## Electronic Journal of Probability

### Stein's method of exchangeable pairs for the Beta distribution and generalizations

Christian Döbler

#### Abstract

We propose a new version of Stein's method of exchangeable pairs, which, given a suitable exchangeable pair $(W,W')$ of real-valued random variables, suggests the approximation of the law of $W$ by a suitable absolutely continuous distribution. This distribution is characterized by a first order linear differential Stein operator, whose coefficients $\gamma$ and $\eta$ are motivated by two regression properties satisfied by the pair $(W,W')$. Furthermore, the general theory of Stein's method for such an absolutely continuous distribution is developed and a general characterization result as well as general bounds on the solution to the Stein equation are given. This abstract approach is a certain extension of the theory developed in previous works, which only consider the framework of the density approach, i.e. $\eta\equiv1$. As an illustration of our technique we prove a general plug-in result, which bounds a certain distance of the distribution of a given random variable $W$ to a Beta distribution in terms of a given exchangeable pair $(W,W')$ and provide new bounds on the solution to the Stein equation for the Beta distribution, which complement the existing bounds from previous works. The abstract plug-in result is then applied to derive bounds of order $n^{-1}$ for the distance between the distribution of the relative number of drawn red balls after $n$ drawings in a Pólya urn model and the limiting Beta distribution measured by a certain class of smooth test functions.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 109, 34 pp.

Dates
Accepted: 19 October 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067215

Digital Object Identifier
doi:10.1214/EJP.v20-3933

Mathematical Reviews number (MathSciNet)
MR3418541

Zentralblatt MATH identifier
1328.60064

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60E99: None of the above, but in this section

Rights

#### Citation

Döbler, Christian. Stein's method of exchangeable pairs for the Beta distribution and generalizations. Electron. J. Probab. 20 (2015), paper no. 109, 34 pp. doi:10.1214/EJP.v20-3933. https://projecteuclid.org/euclid.ejp/1465067215

#### References

• Arratia, R.; Goldstein, L.; Gordon, L. Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Probab. 17 (1989), no. 1, 9–25.
• Barbour, A. D. Stein's method and Poisson process convergence. A celebration of applied probability. J. Appl. Probab. 1988, Special Vol. 25A, 175–184.
• Barbour, A. D.; Holst, Lars; Janson, Svante. Poisson approximation. Oxford Studies in Probability, 2. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. x+277 pp. ISBN: 0-19-852235-5.
• Chatterjee, Sourav; Fulman, Jason; Roellin, Adrian. Exponential approximation by Stein's method and spectral graph theory. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 197–223.
• Chatterjee, Sourav; Shao, Qi-Man. Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie-Weiss model. Ann. Appl. Probab. 21 (2011), no. 2, 464–483.
• Chen, Louis H. Y. Poisson approximation for dependent trials. Ann. Probability 3 (1975), no. 3, 534–545.
• Chen, Louis H. Y.; Goldstein, Larry; Shao, Qi-Man. Normal approximation by Stein's method. Probability and its Applications (New York). Springer, Heidelberg, 2011. xii+405 pp. ISBN: 978-3-642-15006-7
• Chen, Louis H. Y.; Shao, Qi-Man. Stein's method for normal approximation. An introduction to Stein's method, 1–59, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005.
• Daly, Fraser. Upper bounds for Stein-type operators. Electron. J. Probab. 13 (2008), no. 20, 566–587.
• C. Döbler, New developments in Stein's method with applications, (2012), (Ph.D.)-Thesis Ruhr-UniversitÃ¤t Bochum.
• C. Döbler, Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model, arXiv:1207.0533 (2012).
• Eden, Richard; Viquez, Juan. Nourdin-Peccati analysis on Wiener and Wiener-Poisson space for general distributions. Stochastic Process. Appl. 125 (2015), no. 1, 182–216.
• Eichelsbacher, Peter; Loewe, Matthias. Stein's method for dependent random variables occurring in statistical mechanics. Electron. J. Probab. 15 (2010), no. 30, 962–988.
• Fulman, Jason; Ross, Nathan. Exponential approximation and Stein's method of exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 1–13.
• R. E. Gaunt, Rates of convergence in normal approximation under moment conditions via new bounds on solutions of the stein equation, to apper in J. Theoret. Probab. (2014).
• R. E. Gaunt, Variance-gamma approximation via stein's method, Electron. J. Probab. 19 (2014).
• R. E. Gaunt, A. Pickett, and G. Reinert, Chi-square approximation by Stein's method with application to Pearson's statistic, arXiv:1507.01707 (2015).
• Goldstein, Larry; Reinert, Gesine. Stein's method for the beta distribution and the Polya-Eggenberger urn. J. Appl. Probab. 50 (2013), no. 4, 1187–1205.
• Goldstein, Larry; Rinott, Yosef. Multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Probab. 33 (1996), no. 1, 1–17.
• Kusuoka, Seiichiro; Tudor, Ciprian A. Stein's method for invariant measures of diffusions via Malliavin calculus. Stochastic Process. Appl. 122 (2012), no. 4, 1627–1651.
• C. Ley, G. Reinert, and Y. Swan, Approximate computation of expectations: a canonical Stein operator, arXiv:1408.2998 (2014).
• Luk, Ho Ming. Stein's method for the Gamma distribution and related statistical applications. Thesis (Ph.D.) - University of Southern California. ProQuest LLC, Ann Arbor, MI, 1994. 74 pp.
• Nourdin, Ivan; Peccati, Giovanni. Stein's method on Wiener chaos. Probab. Theory Related Fields 145 (2009), no. 1-2, 75–118.
• Nourdin, Ivan; Viens, Frederi G. Density formula and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14 (2009), no. 78, 2287–2309.
• Pekoez, Erol A.; Roellin, Adrian. New rates for exponential approximation and the theorems of Renyi and Yaglom. Ann. Probab. 39 (2011), no. 2, 587–608.
• Pekoez, Erol A.; Roellin, Adrian; Ross, Nathan. Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 (2013), no. 3, 1188–1218.
• Pike, John; Ren, Haining. Stein's method and the Laplace distribution. ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014), no. 1, 571–587.
• Reinert, Gesine. Three general approaches to Stein's method. An introduction to Stein's method, 183–221, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005.
• Roellin, Adrian. A note on the exchangeability condition in Stein's method. Statist. Probab. Lett. 78 (2008), no. 13, 1800–1806.
• Schoutens, Wim. Orthogonal polynomials in Stein's method. J. Math. Anal. Appl. 253 (2001), no. 2, 515–531.
• Stein, Charles. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, pp. 583–602. Univ. California Press, Berkeley, Calif., 1972.
• Stein, Charles. Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes-Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. iv+164 pp. ISBN: 0-940600-08-0.