Norms for copulas

Math. Sci. VOL. 18 NO. 3 (1995) 417-436

417

NORMS

FOR

COPULAS

WILLIAM F. DARSOWandELWOOD T. OLSEN

Department

of Mathematics Illinois Instituteof Technology

Chicago, Illinois 60616

(Received October 19, 1993 and in revised form July 20, 1994)

ABSTRACT. We

considerseveral norms onthespanof the setC ofallcopulas. Dominance and equivalence relationshipsamongthenorms arediscussed, and completenessissues areaddressed. Themotivationfor the studyisdiscussed. Applications to the study ofoneparametersemigroups of copulasarealso addressed.

KEY WORDS AND PHRASES.

Copulas, doubly stochastic measures, Banach algebras, one

parameter semigroups.

1991

AMS SUBJECT

CLASSIFICATION CODES.

46B20,

28A35,

60J25

1.

INTRODUCTION.

A

copula is a function C

[0,1]

2

[0,

1]

_{satisfying the boundary conditions}

C(x,O)

C(O,y)

O,

C(x,

1)

x and

C(1,y)

y, for all x,yE

[0,

1],

and the monotonicity condition

c(,)

c(=, )

c(=,

u)

+

c(,

)

>

0

wheneverxl i_x2andyl

_<

y2. These conditions imply that Cis acontinuous function. Copulas

areofinterestbecausetheylink jointdistributionsto marginal distributions: forany real valued random variables

X

and

X2

with joint distribution

F12

there is acopula C such that

F12(Xl,X2)

C(FI(xl),

F2(x2))

where

Ft

and

F2

denote thecumulative distribution functions of

X

and X2, respectively. For aproof, see

Sklar[6, 7].

Knowledge ofbasicproperties ofcopulas will beassumedin this paper, thoughreferenceswilloften be given when they are used. For a discussionof these properties,

see

Darsow

et al.

[2];

Sklar

[6,

7].

Wedenote by C the setof all copulas. C is closedunder convexcombinations and under a

product operation

A

B defined by

A

_*B

(x,

y)

A,2(x, t)B,z (t,

y)

Hereandin thesequel the subscripts and 2 denote partial derivatives:

aB(,

B,(,)

-x

’(’)

418 W. F. DARSOW AND E. T. OLSEN

The copulas

P(x,y)

xy and

M(x,y)

min{x,y}

are null and unit elements with respect to theproduct operation" that is,

A.P-P.A-P

A.M-M.A-A

for allcopulas

A,

asisreadily verified using the definition of the product.

A

copula Eisidempotent ifE E E; both P and M areidempotent, and therearemany other idempotents.

A

copula

A is left invertibleif there isa copulaB such that B A M and right invertible if thereis a

copula B such that A B M. If A is left invertible, its left inverse is the copula

AT

defined by

AT(x,y)

A(y,x);

similarly, ifA isright invertible,its rightinverse isA

T,

Darsow et al.

[2].

Thereexist copulas which possess one-sidedbut not two-sidedinverses.

ThesetCisalso closed under uniformlimits" if

A,

ECfor allnand

An

A uniformly, then necessarily

A

E C.

In

fact, theset Cis acompact subset of

C([0,

1]

2)

under the uniformnorm. This is auseful property.

However,

theproduct does not behave well under the uniformnorm. If

A,

A

uniformlyandB is anycopula,then

A,

B

A

B andB

A,

B

A,

Darsow

et al.

[2],

sothat the product is continuous ineachplace. But the productisnotjointlycontinuous. It can be shown that

(two-sided)

invertible copulas are dense in C in the topology of uniform convergence. IfA, is a sequenceofinvertible copulas converging to anoninvertible copula, say P, then

AT

.

A,

M for all n but

pT,

p p _M It follows that the product is not jointly continuous. This is an inconvenient

fact,

andapart of the motivationfor the research reported here was to find a natural topology on the set ofcopulas under which the product operation

was not only continuous ineach placebut also jointly continuous. The joint continuity issue is discussedat various points in the paper.

One

parameter semigroups of copulasarisenaturally in thestudyofMarkovprocesses,

Dar-sowetal.

[2].

Let

Cbe anycopula, letEbeanidempotent copulasatisfying C E E C C,

and leta beapositive number. Then

At

defined for

>_

0by

At

e-t

(

E

+

(at)C)k’

k--1

isasemigroupof copulas:

A+,

A.

At.

In

thedefinition,C denotes thek-fold product of C withitself;theseriesconvergesuniformlyfor allaandt, asiseasy toverify. The classicaltheory

ofoneparametersemigroupsis developedin the context of Banach algebras. To investigate the applicability of the classicaltheoryto semigroups ofcopulas, it is natural, first, towork in the spanof

C,

whichwewilldenotespan

(C),

ratherthanC itself, and, second,to seekanormunder which span

(C)

is in fact a Banach algebra. This was a second part of the motivation for the research reported here. The application ofthisresearch tooneparametersemigroupsof copulas is discussedin Section 6 ofthispaper.

The paper is organizedasfollows:

In

Section2, Preliminaries,wediscussspan

(C)

and define thenormstobetreated insubsequent sections.

In

Sections3, 4 and5,wediscuss the Minkowski

norm, Sobolev norms, and the Jordan norm onspan

(C),

respectively. Section 6 is devoted to semigroups ofcopulas and Section 7,to discussion andconclusions; inparticular, weshall make

some commentsabout compactnessinSection 7. 2.

PRELIMINARIES.

By

span

(C)

wemeanthelinear spanofthe copulas the setofall finite real linear combi-nations of elements ofC.

Any

element

A

span

(C)

canbewritten intheform

A-sB-tC

(2.1)

wheres and are nonnegative andB andC arecopulas. To seethis, observe thatif

A

akAk

419

where

Ak

EC and ak is arealnumber, then we can write

A s

Ak

Ak

sB tC

8

a >0 a <0 wherethe last equality defines B and C and

s-

a

and t--

a.

>0 a<0

(In

bothoftheforegoingequations, the sumistakentobezeroif theset summedover is

empty.)

Observe that B andC are copulas, since C isclosed under convex combinations. We will make

repeated useof theobservation

(2.1),

usuallywithoutcomment.

Wenowdefine thenorms wewillinvestigate.

Minkowski norm. The symbol B

-co(-C

U

C)

denotes the convex hull of the set -C

that is, the set of all convex combinations of members of C and -C. It is easy to see that span

(C)

UtRoB.

For

any

A

span

(C)

define

IAM

inf{

>

0

A

e

B}.

(2.2)

IIM

is sometimes referred to as a Minkowskifunctional; it is easy toverify that it is a norm on span

(C),

e.g. Rudin

[5].

Thereisamoreconvenient definitionof the Minkowski functional:

AM

inf{s

+

t

s, 0and

A

sB Cforsome

B,

C

e

C}.

Itis easy to

veN

that

(2.2)

and

(2.a)

areequivalent definitions; weomit theproof.

We

will show in Section

a

that span

(C)

is a Banach algebraunder the norm

1"

I-

The problem with the Minkowski functionalisthatit isdiNcult tocomputeand thusiscumbersome to workwith.

We

includesomeresults inSection

a

which facilitate the calculationofthe MinkowsN normforcertainspecial classes of elements ofspan

(C).

Sobolevnorms. Sincethe product isdefined using first partialderivatives, it seemsnatural towork withnormswhich take derivativesinto account. It isreadily veNfied that

c

,(0,

form 0, and allp

e [1, ].

Wm’()

denotes theSobolevspace

{lI

e

L(a)

and

DI

e

L(a)

for all multi-indices forwhich

I1

m

Here,

C R

,

amulti-index is ad-tuple ofnonnegative integers,

]

denotes thesumof the components of

,

and D denotes a distributional partial derivative.

In

our ce, d 2 and

m -0or1, sothat

(0,0), (1, 0)

or

(0,

1),

and

D(’)I

I,

D(’)I

I,t

and

D(’)

It iswell known, e.g. Adams

[1],

that each of the Sobolevspaces so defined is a Banaeh spree

(i.e.,

is

complete)

under thenormdefinedby

(2.4)

Wewillshow inSection 4 thatC isacompletemetric spaceunder the Sobolevnorms

]l" I[:,p

and that on C theproduct isjointly continuous with respect to

IIx,

for

[1, );

it isan

open issue whether the product is continuous

(jointly

or

separately)

with respect to the norm

420 W. F. DARSOW AND E. T. OLSEN

any ofthe I4

....

P _{spaces. The issue of continuity ofthe product on span}

(C),

_{as opposedto C}

itself, is anopen one.

Jordannorm. Wehave occasionally foundit convenient towork withoneothernorm. It is well known, of.

Darsow

et al.

[2],

that a copula induces a unique doubly stochastic probabilitymeasure ttcon

[0,

1]

viatheassignment

asthemeasureofarectangle. Conversely, foranydoublystochastic measure#, thereis aunique

copula

C,

definedvia

c.(,)

(10,l

I0,l).

Thus, doublystochastic measures on

I0, 1]

2 _{and copulas are in one to one correspondence.}

An

element C E span

(C)

inducesafinite signedmeasure on

[0,

112

viathe definition

(2.5).

Alllinear combinationsof doublystochastic measures areobtained in thismanner.

It is also well known, cf. Italmos

[3],

that for anyfinite signedmeasure _# thereexist mea-surable setsE + andE- such that E +NE-

,

E +t_lE-

[0,

1]

2_{and, for allmeasurable sets}

F,

#(E

+C)

F) _>

0and

#(E-

N

F) _<

0. Itfollows that

#+

and#-defined by

.+(r)-

.(E

+

r)

#-(F)--#(E-NF)

are measures.

Furthermore,

#+

and /z- are unique, though the sets E + and E- need not be. The decomposition #

#+

it- ofa finite signed measure as the difference oftwo measures is called the Jordan decomposition. The set of finite signedmeasures is closed under real linear combinations,and it isaneasy exercisetoshowthat

defines anormonthe set under which the set isa Banach space, Halmos

[3,

p. 123,

Ex.

(4)].

For A

E span

(C)

wedefinethe "Jordan"

(or

total

variation)

norm

We will show in Section 5 that the Jordan norm dominates, but is not equivalent to, the uniform normand isdominated by, butisnot equivalent to, the Minkowski functional.

We

will alsoshow that span

(C)

isnot complete with respect to the Jordannorm. The continuity

(or

lack

thereof)

of the product in thetopologyof the Jordannormisanopenissue.

3.

MINKOWSKI NORM.

We

willshow that span

(C)

is aBanach algebraunder the Minkowskinorm. Theproofwill be by way oftwo preliminary lemmas and two theorems. Then we will state andprove some miscellaneousfactswhich areusefulincomputing theMinkowski normofanelement of span in somespecialcases.

LEMMA

3.1. The Minkowskinormdominates theuniform

(W

,)

normonspan

(C).

PROOF.

Observethatif

A

sB

tC,

where s,

>_

0and

B,

C

C,

then

IlAIIo,o

<

[IBIIo,o

+

tllCIIo,

+

since the uniform norm ofany copula is 1. Thus,

IIAII0,o

is a lower bound ofthe set whose

greatestlower bound isthe Minkowskinorm. The desired conclusionfollows. | The infimum in the definition

(2.2)

of the Minkowski norm is actually achieved:

421

PROOF. By

theproperty oftheinfimum, there existnumberss,,

tn _>

0and copulasBn,

Cn

such that A s,B,- t,C, and s,

+

t,

IIAIM.

But

B,(1,1)

C,(1, 1)

for all n, so s.

tn

A(1, 1)

is aconstant

(and

thus

convergent)

sequence. Itfollows thatthe sequences and

t

both converge; call their limits s and t. Furthermore, sinceC is compact in the uniform topology, there is an increasing sequence ofindices nk such that

B,

and

C.

both converge uniformly;call theiruniformlimits B and C. Clearly A sB tC.

THEOREM

3.1.

Span(C)is

complete under

[[-

JIM.

PROOF.

Thisisaclassicalproof, adaptedtothecaseof copulas.

Suppose

that

An

isCauchy withrespectto

[[-JIM.

Passingtoasubsequence, if needbe,assumethat

[[A,+i-A[[M

2-(+)

for alln. Using Lemma 3.2,we canfindnonnegative numbers

sn

and

t

andcopulas

B

and

Cn

such that

An

snBn

tnCn

and

sn

+

tn

]]AnM.

Observethat

(by

the triangle

inequality)

the sequence of numbers

]]An]]M

sn

+

tn

is Cauchy,

and obsee also that

s-

t

A(1, 1)

is Cauchy, since by

Lemma

3.1 the Minkowski norm dominates the uniformnorm. It follows upon

tang

sums and differences that both

s

and

t

are Cauchy; call their limits s and t. Since C is compact, we can find an increasing sequence nk ofindices such that

Bn

and

C

both converge uniformly; call their limits B and C. Set

A- sB- tC and observe thatA span

(C).

Clearly

An

A

uniformly, and

A

A uniformly follows easily.

To

completetheproof,obsee that we can write

A-AI+(A+I-A).

k=l

This istrue pointwise,by the argument above. But also

IIA

A.IIM

--II

(A+t-

A)IIM

2

-(+1.

hus,

I1

IIM

0, and span

(C)

is completeunder

IIM.

I

The followingtheorem completes theproofthat span

(C),

togetherwith the product,is Banachalgebra.

It

isnot obvious, but definesanassociative binary operation on

C, Darsow

et al.

[2],

and assoeiativity fortheextensionof toall of span

(C)

follows,

so weneednot address this issue.

THEOREM

3.2.

For

allA1,

A2 e

span

(C),

IIA

*

A2IIM

[[AtIIMIIA2IIM.

PROOF.

Wte

Ak

skBk- tkCk

where sk,tk 0, Bk,

Ck

C and sk k--1,2. Then

tt2

C2)

sis2

B1

*

B2

C1

*

A

A2

(ss2

+

tt2)(ss2

tlt2 8182

+

tit2

st2 tls2

C

B2

+

B

(ts2

+

st2)(tls

lt2

tls2

+

sit2

(ss2

+

tlt2)B’-

(ts2

+

st2)C’.

The last expression defines

B

and

C;

clearly both are copulas, since C is closed under the

productand underconvex combinations. Itfollows that

422 W. F. DARSOW AND E. T. OLSEN

This completes theproofof the mainresult concerning the Minkowski norm. Observethat continuity of the product in each place, and also joint continuity, follow immediately from Theorem 3.2. We turn to the computation of

IIM

in specialcases.

THEOREM

3.3. IfB and C areany two distinctleft

(or

right)

invertiblecopulas, ands and

areanynonnegative real numbers, then

IIsB

tell

M s /t.

PROOF. Suppose

that B and C are left invertible and B

#

C, and s,

>

0. Clearly

limb

ClIM

<

+

.

Since B and C are left invertible,

B,1

and

C,1

are 0or almost everywhere,

Darsow

et al.

[2].

It

is easy toseethat the measureof the set where the first partial derivative

B,1

is is

1/2,

and similarly for themeasureof thesetwhere

C,1

1. Since B

#

C, it follows that the sets

F

(x,

y)

lB,(x,

y)

and

C,1 (x,

y)

0}

F2

{(x,

y)

B,i(x,

y)

0and

C,(x,y)

1}

have

(equal)

nonzeromeasure. Now suppose thats

,

>

0 and

D,

EEC aresuch that

sB tC

sD

tE.

Take partialderivativeswithrespecttoxandevaluate at points

(x,

y)

E

F1

where thederivatives of allfour copulas B, C, D andE exist

(this

is almost all points of F1,

[2])

to obtain

s

sB,l(x,y)- tC,

l(x,y)

s’D,l(x,y)- t’E,l(x,y))

<

s’

using the fact that the first partial derivatives ofany copulaliein

[0,

1],

wherevertheyexist,

[2].

It

follows thats

<

s’.

By

similarargument, evaluatingonF2,

<

t’.

Thuss

+

<

This completes theproofforleftinvertiblecopulas. The corresponding result forright invertible

copulas follows bytaking transposes. |

It

follows from the foregoing result that if B and C are any two distinct left

(or

right)

invertiblecopulas, then

lib

-CIIM

2. The topology of the Minkowski norm, restricted to the group ofinvertiblecopulas, isaccordingly thediscretetopology.

The next result concerns the Minkowski norm ofdifferences ofcopulas which are ordinal sums. The ordinalsumconstructionis asfollows:

Let

{(a,, bn)}neZ

bea partitionof

[0,

1],

that

is, a finite or countable set ofdisjoint open intervals the closure ofwhose union is the whole interval.

To

eachn

e

2", assignacopula

A,.

Defineafunction

A

[0,

1]

2

[0, 1]

_via

-

-

-

-

(,)

[.,

.12

A(x,y)

a,

+

(b.

a,)A,(b,

an’

bn

a,

M

(x,

y),

otherwise.

ThenA isacopula, Darsowetal.

[2].

A

iscalled theordinal sumof the copulas

An

withrespect

tothe given partition. Wewillwrite

A

@,ezAr,for theordinalsum.

THEOREM

3.4.

Let

A

,e2"A, and B

,ezB,,

be copulas which areordinal sums withrespecttothesame partition of

[0, 1].

Then

IIA

sll

M max

[IA,

BnllM.

PrtOO.

w

,m

pro

t

rut

wn

Z-

{,}

n

(,)

(0,

a)

nd

(, )

(,)

forsome

,

(E

(0, 1).

The general result thenfollowsviaaninductiveargument.

Thus, suppose thatA1,

A, B1

andBg arecopulas andwrite

c(-,_

-),

o

<

,

<

c(,)

c

c(,)

+

(

)c(

x

1_),

1_))

,X<x,y<l

(3.1)

whereC AorB.

Suppose

A1-

BI

sD1-tD2

wheres,

>

0, D,

D2

e

Cand

[[A1-

Bt[[M

s+t"evaluate at

(1, 1)

andconcludethatnecessarilys t. Similarly,suppose

A2-B2

uEl-vE2

where u, v 0, El,

E2

E C and

JJA2

B2JJM

u -t-v" evaluate at

(1, 1)

and conclude that necessarily u v.

Suppose

thats

>

u

(the

proofis similar ifs

u)

andset

U 3 I1

F

-E

+

5I, k 1,2.

8

Then

A2

B2

sF1

sF2.

It follows that

A- B sDi O

Fi

sD2

F2

so that

IIA

BIIM

_<

2s- _maxk=,2

IlAk-

BalIM.

The opposite inequalityutilizesaconstructionwehave occasionally found usefulintryingto work with the Minkowski norm.

Suppose

that

A

B sD

rE,

wheres,

_>

0,

D,

E EC and

IIA

BIIM

s

+

t; evaluate at

(1, 1)

and conclude that s t; evaluateat

(x,A)

and

(A,y)

and conclude

(using

thefact that, by

(3.1),

A

and Bagree at

(x,A)

and

(A, y))

D(x,

)

E(x,

)

for allx

(3.2)

D(,

y)

E(,

y)

for ally.

(3.3)

We

want to show that

IIAk- BkllM

<_

2S, k 1,2. Wewill show that the result holds for k 1; theproof fork 2 issimilar.

Observe first that

O

(D(x

1y))-

D,(1x

1y)

<

Oz

o

D(

//- D,=( /

<

0--;(

Itfollows that if for any

>

0weset

(x)-

((l+e)x-

(,x,A))/

+e--D(A,A)

thefunctions4) and arestrictly increasingon

[0, 1].

(Observe

that necessarily

D(A, A)

_<

,k; this is a basicproperty of copulas, Darsow et al.

[2].)

To complete the construction, define forall

(,)

[0,

1--D(Ax,

Ay)

b

(, )

()()

+

E(,)

k

(,

)=

()()+

It

is readily verified that

D1

bl/(1

+e)

and

E1

/1/(1

+e)

arecopulas;monotonocityfollows fromthe fact that

D

andE arecopulas and thefactthat and are increasing; theboundary conditionsfor

D1

areverifiedbydirect calculation,and those for

E

follow from

(3.2)

and

(3.3).

Thedesiredresultnowfollowseasily.

For

all x, yE

[0,

1]

wehavebyconstruction 1

B

A(x,y)- B(x,y)

1--A(Ax

Ay)-

(.kx, Ay)

(using

(3.1))

s_

_E

-DS

(Ax,

Ay)

,

(Ax,

Ay)

424 W. F. DARSOW AND E. T. OLSEN

Thus, forany

>

O,

IIA,

BI[IM

<_

(1

+

e)2s

(1

+

t)[IA

BI[M.

The theorem follows.

I

Since span

(C)

is aBanachalgebraunder

I[" I[M,

the Minkowskinorm isanaturalone,and theissueof howtocomputeit becomes important. Wewillreturntotheissueofhow tocompute the Minkowski normagainin Theorem 5.2.

4.

SOBOLEV NORMS.

Wewill workwith theSobolevseminorm

]A ]t,p

givenby

[A[,p-

([A,t(z,y)[+[A,2(x,y)[’)dxdy

pe

[1,o)

(4.1)

ess

sup[A,i(x,y)[,ess

sup[A,2(x,y)[},

p

THEOREM

4.1.

For

all p

{1, ol,

I"

Ii,p

isanorm onspan

(C)

equivalent to

I1"

IIx,-PROOF.

This follows from the fact that an element A span

(C)

necessarily vanishes on the lower and left boundariesof

[0,

1] 2.

The details of theproofcanbefoundin Adams

[1].

One

ofourprincipal resultsisthat the product restricted toCis continuous in eachplace

and jointlycontinuousin thetopologyof

11,,

forp

[1,

oc).

Whetherornot the corresponding result holdsforp o is an openissue,andwhatcanbesaid about continuityofthe product onallof span

(C)

is alsoan open issue. Another principal result is that C is aclosed subset of Wi’’_forallp

[1, o],

so that Cisa completemetric space with respect toeach of theSobolev

norms considered. Puttingthe principal results together, we have that C is a complete metric space under

It,p,

forp

[1,

o)

and that theproductbehaves well withrespect tothe metric. The utility of the Sobolevnormsonspan

(C)

is much lessclear, both because the behavior of the product is not known, as notedabove, and because, as we shall show, span

(C)

is not

complete with respect to any of the Sobolev norms. This latter result will be proved by way

ofaresult ofindependent interest theSobolev normsonspan

(C)

are dominated by, but not

equivalent to,the Minkowskinorm.

LEMMA

4.1. The topologies inducedonCby thenorms

[1,

coincideforallpE

[1,

o).

PROOF. We

willshow that for anyp

>

1, thetopology induced by

Ii,p

coincides with thetopologyinducedby

ll,1.

Since the first partial derivatives ofacopulaliebetween0and almostsurely,Darsowet al.

[2]

wehave

IB-

Cll,v

<

IB-

CI

1/’_{forany copulasB andC Also,}

I,I

by by Hhlder’s inequality,

IB

-C]t,

_< [B

-Cll,.

It

follows thata ball inC in thetopology of

11,1

necessarilycontains aballinthetopologyof

ll,

andvice versa. |

Thetopologyof

]l,o

restrictedtoCis different fromthetopology of

I,l.

Toseethis,

let

Bn

denote theordinal sum

nxy, 0

<_

x,y

<_ l/n

Bn(x,y)

min{x,y},

otherwise

(4.2)

Thenby direct calculation

2

lB.

Mll,

3-

0 whereas

[B.-Mll,oo=

for alln.

Roughly, the topology

of[

[1,,

for p

[1,

o),

is the topology of pointwise convergence of derivatives, since pointwiseconvergenceofderivatives implies convergence in thesenseof

ll,,

forp E

[1,

o),

by the dominated convergence theorem, and convergence inthe senseof

ll,

implies the existenceofasubsequencewhose first partial derivatives converge pointwisea.e. The

topology of

I,o,

onthe otherhand, isthetopology ofuniform convergenceofderivatives.

For

B ECand

f

E

LI([0,

1])

define

425 It iseasy toverify that for

f

e

L

,

B;f

is Lipschitzcontinuous withLipschitz constant and thusis absolutelycontinuous.

d

B’fl

<

d

LEMMA

4.2. Forall B Cand

f

LI([0,

1]),

I-z

--dyB;lfl

a.e.

PROOF.

Sincez

B,9(z,

t)

isalmost surely nondecreasing,

Darsow

et al.

[2],

z

B;f

is nondecreasing whenever

f

isnonnegative. The desired result follows immediately from the fact that for any

f

L1,

Ill-

f

and

Ill

+

f

areboth nonnegative. |

THEOREM

4.2. The product onC is continuousineach place and also jointlycontinuous in thetopology of i,p for allp

[1,

o).

PROOF. By

Lemma4.1, itsuffices to considerthecasep 1. Let B,, B and C becopulas,

and suppose

IB-

Bnll,1

0. Weshow first that

IC.

B-C.

Bnl,l

0and

IB

.C-

Bn *CI1,1

0. This will prove that is continuousin eachplace.

Wewillshow that

II(B

C),1

-(S,

C),1110,1

0 and

II(C

B),I

(C

B,,),1]10,1

0. The

prooffor partial derivatives with respect to thesecondargument issimilar.

Given

>

0, let

C([0,

1]

2)

besuch that

IIC,1

[10,1

<

e/3.

C([0, 1]

2)

is the spaceof C functionsontheunitsquare which vanishontheboundary.

We

canfind sucha since

C

is denseinL

1,

e.g. Adams

[1].

Then

[[(B

C),1

(B,

C),1110,1

[xx

(B

B,),2(x, s)C,1

(s, t)

ds[

dt dx

<

Ixx

(B

B,),2(x,

s)dp(s,

t)

ds[

dt dx

+

l

,(, )(

c,)(,

)l

e e

+

Bn,2(x’s)(-C,’)(s,t)dsldtdx

+

S,(z,s)l(--C,)(s,t)ldsdtdz

+

B,,2(z,

s)J(

c,)(s,

t)J

ds dt d.

Here,

weintegratedby parts in the

st

terminthe second expressiononthe right, andweused

Lemma

4.2 toestimatethe lasttwo terms in the expression. Since 6C

,

we canperformthe integration withrespecttoz in the last two terms toobtaintheestimate

+

2

I(

c,)(, t)l

II,tll0,lB

Blt,t

+

/3,

Choose N solargethatn

>

N implies

Thenforall such n,

426 W. F. DARSOW AND E. ]7. OLSEN To complete theproof, observethat

II(C

B),-(c

Bn),ll0,-

[zz

C,2(x,s)(B-

Bn),t(s,t)dsldtdx

<

C,2(x, s)l(B

B,,),

(s, t)]

ds dt dx

NIB-

B,,l,.

Here,

we used

Lemma

4.2 and performed the integration with respect to x.

It

follows that

II(c

*

B),

(c

B),ll0,

0.

Thus, is continuous in eachplace with respect to the topology

of[

[,p

forp Observe that the argument just given shows that foranythree copulas B, C and

D,

II(c

*

B),- (C

D),llo,

<

Is

Dl.,.

(4.3)

Wewill make use of this observation toshow joint continuity.

To

that end, let B,,

B,

C, and C be copulas, and suppose that

IB

B,,l,l

0and

IC

C,

ll,

0. Wewant to show that

IB

,C B,,*

Crl,

0. WriteB ,C

Bn

,C,-

Bn

_*

(C

-C,,)

+

(B

B,)

,C.Then

II(B

*

C),

(B.

C.),llo,

_<

II[B.

*

(C

C.)l,llo,

+

II[(B

B.)

C],llo,

<_

IC-

C.l,

+

I1[(- B.).

Cl,llo,.

We have used

(4.3)

here. he firs

erm

on he

rih

oes

o

zero byhypothesis and he second term goes tozerobythe onesided continuity of the operation proved above. Thus,

II(U

.C),l-(Bn

Cn),ll0,1

0.

By

an analogous argument,

II(S

C),2

(B,

Cn),2[10,

0. Thus,

IB

*C B,,

C,,I,

0, and joint continuityis established. |

We

turnnowtocompleteness issues.

THEOREM

4.3.

For

all p

[1,

oo]

thetopology

of[

I,p

restrictedtoC isstronger than the

topology of uniform convergence.

PROOF. For

p

>

2,

I1"

II,p,

and thus

I"

I,p,

dominates theuniform norm

II0,oo,

by the

Sobolevembeddingtheorem, Adams

[1].

Thus,

for eachp

>

2,there isaconstant

Cp

such that

IIAII0,oo

_<

CIAI,

(4.)

for all

A

span

(C).

This implies the desired result, forp

>

2. Since by

Lemma

4.1 the topologies of

[,

restricted to Ccoincidefor p

[1,

oe),

the conclusion of thetheorem holds also forpE

[1, 2],

though aninequalityoftheform

(4.4)

neednothold forpE

[1, 2].

|

Observethat the topologies of

I,p

arestrictlystrongerthan that of

II0,o,

sinceif the topologies coincided, the product would be jointly continuous in the latter topology. This is not the case, however.

Thereisasharperstatementwhichimplies Theorem 4.3 and whichwewillmakeuseoflater:

THEOREM

4.4.

For

all p

(1,

oe)

theweaktopology of W

,

restrictedtoCcoincides with the topology of uniform convergence.

PROOF.

A

sequence of copulasis weakly Cauchy in thesense ofW’p _{if and only if it is} uniformly

Cauchy,

and the weak and uniform limits are thesame. This follows easily from the

fact that C is a compact subset ofL and from the fact that the unit ball in L

(which

contains both first partial derivatives ofany

copula)

isweakly sequentially compact. |

THEOREM

4.5.

For

allp

[1, oe],

Cisacompletemetricspace under the metric

427

PROOF.

This isacorollary of Theorem 4.3.

Suppose

B isintheclose,reof(7in W

1.p,

and let

U,6Cbe such that

IB-

B,ll,p

0. Wewant toshowB C.

By

Theorem4.3

lIB-

Bn[[0,

0 andtheuniform limit ofasequenceof copulasisacopula, so BG

.

[

Weturnnow to considerationof theSobolevnorms on allofspan

THEOREM

4.6.

For

p

[1, 1,

the Minkowskinormdominates thenorm

[. [,p

onspan

(C).

PROOF.

Sinceforp G

[1,

),

A,

2/]A,,

it

suces

to establishthe result forp Since thefirst partial derivatives ofa copulalie in

[0,

1]

a.s., the result forp follows viaan

argumentsimilar to thatofLemma 3.1.

Toinvestigate theissue of whetherspan

(6)

iscomplete underanyoftheSobolev norms,we usethefollowingeasyconsequence ofthe open mappingtheorem

(Rudin

[5]):

LEMMA

4.3. Let B be a Banach space under each of two norms. If one ofthe norms dominates theother, thetwo norms areequivalent.

It follows from Theorem 4.6 and Lemma 4.3 that span

()

is complete under one of the

Sobolev norms if and only if that norm is equivalent to

JIM.

A

counterexample given in

Example .1 below shows that thenorms are not equivalent.

By

Theorem 4.6 and

Lemma

4.3, therefore, span

()

is not complete under any of the Sobolev norms. We canalways work with the closure ofspan

(C)

in W

,,

ifwewish, but the subspace span

()

itselfis not closed.

Before leaving the Sobolev norms, wecomment on the diculty of extending Theorem4.2,

concerning continuity of the product, toallofspan

().

Sinceasnoted above thenorms

JIM

and

,

are not equivalent, thereexist sequences

An

sB, t,C,which are Cauchywith respect to

,

for which

s

+

t

is an unbounded sequence. This cannot be accomodated withintheframeworkof theproofwehave given of Theorem 4.2. If, therefore, the product is continuousonallof span

()

inthetopolo

of[

[,,

it willhave to beshownbysomemodified

orcompletely different line ofproof.

5.

JORDAN

NORM

Wecansay less about the Jordannormthan about thenormsconsideredabove. The question

ofhow the product behaves with respect to the topology ofthe Jordan norm is open. The principalinterestofthenormis thatit sheds lighton the behaviorof the othernorms.

THEOREM

5.1. The Jordannormdominatesthe uniformnorm onspan

()

andisdominated

bythe Minkowskinorm.

PROOF. Let A

span

().

Write Aforthe finite

sied

measureinducedby

A

on

[0, 1]

,

write _A for its Jordan decomposition, where

(F)

A(F

E),

and write

IAI-

+

].

Then

]]m]]j

]#A[([0,112).

Observethatforany

(x,

y)

6

[0,

1]

2wehave

IA(x,y)l-

IA([0,

x]

x

[0,])1

<

IAl([0,

x]

X

[0,1)

<

[IallJ.

Take the supremumover

(x,y) e [0,

11

and conclude that

IIAII0,oo

<

IIAIId.

Next,

observe that if

A

sB tC, wheres,

_>

0, B, C

e

C and

[IAIIM

s-}-t,wehave

la([0,

1]

2)

#A(E

+)

SlaB(E

+)

--tlac(E

+)

<_ SlaB(E

+)

<_

s

la([0,

1]

2)

----laA(E-)---SlaB(E-)

+

tlac(E-)

<_

tlaB(E-)

<_

using the fact that lab andlac are probability measures and that #A SlaB

tlaC. Sum

the

foregoinginequalities and conclude that

I[A[Ij

<_

s

+t-[JAIl

_M. |

We havearepresentationofthe Minkowski normwhichmakes useofthe Jordan

428 W. Fo DARSOW AND E. T. OLSEN

THEOREM

5.2. Let A E span

(C),

let #A

#-

#

be the Jordan decomposition of the finite signed measure induced by

A,

as above, and set

A+(x,y)

([0,

x]

[0,

y]),

and

A-(x,y)

#([0,

x]

[0,

y]).

Then

IIAIIM--

max{llA

+

,1(’,

1)11,

IIA,2(1,

+

")11}

+ max{liAr(’,

1)]],

]IA(1,.)I]}.

PROOF. A

+ and

A-

are continuous monotonicfunctions satisfying

A+(x,O)

A+(O,y)

A-(x,O)

A-(O,y)

O.

These facts are easy to verify. They differ from copulas in that they satisfy different boundary conditions ontheupperand right boundaries of the unit square.

If

A

sB tC where B, C EC ands,

_>

0, and

IIAI]M

s

+

t, then forxl

<

x2

using the Lipschitz conditionsatisfiedby the copula B. Sincesectionsof

A

+ arenondecreasing, first partial derivatives exist almosteverywhere, and theforegoinginequalities imply

0

<

A+(x,y)

<

8 a.e.

(5.2a)

By

similar reasoning,

0

<_

A+,2(x,y)

<_

s a.e.

(5.2b)

0_<A(x,y)_<t

a.e.

(5.3a)

0_<A(x,y)_<t

a.e.

(5.3b)

We

usethe notation

"a.e."

in thestatementsaboveasshorthandforthe more precisestatements that, forally,

(5.2a)

and

(5.3a)

hold for almost allx andthat, forall x,

(5.25)

and

(5.3b)

hold for almost ally. It followsfrom

(5.2)

and

(5.3)

that

IIAIIM

s

+

dominatesthe right hand side

of

(5.1),

which in particular must be finite.

Toobtaintheconverseinequality, wemakeuseagain of theconstructionutilizedintheproof ofTheorem 3.4. Observe that, since

A-

A

+-

A

and

A

is linear on the upper and right boundariesof

[0, 1]

2,

we can write

1

(A-(x2, 1)

A-(xl,

1))

(A+(x2,

1)

A+(xl,

1))

A(1, 1)

x2 x x2 x

1

(A-(1,W)

A-(1,Vl))

(A+(1,W)

A+(1,Vl))

A(1,

1)

Y2 Yl Y2 Yl

sothat also

max{ess

sup

{A

5

(x,

1)},

esssup

{A(1,

y)}}

max{ess

sup

{A,+(x,

1)},

ess sup

{A,(1,

y)}}

A(1,

1).

The right hand sideofthis expressionisnecessarily nonnegative.

Let

For any

>

0, set

Then and

p

arestrictly increasingfunctions,and, by reasoningsimilar tothatused in theproof

of Theorem3.4,both B and C arecopulas. Since byconstruction

A

(

+

e)B

(

+e

A(1,

1))C

itfollows that

IIAIIM

_<

2

+

2e-

A(1,

1).

Since

>

0isarbitrarily small,weobtainfinally

IIAIIM

<

2-

A(1,

1)

<max{llA

+,1

(’,

1)11oo,

IIm,2(1,

+

")11oo}

+

max{

A

#("

1)11

IIA

(1,

)11

}

using

(5.4)

and the definitionof

.

Thus, the right handsideof

(5.1)

dominates

IIAIIM.

|

Wenow give twoexamples.

In

the first,weconstruct asequence of elements of span

(C)

on which the norms

I1"

I1’

and

I1"

IIM

are the same, and both diverge to

+oo,

whereas the norm

I1,

isbounded.

In

thesecond, weconstructasequenceofelements of span

(C)

forwhichthe

norms

IIJ

and

I1"

II0,oo

areessentially the same, and both converge tozero, whereas thenorms

I"

11,oo

and

I1"

IIM

diverge to

+oo.

Weshall then commentontheexamples.

EXAMPLE

5.1.

Let

n be aneveninteger. Partition

[0,

1]

2 _{into n}2 _{disjoint squares ofside}

l/n:

letxk Yk

k/n,

k 0,...,nandwrite

l<_j,k<n j=n, l<_k<n k--n,l<_j<n j--k--n

Definea finite signed measure #r on

[0,

1]

2 via theconditions that

#,.,(Rjk)

(-1):+k/n

and

thatthemassof#, bespreaduniformlyoneachRjk. Thatis, definett,via

#n(F n

Rj,)

(-1)J+}A(F

n

R3k)

n(_l)+kA(

F

N

Rjk)

,(R)

foranymeasurable setF,whereA denotes Lebesguemeasure on

[0, 1] 2.

TheJordan decomposition oftt, isasfollows"

Set

E + Uj+kevenRjk E-

U3+koddR3k.

Then

#+(F)

#,(F

VE

+)

and

#-(F)

-tt,(F n

E-).

Observethat

nx

,

([0,

x]

x

[0,

11)---

(5.6)

ny

t+(l

0’

11

[0,

-430 _{W.F. DARSOW AND E.} T. OLSEN so that

B+

definedby

B+(,

)

2

-.

_n +

(10,

l

10, 1)

is acopula. Equations

(5.6)

and

(5.7)

aresatisfied with

#

inplaceof#n, so+ that

B(x,y)-

-2([O,

xl

[0,

yl)

n isa copula. IfwedefineA,via

A,(x,y)

-/n([0,]

[0,

y])

wehave

n ₊ n

An

B

-B-

e

span

(C).

Itiseasy toseethat

I/Znl-

hA,sothat

IIA,IIj

n. Usingformula

(5.1),

togetherwith

(5.5)

and

(5.6)

andtheanalogous formulae

for/,

weobtainalso

IIAnlIM

n. Wewishto compute

IA,ll,o.

For

(x,

y)

R3k

wecalculate, using

(5.5)

-

(-1)

e+’

+

(x

a:_l)

(-1)

+’

=1rn=l rn=l

3--1

+

(y

yk_)

(-1)

t+3

+

n(x

xj_)(y

yk_:t)(-1)

a+k =1

It

follows thatfor

(x, y)

in the interiorof

Rk

wehave

k--1

[A,l(x,y)[--[

-(-1)

j+m

+

n(y-

yk_l)(-1)+[

<

2

since

[y

yk_l[

< 1/n

and

k--1

(-1)J+’n=-1,

0or1.

rn--1

Similarly,

[An,2(,

y)[

_<

2 forall

(x,

y)

interiortooneofthe

Rjk’s. It

followsthat

IAI,

<

2. Thus,asnvaries overtheevenpositive integers,weobtainasequenceofelements in span

(C)

whose Minkowski and Jordannorms areequal, and diverge to

+x,

but whoseW1,o _{norms are}

uniformlybounded above by 2.

EXAMPLE

5.2.

Let

Bn

be the ordinal sumof

(4.2)

above,

and define

An

snM

tnBn

whereSn

tn

n

a,

forsome positive numbera. Thus,

x

>_

lln

ory

>_

lln

O<_x<_y<_l/n

Observe that

An

span

(C)

for alln. Wecalculate

IIAllo,

IAl,oo,

IIAIIJ

and

IIAIIM-An

easycalculation shows that

It

is easy toverify that the maximum valueof

[A(x,y)l

occursat

(x,y)

(1/2n, 1/2n),

sothat

IlAllo,

-/4.

(5.10)

431 L,

{(x,x)10

<

x

< l/n};

and write

A2

for Lebesgue measure on

[0,

1]

2 Then it is easy to verify that the Jordan decomposition of

,

#A,. isgivenby

+(F)-nAI(FML,)

#(F)

na+lA2

(F n

[0,

l/hi2).

It

followsthat

It

follows also that

IIA,IIJ-

I#,l(I0,

112)

2

n’:’-’.

(5.11)

n+([0,

x]

x

[0,

1])=

{

n

-l,nax’

otherwiseX

e I0, /n]

eI0,/n]

n+([0,1]

x

[0, y])=

na_

otheise

Identical formulaeareobtainedfor $.

It follows,

using

(5.1),

that

IIAnIIM

2n

a.

(5.12)

Nowfixa

e

(0, 1).

Observethatby

(5.10)

and

(5.11), IIAll0,

and

IIAnlIJ,

whichdifferbya constant factorof8,both converge to0,whereas by

(5.9)

and

(5.12),

IAnlt,

and

IIAnIIM,

which differbyaconstantfactorof 2, both diverge to

+.

We

turn to theinterpretation ofthese twoexamples. Since the Minkowskinormdominates the Jordannorm, byTheorem5.1, and the twonormscannot be equivalent, by Example5.2, we obtain, by wayof

Lemma

4.3:

THEOREM

5.3.

Span

(C)

isnotcompleteunderthe Jordannorm.

Obseethat Example5.1 completestheargument ofSection4 to similareffect: span

()

is

notaclosed subspace of W

,

for anyp

[1, ].

Finally, obsee that in viewof Example5.2itcannot be true that

II"

Ila

dminates

I"

I,

and that in viewof Example5.1 it cannotbetruethat

I"

It,

dominates

IIg.

6.

ONE

PARAMETER

SEMIGROUPS OF COPULAS

We

say

Ct,

O,

isa oneparametersemigroupif

Cn+t

Cs

Ct

forall s,t.

Obsee that

Ct

*

Co

Co

*

Ct

Ct;

setting 0 we observe that

C0

is necessarily

idempotent.

Thus,

if

Ct

is asemigroup,

C0

is anidempotentwhichcommuteswith

Ct

for

allt.

If

forsome norm

limt0

IIct

c011

0 andthe productis continuous in eachplaceinthe topolo ofthenorm,then

lim

IICt

Co

lim

IICt-to

Cto

Cto

IIC0

Co

co

0.

o

Thatis, continuityat 0 implies fight continuityeye,where.

In

fact,

theexistenceofaright limit at 0implies right continuityeverywhereon

(0,

),

for reasonablenorms:

THEOREM

6.1.

Suppose

isanormon aspace containing span

(C),

and suppose that the product ong is continuous in eachplacein thetopolo ofthe norm and also that norm convergence implies pointwise convergence. Let

C

be asemigroup ofcopulas, and suppose there isafunction

E

such that

lim0

IIc

Ell

0. Then

E

isidempotent,

E

C

C

E

C

forall

>

0and

lim

IIc

coll

0_{for ll 0}

>

0,

PROOF.

The

proof

isanadaptationofaclassical

proof

tothecaseof copulas.

432 W. F. DARSOW AND E. T. OLSEN

Since for

> to

> O,

Ct

Cto

_*

Ct-to

Ct-to

Cto,

onesidedcontinuity of implies lim

IIC,

c,0

’11

0

tl,to

lira

]]Ct-

E

Ct0][

O.

Thus, rightlimits exist everywhere, andalso

limtl.to

Ct

Cto

_*E E

Cto

for all

to

> O.

Itremains toshow that

Cto

Cto

_*E

E,Cto.

Itsufficestoshow that

Ct

isrightcontinuous at asequenceofpoints

tn

converging to

O,

since if

Ct,.,

Ct,,

*E E

Ct,,

and

> tn

then

Ct

Ct-t,,

*

Ct,,

Ct-t,,

*

Ct,

*E

Ct

*E.

Fix

(x,

y)

E

[0, 1]

2,

and define

f: [0, cz)

[0, 1]

via

f(t)

C,(x,

y).

Observethatsince norm convergenceimplies pointwise convergence, right limits

limttt0

.f(t)

exist everywhere. It follows by a classical argument

(an

outline of the argument is given by Hille and Phillips

[4,p.

282])

that the set

7)(x,

y)

ofdiscontinuities of

f

is countable. Now let

(x,y)

vary over the rational pairs

(p,

q)

E

[0, 1}

2,

andset 7)

UT)(p, q).

Then 7)is countable. Since copulasare all Lipsehitz continuouswith Lipsehitz constant 1, Darsow etal.

[2],

hence equieontinuous, itfollowsthat

lim

Ct

Cto

tto

pointwise for all

to

t

7). Since

(0,

cz)

\

7) includes a sequence converging to 0, we have right continuity everywhere.

That E is idempotent now follows easily from thefacts that E

Ct

Ct

*

E

Ct

for all

>

0, that

limtt0

Ct

E and that the productiscontinuous ineach place. |

Note

that Theorem6.1 requires only that the productbe continuous in eachplaceand that it isthe productrestrictedto

C,

ratherthanonallofspan

(C),

forwhichcontinuityisrequired.

Accordingly,we cantake any of

I1"

I[0,oo,

I"

It,v,

P

e [1,

cx)

or

]].

]]M

forthenorminTheorem6.1,

sincethe product is continuousin eachplace with respectto eachof these norms, and sincefor

eachnormconvergence implies pointwise convergence. The theoremthen states that asemigroup

Ct

for whicharight normlimit exists at 0is necessarily right continuouseverywhere in the senseofthenormtopology. Wedo not know whetherwe cantake

Ila

or

{.

11,oo

forthenorm in Theorem 6.1, since we do not know whether the product, restricted to C is continuous in eachplacein thetopologyof thesenorms.

Note

alsothat,ifwe arewilling toassumethat

limt/0 Ct

exists,there is nolossofgenerality in making the stronger assumption that

Ct

is right continuous at 0, sinceby Theorem 6.1, wecanalways redefine

Co

to bethe limit

E

limq0

Ct

and still haveasemigroup.

We

address next the issueofwhether measurability implies continuity fora one parameter

semigroup of copulas. There is a classical argument that this is the ease for one parameter

semigroups inaBanaeh algebra, Hille etal.

[4,

p.

280].

In

the following discussion, measurability ofa function R

B

where /3 is a Banaeh spacemeans

"strong"

measurability, that is,

f

ismeasurable ifforany compactsubset K C R,

flK

is the limit pointwise almost everywhereofasequence of measurable functions with finite range.

A

functionwithfinite range ismeasurable if the inverse imageofeach point in the range ismeasurable.

We

will makeusebelowofthefollowing facts:

1.

Strong

measurability implies "weak" measurability; a function

f

R /3 is weakly

measurable if

u(f(t))

is ameasurablescalar valued functionforeach u/3*.

2.

Strong

and weak measurabilityareequivalent when the Banachspace/3isseparable, Hille et al.

[4,

p.73].

In

this connection, recall that the spaces W",v _{areseparable for p}

e [1, ),

433 3. (Approximation by continuous

functions:)

Let

P(i)

denote the space of integrable

B-valued functions with domain f C R; the set ofcontinuous Banach space valued functions is densein

P()

for allp6

[1,

cx),

Hille et al.

[4,

p.

86].

Observe that if

f

R Bisright continuous, thenit is measurable. Toseethis, let

/):a--t0

<

tl

<

< tn--b

denoteapartition ofaclosedboundedinterval

[a,

b]

anddefine

f’P

via

t=b

where

I[tk_,,t)

denotes thecharacteristicfunctionof the interval

[tk-1, tk).

Then foranysequence of partitions forwhich themaximum subintervalwidth

[P[

goes tozero,

liml,l_.,0

lift

f[[

0

forallt. Accordingly, ifwemake the assumption that asemigroup

Ct

isright continuous at 0 in the senseofsomenorm, then byTheorem6.1 and the preceding observation,

Ct

is measurable in thesenseofthat samenorm.

The classical argument that measurability implies continuity uses the Banach algebra in-equality. Theproofof the following theorem follows theclassical proofbut replacesthe Banach algebra inequality by amore restrictivecondition.

THEOREM

6.2.

Suppose

Ct

is a one parameter semigroup ofcopulas.

Suppose

also that

Ct

is measurable in thesenseofanormsatisfying the followingconditions:

(1)

Thereis anumber

M1

such that

[[Ct[[

< M1

for all t, and

(2)

There is anumber

M2

suchthat forany s, t, andu

llc

*

cu

ct

c,,ll _<

M21lCs

ctll.

(6.1)

Then

Ct

is continuous on

(0,

x).

PROOF. Let

>

0;wewant to show that

[ICt+,-Ctl]

0as/--,0. Let0

<

a

<

b

<

and

let

I1

<

t-b. Thenforalls 6

[a, b]

wehave

c+,

c

(c+,_,

c._,)

c,

c,

(c+,_,

c._,).

The last term in thisexpressionis constant,hence bounded and integrable, andwehave

(6.3)

Condition

(1)

guarantees that the integrandofthe last integral above is bounded.

By

a result mentioned above, togetherwithsome standardarguments, wecanfind a sequenceofuniformly bounded continuousfunctions

g

converging pointwisetothebounded measurable function

C almost everywhereonthe compact interval

[0, t],

that is, for almost all s,

lies

g[I

0 as n cx. With C replaced by

g’

the integral in the last term in

(6.3)

goes to zeroas r/ 0.

It

follows,

viathe triangle inequality and the dominated convergencetheorem, that

434 W.F. DARSOW AND E. T. OLSEN

Wewillshow below that theSobolevnorm

[.

[1,

satisfies conditions

(1)

and

(2)

of Theorem

6.2,and that as aresult measurability ofasemigroup

Ct

of copulas in thesenseof W

1,P,

for p E

[1,

x)),

impliescontinuityofthe semigroup in thesenseofW

1,p,

p E

[1,

o).

Weareunableto show thatcondition

(2)

is satisfiedby the uniformnorm

[[.

[[0,.

Itisanopen question whether

asemigroup whichis measurablein thesenseof the uniform normisnecessarily continuous. Weestablish the inequality

(6.1)

forp l, with

M2

2. Observe first that

by

(4.3).

It follows that

I(C.

c)

Cl,,,

II((C

c)

c.),llo,

/

II((c.

c)

This yields

(6.1)

in the case of

],.

Observe that since

[0,

]2

is compact, for p

>

1, the W

1,P([0,

1]

2)

normdominatesthe W

’([0,

1]

2)

norm,someasurability in the sense of W1,pimplies measurability in the sense of W

t,t.

Accordingly, if

Ct

is measurable in the senseof

W’,

it ismeasurable also in thesenseof W

,1,

whence, by Theorem6.2, it iscontinuous in thesense of

W’.

Since the topologies of Wl’p _{restricted to C coincide forp}

[1, x),

_{Lemma 4.1, we}

conclude that

C

is continuous in thesenseofW1,_{for allp}

[1, x).

Thus, a one parameter semigroup ofcopulas whichis measurable in thesense

of[

[,

is necessarilycontinuousin thesenseof

],p,

forp

[1,

x).

COROLLARY

6.1.

A

semigroup

Ct

ofcopulaswhichis rightcontinuousat 0 in the uniformnormis continuousfor all

(0, x)

in thenorm

I"

It,p

forallpE

[1,

PROOF. Let

C

satisfy

limt/0

IIc-

c0110,o

0, Then by Theorem6.1,

Ct

is fight continuous everywherein the senseof the uniform norm. If is a continuous linear functional on

W’([0,112),

p 5

(1,

x),

(Ct)is

a right continuous real valued function, by Theorem

4.4, hence measurable, by an observation above. Thus,

Ct

is weakly measurable in the senseof W

’’.

Butweakand strong measurability areequivalent for theseparablespaces W

’P,

1

_<

p

<

oo, so,by Theorem

6.2,

Ct

is continuousin thesense

of]

I,

forall

>

0. |

One

mightreasonablyexpect that fight continuity in thesenseoftheuniformnorm would imply measurability in thesense evenofthe Minkowski norm

IIM,

since on heuristic

grounds one expects reasonable functions to be measurable. Then right continuity at 0 in the uniformnormwould imply continuityforall

>

0 in thesenseof

IIM,

directlybythe classical Banach algebra argument, without worrying about how to obtain

(6.1)

in the caseof Sobolev norms. It is an openissuewhetherthis isthecase. Hille andPhillipsraisethesameissuein the related context of transition matricesofMarkov processes with discrete

(but

not

finite)

range,

[4,

p.

635].

We

point out that measurability ofa Banach space valued functiondepends onthe normused,sothat thecopulavaluedfunctions

Ct

hereofinterestmay be measurableornot, dependingonwhat Banach spacewetakeCto beasubset of.

We

nowturn to the issue of analyticityofasemigroup

Ct.

THEOREM

6.3.

Let

C

beasemigroup in

C,

and supposethat

limit0

[IC

E[[M

0, where

E

Co.

Then thereis an

A

E span

(C),

satisfying

A

E

E

A

A,

such that

C

eAt

forall

>

0, whereeAt is definedby its seriesrepresentation, withtheconvention

A

E.

This is aclassicalresultforBanachalgebras;for aproof,seeHilleet al.

[4].

435 It follows from Theorem 6.3 andsomeother elementary arguments that a semigroup

Ct

of copulas possesses an exponential representation

Ct

eAt for some A E span

(C)

if and only if

limt0

IICt-

EIIM

O,

where E

Co.

This says roughly that the Minkowski norm isa natural

onefor oneparametersemigroups withgeneratorsinspan

(C).

It is by no means true that all one parameter semigroups of copulas possess exponential representations. Wenext exhibit aone parameter semigroup

C

which is right continuousat 0 in thesenseof

II,p

forpE

[1,

cx))

but not inthesenseof

II,o

and hence, by Theorem4.6,

also not in thesense of

IIM.

By

Corollary 6.2, therefore, the semigroup does not possess an exponential representation.

EXAMPLE

6.1.

It

wasshown inDarsowetal.

[2]

that thecopulas ofaWienerprocess have theform

wheres

<

and

v

s

Since the Wiener process is aMarkov process,wehave, fors

<

u

<

t,

C,;

C;t

C,;t.

(6.5)

Equation

(6.4)

implies thefollowingscalingproperty: ifr

>

0 ands

<

then

C,Ir;tlr

Cs;t-

(6.6)

Itfollows from

(6.5)

and

(6.6)

that the familyof copulas

Bt

definedby

(6.7)

where

Cs;t

is given by

(6.4),

has the semigroup property for

>

0, as is readily verified.

(We

remark thatit is also possible to go backwards: if

B

isanyoneparametersemigroupof copulas

andwe define

C’;

Bin

t

fors

<

t, then the family

C;

satisfies the Markov condition

(6.5)

and also the scalingproperty

(6.6)

ofaWiener

process.)

Wewillshow that

limt0

list

MI],p

0forp(5

[1,

cx))

but not forp Fromthedefinition,wehave

Bt(x

y)

C

eU2G-i(y)

C-(u).

du.

x/et_l

It

follows,

aftermaking asubstitution in theintegral, that

Since

wehave

/_

et/2y

u _{_ua/2}

B,(C(x),

C(y))

G(

v/e

1

)e

du.

et/2y-

u

f

vf,

y

>

u

limC

()

tO y/et-

O,

y

<

u

Jim

Bt(G(x)

G(y))

X(_oo,u](u)e

-’/2du LO

/min{x,y}

e-u2/2du

436 W. F. DARSOW AND E. T. OLSEN

||ence,

lim0

Bt

M pointwise, and sinceCiscompact in theuniform topology, also uniformly.

By

similar arguments,

limtl0

Bt,l

M,1 pointwisea.e. and

limt0

Bt,2

M,2

pointwisea.e. From the foregoing results and thedominated convergencetheorem, weobtain

lim

11Bt

M

[[1,p

0

t0

forp E

[1,

x).

The result does not hold forp oc, however; the partialderivatives

Bt,

and

Bt,2

arecontinuousforall

>

0,but

M,1

and

M,2

arenotcontinuousfunctions,hence not the uniform limitofcontinuous functions.

That

Bt

possessesnoexponential representationnowfollowsfromanobservationmade above

(see

discussionfollowing Theorem

6.3).

Observethat if

Bt

possessedanexponential representation

Bt

eAt then it would be true that

limt.odBt/dt

A. It can be verified directly that the semigroup

B

ofthis example does not possess an exponential representation, since the latter limit is0 except whenx y, when it isinfinite.

7.

DISCUSSION.

We

conclude with some remarks about compactness. Recall that C is a compact subset of W

’

and alsothat the product fails to be jointly continuous in the topology of uniform convergence. The question iswhetherC itselforany useful subset is compact underany ofthe othernormsaddressed. It iseasy toverify that if isanynormwhich dominatesthe uniform norm, and is compact under thetopology of

II,

then necessarily the product onC is not jointly continuous in thetopology of

I[.

This implies thatC is not acompact subset of WI’p forp E

[1,

oo),

sincethe product isjointlycontinuous onCwith respect to thenorms ofthese Sobolev spaces, Theorem 4.2. Moregenerally, this seems toimply that we cannot obtainboth

compactness ofC and joint continuity of the product in asingle topology. We thought atone time that it might be possible to find a topology under which the group

G

C C of invertible copulas was acompact group. Butthis seemsunlikely, for thefollowing reasons:

7

isdense inC inthe uniform topology, Darsowetah

[2],

and thusis dense in the topology of anynormwhich is dominated by the uniform norm.

Compactness

fails because

7

is not closed.

On

the other

hand, if dominatesthe uniformnorm,and

C,

is asequence of invertiblecopulas converging uniformly toanoninvertiblecopula, say

P,

thenif$ werecompactin thetopologyof

II,

there would beasubsequence converging toanelementof

7,

contradiction, sincethen thesubsequence

would necessarily also convergeuniformly to theelementof

G.

A

normwithrespect to which

G

wascompactcould accordingly neither dominatenorbe dominatedbythe uniformnorm.

We

consider itunlikely thatany normsin which the product behaves well will befoundto possess any usefulcompactnessproperties.

REFERENCES

[1] R.A.

Adams, Sobolev

Spaces

(Academic

Press,

New York,

1975}

[2]

W.F.

Darsow,

B.

Nguyen

and

E.T. Olsen,

"CopulasandMarkov

Processes,"

Ill.

J.Math. 36:600-642

(1992)

[3]

P.R.

Halmos,

Measure

Theory

(Van

Nostrand,Princeton,

1950)

[4]

E. Hille and

R.S.

Phillips, FunctionalAnalysis and Semi-groups

(American

Mathematical Society,New York,

1957)

[5]

W.Rudin, Functional Analysis

(McGraw-Hill,

New

York,

1973)

[6]

A. Sklar,

"Fonctionsde r6partition ndimensions et leur

marges,"

Publ.

Inst.

Statist. Univ. Paris8:229-231

(1959)

[7]

A. Sklar,

"RandomVariables,Joint Distribution Functions and Copulas,"