Functional limit theorems for random walks

Abstract

We survey some geometrical properties of trajectories of $d$-dimensional random walks via the application of functional limit theorems. We focus on the functional law of large numbers and functional central limit theorem (Donsker's theorem). For the latter, we survey the underlying weak convergence theory, drawing heavily on the exposition of Billingsley, but explicitly treat the multidimensional case. Our two main applications are to the convex hull of a random walk and the centre of mass process associated to a random walk. In particular, we establish the limit sets of the convex hull in the two distinct cases of zero and non-zero drift which provides insight into the diameter, mean width, volume and surface area functionals. For the centre of mass process, we find the limiting processes in both the law of large numbers and central limit theorem domains.