THE COMPLEX STATISTICS PARADIGM AND THE LAW OF LARGE NUMBERS

What is it about?

The five basic axioms of Kolmogorov define the probability in the real set R and do not take into consideration the imaginary part which takes place in the complex set C, a problem that we are facing in applied mathematics. Whatever the probability distribution of the random variable in R is, the corresponding probability in the whole set C equals always to one, so the outcome of the random experiment in C can be predicted totally. This is the consequence of the fact that the probability in C is got by subtracting the chaotic factor from the degree of our knowledge of the system. In this study, I will evaluate the complex random vectors and their resultant that represents the whole distribution and system in the complex space C. I will also define imaginary and complex expectations and variances and I will prove the law of large numbers using the concept of the resultant complex vector. In fact, after extending Kolmogorov’s system of axioms, the new axioms encompass the imaginary set of numbers and this by adding to the original five axioms of Kolmogorov an additional three axioms. Hence, the concept of complex random vector becomes clear, evident and it follows directly from the new axioms added. This result will be elaborated throughout this study using discrete probability distributions. Moreover, any experiment executed in the complex set C is the sum of the real set R and the imaginary set M. Therefore, the whole probability distribution of random variables can be represented totally by the resultant complex random vector Z that is used subsequently to prove the very well known law of large numbers. In addition to my previous first paper, this second one elaborates the new field of “Complex Statistics” that considers random variables in the complex set C. Thus, the law of large numbers proves that this complex extension is successful and fruitful.

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Why is it important?

All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, the probability theory is a nondeterministic theory by nature; that means that the outcome of the events is due to chance and luck. By adding new dimensions to the event in R, we make the work deterministic and hence a random experiment will have a certain outcome in the complex set of probabilities C. It is of great importance that the stochastic system, like the problem considered here, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature for example in statistical mechanics or in all stochastic processes. Therefore, the work that should be done is to add to the real set of probabilities R, the contributions of M which is the imaginary set of probabilities which will make the event in C = R +M deterministic. If this is found to be fruitful, then a new theory in statistical sciences and prognostic is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This is what I called ‘The Complex Probability Paradigm (CPP)’, which was initiated and elaborated in my previous papers.

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