  Motivated by second order asymptotic results, we characterize the convergence
in law of double integrals, with respect to Poisson random measures, toward a
standard Gaussian distribution. Our conditions are expressed in terms of
contractions of the kernels. To prove our main results, we use the theory of
stable convergence of generalized stochastic integrals developed by Peccati and
Taqqu. One of the advantages of our approach is that the conditions are
expressed directly in terms of the kernel appearing in the multiple integral
and do not make any explicit use of asymptotic dependence properties such as
mixing. We illustrate our techniques by an application involving linear and
quadratic functionals of generalized Ornstein--Uhlenbeck processes, as well as
examples concerning random hazard rates.
