Abstract
We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and functionals of both continuous and discrete independent random variables. For random variables admitting a continuous density, it recovers classical distance bounds based on absolute third moments, with better and explicit constants. We also apply this method to multiple stochastic integrals that can be used to represent $U$-statistics, and include linear and quadratic functionals as particular cases.
Highlights
The Stein and Chen-Stein methods have been developed together with the Malliavin calculus to derive bounds on the distances between probability laws on the Wiener and Poisson spaces, cf. [9], [12], [13] and for discrete Bernoulli sequences, cf. [10], [4], [5]. The results of these works rely on covariance representations based on the number operator L on multiple Wiener-Poisson stochastic integrals and its inverse L−1
Other covariance representations based on the Clark-Ocone representation formula have been used in [18] on the Wiener and Poisson spaces, and in [19] for Bernoulli processes
This paper focuses on functionals of a countable number of uniformly distributed random variables, and uses the framework of [14], cf. [15], [16], to derive covariance representations from chaos expansions in multiple stochastic integrals, based on a version of the Clark-Ocone formula with finite difference or derivation operators
Summary
The Stein and Chen-Stein methods have been developed together with the Malliavin calculus to derive bounds on the distances between probability laws on the Wiener and Poisson spaces, cf. [9], [12], [13] and for discrete Bernoulli sequences, cf. [10], [4], [5]. The results of these works rely on covariance representations based on the number (or Ornstein-Uhlenbeck) operator L on multiple Wiener-Poisson stochastic integrals and its inverse L−1. We obtain general bounds on the distance of a random functional to the Gaussian and gamma distributions using Stein kernels, see Propositions 3.1-3.3, and we derive specific bounds for multiple stochastic integrals, see Corollary 5.2.
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