianceof theremaindertermdecays fasterthanthecovariance of . Asaconsequence ,thelimit
distributionof M-functionalsandU-statistics based onsuch longmemoryobservationsis obtained.
Statisticalinference forlongmemorytimeseries hasgainedconsiderable attentioninrecent years; see
e.g. Beran (1991),Dehlingand Taqqu(1989), KoulandMukherdjee (1993)and thecomprehensive survey
byBeran(1992)foradditionalreferences. MostofthestudiesdealwithGaussianobservations ,or
(eventually) withinstanteneous functionals = ( ) ofa Gaussianlongmemoryseries , using
well-developp edtechniques ofHermiteexpansions(Major(1981)).
AnaturalgeneralizationofGaussianlongmemoryseriesconstitutelinear(ormovingaverage)pro cesse s
(11) =
withhyp erb olicallydecayingco ecients
(12) = ()
wherethemainparameter (0 1)characterizesthedecayrateofthe"memory", ()isaslowlyvarying
at innity function, and is an iid sequence with zero mean and variance 1. In particular, (1.1)
includes the case of (non-Gaussian) fractional ARMA pro cesses (see Granger and Joyeux (1980)). More
generally,condition(1.2)canb e replacedbythehyp erb olicdecayconditionofthecovariance
(13) () = = = ()
where () ()isaslowlyvaryingat innityfunction, = ( (1 + )) .
Themainobject ofthispap eristhestudyoftheasymptoticsofthe
(14) ( ) =
1
( ) ( )
where isthelinearpro cess of(1.1)-(1.2),and
(15) ( ) = [ ]
D
2
0
Theorem 1.
R
Z
0 1
Z Z Z
X
X
n o
8 9
X D
0 0 0
0
1
0
0
0
3 3
0
D
0
2 2 2
1
0 1
0 0
( +1) 2
1 2 1
=1
2
=1
2
0
1
[ ]
=1
2
0
t
D D D
D
D D =
N
=
N
t;s
D
N
N
t;s
N
iu
n
N
N D
N
Nt
s
s
t
D D 01 1 2
2 01 1 2 )
D
2
j j j j 0j 0 j
j 0 j j j
0
j 0 j
j j j j 2
j j 1 1
2 01 1 2 )
p
2 01 1 2
1 2 2
1 X
; ; g g x;t
x;t ; ;
Z t ;t ;
EZ t Z s t s t s :
c x y dxdy uu du;
d d LN N :
r t s c o d
r t s O d :
u Ee
C> ;>
: u C u ; u :
E < ; n n D ; n D <
: d Nt F x;t x;t ; ; c f x Z t x;t ; ; ;
f x F x
F x
f x
: S t H H X ;
H L F ;EH X : X ;t
H fractional Brownian motion
Assume that the characteristic function of the "noise" in (1.1) satises the
followingcondition: thereexistconstants such that
Moreover, assume that moments for all where is estimated in
Remark1below. Then
where is the marginalprobabilitydensity.
stationarityoftheobservation pro cess (1.1). Ourmainresult (Theorem1b elow)is ageneralizationof
DehlingandTaqqu(1989),Theorem1.1,whoconsideredthecaseofinstanteneousfunctionalsofaGaussian
sequence.
Intro ducethe(two-parameterSkorokho d)space = ([ + ] [0 1])ofrealfunctions = ( ),
( ) [ + ] [0 1],with thesup-top ology,and write = forweakconvergence of randomelements
in (seeDehlingandTaqqu(1989)fordetails).
The () [0 1]isa(continuous)Gaussian pro cess withzero meanand
thecovariance
() ( ) = 1
2
+
Put
= (1+ )
and
= ( )
Thenitiswell-knownthat
( ) = ( + (1))
and
( ) = ( )
( )=
0 0
(16) ( ) (1+ )
( ) 9 ( )
(17) [ ] ( ); ( ) [ + ] [0 1] = ( ) (); ( ) [ + ] [0 1]
( )= ( )
Theorem 1can b e applied to obtainthe limitdistributionof various estimatorsof parameters of the
marginaldistribution ( )oflongmemorymovingaveragesequences (1.1)(see Sect. 2). Condition(1.6)is
aratherweaksmo othnessconditiononthedistributionofthe"noise";hovewer,itguaranteestheexistence
andinnitedierentiabilityofthedensity ( ).
Theempiricalpro cess (1.4)isaparticularcase ofsumsofmoregeneral "instanteneous" functionalsof
theform:
(18) (; ) = ( )
() ( ) ( )=0 Limitdistributionof (1.8)forlongmemorylinearpro cesse s of(1.1)-
(1.2)wasstudied bySurgailis(1981,1983),AvramandTaqqu(1987),GiraitisandSurgailis(1989). There,
it was found that, at least for "nice"analytic functions (), this distribution is basicallythe same as if
0
R
R
R
1 R
1
1
1
0
0
D X
X
Z
Z
Z
X
0 0
1
0 2
0 0
0 0
f 2 g ) f
p
2 g
=0
=0
( )
0
( )
0 0
( )
2
0
1
1 [ ]
=1
([01]) t
k
k k
k k
k k
k
z x z X
k
k
k k
k
k
k
k
k
k
k
k
t t
N Nt
s s
;
D Appellrank
Appell polynomials
not
biorthogonalsystem
: H x a A x =k
A x A x F ;k ;
: z A x=k e =Ee :
a
a EH X H y dF y ;
: a EH X Q X H y Q y f y dy ;
: Q y f y =f y k ; ;:::
L F A y F k
H
H y x y x F x ; y ;
a x
: a x H y x f y dy f x;
: X x F x f xX :::
: d X t ; c Z t t ;
corresp ondingtotheexpansion
(19) ( ) = ( ) !
in ( )= ( ; ) 0 withthegeneratingfunction
(110) ( ) ! =
Hovewer,thefact that App ellp olynomialsare orthogonal ingeneral makes theconvergenceof (1.9)a
verydicultproblemandrestrictstheuseofitbasicallytoanalyticfunctionsonly(seeGiraitisandSurgailis
(1986)). Inthelattercase,theco ecients aregivenby
= ( ) = ( ) ( )
whichcanb eformallyintegratedbyparts, giving
(111) = ( 1 ) ( ) ( ) = ( 1 ) ( ) ( ) ( )
wherethefunctions
(112) ( ) := ( ) ( ); =0 1
forma in ( )totheApp ellsystem ( ; ); 0(seeGiraitisandSurgailis(1986),
Daletskii (1991) ). The ab ove facts help ed us recently to extend the Dobrushin-Major-Taqqu theory to
non-smo oth functionals () of the linear long memory pro cess (1.1)-(1.2), of arbitraryApp ell rank (see
GiraitisandSurgailis(1994)). Inthecaseofthe(centered ) indicatorfunction
( ; ) = ( ) ( )
thecorresp ondingApp ellco ecient ( ) = 0,while
(113) ( ) = ( ; ) ( ) = ( )
whichisnotidenticallyzero. Thisargumentexplainstheleadingtermofthe(App ell)expansion
(114) ( ) ( ) = ( ) +
andtheformofthelimitempiricalpro cess in(1.7),as
(115) ; [0 1] = (); [0 1] ;
seeDavydov(1970)andGoro detskii(1976).
The main step in the pro of of Theorem 1is the following "weak uniform reduction principle" (c.f.
Theorem3.1ofDehlingand Taqqu(1989)).
d
2
0
0 R
R
d
R
R
2
2 0
0 0
0 0
1
jj!1
0 0 j j
2
X
3
b R
Z
b
Z
b
b
Y
b
Y Y
1
=1
3
1 2 1
1 2 1 2 1 2
1 2
1 1 2 1 2
1 2
0
1 2 2
n N
x N
n
j
j j
t t
t
iux
k
t
k
t
t
iuX
j
j
iu
j
J j
J
j
J
j j
i i
1
R
2. Pro ofsof Theorems1and2
R
Lemma A.
R
R
Z
0
!1
2
2
j j j j 1
j j j j 1
2
j j j j
f 2 6 g j j 1
j j j j j j j j
f g f g2
j j
: P d X x F x f xX > CN :
f x ;x X ;X
: f x ;x f x f x r t f x f x o r t t
x ;x
g u e g xdx g g x; x d
k
: u f u du <
: u f u ;u du du < :
f x
f x ;x t
: f u Ee ub ; u ;
u Ee
: u <C= u
J j b k < J < :
f u ub C ub C u ;
b ; b l
: J > k ; i ; ;
For any ,
and
In particular, the marginal probabilitydensity isinnitely dierentiable, and the bivariateprobability
density existsand isjointly continuous in forany suciently large .
Proof.
(116) maxsup ( ( ) ( )+ ( ) ) (1+ )
The pro of ofTheorem 2 uses the chainingargumentof Dehlingand Taqqu(1989),together with the
followingasymptoticsofthejoint(bivariate)probabilitydensity ( )of( ):
(117) ( ) = ( ) ( )+ () ( ) ( )+ ( ()) ( )
uniformly in (see Lemma C b elow). In turn, the pro of of (1.17) uses the factorization of
thecorresp onding bivariatecharacteristic functiondueto theindep endence of thenoise variables, andthe
asymptotics(1.2).
Therestofthepap erisorganizedasfollows. Sect. 2isgiventothepro ofofTheorems1and2andthe
auxiliaryLemmasA,B,C,andD.Sect. 3discusses applicationstothelimitdistributionofsomestatistical
functionalsbased ontheempiricaldistributionfunction.
Write ( )= ( ) for the Fourier transform of a real function = ( ) ( 1),
whenever itiswell-dened.
0
(21) ( )
(22) limsup ( )
( )
( )
Thecharacteristicfunction
(23) ( )= = ( )
where ( )= satises
(24) ( ) (1+ ) ;
see(1.6). Cho ose : =0 suchthat +1 Then
( ) ( ) (1+ ) (1+ )
whichproves(2.1).
Toprove(2.2),let b e tworealsequences such that
(25) +1 =12
2
R
Z
R
k
1 1
1
1 i
i
i
j
j
c Z
Z
Lemma B. Z
R
R
k X
Z
Y
Q
Y Y
Y
Y Y
Z
0
b
1 2 3
b
b
b
X X
Y 2
2
0 0
0 0 j j
0 j j
0
0
0 1
0
0 0
j j j j
2 Assume that,forsome ,
Then
Proof.
j j
j j
i
i
i k
j
j j
i
j J
j j i
J
j j
J
j
j j j j
J
J
i i
i
i J
j j j j t j
i
n
n
n
n
n n
m
J m m
j J k
j k
j J
f 2 j j j jg
f 2 j j j jg
j j j j 1
j j j j j j
j j 0
j j j j j j
j j j j j j
! j j ! 1
j j j j
j j
2
j j 1
j j j j 1
1 2
0
1 1
j j j j
1 1 2
2 1 2
1 2
2
=1
1 1 2 2 1 2
1 2 1 1 2 2
1 1 2 1 1 2 2 1 2
2 1
1 1 2 1 1
2 1 2
1 2
2
=1
1 2
2
=1
1 2 0
1 2 0
0
+
0 +2
( +2)
+2
1
+2
( )
1 =
( )
J j b > b ;
:
J j b < b ;
:
k ;i ; :
: u ;u u u b u b du du < :
u ;u u b u b ; J ;i ;
u ;u C u b u b C u u ;
b =b < >
u ;u C u C u ;
u ;u :
u ;u u ;u C u ;
b b b b b t ; k t >
J ;J J >k ; i ; ; t >t :
t >t
n
E < :
: x f x dx < :
f C
: uf u x f x u
g g L
x f x x f x
m
: f u
m
ub b u ;
= :
(26)
= :
(27)
and 0 =12 Then
(28) 8( ) := ( + )
Indeed,put8( )= ( + ) andassume =12arenite. Then,similarlyasinthe
pro ofof(2.1),
8 ( ) (1+ + ) (1+ + )
where = andmax 1 ,forsome 0. Therefore
8 ( ) (1+ ) = (1+ )
andasimilarestimateholds for8 ( ) Hence
8( ) 8( ) (1+ )
whichproves(2.8).
Now, take = and = . As 0( ) for any 0 there exists 0 such
that dened in(2.6),(2.7),resp ec tively,satisfy +1 =12 forany Moreover,the
corresp ondingestimatesholduniformlyin , hence(2.2)followsfrom(2.8).
(29) (1+ ) ( )
As () ( )(see LemmaA),and
(210) ( ) = ( ) ( )
( ()standsfortheFouriertransformof ()),itsucestoshowthatthelefthandsideof(2.10)isin ( ).
Indeed,thisimpliestheb oundedness of ( ) andtherefore theintegrabilityof (1+ ) ( ).
The -th derivativeofthecharacteristic function(2.3)can b ewrittenas
(211) ( ) = ( ) 8 ( )
+
2
2
2 2
2
c
c
j
j
j
k
Z
k
R
R
R R
R
k Z k
Z Z
Z
Lemma C. R
1
1
1
P
0 1
Q
Y
b
X X Y Y
b b b
Z
b
b
b b b b b
Z
b
Z
b
Z
b
b
Z
b
Thereexists such that,uniformlyin
Proof.
2
2 n
0 0
0 0
j j j j
0
0
0 0 0 0
0 0 1
0 0
0 0 0 0
j j
0 0
1 0 0 0 0
+
+
+ +
+
3
( )
0
(1) 2
0
( +2) 3
1 +2 = +2
: 2 : =1
2
3
( +2) ( +1) ( +2)
2
1 2
1 2 1 2 1 2
1 2
1 2
2
1 2 1 2 1 2 1 2 1 2
3
2 3
j j j
m
j J j
J
j J
j
n;m
J
n
k k
n n
J n n
jk j
k
jk
j
n n n
t
D
t
t
ixu
t
t t
t
D
t
k k
t
k D
u t t
D
ixu
t
D D
2 2 j j
2 2
j j
j j j j
j j j j j jj j
j j j j j j j j
j j
j j j j
j j
2
0
0 0
j j j j
j j j j j j j j !1
j j
j j
k j J ;k ; k m; m= k
u ub :
n m C C
J ; J m
: u C u :
u E k ; u uE ;
f u C u b ub
C u :
uf u n f u uf u
C u :
> x ;x ;
: : f x ;x f x f x r t f x f x Ot :
p x ;x
: p x ;x e p u du;
: p u ;u f u ;u f u f u r t u u f u f u :
>
: p u u >t du O t :
k D =
f u u >t du t u f u du Ct Ot t
>
: p u Ot
e p u u t du t t O t :
overallvectors =( : ) = = = ! !; nally,
8 ( ) = ( )
Similarlyasinthepro ofof LemmaA, forany andany onecan ndaconstant =
such that,uniformlyin = ,
(212) 8 ( ) (1+ )
Furthermore,as ( ) ( 2) ( ) hence,according to(2.11),(2.12),
( ) (1+ )
(1+ )
Consequently,
( ( )) = ( +2) ( )+ ( )
(1+ )
Thisprovesthelemma,according to(2.10)andtheargumentab ove.
0
(213) ( ) ( ) ( )+ () ( ) ( ) = ( )
Write ( )forthelefthandsideof(2.13). Then
(214) ( ) = (2 ) ( )
where
(215) ( ) = ( ) ( ) ( ) ( ) ( ) ( )
AccordingtoLemmaA,forany 0,
(216) ( ) ( ) = ( )
Indeed,cho ose ( + ) ,then
( ) ( ) ( ) = ( ) ( )
by(2.2),and asimilarestimateisvalidforthetwoothertermsontherighthandsideof(2.15).
Itremainstoshowthatthere exists 0suchthat
(217) sup ( ) = ( );
indeed,
( ) ( ) = ( )
0 Z
Z
Z
Z
Z
1 2 3
1 2 3
2
2
2 2
b
Y Y Y Y
b b
Y Y Y Y
b b b
Y Y X
X
X
2 3
X 2
0 0
2
0 0
0 0 0
0 0 0
0 0 0 0 0 0
0
0 0 0
0 0 0 0
0 0
0
0
jj
0 0 0 0
0
0 0
0
0
jj
0 0 0 0
0 0
0
0
0 0
jj j0 j
0 t
j
j t j
J J J
j
j t j
J J J
t
i
i
i
i
D
D
i
i
i
i
i
i
j t
j t j j t j
t j
D
t
x
j t
D D
j t j
t
t
j>t ;t j>t
j t j
1 2 1 2 1 2 3
1 2 1 2
1 2 3
1
2
2
2
3 1 2
1 2 1 2 1 2 3
1 2 3
1
1 2 3
1 2
2 3
1 2 3
3
3
3
3 3
1 2
3
1
1
1 2 1 2
2
2
2
5
1 2
1 2 2 0
1
1
5 3
3
1 2
3
3 3
1 2
1 2 1 2
1 1 1 1
1 1 1 1
f 2 j j g
f 2 j 0 j g
n [
0 0
0 0 0
j j j j
0
0 0 1
j j
j 0 j j 0 j
j j j j
j 0 j j 0 j
j j j j
j j j j
j 0
j j j j j j j
j 0 j j j j j
j j 2
j j j j
0 f g0
f u ;u u b u b ::: ::: ::: a a a ;
f u f u u b u b ::: ::: ::: a a a ;
J j j t ;
J j j t t ;
J J J :
f u ;u f u f u a a a a a a
a a a a a a a a a a a a :
a ; a ;i ; ; ;
a a Ot ; i ; ;
:
a a a u u r t O t ;
:
u t :
b b b b ;
b ; b ;
a a u b u b u b u b ;
u t ; j t ; > ;
u b C t
b D = > D y y
y y Cy C x E
a a O t O t ;
i i
u t j J
u b u b O t ;
< < D t
: a a a Q u ;u ;
Q u ;u u b ;u b
( ) = ( + ) = =:
and,similarly,
( ) ( ) = ( ) ( ) = =:
where
= :
= :
=
We have
( ) ( ) ( ) =
= ( ) + ( ) + ( )
Hence, (2.17)followsfrom 1 1 =12 3 and
= ( ) =12
(218)
= ()+ ( )
(219)
uniformlyin
Usingtheinequality
1 1onehas
( + ) ( ) ( )
where, for and sucientlysmall 0
in viewof the asymptotics (1.2) of , and the inequality ( +1) 2 . Therefore, as ( + )
( ) ( ) with =2sup ( ) 2 ,weobtain
= ( ) = ( )
oreq. (2.18)for =1;thecase =2isanalogous.
Itremainstoprove (2.19). Notethat and imply
+ = ( )
where0 . Therefore,for sucientlylarge ,thelefthandside of(2.19)canb ereprese ntedas
(220) = exp ( ) 1
where
( ) := ( )
2
R
1 0j t0j
t
1 2
1 2
2 2
2
1 2
Z Z
Z Z
2 3
0 1
0
X X
1
0 X
1
X
X
0 1
2
0
0 j j j j
0
j j j j j j
j j j j j j j j
j j
j j
j j
j j j j
0
0
0
00
00
00
0 0
00
0 0 0 0 0 0
jj
0 0
j0 j
0 0
0
0 0
0
j j 0
0 0
0 0
0 0
0
0 0 1 2
0 0
(3)
1 2
0 0
(3)
1 2 1
2
2 1 2
2
1 2 1 2 1 2
1 1 2
2
2
1
2 1
2
2
2 2
1
2 3
2
3 2 3
1 2
2 +3
1 2
( )
1 2
2
1 2
1 2
3
=1
x y
xy
xy
v v
j t j
ub u b
j t j j t j j t j
t
j t
j t j
t j t
j t j
j j
t j j
t j
i
j t t j
D
j j
t j
D
t
D
Q u ;u
t
t
D
N
n
j
j j
x;y
x y
x y
x;y
x; ;y x; ;y ;
;
x y x y :
v ;v x y dxdy
z z
u b ;u b x y z dxdy
u u b b O u b u b u b u b :
Q u ;u r t u u i i ;
i O u u b b b b ;
i O u u u u b b b b :
u t ;i ;
i O t b O t ;
i Ot b b O t
>
Q u ;u O t r t Ot
u ; u t
e Q u ;u OQ u ;u
u u r t O t :
: S n x X x F x f xX :
( ) := log
( + )
( ) ( )
iswell-denedandtwice continuoslydierentiableinaneighb orho o dof( )=0 . Notethat
( 0)= (0 )= ( 0)= (0 )=0
(0 0)= (0)= 1,and
( + ) = (log ) ( + )
Hence
( ) = (log ) ( + )
and,as(log ) ( )isb oundedinaneighb orho o dof =0,
( ) = (log ) (0)+( + )(log ) ( )
= + ( ) + ( )
Consequently,
( ) = () + +
where
= +
= ( + ) +
Here, as =12,so
= ( max ) = ( )
and
= ( ) = ( )
provided 0issucientlysmall. Inparticular,
( ) = ( ()) = ( )
uniformlyin ,whichimplies
1 = ( )+ ( ( ))
= ()+ ( )
By(2.20),thisproves(2.19)and thelemma,to o.
Put
(221) ( ; ) = ( ) ( )+ ( )
+ 0
R
R
R R
2
2 R
1
R
R
1 0
0
0 0
0 0
0
0 0
0
0
0
0
0
0 0 X
X
Z
Z
Z Z
Z Z
Z Z Z Z
Z
Z
N N
N N N
n
j
j
N
n
t;s
t
t
t t
D
t
t
A A
y
x 2
2
1+
=1
2
=1
0
1 2 1 2 1 2
1 2 1 2 1 2 1 2
0
1 2 1 2 1 2
1 2 1 2
1 3
2 3
+
+
3 2
1
0
0 0
0
0
j j j jj j
0 0
f 2 j j
j j j j g n
j j j j j j
j j j j
where isa nitemeasureon .
Proof.
d ES n x;y
n
N
x;y;
S n x;y S n x S n y
H X ;
H z H z x;y y< z x F x;y f x f y z:
ES n x;y t s ;
t t x;y EH X H X :
f z H z dz ;
f z H z dz ;
t
: t H z H z p z ;z dz dz ;
p z ;z f z ;z f z f z r t f z f z
t t>t
: t Ct H z H z p z ;z dz dz ;
D D> >
it A z ;z p z ;z
z z A A :
i t ::: ::: i t i t :
: H z x;y x<z y F x;y z f z dz;
: i t H z z dz C x;y;
( ; ) [ ]
()
We have
( ; ) = ( ; ) ( ; )
= ( )
where
( )= ( ; )= ( ) ( )+( ( ) ( ))
Therefore
( ; ) = ( )
where
()= (; )= ( ) ( )
Usingtherelations
( ) ( ) = 0
( ) ( ) = 0
onecan represe ntthecovariance ()as
(222) () = ( ) ( ) ( )
where
( )= ( ) ( ) ( )+ () ( ) ( );
c.f. (2.13). Assume islargeenough( ). Then,accordingtoLemmaCand (2.22),
(223) () ( ) ( ) ( )
where =( + )(1 ) 0,provided 0ischosen sucientlysmall.
Let us estimate the last integral, denoted by (). Put = ( ) : ( ) (1+
) (1+ ) ; = Then
() = + =: ()+ ()
Usingtheestimate
(224) ( ; ) ( )+ ( )+ ( )
we obtain
(225) () ( )(1 + ) [ ]
R
R
R R
R R
R
R
R
3 3
3 3
3
3
3
3
3
3
3
Proofof Theorem 2.
Z Z Z Z
Z Z
Z Z
Z Z
Z
Z
Z
Z Z
X
Z
2 3
0 1
0 0
0
0 0 0 0
0
3
0
0
0
0 0
0 0
0 0 0 0
0 0
01 0
0
0 0 0
A A
t
n n
t
n n
t
n
t
n
n
A
n
n
A
n
n
D
N N
N n
t;s
D
D
x
x N
N
D
3 3 3
1 2 1
1 2 1 2 +
+ 1 2 1 2 1 2
2
1 1
2
1 2
1 2 1 1
2 +1
1 2
2 +1
+ +
+
2 +1
2 +1
2 +1
0 2 +4
0
0
2 2 2
=1
2 2
1
3 2
j j j j j j j j j j
j j j j j j 2
0
j jj j j j j j
j j j j
j jj j j j
j j j j j j
j j j j
j j j j
j j j j j j j j
j j 1
j j
1
j 0 j
0
01
j j 2
j j
A z dz F A z dz f x dx z z dz
p z ;z z z ; z ;z A ;
n =
i t p H z H z z z dz dz
p H z z dz dz
p z ;z H z z dz dz
f z f z H z z dz:
: i t x;y;
A f z f z z dz
F A f z f z z dz
f x dx z f z f z z dz
E <
t>t
: t t x;y;
t t t
d ES n x;y x;y d t s
x;y n=N =N L N ;
> D
x x x ;x
x F x f y dy ; x ;
: P d S n x > <CN n=N n=N
( ) := (1+ ) + ( ) (1+ ) + ( ) (1+ )
isanite measureon ,accordingtoLemmaB.
Next,as
( ) (1+ ) (1+ ) ( )
with =[3(1 ) ]+1,we obtain
() ( ) ( )(1+ ) (1+ )
( )(1+ )
2 ( ) ( )(1+ )
(4 ( )+2 ( )) ( )(1+ )
Hence, by(2.24),
(226) () [ ]
where
( ) := (4 ( )+2 ( ))(1 + )
+ ( ) (4 ( )+2 ( ))(1 + )
+ ( ) (4 ( )+2 ( ))(1 + )
isanite measure,to o,provided (see LemmaB).Combining(2.25)and(2.26),we obtain
thatforallsucientlylarge
(227) ( ) [ ]
where ()isanite measure,indep endent of . Thesame b oundclearlyholds for0 ,to o,which
canb e shownbythesameargumentasin(2.25),(2.26). Consequently,
( ; ) [ ]
const [ ]( ) ( )
whichprovestherequiredestimate,provided 0ischosensucientlysmall( 1 ).
Let3( )= ( ),where ( )= (( ])isthemeasure ofLemmaD.Then
3( ) ( )+ ( )
andtheargumentofDehlingandTaqqu(1989),Lemma3.2,applieswithsmallchanges,yieldingtheestimate
(228) sup ( ; ) ( )+( )
R
R
3
3
1
01 0
0
0
! 1
6
0
1
01
0
0 0
0
0
P
Z Z Z
e
e
Z
b
X
e b
Z
b e
e
P
e
e e
0
1 2
j j 1 61
6
j 0 j j
j j j j
2
2 2 1 !
0 0 0
3
0
0
j j j
t t s s t
D =
t
N
D
D D =
D = D =
j j
j
N
N
j
j N
N N
N
N
j
j
N
( +1) 2
0 +
+
0
0
1
0 1
0
( +1) 2
( +1) 2
+
( +1) 2
0
0 0 0 0
0
0
0
=1
1
=1
1
Z
3. Applicationsto M-estimatorsandU-statistics
Z
Z R R R
1 ProofofTheorem1
Remark1.
Remark2.
Remark3.
M-estimators.
n D
n D
; <D =
D = D ; = < D<
Y GX ; j N G X ;j
X b b L t t L
L t ; t> l ;l
L t =L t l :
d L N N
c x y dxdy g u g u du;
g u u l u u l u<
X X ; j ;
X ;j
E X E X xf x dx
: :
N
X
x dF x ;
:
F x N X x
X ;:::;X
ofDehlingand Taqqu(1989).
followsfromthe"weakuniformreductionprinciple"(Theorem2)andtheconvergence
(1.8)(c.f. DehlingandTaqqu(1989)).
Fromthepro ofsof LemmasCandD,one canobtainthefollowingestimateof theorder ( )
ofnite"noise"momentsinTheorem1:
( )
9 if0 1 3
(5+3 ) (1 ) if13 1.
Theorems1and2canb egeneralizedfortheempiricalpro cess basedon"nonlinear"observations
= ( ) 1 , where () isa measurable function, and is a linearmovingaverage
pro cess of(1.1)-(1.2).
Theorem1anditspro ofremainvalid,withunimp ortantchanges,foratwo-sidedmovingaverage
= ,where = () and ()variesslowlyat ,sothatthereisaslowlyvarying
function () 0andtwoconstants suchthatthere existthelimits
lim ( ) () =
Insuch acase,(1.7)holdswith = ( ) and
= (1+ ) ( )
where ( ):= if 0;= if 0.
Consider themo del
= +
where satisfytheconditionsofTheorem 1,and isunknownparameter. Let ():
b e acontinuousfunctionsuch that
0( ):= ( ) = ( + )= ( ) ( + )
iswell-denedand1-1inaneighb orho o d 2 and
(31) 0( ) = 0
TheM-estimate of isdened by
0 = 1
( )
= ( ) ( )
(32)
where ( ) = ( ) is the empirical distribution function based on the observations
.
k 1
Theorem 3.
R
R R
R R
R
R R
R R
R R
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0 0
0
0 0
0 0
0
0
1 6
0 ) N
1
) !1
0 0 0 0 0
0
0 0 0 0
0
2 0 )
0 0 0
) 0 0
N
0 0 0 0 0 0
1
0 0
0
1 1 2
0
0
1 1 2
1 1 2
0
0
1 1
1
=1
0
1 2
1
=1
1
1
1 2 1 2
1
=1
1 0 1
1
0 0 0
0
1 1 2
0
1 1 2
0 0
N
=
D N
N
N
N
N
=
D N
N
=
D
N
N N
N N
N N
j
j P
=
D N
N
j j
N
N
=
D
N N
=
D N
N
j
j P
N N N
N
=
D
N
N
=
D
N P
N j j
b
b
b
Z
b 2
e e
3
Z
0
b
1
e
e
Z
2
e e
3 Z
2
e e
3
Z
e
X
e e
P
Z
b 2
e e
3 Z
e b
X
Z
e
Z
Z
0
b
1
e
Z
0
b
1
b
0
b
1 0
b 1
X
Assume, in addition, that has bounded variation and .
Then
Remark3.
Proofof Theorem 3.
weaklyconsistent
U-statistics.
xf x dx
: Nd c ; :
N
: N :
: x d Nd c F F x Nd c x x dF x ;
F x F x
x d Nd F F x Nd F F x d x
f x d x d X o
:
f x F x f x c d X Z
x d Nd c F x F x f x d x c d X o
Z f x d x Z f x d x ;
:
Z ;
x x dF x x f x f x dx :
Nd c Nd c o ;
: U h h X ;:::;X
() 0( ) = ( ) ( ) = 0
(33) ( ) = (0 1)
Theab ove resultimplies,inparticular,thattheasymptoticeciencyofanM-estimatedo esnot
dep end on itskernel (), which is insharp contrast with theiid case. This dierence was rst noted by
Beran(1991,1992)inthecase ofGaussianfunctionals.
M-estimatorsoftheslop eparameterinlinearregressionmo delswithlongmemorymovingaverageerrors
arediscussed inGiraitis,KoulandSurgailis(1994).
It isnotdicult tocheck that,undertheconditionsof thetheorem,the estimate
existsforallsucientlylarge ,andis ,i.e.
(34) = ( )
Accordingto(3.1)and(3.2),
(35) 0 = ( ) ( )( ) + ( ) ( ) ( )
where ( )= ( ). AccordingtoTheorem2,
( ) ( )( ) = ( )( + ) ( )
= ( + ) ( ) + (1)
(36)
uniformlyin 2,where ( )= ( )= ( ). Consequently,as = ,from(3.4)
andthecontinuityof theintegralontherighthand sideof(3.6)in ,
( ) ( ( ) ( )) = ( + ) ( ) + (1)
= ( + ) ( ) = ( ) ( )
(37)
where (0 1). Next,rewrite thesecondintegralontherighthandsideofin(3.5)as
( ) ( ) ( ) = ( ) ( + ) ( ) = 0( ) 0( )
Usingthe fact that 0()is continuouslydierentiableat together withthe meanvaluetheorem andthe
convergence(3.4),weobtain
0( ) 0( ) = 0( )+ (1)
ortherelation(3.3),inviewof (3.5),(3.7).
ConsidertheU-statistic
(38) ( ) = ( )
k R
R
0
0
D
0 0
P
Z
8 9 8
Z
9
!
6 6
1
8 2
)
1
1 2 1 2
[ ]
[01]
1 1 1
N
k
p p q
n k
k
N Nt
;
k
D
k k k
k
R R
Theorem4
R
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1 h
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