REVERSAL FUNCTIONS AND PROBABILITY DISTRIBUTION OF THE PANDOM CROSS-FUNCTIONAL FROM NORMAL MARKOV PROCESS

Abstract

The process, which has the properties of stationary, normality and markovity, is considered. For a given time interval, an energy functional and a cross-functional are studied in this article. In analytic consideration of the problems of probability theory and mathematical statistics, the assumption is widespread that the problem in question has been resolved if the characteristic (generating) function is constructed. However, the operation of the inverse Laplace transform causes the computational difficulties. As a numerical procedure, the inverse Laplace transform is characterized by instability, the degree of which increases with the transformation parameter. An approach based on the application of the reverse functions was proposed and used, which made it possible to obtain an analytical expression for the generating function of the distribution of random values of the cross-functional in this article. The analysis of the statistical properties of the cross functional is carried out. The density of probability distribution and the integral distribution law are obtained numerically using the inverse Laplace transform for the selected values of the observation time , the decrement of the process and its intensity . The dependences of the density and distribution functions for given values of the parameters of functionals are given. It follows from the calculations that an increase in the parameter leads to the expansion of the values of the functional in the peripheral regions of large deviations. Reducing the parameter leads to the localization of the values of the cross-functional in the fluctuation domain . The density is symmetric with respect . It has a unique maximum, two points of inflection, and an exponential asymptotic behavior on the periphery.