A limit theorem for geometric sums of independent nonidentically distributed random variables and its application to the prediction of the probabilities of catastrophes in nonhomogeneous flows of extremal events

The problem of prediction of the probabilities of catastrophes in nonhomogeneous flows of extremal events is considered. The paper develops and generalizes some methods proposed by the authors in their previous works. The flow of extremal events is considered as a marked point stochastic process with not necessarily identically distributed intervals between points (events). The proposed generalizations are based on limit theorems for geometric sums of independent not necessarily identically distributed random variables and the Balkema–Pickands–De Haan theory. Within the framework of the construction under consideration, the Weibull–Gnedenko distribution appears as a limit law for geometric sums of independent not necessarily identically distributed random variables. The efficiency of the proposed methods is illustrated by the example of their application to the problem of prediction the time of the impact of the Earth with a potentially dangerous asteroid based on the data of the IAU (International Astronomical Union) Minor Planet Center.