Iterated-logarithm laws for convex hulls of random walks with drift

Wojciech Cygan, Nikola Sandrić, Stjepan Šebek, and Andrew R. Wade

Transactions of the American Mathematical Society, 377, no. 9, September 2024, 6695–6724. DOI 10.1090/tran/9238

Supported by EPSRC award Anomalous diffusion via self-interaction and reflection (EP/W00657X/1).

Abstract

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero–one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk.