MOMENT MEASURES OF HEAVY-TAILED RENEWAL POINT PROCESSES: ASYMPTOTICS AND APPLICATIONS

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