The Stein and Chen-Stein methods for functionals of non-symmetric Bernoulli processes

Based on a new multiplication formula for discrete multiple stochastic integrals with respect to non-symmetric Bernoulli random walks, we extend the results of Nourdin et al. (2010) on the Gaussian approximation of symmetric Rademacher sequences to the setting of possibly non-identically distributed independent Bernoulli sequences. We also provide Poisson approximation results for these sequences, by following the method of Peccati (2011). Our arguments use covariance identities obtained from the Clark-Ocone representation formula in addition to those usually based on the inverse of the Ornstein-Uhlenbeck operator.