The Paradigm of Complex Probability and The Central Limit Theorem

What is it about?

The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the “real” laboratory in R and therefore to compute all the probabilities in the sets R, M, and C. Accordingly, the probability in the whole set C = R + M is constantly equivalent to one independently of the distribution of the input random variable in R, and subsequently the output of the stochastic experiment in R can be determined absolutely in C. This is the consequence of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. We will apply this innovative paradigm to the well-known Central Limit Theorem and to prove as well its convergence in a novel way.

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Why is it important?

Computing probabilities is all our work in the classical theory of probability. Adding new dimensions to our stochastic experiment is the innovative idea in the current paradigm which will make the study absolutely deterministic. As a matter of fact, the theory of probability is a nondeterministic theory by essence that means that all the random events outcome is due to luck and chance. Hence, we make the study deterministic by adding new imaginary dimensions to the phenomenon occurring in the “real” laboratory which is R, and therefore, a stochastic experiment will have a certain outcome in the complex probabilities set C. It is of great significance that random systems become completely predictable since we will be perfectly knowledgeable to predict the outcome of all stochastic and chaotic phenomena that occur in nature like for example in all stochastic processes, in statistical mechanics, or in the well-established field of quantum mechanics. Consequently, the work that should be done is to add the contributions of M which is the set of imaginary probabilities to the set of real probabilities R that will make the random phenomenon in C = R + M completely deterministic. Since this paradigm is found to be fruitful, then a new theory in prognostic and stochastic sciences is established and this is to understand deterministically those events that used to be stochastic events in R. This is what I coined by the term “The Complex Probability Paradigm” that was elaborated and initiated in my 25 previous papers.

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