Large deviations for the perimeter of convex hulls of planar random walks

Arseniy Akopyan, Vladislav Vysotsky

Published 2016-06-23Version 1

We give logarithmic asymptotic bounds for large deviations probabilities for perimeter of the convex hull of a planar random walk. These bounds are sharp for a wide class of distributions of increments that includes Gaussian distributions and shifted or linearly transformed rotationally invariant distributions. For such random walks, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments. These results on the perimeter are easily extended to mean width of convex hulls of random walks in higher dimensions. Our method also allows to find the logarithmic asymptotics of large deviations probabilities for area of the convex hull of planar random walks with rotationally invariant distributions of increments.