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.
"How long is the convex minorant of a one-dimensional random walk?." Electron. J. Probab. 25 1 - 22, 2020. https://doi.org/10.1214/20-EJP497