Conditions for the emergence of scaling in the inter-event time of uncorrelated and seasonal systems

Inter-event times have been studied across various disciplines in search for correlations. In this paper, we show analytical and numerical evidence that at the population level a power-law can be obtained by assuming Poissonian agents with different characteristic times, and at the individual level by assuming Poissonian agents that change the rates at which they perform an event in a random or deterministic fashion. The range in which we expect to see this behavior and the possible deviations from it are studied by considering the shape of the rate distribution.