Padé approximants of random Stieltjes series

We consider the random continued fraction where s n are independent random variables with the same gamma distribution. Every realization of the sequence defines a Stieltjes function that can be expressed as for some measure σ on the positive half-line. We study the convergence of the finite truncations of the continued fraction or, equivalently, of the diagonal Padé approximants of the function S . Using the Dyson–Schmidt method for an equivalent one-dimensional disordered system and the results of Marklof et al. , we obtain explicit formulae (in terms of modified Bessel functions) for the almost sure rate of convergence of these approximants, and for the almost sure distribution of their poles.