Applied methods of the theory of random functions by Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich

By Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich

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DK(T) dx 37 GENERAL PROPERTIES FIG. 7. On t h e other hand, the first derivatives of the functions a2e~*lxi and a2e~*lz* cos /^r remain continuous for any values of r and accordingly, correlation functions of this t y p e correspond to differentiable random processes. As an additional example of a correlation of a differen­ tiable process consider the expression y (6. 13) o2e-« I TI sin /? 14) a2e~a ITI ( cos j5r+-5-sin/S| r\ K(r) the first derivative of which dK(x) dx a 2 + /S2 P is continuous a t zero and consequently the second derivative exists a t this point.

In the particular case when the function p(t, fx) is a function of the difference of its arguments, that is, P(*,h) =P(*-h)> formula (8) reduces to the form t Y(t) = jpit-tJXtfJdh. 10) to The importance of operators of this form is due to the fact that the finding of the particular integral of a linear homo­ geneous equation the right-hand side of which is the func­ tion X(t) reduces to this. The operator (8) corresponds to the solution of a linear differential equation with variable coef­ ficients and an operator of the form (10) to the solution of a differential equation with constant coefficients.

The nonhomogeneity of the operator means adding an additional term to the result of applying a homogeneous linear operator. This addition does not affect the value of the correlation function and in finding the mathematical expectation it must be taken into account b y additional terms. Hence (33) and (34) obtained for a linear homogeneous operator also apply to a linear nonhomogeneous operator. Therefore the mathematical expectation and correlation function of the result of applying a linear operator to a ran­ dom function are uniquely determined by its mathematical expectation and correlation function irrespective of the na­ ture of the multi-dimensional density distribution laws of this function.

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