Convergence of a sequence of bivariate Archimedean copulas to another Archimedean copula or to the comonotone copula is shown to be equivalent with convergence of the corresponding sequence of Kendall distribution functions.No extra differentiability conditions on the generators are needed.Archimedean copula;generator;Kendall distribution function

Charpentier, A.

Segers, J.J.J.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
No. 2006–28 
 
CONVERGENCE OF ARCHIMEDEAN COPULAS 
 
By Arthur Charpentier, Johan Segers 
 
April 2006 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISSN 0924-7815 Convergence of Archimedean Copulas
Arthur Charpentier a
aENSAE-CREST, Timbre J120, 3 avenue Pierre Larousse, 92240 Malakoﬀ,
France
Johan Segers b,1
bTilburg University, P.O. Box 90153, NL-5000 LE Tilburg, the Netherlands
Abstract
Convergence of a sequence of bivariate Archimedean copulas to another Archime-
dean copula or to the comonotone copula is shown to be equivalent with convergence
of the corresponding sequence of Kendall distribution functions. No extra diﬀeren-
tiability conditions on the generators are needed.
JEL: C14, C16
Key words: Archimedean copula, generator, Kendall distribution function
1 Introduction
Let Cn be a sequence of bivariate Archimedean copulas with generators ψn
and Kendall distribution functions Kn. In this note, we establish necessary and
suﬃcient conditions in terms of ψn or Kn for convergence of the sequence of
copulas Cn to a limiting copula C which is either Archimedean or comonotone.
In particular, we extend results in Genest and MacKay (1986, Proposition 4.2
and 4.3) and Nelsen (1999, Theorems 4.4.7 and 4.4.8) to generators that are
possibly not everywhere diﬀerentiable. Moreover, we show that convergence
of the sequence of copulas is equivalent to convergence of the corresponding
sequence of Kendall distribution functions.
Email addresses: arthur.charpentier@ensae.fr (Arthur Charpentier),
jsegers@uvt.nl (Johan Segers).
1 Supported by a VENI grant of the Netherlands Organization for Scientiﬁc Re-
search (NWO).
Preprint submitted to Elsevier Science 14 April 2006The link between Archimedean copulas and their Kendall distribution func-
tions has already been exploited in the context of statistical inference on Archi-
medean copulas (Genest and Rivest, 1993; Barbe et al., 1996; Wang and Wells,
2000; Genest et al., 2006). Our ﬁndings add perspective to these papers by
showing that closeness of Kendall distribution functions really implies close-
ness of the corresponding Archimedean copulas.
The structure of the paper is as follows. We start with some preliminaries in
section 2. In section 3, we extend to the case of general Archimedean copu-
las C the expression in Genest and MacKay (1986, Proposition 3.3) for the
joint distribution function of the pair of random variables (X,C(X,Y )), where
(X,Y ) is itself a random pair with distribution function C. The main results
in this paper involve characterizations for the convergence of a sequence of
Archimedean copulas to another Archimedean copula or to the comonotone
copula (sections 4 and 5). Extensions to higher dimensions are treated in sec-
tion 6. We conclude in section 7 with a counterexample showing that not every
limit copula of a sequence of Archimedean copulas is necessarily Archimedean
or comonotone.
2 Preliminaries
A function C : [0,1]2 → [0,1] is called a (bivariate) copula if it is the restriction
to the domain [0,1]2 of a bivariate distribution function with uniform margins
on [0,1]. A function ψ : [0,1] → [0,∞] is called a generator if it is convex,
decreasing and ψ(1) = 0. The generalized inverse of ψ is denoted by
ψ
←(t) = inf{u ∈ [0,1] | ψ(u) ≤ t}, t ∈ [0,∞].
A copula C is called Archimedean if there exists a generator ψ such that
C(u,v) = ψ
←{ψ(u) + ψ(v)}, (u,v) ∈ [0,1]
2
(Schweizer and Sklar, 1983; Genest and MacKay, 1986). The conditions im-
posed on ψ are necessary and suﬃcient for the expression in the previous
display to deﬁne a copula. The copula C determines the generator ψ uniquely
up to a multiplicative constant.
The class of Archimedean copulas encompasses many well known bivariate
parametric distributions, such as the Frank, Clayton or Gumbel copulas (Nel-
sen, 1999, Table 4.1). Furthermore, if the inverse of the generator, ψ←, is the
Laplace transform of a nonnegative random variable, then the corresponding
Archimedean copula C reduces to the proportional frailty model in Marshall
and Olkin (1988); Oakes (1989).
2If the generator is a natural way to identify the Archimedean copula, other
functions can be considered as well. The Kendall distribution function K of
a copula C is deﬁned as the distribution function of the random variable
C(X,Y ), where (X,Y ) is a random pair with distribution function C, so
K(t) = Pr[C(X,Y ) ≤ t], t ∈ [0,1].
If the copula C is Archimedean with generator ψ, then K(t) = t − λ(t) with
λ(t) = ψ(t)/ψ0(t) and ψ0 is the right-hand derivative of ψ on [0,1) (Genest
and Rivest, 1993, Proposition 1.1). Conversely, from K or λ it is possible to
reconstruct ψ up to a multiplicative constant via
ψ(u) = ψ(u0)exp
 Z u
u0
1
λ(t)
dt
!
for 0 < u0 < 1 and 0 ≤ u ≤ 1.
3 Auxiliary result
The following result is a useful device to deduce properties of the generator ψ
of an Archimedean copula C from the copula itself. For twice continuously dif-
ferentiable generators, the result can already be found in Genest and MacKay
(1986, Proposition 3.3).
Proposition 1 Let (X,Y ) be a random pair with joint distribution function
C, a bivariate Archimedean copula with generator ψ. Let ψ0 be the right-hand
derivative of ψ on [0,1). Put Z = C(X,Y ). For (z,x) ∈ [0,1]2,
Pr[X ≤ x, Z ≤ z] =

            
            
x if x ≤ z ≤ 1,
z +
ψ(x)
ψ0(z)
−
ψ(z)
ψ0(z)
if 0 < z < x ≤ 1,
ψ(x) − ψ(0)
ψ0(0)
if z = 0 < x and ψ0(0) > −∞,
0 if z = 0 < x and ψ0(0) = −∞.
Proof. Since Z = C(X,Y ) ≤ X, we have Pr[X ≤ x, Z ≤ z] = Pr[X ≤ x] = x
for x ≤ z ≤ 1. Hence we can restrict attention to z < x.
The case z = 0 < x follows from the case 0 < z < x by the fact that
ψ0(0) = limz↓0 ψ0(z) and the fact that limz↓0 ψ(z)/ψ0(z) = 0 if ψ0(z) = −∞,
the latter property following from convexity.
3Hence we can restrict attention to the case 0 < z < x. Since both ψ0 and the
function z 7→ Pr[X ≤ x, Z ≤ z] are right-continuous, it suﬃces to prove the
stated equality for z such that ψ0 is continuous in z.
We have
Pr[X ≤ x, Z ≤ z]=Pr[X ≤ z] + Pr[z < X ≤ x, Z ≤ z]
=z + Pr[z < X ≤ x, Z ≤ z]
We can focus on the last term on the right-hand side. Let n be a positive
integer, and let
z = u0 < u1 < ··· < un = x
be such that
ψ(ui) =

1 −
i
n

ψ(z) +
i
n
ψ(x), i = 0,1,...,n.
We have
Pr[z < X ≤ x, Z ≤ z] =
n X
i=1
Pr[ui−1 < X ≤ ui, Z ≤ z].
If ui−1 < X ≤ ui, then C(ui−1,Y ) ≤ Z ≤ C(ui,Y ). Hence
n X
i=1
Pr[Ui−1 < X ≤ ui, C(ui,Y ) ≤ z]
≤Pr[z < X ≤ x, Z ≤ z] ≤
n X
i=1
Pr[ui−1 < X ≤ ui,C(ui−1,Y ) ≤ z].
Further, for z ≤ u ≤ 1, since ψ and ψ← are decreasing, C(u,Y ) ≤ z is
equivalent to Y ≤ ψ←{ψ(z) − ψ(u)}. We ﬁnd that
Pr[z < X ≤ x, Z ≤ z]
≤
n X
i=1
Pr[ui−1 < X ≤ ui, Y ≤ ψ
←{ψ(z) − ψ(ui−1)}]
=
n X
i=1
(C(ui,ψ
←{ψ(z) − ψ(ui−1)}) − C(ui−1,ψ
←{ψ(z) − ψ(ui−1)}))
=
n X
i=1
(ψ
←{ψ(ui) + ψ(z) − ψ(ui−1)} − ψ
←{ψ(z)}).
Our choice of the grid {ui} is such that
ψ(ui) − ψ(ui−1) = −{ψ(z) − ψ(x)}/n, i = 1,...,n.
4Hence
Pr[z < X ≤ x, Z ≤ z] ≤ n

ψ
←[ψ(z) − {ψ(z) − ψ(x)}/n] − ψ
←{ψ(z)}

Since ψ← is convex with nondecreasing derivative 1/(ψ0 ◦ ψ←),
ψ
←(a) − ψ(b) ≤ (a − b)
1
ψ0{ψ←(a)}
, 0 < a < b < ψ(0).
Combine the two previous displays to ﬁnd
Pr[z < X ≤ x, Z ≤ z] ≤ −
ψ(z) − ψ(x)
ψ0(ψ←[ψ(z) − {ψ(z) − ψ(x)}/n])
Let n tend to inﬁnity and use the fact that z is a continuity point of ψ0 to ﬁnd
Pr[z < X ≤ x, Z ≤ z] ≤ −
ψ(z) − ψ(x)
ψ0(z)
.
The inequality in the other direction follows in a similar fashion. We give the
steps here in full. By the same arguments as above,
Pr[z < X ≤ x, Z ≤ z]
≥
n X
i=1
Pr[ui−1 < X ≤ ui, Y ≤ ψ
←{ψ(z) − ψ(ui)}]
=
n X
i=1
(C(ui,ψ
←{ψ(z) − ψ(ui)}) − C(ui−1,ψ
←{ψ(z) − ψ(ui)}))
=
n X
i=1
(ψ
←{ψ(z)} − ψ
←{ψ(ui−1) + ψ(z) − ψ(ui)})
=n

ψ
←{ψ(z)} − ψ
←[ψ(z) + {ψ(z) − ψ(x)}/n]

≥−
ψ(z) − ψ(x)
ψ0(ψ←[ψ(z) + {ψ(z) − ψ(x)}/n])
.
Let n tend to inﬁnity and use the fact that z is a continuity point of ψ0 to
arrive at
Pr[z < X ≤ x, Z ≤ z] ≥ −
ψ(z) − ψ(x)
ψ0(z)
,
as required. 2
4 Convergence to Archimedean copula
In this section, we investigate necessary and suﬃcient properties of a sequence
of Archimedean copulas Cn with generators ψn to converge to an Archimedean
5copula C with generator ψ. Let ψ0
n and ψ be the right-hand derivatives of ψn
and ψ, respectively, and denote λn = ψn/ψ0
n, λ = ψ/ψ0, Kn(t) = t − λn(t),
and K(t) = t − λ(t).
For twice continuously diﬀerentiable generators, the equivalence of (i) and (ii)
in Proposition 2 below was already established in Genest and MacKay (1986,
Proposition 4.2). The claim that characterizations (iii) and (v) are suﬃcient
for copula convergence seems to be new.
Proposition 2 The following ﬁve conditions are equivalent:
(i) limn→∞ Cn(x,y) = C(x,y) for all (x,y) ∈ [0,1]2
(ii) limn→∞ ψn(x)/ψ0
n(y) = ψ(x)/ψ0(y) for every x ∈ (0,1] and y ∈ (0,1) such
that ψ0 is continuous in y.
(iii) limn→∞ λn(x) = λ(x) for every x ∈ (0,1) such that λ is continuous in x.
(iv) There exist positive constants κn such that limn→∞ κnψn(x) = ψ(x) for
all x ∈ [0,1].
(v) limn→∞ Kn(x) = K(x) for every x ∈ (0,1) such that K is continuous in
x.
Proof. (i) implies (ii). Let (X,Y ) and (Xn,Yn) be pairs of random vari-
ables with joint distribution functions C and Cn, respectively. Also, put Z =
C(X,Y ) and Zn = Cn(Xn,Yn). By (i), (Xn,Yn) converges in distribution to
(X,Y ) as n → ∞. Moreover, since C is a continuous distribution function,
the convergence of Cn to C is necessarily uniform in (x,y) ∈ [0,1]2. Hence
(Xn,Zn) converges in distribution to (X,Z) as n → ∞. By Proposition 1, we
have
lim
n→∞
ψn(x) − ψn(y)
ψ0
n(y)
=
ψ(x) − ψ(y)
ψ0(y)
for all 0 < y < x ≤ 1 such that ψ0 is continuous in y. Choose x = 1 to ﬁnd
lim
n→∞
ψn(y)
ψ0
n(y)
=
ψ(y)
ψ0(y)
.
Combine the two previous displays to get
lim
n→∞
ψn(x)
ψ0
n(y)
=
ψ(x)
ψ0(y)
for every 0 < y ≤ x ≤ 1 such that y < 1 and ψ0 is continuous in y. Let
0 < xi < 1 for i = 1,2 and apply the above display to (x1,y) and (x2,y) for
some 0 < y < min(x1,x2) in which ψ0 is continuous to arrive at
lim
n→∞
ψn(x1)
ψn(x2)
=
ψ(x1)
ψ(x2)
.
Combine the last two displays to arrive at (ii).
6(ii) implies (iii). Trivial.
(iii) implies (iv). For 0 < x < y < 1, we have
logψn(y) − logψn(x) =
Z y
x
ψ0
n(z)
ψn(z)
dz.
Suppose that we can show that the limit of the integral of the right-hand side
of the previous display is equal to the integral of the (almost everywhere) limit
of the integrand. Then we have
lim
n→∞{logψn(y) − logψn(x)} = logψ(y) − logψ(x).
This, in turn, obviously implies (iv).
In order to justify interchanging limit and integral in the previous paragraph,
we will show that (iii) implies
limsup
n→∞
sup
z∈[x,y]


 

ψ0
n(z)
ψn(z)


 
 < ∞.
Let 0 < ε < x be such that
|ψ
0(x − ε)| ≤ ψ(y)/(4ε).
By (iii), we have
lim
n→∞1
 
|ψ0
n(z)|
ψn(z)
> 2
|ψ0(z)|
ψ(z)
!
= 0
for almost every z ∈ [x−ε,y]. Since the above indicator variables are bounded
and converge pointwise to zero, there exists a positive integer nε such that
Z y
x−ε
1
 
|ψ0
n(z)|
ψn(z)
> 2
|ψ0(z)|
ψ(z)
!
dz < ε
for all integer n ≥ nε. Hence, for z ∈ [x,y] and integer n ≥ nε, there exist
z − ε < u < z such that
|ψ0
n(u)|
ψn(u)
≤ 2
|ψ0(u)|
ψ(u)
≤ 2
|ψ0(x − ε)|
ψ(y)
≤
1
2ε
But then, since ψn and |ψ0
n| are both nonincreasing,
ψn(z)
|ψ0
n(z)|
≥
ψn(u) − (z − u)|ψ0
n(u)|
|ψ0
n(u)|
≥ 2ε − ε = ε,
as required.
(iv) implies (i). Let φn = κnψn. Then φn is a generator of Cn. Since each φn is
monotone and since ψ is monotone and continuous, we have limn→∞ φn(xn) =
7ψ(x) whenever limn→∞ xn = x in [0,1]. Hence also limn→∞ φ←
n (tn) = ψ←(t)
whenever limn→∞ tn = t in [0,∞]. Hence, for every (x,y) ∈ [0,1]2,
Cn(x,y) = φ
←
n {φn(x) + φn(y)} → ψ
←{ψ(x) + ψ(y)} = C(x,y),
as n → ∞.
(v) implies (iii) and conversely. Trivial. 2
5 Convergence to comonotone copula
The comonotone copula is itself not an Archimedean copula, so that Propo-
sition 2 is not suitable for deciding whether a sequence of copulas converges
to the comonotone copula. The following resulting, extending Nelsen (1999,
Theorem 4.4.8) to arbitrary generators, gives such a criterion. Let again Cn
(positive integer n) be a sequence of bivariate Archimedean copulas with gen-
erators ψn. Let ψ0
n be the right-hand derivative of ψn, and denote λn = ψn/ψ0
n
and Kn(t) = t − λn(t).
Proposition 3 The following four conditions are equivalent:
(i) limn→∞ Cn(x,y) = min(x,y) for all (x,y) ∈ [0,1]2
(ii) limn→∞ λn(x) = 0 for every x ∈ (0,1).
(iii) limn→∞ ψn(y)/ψn(x) = 0 for every 0 ≤ x < y ≤ 1.
(iv) limn→∞ Kn(x) = x for every x ∈ (0,1).
Proof. (i) implies (ii). Let (Xn,Yn) be a pair of random variables with distri-
bution function Cn. Since the limit of Cn is the comonotone copula, (Xn,Yn)
converges in distribution to (X,X), where X is a uniform random variable
on (0,1). But since the convergence in (i) is necessarily uniform, we ﬁnd
that Zn = Cn(Xn,Yn) converges in distribution to min(X,X) = X, whence
limn→∞ Pr[Zn ≤ z] = z for all z ∈ [0,1]. But by Proposition 1,
Pr[Zn ≤ z] = z +
ψn(z)
ψ0
n(z)
, 0 < z < 1.
Hence we arrive at (ii).
(ii) implies (iii). Let 0 < x < y < 1 (the cases x = 0 or y = 1 follow by
monotonicity of ψn). We have
ψn(x)
ψn(y)
− 1 =
ψn(x) − ψn(y)
ψn(y)
≥
(y − x)|ψ0
n(y)|
ψn(y)
.
By (ii), the right-hand side diverges to inﬁnity as n → ∞.
8(iii) implies (i). Since each Cn is a symmetric copula, it suﬃces to consider 0 <
x ≤ y < 1. Take 0 < w < x. By (ii), we have ψn(w) ≥ 2ψn(x) ≥ ψn(x)+ψn(y)
for all suﬃciently large integer n, whence
w ≤ ψ
←
n {ψn(x) + ψn(y)} = Cn(x,y) ≤ x.
Let ﬁrst n → ∞ and then w ↑ x to ﬁnd that limn→∞ Cn(x,y) = x. 2
6 Extension to higher dimensions
Propositions 2 and 3 can be readily extended to the general multivariate case.
Let d be an integer at least two. A d-variate copula C is the distribution
function of a d-variate random vector (X1,...,Xd),
C(x1,...,xd) = Pr[X1 ≤ x1,...,Xd ≤ xd]
the components of which are uniformly distributed on the interval [0,1], that
is, Pr[Xj ≤ x] = x for j = 1,...,d and x ∈ [0,1]. A d-variate copula C is
called Archimedean if there exists a generator ψ such that
C(x1,...,xd) = ψ
←{ψ(x1) + ··· + ψ(xd)}
for all (x1,...,xd) ∈ [0,1]d. In general, extra conditions on the generator ψ are
required to ensure that the expression in the above display deﬁnes a genuine
copula. A suﬃcient condition is for instance that ψ← is d-times diﬀerentiable
and (−D)jψ← ≥ 0 for every j = 1,...,d; see for instance Kimberling (1974,
Theorems 1 and 2), Schweizer and Sklar (1983), Barbe et al. (1996, Exam-
ple 3), and Nelsen (1999, Section 4.6).
Obviously, if the distribution function of the random vector (X1,...,Xd) is
given by the d-variate Archimedean copula C with generator ψ, then the
distribution function of every bivariate subvector (Xi,Xj), with i 6= j, is given
by the bivariate Archimedean copula with the same generator. This property
can be used to upgrade Propositions 2 and 3 to the general multivariate case.
Let Cn be a sequence of d-variate Archimedean copulas with generators ψn.
On the one hand, if Cn converges to another d-variate Archimedean copula C
with generator ψ or to the d-variate comonotone copula, then the sequence
of bivariate Archimedean copulas with generators ψn must converge to the
bivariate Archimedean copula with generator ψ or to the bivariate comonotone
copula, respectively. Hence, the stated conditions on the sequence of generators
are certainly necessary for convergence of the sequence of copulas. On the other
hand, they are also suﬃcient, as the proofs of the implications “(iv) implies
(i)” in Proposition 2 and “(iii) implies (i)” in Proposition 3 carry over to the
d-variate case with only notational changes.
97 Counterexample
From Propositions 2 and 3, one might get the impression that every limit
copula of a sequence of Archimedean copulas is necessarily Archimedean or
comonotone. This is not true, as is demonstrated by the following example.
For integer n ≥ 2, deﬁne a generator ψn by
ψn(x) =

 
 
n − 2(n − 1)x if 0 ≤ x ≤ 1/2,
2(1 − x) if 1/2 ≤ x ≤ 1.
That is, ψn is piecewise linear with knots ψn(0) = n, ψn(1/2) = 1, and ψn(1) =
0. Denoting the right-hand derivative of ψn with ψ0
n, we have
λn(x) =
ψn(x)
ψ0
n(x)
=

 
 
x − n/{2(n − 1)} if 0 ≤ x < 1/2,
x − 1 if 1/2 ≤ x ≤ 1,
and therefore
Kn(x) = x − λn(x) =

 
 
n/{2(n − 1)} if 0 ≤ x < 1/2,
1 if 1/2 ≤ x ≤ 1.
Let Cn be the Archimedean copula with generator ψn. By direct computation,
one arrives at
lim
n→∞Cn(x,y) = C(x,y) =

        
        
(x + y − 1/2)+ if (x,y) ∈ [0,1/2]2,
x if 0 ≤ x < 1/2 < y ≤ 1,
y if 0 ≤ y < 1/2 < x ≤ 1,
1/2 + (x + y − 3/2)+ if (x,y) ∈ [1/2,1]2.
The copula C corresponds to the uniform distribution, with respect to one-
dimensional Lebesgue measure, on the union of the two line segments {(x,y) ∈
[0,1]2 | x + y = 1/2} and {(x,y) ∈ [0,1]2 | x + y = 3/2}. The copula C is not
Archimedean, because the function
lim
n→∞
|ψ0
n(x)|
ψn(x)
=

 
 
1/(1/2 − x) if 0 ≤ x < 1/2,
1/(1 − x) if 1/2 ≤ x < 1
is not integrable around x = 1/2. Note also that Kn converges towards K as
10n goes to inﬁnity, where
K(x) =

 
 
1/2 if 0 ≤ x < 1/2,
1 if 1/2 ≤ x ≤ 1.
Hence, limx↑1/2 K(x) = 1/2, and from Genest and Rivest (1993, Proposi-
tion 1.2), the associated copula cannot be Archimedean.
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11

A probabilistic interpretation of complete monotonicity.

An Introduction to Copulas.

Bivariate survival models induced by frailties.

Copula convergence theorems for tail events.

Copules archim´ ediennes et familles de lois bidimensionnelles dont les marges sont donn´ ees. La revue canadienne de statistique 14,

Families of multivariate distributions.

Goodness-of-ﬁt procedures for copula models based on the probability integral transformation.

Convergence of Archimedean Copulas

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