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We give necessary and sufficient conditions to characterize the convergencein distribution of a sequence of arbitrary random variables to a probabilitydistribution which is the invariant measure of a diffusion process. This classof target distributions includes the most known continuous probabilitydistributions. Precisely speaking, we characterize the convergence in totalvariation to target distributions which are not Gaussian or Gamma distributed,in terms of the Malliavin calculus and of the coefficients of the associateddiffusion process. We also prove that, among the distributions whose associatedsquared diffusion coefficient is a polynomial of second degree (with somerestrictions on its coefficients), the only possible limits of sequences ofmultiple integrals are the Gaussian and the Gamma laws.

Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions