Formulating Discrete Geometric Random Sums for Facilitating Intelligent Behaviour of a Complex System under a Major Risk

Abstract

The contribution of random sums as strong analytical tools of many areas of probability theory is generally recognized as very important. Infinitely divisible distributions, counting stochastic processes, stochastic integrals, service systems, stochastic processes with stationary and independent increments and branching processes are significant areas of probability theory making extensive use of random sums. Moreover, economics, management, insurance, reliability, quality control and engineering are examples of practical disciplines utilizing random sums as powerful stochastic models. The present paper formulates a geometric random sum of discrete, independent and identically distributed random variables. A stochastic derivation of such a random sum is also provided. Moreover, the paper establishes an interpretation of the formulated discrete geometric random sum as a research tool of computational intelligence for describing and analyzing the evolution of a complex system under the occurrences of a major risk. The paper makes quite clear that research activities on the formulations, stochastic derivations and practical applications of discrete random sums can substantially facilitate the use of computational intelligence principles and methodologies in investigating the structure and evolution of complex systems operating under the negative consequences of a severe risk.