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

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.

21 pages, 1 figure, to appear in the Electronic Journal of Probability

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Cited by in corpus (1)

• Asymptotic shape of the concave majorant of a Lévy process