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Article Dans Une Revue Probability Theory and Related Fields Année : 2011

Strong Central Limit Theorem for isotropic random walks in Rd

Résumé

We prove an optimal Gaussian upper bound for the densities of isotropic random walks on Rd 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 Ef(S~n)→Ef(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.

Dates et versions

hal-03040220 , version 1 (04-12-2020)

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Piotr Graczyk, Jean-Jacques Loeb, Tomasz Żak. Strong Central Limit Theorem for isotropic random walks in Rd. Probability Theory and Related Fields, 2011, 151 (1-2), pp.153 - 172. ⟨10.1007/s00440-010-0295-6⟩. ⟨hal-03040220⟩
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