Recurrence of the twisted planar random walk

TL;DR

This paper demonstrates that twisted planar random walks, formed by summing stationary increments rotated by fixed angles, are recurrent under broad conditions, including alpha-mixing increments with finite second moments, for almost all fixed angles.

Contribution

It establishes recurrence of twisted planar random walks for a wide class of increment processes and almost all fixed angles, extending previous results to more general settings.

Findings

Recurrence holds for alpha-mixing increments with finite second moments.

Recurrence applies to almost all fixed angles in a set of full Lebesgue measure.

The result is robust under various assumptions on the increment process.

Abstract

We show that the "twisted" planar random walk - which results by summing up stationary increments rotated by multiples of a fixed angle - is recurrent under diverse assumptions on the increment process. For example, if the increment process is alpha-mixing and of finite second moment, then the twisted random walk is recurrent for every angle fixed choice of the angle out of a set of full Lebesgue measure, no matter how slow the mixing coefficients decay.