  Consider generalized adapted stochastic integrals with respect to
independently scattered random measures with second moments. We use a
decoupling technique, known as the "principle of conditioning", to study their
stable convergence towards mixtures of infinitely divisible distributions. Our
results apply, in particular, to multiple integrals with respect to
independently scattered and square integrable random measures, as well as to
Skorohod integrals on abstract Wiener spaces. As a specific application, we
establish a Central Limit Theorem for sequences of double integrals with
respect to a general Poisson measure, thus extending the results contained in
Nualart and Peccati (2005) and Peccati and Tudor (2004) to a non-Gaussian
context.
