On Stein's method for products of normal random variables and zero bias couplings

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Abstract

In this paper, we extend Stein’s method to the distribution of the product of nn 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=1n=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.

Bibliographical metadata

Original languageEnglish
Pages (from-to)3311-3345
JournalBernoulli
Volume23
Issue number4B
DOIs
StatePublished - 23 May 2017