Fundamentals of the Markov Processes Theory
The previous sections dealt with the correlation theory of random functions and only the first two moments of a random function — expectation and the correlation function has been considered. Unfortunately, far from all encountered applied problems can be solved by correlation theory methods. A case in point is the problem of determining the probability that the ordinate of a random function will exceed a particular given value, which often arises during the dynamic systems analysis. These problems become solvable if we restrict their treatment to processes not only possessing some special properties, but also interesting in the practical plane. Up to this point we have used correlation theory methods to analyze systems with a linear input-output relation. In this case the correlation theory enables us to obtain the probability characteristics of the differential equations solution, knowing the probability characteristics of perturbations. It is impossible to find a solution of nonlinear equations by correlation theory methods. However, if we confine ourselves to the processes possessing some special properties, we can obtain a solution of nonlinear problems of statistical dynamics. Markov processes, for the exhaustive characterization of which it is sufficient to know only two-dimensional distribution laws, are classified among such processes.
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