In this paper we study ideals arising from moments of local Dirac measures and their mixtures. We provide generators for the case of first order local Diracs and explain how to obtain the moment ideal of the Pareto distribution from them. We then use elimination theory and Prony's method for parameter estimation of finite mixtures and showcase our results with an application in signal processing. We highlight the natural connections to algebraic statistics, combinatorics and applications in analysis throughout the paper.
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