  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.
