Non-asymptotic moment bounds for random variables rounded to non-uniformly spaced sets

We study the effects of rounding on the moments of random variables. Specifically, given a random variable $X$ and its rounded counterpart $\operatorname{rd}(X)$, we study $|\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]|$ for non-negative integer $k$. We consider the case that the rounding function $\operatorname{rd} : \mathbb{R}\to\mathbb{F}$ corresponds either to (i) rounding to the nearest point in some discrete set $\mathbb{F}$ or (ii) rounding randomly to either the nearest larger or smaller point in this same set with probabilities proportional to the distances to these points. In both cases, we show, under reasonable assumptions on the density function of $X$, how to compute a constant $C$ such that $|\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]| < C\epsilon^2$, provided $|\operatorname{rd}(x) - x| \leq \epsilon \: E(x)$, where $E : \mathbb{R} \to \mathbb{R}_{\geq 0}$ is some fixed positive piecewise linear function. Refined bounds for the absolute moments $\mathbb{E}[ |X^k-\operatorname{rd}(X)^{k}|]$ are also given.