  We obtain quantitative Four Moments Theorems establishing convergence of the
laws of elements of a Markov chaos to a Pearson distribution, where the only
assumption we make on the Pearson distribution is that it admits four moments.
While in general one cannot use moments to establish convergence to a
heavy-tailed distributions, we provide a context in which only the first four
moments suffices. These results are obtained by proving a general carr\'e du
champ bound on the distance between laws of random variables in the domain of a
Markov diffusion generator and invariant measures of diffusions. For elements
of a Markov chaos, this bound can be reduced to just the first four moments.
