Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-21T07:41:47.382Z Has data issue: false hasContentIssue false
  • English
  • Français

Relaxation of monotone coupling conditions: Poisson approximation and beyond

Part of: Limit theorems Distribution theory Distribution theory - Probability

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: f.daly@hw.ac.uk
** Postal address: School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. Email address: o.johnson@bristol.ac.uk
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arratia, R. and Baxendale, P. (2015). Bounded size bias coupling: a Gamma function bound, and universal Dickman-function behavior. Prob. Theory Relat. Fields 162, 411–429.Google Scholar
[2]Ball, F. and Barbour, A. D. (1990). Poisson approximation for some epidemic models. J. Appl. Prob. 27, 479–490.Google Scholar
[3]Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press.Google Scholar
[4]Boutsikas, M. V. and Koutras, M. V. (2000). A bound for the distribution of the sum of discrete associated or negatively associated random variables. Ann. Appl. Prob. 10, 1137–1150.Google Scholar
[5]Caputo, P., Dai Pra, P. and Posta, G. (2009). Convex entropy decay via the Bochner-Bakry-Emery approach. Ann. Inst. H. Poincaré Prob. Statist. 45, 734–753.Google Scholar
[6]Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein's Method. Springer, Heidelberg.Google Scholar
[7]Cook, N., Goldstein, L. and Johnson, T. (2018). Size biased couplings and the spectral gap for random regular graphs. Ann. Prob. 46, 72–125.Google Scholar
[8]Daly, F. (2013). Compound Poisson approximation with association or negative association via Stein's method. Electron. Commun. Prob. 18, 30.Google Scholar
[9]Daly, F. and Johnson, O. (2013). Bounds on the Poincaré constant under negative dependence. Statist. Prob. Lett. 83, 511–518.Google Scholar
[10]Daly, F., Lefèvre, C. and Utev, S. (2012). Stein's method and stochastic orderings. Adv. Appl. Prob. 44, 343–372.Google Scholar
[11]Erhardsson, T. (2005). Stein's method for Poisson and compound Poisson approximation. In An Introduction to Stein's Method, Singapore University Press, pp. 61–113.Google Scholar
[12]Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.Google Scholar
[13]Goldstein, L. (2018). Non asymptotic distributional bounds for the Dickman approximation of the running time of the Quickselect algorithm. Electron. J. Prob. 23, 1–13.Google Scholar
[14]Goldstein, L. and Zhang, H. (2011). A Berry–Esseen theorem for the lightbulb process. Adv. Appl. Prob. 43, 875–898.Google Scholar
[15]Johnson, O. (2017). A discrete log-Sobolev inequality under a Bakry-Émery type condition. Ann. Inst. H. Poincaré Prob. Statist. 53, 1952–1970.Google Scholar
[16]Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11, 286–295.Google Scholar
[17]Klaassen, C. A. J. (1985). On an inequality of Chernoff. Ann. Prob. 13, 966–974.Google Scholar
[18]Martin-Löf, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.Google Scholar