Convex hulls of random walks

There are various ways to try to describe the geometry of random walk trajectories in $d$-dimensional space. One is to consider the convex hull of the first $n$ steps of the trajectory, that is, the smallest convex set that contains the first $n$ locations of the random walk. What can we say about the convex hull as $n \to \infty$? Its diameter? Its perimeter length or area in the plane, for example, or analogous quantities in higher dimensions?

In the case where the random walk has zero drift and finite-variance increments, the trajectory converges to Brownian motion and the scaling limit carries across to the convex hull and many associated functionals. In the case where the random walk has a drift, in the planar case scaling arguments also give distributional asymptotics for the area. The perimeter and diameter need a different approach, and, while the distributional limits arising from scaling arguments are non-Gaussian, it turns out that the perimeter and diamater have a Gaussian limit, after centering and scaling.

The shape of the convex hull is quite different in the two cases. In the case with drift, it converges to a line segment. In the zero-drift case, the shape fluctuates across all possible convex compact sets.

Click on the buttons below for some of my work related to convex hulls of random walks.