A note on the multivariate local limit theorem

Abstract

Given a Gaussian distribution G in l2 we construct l2-valued i.i.d. random variables X1,X2,... such that: weakly, F1 is absolutely continuous with respect to G, and the total variation Fn-G[not right arrow]0. This contrasts to Prokhorov's (Dokl. Akad. Nauk SSSR (N.S) 83 (1952) 797) local limit theorem in Rk.Local limit theorem Central limit theorem Total variation Infinite dimension