  We present and develop different approaches to study the asymptotic behavior
of the distribution functions in the odd continued fractions case. Firstly, by
considering the transition operator of the Markov chain associated with these
expansions on a certain Banach space of complex-valued functions of bounded
variation we make a brief survey of the solution in the Gauss-Kuzmin-type
problem. Secondly, we use the method of Sz\"usz to obtain a similar asymptotic
result and to give a good estimate of the convergence rate involved.
