Concave Majorants of Random Walks and Related Poisson Processes

We offer a unified approach to the theory of concave majorants of random walks by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant for a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested compositions that naturally arise from our construction of the concave majorant.