We study higher-order moment measures of heavy-tailed renewal models, including a renewalpoint process with heavy-tailed inter-renewal distribution and its continuous analog, theoccupation measure of a heavy-tailed Lévy subordinator. Our results reveal that theasymptotic structure of such moment measures are given by explicit power-law densityfunctions. The same power-law densities appear naturally as cumulant measures of certainPoisson and Gaussian stochastic integrals. This correspondence provides new and extendedresults regarding the asymptotic fluctuations of heavy-tailed sources under aggregation,and clarifies existing links between renewal models and fractional random processes.
