  A two-dimensional Gauss-Kuzmin theorem for $N$-continued fraction expansions
is shown. More exactly, we obtain a Gauss-Kuzmin theorem related to the natural
extension of the measure-dynamical system corresponding to these expansions.
Then, using characteristic properties of the transition operator associated
with the random system with complete connections underlying $N$-continued
fractions on the Banach space of complex-valued functions of bounded variation
we derive explicit lower and upper bounds for the convergence rate of the
distribution function to its limit.
