Convex Minorants of Random Walks and Brownian Motion

Let $(S_{i})_{i=0}^n$ be the random walk process generated by a sequence of real-valued independent identically distributed random variables $(X_{i})_{i=1}^n$ having densities. We study probability distributions related to the associated convex minorant process. In particular, we investigate the length of a convex minorant's longest segment. Using random permutation theory, we fully characterize the probability distribution of the length of the $r$th longest segment of the convex minorant generated by Brownian motion on finite intervals; we also give an explicit density for the joint distributions of the first $r$ longest segments. In addition, we use the methods developed here to prove Sparre Andersen's formula for the probability of having $m$ segments composing the convex minorant of a random walk of length $N$. We describe analogous statements for random walks with random time increments. The author has recently used these results to solve a problem of adhesion dynamics on the line.