How long is the convex minorant of a one-dimensional random walk?

TL;DR

This paper establishes distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk, revealing different regimes based on the increment distribution and employing permutation representations.

Contribution

It introduces new limit theorems for the convex minorant length of random walks, using permutation-based representations to analyze various distributional regimes.

Findings

Different limit distributions identified for various increment laws.

Representation of convex minorant via uniform permutations.

Limit theorems applicable to diverse regimes of random walk increments.

Abstract

We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized.