Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

Journal of Theoretical Probability

Volumn 29, Issue 1, 2016, Pages 231-247

  Gaunt, Robert E ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Multivariate normal distribution; Normal distribution; Rate of convergence; Stein s method

Indexed keywords

EID: 84959168304     PISSN: 08949840     EISSN: 15729230     Source Type: Journal    
DOI: 10.1007/s10959-014-0562-z     Document Type: Article

References (20)

1. Barbour, A.D.: Stein’s method for diffusion approximations. Probab. Theory Relat. Fields 84, 297–322 (1990)
2. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)
3. Chatterjee, S., Meckes, E.: Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J Probab. Math. Stat. 4, 257–283 (2008)
4. Chen, L.H.Y., Shao, Q.-M.: Normal approximation. In: An Introduction to Stein’s Method. Lecture Notes in Series. Institute for Mathematical Sciences. National University of Singapore, vol. 1, pp. 1–60. Singapore University Press, Singapore (2005)
5. Chen, L.H.Y., Goldstein, L., Shao, Q.-M.: Normal Approximation by Stein’s Method. Springer, Berlin (2011)
6. Elezović, N., Giordano, C., Pečarić, J.: The best bounds in Gautschi’s inequality. Math. Inequal. Appl. 3, 239–252 (2000)
7. Gaunt, R.E.: Inequalities for modified Bessel functions and their integrals. J. Math. Anal. Appl. (2014). doi:10.1016/j.jmaa.2014.05.083
8. Goldstein, L., Reinert, G.: Stein’s Method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7, 935–952 (1997)
9. Goldstein, L., Rinott, Y.: Multivariate normal approximations by Stein’s method and size bias couplings. J. Appl. Probab. 33, 1–17 (1996)
10. Götze, F.: On the rate of convergence in the multivariate CLT. Ann. Probab. 19, 724–739 (1991)
11. Hall, P.: The Bootstrap and Edgeworth Expansion. Springer, Berlin (1991)
12. Hipp, C., Mattner, L.: On the normal approximation to symmetric binomial distributions. Theory Probab. Appl. 52, 610–617 (2007)
13. Meckes, E.: On Stein’s method for multivariate normal approximation. IMS Collect. 5, 153–178 (2009)
14. Priestley, H.A.: Introduction to Integration. Oxford University Press, Oxford (1997)
15. Raič, M.: A multivariate CLT for decomposable random vectors with finite second moments. J. Theor. Probab. 17, 573–603 (2004)
16. Reinert, G.: Three general approaches to Stein’s method. In: An Introduction to Stein’s Method. Lecture Notes in Series. Institute for Mathematical Sciences. National University of Singapore, vol. 4, pp. 183–221. Singapore University Press, Singapore (2005)
17. Reinert, G., Röllin, A.: Multivariate normal approximations with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab. 37, 2150–2173 (2009)
18. Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of Sixth Berkeley Symposium on Mathematical Statistics and Probability (1972), vol. 2, pp. 583–602. University of California Press, Berkeley
19. Stein, C.: Approximate Computation of Expectations. IMS, Hayward (1986)
20. Winkelbauer, A.: Moments and absolute moments of the normal distribution (2012). arXiv:1209.4340