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

2025-07-21

AI Summary

This work investigates the distributional properties of two-dimensional complex random walks constrained to a subset of the unit circle, motivated by modeling requirements in signal processing applications such as over-the-air computation. Methodologically, it integrates complex-domain modeling, probabilistic analysis, numerical integration, and large-deviation asymptotics. Key contributions include: (i) the first exact closed-form expressions for the joint and marginal distributions of the two-step walk; (ii) a complete geometric characterization of the support set for arbitrary numbers of steps; (iii) an efficient numerical algorithm applicable to multi-step scenarios; and (iv) high-accuracy asymptotic approximations for the distribution in the large-step regime, derived via large-deviation theory. The results balance theoretical rigor—establishing precise distributional characterizations and geometric insights—with practical utility, offering computationally tractable tools for system design and performance analysis in wireless and signal processing contexts.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Computational social science

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.

Problem

• Derive distribution of 2D random walk with restricted angles
• Analyze joint and marginal distributions for two steps
• Provide numerical solutions and approximations for multiple steps

Innovation

• Derives exact joint and marginal distributions
• Provides numerical solutions for general steps
• Characterizes exact support for arbitrary steps