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Stein's Method and Stochastic Orderings

Part of: Distribution theory Limit theorems Markov processes

Published online by Cambridge University Press:  04 January 2016

Fraser Daly ,
Claude Lefèvre  and
Sergey Utev
Fraser Daly*
Affiliation:
Universität Zürich
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Sergey Utev*
Affiliation:
University of Nottingham
*
∗ Current address: Heilbronn Institute for Mathematical Research, Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. Email address: [email protected]
∗∗ Postal address: Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine, CP 210, B-1050 Bruxelles, Belgium. Email address: [email protected]
∗∗∗ Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: [email protected]
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Abstract

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A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

Keywords

Stein's methodbirth-death processstochastic orderingtotal variation distance(in)dependent indicators(translated) Poisson approximationtotal negative and positive dependence(approximate) local dependencepolynomial birth-death approximationk-runs

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 343 - 372
Copyright
© Applied Probability Trust 

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