Quasi-copulas and signed measures

• Study of (bivariate) quasi-copulas with fractal mass distributions. • Study of the mass distribution ofW n—the point-wise best-possible lower bound for the set of n-quasi-copulas (and n-copulas). •As a consequence, not every multivariate quasi-copula induces a signed measure on [0, 1]n. 1. Copulas and Quasi-copulas • The importance of copulas: Sklar’s Theorem [6, 8]. The joint distribution func-tion H of a random vector (X1, X2, · · · , Xn) with respective univariate margins F1, F2, · · · , Fn, can be expressed in terms of an n-copula C that is uniquely de-termined on×ni=1RangeFi in the form H(x) = C(F1(x1), F2(x2), · · · , Fn(xn)), x ∈ [−∞,∞]n. • The notion of quasi-copula was introduced [1] in order to characterize opera-tions on distribution functions that can, or cannot, be derived from operations on random variables defined on the same probability space. Theorem 1 [2] An n-dimensional quasi-copula (or n-quasi-copula) is a function Q: [0, 1]n → [0, 1] that satisfies: (Q1) Q(u1, · · · , ui−1, 0, ui+1, · · · , un) = 0 and Q(1, · · · , 1, ui, 1, · · · , 1) = ui for all u ∈ [0, 1]n and for every i ∈ {1, 2, · · · , n}; (Q2) Q is nondecreasing in each variable; and (Q3) the Lipschitz condition: |Q(u)−Q(v) | ≤ n∑ i=1 |ui − vi | ∀u,v ∈ [0, 1]n. • Every n-copula is an n-quasi-copula; and when Q is an n-quasi-copula but not an n-copula, it is said thatQ is a proper n-quasi-copula. • Every n-quasi-copulaQ satisfies that, for every u ∈ [0, 1]n, W n(u) = max 0, n∑ i=1 ui − n + 1) ≤ Q(u) ≤ min(u) = Mn(u). •Mn is an n-copula for all n ≥ 2,W 2 is a 2-copula, andW n (n ≥ 3) is a proper n-quasi-copula. • For an n-quasi-copulaQ and an n-boxB = ×ni=1[ai, bi] in [0, 1]n, theQ-volume of B is defined similarly than for n-copulas, i.e.