Random Walks Whose Concave Majorants Often Have Few Faces

We construct a continuous distribution G such that the number of faces in the smallest concave majorant of the random walk with G-distributed summands will take on each natural number infinitely often with probability one. This investigation is motivated by the fact that the number of faces Fn of the concave majorant of the random walk at time n has the same distribution as the number of records Rn in the sequence of summands up to time n. Since Rn is almost surely asymptotic to log n, the construction shows that despite the equality of all of the one-dimensional marginals, the almost sure behaviors of the sequences {Rn} and {Fn} may be radically different.