  We systematically study the first three terms in the asymptotic expansions of
the moments of the transmission eigenvalues and proper delay times as the
number of quantum channels n in the leads goes to infinity. The computations
are based on the assumption that the Landauer-B\"utticker scattering matrix for
chaotic ballistic cavities can be modelled by the circular ensembles of Random
Matrix Theory (RMT). The starting points are the finite-n formulae that we
recently discovered (Mezzadri and Simm, J. Math. Phys. 52 (2011), 103511). Our
analysis includes all the symmetry classes beta=1,2,4; in addition, it applies
to the transmission eigenvalues of Andreev billiards, whose symmetry classes
were classified by Zirnbauer (J. Math. Phys. 37 (1996), 4986-5018) and Altland
and Zirnbauer (Phys. Rev. B. 55 (1997), 1142-1161). Where applicable, our
results are in complete agreement with the semiclassical theory of mesoscopic
systems developed by Berkolaiko et al. (J. Phys. A.: Math. Theor. 41 (2008),
365102) and Berkolaiko and Kuipers (J. Phys. A: Math. Theor. 43 (2010), 035101
and New J. Phys. 13 (2011), 063020). Our approach also applies to the
Selberg-like integrals. We calculate the first two terms in their asymptotic
expansion explicitly.
