Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gradient maps define novel investment strategies. The gradient maps may also be regarded as optimal transport maps on the simplex. In this paper we construct and study probability measures supported on spaces of concave functions. These measures may serve as prior distributions in Bayesian statistics and Cover’s universal portfolio, and induce distribution-valued random variables via optimal transport. The random concave functions are constructed on the unit simplex by taking a suitably scaled (mollified, or soft) minimum of random hyperplanes. Depending on the regime of the parameters, we show that as the number of hyperplanes tends to infinity there are several possible limiting behaviors. In particular, there is a transition from a deterministic almost sure limit to a nontrivial limiting distribution that can be characterized using convex duality and Poisson point processes.
Leonard Wong’s research is partially supported by NSERC Grant RGPIN-2019-04419.
This project started when Leonard Wong was a postdoc at the University of Southern California. Part of the research was carried out when he was visiting the Faculty of Mathematics at the University of Vienna. He thanks Walter Schachermayer and Christa Cuchiero for many helpful discussions.
We thank the Associate Editor and the anonymous referees for their helpful comments which greatly improved the paper.
"Random concave functions." Ann. Appl. Probab. 32 (2) 812 - 852, April 2022. https://doi.org/10.1214/21-AAP1697