Stein’s method and stochastic analysis of Rademacher functionals
نویسندگان
چکیده
We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein’s method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not require the classical use of exchangeable pairs, we employ a chaos expansion in order to construct an explicit exchangeable pair vector for any random variable which depends on a finite set of Rademacher variables. Among several examples, which include random variables which depend on infinitely many Rademacher variables, we provide three main applications: (i) to CLTs for multilinear forms belonging to a fixed chaos, (ii) to the Gaussian approximation of weighted infinite 2-runs, and (iii) to the computation of explicit bounds in CLTs for multiple integrals over sparse sets. This last application provides an alternate proof (and several refinements) of a recent result by Blei and Janson.
منابع مشابه
Gaussian approximation of functionals: Malliavin calculus and Stein’s method
Combining Malliavin calculus and Stein’s method has recently lead to a new framework for normal and for chi-square approximation, and for second-order Poincaré inequalities. Applications include functionals of Gaussian random fields as well as functionals of infinite Poisson and Rademacher sequences. Here we present the framework and an extension which leads to invariance principles for multili...
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