  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.
