[en] In this work, we study the normal approximation and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we provide a new chain rule that improves the one derived by Nourdin et al. (2010) and then we deduce the bound on Wasserstein distance for normal approximation using the (discrete) Malliavinâ€“Stein approach. Besides, we are able to give the almost sure central limit theorem for a sequence of random variables inside a fixed Rademacher chaos using the Ibragimovâ€“Lifshits criterion