higher order monotonic functions of several...

HIGHER ORDER MONOTONIC FUNCTIONSOF SEVERAL VARIABLES

Paul Ressel

Katholische Universität Eichstätt-Ingolstadt

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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) .

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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..

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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.

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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 .

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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.

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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 .

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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 .

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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} .

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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} .

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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.

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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.

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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

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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

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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 .

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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

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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) .

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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

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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 .

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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 ,

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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.

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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.

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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) .

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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:

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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.

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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} .

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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| .

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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.

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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.

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