Insurance: Mathematics and Economics 37 (2005) 42–48

Copulas with fractal supports
Gregory A. Fredricksa,∗ , Roger B. Nelsena , José Antonio Rodrı́guez-Lallenaba Department of Mathematical Sciences, Lewis & Clark College, Portland, OR 97219, USAb Departamento de Estadı́stica y Matemática Aplicada, Universidad de Almerı́a, 04120 La Cañada de San Urbano, Almerı́a, Spain

Received July 2004 ; received in revised form August 2004; accepted 15 December 2004

Abstract
Using an iterated function system, the authors construct families of copulas whose supports are fractals. In particular, theyshow that the members of one family have supports with arbitrary Hausdorff dimensions in the interval (1,2). They also employ
those copulas to construct more general bivariate distribution functions with fractal supports.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Copula; Fractal; Iterated function system

1. Introduction

2. Preliminaries

In this paper we establish the existence of copulas
whose supports are fractals and prove

A copula is a bivariate distribution function whose
margins are uniform on I = [0, 1]. For an introduction
to copulas see Nelsen (1999). The Borel measure on I 2
corresponding to a copula C is denoted by µC . We use
Mandelbrot’s original definition of a fractal (see Edgar,
1990, p. 179); specifically, a fractal is a set whose topological dimension is less than its Hausdorff dimension.
For an introduction to fractals, including Hausdorff andtopological dimensions, see Edgar (1990).
Most of our results follow from general techniques
relating to iterated function systems (IFSs henceforth), for which Edgar (1990, 1998) offer excellent
introductions. We have chosen these and Barnsley(1989) as our references. A hyperbolic IFS consistsof a complete metric space (X, d) together with afinite set of contraction mappings (wn : X → X). The
hyperbolic IFS {X; (wn )} satisfies Moran’s open set

Theorem 1. For each s ∈ (1, 2), there exists a copula
whose support is a fractal with Hausdorff dimension s.
Section 2 contains background information. We introduce the construction technique in Section 3 andprove Theorem 1 in Section 4. We obtain a generalization of Theorem 1 to distribution functions with moregeneral margins in Section 5.

∗ Corresponding author.

E-mail address: fredrix@lclark.edu (G.A. Fredricks).
0167-6687/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.insmatheco.2004.12.004

G.A. Fredricks et al. / Insurance: Mathematics and Economics 37 (2005) 42–48

condition if there exists a nonempty open subset U
of X for which wn (U) ∩ wm (U) = ∅ whenever n &= mand wn (U) ⊆ U for all n.
Proposition 1 (Barnsley, 1989, p. 131 and Edgar,
1990, pp. 107 and 161). If {X; (wn )} is a hyperbolic
IFS, then there exists a unique! nonempty compact subset K of X for which K = n wn (K). Moreover, if each
wn is a similarity and Moran’s open set condition issatisfied, then the Hausdorff"dimension s of K is given
implicitly by the equation n cns = 1, where cn is thesimilarity ratio of wn .
K is called the invariant set of {X; (wn )}. A hyperbolic IFS with probabilities is a hyperbolic IFS{X; (w1 , . . . , wN )} together with an ordered" set of positive numbers (p1 , . . . , pN ) for which N
i=1 pi = 1.
Each hyperbolic IFS with probabilities {X; (wn ), (pn )},with X compact, defines a Markov operator M :P(X) → P(X), where P(X) denotes the set of normalized Borel measures on X, by#
M(ν) =pn (ν ◦ w−1
n ).
n

Proposition 2(Barnsley, 1989, p. 139). If{X; (wn ), (pn )} is a hyperbolic IFS with probabilities
and X is compact, then there exists a unique µ ∈ P(X)for which M(µ) = µ. Moreover, the support of µ is theinvariant set of {X; (wn )}.
µ is called the invariant measure of {X; (wn ), (pn )}.
3. Transformation matrices and invariant
copulas43

a subdivision of I 2 into subrectangles as follows. Let pi
denote the sum of the entries in the first i columns of T,
let qj denote the sum of the entries in the first j rows of T,
and let p and q denote the partitions of I that they form.
The partitions p on the horizontal axis and q on thevertical axis determine an obvious partition of I 2 into
subrectangles. We let Rij = [pi−1 , pi ] × [qj−1 , qj ].
Example 1. For the transformation matrix$%
0.1 0.2 0.3
T =,0.4 0 0(1)

we have p = (0, .5, .7, 1), q = (0, .4, 1) and the subdivision of I 2 illustrated in Fig. 1 with Rij shaded whenever tij > 0.
For a transformation matrix T and a copula C, let
T(C) be the copula which, for each (i, j), spreads mass
tij on Rij in the same (but rescaled) way in which C
spreads mass on I 2 . Specifically, the copula T (C) is
given on Rij by#u − pi−1 #T (C)(u, v) =ti+ j+ +tij+pi − pi−1 +++i <i, j <j

+

j <j

v − qj−1 #ti+ jqj − qj−1 +i <i'&u − pi−1 v − qj−1,
+ tij C,pi − pi−1 qj − qj−1
where empty sums are defined to be zero.
A direct calculation establishes the following result
on T (C)-volumes.

Definition 1. A transformation matrix is a matrix T
with nonnegative entries, for which the sum of the entries is 1 and no row or column has every entry zero.
We double subscript the entries of T with the column
index first and the rows ordered from bottom to top, e.g.,$%t12 t22T =,t11 t21
to conform with the order of coordinates in I 2 (seeExample 1). Each transformation matrix T determines

Fig. 1. Subdivision of I 2 given by the transformation matrix (1).

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G.A. Fredricks et al. / Insurance: Mathematics and Economics 37 (2005) 42–48

Proposition 3. For each (i, j) and each (u, v) ∈ Rij ,we have()
VT (C) [pi−1 , u] × [qj−1 , v]'&
u − pi−1 v − qj−1
= tij C.
,pi − pi−1 qj − qj−1

Note that the support of T (Π), where Π(u, v) = uv,
is the union of all the rectangles Rij for which tij >0, and that the support of T (C) for any copula C iscontained in the support of T (Π).
Example 2. For T in Example 1 and M(u, v) =min(u, v), T (M) is the copula which, for each (i, j),spreads mass tij uniformly on the diagonal of Rij (seeFig. 2).
If T is a square matrix with exactly one nonzero
entry in each row and each column, then T (M) is a(straight) shuffle of M (see Mikusiński et al., 1992, p.
64). Also, if C is a copula, n is a positive integer, andT is the n × n matrix with coefficients given by
&
'
&
'
&
'
i j
i−1 j
i j−1
tij = C
,
−C
,
−C
,
n n
n n
n
n
&
'
i−1 j−1+C,,nnthen T is a transformation matrix and T (Π) is acheckerboard approximation to C (see Li et al. (1997),
p. 109).

Proposition 4. Let T be a transformation matrix and
let C1 and C2 be copulas. Then(a) T (C1 ) ≤ T (C2 ) whenever C1 ≤ C2 , where ≤ is defined pointwise, and(b) d{T (C1 ), T (C2 )} = max(tij )d(C1 , C2 ), where di,jdenotes the sup metric.
Proof. Part (a) follows from the definition of T (C).
For (b), note that for each (u, v) ∈ Rij ,|T (C1 )(u, v) − T (C2 )(u, v)|* &'*u − pi−1 v − qj−1
= tij **C1,
pi − pi−1 qj − qj−1&'*
u − pi−1 v − qj−1 **− C2.
,pi − pi−1 qj − qj−1 *Thus, sup(u,v)∈Rij |T (C1 )(u, v) − T (C2 )(u, v)| =tij d(C1 , C2 ) and the result follows. !

For a transformation matrix T and a copula C, we now
consider successive iterations of T. Define T 2 (C) =T {T (C)}, T 3 (C) = T {T 2 (C)}, etc. Note+that eachT m (C) +is a copula and that T m (C) = ( m T )(C),
where m T is the tensor product of T with itself mtimes.
Example 3. For the transformation matrix
0.1 0 0.1
T =  0 0.6 0 
0.1 0 0.1(2)(where p = q = (0, 0.2, 0.8, 1)), the supports of T (Π)
and T 2 (Π) are pictured as shaded regions in Fig. 3.

Fig. 2. Support of the copula T (M), where T is the transformation
matrix given by (1).

Fig. 3. Supports of the copulas T (Π) and T 2 (Π), where T is thetransformation matrix given by (2).

G.A. Fredricks et al. / Insurance: Mathematics and Economics 37 (2005) 42–48

45

Note that if T &= [1] (the 1 × 1 matrix with entry 1), then maxi,j (tij ) ∈ (0, 1) and hence, from (b) in
Proposition 4, that T is a contraction mapping. Since theset of copulas endowed with the sup metric is a complete metric space, the Contraction-Mapping Theorem
yield.

has area Am . Note that (σm ) is a nested sequence ofsubsets 2of I 2 and that the support of CT is containedin σ = ∞m=1 σm . Since σ has Lebesgue measure zero,
the support of CT also has Lebesgue measure zero,and thus, by definition, is a singular copula. We have
provenTheorem 2. For each transformation matrix T &=[1], there is a unique copula CT for which T (CT ) =CT . Moreover, CT is the limit of the sequence{C, T (C), T 2 (C), . . . , T m (C), . . .} for any copula C.

Theorem 3. If T is a transformation matrix with atleast one zero entry, then CT is singular.

Definition 2. A copula C is invariant if C = CT forsome transformation matrix T.
In the following example, we see that Π, M andW(u, v) = max(u + v − 1, 0) are invariant.
$%0.25 0.25
Example 4. If T =, then CT = Π.
0.25 0.25
$
%
$
%
0 0.5
0.5 0
If T =, then CT = M. If T =
,
0.5 0
0 0.5
then CT = W. More generally, CT = Π if T is a row
or column matrix or if all the entries of T are the same;
CT = M if T is square and all the nonzero entries ofT lie on the ascending diagonal; and CT = W if T issquare and all the nonzero entries of T lie on the descending diagonal.
In addition, note that if p = (pi ) and q = (qj ) are
any two partitions of I, then the matrix T = (tij ) with
tij = (pi − pi−1 )(qj − qj−1 ) is a transformation matrix which gives rise to the partitions p and q and has
CT = Π.
4. Self-similar copulas and the proof of
Theorem 1
Since the latter part of Theorem 2 tells us that thelimit copula CT is independent of the “seed” copulaused to generate the iterative sequence, we may study
CT as the limit of the sequence {T m (Π)}. For example,
consider a transformation matrix T which has at least
one zero entry. If T has # nonzero entries, then the support σ1 of T (Π) consists of # rectangles and the unionof those rectangles has an area A < 1. For each m ≥ 2,
the support σm of T m (Π) consists of #m rectangles and

For each transformation matrix T = (tij ), define
fij : I 2 → I 2 byfij (u, v)= {pi−1 + (pi − pi−1 )u, qj−1 + (qj − qj−1 )v},
where (pi ) and (qj ) are the partitions of I determinedby T. The following theorem gives two descriptions of
the support of CT .
Theorem 4. If T is a transformation matrix, which is
not a row or column matrix, then the support σ of CT
satisfies both2+(a) σ is the
2∞invariant set of {I , (fij ) }, and(b) σ = m=1 σm ,where σm is the support of T m (Π).
The plus-sign superscript denotes that the collection
is over all (i, j) for which tij > 0. Similar notation will
be used in other settings to denote the same restriction
on (i, j).
Proof. Let r be the maximum of the sidelengths of
the Rij . Note that r < 1 and that |fij (u) − fij (v)| ≤r|u − v| for each u, v ∈ I 2 and each (i, j). Hence{I 2 , (fij )+ , (tij )+ } is a hyperbolic IFS with probabilities. For any copula C we have VT (C) (R) =tij VC {fij−1 (R)} for every rectangle R ⊆ Rij , and hence"−1
VT (C) (R) = +i,j tij VC {fij (R)} for every rectangle
R ⊆ I 2 . ThereforeµT (C) =#+i,jtij (µC ◦ fij−1 ) = M(µC ),where M is the Markov operator of {I 2 , (fij )+ , (tij )+ }.
Now M(µCT ) = µT (CT ) = µCT , so µCT is the invariant measure of {I 2 , (fij )+ , (tij )+ } and (a) follows from

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G.A. Fredricks et al. / Insurance: Mathematics and Economics 37 (2005) 42–48

Proposition 2. It now follows! that σ is the fixed pointof the function F (K) = +i,j fij (K), which is a contraction mapping on the complete metric space of
nonempty compact subsets of I 2 (see Edgar, 1990,
proof of Theorem 4.1.3 on p. 107). The ContractionMapping Theorem insures that σ is also the limit of
the iterative sequence {F m (I 2 )}, which is the nested
sequence (σm ). This establishes (b). !
Corollary 1. If T is a transformation matrix with all
entries nonzero, then the support of CT is I 2 .
Proof. If T is a row or column matrix, then CT =
Π from Example 4. The other cases follow from thepreceding theorem as σm = I 2 for all m. !
Cuculescu and Theodorescu (2001) use a similar
technique to construct copulas, including one whose
support is the Sierpinski carpet. The 2 × 2 case of thepreceding corollary appears in Example 3.3 of the same
paper.
Formulas for computing the Hausdorff dimensionof the types of sets we have encountered require that
the IFSs be similarities. Toward that end, we let T be atransformation matrix that satisfies the following condition:for each nonzero entry of T, the row and column
sumsthrough that entry are equal.
(3)

For a transformation matrix T satisfying (3), let rij
denote the sidelength of Rij whenever tij &= 0, and notethat the hyperbolic IFS {I 2 , (fij )+ } is a system of similarities with similarity ratios (rij )+ . Since this IFS satisfies Moran’s open set condition with open set U equal
to the interior of I 2 , the following theorem is a consequence of Theorem 4 and Proposition 1.
Theorem 5. The support of a self-similar copula CT
has Hausdorff dimension s given by#+r s = 1.
i,j ijIn the following theorem we give sufficient conditions for the support of a self-similar copula to be afractal.
Theorem 6. Let CT be a self-similar copula for which:(a) T has at least one zero entry; and(b) there is at least one row or column of T with twononzero entries.
Then the support σ of CT is a fractal whose Hausdorff
dimension is between 1 and 2.

The preceding condition implies that each Rij is asquare whenever tij &= 0, as the relevant column androw sums are pi − pi−1 and qj − qj−1 , respectively.
It also implies that T is a square matrix. To see this,
suppose that s is a row or column sum and consider thesubmatrix S of T consisting of the intersections of therows and columns with sum s. Note that all entries inthese rows and columns not belonging to S are zero.
If S is m × n, then summing its entries by rows andby columns gives ms = ns, so m = n. It easily follows
that T is square.

" s
Proof. The function f (s) = +i,j rij is continuous andstrictly decreasing on [0, ∞), as the same holds for eachof its terms. Note that f (2) is the sum of the areas ofthe squares in the support σ1 of T (Π) and that f (1) isthe sum of the sidelengths of the squares in σ1 . Hencef (2) < 1 and f (1) > 1 by (a) and (b), respectively. It
follows from Theorem 5 and the Intermediate-Value
Theorem that the Hausdorff dimension of σ is in theinterval (1, 2). Since the topological dimension is an
integer which is less than or equal to the Hausdorff
dimension (see Edgar, 1990, p. 155), we see that the
topological dimension of σ is less than or equal to 1.
Hence σ is a fractal. !

Definition 3. An invariant copula CT is self-similar if
T satisfies (3).

We now consider the family of transformation matrices

The Borel measures corresponding to self-similar
copulas are special cases of self-similar measures (seeEdgar, 1998, p. 133). Note that Π, M and W are selfsimilar.


r/2 0 r/2
Tr =  0 1 − 2r 0 
r/2 0 r/2

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with r ∈ (0, 1/2). T0.2 appears in Example 3. It follows
from Theorems 5 and 6 that the support of CTr is a fractal whose Hausdorff dimension is the unique solutions ∈ (1, 2) to the equation
4rs + (1 − 2r)s = 1.

(4)

Proposition 5. The function s(r) defined implicitly by(4) is a strictly increasing C∞ bijection from (0, 1/2)
to (1, 2).
Proof. For each s ∈ (1, 2) define gs on (0, 1/2)
by gs (r) = 4rs + (1 − 2r)s . Then gs+ (r) = 2s{2rs−1 −(1 − 2r)s−1 }, so gs+ < 0 on (0, rs ) and gs+ > 0
on (rs , 1/2), where rs = 1/{2 + 21/(s−1) } ∈ (0, 1/2).
Since gs (0+) = 1 and gs (1/2 −) = 4/2s > 1, there isa unique solution r ∈ (0, 1/2) to gs (r) = 1, i.e., to(4). Hence s(r) is a bijection from (0, 1/2) to (1, 2).
To see that s(r) is strictly increasing and C∞ on(0, 1/2), consider the function F (r, s) = 4rs + (1 −2r)s , which is C∞ on (0, 1/2) × (1, 2). For each (r, s) ∈(0, 1/2) × (1, 2) we have (∂F/∂s)(r, s) = 4r s ln r +(1 − 2r)s ln(1 − 2r) < 0, so s(r) is C∞ on (0, 1/2) by
the Implicit-Function Theorem. Implicit differentiationin (4) yields−g+ (r)s+ (r) = ∂F s ,∂s (r, s)which is positive at points (r, s) ∈ (0, 1/2) × (1, 2) satisfying (4). !
Theorem 1 follows from the fact that s(r) maps
(0, 1/2) onto (1, 2).

5. A generalization of Theorem 1
Definition 4. A function f : D → Rk with D ⊆ Rk
is bi-Lipschitz on D if there exist c1 , c2 > 0 for
which c1 |x − y| ≤ |f (x) − f (y)| ≤ c2 |x − y| for all
x, y ∈ D.
Note that bi-Lipschitz functions are continuousand one-to-one. Also note that if f : D → R satisfies
0 < c1 ≤ |f + | ≤ c2 on D, then f is bi-Lipschitz on D by
the Mean-Value Theorem. In addition, note that if f isbi-Lipschitz on D, then the inverse of f is bi-Lipschitz
on f (D).

47

Let F, G : R = [−∞, ∞] → I be distributionfunctions such that for some a, b, c, d ∈ R
3
F (a) = 0, F (b) = 1 and F is bi-Lipschitz on [a, b],G(c) = 0, G(d) = 1 and G is bi-Lipschitz on [c, d].
(5)
The following lemma is a straightforward consequenceof the equivalence of the standard and taxi-cab metrics
in the plane.
Lemma 1. If F and G satisfy (5), then the function
F × G defined by (F × G)(x, y) = {F (x), G(y)} is biLipschitz on [a, b] × [c, d].
Recall from Sklar’s Theorem (see Sklar, 1959) that
if H is a bivariate distribution function with continuous
margins F and G, then there is a unique copula C forwhich H = C ◦ (F × G). Hence
VH ([x1 , x2 ] × [y1 , y2 ])= VC ([F (x1 ), F (x2 )] × [G(y1 ), G(y2 )])whenever x1 < x2 , y1 < y2 .

(6)Lemma 2. Let H be a bivariate distribution function
whose margins F and G satisfy (5). If S is the supportof H and T is the support of the copula C associatedwith H, then (F × G)(S) = T .
Proof. Note that S ⊆ [a, b] × [c, d]. Let S c =([a, b] × [c, d])\S, T c = I 2 \T and, for this proof, let
F × G denote the restriction of itself to [a, b] × [c, d].
Recall that F and G are continuous and strictly increasing on [a, b] and [c, d], respectively. If (x, y) isin the interior of a closed rectangle Q in [a, b] × [c, d]with VH (Q) = 0, then (F × G)(x, y) is in the interior of the closed rectangle (F × G)(Q) and VC {(F ×G)(Q)} = 0 by (6). Since S and T are the closures of
their intersections with the interiors of [a, b] × [c, d]
and I 2 , respectively, we see that (F × G)(S c ) ⊆ T c .
Since the inverses of the restrictions of F and G
to [a, b] and [c, d], respectively, are continuous andstrictly increasing on I, a similar argument showsthat (F × G)−1 (T c ) ⊆ S c . The result follows fromthe fact that F × G is a bijection from [a, b] ×[c, d] to I 2 . !

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G.A. Fredricks et al. / Insurance: Mathematics and Economics 37 (2005) 42–48

We have omitted a long and technical proof that
Lemma 2 holds for all continuous margins F and G.
Theorem 7. Let F and G be distribution functions which satisfy (5). For each s ∈ (1, 2) there exists a bivariate distribution function H with margins
F and G, whose support is a fractal of Hausdorff
dimension s.
Proof. Fix s ∈ (1, 2) and choose, by Theorem 1,
a copula C whose support T is a fractal of Hausdorff dimension s. Let H = C ◦ (F × G). Then H isa bivariate distribution function with margins F and
G by Sklar’s Theorem and the support of H is S =(F × G)−1 (T ) by Lemma 2. The function (F × G)−1
is bi-Lipschitz on I 2 as F × G is bi-Lipschitz on[a, b] × [c, d] by Lemma 1. Hence (see Falconer, 1990,
p. 30) the Hausdorff dimension of S is also s. The
topological dimension of S, being an integer, clearlyis less than the Hausdorff dimension of S, so S is afractal. !
Acknowledgements
This work was partially supported by the Ministerio de Ciencia y Tecnologı́a (Spain) and FEDER, underresearch project BFM2003-06522, and also by the Con-sejerı́a de Educacin y Ciencia of the Junta de Andalucı́a(Spain).

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