Convex hulls of planar random walks with drift

Abstract:

Denote by $L_n$ the perimeter length of the convex hull of an $n$-step planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that $n^{-1} L_n$ converges almost surely to a deterministic limit and proved an upper bound on the variance $\mathbb {V}\mathrm {ar} [ L_n] = O(n)$. We show that $n^{-1} \mathbb {V}\mathrm {ar} [L_n]$ converges and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for $L_n$ in the non-degenerate case.