Abstract

Extensions of the Nourdin–Peccati analysis to R n -valued random variables are obtained by taking conditional expectation on the Wiener space. Several proof techniques are explored, from infinitesimal geometry, to quasi-sure analysis (including a connection to Stein's lemma), to classical analysis on Wiener space. Partial differential equations for the density of an R n -valued centered random variable Z = ( Z 1 , … , Z n ) are obtained. Of particular importance is the function defined by the conditional expectation given Z of the auxiliary random matrix ( − D L − 1 Z i | D Z j ) , i , j = 1 , 2 , … , n , where D and L are respectively the derivative operator and the generator of the Ornstein–Uhlenbeck semigroup on Wiener space.

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