RENEWAL AGING IN NON-HOMOGENEOUS POISSON PROCESSES WITH PERIODIC RATE MODULATION

Abstract

The statistical properties of sequences of Waiting Times generated by Non-Homogeneous Poisson Processes are investigated by means of Renewal Aging, i.e. a statistical analysis being able to detect the presence of genuine renewal events with non-Poisson statistics. The Renewal Aging features of two non-homogeneous Poisson models with different prescriptions of the rate modulation are compared. Both prescriptions are essentially periodic. In the first model, the rate is a totally deterministic and smooth harmonic function. In the second model, the rate modulation is deterministic and smooth almost everywhere, except in some singular points occurring with a periodicity affected by weak random fluctuations. The main parameter of both models is the modulation speed, defined as the ratio between the modulation period and the average Waiting Time. This allows to distinguish between slow, intermediate and fast modulation of the rate. As expected, in the very slow modulation case both models have a strongly reduced Renewal Aging, practically zero, because the analysis is affected by local clusters of Waiting Times with the same Poisson rate. In the intermediate range of modulation speeds, both models show non-zero Renewal Aging and the behavior of the two models tends to be more different as the modulation speed decreases. Going towards the fast modulation regime, the behavior of the two models becomes completely different from one another. Model I does not show a definite global Renewal Aging. On the time scale of the modulation period, local oscillations arise in the Probability Density Function and an apparent oscillating aging is also observed. However, this form of aging is not the manifestation of genuine renewal non-Poisson events, but it is related to the local oscillations. On the contrary, Model II displays Renewal Aging features in agreement with a homogeneous Renewal non-Poisson Process. It is argued that this is related to the singular points in the rate modulation of Model II, which, due to their random occurrence, could be identified with genuine critical non-Poisson events. The approach presented here could have interesting applications in problems involving dichotomous noise and, in particular, in the statistical characterization of ON–OFF fluorescence intermittency experimentally observed in complex systems, such as nano-crystals and single bio-molecules stimulated by laser fields.