  Consider the product $X = X_{1}\cdots X_{m}$ of $m$ independent $n\times n$
iid random matrices. When $m$ is fixed and the dimension $n$ tends to infinity,
we prove Gaussian limits for the centered linear spectral statistics of $X$ for
analytic test functions. We show that the limiting variance is universal in the
sense that it does not depend on $m$ (the number of factor matrices) or on the
distribution of the entries of the matrices. The main result generalizes and
improves upon previous limit statements for the linear spectral statistics of a
single iid matrix by Rider and Silverstein as well as Renfrew and the second
author.
