Total variation error bounds for geometric approximation

Details Total variation error bounds for geometric approximation Total variation error bounds for geometric approximation

2010-05-16

arxiv.org/abs/1005.2774v3

Mathematics

Probability

29 pages

Scientific paper

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 nontrivial 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. In the first two examples we obtain error bounds in a metric that is stronger than those available in the literature, and in the final two examples we provide the first explicit bounds.

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