Quantization of the Poisson Type Central Limit Theorem (1)

A sequence of binomial random variables, both classical and algebraic, is modelized in terms of the creation–annihilation operators in a natural way and each of these random variables is a sum of four terms. By taking a proper interacting Fock structure, these random variables verify a certain pre–given (classical, Boolean, free, monotone, anti–monotone, etc) independence and the sum of finite independent binomial random variables formulates the corresponding Bernoulli sequence. With the help of such a structure, the Poisson type central limit theorem is quantized by considering individually the contribution of those four terms to the limit. Moreover, its off–diagonal part gives a quantization of the Laplace–de Moivre type central limit theorem.