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.
"Stein approximation for functionals of independent random sequences." Electron. J. Probab. 23 1 - 34, 2018. https://doi.org/10.1214/17-EJP132