论文标题
随机变量的非反应力矩边界圆形到不均匀间隔的集合
Non-asymptotic moment bounds for random variables rounded to non-uniformly spaced sets
论文作者
论文摘要
我们研究圆形对随机变量矩的影响。具体而言，给定一个随机变量$ x $及其圆形对应物$ \ perepatorName {rd}（x）$，我们研究$ | \ m m马理{e} [x^k] - \ m athbb {e} [\ operatatorname {\ operatatorName {rd}（rd}（rd}（rd}（x）^{k} {k} {k} {k} {k} {k}] | $ for non -n in egeg $ k $。我们考虑以下情况：圆形函数$ \ protatorAtorName {rd}：\ mathbb {r} \ to \ m athbb {f} $对应于（i）在某些离散集$ \ mathbb {f} $或（ii）与较大或较小的概率的较大点的圆形圆形中，在此范围内与较大或较小的点相同，以这些点与这些点相同的范围。在这两种情况下，我们都在合理的假设上表明了$ x $的密度函数，如何计算常数$ c $，以便$ | \ m athbb {e} [x^k] [x^k] - \ Mathbb {e} [\ propatatorName {\ properatorName {rd} {rd}（x）（x）^{k}^{k}] | | <cε^2 $，提供$ | \ operatatorName {rd}（x） - x | \ leqε\：e（x）$，其中$ e：\ mathbb {r} \ to \ mathbb {r} _ {\ geq 0} $是某些固定的正线性线性函数。在绝对矩$ \ mathbb {e} [| x^k- \ operatatorName {rd}（x）^{k} |] $的绝对范围中。
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 $|\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]| < Cε^2$, provided $|\operatorname{rd}(x) - x| \leq ε\: E(x)$, 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.
