Theoretical discussion and Monte-Carlo simulations for a Negative Binomial process paradox

The idea of an over-threshold sampling is to retain all the events of a time-series exceeding a given threshold. The probabilistic analysis implies estimating two statistical models, one describing the occurrence of the events (date of the events), the other describing their magnitude (value of the local maximum). These two models are then combined to obtain the distribution of the annual maxima. A well-known result of a Poisson process is that waiting time, defined as the duration between two successive events exceeding the threshold, is exponentially distributed. The assertion that the waiting time of a Negative Binomial process is also exponentially distributed seems to be in obvious contradiction with the Poisson process properties. A theoretical discussion and Monte-Carlo simulations are presented to solve this apparent paradox.