HIGHER ORDER MONOTONIC FUNCTIONSOF SEVERAL VARIABLES

Paul Ressel

Katholische Universität Eichstätt-Ingolstadt

1

1. Introduction

For an interval I ⊆ R and f : I −→ R we consider

(∆hf)(s) := f(s + h)− f(s)

(h > 0; s, s + h ∈ I), ∆2h := ∆h ◦∆h etc., and call f n-absolutely

monotone if

∆pf ≥ 0 for p = 1, . . . ,n (where defined).

Note that f ≥ 0 is not assumed.Let A1, . . . ,An ⊆ R be non-empty, A := A1 × . . .× An , and letϕ : A −→ R be any function.Then for a = (a1, . . . , an), b = (b1, . . . , bn) ∈ A we put

Dbaϕ := ϕ(b)− ϕ(a1, b2, . . . , bn)− . . .− ϕ(b1, . . . , bn−1, an)

+ ϕ(a1, a2, b3, . . . , bn) + ϕ(a1, b2, a3, b4, . . . , bn)

+ . . .+ ϕ(b1, . . . , bn−2, an−1, an)

− ϕ(a1, a2, a3, b4, . . . , bn)− . . .+− . . .+ (−1)nϕ(a) .

2

Definition.

ϕ is n-increasing iff Dbaϕ ≥ 0 ∀ a < b in A ; ϕ is called fully

n-increasing iff ϕ with k of the variables fixed is(n− k)-increasing in the remaining variables, for every choice ofthese variables, and for every k = 0,1, . . . ,n− 1 . Here a < bmeans aj < bj for all j = 1, . . . ,n .

Theorem 1 (Correspondence Theorem).Suppose sup Ai ∈ Ai for alle i ≤ n . Then ϕ : A −→ R+ is fullyn-increasing and right continuous iff

ϕ(a) = µ([−∞, a] ∩ A) , a ∈ A

for some (unique) µ ∈ M+(A), A denoting the closure in Rn.

That is, ϕ is the distribution function (d.f.) of µ .

Note that here Ai is not assumed to be an interval.3

There is a fundamental connection between the two types ofhigher order monotonicity mentioned so far:

Theorem 2.f : I −→ R is n-absolutely monotone if and only if f ◦ ϕ is fullyn-increasing for each fully n-increasing ϕ with values in I.

On I = [0,1] we can say a little more: let f : I −→ I becontinuous in 1 with value f(1) = 1 . Then there are equivalent:

(i) f is n-absolutely monotone(ii) For every n-dimensional d.f. F of some probability measure

on Rn also f ◦ F is a d.f..

(iii) [0,1]n 3 x 7−→ f

(1n

n∑i=1

xi

)is a d.f..

4

A natural question arises: if F is an m-dimensional d.f., G ann-dimensional d.f., for which functions f on [0,1]2 isf ◦ (F × G) again a d.f.?

Until recently the answer was known only for m = n = 1 : f hasto be a 2-dimensional d.f., and a particularly important class ofsuch functions on [0,1]2 are the bivariate copulas. For m,n ≥ 1the function f has to fulfill stronger (multivariate)monotonicity conditions, and these will now be discussed.

5

Let I1, . . . , Id be any non-degenerate intervals in R ,I := I1 × . . .× Id , and let f : I −→ R be any function. Fors ∈ I, h ∈ [0,∞[d such that also s + h ∈ I put

(Eh f)(s) := f(s + h)

and ∆h := Eh− E0 , i.e. (∆h f)(s) := f(s + h)− f(s) =: −(∇h f)(s).

Since the family of operators {Eh} is commutative (wheredefined), so is the family {∆h} . In particular (with e1, . . . , eddenoting standard unit vectors in Rd ),∆h1e1 , . . . ,∆hded

commute.

Definition. Let n = (n1, . . . ,nd) ∈ Nd . Then f : I −→ Ris called n -↑ (read: n-increasing) iff(

∆ph f)

(s) :=(

∆p1h1e1

∆p2h2e2

. . .∆pdhded

f)

(s) ≥ 0

for all s ∈ I, h = (h1, . . . , hd) ∈ ]0, ∞[d, p = (p1, . . . , pd) ∈Nd

0,0 � p ≤ n such that sj + pjhj ∈ Ij ∀ j ≤ d . If instead6

(∇p

h f)

(s) :=(∇p1

h1e1∇p2

h2e2. . .∇pd

hdedf)

(s) ≥ 0 ,

f is called n -↓ (read: n-decreasing).We’ll say that f is p times (continuously) differentiable if

fp :=∂|p|f

∂sp11 . . . ∂spd

d

(|p| :=

d∑i=1

pi

)exists (and is continuous). We shall use the same symbol also incases where only the right partial derivative(s) exist. If f is ntimes differentiable, then

f is n -↑ ⇐⇒ fp ≥ 0 ∀ 0 � p ≤ n

andf is n -↓ ⇐⇒ (−1)|p|fp ≥ 0 ∀ 0 � p ≤ n .

7

We shall in the following mostly consider the intervals[0,1], [0,1[ or R+ = [0,∞[ , thereby certainly not restricting thegenerality. Since(

∆1dh f)

(s) = Ds+hs f ,

[1d := (1,1, . . . ,1) ∈ Nd

]we see that being fully d-increasing is the same as being 1d -↑ .The property n -↑ for some n ≥ 1d thus corresponds to astronger monotonicity requirement, beyond being just adistribution function.

8

If f is n -↑ then obviously f considered as a function of onlyone variable si is ni -↑ for each i . The converse however is nottrue: f(s1, s2) := (s1s2 − a)+ with a > 0 is 2 -↑ as a function ofs1 or s2 . But(

∆(2,2)(h,h) f

)(0) = 4(h2 − a)+ − 4(2h2 − a)+ + (4h2 − a)+

has the value −a for h =√

a , showing f to be not (2,2) -↑ .That f is (1,1) -↑ can be seen directly, and is also aconsequence of Theorem 1. The function f is not even(1,2) -↑ , since(∆2

he2f)

(s1, s2) = (s1(s2+2h)−a)+−2(s1(s2+h)−a)++(s1s2−a)+

is not increasing in s1 : for a = s1 = s2 = h = 1 this expressionis 0 , and changing s1 to 1

2 yields the value 12 .

9

2. The univariate case

Theorem 3.Let n ≥ 2, f : [0,1[ −→ R+ . Then f is n -↑ if and only if there

exist uniquely determined a0, . . . , an−2 ≥ 0 and a measure µ on[0,1[ such that

f(t) = a0 +a1 t+ . . .+an−2 tn−2 +

∫(t−a)n−1

+ dµ(a), 0 ≤ t < 1 .

The function f is continuous and for n > 2 (n− 2) timescontinuously differentiable on [0,1[ , where f (m) is(n−m)-↑, m = 1, . . . ,n− 2 . The right derivative of f (n−2) existsand equals (n− 1)! · µ([0, t]) , and is therefore right-continuousand increasing; in particular µ({0}) = f (n−1)

r (0)/(n− 1)! . Theconstants aj are given by aj = f (j)(0)/j! , j ≤ n− 2 .

10

Our proof is based on some elementary facts about convexfunctions, and some arguments used by Widder for absolutelymonotone functions (i.e. n-absolutely monotone for all n ∈ N).

We introduce functions fa,0 ≤ a ≤ 1 , on [0,1] :

fa(t) :=(t− a)+

1− afor 0 ≤ a < 1, f1 := 1{1} =: f∞0

(note that f0(t) = t). Furthermore

Kn := { f : [0,1] −→ R+ | f is n -↑, f(1) = 1} .

11

As a relatively easy consequence of Theorem 2 we obtain

Theorem 4.Kn is a Bauer simplex (for n ≥ 2) , and

ex(Kn) ={

f j0 | j = 0,1, . . . ,n− 2

}∪{

f n−1a | a ∈ [0,1]

}.

A function f ∈ Kn is continuous on [0,1[ , and in 1 if and only ifthe measure associated doesn’t charge the singleton { f1} .

12

The extreme points of Kn were already identified in 1964 byMcLachlan (Pac. J. Math. 14), in a direct but rather complicatedway. That Kn is a simplex was not shown there.

Functions which are n -↑ for every n ∈ N have been known fora long time already, they are called absolutely monotone. Wemight abbreviate this property by „∞ -↑ “. LetK∞ := { f : [0,1] −→ R+ | f is∞ -↑, f(1) = 1} . Then we maystate the following properties, due essentially to Widder:

Theorem 5.

(i) K∞ is a Bauer simplex, and ex(K∞) = { f j0 | j ∈ N0} .

(ii) f : [0,1[ −→ R+ is absolutely monotone iff f is analytic withnon-negative coefficients.

13

There are counterparts for n -↓ functions. We consider the directone

Ln := {g : [0,1] −→ R+ | g is n -↓, g(0) = 1}

and also

Hn := {g : R+ −→ R+ | g is n -↓, g(0) = 1} .

Theorem 6.For n ≥ 2 both Ln and Hn are Bauer simplices. Withgc(s) := (1− cs)+ , for c ∈ [0,∞] (where g∞ := 1{0}) we have

ex(Ln) ={

g j1 | j = 0, . . . ,n− 2

}∪ {g n−1

c | c ∈ [1,∞]}

ex(Hn) = {g n−1c | c ∈ [0,∞]} .

Here the structure of Hn goes back to Schoenberg andWilliamson.

14

3. The multivariate case

If F ⊆ RX ,G ⊆ RY for some sets X,Y , then

F ⊗ G := { f ⊗ g | f ∈ F, g ∈ G} ,

not the linear space generated by these functions.

Put En := ex(Kn) = { f j0 | j = 0, . . . ,n− 2} ∪ { f n−1

a | a ∈ [0,1]} .Note that f1 = f n−1

1 is the only discontinuous function in En .An := En r { f1} will also be important.For n = (n1, . . . ,nd) ∈ (Nr 1)d we define

En := En1 ⊗ . . .⊗ End ,

An := An1 ⊗ . . .⊗ And ,

and in analogy to dimension 1

15

Kn :={

f : [0,1]d −→ R+ | f is n -↑, f(1d) = 1},

obviously a compact and convex set of functions.Our main result is the following

Theorem 7.En is a Bauer simplex, and ex(Kn) = En (n ≥ 2d).

We will give a sketch of the proof, consisting of five steps.

I. Let f ∈ Kn be given and assume f is C∞ . Assume firstn = 2d . The partial derivative f1d fulfills ( f1d)1d = f2d ≥ 0 ,hence f1d(t) = ν([0, t]) for some ν ∈ M+([0,1]d) by theCorrespondence Theorem, yielding

16

Ds0d

f =

∫[0,s]

f1d(t) dt =

∫[0,s]

ν([0, t]) dt

=

∫[0,s]

∫[0,1]d

1[0,t](a) dν(a) dt

=

∫[0,1]d

λλd([0, s] ∩ [a,1d]) dν(a)

=

∫[0,1]d

d∏i=1

(si − ai)+ dν(a) =

∫[0,1]d

d∏i=1

fai(si) dν̃(a)

where ν̃ is a finite measure on [0,1]d , concentrated on [0,1[d .

17

For ∅ 6= α ⊆ d := {1, . . . , d} let fα(sα) := f(sα,0αc) , then

f(s) =∑∅6=α⊆d

Dsα0α

fα + f(0d) ,

and for every non-empty α ⊆ d there is a finite measure ν̃α on[0,1[α such that

Dsα0α

fα =

∫[0,1[α

∏i∈α

fai(si) dν̃α(aα) , sα ∈ [0,1]α .

Since si = 0 for i ∈ αc in Dsα0α

fα , we can write (withη := f0

0 ≡ 1)

Dsα0α

fα =

∫ d∏i=1

hi(si) d(ν̃α ⊗ εηαc

)(h1, . . . , hd) .

The measure

18

µ := ν̃ +∑∅6=α$d

ν̃α ⊗ εηαc + f(0d) · εηd

on E2d then fulfills

f(s) =

∫h(s) dµ(h) ∀ s ∈ [0,1]d ,

and it is a probability measure because of

1 = f(1d) =∑∅6=α⊆d

D1α0α

fα + f(0d) = µ([0,1]d) .

19

Assuming the result to hold for some n ≥ 2d it can then bededuced for (n1 + 1,n2, . . . ,nd) , etc. ... .

II. In the second step we only assume f ∈ Kn to be continuous.We now make use of Bernstein polynomials, defined indimension one by

bi,k(t) :=

(ki

)ti(1− t)k−i , i = 0, . . . , k ,

and in higher dimensions by

Bi,k := bi1,k ⊗ . . .⊗ bid,k , i = (i1, . . . , id) ∈ {0, . . . k}d .

It is well known that the Bernstein approximations

fk :=∑

0d≤i≤kd

f(

ik

)· Bi,k

20

converge to f (even uniformly) on [0,1]d . Applying theformula for derivatives of one-dimensional Bernsteinapproximations d times, we get

(fk)p = cp∑

i≤kd−p

∆p1e1/k . . .∆

pded/k f

(ik

)bi1,k−p1 ⊗ . . .⊗ bid,k−pd

with cp :=∏d

i=1 k(k− 1) · . . . · (k− pi + 1) ; hence (fk)p ≥ 0 for0 � p ≤ n . Applying the first part of this proof to fk we have

fk(s) =

∫h(s) dµk(h) , s ∈ [0,1]d ,

for suitable µ1, µ2, . . . ∈ M1+(En), and with a limit point µ of

some convergent subsequence of {µ1, µ2, . . .} we get

f(s) =

∫h(s) dµ(h) , s ∈ [0,1]d .

21

Already at this point the uniqueness of the integralrepresentation (for continuous f ) can and has to be established.

III. In the third step we shall consider a function f defined onlyon [0,1[d and being n -↑ . We show first that f is necessarilycontinuous. Let s ∈ [0,1[d be given and consider

g(r) := f(s + r · 1d) for 0 ≤ r < min1≤i≤d

(1− si) .

g can be shown to be 2 -↑ and hence is continuous. But thefunction f is in particular (simply) increasing, and this propertycombined with the continuity of g (for every choice of s) showsthat f is (everywhere on [0,1[d) continuous.We can now apply part II. to the restriction of f to [0, c]d , for0 < c < 1, i.e.

f(cs) =

∫En

h(s) dνc(h) for s ∈ [0,1]d ,

22

where νc ∈ M+(En) is unique, and is concentrated on An . Thiscan be rewritten with some µc ∈ M+(An) as

f(u) =

∫An

h(u

c

)dνc(h) =

∫An

h(u) dµc(h) , u ∈ [0, c]d ,

and µc turns out to be concentrated on the compact subset

Ac :=

d∏i=1

({0, . . . ,ni − 2} ·∪ [0, c]

)of

An =

d∏i=1

({0, . . . ,ni − 2} ·∪ [0,1[

). Since µc on Ac is

determined by f , we have for 0 < c1 < c2 < 1 the identityµc2 | Ac1 = µc1 . Hence for 0 < ck ↗ 1 the measures µck arecompatible and so determine a Radon measure µ on An forwhich

f(s) =

∫An

h(s) dµ(h) , s ∈ [0,1[d .

Here the unicity of µ is an immediate consequence of the waywe showed its existence.

23

IV. Whereas n -↑ functions on [0,1[d are automaticallycontinuous, and smooth to a certain degree (ni ≥ 2 ∀ i) , thisneed not be the case for such functions on [0,1]d , as we sawalready for d = 1 . In higher dimensions the situation is a lotmore involved, since on each partTα :=

{s ∈ [0,1]d | si < 1⇐⇒ i ∈ α

}of the „upper right

boundary“ [0,1]d r [0,1[d, ∅ 6= α $ d , one may add„independently“ functions of |α| variables with sufficiently highmonotonicity. For a given n -↑ function f on [0,1]d this„procedure“ has to be reversed, i.e. these different parts of thefunction have to be identified.This part of the proof is rather involved and I’ll omit its details.

24

V. Here the uniqueness of the representation is proved in thegeneral case. Let f ∈ Kn be represented as

f(s) =

∫En

h(s) dν(h) , s ∈ [0,1]d ;

we have to show that ν = µ , the measure found before. Weintroduce the parts

Qα := {h = (h1, . . . , hd) ∈ En | hi = ϑ⇐⇒ i ∈ αc}

=∏i∈α

Ani × {ϑαc}

of En for α ⊆ d , where Q∅ = {ϑd} and Qd = An .The restriction ν|Qα , i.e. the measure B 7−→ ν(B ∩ Qα) , has for∅ 6= α $ d the form να ⊗ εϑαc for some να ∈ M+(Anα) ,ν|Q∅ = c′ · εϑd for some c′ ≥ 0 , and ν|Qd =: νd ∈ M+(An) .

25

The (restricted) unicity from III. shows νd = µd , and then a„backward induction“ (over the size |α| of α ⊆ d) shows finallyνα = µα ∀ α ⊆ d , i.e. µ = ν . 2

Corollary 8.If f ∈ Kn is continuous at 1d , it is every where continuous.

Part III of the proof just given is in fact a multivariate analogueof Theorem 2:

26

Theorem 9.

Let n ≥ 2d, f : [0,1[d −→ R+ . Then f is n -↑ iff

f(s) =

∫h(s) dµ(h) , s ∈ [0,1[d ,

for a uniquely determined Radon measure µ on An . The functionf is continuous and (n− 2d) times continuously differentiable.For m ≤ n− 2d the derivative fm is (n−m) -↑ . The rightderivative of fn−2d with respect to each variable exists, is given by(n− 1d)! · µ([0, s]) , and is therefore still right-continuous and1d -↑ . For j ≤ n− 2d we have

µ({

f j10 ⊗ . . .⊗ f jd

0

})= fj(0d)/j!

and this holds also if ji = ni − 1 for some (or all) i ≤ d , fjdenoting then the right derivative in those coordinates.

27

If a function of d variables is n -↑ for every n ∈ Nd , we call itagain absolutely monotone.

Theorem 10.

(i) f : [0,1[d −→ R+ is absolutely monotone if and only if f isanalytic with non-negative coefficients.

(ii) K∞d is a Bauer simplex, andex(K∞d) = {f i1

0 ⊗ . . .⊗ f id0 | i1, . . . , id ∈ N0} .

28

4. COMPOSITION OF HIGHER ORDERMONOTONIC FUNCTIONS

We have the following multivariate generalisation of Theorem 2:

Theorem 11.

Let f : [0,1]d −→ R+ and n ∈ Nd be given. Then there areequivalent:

(i) f is n -↑(ii) For any fully ni-increasing

ϕi : {0,1}ni −→ [0,1], 1 ≤ i ≤ d , the compositionf ◦ (ϕ1 × · · · × ϕd) is fully |n|-increasing on {0,1}|n| .

29

We indicate the proof of „(i) =⇒ (ii)“ for n ≥ 2d . Since f isn -↑ it has by Theorem 7 (our main result) the representation

f(s) =

∫ d∏i=1

hi(si) dµ(h1, . . . , hd) , s ∈ [0,1]d ,

for some measure µ on En . This leads to

f ◦ (ϕ1 × . . .× ϕd) =

∫ d⊗i=1

(hi ◦ ϕi) dµ(h1, . . . , hd) ,

where each hi ◦ ϕi is 1ni -↑ by Theorem 2, hence⊗d

i=1(hi ◦ ϕi)is 1|n| -↑ , and so is f ◦ (ϕ1 × . . .× ϕd) as a mixture of thesefunctions.

30

We close with our second main result:

Theorem 12.

Given n = (n1, . . . ,nd) ∈ Nd,mi ∈ Nni , i ≤ d , put`i := |mi|, ` := (`1, . . . , `d) and m := (m1, . . . ,md) ∈ N|n| . Letgi : [0,1]ni −→ [0,1] be mi -↑, i ≤ i ≤ d , and supposef : [0,1]d −→ R to be ` -↑ , then f ◦ (g1 × . . .× gd) is m -↑ .

In dimension d = 2 this means:

If g1 × g2 : [0,1]n1 × [0,1]n2 −→ [0,1]2f−→ R , where gi is

mi -↑, i = 1,2 , and f is (|m1|, |m2|) -↑ , then f ◦ (g1 × g2) is(m1,m2) -↑ .

The special case mi = 1ni (hence |mi| = ni) answers ourquestion from the beginning:

Precisely the (n1,n2) -↑ functions f on [0,1]2 lead to1n1+n2 -↑ compositions f ◦ (g1 × g2) , i.e. to newdistribution functions.

31