uerlenma@p opix.urz.uni-heidelb erg.de Let b e an arbitrary sto chastic pro cess on the real line and

(1)

0

Date

r s t

s;t I

00

0

00

2

00 ULRICH ERLENMAIER

: Septemb er1997.

fj 0 j^j 0 j 2 g

fj 0 j^j 0 j g

2

f g

fj 0 jg

U I a;b

! U;I

U I

! U;I U r U s U s U t r;s;t I :

U

U r U s U s U t r;t

r s t I > >

K K ;

! U;I >

K

a;b :

U U

Z

! U;I U s U t

! U;I ! U;I ! U;I Z

f Abstract.

Correspondence to: Ulrich Erlenmaier, Institut f ur Angewandte Mathematik,

Universitat Heidelb erg, Im Neuenheimer Feld 294, D-69120 Heidelb erg, Germany;

uerlenma@p opix.urz.uni-heidelb erg.de

Let b e an arbitrary sto chastic pro cess on the real line and = [ ]an

interval onIR. We proveasto chasticinequality forthemo dulus of continuity ( )of

on :

( ):= sup ( ) ( ) ( ) ( ) :

Supp ose that hasrighthand (orlefthand)continuouspaths and that theincrementsof

can b e sto chastically b ounded in thefollowing way:

IP ( ) ( ) ( ) ( ) (] ])

forall ,forall 0,with a realnumb er 0 and asetfunction whichis

subadditive in a certainsense. Thenexists aconstant = ( )with:

IP ( ) (] ])

Ifin additionthepathsof arecadlag and thejumps of can b eb oundedbyarandom

variable wecanextendthisresulttoasto chasticinequalityforthemo dulusofcontinuity

( ):= sup ( ) ( )

using that ( )isb ounded in thefollowing way: ( ) 4 ( )+ .

Thisresultisusedtoprovetheweakconvergenceofa go o dness-of-tteststatisticforhy-

p othesesab outtheconditional medianfunction ofastationary,real-valued, Markovian

timeseries.

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1.

0 1

X

0 1

0 0

2

Introduction

L 0 j

p f g 0

j 0 j

1

=1

0 1 1

0

=1

1 1

IR

1

0 t

n

t

t s

t t t

n

t

t t t

n

s;t

n n

n

n n n

n X

X X s <t

X f X X t ;::: ;n

f f

W s;f

n

X s X f X

T f W s;f W t;f :

T f

X

W W s W s;f

T f This pap er is concerned with stationary, real-valued, Markovian time series with

stationary transition probabilities. Let ( ) b e such a timeseries. The median

functionoftheconditionallawof given ( : )isimplicitlydenedthrough

the following condition:

Med ( ( ) ) =0 ( =1 )

Weprop ose aKuip ertyp estatisticto testthe hyp otheses = . Itisbasedon the

pro cess

( ):=

1

1 sign ( )

and has the following form:

( ):= sup ( ) ( )

The test willreject the hyp otheses if ( ) is to o large.

In a simular context Hong-zhi and Bing (1991) have intro duced a Kolmogorov-

Smirnov typ e statistic to test hyp otheses ab out the conditional mean function.

But for the weak convergence of their statistic they need quite restrictive mixing-

prop erties of the time series and also (4+ )-moments of . In my opinion the

assumptions that are made here are muchweaker. The idea of the pro of of tight-

ness is dierent and is due to Koul and Stute (1996). In their pro of seems to b e a

gapwhich -undersomeslightly stronger assumptions-is closedhere generalizinga

result ofBillingsley (1969, pp.98).

In section2 we formulatethis generalizationand show how it can b e used to prove

tightness. In section 3 we apply this metho d to show the weak convergence of the

pro cess () with ( ) := ( ) under some quite weak assumptions. By

the Continuos MappingTheoremthis resultimpliesthe weakconvergenceof ( ).

(3)

2.

1 2

00

Denition 1.

r s t

s;t

A

b b

A

A Tightness Criterion

fj 0 j^j 0 j 2 g

D

fj 0 j 2 g

0 2

I

I f 2 g

I 2

I A

!

U A

! U;A U r U s U s U t r;s;t A

! U

! U;A U s U t s;t A

! U;a

! U;A

U s U t s;t I

A

I a;b a;b A

I a;c b A a < b < c

I ;I I

I a;b I b;c

J d;e I J c d

(intervals in with coordinates in A)

The set of all left open intervals with coordinates in will be denoted by :

For be an interval in . For every with we dene a

subdivision ( of by:

and

Let be another interval in . and are called neighbours if . In this section we formulate a Prop osition that generalizes a result of Billingsley.

Given an arbitrary subset of IR we want to state conditions under whichwe can

get a sto chastic inequationfor the mo dulus of continuity ofa sto chastic pro cess

on . It isdened inthe following way:

( ):= sup ( ) ( ) ( ) ( ) :

If weak convergence is studied in the context of the function space with the

Skohoro d-top o logy such inequations for ( ) are needed to prove tightness (see

Billingsley,pp.109). Within the theorythat isused here (seePollard1990) wehave

to treat with the following mo dulusof continuity:

( ):=sup ( ) ( ) :

The sto chastic inequation for ( ) will b e concluded from the inequality for

( ).

In order to formulate the conditions on the increments ( ) ( ) ( ) that

we need forthe Prop osition we haveto makesomeprep erations:

IR

:= =] ]:

:=] ]

)

:=] ] :=] ]

:=] ] =

(4)

0

00

1 2

U A

U

A

A A

A

b b

A

Denition 2.

Denition 3.

Proposition 1.

I !

0

I

M I f I ! 1 g

2 I

2 M I

fj j^j j g [ g

2I

f g

U

I a;b U b U a

U I I

<

B;C B C

B C

M M <

I I > I ;I I

I I M I

A a

b U A

M < M

>

U I U J I J >

I;J :

KM;

! U;A

KM;

a;b

:

IR =] ] ( ) ( )

( ):= ( )

( ):= : IR: 0 ;

( ) ( )

1

( ) 0 ( )

( )+ ( ) ( )

Now we can state the Prop osition:

1 ( )

0

IP ( ) ( ) ( ) 0

( )

IP ( )

( )

(] ]) ( as distribution function of a set function)

One can think of theincrements of as a signed, additive set function

which maps intervals to . In order to simplify notation we

write .

(M - subadditive set functions)

Correspondingtothe signed set functions we neednonnegative nondecreasing set

functions on :

nondecreasing

There a set function is called nondecreasing if for arbitrary sets

implies .

Let now be a real number, . A set function is called M - subadditive if

for every interval with there exists a subdivision of so

that the following holds:

Let bea nitesubset ofthereal line withsmallest element and

largest element . Further let be a stochastic process on . Suppose that there

exists a real number , a -subadditive set function and a real

number which satise

for all

for all neighbouring intervalls

Then exists a constant with:

(5)

0 1

00

2 2 2

00

00 Remark1.

Remark2.

Remark3.

l A i A;i<l i A;i l

n

m

m m m m m m m

m

1 1

1

j 0 j_ j 0 j

f g

f

0

g

2

j 0 j^j 0 j

2 # # #

>

! !

! U;A U i U a U b U i

W

I a;b

U I U

I

! U;I >

KM;

a;b

KM;

U

I I

I a

i b a

i

> r<s<t I

! U;I U r U s U s U t

r;s;t r ;s ;t I r r;s s;t t

U ! U;I

()= ()

1 ()

~

~( ):=min max ( ) ( ) max ( ) ( )

=[ ]

IP ( )

( )

(] ])

( )

W.l.o.g. let have righthand continuous paths. We cho ose the following

dyadic subsets of :

:= +

( )

2

: 0 2

For every 0 we nd real numb ers with:

( ) ( ) ( ) ( ) ( ) +

Now we can approximate by elements : .

Using the righthand continuity of the paths of we nd therefore that ( ) In Billing sley's result there is a m ore restrictive condition on the set

function , namely,that:

with a real number and a nite measure .

It follows immediately fromthe proof that the statement of the Propo-

sition remains if we replace the functional by the functional which is dened

in the following way:

We will use this fact in one part of the proof of tightness for the process .

Let bean arbitraryinterval onthe real line. Suppose that the

pathsof arerighthand (or lefthand) continuous on and fulllls the conditions

of the Proposition on every nite subset of . Then the following inequality holds:

with the same constant as above.

Proof.

(6)

1

2 Remark4.

m

cadlag

s s

t s

1 2

1 1

2 2

1 2 1 2

1 2

2 2

2 1

1 2 1 1 1 1 1 2

00

!1 00

!1

00

00 00

00

0 0

f g f g

f g

F f ! g

F

2

f 2 j 0 j g

j 0 j j 0 j

j 0 j j 0 j j 0

j

j 0 jj 0 j j 0 j

j 0 j j 0 0 j j 0 0 j j 0 j

! U;I > ! U;I >

! U;I >

KM;

a;b

I a;b

I f I f

U I U

I Z

! U;I ! U;I Z

s <s I

I U s U s ! U;I

I U t U s ! U;I

< s;t s <t <

U s U s > ! U;I U t U s > ! U;I :

U s U s >! U;I U s U s ! U;I U s

U t ! U;I

U t U s U t U s U s U s ! U;I

U s U s U s U U U U U s

! U;I Z

For anarbitrary interval on we dene:

is righthand continuous and has existing lefthand limits

Suppose that the paths of are elements of and that the jumps of are

bounded uniformly on by a randomvariable .

Then we can conclude:

Proof.

Theoremwe can conclude:

IP ( ) =IE1 lim ( )

=IE lim 1 ( )

= lim IE1 ( )

( )

(] ])

=[ ] IR

( ):= : IR:

( )

( ) 4 ( )+

For arbitrary we dene:

:=sup : sup ( ) ( ) 2 ( )

:=inf : sup ( ) ( ) 2 ( )

If , then there existreal numb ers ( ) with:

( ) ( ) 2 ( ) and ( ) ( ) 2 ( )

But from ( ) ( ) ( )follows ( ) ( ) ( ) and ( )

( ) ( ). Thereforewecan conclude:

( ) ( ) ( ) ( ) + ( ) ( ) 2 ( )

which contradicts the assumption made ab ove and hence implies that .

Finallyweget:

( ) ( ) ( ) ( ) + ( ) ( +) + ( +) ( )

4 ( )+

(7)

W

t

1 3.

K K

P

p

2

0

1

01

0

IR

1

=0

1

+

=

q

c

x c

n

t X

n

n w

t

c

i

g

n

g Remark5.

K

K K

Theorem1.

1 1

!

j j

1

01 1

j 0 j

2 01 1

1

01 1

^

Weak Convergence of the Process

U

n

P ;

P A P A;x

A P =n

W

P P P

X

> P I I I

P i;i <

;

s;t g s g t g

g t > t ; ;

W

; ;

W Ks;t F s t F

There isalsoa multidimensional version of this Proposition,thatdeals

with stochastic processes on nite subsets of . It can in a simulare way be

extended to nonnite subsets of the (see Erlenmaier 1997).

Iftheconditions 1- 3aresatised, thesequence of stochas-

tic processes on convergesweakly toa pathwise uniformlycontinuous,

centered Gaussian process with covariance function , where IR

IR

We denote by ( )the kernelof the stationary transition probability of the time

series and dene:

( ):=sup ( )

for allmeasurable sets IRand

^

:=1 .

Now we are able to state the conditions under which the pro cess converges

weakly:

1

^

(in probability) where is the stationary measure of the time

series ( )

2 Thereexistsarealnumb er 0sothat ( ) forallintervalls IR.

3 ([ +1])

Condition 1 is needed for the convergence of the nite-dimensional distributions

(dis), 2 and 3 willb eused to prove tightness.

The pro of of tightnessuses a functional CLT (seePollard 1990). In order to apply

this Theoremwe rst of allhave to dene a pseudometric on [ ]. Wewrite

( ) := ( ) ( ), where is a continuously dierentiable function on the

compact real line with ( ) 0 for every IR. That implies that ([ ] )

isa totally b ounded pseudometricspace.

()

([ ] )

( ):= ( )

(8)

K

K K

X

2 3

2 2

2

0

K

Corollary 1.

Remark6.

0

IR [01]

[01]

=1

1

=1

0

=0

2

1

! j 0 j j 0 j

j 0 j

p

f g

j 1 2

p

p f 2 gj j

n w

s;t s;t ;

s;t ;

n;Z

n

t

t n;t

n;t n

t

t n

t

n;t

n;t t

n;Z

n

t n

t n;t

W

I

B

;

T f W s W t B F s B F t

B s B t

W s

n

X s Z

Z

X ;:::X

Z

Z X K < n t ;:::;n

W = n

I

Z X I Z

IR

The convergenceofthedis follows fromcondition 1andthe CLTfor mar-

tingaledierencearrays(seePollard1984). Thepro ofofthesto casticequicontinuity

isdefered to section4.

[0 1]

( ) sup ( ) ( ) = sup ( ( )) ( ( ))

= sup ( ) ( )

( ):=

1

( )

IE IN; =1

Mostoftheargumentationisidentical. Theonlydierenceis,thatthejumps

of their pro cess cannot b e b ounded by 1 . But it is sucient to observe

that on an interval they are b oundedby

:= max 1

1

If condition 3 isnot satised at least the sequence converges to on every nite

intervall .

Proof.

If 1 - 3 hold the Continuous Mapping Theoremimpliesthe weak

convergence of the teststatistic to a functional of the standard Brownian motion

on :

in law

Thestochasticequicontinuityof themoregeneralmarkedempiricalpro-

cesses

whichKoulandStuteinvestigate insection3oftheirpapercanbeprovedinasimilar

way. There the series of randomvariables forma martingale dierence

arraywith respect to the series of -algebra (for details see Koul

and Stute 1996). The assumption about the randomvariable can be replaced by

the weaker assumption:

for all a.s

Proof.

(9)

. 4.

X

2 3

Some Proofs

0

0 0

00

=1

1

=1 2

1

2

1

2

1 1

1 Proof of Proposition 1

n

t

t n;t

n

t

n;t t

i

l m i<l i

l i m

m i

f g

fp

f 2 gj j g

f 2 g j

f g

j 0 j_ j 0 j

0

Z

n

X I Z

n

X I Z X

K P I

A ;:::;m U U i

i ;::: ;m

! U

! U U U U U

! U ! U

!

m

m ; ! m m

h ;h h ;m M ;m

Thereforewecan conclude:

IP IP

1

IE1 IE

1

( )

Therefore in this context we end up with the same tightness result as in the pro of

of Theorem1.

Withinsomepro ofs therearestatementswhich-forthesakeofclearity-arepro ofed

afterthe mainargumentationis nished.

In most partsof thepro of we followthe argumentation

of Billingsley(1968).

W.l.o.g. supp ose that = 1 . In the following we write := ( )

( =1 ).

First of allwe recall the denition of ~( ):

~( ):= min max max

The following inequality ispro ofed b elow:

( ) 2~( ) (4.1)

Therefore itis sucientto show the result of the prop osition for ~. We do this by

induction over .

For =1 2 ~ equals0. Hereis the induction step from 1to :

Cho ose an integer which satises (]1 1])+ (] +1 ]) (]1 ]) and

dene

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1

.

00

0

1 2 1 2

1 2

1 1

2

1

+2

2

+2

( +2)

+2 1

1 1

1 1 +1

1 1

r s s t

=

r s r s

s t s m m t

h m h

h

Proof of inequality 4.1

j 0 j^j 0 j

0 _ 0 _ 0 _

_

f g f _ g f g

f g f g f g

0

j 0 jj 0 j j 0 j

j 0 j j 0 j

j 0 j

! Y ! Z

r;s;t U U U U

B ;h ;m ;h ;h h ;h ;m ;h ;m

! U ! Y ! Z B

; > =

! U ! Y ! Z B

! Y ! Z B

KM;

;h h ;m

a;b

KM;

M a;b

a;b

MKM;

KM; = M

! ! r;s;t

! l

!

U U U U U U ! l>r;s

U U U U U U ! l s;t

U U B U U B

U U >B

Later on we want to apply the induction hyp otheses to ~( ) and ~( ). Now we

dene ( ):= and nally

:= (1 1 ) (1 1 +1) ( 1 +1 ) (1 +1 )

to conclude:

~( ) ~( ) ~( )+2 (pro ofed b elow) (4.2)

Let 0b earbitrarynumb erswith + =1and dene :=1 (1+ ). From

condition (a) and induction hyp otheses follows:

IP ~( ) IP ~( ) ~( ) +IP 2

IP ~( ) +IP ~( ) +IP

2

( )

(]1 1])+ (] +1 ]) + 2

(] ])

( )

(] ]) + 2

(] ])

( ) +2

Cho osing ( ):=2 (1 ) completesthe pro of.

We have to show that 2~. Let b e the

integers that minimize the expression for and the one that maximizes the

expressionfor ~. We observe:

+ 2~ ( )

We distinguish the followingthree cases

a) und

b)

c)

(11)

.

0 0

Proof of Theorem 1 1

1

1 2 1 2

1 1

1 1 1 1 1

+1 +1 2

2

1 +1 1

1 1

1 1 1

1 1 +1 +1 2

2

0

0 0

i<l i

l i m

m i

i

i i h h

m i m h h i

m i

m h h h

i

m i m h h i

m i m h h h h i

m i

n g

k k ;m

m

j 0 j_ j 0 j _

j 0 j

j 0 j j 0 j j 0 j

j 0 j

j 0 j j 0 j

j 0 j

j 0 j j 0 j j 0 j

j 0 j j 0 j j 0 j j 0 j

j 0 j

f j g

0

f

0

g

l

U U U U ! Y ! Z B

l h l l l l l l

! Y ! Z

U U ! Y i<l

U U U U U U ! Y B l i<l h

U U U U U U B ! Z h l i<l

U U ! Z l i m

U U B U U B

U U ! Y i<l l

U U U U U U ! Y B l l i<h

U U U U U U U U B ! Z h i<l

U U ! Z l i m

n

>

! W ; > n n

I k ;k I

k ; ;::: m ; ::: I a;b

I

I a

i b a

i In eachcase wewantto cho ose a numb er which satises

max max ~( ) ~( )+2

In case a) we can cho ose = , in b) = and in case c) = , where and

are supp osed to b e the integers that minimizethe expressions for ~( ) and ~( )

resp ectively. Imcase a) we provethis by the followinglines:

~( ) (1 )

+ ~( )+ ( = )

+ +~( ) ( = )

~( ) ( )

Case b)implies:

and

Now we can argue as ab ove:

~( ) (1 = )

+ ~( )+ ( = )

+ + 2 +~( ) ( )

~( ) ( )

Case c)can b e treatedsimularly.

We have to show that there is an integer and a real

numb er 0 so that the following holds:

IP ( ) if and

Firstofallwedeneintervals :=[ 1 ]and niteapproximations ofthem

( = 1 2 ; = 1 2 ) , where for an arbitrary intervall = [ ] its nite

approximation is denedas follows:

:= +

( )

: 0 2

(12)

00 00

00

1

00

X

X X

X p

X

p

X

X X

X

p +1

= +2

4

= +2 = +2

= +2

2

= +2

+1

= +2

= +1 = +2

4

= +2

0

2 2

j 0 j

f g

0

j 0 j j 0 j

0

f g f g

f g

f g f j 0 j g

1

0

2

f 1 g 8 2

l

n l;m

a k l

n k ;m

l

k a n

n k ;m

l

k a n

l

k a

n

l

k a

n

l

k a

n l n l;m

n

n l;m

a k l

n k ;m

l

k a n

l

k a

n k ;m

l

k a n

k a

n

a;l l >a J a;l n;m

! W ;J ! W ;I W k ;k

! W ;I

KP k ;k

K

W k ;k W k ;k

W k ;k

P k ;k

! W ;J > ! W ;J

W

! W ;J

! W ;I

W k ;k

! W ;I

W k ;k

KP a;

P k ;k

n;m

W

a

! W ; a ; = n;m

With IN ( ) and :=[ ] one can show for all IN:

~( ) max ~( ) + (] 1 ]) (pro ofed b elow)

(4.3)

and

IP ~( )

(] 1 ])

(pro ofed b elow) (4.4)

with auniversal constant . If we use that

IE (] 1 ]) = IE (] 1 ])

IE (] 1 ])

= (] 1 ])

we can conclude:

IP ( ) IP ( )

(b ecause has cadlag paths)

IP 2~( )

(seeinequality 4.1)

IP max ~( )

4

+IP (] 1 ])

4

IP ( )

2

+IP (] 1 ])

2

4 (] ])

+ 4

(] 1 ])

for all IN.

Togetherwith thefact, thatthe pathsof has only aniteamountofvaluesthis

impliesthat wecan nd areal numb er so that the followingholds:

IP ( [ ]) 5 IN

(13)

1

0 1

X X

n

=

k k

n j

k

j

k

g

g g

g

j

k

n g

u

k

n

k

u

k

n

k

n n

0 0

2

1

1 0

0 1

4 4 1

1 2

1 0

0

0 1

0 0 1

0

0 1

0

=0

1

=0

2

0 1

0 p

f 1 g

f 01 g

^

0

d 0 e

f g

j 0 j 2

^ 01 ^ 1

2

01 1

f j g f g f g

f 01 g f 1 g

W

= n

! W ; a ; n n = :

a

! W ; ;a n n

a ;a

= K

I a k ;a k I a

k

;a

k

k u a a =

! W ;I P I n n k u j ;

g L

s;t Ls t s;t a ;a

L

;a a ;

s;t s;t I

k u ;a a ;

! W ; ! W ;I ! W ;I

! W ; ;a ! W ; a ;

n n

Nowweuse thefactthatthejumpsofalmostallpathsof areuniformlyb ounded

by 1 and apply Remark4. That leads to the inequality:

IP ( [ ]) for all :=1

In the sameway one can nd a real numb er for the left edge with the prop erty:

IP ( [ ]) for all

Thereforeitjust remainsto treat themo dulus ofcontinuityon the compact[ ].

We dene:

~

:= ( ) 5 1 and intervals

:=[ +

~

+( +1)

~

] :=[ +

(2 1)

~

2

+

(2 +1)

~

2

] (0 := ( )

~

)

We use Prop osition 1and the Remarks3 and 4to conclude:

IP ( ) ( ) for all (0 ; =1 2)

(detailscan b e found b elow). Now by the prop ertiesof we can nd aconstant

with ( ) for all [ ]. Ifwedene

:=(

~

2

) ( ) ( + )

we know that two p oints IR with ( ) lie in one of the intervals

(0 ), [ ] or [ ]. Therefore we can write:

IP ( ) IP ( ) + IP ( )

+IP ( [ ]) +IP ( [ ])

4

(for all )

(14)

.

0 0

(

X

)

X

X X

_ j 0 j

j 0 j _ j 0 j

j 0 j

F

p

I

f 2 g f 2 g

f ^ g

+1 +1

+1

= +2

+1

+1 +1

+1

= +2

IR

1 1

4

2 2

4 2

1

4 2

2 2

l

! W ;J ! W ;J ! W ;I W k ;k

! W ;I W k ;k ! W ;I W k ;k

! W ;I W k ;k

W

= n

I J

U X I V X J

W I W J

n

UU V V

n

UU V V V U

n l ;m n l;m n l ;m n

a k l

n k ;m

l

k a

n n l ;m n

a k l

n k ;m

l

k a n

n cadlag

i i i i i i

n n n n

i;j;k ;l n

i j k l

i j

k

i j

k We use induction over . The start of induction is

trivial. Supp ose nowthat the resultholds for each integer up to . That implies:

~( ) ~( ) ~( )+ (] 1 ])

(seethe pro of ofinequality 4.2)

max ~( ) + (] 1 ]) ~( )+ (] 1 ])

max ~( ) + (] 1 ])

Weknowthatthepathsof are elementsof

and that the jumpsof almost allof themare b ounded by 1 . Therefore we just

haveto show that Prop osition 1 is applicable on arbitrary nite subsets of the real

line.

Thussupp ose that and are intervalsin . With

:=1 sign :=1 sign

we get:

IP ( ) ( )

1

IE ( ) ( )

= 1

IE

= 1

IE +IE

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1

1 1

0

Acknowledgements X

X

j j

0

f ^ g _ [

j [ j [

jj 1

2

1

=1 2

2

=1 2

2 2

1 2

2

=1 2

2

1 2

2

=1 2

2

1

2

=1 2

1

2

4

4 i;j<k

i j

k

i i

k

i i

k k k

k

i i

k k

k

c

k

i

i c

k

c k

i

c

i;j<k i j

k

c

n n c c

c c

k n

UU V U V

U V U V

U V X U V X

P J U P J U

P J U P J P I

k P J P I

V V U k P I P J

k

W I W J

P I P J P I J

P J P I P I P J

I J P I J

: P : M M

For xed wecan conclude:

IE =IE

2IE +2IE

=2IE IE( )+2IE IE( )

2 ( )IE +2 ( )IE

2 ( ) IE +2 ( ) ( )

2( 1) ( ) ( )

The sameargumentationapplied to the second sumleads to:

IE 2( 1) ( ) ( )

Summing over we nd

IP ( ) ( )

1

( ) ( ) ( )

1

( ) ( )+ ( ) ( )

1

( )

The setfunction ()is -subadditivewith :=2 . One gets thisconstant

by halving the intervallsaccordingto Leb esgues measure.

The content of this pap er is part of my diploma thesis on which I am working

right now. I want to thank my advisor, Lutz D umbgen, for his patience with my

sillyquestions,formanyhelpfulsuggestionsandingeneralforhisp erfectadvice. In

addition I would like to thank Hira Koul for drawing my attention to his preprint

(16)

59 W

A

I

References

[1] Billingsley,P.(1969).Convergenceofprobabilitymeasures. iley,NewYork.

[2] Erlenmaier,U.(1997).Diplomathesis.Inpreparation.

[3] Hall,P.andHeyde,C.C.Martingalelimittheory anditsapplication. cademicPress,Inc.

[4] Hong-zhi,A. and Cheng,B. (1991).A Kolmogorov-SmirnovTyp e Statistic with Application

to TestofNonlinarityinTimeSeries. ntern.StatisticalReview, ,287-307.

[5] Koul,H.L.andStute,W.(1991).Nonparametricmo delchecksfortimeseries.Preprint.

[6] Pollard,D.(1984).Convergenceofsto chasticpro cesses . SpringerVerlag.NewYork.

[7] Pollard,D.(1990).Empiricalpro cesse s: theoryandapplication.NSF-CBMSRegionalConfer-

enceSeries inProbabilityandStatistics,Volume2.