It is well known that the Gauss dynamical system of regular continued
fractions satisfies many strong mixing properties, in fact, it is
Bernoulli. In addition to the standard mixing properties in dynamical
system, the Gauss dynamical system also satisfies the $\psi$-mixing
property from the viewpoint of stochastic processes. This $\psi$-mixing
property, introduced by J.R. Blum, D.L. Hanson and L.H. Koopmans (1963),
implies strong mixing, weak mixing and ergodicity in the sense of
dynamical system. More importantly, the $\psi$-mixing property gives an
upper bound for the $L^2$-convergence of the ergodic average. We use
this upper bound to develop a quantitative $L^2$-ergodic theorem and
apply the theorem to refine classical results of metric theory of
continued fractions. For example, we show that the geometric mean of the
partial quotients of the continued fraction expansion converges to the
Khinchin's constant with an error term of order $o(n^{-1/2}(\log
n)^{3/2}(\log(\log n))^{1/2 +\epsilon})$. It is worth noting that our
method gives an improvement on the error term derived from the classical
method of I.S. Gal and J.F. Koksma (1950). This is a joint work with J.
Hancl, A. Haddley and R. Nair.