Strong Central Limit Theorem for isotropic random walks in {mathbb{R}^d}

We prove an optimal Gaussian upper bound for the densities of isotropic random walks on ${mathbb{R}^d}$ in spherical case (d ?? 2) and ball case (d ?? 1). We deduce the strongest possible version of the Central Limit Theorem for the isotropic random walks: if ${tilde S_n}$ denotes the normalized random walk and Y the limiting Gaussian vector, then ${mathbb{E} f(tilde S_{n}) rightarrow mathbb{E} f(Y)}$ for all functions f integrable with respect to the law of Y. We call such result a ??Strong CLT??. We apply our results to get strong hypercontractivity inequalities and strong Log-Sobolev inequalities.