An invariance principle under the total variation distance

Let X1,X2,…X1,X2,… be a sequence of i.i.d. random variables, with mean zero and variance one and let Sn=(X1+⋯+Xn)/n. An old and celebrated result of Prohorov (1952) asserts that SnSn converges in total variation to the standard Gaussian distribution if and only if Sn0Sn0 has an absolutely continuous component for some integer n0≥1. In the present paper, we give yet another proof of Prohorov’s Theorem, but, most importantly, we extend it to a more general situation. Indeed, instead of merely SnSn, we consider a sequence of homogeneous polynomials in the XiXi. More precisely, we exhibit conditions under which some nonlinear invariance principle, discovered by Rotar (1979) and revisited by Mossel et al. (2010), holds in the total variation topology. 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 XiXi with degree strictly greater than one.