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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Extra resources for Applied methods of the theory of random functions

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K(T) FIG. 4. 30 RANDOM FUNCTIONS K(x) FIG. 5. Functions of the form shown in Fig. 21) and a function of the form shown in Fig. 5 by the expression K(r) ^ o2e~« ITI cos £T or by the expression K(r) ^ a2e-^z cos &r. 23) In what follows we shall find it necessary to approximate the correlation function by a sufficiently simple analytic ex­ pression when solving a number of problems; the accuracy of the approximation required will be discussed. In conclusion we note that the addition to a random func­ tion of a non-random variable or non-random function does not change the value of the correlation function.

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. 13') obtained from (13) b y passing to the limit as /? -+ 0 also corresponds to a differentiable process. 38 RANDOM FUNCTIONS Let us continue the study of the derivative of a random function. If X(t) is differentiable then its derivative in its turn is a random time function, possessing its own multi­ dimensional distribution law and its own correlation func­ tion.

The result of the application of the oper­ ator to the sum of functions is equal to the sum of the results of the application of this operator to each term separately, that is, if cpjjjt) and y2{t) are arbitrary functions, then H

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