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

TL;DR

This paper derives non-asymptotic bounds on the differences in moments between a random variable and its rounded version, considering non-uniform rounding schemes and under certain density assumptions.

Contribution

It introduces explicit bounds for moment differences when rounding to non-uniform sets, extending previous asymptotic analyses to finite-sample, non-asymptotic contexts.

Findings

Bound on the difference of moments is proportional to psilon^2.

Provides explicit constant C for moment difference bounds.

Refined bounds for absolute moments of the rounding error.

Abstract

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.