Principles of Copula Theory

Copulas: Basic Definitions and Properties Notations Preliminaries on random variables and distribution functions Definition and first examples Characterization in terms of properties of d.f.s Continuity and absolutely continuity The derivatives of a copula The space of copulas Graphical representations Copulas and Stochastic Dependence Construction of multivariate stochastic models via copulas Sklar's theorem Proofs of Sklar's theorem Copulas and risk-invariant property Characterization of basic dependence structures via copulas Copulas and order statistics Copulas and Measures Copulas and d-fold stochastic measures Absolutely continuous and singular copulas Copulas with fractal support Copulas, conditional expectation, and Markov kernel Copulas and measure-preserving transformations Shuffles of a copula Sparse copulas Ordinal sums The Kendall distribution function Copulas and Approximation Uniform approximations of copulas Application to weak convergence of multivariate d.f.s Markov kernel representation and related distances Copulas and Markov operators Convergence in the sense of Markov operators The Markov Product of Copulas The Markov product Invertible and extremal elements in C2 Idempotent copulas, Markov operators, and conditional expectations The Markov product and Markov processes A generalization of the Markov product A Compendium of Families of Copulas What is a family of copulas? Frechet copulas EFGM copulas Marshall-Olkin copulas Archimedean copulas Extreme-value copulas Elliptical copulas Invariant copulas under truncation Generalizations of Copulas: Quasi-Copulas Definition and first properties Characterizations of quasi-copulas The space of quasi-copulas and its lattice structure Mass distribution associated with a quasi-copula Generalizations of Copulas: Semi-Copulas Definition and basic properties Bivariate semi-copulas, triangular norms, and fuzzy logic Relationships among capacities and semi-copulas Transforms of semi-copulas Semi-copulas and level curves Multivariate aging notions of NBU and IFR Bibliography Index