Random concave functions

TL;DR

This paper develops a probabilistic framework for random concave functions on the simplex, exploring their limiting behaviors and potential applications in Bayesian statistics and portfolio theory.

Contribution

It introduces a novel construction of probability measures on spaces of concave functions and analyzes their asymptotic properties using convex duality and Poisson processes.

Findings

Transition from deterministic to distributional limits as hyperplanes increase

Characterization of limits via convex duality and Poisson processes

Potential applications in Bayesian priors and portfolio strategies

Abstract

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 non-trivial limiting distribution that can be characterized using convex duality and Poisson point processes.