Let 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 Sn converges in total variation to the standard Gaussian distribution if and only if Sn0 has an absolutely continuous...

Let {Fn} be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards F∞ satisfying V ar(F∞)0. Our first result is a sequential version of a theorem by Shigekawa (1980) [23]. More precisely, we prove, without...