The Annals of Probability

New rates for exponential approximation and the theorems of Rényi and Yaglom

Erol A. Peköz and Adrian Röllin

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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.

Article information

Ann. Probab., Volume 39, Number 2 (2011), 587-608.

First available in Project Euclid: 25 February 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

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


Peköz, Erol A.; Röllin, Adrian. New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab. 39 (2011), no. 2, 587--608. doi:10.1214/10-AOP559.

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