Krasnitskiy, S. M.

Silvestrov, D. S.

Electronic Repository Kyiv National University of Technologies and Design

III Міжнародна науково-практична конференція «Мехатронні системи: інновації та інжиніринг» Мехатронні інформаційні пристрої та системи  113 UDC 519.21 CONDITION OF ORTHOGONALITY OF PROBABILISTIC MEASURES CORRESPONDING TO GAUSSIAN GENERALIZED RANDOM PROCESSES WITH INDEPENDENT VALUES S.M. Krasnitskiy, Doctor of Science, Professor Kyiv National University of Technology ang Designe D.S. Silvestrov, Doctor of Science, Professor Stockholm University Keywords: generalized Gaussian random processes, generalized processes with independent values, orthogonality (singularity) of probabilistic measures.  Statistical analysis systems subject to random perturbations action substantially uses information about the relationship between probability measures in certain statistical structures [1] corresponding to the random functions describing perturbation mentioned [2]. In particular, the Gaussian functions are possible only their equivalence or orthogonality (a singularity). In the latter case, the aforementioned measures comply with incompatible observable events that makes it possible to uniquely identify the current stochastic perturbation.  In this paper we present the orthogonality condition of probabilistic measures corresponding to the generalized Gaussian random processes with independent values in terms of the covariance functions of these processes. Under the generalized random process = () we mean the set of real random variables  indexed by the elements of a linear space A offunctions on the real line R, which introduced a topology or convergence. It is supposed that the correspondence is linear with probability 1, and continuous. The latter means that for (in the sense of convergence in A) we have the convergence (in one or another probabilistic sense).HereA is the set of infinitely differentiable functions on having compact carriers in the interval (0,1). (For the convergence of sequences in such spaces seefor example [3]). All finite-dimensional distributions of these processes assumed to be normal (Gaussian). Independence of process valuesmeans that the random variables are independent, if it’s supports do not have a common interior points. Such processes play an important role, for example, in the theory of stochastic differential equations A number of properties of these processes are given in [2, 3].Next E is a symbol of expectation. According to [3], the functional covariance   of the process is represented as  III Міжнародна науково-практична конференція «Мехатронні системи: інновації та інжиніринг» Мехатронні інформаційні пристрої та системи  114 where are continuous functions, the number of terms is finite.Further (without loss of generality) the order of the differential form in (1), i.e. value is considered an even number.  The main result of the work is the following statement Theorem. Let be a statistical structure. Suppose that, with respect to probability measures , the process  (φ), φ∈D((0,1)) is Gaussian with zero mean values and the corresponding  covariance functions admitting (according to (1))    Then if , then the measures and  are orthogonal. If , then and are orthogonal when the inequality takes place  Note that in the case when the functions are constant values, the above theoremgives also necessary conditions for the orthogonality of the measures and .This follows from the results of [4,5]. References 1. Соле Ж.Л. Основные структуры математической статистики / Ж.Л. Соле. – М.: Мир, 1972. – 102 с.  2. Rozanov, Yu.A.: Random Fields and Stochastic Partial Differential Equations. Nauka,Moscow  (1995). 3. Gelfand, I.M., Vilenkin, N.Ya.: Applications of Harmonic Analysis. Equipped Hilbert Spaces, Fizmatgiz, Moscow (1961). 4. S.M.Krasnitskiy. On the necessary and sufficient conditions for the equivalence of probability measures corresponding to homogeneous random fields having spectral densities. - Cybernetics and system analysis. - 1997. - №5. - P. 37 - 44. 5. Springer Proceedings in Mathematics & Statistics book series, V. 271 / Stochastic Processes and Applications, S. Silvestrov et. al. (eds): book / Chapter 4.  On Baxter Type Theorems for Generalized Random Gaussian Fields – S. Krasnitskiy and O. Kurchenko, 91-103/ Springer International Publishing, 2018. – 475 p. 

Condition of orthogonality of probabilistic measures corresponding to gaussian generalized random processes with independent values

https://er.knutd.edu.ua/bitstream/123456789/14620/1/MSIE2019_P113-114.pdf

Condition of orthogonality of probabilistic measures corresponding to gaussian generalized random processes with independent values

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