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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

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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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