  In this paper, we continue Voiculescu's recent work on the analogous extreme
value theory in the context of bi-free probability theory. We derive various
equivalent conditions for a bivariate distribution function to be bi-freely
max-infinitely divisible. A bi-freely max-infinitely divisible distribution
function can be expressed in terms of its marginals and a special form of
copulas. Such a distribution function is shown to be also max-infinitely
divisible in the classical sense. In addition, we characterize the set of
bi-free extreme value distribution functions. A distribution function of this
type is also bi-free max-stable and represented by its marginals and one copula
composing of a Pickands dependence function, as in the classical extreme value
theory. As a consequence, the determination of its bi-free domain of attraction
is the same as the criteria in the classical theory. To illustrate these
connections, some concrete examples are provided.
