We study the effects of rounding on the moments of random variables. Specifically, given a random variable X and its rounded counterpart $\text{rd}(X)$, we study $|𝔼[X^k]-𝔼[\text{rd}{(X)}^k]|$ for non-negative integer k. We consider the case that the rounding function $\text{rd}:ℝ\to 𝔽$ corresponds either to (i) rounding to the nearest point in some discrete set $𝔽$ 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 $|𝔼[X^k]-𝔼[\text{rd}{(X)}^k]|<C{ϵ}^2$, provided $|\text{rd}(x)-x|\le ϵE(x)$, where $E:ℝ\to {ℝ}_{\ge 0}$ is some fixed positive piecewise linear function. Refined bounds for the absolute moments $𝔼\left[|X^k-\text{rd}{(X)}^k|\right]$ are also given.

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随机变量的非渐近矩界限四舍五入到非均匀间隔集

我们研究四舍五入对随机变量矩的影响。具体来说，给定一个随机变量X及其四舍五入的对应物$\text{路}(X)$， 我们学习 $|𝔼[X^{克}]-𝔼[\text{路}{(X)}^{克}]|$对于非负整数k。我们考虑舍入函数的情况$\text{路}：ℝ\to 𝔽$ 对应于（i）四舍五入到某个离散集中的最近点 $𝔽$或 (ii) 随机四舍五入到同一组中最近的较大或较小的点，概率与到这些点的距离成正比。在这两种情况下，我们展示了在X的密度函数的合理假设下，如何计算常数C使得$|𝔼[X^{克}]-𝔼[\text{路}{(X)}^{克}]|<C{\epsilon }^2$， 假如 $|\text{路}(X)-X|\le \epsilon 乙(X)$， 在哪里 $乙：ℝ\to {ℝ}_{\ge 0}$是一些固定的正分段线性函数。绝对矩的细化边界$𝔼\left[|X^{克}-\text{路}{(X)}^{克}|\right]$ 也给了。