On the regularity and approximation of invariant densities for random continued fractions

We study perturbations of piecewise smooth random dynamical systems whose associated annealed transfer operators admit a uniform spectral gap on C-l. We provide a kth-order approximation for the invariant density of the associated random dynamical system. We apply our result to random continued fractions. In particular, we show that probability density function preserved by the randommap associated with random continued fractions belongs to C-l, for any l >= 1; more importantly, we provide a k-order approximation of this invariant density and consequently provide a kth-order approximation to the distribution of the nth digit of a random continued fraction expansion as n tends to infinity.