Hausdorff Moment Transforms and Their Applications to Wireless Networks

In the analysis of wireless networks, the standard signal-to-interference (SIR) distribution does not capture the performance at the individual link level. The meta distribution (MD) of the SIR resolves this problem by separating different sources of randomness, such as fading and point process(es). While it allows for a much sharper performance characterization, it can in most cases only be calculated based on the moments of the underlying conditional distribution, i.e., by solving a Hausdorff moment problem. Several methods to reconstruct MDs from the moments have been proposed but a rigorous analysis, comparison of their performance, and practical implementations are missing. This dissertation focuses on three parts: the reconstruction of MDs, the fundamental problem behind it, i.e., the truncated Hausdorff moment problem, and the sensitivity issues related to the moments subject to perturbations. As for the truncated Hausdorff moment problem, we establish a method of comparison for the performance of the approximations. Three ways of producing random moment sequences are discussed and applied. Also, some of the approximations have been rewritten as linear transforms and detailed accuracy requirements are analyzed. Our finding shows that the performance of the approximations differ significantly in their convergence properties, accuracy, and numerical complexity, and that the decay type of the moment sequence strongly affects the accuracy requirement. As for the reconstruction of MDs, we introduces a tweaking mapping for adjusting approximations, presents terminology to categorize the quality of approximations, proposes the use of the Fourier-Legendre method, which has not previously been applied to MDs, and provides the achievable lower and upper bounds on the MD given the first $n$ moments. Further, to facilitate the use of MDs, we give comprehensive guidance on the selection of the best method to determine MDs, and we offer ready-to-use implementations of the proposed algorithms. This study fills an important gap in the literature by rigorously analyzing the MDs, comparing the performance of different methods, and offering user-friendly implementations for recovering MDs from moments. As for the sensitivity issues, we explore the reliability and robustness of these techniques. We analyze the sensitivity of commonly used MD reconstruction methods in the presence of perturbations to moments and provide valuable guidelines for the application of these methods. Furthermore, we quantify the impact of inaccurate moments on MD reconstructions, examining the validity of perturbed moment sequences and demonstrating the critical importance of moment accuracy. Our investigation demonstrates the necessity for precise moment computation. Succinctly put, moment quality is preferred over moment quantity.