@article {ua127,
title = {Strong Central Limit Theorem for isotropic random walks in Rd},
journal = {Probability Theory and Related Fields},
volume = {151},
number = {1-2},
year = {2011},
month = {2011/10/01},
pages = {153 - 172},
publisher = {Springer},
type = {1},
address = {Berlin ; Heidelberg},
abstract = {We prove an optimal Gaussian upper bound for the densities of isotropic random walks on 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 S~_{n} denotes the normalized random walk and Y the limiting Gaussian vector, then E*f*(S~_{n}){\textrightarrow}E*f*(Y) for all functions *f *integrable with respect to the law of Y. We call such result a {\textquotedblleft}Strong CLT{\textquotedblright}. We apply our results to get strong hypercontractivity inequalities and strong Log-Sobolev inequalities.

},
keywords = {Analyse, Probabilit{\'e}s et Statistique, Central, Gaussian, Logarithmic, Mathematical, Operations, Probability, Quantitative, Random, Statistics, Theoretical, Mathematical and Computational Physics},
issn = {1432-2064},
doi = {10.1007/s00440-010-0295-6},
url = {http://okina.univ-angers.fr/publications/ua127},
author = {Graczyk, Piotr and Loeb, Jean-Jacques and {\.Z}ak, Tomasz}
}