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March 2011 New rates for exponential approximation and the theorems of Rényi and Yaglom
Erol A. Peköz, Adrian Röllin
Ann. Probab. 39(2): 587-608 (March 2011). DOI: 10.1214/10-AOP559

Abstract

We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Rényi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classical theorem of Yaglom on the exponential asymptotic behavior of a critical Galton–Watson process conditioned on nonextinction. The primary tools are an adaptation of Stein’s method, Stein couplings, as well as the equilibrium distributional transformation from renewal theory.

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Erol A. Peköz. Adrian Röllin. "New rates for exponential approximation and the theorems of Rényi and Yaglom." Ann. Probab. 39 (2) 587 - 608, March 2011. https://doi.org/10.1214/10-AOP559

Information

Published: March 2011
First available in Project Euclid: 25 February 2011

zbMATH: 1213.60049
MathSciNet: MR2789507
Digital Object Identifier: 10.1214/10-AOP559

Subjects:
Primary: 60F05
Secondary: 60J10 , 60J80

Keywords: critical Galton–Watson branching process , equilibrium and size-biased distribution , exponential approximation , first passage times , geometric convolution , Stein’s method

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 2 • March 2011
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