$L^{q}$-error estimates for approximation of irregular functionals of random vectors

Avikainen showed that, for any $p,q in [1,infty)$, and any function $f$ of bounded variation in $mathbb{R}$, it holds that $mathbb{E}[|f(X)-f(widehat{X})|^{q}] leq C(p,q) mathbb{E}[|X-widehat{X}|^{p}]^{frac{1}{p+1}}$, where $X$ is a one-dimensional random variable with a bounded density, and $widehat{X}$ is an arbitrary random variable. In this article, we will provide multi-dimensiona