Concentration of Randomized Functions of Uniformly Bounded Variation

Abstract: 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.