Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T05:03:36.893Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© Applied Probability Trust 2020

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

Arratia, R. andGoldstein, L. (2010). Size bias, sampling, the waiting time paradox, and infinite divisibility: when is the increment independent? Available at arXiv:1007.3910.Google Scholar
Arratia, R., Goldstein, L. andGordon, L. (1989). Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Prob. 17, 9–25.CrossRefGoogle Scholar
Arratia, R., Goldstein, L. andKochman, F. (2013). Size bias for one and all. Available at arXiv:1308.2729.Google Scholar
Barbour, A. D. (1988). Stein’s method and Poisson process convergence. J. Appl. Prob 25 (A), 175–184.10.2307/3214155CrossRefGoogle Scholar
Barbour, A. D. andBrown, T. C. (1992). Stein’s method and point process approximation. Stoch. Proc. Appl. 43, 9–31.10.1016/0304-4149(92)90073-YCrossRefGoogle Scholar
Barbour, A. D. andEagleson, G. K. (1984). Poisson convergence for dissociated statistics. J. R. Statist. Soc. B [Statist. Methodology] 46, 397–402.Google Scholar
Barbour, A. D. andXia, A. (1999). Poisson perturbations. ESAIM Prob. Statist. 3, 131–150.CrossRefGoogle Scholar
Barbour, A. D., Chen, L. H. Y. andChoi, K. P. (1995). Poisson approximation for unbounded functions, I: Independent summands. Statist. Sinica 2, 749–766.Google Scholar
Barbour, A. D., Holst, L. andJanson, S. (1992). Poisson Approximation. Oxford University Press.Google Scholar
Borovkov, K. A. (1988). Refinement of Poisson approximation. Theory Prob. Appl 33, 343–347.CrossRefGoogle Scholar
Borovkov, K. andPfeifer, D. (1996). On improvements of the order of approximation in the Poisson limit theorem. J. Appl. Prob. 33, 146–155.10.2307/3215272CrossRefGoogle Scholar
Brown, T. C. andXia, A. (2001). Stein’s method and birth–death processes. 1373–1403.Google Scholar
Čekanavičius, V. andVellaisamy, P. (2019). On large deviations for sums of discrete m-dependent random variables. Stochastics 91 (8), 1092–1108.CrossRefGoogle Scholar
Chatterjee, S., Diaconis, P. andMeckes, E. (2005). Exchangeable pairs and Poisson approximation. Prob. Surv. 2, 64–106.CrossRefGoogle Scholar
Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.10.1214/aop/1176996359CrossRefGoogle Scholar
Chen, L. H. Y. andChoi, K. P. (1992). Some asymptotic and large deviation results in Poisson approximation. Ann. Prob. 20, 1867–1876.CrossRefGoogle Scholar
Chen, L. H. Y. andShao, Q.-M. (2004). Normal approximation under local dependence. Ann. Prob. 32, 1985–2028.CrossRefGoogle Scholar
Chen, L. H. Y., Fang, X. andShao, Q.-M. (2013). Moderate deviations in Poisson approximation: a first attempt. Statist. Sinica. 23, 1523–1540.Google Scholar
Chen, L. H. Y., Fang, X. andShao, Q.-M. (2013). From Stein identities to moderate deviations. Ann. Prob. 41, 262–293.CrossRefGoogle Scholar
Chung, F. andLu, L. (2006). Complex Graphs and Networks (CBMS Regional Conference Series in Mathematics 107). American Mathematical Society.Google Scholar
Cochran, W. (1977). Sampling Techniques. John Wiley & Sons.Google Scholar
Deheuvels, P. andPfeifer, D. (1988). On a relationship between Uspensky’s theorem and Poisson approximations. Ann. Inst. Statist. Math. 40 (4), 671–681.CrossRefGoogle Scholar
Dwass, M. (1960). Some k-sample rank order tests. In Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling, ed. I. Olkin, pp. 198–202. Stanford University Press, CA.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, vols 1 and 2, 3rd edn. John Wiley & Sons.Google Scholar
Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Ann. of Math. 44, 423–453.CrossRefGoogle Scholar
Goldstein, L. andXia, A. (2006). Zero biasing and a discrete central limit theorem. Ann. Prob 34, 1782–1806.CrossRefGoogle Scholar
Hoeffding, W. (1956). On the distribution of the number of successes in independent trials. Ann. Math. Statist. 27, 713–721.CrossRefGoogle Scholar
Mattner, L. andRoos, B. (2007). A shorter proof of Kanter’s Bessel function concentration bound. Prob. Theory Rel. Fields 139, 191–205.10.1007/s00440-006-0043-0CrossRefGoogle Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer.Google Scholar
Rényi, A. (1962). Théorie des éléments saillants d’une suite d’observations. Ann. Fac. Sci. Univ. Clermont-Ferrand No. 8, 7–13.Google Scholar
Röllin, A. (2005). Approximation of sums of conditionally independent variables by the translated Poisson distribution. Bernoulli 11, 1115–1128.CrossRefGoogle Scholar
Röllin, A. (2007). Translated Poisson approximation using exchangeable pair couplings. Ann. Appl. Prob. 17, 1596–1614.10.1214/105051607000000258CrossRefGoogle Scholar
Ross, N. (2011). Fundamentals of Stein’s method. Prob. Surv. 8, 210–293.CrossRefGoogle Scholar
Tan, Y., Lu, Y. andXia, C. (2018). Relative error of scaled Poisson approximation via Stein’s method. Available at arXiv:1810.04300.Google Scholar