Contributions to the diagonal expansion of a bivariate copula with continuous extensions

• Some properties of integral operators related to copulas are obtained.
• The structure of the eigenvalues and eigenfunctions of two kernels is obtained.
• The eigenfunctions related to some families of copulas can have zero norm.
• Lancaster’s expansion in series is generalized in terms of integrals.
• This gives rise to a general family of canonical copulas.

We find some properties and eigendecompositions of two integral operators related to copulas. By using an inner product between two functions via an extension of the covariance, we study the countable set of eigenpairs, which is related to the set of canonical correlations and functions. Then a canonical analysis on the so-called Cuadras–Augé family of copulas is performed, showing the continuous dimensionality of this distribution. A diagonal expansion in terms of an integral is obtained. As a consequence, this continuous expansion allows us to generate a wide family of copulas.