Fast Hausdorff Moment Transforms for Meta Distributions in 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. In addition, a standard is needed for a consistent and objective comparison. This paper addresses the above-mentioned important shortcomings, 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.