Convex Hulls of Random Walks

Features related to the perimeter of the convex hull C„ of a random walk in R2 are studied, with particular attention given to its length L„. Bounds on the variance of Ln are obtained to show that, for walks with drift, L„ obeys a strong law. Exponential bounds on the tail probabilities of L„ under special conditions are also obtained. We then develop simple expressions for the expected values of other features of Cn, including the number of faces, the sum of the lengths and squared lengths of the faces, and the number of faces of length t or less.