ON CONVERGENCE AND MASS DISTRIBUTIONS OF MULTIVARIATE ARCHIMEDEAN COPULAS AND THEIR INTERPLAY WITH THE WILLIAMSON TRANSFORM

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ON CONVERGENCE AND MASS DISTRIBUTIONS OF

MULTIVARIATE ARCHIMEDEAN COPULAS AND THEIR

INTERPLAY WITH THE WILLIAMSON TRANSFORM

THIMO M. KASPER, NICOLAS DIETRICH AND WOLFGANG TRUTSCHNIG

Abstract. Motivated by a recently established result saying that within the class of

bivariate Archimedean copulas standard pointwise convergence implies weak conver-

gence of almost all conditional distributions this contribution studies the class Cd

ar of

all d-dimensional Archimedean copulas with d≥3 and proves the afore-mentioned

implication with respect to conditioning on the ﬁrst d−1 coordinates. Several proper-

ties equivalent to pointwise convergence in Cd

ar are established and - as by-product of

working with conditional distributions (Markov kernels) - alternative simple proofs for

the well-known formulas for the level set masses µC(Lt) and the Kendall distribution

function Fd

Kas well as a novel geometrical interpretation of the latter are provided.

Viewing normalized generators ψof d-dimensional Archimedean copulas from the per-

spective of their so-called Williamson measures γon (0,∞) is then shown to allow not

only to derive surprisingly simple expressions for µC(Lt) and Fd

Kin terms of γand to

characterize pointwise convergence in Cd

ar by weak convergence of the Williamson mea-

sures but also to prove that regularity/singularity properties of γdirectly carry over

to the corresponding copula Cγ∈ Cd

ar. These results are ﬁnally used to prove the fact

that the family of all absolutely continuous and the family of all singular d-dimensional

copulas is dense in Cd

ar and to underline that despite of their simple algebraic struc-

ture Archimedean copulas may exhibit surprisingly singular behavior in the sense of

irregularity of their conditional distribution functions.

Contents

1. Introduction 1

2. Notation and preliminaries 3

3. Markov kernel, mass distribution and Kendall distribution function of

multivariate Archimedean copulas 5

4. Characterizing pointwise convergence in Cd

ar and the interrelation with weak

conditional convergence 8

5. Archimedean copulas and the Williamson transform 14

6. Singular Archimedean copulas with full support 22

Appendix A. Level set mass and Kendall distribution function: Calculations 24

Appendix B. Approximations by discrete, absolutely continuous and singular

Williamson measures 26

Acknowledgements 28

References 28

1. Introduction

Archimedean copulas are a well-known family of dependence models whose popular-

ity is mainly due to their simple algebraic structure: given a so-called (Archimedean,

arXiv:2210.11868v1 [math.ST] 21 Oct 2022

2 CONVERGENCE AND MASS DISTRIBUTIONS OF MULTIVARIATE ARCHIMEDEAN COPULAS

suﬃciently monotone) generator ψ: [0,∞)→[0,1] =: Iand letting ϕdenote its pseudo-

inverse the Archimedean copula Cψis deﬁned by

Cψ(x1, . . . , xd) = ψ(ϕ(x1) + · · · +ϕ(xd)).

As a consequence, analytic, dependence, and convergence properties of Archimedean

copulas can be characterized in terms of the corresponding generators (see, e.g.,

[2, 4, 14, 21, 22]). In particular, it was recently shown in [14] that within the class

of bivariate Archimedean copulas pointwise convergence is equivalent to uniform

convergence of the corresponding generators and, more importantly, even implies weak

convergence of almost all conditional distributions (a.k.a weak conditional convergence,

a concept generally much stronger than pointwise convergence). This result is surprising

insofar that given samples (X1, Y1),(X2, Y2), . . . from some bivariate copula Cthe

corresponding sequence of empirical copulas ( ˆ

En)n∈Ndoes not necessarily converge

weakly conditional to C.

The focus of the current paper is twofold: on the one hand we study convergence in the

class Cd

ar of all d-dimensional Archimedean copulas, d≥3, and show that most results

from the bivariate setting as established in [14] also hold in Cd

ar, including the surprising

fact that pointwise convergence implies weak conditional convergence (whereby we

consider conditioning on the ﬁrst d−1 coordinates). As a nice by-product of working

with Markov kernels (conditional distributions) we obtain simple, alternative proofs

of the well-known formulas for the Kendall distribution function Fd

Kand the level set

masses of Archimedean copulas. To the best of the authors’ knowledge, working with

so-called `1-norm symmetric distributions (as studied in [21]) nowadays seems to be the

standard approach for deriving these formulas in the multivariate setting - we show that

working with Markov kernels constitutes an interesting alternative and may provide

additional insight. Additionally, motivated by [22, Chapter 4.3] and [4, Section 3] we

oﬀer a seemingly novel geometric interpretation of the level set masses in terms of ψ.

And, on the other hand, we revisit the close interrelation between Archimedean copulas

and probability measures γon (0,∞) via the so-called Williamson transform as studied

in [21] (also see [25]), characterize properties of the generator ψin terms of normalized

γ(to which we will refer to as Williamson measure) and then prove the fact that

pointwise convergence in Cd

ar is equivalent to weak convergence of the corresponding

probability measures on (0,∞). Moreover, we derive surprisingly simple expressions

for the level set masses and the Kendall distribution functions in terms of γand then

show that singularity/regularity properties of γdirectly carry over to the corresponding

Archimedean copula Cγ∈ Cd

ar. This very property is then used, ﬁrstly, to show that

the family of absolutely continuous as well as two disjoint subclasses of the family

of all singular Archimedean copulas are dense in Cd

ar and, secondly, to illustrate the

fact that despite their simple and handy algebraic structure Archimedean copulas

may exhibit surprisingly irregular behavior by constructing elements of Cd

ar which have

full support although being singular (see [4] for the already established bivariate results).

The rest of this contribution is organized as follows: Section 2 contains notation and

preliminaries that are used throughout the text. Section 3 starts with deriving an explicit

expression for the Markov kernel of d-dimensional Archimedean copulas, then restates

the well-known formulas for the masses of level sets as well as the Kendall distribution

function, and provides a geometric interpretation for the latter in terms of the generator

ψ. In Section 4 we derive various characterizations of pointwise convergence in Cd

ar in

several steps and prove the main result saying that pointwise convergence implies weak

conditional convergence (with respect to the ﬁrst d−1 coordinates). Turning to the

CONVERGENCE AND MASS DISTRIBUTIONS OF MULTIVARIATE ARCHIMEDEAN COPULAS 3

Williamson transform, in Section 5 we ﬁrst establish some complementing results on

the interrelation of the generator ψand the Williamson measure γ, then characterize

pointwise convergence in Cd

ar in terms of weak convergence of the Williamson measures,

and ﬁnally show how regularity/singularity properties of γcarry over to Cγ. These

properties are then used to prove the afore-mentioned denseness results and to construct

singular Archimedean copulas with full support. Several examples and graphics illustrate

the obtained results.

2. Notation and preliminaries

In the sequel we will let Cddenote the family of all d-dimensional copulas for some

ﬁxed integer d≥3 and write vectors in bold symbols. For each copula C∈ Cdthe

corresponding d-stochastic measure will be denoted by µC, i.e., µC([0,x]) = C(x) for all

x∈Id, whereby [0,x] := [0, x1]×[0, x2]×. . . ×[0, xd] and I:= [0,1]. To notation as

simple as possible we will frequently write x1:m= (x1, . . . , xm) for x∈Idand m≤d.

Considering 1 ≤i < j ≤d, the i-j-marginal of Cwill be denoted by Cij, i.e., we have

Cij(xi, xj) = C(1,...,1, xi,1,...,1, xj,1,...,1). In order to keep notation as simple as

possible for every m < d the marginal copula of the ﬁrst mcoordinates will be denoted

by C1:m, i.e., C1:m(x1, x2, . . . , xm) = C(x1, x2, . . . , xm,1,...,1). Considering the uniform

metric d∞on Cdit is well-known that (Cd, d∞) is a compact metric space and that in

Cdpointwise and uniform convergence are equivalent. For more background on copulas

and d-stochastic measures we refer to [3, 22].

For every metric space (S, d) the Borel σ-ﬁeld on Swill be denoted by B(S) and P(S)

denotes the family of all probability measures on B(S). The Lebesgue measure on B(Id)

will be denoted by λor (whenever particular emphasis to the dimension dis required)

by λd. Furthermore δxdenotes the Dirac measure in x∈S.

In what follows Markov kernels will play a prominent role. For m < d an m-Markov

kernel from Rmto Rd−mis a mapping K:Rm× B(Rd−m)→Ifulﬁlling that for every

ﬁxed E∈ B(Rd−m) the mapping x7→ K(x, E) is B(Rm)-B(Rd−m)-measurable and for

every ﬁxed x∈Rmthe mapping E7→ K(x, E) is a probability measure on B(Rd−m).

Given a (d−m)-dimensional random vector Yand an m-dimensional random vector

Xon a probability space (Ω,A,P) we say that a Markov kernel K(·,·) is a regular

conditional distribution of Ygiven Xif for every ﬁxed E∈ B(Rd−m) the identity

K(X(ω), E) = E(1E◦Y|X)(ω)

holds for P-almost every ω∈Ω. It is well-known that for each pair of random vectors

(X,Y) as above, a regular conditional distribution K(·,·) of Ygiven Xexists and is

unique for PX-almost all x∈Rm, whereby as usual PXdenotes the push-forward of

Pvia X. In case (X,Y) has C∈ Cdas distribution function (restricted to Id) we let

KC:Im× B(Id−m)→Idenote (a version of) the regular conditional distribution of Y

given Xand simply refer to it as m-Markov kernel of C. Deﬁning the x-section Gxof a

set G∈ B(Id) w.r.t. the ﬁrst mcoordinates by Gx:= {y∈Id−m: (x,y)∈G}∈B(Id−m)

the well-known disintegration theorem implies

µC(G) = ZIm

KC(x, Gx) dµC1:m(x).

It is well-known that the disintegration theorem also holds for general ﬁnite measures

in which case the conditional measures K(·,·) are not necessarily probability measures

but general ﬁnite measures (sub- or super Markov kernels). For more background on

conditional expectation and disintegration we refer to the [11] and [17].

4 CONVERGENCE AND MASS DISTRIBUTIONS OF MULTIVARIATE ARCHIMEDEAN COPULAS

An Archimedean generator ψis a continuous, non-increasing function ψ: [0,∞)→

[0,1] fulﬁlling ψ(0) = 1, limz→∞ ψ(z) = 0 =: ψ(∞) and being strictly decreasing on

[0,inf{z∈[0,∞] : ψ(z)=0}] (with the convention inf ∅:= ∞). For every Archimedean

generator ψwe will let ϕ: [0,1] →[0,∞] denote its pseudo-inverse deﬁned by ϕ(y) =

inf{z∈[0,∞] : ψ(z) = y}for every y∈[0,1]. Obviously ϕis strictly decreasing on [0,1]

and fulﬁlls ϕ(1) = 0, moreover it is straightforward to verify that ϕis right-continuous

at 0 (for a short discussion of this property see Section 4 in [14]). If ϕ(0) = ∞we refer to

ψ(and ϕ) as strict and as non-strict otherwise. A copula C∈ Cdis called Archimedean

(in which case we write C∈ Cd

ar) if there exists some Archimedean generator ψwith

Cψ(x) = ψ(ϕ(x1) + · · · +ϕ(xd))

for every x∈Id. Following [21], Cψ(x) = ψ(ϕ(x1) + · · · +ϕ(xd)) is a d-dimensional

copula if, and only if, ψis a d-monotone Archimedean generator on [0,∞), i.e., if, and

only if, ψis an Archimedean generator fulﬁlling that (−1)d−2ψ(d−2) exists on (0,∞), is

non-negative, non-increasing and convex on (0,∞), whereby, as usual, g(m)denotes the

m-th derivative of a function g. Moreover (again see [21]) it is straightforward to verify

that in the latter case (−1)mψ(m)exists on (0,∞), is non-negative, non-increasing and

convex on (0,∞) for every m∈ {0, . . . , d −2}. In the sequel we will sometimes simply

write Cinstead of Cψwhen no confusion may arise. Furthermore in the sequel we will

simply refer to Archimedean generators as ‘generators’.

Letting D−gand D+gdenote the left- and right- hand derivative of a function g,

respectively, convexity of (−1)d−2ψ(d−2) implies that both, D−ψ(d−2)(z) and D+ψ(d−2)(z)

exist for every z∈(0,∞) and that the two derivatives coincide outside an at most

countable set (see [12, 23]) - in fact, for every continuity point zof D−ψ(d−2) we

have D−ψ(d−2)(z) = D+ψ(d−2)(z). Moreover, every d-monotone generator ψfulﬁlls

limz→∞ ψ(m)(z) = 0 for every m∈ {0, . . . , d −2}as well as limz→∞ D−ψ(d−2)(z) = 0.

Indeed, limz→∞ ψ0(z) = 0 directly follows from monotonicity and convexity of ψsince

d-monotonicity implies that −ψ0is decreasing and convex too; proceeding iteratively

yields the assertion. Notice that Lemma 5.3 yields an even simpler direct proof of this

assertion. In the sequel we will also use the fact that (again by convexity, see [12, 23]) we

can reconstruct the generator ψfrom its derivatives in the sense that (m∈ {1, . . . , d−2})

ψ(m−1)(z) = Z[z,∞)

−ψ(m)(s) dλ(s), ψ(d−2)(z) = Z[z,∞)

−D−ψ(d−2)(s) dλ(s).(2.1)

In order to have a one-to-one correspondence between copulas and their generator we

follow [14] and from now on implicitly assume that all generators are normalized in the

sense that ϕ(1

2) = 1, or equivalently, ψ(1) = 1

2holds.

According to [21], an Archimedean copula C∈ Cd

ar is absolutely continuous if, and

only if, ψ(d−1) exists and is absolutely continuous on (0,∞). In this case a version of the

density cof Cis given by

c(x) = 1(0,1)d(x)

i=1

ϕ0(xi)·D−ψ(d−1)ϕ(x1) + · · · +ϕ(xd).(2.2)

In the sequel we will use the handy consequence that lower dimensional marginals of

d-dimensional Archimedean copulas are absolutely continuous ([21, Proposition 4.1]).

CONVERGENCE AND MASS DISTRIBUTIONS OF MULTIVARIATE ARCHIMEDEAN COPULAS 5

3. Markov kernel, mass distribution and Kendall distribution function

of multivariate Archimedean copulas

In the proceedings contribution [15] the authors derive an explicit expression for (a

version of) the (d−1)-Markov kernel of d-variate Archimedean copulas which, in turn,

allows to derive the well-known formulas for level-set mass and the Kendall distribu-

tion function of d-variate Archimedean copulas in an alternative way. Considering that

Markov kernels are key for the rest of this paper we introduce a (slightly modiﬁed) version

of the Markov kernel considered in [15] and prove its properties in detail. Furthermore,

we restate the already known formulas for the Kendall distribution function and level-set

mass, add a new geometric interpretation, and, for the sake of completeness, include the

corresponding purely Markov kernel-based proofs in the Appendix.

For every t∈(0,1] deﬁne the t-level set of C∈ Cd

ar by

(3.1)

Lt:= (x, y)∈Id−1×I:C(x, y) = t=((x, y)∈Id−1×I:

d−1

i=1

ϕ(xi) + ϕ(y) = ϕ(t))

and for t= 0 set

(3.2)

L0:= (x, y)∈Id−1×I:C(x, y)=0=((x, y)∈Id−1×I:

d−1

i=1

ϕ(xi) + ϕ(y)≥ϕ(0)).

Subsequently we will work with the level sets of the (d−1)-dimensional marginal of

Cdeﬁned analogously and denote them by L1:d−1

tand L1:d−1

0, respectively. As in the

bivariate setting (see [14]) we can deﬁne functions ftwhose graph coincides with Ltfor

t∈(0,1] and with the boundary of L0for t= 0: In fact, for t= 0 deﬁning the function

f0:Id−1→Iby

f0(x) = (1 if x∈L1:d−1

ψϕ(0) −Pd−1

i=1 ϕ(xi)if x6∈ L1:d−1

with the conventions ψ(∞) = 0 as well as ψ(u) = 1 for all u < 0 and for t∈(0,1], deﬁning

the upper t-cut of C1:d−1by [C1:d−1]t={x∈Id−1:C1:d−1(x)≥t}and considering the

function ft: [C1:d−1]t→Igiven by

ft(x) := ψ ϕ(t)−

d−1

i=1

ϕ(xi)!.

yields the above-mentioned property. It is straightforward to verify that for x6∈ L1:d−1

and y < f0(x) we have (x, y)∈L0and that for strict Archimedean copulas x∈L1:d−1

if, and only if, M(x) = 0 where Mdenotes the d-dimensional minimum copula.

Theorem 3.1. Suppose that C∈ Cd

ar has generator ψ. Then setting

KC(x,[0, y]) := 









1, M(x) = 1 or x∈L1:d−1

0, M(x)<1,x6∈ L1:d−1

0, y < f0(x)

D−ψ(d−2) d−1

i=1

ϕ(xi)+ϕ(y)!

D−ψ(d−2) d−1

i=1

ϕ(xi)!, M(x)<1,x6∈ L1:d−1

0, y ≥f0(x)

(3.3)

yields (a version of) the (d−1)-Markov kernel of C.

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ONCONVERGENCEANDMASSDISTRIBUTIONSOFMULTIVARIATEARCHIMEDEANCOPULASANDTHEIRINTERPLAYWITHTHEWILLIAMSONTRANSFORMTHIMOM.KASPER,NICOLASDIETRICHANDWOLFGANGTRUTSCHNIGAbstract.MotivatedbyarecentlyestablishedresultsayingthatwithintheclassofbivariateArchimedeancopulasstandardpointwiseconvergenceimpliesweakconv...

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