On bivariate Archimedean copulas with fractal support

1 Introduction

Considering Lipschitz continuity and the fact that bivariate copulas are distributions functions (restricted to [0, 1]²) of random vectors ( X , Y ) with X , Y being uniformly distributed on [ 0 , 1 ] , it seems somehow natural to conjecture that copulas distribute their mass in a fairly regular way. Using iterated function systems (IFSs) in 2005, Fredricks et al. [15] falsified this very conjecture by constructing bivariate copulas with fractal support. In fact, the authors showed the existence of families ( A r ) r ∈ ( 0 , 1 ∕ 2 ) of bivariate copulas fulfilling the following property: for every s ∈ (1, 2), there exists some r s ∈ ( 0 , 1 ∕ 2 ) such that the Hausdorff dimension of the support Z r s of A r s is exactly s ; in other words, the smallest closed subset of [0, 1]² with full mass 1 has Hausdorff dimension s .

Since then various papers on copulas with fractal support have appeared in the literature, each of them underlining the fact that analytically nice objects (Lipschitz continuous, common marginals, etc.) like copulas may exhibit surprisingly irregular/pathological analytic behavior: In [28], we showed that the result by Fredricks et al. also holds for the subclass of the so-called idempotent copulas (idempotent with respect to the star-product going back to Darsow et al. in [7] and corresponding to the standard composition of transition probabilities well known from the Markov chain setting) and generalized the IFS construction to arbitrary dimensions d ≥ 3 . Families ( A r ) r ∈ ( 0 , 1 ∕ 2 ) of copulas with fractal support were also studied by de Amo et al. in [1,2], moments of these copulas were calculated in [3]. Some exotic properties of homeomorphisms between fractal supports of copulas were studied in [4], an alternative proof for the result by Durante et al. via the so-called spatially homogeneous copulas was provided in [10], and Kendall’s τ of mutually completely dependent copulas with self-similar support was determined in [14].

While each of the afore-mentioned contributions illustrates that the family C of all bivariate copulas contains analytically highly irregular elements, one might still conjecture that standard, commonly used subclasses like the bivariate extreme-value and the bivariate Archimedean family do not allow for such pathological behavior. The results given in [23,29] verify this conjecture in the extreme-value setting – in this case, the support of the copula has integer Hausdorff dimension 1 or 2 and coincides with the area between the graphs of two nondecreasing functions. As we will show in this note, however, in the Archimedean setting, it is indeed possible to establish a result analogous to the one by Fredrick et al. In fact, for every fixed s ∈ [1, 2], we will construct an Archimedean copula A s whose support has Hausdorff dimension s , i.e., dim H ( supp ( A s ) ) = s holds. Contrary to the afore mentioned papers, we do not work with the IFS approach directly in the class C of bivariate copulas but use it to construct Archimedean generators φ of sufficient irregularity, which is then shown to propagate to the corresponding Archimedean copula A φ . As a nice by-product of the studied construction, we derive an analogous result for the (measure corresponding to the) Kendall distribution function F A φ Kendall , i.e., we show that for every r ∈ [ 0 , 1 ] , there exists some copula A r whose Kendall distribution function has support with Hausdorff dimension r .

The remainder of this note is organized as follows: Section 2 gathers notation and preliminaries, Section 3 presents some auxiliary results on Cantor functions needed in the sequel. All main results are presented in Section 4. Several graphics and an example corresponding to the classical middle third Cantor set illustrate the chosen procedures and underlying ideas.

2 Notation and preliminaries

For every metric space ( Ω , ρ ) , the Borel σ -field on Ω will be denoted by ℬ ( Ω ) , the family of all probability measures on ℬ ( Ω ) by P ( Ω ) . The support of a measure ϑ ∈ P ( Ω ) , defined as the set of all points x ∈ Ω such that every open ball B ( x , r ) of radius r > 0 around x fulfills ϑ ( B ( x , r ) ) > 0 , will be denoted by supp ( ϑ ) . It is well-known [26] that supp ( ϑ ) is closed and that supp ( ϑ ) is the complement of the union of all open sets U ⊆ Ω fulfilling ϑ ( U ) = 0 .

As already mentioned, C will denote the family of all two-dimensional copulas, i.e., the family of distribution functions (restricted to [0, 1]²) of random vectors ( X , Y ) on a probability space ( Ω , A , P ) , fulfilling that the marginal distributions P X , P Y coincide with the Lebesgue measure λ on [ 0 , 1 ] . Letting d ∞ denote the uniform metric on C , it is well known that ( C , d ∞ ) is a compact metric space. M will denote the minimum copula, Π the product copula, and W the lower Fréchet-Hoeffding bound. For every C ∈ C , the corresponding doubly stochastic measure will be denoted by μ C and P C ⊂ P ( [ 0 , 1 ] 2 ) will denote the family of all doubly stochastic measures. By definition, the support of a copula C is the support of its corresponding doubly stochastic measure μ C . Considering compactness of [0, 1]², the support of every copula is (as closed subset of a compact set) compact. For general background on copulas and doubly stochastic measure, we refer to the textbooks [9,24].

A Markov kernel from R to ℬ ( R ) is a mapping K : R × ℬ ( R ) → [ 0 , 1 ] such that x ↦ K ( x , B ) is measurable for every fixed B ∈ ℬ ( R ) and B ↦ K ( x , B ) is a probability measure for every fixed x ∈ R . Given real-valued random variables X , Y on a joint probability space ( Ω , A , P ) , then a Markov kernel K : R × ℬ ( R ) → [ 0 , 1 ] is called a regular conditional distribution of Y given X if for every B ∈ ℬ ( R )

holds for P -almost every ω ∈ Ω . It is well known that for each pair ( X , Y ) of real-valued random variables a regular conditional distribution K ( ⋅ , ⋅ ) of Y given X exists, that K ( x , ⋅ ) is unique P X -almost everywhere (i.e., unique for P X – almost all x ∈ R ) and that K ( ⋅ , ⋅ ) only depends on the joint distribution P ( X , Y ) of ( X , Y ) . Hence, given C ∈ C , we will denote (a version of) the regular conditional distribution of Y given X by K C ( ⋅ , ⋅ ) , view it directly as a function mapping [ 0 , 1 ] × ℬ ( [ 0 , 1 ] ) → [ 0 , 1 ] and refer to K C ( ⋅ , ⋅ ) simply as regular conditional distribution of C or as Markov kernel of C. Note that for every C ∈ C , its Markov kernel K C ( ⋅ , ⋅ ) , and every Borel set G ∈ ℬ ( [ 0 , 1 ] 2 ) , we have ( G x ≔ { y ∈ [ 0 , 1 ] : ( x , y ) ∈ G } denoting the x -section of G for every x ∈ [ 0 , 1 ] )

As a special case, the latter yields that

holds for every F ∈ ℬ ( [ 0 , 1 ] ) . On the other hand, every Markov kernel K : [ 0 , 1 ] × ℬ ( [ 0 , 1 ] ) → [ 0 , 1 ] fulfilling equation (3) induces a unique element μ ∈ P C via equation (2). For every C ∈ C and x ∈ [ 0 , 1 ] , the function y ↦ G x C ( y ) ≔ K C ( x , [ 0 , y ] ) will be called conditional distribution function of C at x. Moreover, considering that K C ( x , ⋅ ) is a probability measure for every x ∈ [ 0 , 1 ] , the afore-mentioned definition of the support of a measure directly carries over to K C ( x , ⋅ ) . For more details and properties of conditional expectation, regular conditional distributions, Markov kernels, and disintegration, we refer to the excellent textbooks [17,20].

Archimedean copulas can be expressed via generators φ : [ 0 , 1 ] → [ 0 , ∞ ] (see, e.g., [13,24]) or, equivalently, via generators ψ : [ 0 , 1 ] → [ 0 , ∞ ) (see, e.g., [19,22]). Here, we use the former approach since it facilitates our construction. Following [24], a function φ : [ 0 , 1 ] → [ 0 , ∞ ] is called generator if φ is convex on ( 0 , 1 ] , continuous and strictly decreasing on [ 0 , 1 ] and fulfills φ ( 1 ) = 0 . A generator φ is called strict if φ ( 0 ) = ∞ holds; in case of φ ( 0 ) < ∞ , we will refer to φ as nonstrict. For every generator φ , we will let ψ : [ 0 , ∞ ) → [ 0 , 1 ] denote its pseudo-inverse, defined by

To simplify notation in what follows, we will work with the convention ψ ( ∞ ) ≔ 0 . If φ is strict, then obviously ψ coincides with the standard inverse. Every (strict or nonstrict) generator φ induces a symmetric copula A φ via

to which we will refer as the (strict or nonstrict) Archimedean copula induced by φ . The family of all bivariate Archimedean copulas will be denoted by C a r . Since for our construction of Archimedean copulas with fractal support we will only work with nonstrict generators, in the sequel, we only focus on the nonstrict case without explicit mention and refer to [19,22]) for the general setting.

In what follows, we will let [ A φ ] t denote the lower t -cut of A φ , i.e.,

According to [24] for every Archimedean copula A φ , the level set L t ≔ { ( x , y ) ∈ [ 0 , 1 ] 2 : A φ ( x , y ) = t } is a convex curve for every t ∈ ( 0 , 1 ] . For t = 0 , the set L 0 has positive area. Defining the function f t : [ t , 1 ] → [ 0 , 1 ] for t ∈ ( 0 , 1 ] by

have that f t is continuous and that

for every t ∈ ( 0 , 1 ] , i.e., the graph Γ ( f t ) of f t coincides with the level curve L t . For t = 0 , we define f 0 analogous to equation (4) for every x > 0 and set f 0 ( 0 ) = 1 . Then L 0 = { ( x , y ) ∈ [ 0 , 1 ] 2 : y ≤ f 0 ( x ) } holds, i.e., the graph of f 0 is the upper bound of L 0 . As a direct consequence, for arbitrary 0 ≤ s < t ≤ 1 and every x ∈ [ t , 1 ] we have f t ( x ) > f s ( x ) . Moreover, it is straightforward to verify that for every A φ ∈ C a r and every x ∈ ( 0 , 1 ] , the function y ↦ A φ ( x , y ) is strictly increasing on [ f 0 ( x ) , 1 ] .

Before providing an explicit expression of the Markov kernel of a general (nonstrict) Archimedean copula, we first recall some analytic properties of generators that will be used in the sequel: For every generator φ : [ 0 , 1 ] → [ 0 , ∞ ] , we will let D + φ ( x ) ( D − φ ( x ) ) denote the right-hand (left-hand) derivative of φ at x ∈ ( 0 , 1 ) . Convexity of φ implies that D + φ ( x ) = D − φ ( x ) holds for all but at most countably many x ∈ ( 0 , 1 ) , i.e., φ is differentiable outside a countable subset of (0, 1), and that D + φ is nondecreasing and right-continuous while D − φ is nondecreasing and left continuous (see [18,25]). Setting D + φ ( 1 ) = 0 allows to view D + φ as nondecreasing and right-continuous function on the full unit interval [ 0 , 1 ] . In addition, (again see [18,25]), we have D − φ ( x ) = D + φ ( x − ) for every x ∈ ( 0 , 1 ) .

To simplify notation, for every a ∈ R , expressions of the from a − ∞ will be interpreted as zero in what follows. According to [13], for nonstrict φ the mapping K A φ ( ⋅ , ⋅ ) , defined by

is a Markov kernel of A φ . For every generator φ [13,24], the following identity holds for the mass of the level sets L t = Γ ( f t ) :

Moreover we have μ A φ ( L 0 ) = − φ ( 0 ) D + φ ( 0 ) , so in the latter case L 0 may or may not have mass depending on whether D + φ ( 0 ) is unbounded or not. Notice that formula (7) offers a nice geometric interpretation (see [13, Figure 1]): μ A φ ( L t ) coincides with the length of the line segment on the x -axis generated by left- and right-hand tangents of φ at t .

As shown in [13,24], the Kendall distribution function F A φ Kendall of a nonstrict Archimedean copula A φ is given by

We will directly use these expressions in the next sections in order to show that the probability measure κ A φ corresponding to the Kendall distribution function, i.e., the measure fulfilling κ A φ ( ( a , b ] ) = F A φ Kendall ( b ) − F A φ Kendall ( a ) for all intervals ( a , b ] ⊆ [ 0 , 1 ] , has fractal support.

As a last key component, we recall the definition of an IFS and some main results about IFSs [5,11,12,21]. Suppose for the following that ( Ω , ρ ) is a compact metric space and let δ H denote the Hausdorff metric on the family K ( Ω ) of all nonempty compact subsets of Ω . A mapping w : Ω → Ω is called contraction if there exists a constant L < 1 such that ρ ( w ( x ) , w ( y ) ) ≤ L ρ ( x , y ) holds for all x , y ∈ Ω . A family ( w l ) l = 1 N of N ≥ 2 contractions on Ω is called IFS and will be denoted by { Ω , ( w l ) l = 1 N } . Every IFS induces the so-called Hutchinson operator ℋ : K ( Ω ) → K ( Ω ) , defined by

It can be shown [5,12,21] that ℋ is a contraction on the compact metric space ( K ( Ω ) , δ H ) , so Banach’s fixed point theorem implies the existence of a unique, globally attractive fixed point Z ⋆ ∈ K ( Ω ) of ℋ . Hence, for every R ∈ K ( Ω ) , we have

The attractor Z ⋆ will be called self-similar if all contractions in the IFS are similarities, i.e., if for every w l , there exists some constant L l ∈ ( 0 , 1 ) such that ρ ( w l ( x ) , w l ( y ) ) = ρ ( x , y ) holds for all x , y ∈ Ω . An IFS { Ω , ( w l ) l = 1 N } is called totally disconnected (or disjoint) if the sets w 1 ( Z ⋆ ) , w 2 ( Z ⋆ ) , … , w N ( Z ⋆ ) are pairwise disjoint.

An IFS together with a vector ( p l ) l = 1 N ∈ ( 0 , 1 ] N fulfilling ∑ l = 1 N p l = 1 is called iterated function system with probabilities (IFSP) and will be denoted by { Ω , ( w l ) l = 1 N , ( p l ) l = 1 N } . In addition to the operator ℋ every IFSP also induces a (Markov) operator V : P ( Ω ) → P ( Ω ) , defined by ( ϑ w i denoting the push-forward of ϑ via w i )

The so-called Hutchison metric h (sometimes also called Kantorovich or Wasserstein metric) on P ( Ω ) is defined by

where Lip 1 ( Ω , R ) is the class of all nonexpanding functions f : Ω → R , i.e., functions fulfilling ∣ f ( x ) − f ( y ) ∣ ≤ ρ ( x , y ) for all x , y ∈ Ω . It is not difficult to show that V is a contraction on ( P ( Ω ) , h ) , that h is a metrization of the topology of weak convergence on P ( Ω ) and that ( P ( Ω ) , h ) is a compact metric space [5,8]. Consequently, again by Banach’s fixed point theorem, it follows that there is a unique, globally attractive fixed point ϑ ⋆ ∈ P ( Ω ) of V , i.e., for every ν ∈ P ( Ω ) , we have

The fixed point ϑ ⋆ will be called invariant measure. It is well known that the support supp ( ϑ ⋆ ) of ϑ ⋆ is exactly the attractor Z ⋆ [5,11,12,21]. The measure ϑ ⋆ will be called self-similar if Z ⋆ is self-similar, i.e., if all contractions in the IFSP are similarities.

Attractors of IFSs are strongly interrelated with symbolical dynamical systems via the so-called address map [5,21]: For every N ∈ N , the code space of N symbols will be denoted by Σ N , i.e.,

To simplify notation in what follows, we will write k = ( k 1 , k 2 , … ) for element of Σ N . Moreover, the (left-) shift operator on Σ N will be denoted by σ , i.e., σ ( ( k 1 , k 2 , … ) ) = ( k 2 , k 3 , … ) . Defining a metric m on Σ N by setting

it is straightforward to verify that ( Σ N , m ) is a compact ultrametric space and that m is a metrization of the product topology.

Suppose now that { Ω , ( w l ) l = 1 N } is an IFS with attractor Z ⋆ , fix an arbitrary x ∈ Ω and define the address map G : Σ N → Ω by

then [21] G ( k ) is independent of x , G : Σ N → Ω is Lipschitz continuous and G ( Σ N ) = Z ⋆ . Furthermore, G is injective (and hence a homeomorphism) if, and only if the IFS is totally disconnected. Given z ∈ Z ⋆ every element of the preimage G − 1 ( { z } ) will be called address of z . Considering a IFSP { Ω , ( w l ) l = 1 N , ( p l ) l = 1 N } with attractor Z ⋆ and invariant measure μ ⋆ , we can further define a probability measure P on ℬ ( Σ N ) by setting

and extending in the standard way to full ℬ ( Σ N ) . According to [21], μ ⋆ is the push-forward of P via the address map, i.e., P G ( B ) = P ( G − 1 ( B ) ) = μ ⋆ ( B ) holds for each B ∈ ℬ ( Z ⋆ ) .

3 Auxiliary results on Cantor functions

Since for the construction of Archimedean copulas with fractal support, we will work with Cantor functions, we recall their construction via IFSs (only consisting of two functions) and then derive some properties needed in the sequel. For every r ∈ ( 0 , 1 2 ) , let the similarities w 1 r , w 2 r : [ 0 , 1 ] → [ 0 , 1 ] be defined by

set p 1 = p 2 = 1 2 , consider the totally disconnected IFSP { [ 0 , 1 ] , ( w l r ) l = 1 2 , ( p l ) l = 1 2 } and denote the corresponding Hutchinson and Markov operator by ℋ r and V r , respectively. As mentioned earlier, both ℋ r and V r have unique fixed points Z r ⋆ ∈ K ( [ 0 , 1 ] ) and ϑ r ⋆ ∈ P ( [ 0 , 1 ] ) , respectively, which are linked via

Obviously Z r ⋆ and ϑ r ⋆ are self-similar, so [5,12] the Hausdorff dimension dim H ( Z r ⋆ ) of Z r ⋆ is the unique solution s of the equation 2 r s = 1 , i.e.,

holds. Defining ι : ( 0 , 1 2 ) → ( 0 , 1 ) by ι ( r ) = − log ( 2 ) log ( r ) , obviously ι is a bijection. Considering that the IFS { [ 0 , 1 ] , ( w l r ) l = 1 2 } is totally disconnected, the address map G , defined according to equation (13), is a homeomorphism. Hence, using the fact that the product measure P on Σ 2 has no atoms, it follows that the invariant measures ϑ r ⋆ does not have any point masses. Letting G r ⋆ denote the distribution function (restricted to [ 0 , 1 ] ) corresponding to ϑ r ⋆ shows that G r ⋆ : [ 0 , 1 ] → [ 0 , 1 ] is continuous. The construction of Z r ⋆ implies that [ 0 , 1 ] \ Z r ⋆ can be expressed as follows:

with J 1 , J 2 , … denoting pairwise disjoint, nondegenerated open intervals, given by

Note that by using the Hutchinson operator, we obviously have

Considering that the IFSP construction of ϑ r ⋆ implies that G r ⋆ is constant on each J i , it follows immediately that G r ⋆ is a singular distribution function, i.e., G r ⋆ is a continuous distribution function fulfilling that the derivative ( G r ⋆ ) ′ is identical to zero λ -almost everywhere in [ 0 , 1 ] . Moreover, the property supp ( ϑ ⋆ ) = Z r ⋆ implies that for every x ∈ [ 0 , 1 ] , the following equivalence holds (extend G r ⋆ to R by setting G ( x ) = 0 for x < 0 and G ( x ) = 1 for x > 0 to assure that all expressions are well defined):

For confirming that the function φ r considered in the next section is indeed an Archimedean generator, we need the property

which either follows geometrically by using symmetry (the endo- and the hypergraph of G r ⋆ have the same area) or, alternatively, as follows: the measure ϑ r ⋆ obviously is symmetric in the sense that ( ϑ r ⋆ ) 1 − i d = ϑ r ⋆ holds, whereby ( ϑ r ⋆ ) 1 − i d is the push-forward of ϑ via the transformation ( 1 − i d ) : [ 0 , 1 ] → [ 0 , 1 ] with i d denoting the identity on [ 0 , 1 ] . In fact, for every measure ϑ ∈ P ( [ 0 , 1 ] ) , the measure V r ( ϑ ) has this property. Using change of coordinates shows

implying ∫ [ 0 , 1 ] t d ϑ r ⋆ ( t ) = 1 2 . Finally, by using the fact that for nonnegative, integrable random variables X with distribution function F , we have E ( X ) = ∫ [ 0 , ∞ ) ( 1 − F ( t ) ) d λ ( t ) yields equation (19).

Figure 1

Approximation of the classical (middle third) Cantor function G r 0 ⋆ as considered in Examples 3.1 and 4.3 (gray line). The black line depicts (an approximation of) the Kendall distribution function of the copula A r 0 , the solid magenta line is (an approximation of) the generator φ r 0 , and the dashed line is the corresponding pseudo-inverse ψ r 0 .

4 Constructing bivariate Archimedean copulas with fractal support

Let r ∈ ( 0 , 1 2 ) be arbitrary but fixed and define the function φ r : [ 0 , 1 ] → R by

Then obviously we have φ r ( 0 ) = 1 and, using equation (19), φ r ( 1 ) = 0 follows. Moreover, considering that the integrand in equation (20) is negative on [ 0 , 1 ) and nondecreasing on [ 0 , 1 ] , it follows that φ r is convex (see [18,25]) and strictly decreasing on [ 0 , 1 ] . Altogether, φ r is a nonstrict generator with right-hand derivative given by D + φ r ( t ) = − 2 + 2 G r ⋆ ( t ) . The magenta line in Figure 1 depicts the generator φ r for the case r = 1 3 , and the dashed magenta line is the corresponding pseudo-inverse ψ r .

Letting A r ≔ A φ r ∈ C a r denote the induced Archimedean copula (see Figure 2 for the case r = 1 3 ), using the results from Section 2, it follows that

i.e., the copula A r assigns half of its mass to the graph of f 0 . Using continuity of D + φ r and equation (7), all other level sets L t carry no mass, i.e., μ A r ( L t ) = 0 holds for every t ∈ ( 0 , 1 ] . Moreover, the Kendall distribution function F A r Kendall fulfills F A r Kendall ( 0 ) = 1 2 as well as

for every t ∈ ( 0 , 1 ] .

Figure 2

3D-plot of (an approximation of) the copula A r 0 considered in Examples 3.1 and 4.3; the lines depict the graphs of the function f t with t ∈ 0 , 1 6 , 1 3 , 1 2 , 2 3 , 5 6 .

We are now going to show that the Hausdorff dimension of the support of μ A r is given by supp ( μ A r ) = 1 − log ( 2 ) log ( r ) and the one of the support of the measure κ A r by − log ( 2 ) log ( r ) . Doing so, we proceed in several steps formalized as lemmas and work with the sets L J ⊆ [ 0 , 1 ] 2 , defined by

for every interval J ⊆ [ 0 , 1 ] .