  In \cite{BNT}, a framework to prove almost sure central limit theorems for
sequences $(G_n)$ belonging to the Wiener space was developed, with a
particular emphasis of the case where $G_n$ takes the form of a multiple
Wiener-It\^o integral with respect to a given isonormal Gaussian process. In
the present paper, we complement the study initiated in \cite{BNT}, by
considering the more general situation where the sequence $(G_n)$ may not need
to converge to a Gaussian distribution. As an application, we prove that
partial sums of Hermite polynomials of increments of fractional Brownian motion
satisfy an almost sure limit theorem in the long-range dependence case, a
problem left open in \cite{BNT}.
