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We analyze from the viewpoint of an abstract Markov operator recent resultsby Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as acondition on a Wiener chaos to have a distribution close to Gaussian. Inparticular, we are led to introduce a notion of chaos associated to a Markovoperator through its iterated gradients and present conditions on the (pure)point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussiandistribution. Convergence to gamma distributions may be examined similarly.

Chaos of a Markov operator and the fourth moment condition