  Let $X_1,X_2,\ldots$ be a sequence of i.i.d. random variables, with mean zero
and variance one. Let $W_n=(X_1+\ldots+X_n)/\sqrt{n}$. An old and celebrated
result of Prohorov asserts that $W_n$ converges in total variation to the
standard Gaussian distribution if and only if $W_{n_0}$ has an absolutely
continuous component for some $n_0$. In the present paper, we give yet another
proof and extend Prohorov's theorem to a situation where, instead of $W_n$, we
consider more generally a sequence of homogoneous polynomials in the $X_i$.
More precisely, we exhibit conditions for a recent invariance principle proved
by Mossel, O'Donnel and Oleszkiewicz to hold under the total variation
distance. There are many works about CLT under various metrics in the
literature, but the present one seems to be the first attempt to deal with
homogeneous polynomials in the $X_i$ with degree strictly greater than one.
