Copulas Approximation and New Families

In this paper, we study the approximation procedures introduced by Li, Mikusinski, Sherwood and Taylor [1997]. We show that there exists a bijection between the set of the discretized copulas and the set of the doubly stochastic matrices. For the Bernstein and checkerboard approximations, we then provide analytical formulas for the Kendall's tau and Spearman's rho concordance measures. Moreover, we demonstrate that these approximations do not exhibit tail dependences. Finally, we consider the general case of approximations induced by partitions of unity. Moreover, we show that the set of copulas induced by partition of unity is a Markov sub - algebra with respect to the *-product of Darsow, Nguyen and Olsen [1992]