Rounding random variables to finite precision.

In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable $X$ and a perturbation rd$(X)$, when $|{\rm rd}(x)-x| \leq \epsilon |x|$ or $|{\rm rd}(x)-x| \leq \delta$ which depend linearly on $\epsilon$ and $\delta$. We then show that if the perturbation corresponds to rounding to the nearest point in some fixed discrete set, the bounds on the difference of moments can be improved to quadratic in many cases. When the points in this fixed set are uniformly spaced, our analysis can be viewed as a generalization of Sheppard's corrections. We discuss how our bounds can be used to balance measurement error with sample error in a rigorous way, as well as how they can be used to generalize classical numerical analysis results. The frameworks developed in our analysis can be applied to a wider range of applications than those studied in this paper and may be of general interest.