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

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

2009-07-12

arxiv.org/abs/0907.2009v2

Annals of Probability 2011, Vol. 39, No. 2, 587-608

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP559 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP559

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 R\'{e}nyi 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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