Abstract
For the perimeter length and the area of the convex hull of the first n steps of a planar random walk, we study n→∞ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random walks with drift (for the area) and walks with no drift (for both area and perimeter length) under mild moments assumptions on the increments. These results complement and contrast with previous work which showed that the perimeter length in the case with drift satisfies a central limit theorem. We deduce these results from weak convergence statements for the convex hulls of random walks to scaling limits defined in terms of convex hulls of certain Brownian motions. We give bounds that confirm that the limiting variances in our results are non-zero.
Highlights
Random walks are classical objects in probability theory
Many of the questions of stochastic geometry, traditionally concerned with functionals of independent random points, are of interest for point sets generated by random walks
We examine the asymptotic behaviour of the convex hull of the first n steps of a random walk in R2, a natural geometrical characteristic of the process
Summary
Random walks are classical objects in probability theory. Recent attention has focussed on various geometrical aspects of random walk trajectories. Armed with these weak convergence results, we present asymptotics for expectations and variances of the quantities Ln and An in Section 3; the arguments rely in part on the scaling limit apparatus, and in part on direct random walk computations. Appendix A collects some auxiliary results on random walks that we use
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