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On moderate deviations in Poisson approximation

Part of: Limit theorems Distribution theory - Probability

Published online by Cambridge University Press:  04 September 2020

Qingwei Liu  and
Aihua Xia
Qingwei Liu*
Affiliation:
University of Melbourne
Aihua Xia*
Affiliation:
University of Melbourne
*
*Postal address: School of Mathematics and Statistics, University of Melbourne, VIC3010, Australia.
*Postal address: School of Mathematics and Statistics, University of Melbourne, VIC3010, Australia.
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Abstract

In this paper we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of a Poisson distribution than those of the normal distribution. We then show that the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in [18]. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems via six applications: Poisson-binomial distribution, the matching problem, the occupancy problem, the birthday problem, random graphs, and 2-runs. The paper complements the works [16], [8], and [18].

Keywords

Stein–Chen methodPoisson approximationmoderate deviation

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 1005 - 1027
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Copyright
© Applied Probability Trust 2020

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