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Relaxation of monotone coupling conditions: Poisson approximation and beyond

Part of: Distribution theory Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  16 November 2018

Fraser Daly  and
Oliver Johnson
Fraser Daly*
Affiliation:
Heriot-Watt University
Oliver Johnson*
Affiliation:
University of Bristol
*
* Postal address: Department of Actuarial Mathematics and Statistics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: [email protected]
** Postal address: School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. Email address: [email protected]
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Abstract

It is well known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also state explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincaré inequality and in a normal approximation result.

Keywords

Monotone coupling(negative) associationPoisson approximationPoincaré inequalitynormal approximationsize biasingstochastic ordering

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60E15: Inequalities; stochastic orderings 60F05: Central limit and other weak theorems 62E10: Characterization and structure theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 742 - 759
Copyright
Copyright © Applied Probability Trust 2018 

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