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的随机函数，在满足一致有界变差假设条件下，我们证明了关于均值函数集中性的一个尖锐、无分布假设的非渐近结果。本研究的具体动机源于对社会政策干预的分布效应进行推断的需求。然而，我们研究的随机函数族具有足够广泛的适用性，可覆盖多种应用场景。例如，我们为几乎必然利普希茨连续的随机函数提供了类似科尔莫戈罗夫-斯米尔诺夫检验的方法，并提出了处理异质性处理效应推断的新工具。此外，针对几乎必然单调随机函数之和，我们推导出类似德沃雷茨基-基弗-沃尔福威茨不等式的不等式，将马萨尔针对独立同分布数据生成的累积分布函数的著名非渐近结果推广至卡奈、桑托斯和谢赫所提出的无微簇设定。通过实证应用，我们展示了理论结果对实际工作的相关性。值得注意的是，我们主要集中性结果的证明依赖于德布罗基本定理（通常称为"间隙引理"）的一个新颖随机版本，该定理将前序关系的非连续效用表示转化为连续效用表示，同时依赖于我们精心构造的无限维优化问题的包络定理。