  Through a reformulation of the local limit theorem and law of small numbers,
which is obtained by working in the spaces naturally associated to the limiting
distributions, we discover a general and abstract framework for the
investigation of that type of limit theorems. From this new perspective, the
convolution and scaling operators utilized in the classical results mentioned
before will be identified with the Wick product and second quantization
operators, respectively. And here is the advantage of our approach: definitions
and most of the properties of Wick products and second quantization operators
do not depend (mutatis mutandis) on the underlying probability measure. Then,
with the help of Holder-Young-type inequalities for Gaussian and Poisson Wick
products proved in previous papers, we show the L^1 convergence of the
densities towards the desired limit. We remark that our approach extends
without additional assumptions to infinite dimensional Gaussian spaces.
