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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. 

Meta distributions (MDs) have emerged as a powerful tool in the analysis of wireless networks. Compared to standard distributions, they enable a clean separation of the different sources of randomness, resulting in sharper, more refined results. In particular, they capture the disparity of the performances of individual links or users. In this first part of a two-letter series, we start from first principles and give the formal definition of MDs and present several simple yet illustrative examples. Part 2 [1] explores the properties of the MD in more depth and offers multiple interpretations and applications. 

In the companion letter [1], we have defined and exemplified meta distributions (MDs) as a natural extension of the concepts of the mean and distribution of a random variable. Here we provide an in-depth discussion of the properties and interpretations of MDs. It includes original results on the calculation of MDs in the monotone case and two applications to simple Poisson wireless networks models. 

This work studies the signal-to-interference-plus-noise ratio (SINR) meta distribution (MD) in cellular networks with a focus on the Poisson model. Firstly, we show that for stationary base station point processes, arbitrary fading, and power-law path loss with exponent α, the base station density λ and the noise power σ2 impact the SINR MD only through ηλα/2/σ2, termed the network signal-to-noise ratio (NSNR). Next, we show that for Poisson cellular networks, the SINR MD can be written as g(x)θ-2/α when the target SINR θ and the target reliability x jointly satisfy a constraint. We derive this constraint and the integral of g(x). Lastly, we discuss several extensions of the results to more general models and architectures.