Rounding of discrete variables

TL;DR

The paper addresses rounding of a discrete random variable $X$ with values in $\frac{1}{q}\mathbb Z$ to integers, deriving general formulas for the characteristic function and moments of the rounded variables $\lfloor X\rfloor$ and $\langle X\rangle$. Using Fourier-analytic tools and auxiliary uniform variables $U_q$ and $\widetilde{U}_q$, it expresses $\varphi_{\lfloor X\rfloor}$ and $\varphi_{\langle X\rangle}$ through $\varphi_X$ and $h_q$, $\widetilde{h}_q$, and provides explicit mean and second-moment formulas that depend on the parity of $q$. The results extend the previously studied continuous-case formulas (SJ175) to discrete-valued $X$, and are illustrated with detailed examples, including a Sheppard-type correction and bounds for the variance error in odd $q$. Overall, the findings yield computable expressions for distributional characteristics of rounded values and connect discrete rounding to modular Fourier analysis on $\mathbb Z_q$, with practical implications for translating rounding to multiples of $q$ in applications.

Abstract

Let $X$ be a random variable that takes its values in $\frac{1}{q}\mathbb{Z}$, for some integer $q\ge2$, and consider $X$ rounded to an integer, either downwards or upwards or to the nearest integer. We give general formulas for the characteristic function and moments of the rounded variable. These formulas complement the related but different formulas in the case that $X$ has a continuous distribution, which was studied by Janson (2006).

Theorems & Definitions (14)

• Remark 1.1
• Remark 2.1
• Theorem 3.1
• proof
• Remark 3.2
• Theorem 4.1
• proof
• Theorem 5.1
• proof
• Theorem 6.1