Journal of Theoretical Probability

, Volume 29, Issue 1, pp 231–247 | Cite as

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



New bounds for the \(k\)th-order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test function than those in the existing literature. We apply these bounds and local approach couplings to obtain an order \(n^{-(p-1)/2}\) bound, for smooth test functions, for the distance between the distribution of a standardised sum of independent and identically distributed random variables and the standard normal distribution when the first \(p\) moments of these distributions agree. We also obtain a bound on the convergence rate of a sequence of distributions to the normal distribution when the moment sequence converges to normal moments.


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

Mathematics Subject Classification (2010):




During the course of this research the author was supported by an EPSRC D.Phil Studentship, an EPSRC Doctoral Prize and EPSRC research Grant AMRYO100. The author would like to thank Gesine Reinert for some productive discussions. The author would also like to thank two anonymous referees for their helpful comments and suggestions, which helped me to prepare an improved manuscript.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of OxfordOxford UK

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