Functional generalizations of Hoeffding's covariance lemma and a formula for Kendall's tau

- Generalizations of Hoeffding's lemma to functions of random variables are developed.
- Existing generalizations are clarified and limitations overcome.
- A closed-form formula for Kendall's tau for general random variables is derived.

This note extends Hoeffding's lemma from the covariance between two random variables to that between transformations of random variables, and provides clarification of several existing functional generalizations. In the same spirit as these results, an explicit integral formula of Kendall's tau for general, possibly discontinuous random variables is also determined.