Baire category results for quasi–copulas

Abstract The aim of this manuscript is to determine the relative size of several functions (copulas, quasi– copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi– copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are obtained via a checkerboard approximation of quasi–copulas.


Introduction
The concept of a quasi-copula was introduced by Alsina et al. in [1] in order to characterize those operations on random variables that can be derived from operations on the related distribution functions (see also [5,14,20]). Since then, it also has appeared in various problems related to stochastic dependence [4], multicriteria decision making [15][16][17]19] and probabilistic metric spaces [27,28]. For more details, see [6, chapter 7] and the references therein.
It is well known that the class of quasi-copulas includes the class of copulas [3,6], which are nowadays standard tools for dealing with a variety of problems related to stochastic models. From a measure-theoretic viewpoint, however, these two objects di er in a signi cant way. While copulas are in one-to-one correspondence with (σ-additive) probability measures on [ , ] , quasi-copulas may, in general, neither be associated to positive measures (since they can assign negative mass to speci c rectangles in [ , ] ) nor to signed measures, as recently proved by [9,10,23].
The aim of this note is to clarify the relationships among several sub-classes of quasi-copulas that are characterized in terms of properties of their induced measures. In particular, we will describe the relative size of these classes in the language of Baire categories (see, e.g. [24]). In order to prove our main results, we will de ne a powerful methods to approximate quasi-copulas by means of special matrices.
We would like to stress that the use of Baire categories to characterize special classes of copulas (and related functions) originated from an open problem proposed by [2, Problem 10] (see also [8]), and has been recently applied to various contexts by [7].

Quasi-transformation matrices and checkerboard approximations
In the sequel Q denotes the class of all two-dimensional quasi-copulas, C the family of all two-dimensional copulas, Π denotes the independence copula and M the Fréchet-Hoe ding upper bound copula. All elements of Q \ C are called proper quasi-copulas. Letting d∞ denote the uniform metric on Q, Ascoli-Arzelá theorem implies compactness of (Q, d∞) as well as of (C, d∞). For further properties of copulas and quasi-copulas we refer to [6,21].
It is well-known that every copula A ∈ C corresponds to a unique doubly stochastic measure µ A on the Borel σ-eld B([ , ] ) but not all quasi-copulas correspond to a doubly stochastic signed measure on B([ , ] ), i.e. there are quasi-copulas Q ∈ Q for which there exists no (doubly stochastic) signed measure µ (see [18,26] . As usual we will refer to V Q (R) as Q-volume of the compact rectangle R. In the sequel Q S will denote the family of all quasi-copulas Q for which there exists a doubly stochastic signed measure µ Q ful lling eq.
. One possible construction of elements in Q c S is via so-called quasi-transformation matrices (which are natural generalizations of transformation matrices as studied in [11,13]) and the induced Iterated Functions Systems. We will rst recall the construction going back to [9], discuss basic properties and as well as some examples that will be used later on, and then concentrate on checkerboard approximations which are, in fact, strongly related.    Notice that these three conditions (using arguments similar to the proof of Theorem 7.4.3 in [6]) are easily seen to imply τ ∈ [− , ] m×m . In the sequel T will denote the family of all quasi-transformation matrices. τ ∈ T will be called proper if at least one entry is strictly negative.
Given a quasi-transformation matrix τ ∈ [− , ] m×m , let p i (respectively, q j ) denote the sum of the entries in the rst i columns (respectively, rst j rows) of τ, for every i (respectively, j) in { , . . . , m} and set p = q = . Having this and de ning R ij := [p i− , p i ] × [q j− , q j ] for all i, j ∈ { , . . . , m} yields a family of m non-empty compact rectangles with pairwise disjoint interior whose union is [ , ] . Every transformation matrix τ ∈ [ , ] m×m induces an operator Wτ on the family Q of all two-dimensional quasi-copulas. In fact, considering (x, y) ∈ R i ,j and setting (empty sums are zero by de nition) it is straightforward to verify that Wτ(B) ∈ Q for every B ∈ Q and that Wτ(B) ∈ Q \ C for proper τ and arbitrary B ∈ Q. Furthermore (see again [9]) it can be shown that Wτ is a contraction on the compact metric space (Q, d∞) whenever τ ∈ T ful lls L := max{|t ij | : ≤ i, j ≤ m} < . In the latter case, Banach's xed point theorem implies the existence of a unique, globally attractive xed point B * τ ∈ Q. Whenever τ is proper, according to [9] the quasi-copula B * τ has no corresponding doubly stochastic signed measure, i.e. B * τ ̸ ∈ Q S . In fact, if µ + n − µ − n denotes the Hahn decomposition (see [26]) of the signed measure induced by W n τ (Π) ∈ D S , then both sequences (µ + n ([ , ] ) n∈N and (µ − n ([ , ] ) n∈N are strictly increasing and unbounded whenever τ is proper. One obvious, but essential property of the operator Wτ that we will use in the sequel is the fact that Wτ(Q c S ) ⊆ Q c S as well as Wτ(Q S ) ⊆ Q S . Additionally, letting Qac the family of all elements on Q S for which the corresponding doubly stochastic signed measure is absolutely continuous with bounded density, we also have Wτ(Qac) ⊆ Qac. As in the case of (non-negative) transformation matrices (see [11,13]) it seems natural to extend the notion of a quasi-transformation matrix to arbitrary dimensions d ≥ . The exact conditions which such a d-dimensional quasi-transformation matrix should ful ll, however, are unclear since the property that the Q-volume V Q ful lls that V Q (R) ≥ for all d-dimensional rectangles R with at least d − adjacent faces being contained in the boundary of [ , ] d together with the standard boundary conditions is known to be su cient but not necessary for Q to be a d-dimensional quasi-copula (see [25]).     ((x, y), r). In the sequel Q loc S will denote the class of all locally extendable quasi-copulas.
We will see in the next section that (topologically speaking) typical quasi-copulas are not even locally extendable.
Suppose now that Q ∈ Q, x m ≥ and set for all i, j ∈ { , . . . , m }. Then τ Q := (t Q ij ) m i,j= is a m × m quasi-transformation matrix inducing the operator W Q m := W τ Q on Q.

De nition 2.2.
Given quasi-copulas Q and B, the quasi-copula W Q m (B) is called the B-checkerboard (quasicopula) of Q of order m. In case of B = Π we will refer to W Q m (B) simply as checkerboard of Q of order m.

Example 2.3.
We consider checkerboards of A δ with A δ being the maximal quasi-copula (MQC, for short) with given diagonal δ as studied in [22] and [29]. The MQC A δ is given by for all x, y ∈ [ , ]. According to [12] A δ ∈ Q S and the positive as well as the negative part of the Hahn decomposition of A δ are singular and concentrate their mass on the graphs of (at most) three functions. Choosing δ as in the left part of Figure 3 A δ has the form shown in Figure 4. The right part of Figure 3 depicts the (signed) density of the checkerboard W A δ (Π). Since W Q m maps Q c S into Q c S , Q S into Q S , and Qac into Qac for every m ∈ N, Theorem 2.1 has the following direct consequence: In the next section we will prove a much stronger result and show that Q c S is even co-meager in (Q, d∞).

Category results for some subsets of (Q, d ∞ )
We recall that a subset N of a (complete) metric space (Ω, d) is called nowhere dense if it is not dense in any open ball B(x, r) of radius r > (equivalently, if its closure has empty interior). A set A ⊆ Ω is called meager or of rst category in (Ω, d) if it can be expressed as (or covered by) a countable union of nowhere dense sets.
A is called of second category if it is not meager. Finally, A is called co-meager (or residual) if A c = Ω \ A is meager. Loosely speaking, we will also refer to the elements of a co-meager set as typical and to the elements of a meager set as atypical in Ω.
A typical quasi-copula is proper -the following even stronger result holds: Proof. For every C ∈ C there exists a sequence (Qn) n∈N of proper quasi-copulas that converges to C with respect to d∞. In fact, setting Bn := ( − n )C + n Qn with Qn ∈ Q S being a proper quasi-copula for which the corresponding doubly stochastic signed measure µ Qn ful lls µ Qn ( M m= Rm) < − n for some rectangles R , . . . , R M ⊆ [ , ] directly yields Bn ∈ Q \ C. Notice that such copulas Qn can be constructed by considering W l τ (Π) for su ciently large l ∈ N. As immediate consequence (the closed set) C cannot contain any nonempty open subset of (Q, d∞).
The next theorem shows that typical quasi-copulas cannot be associated with doubly stochastic signed measures on B([ , ] ): Proof. For every M ∈ N consider the set Q M S of all quasi-copulas Q ∈ Q S ful lling that for every k ∈ N and every nite collection (R i ) k i= of compact rectangles with pairwise disjoint interior ). Finally, since for every Q ∈ Q S the corresponding doubly stochastic signed measure µ Q is nite (in fact we must have µ([ , ] ) = ) we get Q S ⊆ ∞ M= Q M S , implying that Q S is of rst category too.
We now concentrate on the class Q loc S which is strictly greater than Q S . Nevertheless even Q loc S is topologically very small, implying that (topologically speaking) typical quasi-copulas do not even locally induce signed measures and underlining the fact that there is only a very weak connection between quasi-copulas and doubly stochastic signed measures.

Conclusions
Inspired by an open problem related to the study of Baire category for copulas [2], we study the category of special subsets of quasi-copulas. Interestingly, the class of quasi-copulas that are (locally) associated with a doubly stochastic signed measure is a set of rst category in the class of all quasi-copulas. In other words, a typical copula cannot be associated (even locally) with a signed measure. The results are proved by using a special approximation of quasi-copulas, a tool of general interest in copula theory. Although the results are formulated in the two-dimensional case, they can be extended to higher dimensions.