Stein's Method and Stochastic Analysis of Rademacher Functionals

By Ivan Nourdin, Giovanni Peccati, Gesine Reinert. 2008

73 citations recorded by the source corpus.

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. As the bounds are given in terms of Malliavin operators, no coupling construction is required. When the functional depends only on the first d coordinates of the Rademacher sequence, a simple sufficient condition for convergence to a normal distribution is derived. For finite quadratic forms, we obtain necessary and sufficient conditions. Although our approach does not require the classical use of exchangeable pairs, when the functional depends only on the first d coordinates of the Rademacher sequence we employ chaos expansion in order to construct an explicit exchangeable pair vector; the elements of the vector relate to the summands in the chaos decomposition and satisfy a linearity condition for the conditional expectation. Among several examples, such as 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.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6b286217c7774fc70a65c54171320e9c8de2c648

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Citations in the corpus

Stein's method meets Malliavin calculus: a short survey with new estimates (2009, methodology): “- In [21], by Nourdin, Peccati and Reinert, one can find an extension of the ideas introduced in [18] to the framework of functionals of Rademacher sequences.”
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos (2009, methodology): “References [21] and [15] use appropriate versions of the non-Gaussian Malliavin calculus in order to deal with the one-dimensional normal approximation, respectively, of functionals of Poisson measures and of functionals of infinite Rademacher sequences.”
Chaos of a Markov operator and the fourth moment condition (2012, methodology): “This analysis has been recently achieved in [16].”