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The Stein and Chen-Stein methods for functionals of non-symmetric Bernoulli processes. (English) Zbl 1329.60079
Summary: Based on a new multiplication formula for discrete multiple stochastic integrals with respect to non-symmetric Bernoulli random walks, we extend results of G. D. Reinert et al. [Electron. J. Probab. 15, 1703–1742 (2010; Zbl 1225.60046)] 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 a method of G. Peccati [“The Chen-Stein method for Poisson functionals”, Preprint (2011), arXiv: 1112.5051]. 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.

MSC:
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60H05 Stochastic integrals
60G50 Sums of independent random variables; random walks
60H07 Stochastic calculus of variations and the Malliavin calculus
60G57 Random measures
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