Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions by A. A. Sveshnikov

Author:A. A. Sveshnikov
Language: eng
Format: epub
Publisher: Dover Publications
Published: 1968-03-14T16:00:00+00:00

where ajl are constants, Xj(t) are stationary random functions and time t is sufficiently large. Its solutions are stationary random functions, whose spectral densities and mutual spectral densities can be expressed in terms of the spectral densities and mutual spectral densities of the right-hand sides of the equations as follows:

Here Δ(ω) is the determinant formed from the coefficients appearing on the left-hand sides of the equations :

where Aij(ω) is the cofactor of the element located at the intersection of the ith row and the jth column, and Sxjxj(ω) ≡ Sxj(ω).

The distribution law for the solution of a linear equation (system of linear equations), whose right-hand side contains normal random functions and variables, is also normal. If the equation is linear but the distribution law of the random functions on the right-hand side is not normal, the distribution law for the solution also will not be normal. The expectation and the central moments μj of this distribution law for any t are determined by the formulas

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