Pade approximants of random Stieltjes series
Mathematical Physics
2009-11-13 v1 math.MP
Numerical Analysis
Abstract
We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 + >...))) where the s_n are independent random variables with the same gamma distribution. For every realisation of the sequence, S(t) defines a Stieltjes function. We study the convergence of the finite truncations of the continued fraction or, equivalently, of the diagonal Pade approximants of the function S(t). By using the Dyson--Schmidt method for an equivalent one-dimensional disordered system, and the results of Marklof et al. (2005), 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.
Keywords
Cite
@article{arxiv.0708.1245,
title = {Pade approximants of random Stieltjes series},
author = {Jens Marklof and Yves Tourigny and Lech Wolowski},
journal= {arXiv preprint arXiv:0708.1245},
year = {2009}
}
Comments
To appear in Proc Roy Soc