  This paper deals with sequences of random variables belonging to a fixed
chaos of order $q$ generated by a Poisson random measure on a Polish space. The
problem is investigated whether convergence of the third and fourth moment of
such a suitably normalized sequence to the third and fourth moment of a centred
Gamma law implies convergence in distribution of the involved random variables.
A positive answer is obtained for $q=2$ and $q=4$. The proof of this four
moments theorem is based on a number of new estimates for contraction norms.
Applications concern homogeneous sums and $U$-statistics on the Poisson space.
