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.

翻译：本文推导了二维（复数）随机游走的分布，其中每一步的角度限制在圆周的某个子集内。这种设定出现在信号处理中的空中计算等多个领域。具体而言，我们推导了两步时的精确联合分布和边缘分布，一般步数时的数值解，以及大步数时的近似解。此外，我们给出了任意步数下支撑集的精确刻画。本文的研究结果为未来涉及此类问题的工作提供了参考依据。