On the Distribution of a Two-Dimensional Random Walk with Restricted Angles

TL;DR

This paper derives the exact and approximate distributions of a 2D random walk with restricted step angles, relevant for applications like over-the-air computation in signal processing.

Contribution

It provides the first comprehensive derivation of joint, marginal, and support distributions for such constrained random walks.

Findings

Exact joint and marginal distributions for two steps

Numerical solutions for multiple steps

Support characterization for any number of steps

Abstract

In this paper, we derive the distribution of a two-dimensional (complex) random walk in which the angle of each step is restricted to a subset of the circle. This setting appears in various domains, such as in over-the-air computation in signal processing. In particular, we derive the exact joint and marginal distributions for two steps, numerical solutions for a general number of steps, and approximations for a large number of steps. Furthermore, we provide an exact characterization of the support for an arbitrary number of steps. The results in this work provide a reference for future work involving such problems.