Padé approximants of random Stieltjes series

We consider the random continued fractionS(t)≔1/s1+t/s2+(t/(s3+…)), t∈C\R−,where sn are independent random variables with the same gamma distribution. Every realization of the sequence defines a Stieltjes function that can be expressed as S(t)=∫0∞σ(dx)/1+xt, t∈C\R−, 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.