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

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**Extra resources for Applied methods of the theory of random functions**

**Sample text**

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