The Paradigm of Complex Probability and Markov Chains, Edition 1

What is it about?

The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Hence, we will apply this novel paradigm to Markov Chains Theory in order to express it totally and absolutely deterministically in the complex universe C of probabilities.

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Why is it important?

Calculating probabilities is the main task of the theory of classical probability. In fact, if we add new dimensions to a random phenomenon it will result to a deterministic expression of the theory of probability. This is the original and novel idea at the foundations of “The Complex Probability Paradigm (or CPP for short)”. As a matter of fact, probability theory is a nondeterministic theory in its core; that means that the outcomes of events are due to chance and randomness. If we add imaginary and new dimensions to a random experiment occurring in the set R it will result to a deterministic experiment and thus a nondeterministic phenomenon will have a certain outcome in the complex probability set C. If the random event becomes completely predictable then we will have perfect knowledge to predict the outcome of random experiments that arise in the real world in all random processes. Consequently, the work that has been accomplished in CPP was to extend the set R of real probabilities to the set C = R + M of deterministic complex probabilities by incorporating the contributions of the set M which is the imaginary probabilities set. Therefore, because this extension was found to be fruitful, then a novel paradigm of prognostic and nondeterministic sciences was established in which all random phenomena in R was defined deterministically. I called this original model "the Complex Probability Paradigm" that was initiated and illustrated in my numerous earlier research publications.

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