On the Distribution of a Two-Dimensional Random Walk with Restricted Angles
arXiv:2507.15475 · doi:10.1109/TSP.2026.3694607
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.
14 pages, 14 figures