The variance of the discrepancy distribution of rounding procedures, and sums of uniform random variables

Abstract

When $\ell$ probabilities are rounded to integer multiples of a given accuracy n, the sum of the numerators may deviate from n by a nonzero discrepancy. It is proved that, for large accuracies $n \rightarrow \infty$, the limiting discrepancy distribution has variance $\ell /12$. The relation to the uniform distribution over the interval $[-1/2, 1/2]$, whose variance is 1 / 12, is explored in detail.