On Bertino copulas and the Markov product

We revisit the family of bivariate Bertino copulas, first studied by Fredricks and Nelsen in 2002 and characterized as the minimal elements in the family of all copulas with given diagonal, and derive several (partially surprising) results on their behavior under the so-called Markov product (also known as star product) of copulas. Using tools from measure theory and Markov chains in discrete time we first show that for every Bertino copula B δ with diagonal δ all Markov product iterates ⁎ ⁎ ⁎ B δ ⁎ n = B δ ⁎ ⋯ ⁎ B δ are Bertino copulas too. Having this, we then prove that for each Bertino copula B δ whose diagonal fulfills a simple but essential regularity/slope condition, the sequence ⁎ ( B δ ⁎ n ) n ∈ B converges uniformly (in fact even weakly conditional) to some copula C , and show that C again is a Bertino copula which, additionally, is idempotent. Rounding off these results we provide a handy characterization of idempotence in the Bertino family.