  We study the convergence in total variation distance for series of the form
$$
S_{N}(c,Z)=\sum_{l=1}^{N}\sum_{i_{1}<\cdots<i_{l}}c(i_{1},...,i_{l})Z_{i_{1}}\cdots
Z_{i_{l}}, $$ where $Z_{k},k\in {\mathbb{N}}$ are independent centered random
variables with ${\mathbb{E}}(Z_{k}^{2})=1$. This enters in the framework of the
$U$--statistics theory which plays a major role in modern statistic. In the
case when $Z_{k},k\in {\mathbb{N}}$ are standard normal, $S_{N}(c,Z)$ is an
element of the sum of the first $N$ Wiener chaoses and, starting with the
seminal paper of D. Nualart and G. Peccati, the convergence of such functionals
to the Gaussian law has been extensively studied. So the interesting point
consists in studying invariance principles, that is, to replace Gaussian random
variables with random variables with a general law. This has been done in
several papers using the Fortet--Mourier distance, the Kolmogorov distance or
the total variance distance. In particular, estimates of the total variance
distance in terms of the fourth order cumulants has been given in the part I
(Gaussian limits) of the present paper. But, as the celebrated Fourth Moment
Theorem of Nualart and Peccati shows, such estimates are pertinent to deal with
Gaussian limits. In the present paper we study the convergence to general
limits which may be non Gaussian, and then the estimates of the error has to be
done in terms of the low influence factor only.
