A sharp, distribution free, non-asymptotic result is proved for the concentration of a random function around the mean function, when the randomization is generated by a finite sequence of independent data and the random functions satisfy uniform bounded variation assumptions. The specific motivation for the work comes from the need for inference on the distributional impacts of social policy intervention. However, the family of randomized functions that we study is broad enough to cover wide-ranging applications. For example, we provide a Kolmogorov-Smirnov like test for randomized functions that are almost surely Lipschitz continuous, and novel tools for inference with heterogeneous treatment effects. A Dvoretzky-Kiefer-Wolfowitz like inequality is also provided for the sum of almost surely monotone random functions, extending the famous non-asymptotic work of Massart for empirical cumulative distribution functions generated by i.i.d. data, to settings without micro-clusters proposed by Canay, Santos, and Shaikh. We illustrate the relevance of our theoretical results for applied work via empirical applications. Notably, the proof of our main concentration result relies on a novel stochastic rendition of the fundamental result of Debreu, generally dubbed the "gap lemma," that transforms discontinuous utility representations of preorders into continuous utility representations, and on an envelope theorem of an infinite dimensional optimisation problem that we carefully construct.

翻译：针对由有限独立数据序列生成随机化且随机函数满足均匀有界变差假设的情形，本文证明了随机函数围绕均值函数集中性的一个尖锐、无分布假设、非渐进性结果。该研究的具体动机源于社会政策干预对分配影响进行推断的需求，但所研究的随机函数族具有足够广泛性以覆盖多类应用。例如，我们为几乎必然利普希茨连续的随机函数提供了柯尔莫哥洛夫-斯米尔诺夫型检验，并为异质性处理效应的推断提供了新型工具。同时，针对几乎必然单调随机函数之和，本文推导出德沃雷茨基-基弗-沃尔福威茨型不等式，将马萨特关于独立同分布数据生成经验累积分布函数的著名非渐近性成果，推广至卡奈、桑托斯和谢赫提出的无微聚类情境。通过实证应用，我们展示了理论结果对应用研究的实际价值。值得注意的是，本文主要集中性结果的证明依赖于对德布鲁基本定理（通常称为“缺口引理”）的随机化创新诠释——该定理将预序的不连续效用表示转化为连续效用表示，同时依赖我们精心构建的无穷维优化问题包络定理。