The Annals of Probability
- Ann. Probab.
- Volume 39, Number 2 (2011), 587-608.
New rates for exponential approximation and the theorems of Rényi and Yaglom
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.
Ann. Probab., Volume 39, Number 2 (2011), 587-608.
First available in Project Euclid: 25 February 2011
Permanent link to this document
Digital Object Identifier
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.)
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. https://projecteuclid.org/euclid.aop/1298669174