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## Abstract

We consider the random continued fractionS(t)â‰”1/s1+t/s2+(t/(s3+â€¦)), tâˆˆC\Râˆ’,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)=âˆ«0âˆžÏƒ(dx)/1+xt, tâˆˆC\Râˆ’, for some measure Ïƒ on the positive half-line. We study the convergence of the finite truncations of the continued fraction or, equivalently, of the diagonal PadÃ© approximants of the function S. Using the Dysonâ€“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.

Translated title of the contribution | Padé approximants of random Stieltjes series |
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Original language | English |

Pages (from-to) | 2813 - 2832 |

Number of pages | 20 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 463 (2087) |

DOIs | |

Publication status | Published - Nov 2007 |

### Bibliographical note

Publisher: The Royal Society## Fingerprint

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