Bernoulli

  • Bernoulli
  • Volume 19, Number 2 (2013), 610-632.

Total variation error bounds for geometric approximation

Erol A. Peköz, Adrian Röllin, and Nathan Ross

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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.

Article information

Source
Bernoulli Volume 19, Number 2 (2013), 610-632.

Dates
First available in Project Euclid: 13 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1363192040

Digital Object Identifier
doi:10.3150/11-BEJ406

Mathematical Reviews number (MathSciNet)
MR3037166

Zentralblatt MATH identifier
06168765

Keywords
discrete equilibrium distribution geometric distribution preferential attachment model Stein’s method Yaglom’s theorem

Citation

Peköz, Erol A.; Röllin, Adrian; Ross, Nathan. Total variation error bounds for geometric approximation. Bernoulli 19 (2013), no. 2, 610--632. doi:10.3150/11-BEJ406. https://projecteuclid.org/euclid.bj/1363192040


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