Angular asymptotics for random walks
Alejandro López Hernández and Andrew R. Wade
In:
A Lifetime of Excursions Through Random Walks and Lévy Processes:
A Volume in Honour of Ron Doney’s 80th Birthday,
Eds. L. Chaumont and A.E. Kyprianou.
Progress in Probability, vol 78. Birkhäuser, 2021. DOI: 10.1007/978-3-030-83309-1_17
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Abstract
We study the set of directions asymptotically explored by a spatially homogeneous random walk in $d$-dimensional Euclidean space. We survey some pertinent results of Kesten and Erickson, make some further observations, and present some examples. We also explore links to the asymptotics of one-dimensional projections, and to the growth of the convex hull of the random walk.
Addendum (March 2023)
Section 8 (on "Projection asymptotics") of the paper raises the question of existence of exceptional projections for a random walk, and establishes that there are none for dimensions 1 and 2. Subsequent to the paper being published, we learned of the 1998 paper "Sets avoided by Brownian motion" [MR1626170] by Adelman, Burdzy, and Pemantle, which shows, by contrast, that exceptional projections do exist for three-dimensional Brownian motion.