# On convergence and singularity of conditional copulas of multivariate
Archimedean copulas, and estimating conditional dependence

Thimo M. Kasper  Thimo M. Kasper
University of Salzburg
Department for Artificial Intelligence and Human Interfaces
Hellbrunnerstrasse 34, 5020 Salzburg<EMAIL_ADDRESS>
###### Abstract.

The present contribution derives an explicit expression for (a version of)
every uni- and multivariate conditional distribution (i.e., Markov kernel) of
Archimedean copulas and uses this representation to generalize a recently
established result, saying that in the class of multivariate Archimedean
copulas standard uniform convergence implies weak convergence of almost all
univariate Markov kernels, to arbitrary multivariate Markov kernels. Moreover,
we prove that an Archimedean copula is singular if, and only if, almost all
uni- and multivariate Markov kernels are singular. These results are then
applied to conditional Archimedean copulas which are reintroduced largely from
a Markov kernel perspective and it is shown that convergence, singularity and
conditional increasingness carry over from Archimedean copulas to their
conditional copulas. As consequence the surprising fact is established that
estimating (the generator of) an Archimedean copula directly yields an
estimator of (the generator of) its conditional copula. Building upon that, we
sketch the use and estimation of a conditional version of a recently
introduced dependence measure as alternative to well-known conditional
versions of association measures in order to study the dependence behaviour of
Archimedean models when fixing covariate values.

##  1\. Introduction

Archimedean copulas pose a popular class of dependence structures. As the
class includes various parametric families covering a wide range of dependence
models, Archimedean copulas are applied in numerous fields such as hydrology
(see, e.g., [29]), finance (see, e.g., [14]) or actuarial sciences (see, e.g.,
[8]) among many others. Archimedean copulas are also attractive from a
theoretical point of view as they enjoy the property of being generated by a
single univariate function $\psi$ which fully characterizes analytic and
dependence aspects of Archimedean copulas ([2, 5, 7, 21, 20, 25, 28]). The
current paper is particularly concerned with convergence as well as
singularity of multivariate Archimedean copulas and it turns out that studying
them via their conditional distributions (i.e., Markov kernels) leads to
simple, alternative derivations of important results in the field (see [7, 21,
22, 20]). In the two-dimensional case it is well-known that within the class
of multivariate Archimedean copulas pointwise convergence of the copulas
coincides with pointwise convergence of the corresponding generators as well
as with weak convergence of almost all Markov kernels ([21]) while in [20] it
was recently shown that pointwise convergence of _multivariate_ Archimedean
copulas implies convergence of almost all univariate Markov kernels.

In this contribution we derive an explicit expression for all uni- and
multivariate Markov kernels of Archimedean copulas, hence allowing for an
arbitrary number of conditioning coordinates. Utilizing this representation we
prove that, in addition to several well-known characterizing properties,
pointwise convergence of Archimedean copulas is equivalent to weak convergence
of almost all $(d-1)$-dimensional Markov kernels and that any of these
properties implies weak convergence of almost all uni- and multivariate Markov
kernels. Extending [20] in another aspect, we show the surprising fact that an
Archimedean copula is singular w.r.t. $\lambda_{d}$ if, and only if, almost
all uni- and multivariate conditional distributions are singular w.r.t. the
appropriate Lebesgue measure.

These results are then applied to conditional Archimedean copulas. In general,
conditional copulas allow for a means to study the dependence of a random
vector when influenced by a (vector of) covariate(s). In [11, 12], for
instance, the authors study the estimation of conditional copulas from data
and define conditional versions of the concordance measures Kendall’s tau and
Spearman’s rho. Conditional copulas of Archimedean copulas, in particular, are
studied, e.g., in [26, 31]. In [26] the authors study tail behaviour,
conditional versions of tail dependence indices, positive dependence and show
the surprising fact that the conditional copula of an Archimedean copula is
again Archimedean (also called the truncation invariance property). In the
present paper we will reintroduce conditional copulas largley from a Markov
kernel perspective, provide an alternative proof of the afore-mentioned
truncation invariance property and demonstrate that the Markov kernel
representation allows for several beautiful relations concerning conditional
Archimedean generators. Using our established results we will then show that
convergence, singularity and conditional increasingness carry over from the
original Archimedean copula to its conditional copula. As consequence we
obtain the surprising fact that for estimating the conditional copula of an
Archimedean copula $C$ it suffices to estimate (the generator of) $C$ which
can be done using well-known methods (see, e.g., [10, 15, 17] and the
references therein). Going a step further and utilizing this observation we
sketch the use and estimation of a conditional version of the multivariate
dependence measure $\zeta_{1}$ ([13]) as a new tool to investigate dependence
behaviour of Archimedean data given covariate values, thus providing a
potentially powerful alternative to the well-known conditional Kendall’s tau
([12]) or general tail concentration functions (see, e.g., [4]).

The remainder of this paper is organized as follows: Section 2 gathers
notation and preliminaries that are used throughout the text. Focussing on
Archimedean copulas, Section 3 derives an explicit expression for (a version
of) every uni- and multivariate Markov kernel. This representation is then
used in Section 4 to characterize pointwise/uniform convergence of Archimedean
copulas in terms of their $(d-1)$-dimensional Markov kernels and to show that,
in this setting, uniform convergence even implies weak convergence of almost
all uni- _and_ multivariate Markov kernels. The main result of Section 5 is
that singularity of Archimedean copulas is equivalent to singularity of almost
all uni- and multivariate Markov kernels. Following up on an introduction to
conditional copulas of Archimedean copulas via Markov kernels, in Section 6 we
prove that convergence as well as singularity transfers from Archimedean
copulas to their conditional copulas, and in addition, sketch the use and
estimation of a conditional version of the dependence measure $\zeta_{1}$ to
obtain additional insights into the dependence behaviour of Archimedean data
given covariate values. We conclude the paper by remarking that, at last,
conditional increasingness carries over to the conditional Archimedean
copulas, too.

##  2\. Notation and preliminaries

This paper considers the class $\mathcal{C}^{d}$ of all $d$-dimensional
copulas for some integer $d\geq 2$. Given $C\in\mathcal{C}^{d}$ the
corresponding $d$-stochastic measure will be denoted by $\mu_{C}$, i.e.,
$\mu_{C}([\mathbf{0},\mathbf{x}])=C(\mathbf{x})$ for all
$\mathbf{x}\in\mathbb{I}^{d}$, where $\mathbb{I}:=[0,1]$ and
$[\mathbf{0},\mathbf{x}]:=[0,x_{1}]\times\ldots\times[0,x_{d}]$. Considering
$1\leq i<j\leq d$, the $i$-$j$-marginal $C^{ij}$ of $C$ is given by
$C^{ij}(x_{i},x_{j})=C(1,\ldots,1,x_{i},1,\ldots,1,x_{j},1,\ldots,1)$ and
similarly for $1<\ell<d$ the marginal copula $C^{1:\ell}$ of the first $\ell$
coordinates we have
$C^{1:\ell}(\mathbf{x})=C(x_{1},x_{2},\ldots,x_{\ell},1,\ldots,1)$ for every
$\mathbf{x}\in\mathbb{I}^{\ell}$. The uniform metric on $\mathcal{C}^{d}$ is
defined by

$\displaystyle
d_{\infty}(C_{1},C_{2}):=\max_{\mathbf{x}\in\mathbb{I}^{d}}|C_{1}(\mathbf{x})-C_{2}(\mathbf{x})|$

and it is well-known that $(\mathcal{C}^{d},d_{\infty})$ is a compact metric
space. For more background on multivariate copulas and $d$-stochastic
probability measures we refer to [28, 5].

For every metric space $(\mathbb{S},d_{\mathbb{S}})$ the Borel $\sigma$-field
on $\mathbb{S}$ will be denoted by $\mathcal{B}(\mathbb{S})$ and we write
$\lambda_{d}$ for the Lebesgue measure on $\mathcal{B}(\mathbb{I}^{d})$. In
the sequel Markov kernels will play the key role. For $1<\ell<d$ an
$\ell$-_Markov kernel_ from $\mathbb{R}^{\ell}$ to $\mathbb{R}^{d-\ell}$ is a
mapping
$K:\mathbb{R}^{\ell}\times\mathcal{B}(\mathbb{R}^{d-\ell})\rightarrow\mathbb{I}$
such that for every fixed $E\in\mathcal{B}(\mathbb{R}^{d-\ell})$ the mapping
$\mathbf{x}\mapsto K(\mathbf{x},E)$ is
$\mathcal{B}(\mathbb{R}^{\ell})$-$\mathcal{B}(\mathbb{R}^{d-\ell})$-measurable
and for every fixed $\mathbf{x}\in\mathbb{R}^{\ell}$ the mapping $E\mapsto
K(\mathbf{x},E)$ is a probability measure on
$\mathcal{B}(\mathbb{R}^{d-\ell})$. Given a $(d-\ell)$-dimensional random
vector $\mathbf{Y}$ and a $\ell$-dimensional random vector $\mathbf{X}$ on a
probability space $(\Omega,\mathcal{A},\mathbb{P})$ we say that a Markov
kernel $K$ is a regular conditional distribution of $\mathbf{Y}$ given
$\mathbf{X}$ if
$K(\mathbf{X}(\omega),E)=\mathbb{E}(\mathbf{1}_{E}\circ\mathbf{Y}|\mathbf{X})(\omega)$
holds $\mathbb{P}$-almost surely for every
$E\in\mathcal{B}(\mathbb{R}^{d-\ell})$. It is well-known that for each pair of
random vectors $(\mathbf{X},\mathbf{Y})$ as above, a regular conditional
distribution $K(\cdot,\cdot)$ of $\mathbf{Y}$ given $\mathbf{X}$ always exists
and is unique for $\mathbb{P}^{\mathbf{X}}$-a.e.
$\mathbf{x}\in\mathbb{R}^{\ell}$ whereby $\mathbb{P}^{\mathbf{X}}$ denotes the
push-forward of $\mathbb{P}$ via $\mathbf{X}$. In case
$(\mathbf{X},\mathbf{Y})$ has distribution function $H$ we let
$K_{H}:\mathbb{R}^{\ell}\times\mathcal{B}(\mathbb{R}^{d-\ell})\to\mathbb{I}$
denote (a version of) the regular conditional distribution of $\mathbf{Y}$
given $\mathbf{X}$ and refer to it as $\ell$-_Markov kernel_ of $H$. Defining
the $\mathbf{x}$-section of a set $G\in\mathcal{B}(\mathbb{R}^{d})$ w.r.t. the
first $\ell$ coordinates by
$G_{\mathbf{x}}:=\\{\mathbf{y}\in\mathbb{R}^{d-\ell}:(\mathbf{x},\mathbf{y})\in
G\\}\in\mathcal{B}(\mathbb{R}^{d-\ell})$ the so-called _disintegration
theorem_ implies

$\displaystyle\mu_{H}(G)=\int_{\mathbb{R}^{\ell}}K_{H}(\mathbf{x},G_{\mathbf{x}})\
\mathrm{d}\mu_{H^{1:\ell}}(\mathbf{x}).$

For more background on conditional expectation and disintegration we refer to
[19] and [23].

Markov kernels allow to define metrics stronger than $d_{\infty}$. Indeed,
following [32] and defining for two copulas $A,B\in\mathcal{C}^{d}$

$\displaystyle D_{1}(A,B)$
$\displaystyle:=\int_{\mathbb{I}^{d}}|K_{A}(x,[\mathbf{0},\mathbf{y}])-K_{B}(x,[\mathbf{0},\mathbf{y}])|\
\mathrm{d}\lambda_{d}(x,\mathbf{y}),$ $\displaystyle D_{2}^{2}(A,B)$
$\displaystyle:=\int_{\mathbb{I}^{d}}(K_{A}(x,[\mathbf{0},\mathbf{y}])-K_{B}(x,[\mathbf{0},\mathbf{y}]))^{2}\
\mathrm{d}\lambda_{d}(x,\mathbf{y}),$ $\displaystyle D_{\infty}(A,B)$
$\displaystyle:=\sup_{\mathbf{y}\in\mathbb{I}^{d-1}}\int_{\mathbb{I}}|K_{A}(x,[\mathbf{0},\mathbf{y}])-K_{B}(x,[\mathbf{0},\mathbf{y}])|\
\mathrm{d}\lambda_{1}(x)$

yield $1$-Markov kernel based metrics on $\mathcal{C}^{d}$ and it is well-
known that convergence w.r.t. either of these metrics implies uniform
convergence (and not vice versa).

In this paper we will mainly consider the class $\mathcal{C}^{d}_{\text{ar}}$
of Archimedean copulas. Following [25], $C\in\mathcal{C}^{d}_{\text{ar}}$ if
there exists a continuous, non-increasing function
$\psi:[0,\infty)\to\mathbb{I}$ fulfilling
$\psi(0)=1,\lim_{z\to\infty}\psi(z)=0$ and being strictly decreasing on
$[0,\inf\\{z\in[0,\infty]:\psi(z)=0\\}]$ (where we set $\psi(\infty)=0$) such
that for every $\mathbf{x}\in\mathbb{I}^{d}$ we have

$\displaystyle C(\mathbf{x})=\psi(\varphi(x_{1})+\cdots+\varphi(x_{d})),$

where $\varphi:(0,1]\to[0,\infty)$ denotes the pseudoinverse of $\psi$.
Setting $\varphi(0)=\inf\\{z\in[0,\infty]:\psi(z)=0\\}$ the function $\varphi$
is strictly decreasing and satisfies $\varphi(1)=0$. In case of
$\varphi(0)=\infty$ we call $C$ strict and non-strict otherwise. Conversely,
the function $\psi(\varphi(x_{1})+\cdots+\varphi(x_{d}))$ for
$\mathbf{x}\in\mathbb{I}^{d}$ returns a $d$-dimensional copula if, and only
if, $\psi$ is $d$-monotone on $[0,\infty)$ meaning that $\psi$ is continuous
on $[0,\infty)$, has derivatives up to order $(d-2)$ on $(0,\infty)$, fulfills
$(-1)^{m}\psi^{(m)}(z)\geq 0$ for every $z\in(0,\infty)$, $m=0,1,\ldots,d-2$,
and it holds that $(-1)^{d-2}\psi^{(d-2)}$ is non-increasing and convex on
$(0,\infty)$. Thereby $\psi^{(m)}$ denotes the $m$-th derivative of $\psi$.
Letting $D^{-}\psi$ ($D^{+}\psi$) denote the left (right) derivative of $\psi$
and considering convexity of $(-1)^{d-2}\psi^{(d-2)}$ we have
$D^{-}\psi=D^{+}\psi$ almost everywhere on $(0,\infty)$. As direct consequence
the set of discontinuity points $(0,\infty)\setminus$
Cont$(D^{-}\psi^{(d-2)})$ of $D^{-}\psi^{(d-2)}$ is at most countably infinite
and it follows directly from the properties of $\psi$ that
$\lim_{z\to\infty}D^{-}\psi^{(d-2)}(z)=0$ holds (see [20]). As generators of
Archimedean copulas are not unique, from now on we will assume that generators
are normalized in the sense that $\varphi(\frac{1}{2})=1$, or equivalently,
$\psi(1)=\frac{1}{2}$.

##  3\. Markov kernels of multivariate Archimedean copulas

It is well-known (cf. [25]) that in case of $C\in\mathcal{C}_{\text{ar}}^{d}$
being absolutely continuous (a version of) its density is given by

(3.1) $\displaystyle
c(\mathbf{x})=\mathds{1}_{(0,1)^{d}}(\mathbf{x})\prod_{i=1}^{d}\varphi^{\prime}(x_{i})\cdot
D^{-}\psi^{(d-1)}\big{(}\varphi(x_{1})+\cdots+\varphi(x_{d})\big{)}$

and that all lower dimensional marginals of Archimedean copulas are absolutely
continuous. Combining the latter property with Theorem A.1, Lemma A.2 in A and
[20, Theorem 3.1] allows to derive an explicit expression for (a version of)
_every_ Markov kernel of an Archimedean copula. To that end, define the zero
set $L_{0}$ of $C\in\mathcal{C}^{d}_{\text{ar}}$ as

$\displaystyle L_{0}$
$\displaystyle=\\{(\mathbf{x},\mathbf{y})\in\mathbb{I}^{\ell}\times\mathbb{I}^{d-\ell}:C(\mathbf{x},\mathbf{y})=0\\}$
$\displaystyle=\left\\{(\mathbf{x},\mathbf{y})\in\mathbb{I}^{\ell}\times\mathbb{I}^{d-\ell}:\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{i})\geq\varphi(0)\right\\}.$

In what follows we will work with the zero level set of the $\ell$-dimensional
marginal of $C$ which is defined analogously and denoted by $L_{0}^{1:\ell}$.
Writing int$S$ for the interior of a set $S$, notice that
$(\mathbf{x},\mathbf{y})\in$ int$L_{0}$ if, and only if,
$\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{j})>\varphi(0)$
and that for strict $C\in\mathcal{C}^{d}_{\text{ar}}$ we have $\mathbf{x}\in
L_{0}^{1:\ell}$ if, and only if, $\min(\mathbf{x})=0$

######  Theorem 3.1.

Suppose that $C_{\psi}$ is a $d$-dimensional Archimedean copula with generator
$\psi$. Then for $\ell\in\\{1,\ldots,d-1\\}$ setting

(3.2) $\displaystyle
K_{\psi}(\mathbf{x},[\mathbf{0},\mathbf{y}])=\begin{cases}1&\emph{ if
}\min(\mathbf{x})=1\emph{ or }\mathbf{x}\in L_{0}^{1:\ell}\\\ 0&\emph{ if
}\min(\mathbf{x})\in(0,1),\mathbf{x}\not\in
L_{0}^{1:\ell},(\mathbf{x},\mathbf{y})\in\emph{int}L_{0}\\\
\frac{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{i})\right)}{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}&\emph{
if }\min(\mathbf{x})\in(0,1),\mathbf{x}\not\in
L_{0}^{1:\ell},(\mathbf{x},\mathbf{y})\not\in\emph{int}L_{0}\end{cases}$

yields (a version of) the $\ell$-Markov kernel of $C_{\psi}$.

###### Proof.

For $\min(\mathbf{x})\in\\{0,1\\}$ or $\mathbf{x}\in$ $L_{0}^{1:\ell}$ the
mapping $F_{\mathbf{x}}:$ $\mathbf{y}\mapsto
K_{\psi}(\mathbf{x},[\mathbf{0},\mathbf{y}])$ is obviously a
$(d-\ell)$-dimensional distribution function so fix $\mathbf{x}\not\in$
$L_{0}^{1:\ell},\min(\mathbf{x})\in(0,1)$. First notice that right-continuity
of $F_{\mathbf{x}}$ in every coordinate is a consequence of the continuity of
$\psi^{(\ell)}$ (if $\ell=d-1$, $\psi^{(\ell)}$ is replaced be the left-hand
derivative $D^{-}\psi^{(d-2)}$ whose left-continuity yields right-continuity
of $F_{\mathbf{x}}$). Concerning $(d-\ell)$-increasingness suppose
$\mathbf{a},\mathbf{b}\in\mathbb{I}^{d-\ell}$ such that $b_{j}\geq a_{j}$ for
$j=1,\ldots,d-\ell$. As in the previous lemma we have to verify that
$V_{F_{\mathbf{x}}}((\mathbf{a},\mathbf{b}])\geq 0$ holds. We distinguish the
following cases: (i) $(\mathbf{x},\mathbf{a}),(\mathbf{x},\mathbf{b})\not\in$
int$L_{0}$. Then
$F_{\mathbf{x}}(\mathbf{y})=\frac{\partial^{\ell}C(\mathbf{x},\mathbf{y})}{\partial
x_{1}\partial x_{2}\cdots\partial x_{\ell}}\cdot c^{1:\ell}(\mathbf{x})^{-1}$
and Lemma A.2 yields $(d-\ell)$-increasingness. (ii)
$(\mathbf{x},\mathbf{b})\in$ int$L_{0}$. Then also
$(\mathbf{x},\mathbf{a})\in$ int$L_{0}$ so that
$V_{F_{\mathbf{x}}}((\mathbf{a},\mathbf{b}])=0$ holds. (iii)
$(\mathbf{x},\mathbf{a})\in$ int$L_{0}$ and $(\mathbf{x},\mathbf{b})\not\in$
int$L_{0}$. Whenever a vertex $\mathbf{v}$ of $(\mathbf{a},\mathbf{b}]$
fulfills $\mathbf{v}\in$ int$L_{0}$ we have
$\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(v_{j})>\varphi(0)$
and hence $F_{\mathbf{x}}(\mathbf{v})=0$. Moreover, we already know that
$\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(v_{j})\right)=0$
holds too, so again we have

$F_{\mathbf{x}}(\mathbf{v})=0=\frac{\partial^{\ell}C(\mathbf{x},\mathbf{v})}{\partial
x_{1}\partial x_{2}\cdots\partial x_{\ell}}\cdot c^{1:\ell}(\mathbf{x})^{-1}$

and Lemma A.2 yields the desired property.

Altogether, for every fixed $\mathbf{x}\in\mathbb{I}^{\ell}$ the map
$F_{\mathbf{x}}:\mathbf{y}\mapsto
K_{\psi}(\mathbf{x},[\mathbf{0},\mathbf{y}])$ is a $(d-\ell)$-dimensional
distribution function and it follows that $K_{\psi}(\mathbf{x},\cdot)$ is a
probability measure on $\mathcal{B}(\mathbb{I}^{d-\ell})$. On the other hand,
for every fixed $\mathbf{y}\in\mathbb{I}^{d-\ell}$ the mapping
$\mathbf{x}\mapsto K_{\psi}(\mathbf{x},[\mathbf{0},\mathbf{y}])$ is
measurable. Since the collection
$\\{F\in\mathcal{B}(\mathbb{I}^{d-\ell}):\mathbf{x}\mapsto
K_{\psi}(\mathbf{x},F)\text{ is measurable}\\}$ is a Dynkin system and
contains a generating class of $\mathcal{B}(\mathbb{I}^{d-\ell})$ it follows
that $\mathbf{x}\mapsto K_{\psi}(\mathbf{x},E)$ is measurable for every
$E\in\mathcal{B}(\mathbb{I}^{d-\ell})$. Consequently, $K_{\psi}(\cdot,\cdot)$
is a Markov kernel from $\mathbb{I}^{\ell}$ to $\mathbb{I}^{d-\ell}$ and it
remains to show that $K_{\psi}(\cdot,\cdot)$ is an $\ell$-Markov kernel of
$C_{\psi}$ which can be done in a similar manner as in [20, Theorem 3.1]
(included hereafter for the sake of completeness): We have to prove the
identity

$\int_{[\mathbf{0},\mathbf{x}]}K_{\psi}(\mathbf{s},[\mathbf{0},\mathbf{y}])\
\mathrm{d}\mu_{C^{1:\ell}_{\psi}}(\mathbf{s})=C_{\psi}(\mathbf{x},\mathbf{y})$

for every
$(\mathbf{x},\mathbf{y})\in\mathbb{I}^{\ell}\times\mathbb{I}^{d-\ell}$. If
$\min(\mathbf{y})=0$ we have $(\mathbf{x},\mathbf{y})\in L_{0}$ whereas
$\min(\mathbf{y})=1$ gives
$\int_{[\mathbf{0},\mathbf{x}]}K_{\psi}(\mathbf{s},[\mathbf{0},\mathbf{1}])\
\mathrm{d}\mu_{C^{1:\ell}_{\psi}}(\mathbf{s})=\mu_{C^{1:\ell}}([\mathbf{0},\mathbf{x}])=C_{\psi}(\mathbf{x},\mathbf{1})$
so fix $\mathbf{y}\in(\mathbf{0},\mathbf{1})$. As
$\mu_{C^{1:\ell}_{\psi}}(L_{0}^{1:\ell})=0$ and considering absolute
continuity of $\mu_{C^{1:\ell}}$ we get

$\displaystyle(\star)$
$\displaystyle:=\int_{[\mathbf{0},\mathbf{x}]}K_{\psi}(\mathbf{s},[\mathbf{0},\mathbf{y}])\
\mathrm{d}\mu_{C^{1:\ell}_{\psi}}(\mathbf{s})=\int_{[\mathbf{0},\mathbf{x}]\setminus
L_{0}^{1:\ell}}K_{\psi}(\mathbf{s},[\mathbf{0},\mathbf{y}])c^{1:\ell}(\mathbf{s})\
\mathrm{d}\lambda_{\ell}(\mathbf{s})$
$\displaystyle=\int_{[\mathbf{0},\mathbf{x}]\cap\\{\mathbf{t}\in\mathbb{I}^{\ell}\setminus
L_{0}^{1:\ell}:(\mathbf{t},\mathbf{y})\not\in\text{int}L_{0}\\}}\prod_{i=1}^{\ell}\varphi^{\prime}(s_{i})\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(s_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)\
\mathrm{d}\lambda_{\ell}(\mathbf{s}).$

If $\mathbf{s}\in L_{0}^{1:\ell}$ then $(\mathbf{s},\mathbf{y})\in L_{0}$ if,
and only if,
$\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{j})>\varphi(0)$
and hence
$\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)=0$
and whenever $\mathbf{s}\not\in L_{0}^{1:\ell}$ we have
$\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)=0$,
too. As direct consequence,

$\displaystyle(\star)$
$\displaystyle=\int_{(\mathbf{0},\mathbf{x}]}\prod_{i=1}^{\ell}\varphi^{\prime}(s_{i})\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(s_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)\
\mathrm{d}\lambda_{\ell}(\mathbf{s})$
$\displaystyle=\int_{(\mathbf{0},\mathbf{x}_{1:\ell-1}]}\prod_{i=1}^{\ell-1}\varphi^{\prime}(s_{i})\int_{(0,x_{\ell}]}\varphi^{\prime}(s_{\ell})\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(s_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)\
\mathrm{d}\lambda(s_{\ell})\mathrm{d}\lambda_{\ell-1}(\mathbf{s}_{1:\ell-1})$

Applying change of coordinates and considering that
$\psi^{(r)}(\infty):=\psi^{(r)}(\lim_{z\to\infty}z)=0$ for every
$r\in\\{1,\ldots,d-2\\}$ yields

$\displaystyle(\star)$
$\displaystyle=\int_{(\mathbf{0},\mathbf{x}_{1:\ell-1}]}\prod_{i=1}^{\ell}\varphi^{\prime}(s_{i})\lim_{\delta\to
0}\Biggl{(}\psi^{(\ell-1)}\left(\sum_{i=1}^{\ell-1}\varphi(s_{i})+\varphi(x_{\ell})+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)$
$\displaystyle\qquad-\psi^{(\ell-1)}\left(\sum_{i=1}^{\ell-1}\varphi(s_{i})+\varphi(\delta)+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)\Biggr{)}\
\mathrm{d}\lambda_{\ell-1}(\mathbf{s}_{1:\ell-1})$
$\displaystyle=\int_{(\mathbf{0},\mathbf{x}_{1:\ell-1}]}\prod_{i=1}^{\ell}\varphi^{\prime}(s_{i})\psi^{(\ell-1)}\left(\sum_{i=1}^{\ell-1}\varphi(s_{i})+\varphi(x_{\ell})+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)\
\mathrm{d}\lambda_{\ell-1}(\mathbf{s}_{1:\ell-1}).$

Iterating this procedure another $\ell-2$ times finally results in

$\displaystyle(\star)$
$\displaystyle=\int_{(0,x_{1}]}\varphi^{\prime}(s_{1})\psi^{\prime}\left(\varphi(s_{1})+\sum_{i=2}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)$
$\displaystyle=\psi\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(y_{j})\right)=C_{\psi}(\mathbf{x},\mathbf{y})$

as required. ∎

##  4\. Convergence in $\mathcal{C}_{\text{ar}}^{d}$

In this section we extend the results in [21] characterizing uniform
convergence in the space of two-dimensional Archimedean copulas to the
multivariate setting. There already exist several characterizations in terms
of the corresponding generators, inverse generators, generating probability
measures, marginal copulas and marginal densities ([20], also see [2]). In
what follows, we contribute the items of convergence w.r.t. the metric $D_{1}$
(and $D_{2},D_{\infty}$) as well as $1$-weak conditional convergence to the
list of characterizing properties. Thereby, given $d$-dimensional copulas
$C,C_{1},C_{2},\ldots$ with corresponding $\ell$-Markov kernels
$K_{C},K_{C_{1}},K_{C_{2}},\ldots$, where $\ell\in\\{1,\ldots,d-1\\}$, we say
that $(C_{n})_{n\in\mathbb{N}}$ converges $\ell$-_weakly conditional_ to $C$
if, and only if, there exists a set $\Lambda\in\mathcal{B}(\mathbb{I}^{\ell})$
with $\mu_{C^{1:\ell}}(\Lambda)=1$ such that for every $\mathbf{x}\in\Lambda$
the sequence $(K_{C_{n}}(\mathbf{x},\cdot))_{n\in\mathbb{N}}$ of probability
measures on $\mathcal{B}(\mathbb{I}^{d-\ell})$ converges weakly to the
probability measure $K_{C}(\mathbf{x},\cdot)$ (cf. [13, Section 4]).
Considering $\ell>1$, however, this notion may fail to be reasonable in full
generality as for arbitrary $A,B\in\mathcal{C}^{d}$ it is possible that
$\mu_{A^{1:\ell}}$ and $\mu_{B^{1:\ell}}$ are singular with respect to each
other, and being only defined uniquely $\mu_{A^{1:\ell}}$-almost everywhere,
$\mu_{B^{1:\ell}}$-almost everywhere, respectively, comparing
$K_{A}(\mathbf{x},\cdot)$ and $K_{B}(\mathbf{x},\cdot)$ is nonsensical. As
pointed out in [20], however, for special classes of copulas such as the class
of copulas with identical marginals or Archimedean copulas this notion does
make sense. Since every Archimedean copula $C$ has absolutely continuous
marginals and its $\ell$-marginal density $c^{1:\ell}$ fulfills
$c^{1:\ell}(\mathbf{x})>0$ for every $\mathbf{x}\in\mathbb{I}^{\ell}\setminus
L_{0}^{1:\ell}$, the marginal measures $\mu_{C^{1:\ell}_{1}}$,
$\mu_{C^{1:\ell}_{2}}$ of two Archimedean copulas $C_{1},C_{2}$ can not be
singular with respect to each other.

The following characterization of uniform convergence for multivariate
Archimedean copulas is due to [20]:

######  Theorem 4.1.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas
with generators $\psi,\psi_{1},\psi_{2},\ldots$, respectively. Then the
following assertions are equivalent:

  1. (1)

$(C_{n})_{n\in\mathbb{N}}$ converges uniformly to $C$,

  2. (2)

$(C_{n}^{i:j})_{n\in\mathbb{N}}$ converges uniformly to $C^{i:j}$ for any
$1\leq i<j\leq d$,

  3. (3)

$(\varphi_{n})_{n\in\mathbb{N}}$ converges pointwise to $\varphi$ on $(0,1]$,

  4. (4)

$(\psi_{n})_{n\in\mathbb{N}}$ converges uniformly to $\psi$ on $[0,\infty)$,

  5. (5)

$(\psi_{n}^{(m)})_{n\in\mathbb{N}}$ converges pointwise to $\psi^{(m)}$ on
$(0,\infty)$ for $m\in\\{1,\ldots,d-2\\}$ and
$(D^{-}\psi_{n}^{(d-2)})_{n\in\mathbb{N}}$ converges pointwise to
$D^{-}\psi^{(d-2)}$ on _Cont_ $(D^{-}\psi^{(d-2)})$,

  6. (6)

$(c_{n}^{1:m})_{n\in\mathbb{N}}$ converges to $c^{1:m}$ a.e. in
$\mathbb{I}^{m}$, $m\in\\{2,\ldots,d-1\\}$.

As direct consequence of Theorem 3.1 (a version of) the 1-Markov kernel of an
Archimedean copula $C_{\psi}$ is given by

$\displaystyle
K_{\psi}(x_{1},\bigtimes_{i=2}^{d}[0,x_{i}])=\begin{cases}1&\text{ if
}x_{1}\in\\{0,1\\}\\\ 0&\text{ if
}x_{1}\in(0,1),\mathbf{x}\in\text{int}L_{0}\\\
\frac{\psi^{\prime}\left(\sum_{i=1}^{d}\varphi(x_{i})\right)}{\psi^{\prime}\left(\varphi(x_{1})\right)}&\text{
if }x_{1}\in(0,1),\mathbf{x}\not\in\text{int}L_{0}\end{cases}.$

The following lemma contributes a characterization of pointwise/uniform
convergence of Archimedean copulas in terms of $1$-Markov kernels (i.e.,
$(d-1)$-dimensional conditional distributions):

######  Lemma 4.2.

Suppose that $C_{\psi},C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean
copulas with generators $\psi,\psi_{1},\psi_{2},\ldots$ and 1-Markov kernels
$K_{\psi},K_{1},K_{2},\ldots$, respectively. If $(C_{n})_{n\in\mathbb{N}}$
converges uniformly to $C$ then for every $x_{1}\in(0,1)$

$\displaystyle\lim_{n\to\infty}K_{n}(x_{1},\bigtimes_{i=2}^{d}[0,x_{i}])=K_{\psi}(x_{1},\bigtimes_{i=2}^{d}[0,x_{i}])$

holds for every $(x_{2},\ldots,x_{d})\in\mathbb{I}^{d-1}$.

###### Proof.

Suppose $C$ is non-strict and fix $x_{1}\in(0,1)$. First, consider the
$x_{1}$-section of $L_{0}$ and let
$(x_{2},\ldots,x_{d})\in(0,1)^{d-1}\cap(\text{int}L_{0}$)${}_{x_{1}}$, i.e.,
$\sum_{i=1}^{d}\varphi(x_{i})>\varphi(0)$. Defining
$z_{n}:=\sum_{i=1}^{d}\varphi_{n}(x_{i})$, $z:=\sum_{i=1}^{d}\varphi(x_{i})$
and applying Theorem 4.1, the sequence $\left(z_{n}\right)_{n\in\mathbb{N}}$
converges to $z>\varphi(0)$ and according to [20, Lemma 4.3, Lemma 4.4] we
have $\psi^{\prime}_{n}\left(z_{n}\right)\to\psi^{\prime}\left(z\right)=0$ for
$n\to\infty$. Additionally,
$\psi^{\prime}_{n}(\varphi_{n}(x_{1}))\to\psi^{\prime}(\varphi(x_{1}))$ as
$n\to\infty$ and taking together we get
$K_{n}(x_{1},\bigtimes_{i=2}^{d}[0,x_{i}])\to
K_{\psi}(x_{1},\bigtimes_{i=2}^{d}[0,x_{i}])$ for $n\to\infty$. Second,
consider
$(x_{2},x_{3},\ldots,x_{d})\in(0,1)^{d-1}\setminus(\text{int}L_{0}$)${}_{x_{1}}$.
Again using $z_{n}$, $z$ defined as above (in this case we have
$z<\varphi(0)$) and [20, Lemma 4.3] we directly obtain
$\psi_{n}^{\prime}(z_{n})\to\psi^{\prime}(z)$ as well as
$\psi^{\prime}_{n}(\varphi_{n}(x_{1}))\to\psi^{\prime}(\varphi(x_{1}))$ as
$n\to\infty$ from which convergence of the $1$-Markov kernels follows. For
strict limit copula $C$ we have int$L_{0}=\varnothing$ and the assertion
results in the same way as for the second case above. ∎

Furthermore, according to [6, Lemma 3, Lemma 5] $1$-weak conditional
convergence implies convergence w.r.t. $D_{1}$, $D_{2}$ and w.r.t.
$D_{\infty}$ which in turn all imply uniform convergence. Incorporating these
facts directly yields the following extension of Theorem 4.1:

######  Theorem 4.3.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas.
Then any of the statements in Theorem 4.1 is equivalent to any of the
following:

  1. (1)

$(C_{n})_{n\in\mathbb{N}}$ converges to $C$ w.r.t. $D_{i}$, $i=1,2$,

  2. (2)

$(C_{n})_{n\in\mathbb{N}}$ converges to $C$ w.r.t. $D_{\infty}$,

  3. (3)

$(C_{n})_{n\in\mathbb{N}}$ converges $1$-weakly conditional to $C$.

Having all Markov kernels of Archimedean copulas at hand, the next corollary
says that within the class of $d$-dimensional Archimedean copulas uniform
convergence (and thus any of the equivalent conditions) implies $\ell$-weak
conditional convergence for every $\ell\in\\{1,\ldots,d-1\\}$, constituting an
additional way of extending [20, Theorem 4.1].

######  Corollary 4.4.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas
with generators $\psi,\psi_{1},\psi_{2},\ldots$, respectively. If
$(C_{n})_{n\in\mathbb{N}}$ converges uniformly to $C$ then for any
$\ell\in\\{1,\ldots,d-1\\}$ there exists a set
$\Lambda\in\mathcal{B}(\mathbb{I}^{\ell})$ with $\mu_{C^{1:\ell}}(\Lambda)=1$
such that for all $\mathbf{x}\in\Lambda$ we have

$\displaystyle\lim_{n\to\infty}K_{n}(\mathbf{x},[\mathbf{0},\mathbf{y}])=K_{\psi}(\mathbf{x},[\mathbf{0},\mathbf{y}])$

for every $\mathbf{y}\in U^{\mathbf{x}}$, where $U^{\mathbf{x}}$ is dense in
$\mathbb{I}^{d-\ell}$.

###### Proof.

The case $\ell=d-1$ can be found in [20] while Lemma 4.2 deals with the case
$\ell=1$. Suppose $\ell\in\\{2,\ldots,d-2\\}$ then considering that
$\psi^{(\ell)}$ is continuous on $(0,\infty)$ and setting
$\Lambda:=\mathbb{I}^{\ell}\setminus$ $L_{0}^{1:\ell}$ yields
$\mu_{C^{1:\ell}}(\Lambda)=1$. Now fix $\mathbf{x}\in\Lambda$. Proceeding as
in Lemma 4.2 and distinguishing the cases
$\mathbf{y}\in(\text{int}L_{0})_{\mathbf{x}}$ and
$\mathbf{y}\in\mathbb{I}^{d-\ell}\setminus(\text{int}L_{0})_{\mathbf{x}}$ the
result follows. ∎

##  5\. Singularity in $\mathcal{C}_{\text{ar}}^{d}$

Turning to singularity aspects, the main goal of this short section is to show
that a $d$-dimensional copula $C$ with absolutely continuous $\ell$-marginal
is singular if, and only if, its $\ell$-Markov kernel is singular w.r.t.
$\lambda_{d-\ell}$. As direct consequence we obtain the fact that an
Archimedean copula is singular if, and only if, its $\ell$-Markov kernel is
singular w.r.t. $\lambda_{d-\ell}$ for _every_ $\ell\in\\{1,\ldots,d-1\\}$.
According to [24] the $\ell$-Markov kernel $K_{C}(\mathbf{x},\cdot)$ of a
$d$-dimensional copula $C$ can be decomposed into the sum of two substochastic
kernels $K_{C}^{\text{abs}}(\mathbf{x},\cdot)$ and
$K_{C}^{\text{sing}}(\mathbf{x},\cdot)$, i.e. for every
$\mathbf{x}\in\mathbb{I}^{\ell}$ and $E\in\mathcal{B}(\mathbb{I}^{d-\ell})$ it
holds that

$\displaystyle
K_{C}(\mathbf{x},E)=K_{C}^{\text{abs}}(\mathbf{x},E)+K_{C}^{\text{sing}}(\mathbf{x},E),$

where the measure $K_{C}^{\text{abs}}(\mathbf{x},\cdot)$ is absolutely
continuous w.r.t. $\lambda_{d-\ell}$ and the measure
$K_{C}^{\text{sing}}(\mathbf{x},\cdot)$ is singular w.r.t. $\lambda_{d-\ell}$.

The next theorem is this section’s main result:

######  Theorem 5.1.

Suppose that $C$ is a $d$-dimensional copula with absolutely continuous
$\ell$-marginal for some $\ell\in\\{2,\ldots,d-1\\}$. Then $C$ is singular
(w.r.t. to $\lambda_{d}$) if, and only if, there exists a set
$\Lambda\in\mathcal{B}(\mathbb{I}^{\ell})$ with $\mu_{C^{1:\ell}}(\Lambda)=1$
such that for all $\mathbf{x}\in\Lambda$ we have $K_{C}(\mathbf{x},\cdot)$ is
singular w.r.t. $\lambda_{d-\ell}$.

###### Proof.

Assuming singularity of $C$ there exists a set
$E\in\mathcal{B}(\mathbb{I}^{d})$ such that $\mu_{C}$ concentrates its mass on
$E$ while $E$ is a $\lambda_{d}$-null set. Using disintegration twice yields,
on the one hand, the existence of a set $\Upsilon_{1}$ with
$\mu_{C^{1:\ell}}(\Upsilon_{1})=1$ such that for all
$\mathbf{x}\in\Upsilon_{1}$ we have $K_{C}(\mathbf{x},E_{\mathbf{x}})=1$ and,
on the other hand, a set $\Upsilon_{2}$ with $\lambda_{\ell}(\Upsilon_{2})=1$
such that for all $\mathbf{x}\in\Upsilon_{2}$ we have
$\lambda_{d-\ell}(E_{\mathbf{x}})=0.$

Further, $\mu_{C^{1:\ell}}(\Upsilon_{2})=\int_{\Upsilon_{2}}c^{1:\ell}\
\mathrm{d}\lambda_{\ell}=\int_{\mathbb{I}^{\ell}}c^{1:\ell}\
\mathrm{d}\lambda_{\ell}=1.$ Considering that
$\mu_{C^{1:\ell}}(\Upsilon_{1}\cap\Upsilon_{2})=1$ the desired property
follows.

Assuming the converse, for every $\mathbf{x}\in\Lambda$ we have
$K_{C}(\mathbf{x},\cdot)\perp\lambda_{d-\ell}$ and hence also
$K_{C}^{\text{abs}}(\mathbf{x},\mathbb{I}^{d-\ell})=0$. We show that the
absolutely continuous part $\mu_{C}^{\text{abs}}$ of the Lebesgue
decomposition of $\mu_{C}$ vanishes. Again denoting by $c^{1:d}$ the Radon-
Nikodym derivative of $\mu_{C}$ w.r.t. $\lambda_{d}$, Lemma B.1 in B directly
yields

$\displaystyle 0$
$\displaystyle=\int_{\Lambda}K_{C}^{\text{abs}}(\mathbf{x},\mathbb{I}^{d-\ell})\
\mathrm{d}\mu_{C^{1:\ell}}(\mathbf{x})=\int_{\Lambda}\int_{\mathbb{I}^{d-\ell}}\frac{c^{1:d}(\mathbf{x},\mathbf{y})}{c^{1:\ell}(\mathbf{x})}\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y})\mathrm{d}\mu_{C^{1:\ell}}(\mathbf{x})$
$\displaystyle=\int_{\mathbb{I}^{\ell}}\int_{\mathbb{I}^{d-\ell}}\frac{c^{1:d}(\mathbf{x},\mathbf{y})}{c^{1:\ell}(\mathbf{x})}\cdot
c^{1:\ell}(\mathbf{x})\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y})\mathrm{d}\lambda_{\ell}(\mathbf{x})=\int_{\mathbb{I}^{d}}c^{1:d}(\mathbf{x},\mathbf{y})\
\mathrm{d}\lambda_{d}(\mathbf{x},\mathbf{y})=\mu_{C}^{\text{abs}}(\mathbb{I}^{d})$

completing the proof. ∎

The following corollary is immediate:

######  Corollary 5.2.

A $d$-dimensional Archimedean copula $C$ is singular if, and only if, every
uni- and multivariate Markov kernel is singular w.r.t. the appropriate
Lebesgue measure.

######  Remark 5.3.

The result in Corollary 5.2 is in strong contrast to the general case:
Consider, for instance, the simple example of the $3$-dimensional copula
$B(x,y,z)=z\cdot\min(x,y)$ with marginals $B^{12}=M,B^{13}=B^{23}=\Pi_{2}$ as
in Figure 1. Then for $\lambda_{1}$-a.e. $x\in\mathbb{I}$ the probability
measure $K_{B}(x,\cdot)$ on $\mathcal{B}(\mathbb{I}^{2})$ is singular w.r.t.
$\lambda_{2}$ while for $\mu_{M}$-a.e. $(x,y)\in\mathbb{I}^{2}$ the
probability measure $K_{B}(x,y,\cdot)$ on $\mathcal{B}(\mathbb{I})$ is
absolutely continuous w.r.t. $\lambda_{1}$.

Figure 1. Sample of size $n=2000$ of the $3$-dimensional copula $B$ as
considered in Remark 5.3.

##  6\. Conditional Archimedean copulas from a Markov kernel perspective

As the dependence structure of a random vector may be heavily impacted by a
(vector of) covariate(s) the introduction of conditional copulas is natural to
determine the variation in dependence when changing the values of the
covariate(s). We particularly follow [26] where the authors study various
properties of conditional Archimedean copulas. In contrast, however, we will
introduce these conditional dependence structures largely from a Markov kernel
perspective which leads to several beautiful relations, and present an
alternative proof of the truncation invariance property while working with
general strict $d$-monotone generators instead of requiring complete
monotonicity. Furthermore, the results of the previous sections are used to
show that convergence behaviour, singularity as well as conditional
increasingness carry over from the original Archimedean copulas to their
conditional copulas. Additionally, we demonstrate that in this scenario
estimating (the generator of) the original copula suffices to get an estimator
of its conditional copula. From there, we propose a ‘conditional dependence’
measure (in analogy to conditional Kendall’s tau or conditional Spearman’s
rho, see, e.g., [11, 12]) as an alternative way to obtain additional
information of the dependence in data from an Archimedean model given
covariate values.

We start with the following simple lemma where we conduct a slight abuse of
notation and simply set

$\displaystyle
K_{C}(\mathbf{x},[0,y])=K_{C}(\mathbf{x},\mathbb{I}\times\ldots\times\mathbb{I}\times[0,y]\times\mathbb{I}\ldots\times\mathbb{I})$

for $\mathbf{x}\in\mathbb{I}^{\ell},y\in\mathbb{I}$.

######  Lemma 6.1.

Suppose that $C$ is a $d$-dimensional Archimedean copulas with strict
generator $\psi$ and let $\ell\in\\{1,\ldots,d-1\\}$. Then

$\displaystyle K_{C}(\mathbf{x},[0,y_{\ell+1}]\times\ldots\times[0,y_{d}])$
$\displaystyle=K_{C}(\mathbf{x},[0,C^{\ell+1:\ell+2}(y_{\ell+1},y_{\ell+2})]\times\ldots\times[0,C^{d-1:d}(y_{d-1},y_{d})])$
$\displaystyle=\ldots$
$\displaystyle=K_{C}(\mathbf{x},[0,y_{\ell+1}]\times[0,C^{\ell+2:d}(y_{\ell+2},\ldots,y_{d})])$
$\displaystyle=\ldots$
$\displaystyle=K_{C}(\mathbf{x},[0,C^{\ell+1:d}(y_{\ell+1},\ldots,y_{d})]).$

Therefore, consolidating the rectangles via the appropriate marginal copulas
we can vary the dimension of the measured set of the Markov kernel.

###### Proof.

Since $\psi$ is strict we have $\varphi(\psi(z))=z$ for all $z\in[0,\infty)$
so is is straightforward to verify

$\displaystyle K_{C}(\mathbf{x},[0,y_{\ell+1}]\times\ldots\times[0,y_{d}])$
$\displaystyle=\frac{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=\ell+1}^{d}\varphi(y_{j})\right)}{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}$
$\displaystyle=\frac{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\varphi\left(C^{\ell+1:d}(y_{\ell+1},\ldots,y_{d})\right)\right)}{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}$
$\displaystyle=K_{C}(\mathbf{x},[0,C^{\ell+1:d}(y_{\ell+1},\ldots,y_{d})])$

and similarly for all other expressions. Note that for $\ell=d-1$ the same
holds true by using the left-hand derivative $D^{-}\psi^{(d-2)}$ of the
$(d-2)$-nd derivative of $\psi$. ∎

From now on we consider a strict $d$-dimensional Archimedean copula $C$ and
let $\ell\in\\{1,\ldots,d-2\\}$ (notice that therefore $\psi^{(\ell)}$ is
continuous and invertible). The $\ell$-Markov kernel of $C$ (as given in eq.
(3.2)) being a $(d-\ell)$-dimensional distribution function for any fixed
$\mathbf{x}\in(0,1)^{\ell}$, Sklar’s Theorem (see, e.g., [5]) yields the
existence of some copula $C^{\mathbf{x}}\in\mathcal{C}^{d-\ell}$ such that
both,

$\displaystyle
K_{C}(\mathbf{x},[\mathbf{0},\mathbf{y}])=C^{\mathbf{x}}(K_{C}(\mathbf{x},[0,y_{1}]),\ldots,K_{C}(\mathbf{x},[0,y_{d-\ell}])),$
$\displaystyle
C^{\mathbf{x}}(u_{1},\ldots,u_{d-\ell})=K_{C}(\mathbf{x},[0,g_{\mathbf{x}}^{-1}(u_{1})]\times\ldots\times[0,g_{\mathbf{x}}^{-1}(u_{d-\ell})]),$

where, again slightly abusing notation,
$g_{\mathbf{x}}(y)=K_{C}(\mathbf{x},[0,y])$, $y\in[0,1]$, is the marginal
distribution of $K_{C}(\mathbf{x},\cdot)$ and $g_{\mathbf{x}}^{-1}$ its
inverse. Knowing the form of all Markov kernels, we compute
$g_{\mathbf{x}}^{-1}$ as

$\displaystyle
g_{\mathbf{x}}^{-1}(u)=\psi\left(\psi^{(\ell)^{-1}}\left(u\cdot\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)\right)-\sum_{i=1}^{\ell}\varphi(x_{i})\right)$

for $u\in\mathbb{I}$ and

$\displaystyle
C^{\mathbf{x}}(\mathbf{u})=\frac{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(g_{\mathbf{x}}^{-1}(u_{j}))\right)}{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}$

for $\mathbf{u}\in\mathbb{I}^{d-\ell}$. Notice that according to Lemma 6.1 we
can additionally write

$\displaystyle
C^{\mathbf{x}}(\mathbf{u})=K_{C}(\mathbf{x},[0,C^{1:d-\ell}(g_{\mathbf{x}}^{-1}(u_{1}),\ldots,g_{\mathbf{x}}^{-1}(u_{d-\ell}))])$

which will turn out to be useful in the sequel.

In [26] the surprising fact is stated that $C^{\mathbf{x}}$ is again of
Archimedean type if $C$ is induced by a completely monotone generator (the so-
called truncation invariance property). Not only will we get rid of the
restriction to completely monotone generators but offer an alternative
approach to deriving this assertion. We pick up the idea in [18] where the
authors used the well-known result [28, Theorem 4.1.6], saying that a
_bivariate_ associative copula $B$ fulfilling $B(x,x)<x$ for all $x\in(0,1)$
is necessarily Archimedean, to prove that the so-called lower tail dependence
copula of Archimedean copulas is again Archimedean. Similar conditions hold in
the general case. In fact, according to [30] a $d$-dimensional copula $C$ that
is associative and satisfies $C(x,\ldots,x)<x$ for every $x\in(0,1)$ is
necessarily Archimedean. Thereby, a $d$-variate function
$f:\mathbb{I}^{d}\to\mathbb{I}$ is defined to be associative if, and only if,
for all $x_{1},x_{2},\ldots,x_{2d-1}\in\mathbb{I}$ we have

$\displaystyle f(f(x_{1},\ldots,x_{d}),x_{d+1},\ldots,x_{2d-1})$
$\displaystyle=f(x_{1},f(x_{2},\ldots,x_{d+1}),x_{d+2},\ldots,x_{2d-1})$
$\displaystyle=\ldots$
$\displaystyle=f(x_{1},\ldots,x_{d-1},f(x_{d},\ldots,x_{2d-1})).$

We will now use this characterization to obtain a nice and elementary
derivation of the fact that the conditional copula of an Archimedean copula is
again Archimedean:

######  Proposition 6.2.

Suppose $C$ is a strict Archimedean copula and $\ell\in\\{1,\ldots,d-2\\}$.
For any $\mathbf{x}\in(0,1)^{\ell}$ the conditional copula $C^{\mathbf{x}}$ of
$C$ is a $(d-\ell)$-dimensional Archimedean copula.

###### Proof.

We start with associativity and suppose
$u_{1},\ldots,u_{2(d-\ell)-1}\in\mathbb{I}$. Plugging in and cancelling gives

$\displaystyle\varphi(g_{\mathbf{x}}^{-1}$
$\displaystyle(C^{\mathbf{x}}(u_{1},\ldots,u_{d-\ell})))$
$\displaystyle=\varphi\left(\psi\left(\psi^{(\ell)^{-1}}\left(\tfrac{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(g_{\mathbf{x}}^{-1}(u_{j}))\right)}{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}\cdot\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)\right)-\sum_{i=1}^{\ell}\varphi(x_{i})\right)\right)$
$\displaystyle=\sum_{j=1}^{d-\ell}\varphi(g_{\mathbf{x}}^{-1}(u_{j})).$

Thus, it follows that

$\displaystyle C^{\mathbf{x}}(C^{\mathbf{x}}(u_{1},\ldots,u_{d-\ell}),$
$\displaystyle u_{d-\ell+1},\ldots,u_{2(d-\ell)-1})$
$\displaystyle=\tfrac{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(g_{\mathbf{x}}^{-1}(u_{j}))+\sum_{r=d-\ell+1}^{2(d-\ell)-1}\varphi(g_{\mathbf{x}}^{-1}(u_{r}))\right)}{\varphi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}$
$\displaystyle=\tfrac{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{2(d-\ell)-1}\varphi(g_{\mathbf{x}}^{-1}(u_{j}))\right)}{\varphi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}$
$\displaystyle=C^{\mathbf{x}}(u_{1},C^{\mathbf{x}}(u_{2},\ldots,u_{d-\ell+1}),u_{d-\ell+2},\ldots,u_{2(d-\ell)-1})$
$\displaystyle=\ldots$
$\displaystyle=C^{\mathbf{x}}(u_{1},\ldots,u_{d-\ell-1},C^{\mathbf{x}}(u_{d-\ell},\ldots,u_{2(d-\ell)-1}))$

and $C^{\mathbf{x}}$ is associative. For the diagonal property let $u\in(0,1)$
and assume $\ell$ is even, so that $\psi^{(\ell)}\geq 0$ and decreasing (the
odd case follows analogously). Then it is straightforward to verify that
$C^{\mathbf{x}}(u,\ldots,u)<u$ holds if, and only if,

$\displaystyle\psi^{(\ell)}$
$\displaystyle\left(\sum_{i=1}^{\ell}\varphi(x_{i})+(d-\ell)\psi^{{(\ell)}^{-1}}\left(u\cdot\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)\right)-(d-\ell)\sum_{i=1}^{\ell}\varphi(x_{i})\right)$
$\displaystyle<u\cdot\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)$

which is the case if, and only if, $u<1$ as desired. Now, as direct
consequence of [30, Theorem 4.2] we have
$C^{\mathbf{x}}\in\mathcal{C}^{d-\ell}_{\text{ar}}$. ∎

In contrast to [26], we use a multivariate analogue of Nelsen’s method [28,
Theorem 4.3.8] to determine the (inverse) generator of $C^{\mathbf{x}}$ and
give a representation in terms of the Markov kernel which will lead to nice
interrelations between (conditional) Archimedean generators:

######  Lemma 6.3.

Suppose $C$ is a strict $d$-dimensional Archimedean copula with generator
$\psi$ and let $\ell\in\\{1,\ldots,d-2\\}$. For any
$\mathbf{x}\in(0,1)^{\ell}$ the Archimedean generator of the conditional
copula $C^{\mathbf{x}}$ of $C$ is given by

$\displaystyle\psi^{\mathbf{x}}(z)=K_{C}(\mathbf{x},[0,\psi(z)])$

for every $z\in[0,\infty)$.

###### Proof.

Similarly to [28, Theorem 4.3.8] it holds that for
$\mathbf{u}\in\mathbb{I}^{d-\ell}$ we have

$\displaystyle\frac{(\varphi^{\mathbf{x}}(u_{1}))^{\prime}}{(\varphi^{\mathbf{x}}(u_{2}))^{\prime}}=\frac{\varphi^{\prime}(g_{\mathbf{x}}^{-1}(u_{1}))\cdot(g_{\mathbf{x}}^{-1})^{\prime}(u_{1})}{\varphi^{\prime}(g_{\mathbf{x}}^{-1}(u_{2}))\cdot(g_{\mathbf{x}}^{-1})^{\prime}(u_{2})}$

for $\lambda$-almost all $u_{1},u_{2}\in(0,1)$. Hence,
$(\varphi^{\mathbf{x}})^{\prime}(s)=\alpha_{\mathbf{x}}\varphi^{\prime}(g_{\mathbf{x}}^{-1}(s))(g_{\mathbf{x}}^{-1})^{\prime}(s)$
where $\alpha_{\mathbf{x}}$ is a constant only depending on $\mathbf{x}$.
Integrating and substituting $t=g_{\mathbf{x}}^{-1}(s)$ yields

$\displaystyle\varphi^{\mathbf{x}}(s)=-\int_{[s,1]}(\varphi^{\mathbf{x}})^{\prime}(t)\
\mathrm{d}\lambda(t)=\alpha_{\mathbf{x}}\int_{[g_{\mathbf{x}}^{-1}(s),1]}\varphi^{\prime}(t)\
\mathrm{d}\lambda(t)=\alpha_{\mathbf{x}}\varphi(g_{\mathbf{x}}^{-1}(s)).$

As Archimedean generators are only unique up to a multiplicative constant
(see, e.g., [28]) we set $\alpha_{\mathbf{x}}=1$ and arrive at

$\displaystyle\psi^{\mathbf{x}}(z)=K_{C}(\mathbf{x},[0,\psi(z)]).$

It follows from the properties of $K_{C}(\mathbf{x},\cdot)$ and
$\psi^{(\ell)}$ that $\psi^{\mathbf{x}}$ is strictly decreasing and continuous
on $[0,\infty)$, fulfills $\psi^{\mathbf{x}}(0)=1$ as well as
$\lim_{z\to\infty}\psi^{\mathbf{x}}(z)=0$. Thus, for
$\mathbf{u}\in\mathbb{I}^{d-\ell}$ Lemma 6.1 gives

$\displaystyle\psi^{\mathbf{x}}\left(\sum_{i=1}^{d-\ell}\varphi^{\mathbf{x}}(u_{i})\right)$
$\displaystyle=K_{C}\left(\mathbf{x},\left[0,\psi\left(\sum_{j=1}^{d-\ell}\varphi(g_{\mathbf{x}}^{-1}(u_{j}))\right)\right]\right)$
$\displaystyle=K_{C}\left(\mathbf{x},\left[0,C^{1:d-\ell}(g_{\mathbf{x}}^{-1}(u_{1}),\ldots,g_{\mathbf{x}}^{-1}(u_{d-\ell}))\right]\right)$
$\displaystyle=K_{C}(\mathbf{x},[0,g_{\mathbf{x}}^{-1}(u_{1})]\times\ldots\times[0,g_{\mathbf{x}}^{-1}(z_{d-\ell})])=C^{\mathbf{x}}(\mathbf{u}).$

Observe that $(d-\ell)$-monotonicity of $\psi^{\mathbf{x}}$ is also inherited
from $\psi$: In fact, according to [25, Proposition 2.3] it suffices to check
that $(-1)^{d-\ell-2}(\psi^{\mathbf{x}})^{(d-\ell-2)}$ exists on $(0,\infty)$,
is non-negative, decreasing and convex. Writing

$\displaystyle(-1)^{d-\ell-2}(\psi^{\mathbf{x}})^{(d-\ell-2)}(z)$
$\displaystyle=(-1)^{d-\ell-2}\frac{\psi^{(d-2)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+z\right)}{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}$
$\displaystyle=\frac{(-1)^{d-2}\psi^{(d-2)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+z\right)}{(-1)^{\ell}\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}$

all required properties follow directly. ∎

######  Example 6.4.

We consider the $3$-dimensional Gumbel copula with parameter $\vartheta$ which
is generated by $\tilde{\psi}(z)=\exp(-z^{1/\vartheta})$. Thus, a normalized
generator is given by
$\psi_{\vartheta}(z)=\exp\left(-(z\cdot(-\log(1/2))^{\vartheta})^{1/\vartheta}\right)=2^{-z^{1/\vartheta}}$.
Fixing $\vartheta=3$ we calculated $\psi^{x}_{5}$ for several $x\in(0,1)$ and
illustrated the results in Figure 2.

Figure 2. Archimedean generators of the original Gumbel copula $C_{3}$ and
conditional generators $\psi_{3}^{x}$ for $x\in\\{0.05,0.15,\ldots,0.85\\}$
(left panel) and samples of size $n=2000$ of the conditional copulas
$C_{3}^{0.05}$ and $C_{3}^{0.85}$ (middle and right panel) as considered in
Example 6.4.

The left panel shows the original as well as the normalized conditional
generators while the middle and right panels depict samples of $C^{0.05}_{3}$
and $C^{0.95}_{3}$, respectively.

######  Remark 6.5.

Having the Markov kernel representation, the following connection between
generator and conditional generator now is an immediate consequence of
disintegration (cf. [26, Section 4] for the $3$-dimensional case):

(6.1) $\displaystyle\psi(z)=\int_{\mathbb{I}^{\ell}}\psi^{\mathbf{x}}(z)\
\mathrm{d}\mu_{C^{1:\ell}}(\mathbf{x}).$

The authors in [26] state that the conditional copula w.r.t. any conditioning
event $\mathbf{X}\in A\subseteq\mathbb{I}^{\ell}$ of an Archimedean copula is
again Archimedean and, in particular, show that the corresponding inverse
generator $\varphi^{[\mathbf{0},\mathbf{x}]}$ of the conditional copula
$C^{[\mathbf{0},\mathbf{x}]}$ w.r.t. the conditioning
$\mathbf{X}\leq\mathbf{x}$ is given by

$\displaystyle\varphi^{[\mathbf{0},\mathbf{x}]}(z)=\varphi\left(z\cdot\psi\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)\right)-\varphi\left(\psi\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)\right).$

Building upon that and rewriting $\varphi^{[\mathbf{0},\mathbf{x}]}$ we get
$\varphi^{[\mathbf{0},\mathbf{x}]}(z)=\varphi\left(z\cdot
C^{1:\ell}(\mathbf{x})\right)-\varphi(C^{1:\ell}(\mathbf{x}))$ and hence

$\displaystyle\psi^{[\mathbf{0},\mathbf{x}]}$
$\displaystyle=\frac{\psi(z+\varphi(C^{1:\ell}(\mathbf{x})))}{C^{1:\ell}(\mathbf{x})}=\frac{C^{1:\ell+1}(\mathbf{x},\psi(z))}{C^{1:\ell}(\mathbf{x})}.$

In analogy to eq. (6.1) disintegration now also yields the nice relation

$\displaystyle\psi^{[\mathbf{0},\mathbf{x}]}(z)=\frac{1}{C^{1:\ell}(\mathbf{x})}\int_{[\mathbf{0},\mathbf{x}]}\psi^{\mathbf{s}}(z)\
\mathrm{d}\mu_{C^{1:\ell}}(\mathbf{s}).$

It is well-known that imposing some regularity assumption (regular variation
at infinity) on the generator $\psi$ of the Archimedean copula $C$, the
conditional generator $\psi^{\mathbf{x}}$ converges to the Clayton copula with
parameter $\rho/(1+\rho\cdot\ell)$ as $\mathbf{x}\to\mathbf{0}$. From a
different angle, given a sequence $(C_{n})_{n\in\mathbb{N}}$ converging
uniformly to $C$ within $\mathcal{C}^{d}_{\text{ar}}$ we now have the tools to
answer the natural question whether the corresponding conditional copulas
converge in that sense, too.

######  Theorem 6.6.

Suppose that $C_{\psi},C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean
copulas with strict, normalized generators $\psi,\psi_{1},\psi_{2},\ldots$ and
Markov kernels $K_{\psi},K_{1},K_{2},\ldots$, respectively. If
$(C_{n})_{n\in\mathbb{N}}$ converges pointwise to $C$ then for
$\ell\in\\{1,\ldots,d-2\\}$ and any fixed $\mathbf{x}\in(0,1)^{\ell}$ we have
that for every $\mathbf{u}\in\mathbb{I}^{d-\ell}$ it holds

$\displaystyle\lim_{n\to\infty}C^{\mathbf{x}}_{n}(\mathbf{u})=C^{\mathbf{x}}_{\psi}(\mathbf{u}).$

###### Proof.

Recall that according to Corollary 4.4 we already have

$\displaystyle\lim_{n\to\infty}K_{n}(\mathbf{x},[0,y])=K_{\psi}(\mathbf{x},[0,y])$

for every $y\in\mathbb{I}$. Considering that $\psi^{(\ell)}$ is continuous on
$(0,\infty)$ and applying [20, Lemma 4.4] it directly follows that

$\displaystyle\lim_{n\to\infty}\psi_{n}^{\mathbf{x}}(z)=\lim_{n\to\infty}K_{n}(\mathbf{x},[0,\psi_{n}(z)])=K_{\psi}(\mathbf{x},[0,\psi(z)])=\psi^{\mathbf{x}}(z)$

for every $z\in[0,\infty)$. As continuous monotone functions the convergence
is even uniform (see [22]) and applying Theorem 4.3 completes the proof. ∎

Triggered by [11] where the authors study a conditional version of Kendall’s
tau, we propose a conditional version of the multivariate, Markov kernel based
dependence measure $\zeta_{1}=\zeta_{1}^{d}$ introduced in [13]. For
$(\mathbf{X},Y)\sim C\in\mathcal{C}^{d}$ this dependence measure is defined by

$\displaystyle\zeta_{1}^{d}(C):=3\int_{\mathbb{I}}\int_{\mathbb{I}^{d-1}}\left|K_{C}(\mathbf{s},[0,y])-y\right|\
\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{s})\mathrm{d}\lambda(y).$

Considering a random vector $(\mathbf{X},\mathbf{Y})$, where the subvector
$\mathbf{X}$ of covariates is $\ell$-dimensional, and setting

$\displaystyle\zeta_{1}^{\mathbf{x}}(C)=\zeta_{1}^{d-\ell}(C^{\mathbf{x}})$

yields a conditional multivariate dependence measure quantifying the degree of
dependence of the conditional copula $C^{\mathbf{x}}$ of $C$ when fixing the
values of the covariate $\mathbf{X}$.

Together with Lemma B.2 in B we now have the tools to show the remarkable fact
that for fixed $\mathbf{x}\in(0,1)^{\ell}$ uniform convergence of a sequence
$(C_{n})_{n\in\mathbb{N}}$ of Archimedean copulas converging to
$C\in\mathcal{C}^{d}_{\text{ar}}$ already implies convergence of the
conditional dependence measures
$(\zeta_{1}^{\mathbf{x}}(C_{n}))_{n\in\mathbb{N}}$ to
$\zeta_{1}^{\mathbf{x}}(C)$.

######  Corollary 6.7.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas.
If $(C_{n})_{n\in\mathbb{N}}$ converges uniformly to $C$ then for every fixed
$\mathbf{x}\in(0,1)^{\ell}$ the sequence of conditional copulas
$(C_{n}^{\mathbf{x}})_{n\in\mathbb{N}}$ converges $(d-\ell-1)$-weakly
conditional to $C^{\mathbf{x}}$. Furthermore we have

(6.2)
$\displaystyle\lim_{n\to\infty}|\zeta_{1}^{\mathbf{x}}(C_{n})-\zeta_{1}^{\mathbf{x}}(C)|=0.$

###### Proof.

Combining Theorems 4.3, 6.6 as well as Lemma B.2 yields both,
$(d-\ell-1)$-weak conditional convergence of the conditional copulas and
convergence of the conditional dependence measures. ∎

The practical relevance of our study is twofold and entailed by Theorem 6.6
and Corollary 6.7. First, in strong contrast to the estimation of general
conditional copulas (see, e.g., [1, 12]) in case of an Archimedean copula $C$
it suffices to estimate (the generator of) $C$ in order to obtain an estimator
of (the generator of) its conditional copula $C^{\mathbf{x}}$. Additionally,
an estimator of (the generator of) $C$ also gives rise to an estimation of the
conditional dependence measure $\zeta_{1}^{\mathbf{x}}(C)$ which might be a
powerful alternative to conditional association measures (as studied, e.g., in
[11, 12]).
Regarding the estimation of multivariate Archimedean copulas we refer to [15,
17], and the references therein, for parametric methodologies and to [9] for a
non-parametric procedure. It is worth remarking, however, that the approach
from [9] is only proven to be valid in dimensions $2,3$ and $4$ whereas,
despite strong evidence, the general case remains an open problem.

The following example provides a small simulation in the $3$-dimensional case
illustrating the estimation of $\zeta_{1}^{x}$ from data of an Archimedean
parametric family for several choices of conditioning and different sample
sizes.

######  Example 6.8.

Consider the $3$-dimensional Gumbel copula $C_{\vartheta}$ with generator
$\psi(z)=\exp(-z^{1/\vartheta})$ and parameter $\vartheta=5$. Working with the
R-package ‘copula’ ([16]) we generate samples of $C_{5}$ of size
$n\in\\{50,100,500,1000\\}$ based on which we estimate its parameter to obtain
an estimated generator $\hat{\psi}_{n}$ of $\psi$. Fixing covariate values
$x\in\\{0.05,0.15,\ldots,0.95\\}$ we directly compute $\hat{\psi}^{x}_{n}$ and
consequently $\zeta_{1}^{x}(\hat{C}_{n})=\zeta_{1}(\hat{C}_{n}^{x})$. We
repeated this process $R=300$ times and compared the real values of
$\zeta_{1}^{x}(C_{5})$ to the average values of $\zeta_{1}^{x}(\hat{C}_{n})$
over all runs. The results are plotted in Figure 3. One can observe that for
increasing $x$ the dependence increases and that the estimation works very
well, with small deviations for higher values of $x$ which get better for
higher sample sizes.

Figure 3. Values of $\zeta_{1}^{x}(C)$ and averages of
$\zeta_{1}^{x}(\hat{C}_{n})$ over $R=30=$ runs for
$x\in\\{0.05,0.15,\ldots,0.85\\}$ and sample sizes $n=\\{50,100,500,1000\\}$
as considered in Example 6.8.

The quality of the estimation stems from our choice of simulating data from a
parametric family and using parametric estimation. The more general non-
parametric estimation is more difficult even in dimension $3$ and left for
future work on the topic.

We conculde the paper by showing that, in addition to convergence, singularity
as well as conditional increasingness of an Archimedean copula carry over to
its conditional copulas, too.

######  Theorem 6.9.

Suppose that $C$ is a $d$-dimensional Archimedean copula with strict generator
$\psi$. If $C$ is singular w.r.t. $\lambda_{d}$ then for every
$\ell\in\\{1,\ldots,d-2\\}$ there exists a set
$\Lambda\in\mathcal{B}(\mathbb{I}^{\ell})$ with $\lambda_{\ell}(\Lambda)=1$
such that for all $\mathbf{x}\in\Lambda$ the conditional copula
$C^{\mathbf{x}}$ is singular w.r.t. $\lambda_{d-\ell}$.

###### Proof.

According to [5, Section 3.2] it suffices to show that the $(d-\ell)$-th
derivative vanishes $\lambda_{\ell}$-almost everywhere. Straightforward
calculations yield

$\displaystyle\frac{\partial^{d-\ell}}{\partial u_{1}\cdots\partial
u_{d-\ell}}C^{\mathbf{x}}(\mathbf{u})=\tfrac{\psi^{(d)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(g_{\mathbf{x}}^{-1}(u_{j}))\right)}{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}\prod_{j=1}^{d-\ell}\varphi^{\prime}(g_{\mathbf{x}}^{-1}(u_{j}))(g_{\mathbf{x}}^{-1})^{\prime}(u_{j})$
$\displaystyle=\underbrace{\tfrac{\psi^{(d)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(u_{j})\right)}{\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}}_{=\frac{\partial^{d-\ell}}{\partial
u_{1}\cdots\partial
u_{d-\ell}}K_{C}(\mathbf{x},[\mathbf{0},\mathbf{u}])\prod_{j=1}^{d-\ell}\frac{1}{\varphi^{\prime}(u_{j})}}\cdot\tfrac{\psi^{(d)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(g_{\mathbf{x}}^{-1}(u_{j}))\right)}{\psi^{(d)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+\sum_{j=1}^{d-\ell}\varphi(u_{j})\right)}\prod_{j=1}^{d-\ell}\varphi^{\prime}(g_{\mathbf{x}}^{-1}(u_{j}))(g_{\mathbf{x}}^{-1})^{\prime}(u_{j}).$

Since $C$ is singular, however, Theorem 5.1 yields a set
$\Lambda\in\mathcal{B}(\mathbb{I}^{\ell})$ with $\lambda_{\ell}(\Lambda)=1$
such that for all $\mathbf{x}\in\Lambda$ we have that
$K_{C}(\mathbf{x},\cdot)$ is singular w.r.t. $\lambda_{d-\ell}$. As direct
consequence, $\frac{\partial^{d-\ell}}{\partial u_{1}\cdots\partial
u_{d-\ell}}K_{C}(\mathbf{x},[\mathbf{0},\mathbf{u}])=0$ for all
$\mathbf{x}\in\Lambda$ so the above expression evaluates to $0$ as well and
singularity of $C^{\mathbf{x}}$ w.r.t. $\lambda_{d-\ell}$ follows. ∎

Following [27], $C$ is conditionally increasing if the random vector
$\mathbf{X}\sim C$ and $X_{i}$ is stochastically increasing in $(X_{p},p\in
J)$ for all $i\not\in J$ and $J\subset\\{1,\ldots,d\\}$. Translating to the
Markov kernel setup, $C$ is conditionally increasing if for every
$\ell\in\\{1,\ldots,d-1\\}$ we have that $\mathbf{x}\mapsto
K_{C}(\mathbf{x},[0,y])$ is decreasing in $\mathbf{x}$ for all
$y\in\mathbb{I}$ (w.l.o.g., we take the first $\ell$ coordinates).

######  Proposition 6.10.

Suppose that $C$ is a $d$-dimensional Archimedean copula with generator $\psi$
and let $\ell\in\\{1,\ldots,d-2\\}$. If $C$ is conditionally increasing, then
so is $C^{\mathbf{x}}$ for any $\mathbf{x}\in(0,1)^{\ell}$.

###### Proof.

According to [27, Theorem 2.8] we have to prove that
$(-1)^{d-\ell-1}(\psi^{\mathbf{x}})^{(d-\ell-1)}$ is log-convex and writing

$\displaystyle(-1)^{d-\ell-1}(\psi^{\mathbf{x}})^{(d-\ell-1)}(z)$
$\displaystyle=\frac{(-1)^{d-1}D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})+z\right)}{(-1)^{\ell}\psi^{(\ell)}\left(\sum_{i=1}^{\ell}\varphi(x_{i})\right)}$

we directly observe that log-convexity follows immediately from log-convexity
of $(-1)^{d-1}D^{-}\psi^{(d-2)}$. ∎

## Acknowledgments

The author gratefully acknowledges the financial support from Porsche Holding
Austria and Land Salzburg within the WISS 2025 project ‘KFZ’ (P1900123).

##  Appendix A Multivariate conditional distributions

This section derives several results concerning the connection of derivatives
and Markov kernels of general multivariate distribution functions with
absolutely continuous marginals which are applied to our setting of
Archimedean copulas in the main text. Suppose $H$ is a $d$-dimensional
distribution function and let $\ell\in\\{1,\ldots,d-1\\}$. Considering the
disintegration theorem, (a version of) the $\ell$-Markov kernel $K_{H}$ is
given by the Radon-Nikodym derivative of $\mu_{H}$ w.r.t. the marginal measure
$\mu_{H^{1:\ell}}$, i.e.,
$K_{H}=\frac{\mathrm{d}\mu_{H}}{\mathrm{d}\mu_{H^{1:\ell}}}$, whence deriving
it is generally highly non-trivial.

For some special cases, however, it is well-known that more can be said. For
instance, if $H$ is absolutely continuous then there exists a density
$h^{1:d}$ of $H$ and disintegration directly yields

$\displaystyle
H(\mathbf{x},\mathbf{y})=\int_{(\mathbf{-\infty},\mathbf{x}]}K_{H}(\mathbf{s},(\mathbf{-\infty},\mathbf{y}])\
\mathrm{d}\mu_{H^{1:\ell}}(\mathbf{s})=\int_{(\mathbf{-\infty},\mathbf{x}]}\int_{(\mathbf{-\infty},\mathbf{y}]}h^{1:d}(\mathbf{s},\mathbf{t})\
\mathrm{d}\lambda_{d-\ell}(\mathbf{t})\mathrm{d}\lambda_{\ell}(\mathbf{s}).$

Considering that $\mu_{H^{1:\ell}}$ is also absolutely continuous, there
exists $\Lambda\in\mathcal{B}(\mathbb{R}^{\ell})$ with
$\lambda_{\ell}(\Lambda)=1$ such that for all $\mathbf{x}\in\Lambda$ we have

$\displaystyle
K_{H}(\mathbf{x},(\mathbf{-\infty},\mathbf{y}])=\int_{(\mathbf{-\infty},\mathbf{y}]}\frac{h^{1:d}(\mathbf{x},\mathbf{t})}{h^{1:\ell}(\mathbf{x})}\
\mathrm{d}\lambda_{d-\ell}(\mathbf{t})=\int_{(\mathbf{-\infty},\mathbf{y}]}h^{1:d|1:\ell}(\mathbf{t}|\mathbf{x})\
\mathrm{d}\lambda_{d-\ell}(\mathbf{t}),$

where $h^{1:d|1:\ell}(\cdot|\mathbf{x})$ is the conditional density of $H$
given $\mathbf{x}$ (cf. [23, Section 8.3] for the two dimensional case).

Furthermore, in dimension two it is well-known that for every $y\in\mathbb{R}$
there exists a set $\Gamma^{y}$ with $\lambda(\Gamma^{y})=1$ such that for
every $x\in\Gamma^{y}$ it holds that

$\displaystyle K_{H}(x,(-\infty,y])=\frac{\partial H(x,y)}{\partial x}$

regardless of $H$ being purely absolutely continuous, singular or mixed (see,
e.g. [5, Theorem 3.4.4] for a proof in the copula setting). The next theorem
extends this assertion to multivariate disitribution functions with absolutely
continuous marginals:

######  Theorem A.1.

Suppose that $H$ is a $d$-dimensional distribution function with absolutely
continuous $\ell$-marginal for some $\ell\in\\{2,\ldots,d-1\\}$. Then for
every $\mathbf{y}\in\mathbb{R}^{d-\ell}$ there exists
$\Gamma^{\mathbf{y}}\in\mathcal{B}(\mathbb{R}^{\ell})$ with
$\lambda_{\ell}(\Gamma^{\mathbf{y}})=1$ such that

(A.1) $\displaystyle
K_{H}(\mathbf{x},(\mathbf{-\infty},\mathbf{y}])=\frac{\partial^{\ell}H(\mathbf{x},\mathbf{y})}{\partial
x_{1}\partial x_{2}\cdots\partial x_{\ell}}\cdot h^{1:\ell}(\mathbf{x})^{-1}$

holds for all $\mathbf{x}\in\Gamma^{\mathbf{y}}$.

The proof of this theorem is essentially carried out in [13, Supplementary]
where the authors show that for a $d$-dimensional copula $C$ with only
independent $(d-1)$-marginals, i.e. $C^{1:d-1}=\Pi_{d-1}$, it holds that for
every $y\in\mathbb{I}$ we have

$\displaystyle K_{C}(x_{1},x_{2},\ldots
x_{d-1},[0,y])=\frac{\partial^{d-1}C(\mathbf{x},y)}{\partial x_{1}\partial
x_{2}\cdots\partial x_{d-1}}$

for $\lambda_{d-1}$-almost every $\mathbf{x}\in\mathbb{I}^{d-1}$. The proof
can be readily adapted to our setting.

Following [5] a $d$-dimensional distribution function $G$ is $d$-increasing if
for $\mathbf{a},\mathbf{b}\in\mathbb{R}^{d}$ such that $b_{i}\geq a_{i}$ for
$i=1,2,\ldots,d$ it holds that

$\displaystyle
V_{G}((\mathbf{a},\mathbf{b}]):=\sum_{\mathbf{v}\in\text{ver}((\mathbf{a},\mathbf{b}])}\text{sign}(\mathbf{v})G(\mathbf{v})\geq
0,$

where sign$(\mathbf{v})=1$ if $v_{j}=a_{j}$ for an even number of indices and
sign$(\mathbf{v})=-1$ otherwise and
ver$((\mathbf{a},\mathbf{b}])=\\{a_{1},b_{1}\\}\times\ldots\times\\{a_{d},b_{d}\\}$.
Using the finite difference operator $\Delta_{a_{i},b_{i}}^{j}$ given by

$\displaystyle\Delta_{a_{i},b_{i}}^{j}G(\mathbf{s}):=G(s_{1},\ldots,s_{j-1},b_{j},s_{j+1},\ldots,s_{d})-G(s_{1},\ldots,s_{j-1},a_{j},s_{j+1},\ldots,s_{d})$

allows to alternatively compute the $G$-volume $V_{G}$ of
$(\mathbf{a},\mathbf{b}]$ via

$\displaystyle
V_{G}((\mathbf{a},\mathbf{b}])=\Delta^{d}_{a_{d},b_{d}}\Delta^{d-1}_{a_{d-1},b_{d-1}}\cdots\Delta^{1}_{a_{i},b_{1}}G(\mathbf{s}).$

This expression will be useful in the next lemma saying that if it exists, the
$\ell$-times iterated partial derivatives of a $d$-dimensional distribution
function is $(d-\ell)$-increasing.

######  Lemma A.2.

Suppose that $H$ is a $d$-dimensional distribution function and
$\ell\in\\{1,\ldots,d-1\\}$. If the $\ell$-th order partial derivatives exist
and are continuous then the function

(A.2) $\displaystyle
F_{\mathbf{x}}(\mathbf{y})=\frac{\partial^{\ell}H(\mathbf{x},\mathbf{y})}{\partial
x_{1}\partial x_{2}\cdots\partial x_{\ell}}$

is $(d-\ell)$-increasing.

###### Proof.

Applying the notation $\mathbf{s}_{i:j}=(s_{i},s_{i+1},\ldots,s_{j})$ for
$i,j\in\mathbb{N}$ with $i<j$ we fix
$\mathbf{a}_{\ell+1:d},\mathbf{b}_{\ell+1:d}$ such that $b_{i}\geq a_{i}$ for
$i\in\\{\ell+1,\ldots,d\\}$.

Using the expression in terms of the finite difference operator and its
relation to partial derivatives we get

$\displaystyle
V_{F_{\mathbf{x}}}((\mathbf{a}_{\ell+1:d},\mathbf{b}_{\ell+1:d}])$
$\displaystyle=\Delta^{d}_{a_{d},b_{d}}\cdots\Delta^{\ell+1}_{a_{\ell+1},b_{\ell+1}}F_{\mathbf{x}}(\mathbf{s})$
$\displaystyle=\Delta^{d}_{a_{d},b_{d}}\cdots\Delta^{\ell+1}_{a_{\ell+1},b_{\ell+1}}\lim_{h\downarrow
0}\frac{1}{h^{\ell}}\Delta^{\ell}_{x_{\ell}+h,x_{\ell}}\cdots\Delta^{1}_{x_{1}+h,x_{1}}H(\mathbf{x},\mathbf{s})$
$\displaystyle=\lim_{h\downarrow
0}\frac{1}{h^{\ell}}\underbrace{V_{H}((\mathbf{a},\mathbf{b}])}_{\geq 0}\geq
0,$

where for $h>0$ we set
$\mathbf{a}=(x_{1},\ldots,x_{\ell},a_{\ell+1},a_{\ell+2},\ldots,a_{d})$ and
$\mathbf{b}=(x_{1}+h,\ldots,x_{\ell}+h,b_{\ell+1},b_{\ell+2},\ldots,b_{d})$. ∎

##  Appendix B Auxiliary results for multivariate copulas

Instead of general multivariate distributions, this sections considers
$d$-dimensional copulas and provides useful lemmata which are relevant for our
main results.

######  Lemma B.1.

Suppose $C$ is a $d$-dimensional copula and denote by $c^{1:d}$ ($c^{1:\ell}$)
the Radon-Nikodym derivative of $\mu_{C}$ w.r.t. $\lambda_{d}$
($\mu_{C^{1:\ell}}$ w.r.t. $\lambda_{\ell}$). Then there exists a set
$\Lambda\in\mathcal{B}(\mathbb{I}^{\ell})$ with $\lambda_{\ell}(\Lambda)=1$
such that for all $\mathbf{x}\in\Lambda$ we have

$\displaystyle
K_{C}^{\text{abs}}(\mathbf{x},F)=\int_{F}\frac{c^{1:d}(\mathbf{x},\mathbf{y})}{c^{1:\ell}(\mathbf{x})}\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y}).$

###### Proof.

Using Lebesgue decomposition of $\mu_{C}$, disintegration and absolute
continuity, on the one hand we have

(B.1) $\displaystyle\mu_{C}(E\times F)$
$\displaystyle=\mu_{C}^{\text{abs}}(E\times F)+\mu_{C}^{\text{sing}}(E\times
F)$ $\displaystyle=\int_{E}\int_{F}c^{1:d}(\mathbf{x},\mathbf{y})\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y})\mathrm{d}\lambda_{\ell}(\mathbf{x})+\int_{E}K_{C^{\text{sing}}}(\mathbf{x},F)\
\mathrm{d}(\mu_{C}^{\text{sing}})^{1:\ell}$

and, on the other hand,

(B.2) $\displaystyle\mu_{C}(E\times F)$
$\displaystyle=\int_{E}K_{C}(\mathbf{x},F)\
\mathrm{d}\mu_{C^{1:\ell}}(\mathbf{x})$
$\displaystyle=\int_{E}K_{C}^{\text{abs}}(\mathbf{x},F)\
\mathrm{d}\mu_{C^{1:\ell}}(\mathbf{x})+\int_{E}K_{C}^{\text{sing}}(\mathbf{x},F)\
\mathrm{d}\mu_{C^{1:\ell}}(\mathbf{x})$
$\displaystyle=\int_{E}K_{C}^{\text{abs}}(\mathbf{x},F)\
\mathrm{d}(\mu_{C^{1:\ell}})^{\text{abs}}(\mathbf{x})+\int_{E}K_{C}^{\text{abs}}(\mathbf{x},F)\
\mathrm{d}(\mu_{C^{1:\ell}})^{\text{sing}}(\mathbf{x})$
$\displaystyle\qquad+\int_{E}K_{C}^{\text{sing}}(\mathbf{x},F)\
\mathrm{d}(\mu_{C^{1:\ell}})^{\text{abs}}(\mathbf{x})+\int_{E}K_{C}^{\text{sing}}(\mathbf{x},F)\
\mathrm{d}(\mu_{C^{1:\ell}})^{\text{sing}}(\mathbf{x})$
$\displaystyle=\underbrace{\int_{E}K_{C}^{\text{abs}}(\mathbf{x},F)c^{1:\ell}(\mathbf{x})\
\mathrm{d}\lambda_{\ell}(\mathbf{x})}_{(a)}+\underbrace{\int_{E}K_{C}^{\text{abs}}(\mathbf{x},F)\
\mathrm{d}(\mu_{C^{1:\ell}})^{\text{sing}}(\mathbf{x})}_{(b)}$
$\displaystyle\qquad+\underbrace{\int_{E}K_{C}^{\text{sing}}(\mathbf{x},F)c^{1:\ell}(\mathbf{x})\
\mathrm{d}\lambda_{\ell}(\mathbf{x})}_{(c)}+\underbrace{\int_{E}K_{C}^{\text{sing}}(\mathbf{x},F)\
\mathrm{d}(\mu_{C^{1:\ell}})^{\text{sing}}(\mathbf{x})}_{(d)}$

via disintegration, decomposition of the Markov kernel and then Lebesgue
decomposition of the marginal measure $\mu_{C^{1:\ell}}$. Notice additionally
that we have both

$\displaystyle\mu_{C}(E\times\mathbb{I}^{d-\ell})$
$\displaystyle=\mu_{C^{1:\ell}}(E)=\mu_{C^{1:\ell}}^{\text{abs}}(E)+\mu_{C^{1:\ell}}^{\text{sing}}(E),$
$\displaystyle\mu_{C}(E\times\mathbb{I}^{d-\ell})$
$\displaystyle=\mu_{C}^{\text{abs}}(E\times\mathbb{I}^{d-\ell})+\mu_{C}^{\text{sing}}(E\times\mathbb{I}^{d-\ell})=(\mu_{C}^{\text{abs}})^{1:\ell}(E)+(\mu_{C}^{\text{sing}})^{1:\ell}(E)$

and therefore
$\mu_{C^{1:\ell}}^{\text{abs}}(E)=(\mu_{C}^{\text{abs}})^{1:\ell}(E)$ as well
as $\mu_{C^{1:\ell}}^{\text{sing}}(E)=(\mu_{C}^{\text{sing}})^{1:\ell}(E)$.
(a): The measure $\Gamma_{\text{a}}(E\times
F):=\int_{E}K_{C}^{\text{abs}}(\mathbf{x},F)c^{1:\ell}(\mathbf{x})\
\mathrm{d}\lambda_{\ell}(\mathbf{x})$ is absolutely continuous w.r.t.
$\lambda_{d-\ell}$ as every $\lambda_{d-\ell}$-null set is also a
$\Gamma_{\text{a}}$-null set.
(b): Defining $\Gamma_{\text{b}}(E\times
F):=\int_{E}K_{C}^{\text{abs}}(\mathbf{x},F)\
\mathrm{d}(\mu_{C^{1:\ell}})^{\text{sing}}(\mathbf{x})$ there exists a set
$S\in\mathcal{B}(\mathbb{I}^{\ell})$ such that
$\mu_{C^{1:\ell}}^{\text{sing}}(S)=1$ while $\lambda_{\ell}(S)=0$. Considering
that $K_{C}^{\text{abs}}$ is a subkernel (i.e.,
$K_{C}^{\text{abs}}(\mathbf{x},\mathbb{I}^{d-\ell})\leq 1$) we have

$\displaystyle\Gamma_{\text{b}}(\mathbb{I}^{d})$
$\displaystyle=\int_{\mathbb{I}^{\ell}}K_{C}^{\text{abs}}(\mathbf{x},\mathbb{I}^{d-\ell})\
\mathrm{d}\mu_{C^{1:\ell}}^{\text{sing}}(\mathbf{x})=\int_{S}K_{C}^{\text{abs}}(\mathbf{x},\mathbb{I}^{d-\ell})\
\mathrm{d}\mu_{C^{1:\ell}}^{\text{sing}}(\mathbf{x})=\parallel\Gamma_{\text{b}}\parallel=\Gamma_{\text{b}}(S\times\mathbb{I}^{d-\ell})$

whereas

$\displaystyle\lambda_{d}(S\times\mathbb{I}^{d-\ell})=\int_{S}\lambda_{d-\ell}(\mathbb{I}^{d-\ell})\
\mathrm{d}\lambda_{\ell}=0.$

The same argument yields that the measure defined according to (d) is singular
w.r.t. $\lambda_{d}$, too.
For part (c) we start by defining $\Gamma_{\text{c}}(E\times
F):=\int_{E}K_{C}^{\text{sing}}(\mathbf{x},F)c^{1:\ell}(\mathbf{x})\
\mathrm{d}\lambda_{\ell}(\mathbf{x})$. Observe that the marginal
$\Gamma_{\text{c}}^{1:\ell}$ is absolutely continuous w.r.t. $\lambda_{\ell}$.
In fact,

$\displaystyle\Gamma_{\text{c}}^{1:\ell}(E)=\Gamma_{\text{c}}(E\times\mathbb{I}^{d-\ell})=\int_{E}K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})c^{1:\ell}(\mathbf{x})\
\mathrm{d}\lambda_{\ell}(\mathbf{x})$

so disintegration yields

$\displaystyle\Gamma_{\text{c}}(E\times
F)=\int_{E}K_{\Gamma_{\text{c}}}(\mathbf{x},F)\
\mathrm{d}\Gamma_{\text{c}}^{1:\ell}(\mathbf{x})=\int_{E}K_{\Gamma_{\text{c}}}(\mathbf{x},F)K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})c^{1:\ell}(\mathbf{x})\
\mathrm{d}\lambda_{\ell}(\mathbf{x}).$

Consequently, there exists a set
$\Upsilon_{1}\in\mathcal{B}(\mathbb{I}^{\ell})$ with
$\lambda_{\ell}(\Upsilon_{1})=1$ such that for all $\mathbf{x}\in\Upsilon_{1}$
we have

$\displaystyle
K_{C}^{\text{sing}}(\mathbf{x},F)c^{1:\ell}(\mathbf{x})=K_{\Gamma_{\text{c}}}(\mathbf{x},F)K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})c^{1:\ell}(\mathbf{x})$

and hence

$\displaystyle
K_{\Gamma_{\text{c}}}(\mathbf{x},F)=\frac{K_{C}^{\text{sing}}(\mathbf{x},F)}{K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})}$

which is the normalization of the subkernel. It directly follows that
$K_{\Gamma_{\text{c}}}(\mathbf{x},\cdot)$ is singular w.r.t.
$\lambda_{d-\ell}$ for all $\mathbf{x}\in\Upsilon_{1}$. We now proceed as
before and decompose $\Gamma_{\text{c}}$ twice:

$\displaystyle\Gamma_{\text{c}}(E\times F)$
$\displaystyle=\Gamma_{\text{c}}^{\text{abs}}(E\times
F)+\Gamma_{\text{c}}^{\text{sing}}(E\times F)$
$\displaystyle=\int_{E}\int_{F}g^{1:d}(\mathbf{x},\mathbf{y})\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y})\mathrm{d}\lambda_{\ell}(\mathbf{x})+\int_{E}K_{\Gamma_{\text{c}}^{\text{sing}}}(\mathbf{x},F)\
\mathrm{d}\Gamma_{\text{c}}^{1:\ell}(\mathbf{x})$
$\displaystyle=\int_{E}\int_{F}g^{1:d}(\mathbf{x},\mathbf{y})\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y})\mathrm{d}\lambda_{\ell}(\mathbf{x})$
$\displaystyle\qquad+\int_{E}K_{\Gamma_{\text{c}}^{\text{sing}}}(\mathbf{x},F)K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})c^{1:\ell}(\mathbf{x})\
\mathrm{d}\lambda_{\ell}(\mathbf{x}),$

where $g^{1:d}$ denotes the Radon-Nikodym density of
$\Gamma_{\text{c}}^{\text{abs}}$ w.r.t. $\lambda_{d}$. Additionally,

$\displaystyle\Gamma_{\text{c}}(E\times F)$
$\displaystyle=\int_{E}K_{\Gamma_{\text{c}}}(\mathbf{x},F)\
\mathrm{d}\Gamma_{\text{c}}^{1:\ell}(\mathbf{x})$
$\displaystyle=\int_{E}\frac{K_{C}^{\text{sing}}(\mathbf{x},F)}{K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})}\cdot
K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})c^{1:\ell}(\mathbf{x})\
\mathrm{d}\lambda_{\ell}(\mathbf{x}).$

Consequently, there exists a set
$\Upsilon_{2}\in\mathcal{B}(\mathbb{I}^{\ell})$ with
$\lambda_{\ell}(\Upsilon_{2})=1$ such that for all $\mathbf{x}\in\Upsilon_{2}$
we have

$\displaystyle
K_{C}^{\text{sing}}(\mathbf{x},F)c^{1:\ell}(\mathbf{x})=\int_{F}g^{1:d}(\mathbf{x},\mathbf{y})\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y})+K_{\Gamma_{\text{c}}^{\text{sing}}}(\mathbf{x},F)\cdot
K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})c^{1:\ell}(\mathbf{x})$

and considering that
$\\{\mathbf{x}\in\mathbb{I}^{\ell}:c^{1:\ell}(\mathbf{x})>0\\}$ is of full
$\mu_{C^{1:\ell}}$-measure and by absolute continuity of full
$\lambda_{\ell}$-measure too, yields the existence of a set
$\Upsilon_{3}\in\mathcal{B}(\mathbb{I}^{\ell})$ with
$\lambda_{\ell}(\Upsilon_{3})=1$ such that for all $\mathbf{x}\in\Upsilon_{3}$
we have

$\displaystyle\frac{K_{C}^{\text{sing}}(\mathbf{x},F)}{K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})}=\int_{F}\frac{g^{1:d}(\mathbf{x},\mathbf{y})}{c^{1:\ell}(\mathbf{x})K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})}\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y})+K_{\Gamma_{\text{c}}^{\text{sing}}}(\mathbf{x},F).$

However, we already know that
$K_{\Gamma_{\text{c}}}^{\text{sing}}(\mathbf{x},F)=K_{\Gamma_{\text{c}}}(\mathbf{x},F)=\frac{K_{C}^{\text{sing}}(\mathbf{x},F)}{K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})}$
so that

$\displaystyle\int_{F}\frac{g^{1:d}(\mathbf{x},\mathbf{y})}{c^{1:\ell}(\mathbf{x})K_{C}^{\text{sing}}(\mathbf{x},\mathbb{I}^{d-\ell})}\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y})=0$

and therefore $\Gamma_{\text{c}}^{\text{abs}}(E\times F)=0$ holds which
implies that $\Gamma_{\text{c}}$ is purely singular.
Having this we can finally compare the two absolutely continuous components of
eq. (B.1) and eq. (B.2) so that for $\mu_{C}(E\times F)$ we get the existence
of a set $\Lambda\in\mathcal{B}(\mathbb{I}^{\ell})$ with
$\lambda_{\ell}(\Lambda)=1$ such that for all $\mathbf{x}\in\Lambda$ we have

$\displaystyle\int_{F}c^{1:d}(\mathbf{x},\mathbf{y})\
\mathrm{d}\lambda_{d-\ell}(\mathbf{y})=K_{C}^{\text{abs}}(\mathbf{x},F)c^{1:\ell}(\mathbf{x})$

from which the assertion follows. ∎

The subsequent lemma concerns the approximation of $\zeta_{1}^{d}$. Recall
that the total variation metric of two probability measures $\mu,\nu$ on
$(\Omega,\mathcal{A})$ is given by
$d_{TV}(\mu,\nu)=\sup_{A\in\mathcal{A}}|\mu(A)-\nu(A)|$ and that it is well-
known that for absolutely continuous probability measures, convergence w.r.t.
$d_{TV}$ is equivalent to a.e.-convergence of the corresponding densities
(see, e.g., [3]).

######  Lemma B.2.

Suppose $C,C_{1},C_{2},\ldots$ are $d$-dimensional copulas with $(d-1)$-Markov
kernels $K_{C},K_{C_{1}},K_{C_{2}},\ldots$, respectively. If
$(C_{n})_{n\in\mathbb{N}}$ converges $(d-1)$-weakly conditional to $C$ and
$(\mu_{C_{n}^{1:d-1}})_{n\in\mathbb{N}}$ converges in total variation to
$\mu_{C^{1:d-1}}$ then

$\displaystyle\lim_{n\to\infty}\zeta_{1}^{d}(C_{n})=\zeta_{1}^{d}(C).$

###### Proof.

Defining $\beta_{n}(\mathbf{x},y):=K_{C_{n}}(\mathbf{x},[0,y])-y$,
$\mathbf{x}\in\mathbb{I}^{d-1},y\in\mathbb{I}$ (and $\beta$ analogously), we
get

$\displaystyle\left|\int_{\mathbb{I}}\int_{\mathbb{I}^{d-1}}\left|\beta_{n}\right|\
\mathrm{d}\mu_{C^{1:d-1}}\mathrm{d}\lambda-\int_{\mathbb{I}}\int_{\mathbb{I}^{d-1}}\left|\beta\right|\
\mathrm{d}\mu_{C^{1:d-1}}\mathrm{d}\lambda\right|$
$\displaystyle\leq\int_{\mathbb{I}}\left|\int_{\mathbb{I}^{d-1}}|\beta_{n}|\
\mathrm{d}\mu_{C_{n}^{1:d-1}}-\int_{\mathbb{I}^{d-1}}|\beta_{n}|\
\mathrm{d}\mu_{C^{1:d-1}}\right|\mathrm{d}\lambda$
$\displaystyle\qquad+\int_{\mathbb{I}}\left|\int_{\mathbb{I}^{d-1}}|\beta_{n}|\
\mathrm{d}\mu_{C^{1:d-1}}-\int_{\mathbb{I}^{d-1}}|\beta|\
\mathrm{d}\mu_{C^{1:d-1}}\right|\mathrm{d}\lambda$

For the first summand (I) we use the well-known fact (see, e.g., [3]) that
$d_{TV}(C_{n}^{1:d-1},C^{1:d-1})\xrightarrow{n\to\infty}0$ if, and only if,

$\displaystyle\lim_{n\to\infty}\frac{1}{2}\sup_{\|g\|_{\infty}\leq
1}\left|\int_{\mathbb{I}^{d-1}}g\
\mathrm{d}\mu_{C_{n}^{1:d-1}}-\int_{\mathbb{I}^{d-1}}g\
\mathrm{d}\mu_{C^{1:d-1}}\right|=0$

and estimate

$\displaystyle(I)\leq\int_{\mathbb{I}}2\cdot\frac{1}{2}\sup_{\|g\|_{\infty}\leq
1}\left|\int_{\mathbb{I}^{d-1}}g\
\mathrm{d}\mu_{C_{n}^{1:d-1}}-\int_{\mathbb{I}^{d-1}}g\
\mathrm{d}\mu_{C^{1:d-1}}\right|\xrightarrow{n\to\infty}0.$

For the second summand (II) we compute

$\displaystyle(II)$
$\displaystyle=\int_{\mathbb{I}}\left|\int_{\mathbb{I}^{d-1}}|\beta_{n}|-|\beta|\
\mathrm{d}\mu_{C^{1:d-1}}\right|\
\mathrm{d}\lambda\leq\int_{\mathbb{I}}\int_{\mathbb{I}^{d-1}}\left||\beta_{n}|-|\beta|\right|\
\mathrm{d}\mu_{C^{1:d-1}}\mathrm{d}\lambda$
$\displaystyle\leq\int_{\mathbb{I}}\int_{\mathbb{I}^{d-1}}\left|\beta_{n}-\beta\right|\
\mathrm{d}\mu_{C^{1:d-1}}\mathrm{d}\lambda=\int_{\mathbb{I}^{d-1}}\int_{\mathbb{I}}\left|\beta_{n}-\beta\right|\
\mathrm{d}\lambda\mathrm{d}\mu_{C^{1:d-1}}$
$\displaystyle=\int_{\mathbb{I}^{d-1}}\int_{\mathbb{I}}\left|K_{C_{n}}(\mathbf{x},[0,y])-K_{C}(\mathbf{x},[0,y])\right|\
\mathrm{d}\lambda(y)\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{x}).$

By $(d-1)$-weak conditional convergence there exists
$\Lambda\in\mathcal{B}(\mathbb{I}^{d-1})$ with $\mu_{C^{1:d-1}}(\Lambda)=1$
such that for all $\mathbf{x}\in\Lambda$ we have that
$|K_{C_{n}}(\mathbf{x},[0,y])-K_{C}(\mathbf{x},[0,y])|\to 0$ as $n\to\infty$
for every continuity point $y$ of $K_{C}(\mathbf{x},\cdot)$ so using Dominated
Convergence it follows that (II) $\to 0$ as $n\to\infty$ which completes the
proof. ∎

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