Total variation error bounds for geometric approximation

Abstract

We develop a new formulation of Stein's method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework reduces the problem to the construction of a coupling between the original distribution and the "discrete equilibrium" distribution from renewal theory. We illustrate the approach in four non-trivial examples: the geometric sum of independent, non-negative, integer-valued random variables having common mean, the generation size of the critical Galton-Watson process conditioned on non-extinction, the in-degree of a randomly chosen node in the uniform attachment random graph model and the total degree of both a fixed and randomly chosen node in the preferential attachment random graph model.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ406 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm