A unified approach to local limit theorems in Gaussian spaces and the law of small numbers

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 these limit theorems. From this new perspective, the convolution and scaling operators employed in the classical results mentioned before will be identified with the Wick product and second quantization operators, respectively. And this 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 H¨older-Young-type inequalities for Gaussian and Poisson Wick products proved in previous papers, we show the L1 convergence of the densities towards the desired limit. We remark that our approach extends without additional assumptions to infinite dimensional Gaussian spaces.