When developing large-sample statistical inference for quantiles, also known as Values-at-Risk in finance and insurance, the usual approach is to convert the task into sums of random variables. The conversion procedure requires that the underlying cumulative distribution function (cdf) would have a probability density function (pdf), plus some minor additional assumptions on the pdf. In view of this, and in conjunction with the classical continuous-mapping theorem, researchers also tend to impose the same pdf-based assumptions when investigating (functionals of) integrals of the quantiles, which are natural ingredients of many risk measures in finance and insurance. Interestingly, the pdf-based assumptions are not needed when working with integrals of quantiles, and in this paper we explain and illustrate this remarkable phenomenon.

翻译：当为四分位数(又称 " 金融和保险中的风险值 " )开发大型抽样统计推论时,通常的做法是将任务转换成随机变量的总和。转换程序要求基本累积分布函数(cdf)具有概率密度功能(pdf),加上一些对pdf的微小额外假设。 有鉴于此,结合传统的连续绘制定理法,研究人员在调查四分位数整体体(即金融和保险中许多风险计量的自然成分)时,往往会采用相同的pdf假设。有趣的是,在与四分位数组合体合作时,并不需要基于pdf的假设,因此在本文件中我们解释和说明这一显著现象。