Open Access
November 2017 On Stein’s method for products of normal random variables and zero bias couplings
Robert E. Gaunt
Bernoulli 23(4B): 3311-3345 (November 2017). DOI: 10.3150/16-BEJ848


In this paper, we extend Stein’s method to the distribution of the product of $n$ independent mean zero normal random variables. A Stein equation is obtained for this class of distributions, which reduces to the classical normal Stein equation in the case $n=1$. This Stein equation motivates a generalisation of the zero bias transformation. We establish properties of this new transformation, and illustrate how they may be used together with the Stein equation to assess distributional distances for statistics that are asymptotically distributed as the product of independent central normal random variables. We end by proving some product normal approximation theorems.


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Robert E. Gaunt. "On Stein’s method for products of normal random variables and zero bias couplings." Bernoulli 23 (4B) 3311 - 3345, November 2017.


Received: 1 June 2014; Revised: 1 March 2016; Published: November 2017
First available in Project Euclid: 23 May 2017

zbMATH: 06778288
MathSciNet: MR3654808
Digital Object Identifier: 10.3150/16-BEJ848

Keywords: coupling , distributional transformation , products of normal random variables , Stein’s method , zero biasing

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 4B • November 2017
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