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Article

On the Size of Subclasses of Quasi-Copulas and Their Dedekind–MacNeille Completion

1
Dipartimento di Scienze dell’Economia, Università del Salento, 73100 Lecce, Italy
2
Research Group of Theory of Copulas and Applications, University of Almería, 04120 Almería, Spain
3
Department for Mathematics, University of Salzburg, A-5020 Salzburg, Austria
4
Department of Mathematics, University of Almería, 04120 Almería, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(12), 2238; https://doi.org/10.3390/math8122238
Received: 11 November 2020 / Revised: 11 December 2020 / Accepted: 14 December 2020 / Published: 18 December 2020
(This article belongs to the Special Issue Stochastic Models with Applications)

Abstract

We study some topological properties of the class of supermodular n-quasi-copulas and check that the topological size of the Dedekind–MacNeille completion of the set of n-copulas is small, in terms of the Baire category, in the Dedekind–MacNeille completion of the set of the supermodular n-quasi-copulas, and in turn, this set and the set of n-copulas are small in the set of n-quasi-copulas.
Keywords: copula; Dedekind–MacNeille completion; quasi-copula; supermodularity copula; Dedekind–MacNeille completion; quasi-copula; supermodularity

1. Introduction

The classical procedure in multivariate stochastic models assumes the existence of a suitable (joint) probability distribution, expressed in terms of the univariate marginals and the copula, that provides an accurate description of the phenomenon under consideration, and all subsequent computations (i.e., risk calculations) are based on this model. However, in most practical cases, the joint distribution is only partially known, and this model uncertainty should be taken into consideration. In particular, various investigations have considered that we only know the marginal distributions, but do not know the dependence structure, i.e., we do not know the copula.
These latter studies typically provide dependence uncertainty bounds for the copula and/or for related functionals of the multivariate distributions. Their origin has a long history going back to earlier works by Hoeffding and Fréchet (see, for instance, [1] and the references therein), but they have been extended to many kinds of functionals of the joint distribution, especially related to quantile estimation and risk calculation in finance and insurance (see, e.g., [2,3] and the references therein). As a matter of fact, these bounds are in general not copulas, but quasi-copulas. Such objects were introduced in [4,5] and characterized in analytical terms in [6,7] (for a complete overview, we refer to [8,9]). Noticeably, the suprema and infima of copulas are actually quasi-copulas, although not all quasi-copulas can be expressed in terms of lattice operations on copulas in dimensions greater than three (see [10,11]).
Quasi-copulas have recently appeared in the study of the model-free procedure for pricing financial instruments (see [3,12,13,14]). In fact, as underlined in [3], since the bounds on the dependence structure of a given random vector X may not be copulas, then the results from stochastic order theory that translate the bounds on the copula of X into the bounds on the expectation of f ( X ) do not apply directly, and some concepts need to be extended into the quasi-copula framework.
Motivated by the study of bounds for special classes of (quasi-)copulas, here, we focus our attention on copulas, supermodular quasi-copulas (see, for instance, [15]), as well as their Dedekind–MacNeille completion. In fact, these sets may represent natural frameworks where lattice operations on copulas naturally lie. Our purpose is, hence, to study the relative size of these sets by using the concept of the Baire category as considered, for instance, in [16,17,18,19]. Specifically, we will determine the size of the Dedekind–MacNeille completion of the sets of copulas and supermodular quasi-copulas with respect to the topology induced by the distance (or metric) d , i.e., the uniform convergence in the set of quasi-copulas.
The paper is organized as follows. After recalling some preliminary definitions and notations (Section 2), in Section 3, we present some properties related to supermodular quasi-copulas and show that the set of supermodular quasi-copulas is compact under the metric d . In Section 4, we show that the Dedekind–MacNeille completion of the set of copulas is small, in terms of the Baire category, in the Dedekind–MacNeille completion of the set of the supermodular quasi-copulas, and in turn, this set and the set of copulas are small in the set of quasi-copulas. Finally, Section 5 concludes.

2. Preliminaries

First, we recall some basic aspects about copulas and quasi-copulas (see, e.g., [20,21]).
Let n 2 be a natural number. We recall that an n-dimensional quasi-copula (briefly, n-quasi-copula) is a function Q from [ 0 , 1 ] n to [ 0 , 1 ] satisfying:
(Q1)
Boundary conditions: For every u = ( u 1 , u 2 , , u n ) [ 0 , 1 ] n , Q ( u ) = 0 if at least one coordinate of u is equal to zero; and Q ( u ) = u k whenever all coordinates of u are equal to one, except maybe u k .
(Q2)
Monotonicity: Q is nondecreasing in each variable.
(Q3)
Lipschitz condition: For every u , v [ 0 , 1 ] n , it holds that | Q ( u ) Q ( v ) | i = 1 n | u i v i | .
The set of n-quasi-copulas will be denoted by Q n .
Quasi-copulas are generalizations of the concept of a copula, which is recalled here. An n-copula is a function C from [ 0 , 1 ] n to [ 0 , 1 ] that satisfies the condition (Q1) for n-quasi-copulas and, in place of (Q2) and (Q3), the stronger condition:
(Q4)
n-increasing property: V C ( B ) : = ( 1 ) k ( c ) C ( c ) 0 for every n-box B = × i = 1 n [ a i , b i ] in [ 0 , 1 ] n , where the sum is taken over all the vertices c = ( c 1 , c 2 , , c n ) of B (i.e., each c k is equal to either a k or b k ), and k ( c ) is the number of indices k’s such that c k = a k .
The set of n-copulas will be denoted by C n .
Every n-copula is an n-quasi-copula, and a proper n-quasi-copula is an n-quasi-copula, which is not an n-copula—the set of proper n-quasi-copulas will be denoted by Q n C n .
If we consider the standard partial order among real-valued functions in the space of quasi-copulas, then we can provide upper and lower bounds in Q n . In fact, every n-quasi-copula Q satisfies the following condition:
W n ( u ) : = max i = 1 n u i n + 1 , 0 Q ( u ) min ( u 1 , u 2 , , u n ) = : M n ( u ) for all u [ 0 , 1 ] n
It is known that: (a) M n is an n-copula for every n 2 , (b) W 2 is a two-copula, and (c) W n is a proper n-quasi-copula for every n 3 . For several interesting similarities and differences between copulas and proper quasi-copulas, see, for example, [22,23,24,25,26,27].
In the following, we will also consider some notions from lattice theory (see, e.g., [28]), which are recalled here.
Given two elements x and y of a partially ordered set (i.e., poset) ( P , ) , let x y denote the join (or the least upper bound) of x and y (when it exists); similarly for S , where S is a subset of P; x y denotes the meet (or the greatest lower bound) of x and y (when it exists); and similarly for S . If the join or meet is found within a particular poset P, we subscript P S . Given two posets A and B, we say that A is join-dense (respectively, meet-dense) in B if, for every d in B, there exists a set S A such that d = B S (respectively, d = B S ). A poset P is a lattice if for every x , y in P, x y and x y are in P; and P is a complete lattice if, for every S P , S and S are in P.
If φ : P L is an order-embedding (i.e., order-preserving injection) of a poset P into a complete lattice L, then we say that L is a completion of P. Moreover, if φ maps P onto L, then φ is referred to as an order-isomorphism (i.e., order-preserving bijection). We also have the following definition (see [28]).
Definition 1.
A completion P of a lattice L is called a Dedekind–MacNeille completion of L if P is join-dense and meet-dense in L.

3. Quasi-Copulas and Related Subclasses

Now, we consider the class of quasi-copulas Q n equipped with the standard partial order among real-valued functions. For any pair Q 1 and Q 2 of quasi-copulas (or copulas), Q 1 Q 2 = inf { Q Q n | Q 1 Q , Q 2 Q } and Q 1 Q 2 = sup { Q Q n | Q Q 1 , Q Q 2 } . As is known (see [10,11]), Q n is a complete lattice; however, C n (respectively, Q n C n ) is not even a lattice. We denote by DM ( C n ) the Dedekind–MacNeille completion of the set of n-copulas in Q n .
It is known that:
  • the set of two-quasi-copulas Q 2 is order-isomorphic to the Dedekind–MacNeille completion of the set of two-copulas C 2 (see [11]);
  • for n 3 , Q n is not order-isomorphic to DM ( C n ) (see [10]).
In the quest for a suitable subset of quasi-copulas that may be order-isomorphic to DM ( C n ) , supermodular quasi-copulas were considered in [15]. Here, we recall the definition of this concept (see, e.g., [29]).
Definition 2.
A function f : [ 0 , 1 ] n [ 0 , 1 ] is called supermodular if, for all x , y [ 0 , 1 ] n , it holds that f ( x y ) + f ( x y ) f ( x ) + f ( y ) .
The following result is a useful characterization of supermodular functions (see [30,31]). We recall that for any u [ 0 , 1 ] n and any set of indices A { 1 , 2 , , n } with 0 < c a r d ( A ) = k < n , the k-dimensional section of a function f : [ 0 , 1 ] n [ 0 , 1 ] with fixed values given by u at the positions not in A is the function f u , A : [ 0 , 1 ] k [ 0 , 1 ] defined by f u , A ( x ) = f ( y ) , where y j = x j if j A and y j = u j if j A .
Proposition 1.
A function f : [ 0 , 1 ] n [ 0 , 1 ] is supermodular if, and only if, all of its two-dimensional sections are supermodular.
For n = 2 , supermodularity and two-increasingness are equivalent. However, this is no longer true for n 3 (see [22]). Furthermore, supermodularity together with the boundary conditions (Q1) implies increasingness and one-Lipschitz continuity, whence the following result is obtained (see [22]).
Proposition 2.
If S : [ 0 , 1 ] n [ 0 , 1 ] is a supermodular function satisfying the condition (Q1) of an n-quasi-copula, then S is an n-quasi-copula.
Let SQ n denote the set of supermodular n-quasi-copulas. As can be easily seen, for n 3 , C n SQ n (see [22]). Moreover, C 2 = SQ 2 . A relevant subset of SQ n is formed by all Archimedean n-quasi-copulas (see [22]), as introduced in [32]. In particular, W n is supermodular. For other examples of supermodular n-quasi-copulas, see [33].
Now, we consider the lattice structure of SQ n and its related Dedekind–MacNeille completion, denoted by DM ( SQ n ) . The following result follows from [15].
Proposition 3.
For n 3 , the following results hold:
(a) 
SQ n is join-dense in Q n ;
(b) 
SQ n is not meet-dense in Q n , i.e., there exists an n-quasi-copula Q L such that for any A SQ n , Q L Q n A .
Thus, SQ n is not a complete lattice.
As a consequence of Proposition 3, for n 3 , Q n is not order-isomorphic to DM ( SQ n ) . Furthermore, in the following example, we show that neither SQ n DM ( C n ) nor DM ( C n ) SQ n hold.
Example 1.
Let n 3 . We know that W n SQ n ; however, W n DM ( C n ) since there does not exist an n-copula C such that C W n .
On the other hand, consider the following two n-copulas: C i * ( u ) = C i ( u 1 , u 2 ) j = 3 n u j for i = 1 , 2 and for all u = ( u 1 , u 2 , , u n ) [ 0 , 1 ] n , where C 1 and C 2 are the two-copulas given by C 1 ( u 1 , u 2 ) = min ( u 1 , u 2 , max ( 0 , u 1 2 / 3 , u 2 1 / 3 , u 1 + u 2 1 ) ) and C 2 ( u 1 , u 2 ) = C 1 ( u 2 , u 1 ) , respectively. Then, we have that Q = C 1 * C 2 * is a proper n-quasi-copula such that Q DM ( C n ) (see [10]); furthermore, Q is not supermodular, since C 1 C 2 is a proper two-quasi-copula [11], i.e., it is not supermodular.
We conclude this section with two additional properties about the structure of the set SQ n .
It is known that the sets C n and Q n are compact under the metric d (see [20,34]). This is also the case for the set SQ n , as the next result shows.
Proposition 4.
The set SQ n is compact under the metric d .
Proof. 
Let Q r r N be a sequence in SQ n that converges pointwise to an n-quasi-copula Q n . Since Q r ( x y ) + Q r ( x y ) Q r ( x ) + Q r ( y ) for every r N and for all x , y [ 0 , 1 ] n , taking the limits on both sides of the inequality, we have:
Q ( x y ) + Q ( x y ) = lim r + Q r ( x y ) + lim r + Q r ( x y ) = lim r + Q r ( x y ) + Q r ( x y ) lim r + Q r ( x ) + Q r ( y ) = lim r + Q r ( x ) + lim r + Q r ( y ) = Q ( x ) + Q ( y ) ,
i.e., SQ n is closed in Q n . Since Q n is compact under the metric d , the set SQ n is also compact, which completes the proof. □
For the next result, we need to recall the concept of an ordinal sum for quasi-copulas. Let J be a finite or countably infinite subset of N , and let F k : [ 0 , 1 ] n [ 0 , 1 ] , k J , be a collection of functions. The ordinal sum F of ( F k ) k J with respect to the family of pairwise disjoint intervals ( ] a k , b k [ ) k J is defined, for all u [ 0 , 1 ] n , by:
F ( u ) : = a k + ( b k a k ) F k min ( u 1 , b k ) a k b k a k , , min ( u n , b k ) a k b k a k , if min ( u 1 , , u n ) ] a k , b k [ for some k J , min ( u 1 , , u n ) , otherwise .
The sets C n and Q n are closed under ordinal sums (see [20,35,36]). In the next result, we study the ordinal sum of two supermodular n-quasi-copulas in intervals of the form ] 0 , α [ and ] α , 1 [ , with α ] 0 , 1 [ .
Proposition 5.
The ordinal sum of two n-quasi-copulas of SQ n with respect to the intervals ] 0 , α [ and ] α , 1 [ , with α ] 0 , 1 [ , is in SQ n .
Proof. 
Let Q be the ordinal sum of Q 1 , Q 2 SQ n with respect to the intervals ] 0 , α [ , ] α , 1 [ . Let u i j : = ( u 1 , , u i 1 , u i + 1 , , u j 1 , u j + 1 , , u n ) [ 0 , 1 ] n 2 be a fixed point in [ 0 , 1 ] n 2 , with 1 i < j n 2 , and define the function G ( x , y ) : = Q u 1 , , u i 1 , x , u i + 1 , , u j 1 , y , u j + 1 , , u n on [ 0 , 1 ] 2 . We check that G is two-increasing (note that, as a consequence of Proposition 1, we would have G SQ n ). Observe that, for all x 1 , x 2 , y 1 , y 2 [ 0 , 1 ] such that x 1 x 2 and y 1 y 2 , we have:
G ( x 2 , y 2 ) + G ( x 1 , y 1 ) G ( x 2 , y 1 ) G ( x 1 , y 2 ) = Q u 1 , , u i 1 , x 2 , u i + 1 , , u j 1 , y 2 , u j + 1 , , u n + Q u 1 , , u i 1 , x 1 , u i + 1 , , u j 1 , y 1 , u j + 1 , , u n Q u 1 , , u i 1 , x 2 , u i + 1 , , u j 1 , y 1 , u j + 1 , , u n Q u 1 , , u i 1 , x 1 , u i + 1 , , u j 1 , y 2 , u j + 1 , , u n = V Q × h = 1 i 1 [ 0 , u h ] × [ x 1 , y 1 ] × h = i + 1 j 1 [ 0 , u h ] × [ x 2 , y 2 ] × h = j + 1 n [ 0 , u h ] .
Note also that the rectangle [ x 1 , x 2 ] × [ y 1 , y 2 ] can be decomposed into a union of rectangles, namely R k : = [ x k 1 , x k 2 ] × [ y k 1 , y k 2 ] , of disjoint interiors so that, if ] x k 1 , x k 2 [ × ] y k 1 , y k 2 [ ] 0 , α [ 2 (respectively, ] x k 1 , x k 2 [ × ] y k 1 , y k 2 [ ] α , 1 [ 2 ), then R k [ 0 , α ] 2 (respectively, R k [ α , 1 ] 2 ). We consider three cases:
  • R k ] 0 , α [ 2 ] α , 1 [ 2 = . We consider two subcases:
    1a.
    R k [ 0 , α ] × [ α , 1 ] . Then, we have 0 x k 1 < x k 2 α y k 1 < y k 2 1 ; thus:
    Q u 1 , , u i 1 , x i r , u i + 1 , , u j 1 , y i s , u j + 1 , , u n = α Q 1 ( min ( u 1 , α ) α , , min ( u i 1 , α ) α , x i r α , min ( u i + 1 , α ) α , , min ( u j 1 , α ) α , 1 , min ( u j + 1 , α ) α , , min ( u n , α ) α ) ,
    with r , s { 1 , 2 } , whence V G R k = 0 .
    1b.
    R k [ α , 1 ] × [ 0 , α ] . In this case, we have 0 y k 1 < y k 2 α x k 1 < x k 2 1 ; thus:
    Q u 1 , , u i 1 , x i r , u i + 1 , , u j 1 , y i s , u j + 1 , , u n = α Q 1 ( min ( u 1 , α ) α , , min ( u i 1 , α ) α , 1 , min ( u i + 1 , α ) α , , min ( u j 1 , α ) α , y i s α , min ( u j + 1 , α ) α , , min ( u n , α ) α ) ,
    with r , s { 1 , 2 } , whence V G R k = 0 .
  • R k ] 0 , α [ 2 . We need to consider two subcases:
    2a.
    min u i j α . Then, we have:
    Q u 1 , , u i 1 , x i r , u i + 1 , , u j 1 , y i s , u j + 1 , , u n = α Q 1 ( min ( u 1 , α ) α , , min ( u i 1 , α ) α , x i r α , min ( u i + 1 , α ) α , , min ( u j 1 , α ) α , y i s α , min ( u j + 1 , α ) α , , min ( u n , α ) α ) ,
    with r , s { 1 , 2 } . Therefore, V G R k = α V Q 1 R k 0 , where:
    R k = × h = 1 i 1 0 , min ( u h , α ) α × x k 1 α , y k 1 α × h = i + 1 j 1 0 , min ( u h , α ) α × x k 2 α , y k 2 α × h = j + 1 n 0 , min ( u h , α ) α .
    2b.
    min u i j > α . We separately study the values:
    V Q × h = 1 i 1 I h × [ x k 1 , x k 2 ] × h = i + 1 j 1 I h × [ y k 1 , y k 2 ] × h = j + 1 n I h ,
    where I j { [ 0 , α ] , [ α , u j ] } . Unless I j = [ α , u j ] for all j, all the cases correspond to Case 2a; thus, their Q-volumes are non-negative, and:
    V Q × h = 1 i 1 [ α , u h ] × [ x k 1 , x k 2 ] × h = i + 1 j 1 [ α , u h ] × [ y k 1 , y k 2 ] × h = j + 1 n [ α , u h ] = 0 .
  • R k ] α , 1 [ 2 . We have two subcases.
    3a.
    min u i j α . Then, we have:
    Q u 1 , , u i 1 , x i r , u i + 1 , , u j 1 , y i s , u j + 1 , , u n = α Q 1 ( min ( u 1 , α ) α , , min ( u i 1 , α ) α , 1 , min ( u i + 1 , α ) α , , min ( u j 1 , α ) α , 1 , min ( u j + 1 , α ) α , , min ( u n , α ) α ) ,
    with r , s { 1 , 2 } . Therefore, V G R k = 0 .
    3b.
    min u i j > α . This case is similar to Case 2b.
Therefore, we have that G is two-increasing, and the result follows. □
Notice that the previous result can be extended to the ordinal sum of a finite (or countable) set of copulas by using a similar procedure as in [36]. Specifically, the ordinal sum of k copulas, namely C 1 , , C k , can be interpreted as the ordinal sum of two copulas, C 1 and C 1 , where C 1 is an ordinal sum of the copulas C 2 , , C k (with respect to suitable intervals).

4. Baire Category Results for Subclasses of Quasi-Copulas

In this section, we check that the “size” of the set DM ( SQ n ) is “small”—in terms of the Baire category—in Q n and, in turn, DM ( C n ) is “small” in DM ( SQ n ) . We recall that a subset of a (complete) metric space is called nowhere dense if it is not dense in any open ball B ( x , r ) of centre x and radius r > 0 (equivalently, if its closure has an empty interior). Thus, for example, the set C 2 is nowhere dense in the class Q 2 (see [18]). Using the same techniques as those used in [18], it can be proven that, for n 3 , C n is nowhere dense Q n .
To study the Baire category results for the set DM ( SQ n ) in Q n , we need some preliminary results.
Lemma 1.
Let Q Q n . Then, Q DM ( SQ n ) if, and only if, for every x [ 0 , 1 ] n , there exist Q x and Q x in SQ n such that Q x ( x ) = Q x ( x ) = Q ( x ) and Q x ( y ) Q ( y ) Q x ( y ) for all y [ 0 , 1 ] n .
Proof. 
Suppose Q DM ( SQ n ) in Q n . Then, there exists a set A SQ n such that, for all x [ 0 , 1 ] n , Q ( x ) = sup Q * ( x ) : Q * A . Fixing x , there exists a sequence Q k k N in SQ n such that Q k ( y ) Q ( y ) for all y 0 , 1 n and for every k N , and Q k ( x ) converges to Q ( x ) as k goes to .
Since SQ n is compact with the topology induced by the metric d (recall Proposition 4), there exists a subsequence of Q k k N convergent to Q x SQ n . The n-quasi-copula Q x satisfies that Q x ( x ) = Q ( x ) and Q x ( y ) Q ( y ) for all y [ 0 , 1 ] n . Analogously, we can obtain that Q x ( x ) = Q ( x ) and Q ( y ) Q x ( y ) for all y [ 0 , 1 ] n .
Conversely, note that Q ( x ) = sup Q y ( x ) : y 0 , 1 n = inf Q y ( x ) : y 0 , 1 n , so that Q belongs to DM ( SQ n ) , and this completes the proof. □
Proposition 6.
The set DM ( SQ n ) is compact in Q n with respect to the metric d .
Proof. 
Let Q be an n-quasi-copula in the closure of DM ( SQ n ) . Then, there exists a sequence Q k k N in DM ( SQ n ) such that Q k Q < 1 / k for every k N . Let x [ 0 , 1 ] n be fixed. From Lemma 1, for every k N , there exists an n-quasi-copula Q k x such that Q k x ( x ) = Q k ( x ) and Q k x ( y ) Q k ( y ) for all y [ 0 , 1 ] n . Then, there exists a subsequence Q σ k x k N that converges to Q x such that Q σ k x ( x ) = Q σ k ( x ) and Q σ k x ( y ) Q σ k ( y ) for all y [ 0 , 1 ] n . Therefore, we have Q x ( x ) = Q ( x ) , and since Q σ k x ( y ) Q ( y ) + 1 / σ k , then Q x ( y ) Q ( y ) for all y [ 0 , 1 ] n . Since SQ n is closed, then Q x is supermodular. From Lemma 1, we conclude that Q DM ( SQ n ) . □
In what follows, we will need additional notation. For any Q Q n , let Q L , k denote the ordinal sum of Q and the n-quasi-copula Q L given in Proposition 3, with intervals ] 0 , 1 1 / k [ and ] 1 1 / k , 1 [ , where k N , k 2 .
Lemma 2.
For any Q Q n , we have Q L , k DM ( SQ n ) . Moreover, Q L , k converges pointwise to Q as k goes to ∞.
Proof. 
Suppose Q L , k DM ( SQ n ) . Then, from Lemma 1, for every x 1 1 / k , 1 n , there exists Q x SQ n such that Q x x = Q L , k x and Q x y Q L , k y for all y [ 0 , 1 ] n . This implies:
Q x k 1 k , , k 1 k n = k 1 k ,
i.e., Q x is an ordinal sum. Thus, the bijection from 0 , k 1 k n on [ 0 , 1 ] n defined by:
x 1 , , x n k x 1 k 1 , , k x n k 1
provides a family Q * x of supermodular n-quasi-copulas such that, for a fixed x 0 , 1 n , we have Q * x x = Q L , k x and Q * x y Q L , k y for all y 0 , 1 n . Therefore, we obtain a contradiction and, hence, Q L , k DM ( SQ n ) .
On the other hand, from the definition of Q L , k , we have:
Q L , k ( x ) Q ( x ) i = 1 n min 1 x i , x i / k .
This guarantees that Q L , k converges pointwise to Q as k goes to . □
We are now in a position to provide the main result about the “size” of the set DM ( SQ n ) in Q n .
Theorem 1.
The set DM ( SQ n ) is nowhere dense in Q n .
Proof. 
Suppose the set DM ( SQ n ) is not nowhere dense in Q n . Since, from Proposition 6, DM ( SQ n ) is closed in Q n , it must contain an open ball B, and the n-quasi-copula Q is in its interior. For a sufficiently large k, we have that Q L , k B . However, this is a contradiction since, from Lemma 2, we have Q L , k DM ( SQ n ) . Therefore, DM ( SQ n ) is nowhere dense in Q n . □
Since C n SQ n , it follows that DM ( C n ) is nowhere dense in Q n .
In the sequel, we show that DM ( C n ) is nowhere dense in DM ( SQ n ) . In order to prove it, we need some preliminary results.
Lemma 3.
Let Q Q n . Then, Q DM ( C n ) if, and only if, for every x [ 0 , 1 ] n , there exist C x and C x in C n such that C x ( x ) = C x ( x ) = Q ( x ) and C x ( y ) Q ( y ) C x ( y ) for all y [ 0 , 1 ] .
The proof of Lemma 3 is similar to the proof of Lemma 1, and we omit it.
As a consequence of Lemma 3, we have the following result, whose proof is similar to the proof of Proposition 6, and we omit it.
Proposition 7.
The set DM ( C n ) is closed in Q n with respect to the metric d , and hence, it is also compact.
For any Q Q n , let Q W n , k denote the ordinal sum of Q and W n with respect to the intervals ] 0 , 1 1 / k [ and ] 1 1 / k , 1 [ , where k N , k 2 .
Lemma 4.
For any Q Q n , we have Q W n , k DM ( C n ) . Moreover, Q W n , k converges pointwise to Q as k goes to ∞.
Proof. 
Suppose Q W n , k DM ( C n ) . Then, from Lemma 3, there exists C k C n such that:
C k k 1 k , , k 1 k n = Q W n , k k 1 k , , k 1 k n = k 1 k
and C k y Q W n , k y for all y [ 0 , 1 ] n , i.e., C k is an ordinal sum. Thus, there exists n-copulas C , C such that C k can be represented as the ordinal sum of ( C , C ) with respect to the intervals ] 0 , 1 1 / k [ and ] 1 1 / k , 1 [ . Since C k y Q W n , k y , it follows that C W n , which is absurd for n 3 . Therefore, Q W n , k DM ( C n ) .
In order to prove the convergence of Q W n , k , we just have to follow the same steps as the ones given in the proof of Lemma 2, which completes the proof. □
We are now in a position to check the “size” of the set DM ( C n ) in DM ( SQ n ) . As a consequence of Proposition 7 and Lemma 4, we have the following result, whose proof is similar to the proof of Theorem 1, and we omit it.
Theorem 2.
The set DM ( C n ) is nowhere dense in DM ( SQ n ) .
From the results presented here and mimicking the same arguments used in their proofs, it is possible to show the following two results, whose proofs are omitted.
Theorem 3.
The set C n is nowhere dense in SQ n .
For the next result, we recall that a function F : [ 0 , 1 ] n [ 0 , 1 ] is called k-dimensionally-increasing, with k { 1 , , n } , if any of its k-dimensional sections is k-increasing. Let DQ n , k denote the class of all k-dimensionally-increasing n-quasi-copulas. In [22], it was shown that C n DQ n , n 1 DQ n , n 2 DQ n , 3 SQ n Q n . Then, we have:
Theorem 4.
For every k = 3 , , n 1 , the set DQ n , k is nowhere dense in DQ n , k 1 .

5. Conclusions

In this paper, we considered the set of quasi-copulas Q n equipped with the natural ordering among real functions and the metric d . As is known, Q n is compact, and it is a complete lattice. We consider the subsets given by the class of copulas C n and of supermodular quasi-copulas SQ n , with C n SQ n . The following results hold:
  • C n and SQ n are compact (see, respectively, [20], Theorem 1.7.7, and Proposition 4;
  • C n is nowhere dense in SQ n (see Theorem 3);
  • SQ n is nowhere dense in Q n (it is an immediate consequence of Theorem 1).
Moreover, by considering the Dedekind–MacNeille completion DM ( C n ) and DM ( SQ n ) , we have that:
  • DM ( C n ) and DM ( SQ n ) are compact (see, respectively, Propositions 6 and 7);
  • DM ( C n ) is nowhere dense in DM ( SQ n ) (see Theorem 2);
  • DM ( SQ n ) is nowhere dense in Q n (see Theorem 1).
In brief, both the sets of copulas and supermodular quasi-copulas (and their Dedekind–MacNeille completions) are small, in the sense of the Baire category, in the set of n-quasi-copulas.

Author Contributions

Investigation, F.D., J.F.-S., W.T., M.Ú.-F. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

Funding

Fabrizio Durante was supported by MIUR–PRIN 2017, Project “Stochastic Models for Complex Systems” (No. 2017JFFHSH). Wolfgang Trutschnig gratefully acknowledges the support of the WISS 2025 project “IDA-lab Salzburg” (20204-WISS/225/197-2019 and 0102-F1901166-KZP).

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Rüschendorf, L. Improved Hoeffding-Fréchet bounds and applications to VaR estimates. In Copulas and Dependence Models with Applications; Úbeda-Flores, M., de Amo Artero, E., Durante, F., Sánchez, J.F., Eds.; Springer: Cham, Switzerland, 2017; pp. 181–202. [Google Scholar]
  2. Embrechts, P.; Puccetti, G.; Rüschendorf, L. Model uncertainty and VaR aggregation. J. Bank. Financ. 2013, 58, 2750–2764. [Google Scholar] [CrossRef]
  3. Lux, T.; Papapantoleon, A. Improved Fréchet-Hoeffding bounds on d-copulas and applications in model-free finance. Ann. Appl. Probab. 2017, 27, 3633–3671. [Google Scholar] [CrossRef]
  4. Nelsen, R.B.; Quesada-Molina, J.J.; Schweizer, B.; Sempi, C. Derivability of some operations on distribution functions. In Distributions with Fixed Marginals and Related Topics; Rüschendorf, L., Schweizer, B., Taylor, M.D., Eds.; IMS Lecture Notes-Monograph Series No. 28; Institute of Mathematical Statistics: Hayward, CA, USA, 1996; pp. 233–243. [Google Scholar]
  5. Alsina, C.; Nelsen, R.B.; Schweizer, B. On the characterization of a class of binary operations on distribution functions. Stat. Probab. Lett. 1993, 17, 85–89. [Google Scholar] [CrossRef]
  6. Cuculescu, I.; Theodorescu, R. Copulas: Diagonals and tracks. Rev. Roum. Math. Pures Appl. 2001, 46, 731–742. [Google Scholar]
  7. Genest, C.; Quesada-Molina, J.J.; Rodríguez-Lallena, J.A.; Sempi, C. A characterization of quasi-copulas. J. Multivar. Anal. 1999, 69, 193–205. [Google Scholar] [CrossRef]
  8. Arias-García, J.J.; Mesiar, R.; De Baets, B. A hitchhiker’s guide to quasi-copulas. Fuzzy Sets Syst. 2020, 393, 1–28. [Google Scholar] [CrossRef]
  9. Sempi, C. Quasi-copulas: A brief survey. In Copulas and Dependence Models with Applications; Úbeda-Flores, M., de Amo Artero, E., Durante, F., Sánchez, J.F., Eds.; Springer: Cham, Switzerland, 2017; pp. 203–224. [Google Scholar]
  10. Fernández-Sánchez, J.; Nelsen, R.B.; Úbeda-Flores, M. Multivariate copulas, quasi-copulas and lattices. Stat. Probab. Lett. 2011, 81, 1365–1369. [Google Scholar] [CrossRef]
  11. Nelsen, R.B.; Úbeda-Flores, M. The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas. C. R. Acad. Sci. Paris Sér. I 2005, 341, 583–586. [Google Scholar] [CrossRef]
  12. Bernard, C.; Liu, C.Y.; MacGillivray, N.; Zhang, J. Bounds on capital requirements for bivariate risk with given marginals and partial information on the dependence. Depend. Model. 2013, 1, 37–53. [Google Scholar]
  13. Lux, T.; Papapantoleon, A. Model–free bounds on Value-at-Risk using extreme value information and statistical distances. Insur. Math. Econ. 2019, 86, 73–83. [Google Scholar] [CrossRef]
  14. Tankov, P. Improved Fréchet bounds and model-free pricing of multi-asset options. J. Appl. Probab. 2011, 48, 389–403. [Google Scholar] [CrossRef]
  15. Arias-García, J.J.; De Baets, B. On the lattice structure of the set of supermodular quasi-copulas. Fuzzy Sets Syst. 2019, 354, 74–83. [Google Scholar] [CrossRef]
  16. Durante, F.; Fernández-Sánchez, J.; Trutschnig, W. A typical copula is singular. J. Math. Anal. Appl. 2015, 430, 517–527. [Google Scholar] [CrossRef]
  17. Durante, F.; Fernández-Sánchez, J.; Trutschnig, W. Baire category results for exchangeable copulas. Fuzzy Sets Syst. 2016, 284, 146–151. [Google Scholar] [CrossRef]
  18. Durante, F.; Fernández-Sánchez, J.; Trutschnig, W. Baire category results for quasi-copulas. Depend. Model. 2016, 4, 215–223. [Google Scholar] [CrossRef]
  19. Kim, C. Uniform approximation of doubly stochastic operators. Pac. J. Math. 1968, 26, 515–527. [Google Scholar] [CrossRef]
  20. Durante, F.; Sempi, C. Principles of Copula Theory; Chapman & Hall/CRC: Boca Raton, FL, USA, 2016. [Google Scholar]
  21. Nelsen, R.B. An Introduction to Copulas, 2nd ed.; Springer: New York, NY, USA, 2006. [Google Scholar]
  22. Arias-García, J.J.; Mesiar, R.; De Baets, B. The unwalked path between quasi-copulas and copulas: Stepping stones in higher dimensions. Int. J. Approx. Reason. 2017, 80, 89–99. [Google Scholar] [CrossRef]
  23. De Baets, B.; De Meyer, H.; Úbeda-Flores, M. Extremes of the mass distribution associated with a trivariate quasi-copula. C.R. Acad. Sci. Paris Ser. I 2007, 344, 587–590. [Google Scholar] [CrossRef]
  24. Fernández-Sánchez, J.; Úbeda-Flores, M. A note on quasi-copulas and signed measures. Fuzzy Sets Syst. 2014, 234, 109–112. [Google Scholar] [CrossRef]
  25. Nelsen, R.B.; Quesada-Molina, J.J.; Rodríguez-Lallena, J.A.; Úbeda-Flores, M. Some new properties of quasi-copulas. In Distributions with Given Marginals and Statistical Modelling; Cuadras, C., Fortiana, J., Rodríguez-Lallena, J.A., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2002; pp. 187–194. [Google Scholar]
  26. Nelsen, R.B.; Quesada-Molina, J.J.; Rodríguez-Lallena, J.A.; Úbeda-Flores, M. Quasi-copulas and signed measures. Fuzzy Sets Syst. 2010, 161, 2328–2336. [Google Scholar] [CrossRef]
  27. Rodríguez-Lallena, J.A.; Úbeda-Flores, M. Some new characterizations and properties of quasi-copulas. Fuzzy Sets Syst. 2009, 160, 717–725. [Google Scholar] [CrossRef]
  28. Davey, B.A.; Priestley, H.A. Introduction to Lattices and Order, 2nd ed.; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
  29. Klement, E.P.; Manzi, M.; Mesiar, R. Ultramodular aggregation functions. Inf. Sci. 2011, 181, 4101–4111. [Google Scholar] [CrossRef]
  30. Block, H.W.; Griffith, W.S.; Savits, T.H. L-superadditive structure functions. Adv. Appl. Probab. 1989, 21, 919–929. [Google Scholar] [CrossRef]
  31. Kemperman, J.H.B. On the FKG-inequality for measures on a partially ordered space. K. Ned. Akad. Wet. Proc. Ser. A 1977, 39, 313–331. [Google Scholar] [CrossRef]
  32. Nelsen, R.B.; Quesada-Molina, J.J.; Rodríguez-Lallena, J.A.; Úbeda-Flores, M. Multivariate Archimedean quasi-copulas. In Distributions with Given Marginals and Statistical Modelling; Cuadras, C., Fortiana, J., Rodríguez-Lallena, J.A., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2002; pp. 179–185. [Google Scholar]
  33. Arias-García, J.J.; Mesiar, R.; Klement, E.P.; Saminger-Platz, S.; De Baets, B. Extremal Lipschitz continuous aggregation functions with a given diagonal section. Fuzzy Sets Syst. 2018, 346, 147–167. [Google Scholar] [CrossRef]
  34. Grabisch, M.; Marichal, J.L.; Mesiar, R.; Pap, E. Aggregation Functions; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
  35. Alsina, C.; Frank, M.J.; Schweizer, B. Associative Functions. Triangular Norms and Copulas; World Scientific: Singapore, 2006. [Google Scholar]
  36. Mesiar, R.; Sempi, C. Ordinal sums and idempotents of copulas. Aequationes Math. 2010, 79, 39–52. [Google Scholar] [CrossRef]
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