An invariance principle under the total variation distance

Ivan Nourdin (1) , Guillaume Poly (2)

1 IECL - Institut Élie Cartan de Lorraine
2 FSTC - Faculté des Sciences, de la Technologie et de la Communication

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.

Mots clés: absolute continuity; convergence in total variation; Convergence in law; invariance principle

Domaines: Probabilités [math.PR]

Ivan Nourdin, Guillaume Poly. An invariance principle under the total variation distance. Stochastic Processes and their Applications, 2015, 125 (6), pp.2190-2205. ⟨10.1016/j.spa.2014.12.010⟩. ⟨hal-00873311⟩