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.
Publié le : 2011-03-15
Classification:
Exponential approximation,
geometric convolution,
first passage times,
critical Galton–Watson branching process,
Stein’s method,
equilibrium and size-biased distribution,
60F05,
60J10,
60J80
@article{1298669174,
author = {Pek\"oz, Erol A. and R\"ollin, Adrian},
title = {New rates for exponential approximation and the theorems of R\'enyi and Yaglom},
journal = {Ann. Probab.},
volume = {39},
number = {1},
year = {2011},
pages = { 587-608},
language = {en},
url = {http://dml.mathdoc.fr/item/1298669174}
}
Peköz, Erol A.; Röllin, Adrian. New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab., Tome 39 (2011) no. 1, pp. 587-608. http://gdmltest.u-ga.fr/item/1298669174/