Informatics and Applications

December 2013, Volume 7, Issue 4, pp 11-19

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

  • M. E. Grigor’eva
  • V. Yu. Korolev
  • I.A. Sokolov

Abstract

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 amarked point stochastic processwith 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 proposedmethods is illustrated by the example of their application to the problemof 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.

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