  Stein's method allows to prove distributional convergence of a sequence of
random variables and to quantify it with respect to a given metric such as
Kolmogorov's (a Berry-Ess\'een type theorem). Mod-* convergence quantifies the
convergence of a sequence of random variables to a given distribution in a
sense unusual in probability theory, a priori unrelated to a metric on
probability measures.
  This article gives a connection between these two notions. It shows that
mod-* convergence can be understood as a higher order approximation in
distribution when the limiting function is integrable and proves a refined
Berry-Ess\'een type theorem for sequences converging in the mod-Gaussian sense.
