By Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich
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27) -cp(t) = x(t). 29) Let us p u t Then t h e function which has zero mathematical expectation, will possess t h e same correlation function as X(t). Accordingly, in t h e deriva tion of the general properties of correlation functions it is possible without loss of generality to consider the mathemat ical expectation of the random function as equal to zero. We will make extensive use of this fact later in order to shorten proofs. § 6. T H E DIFFERENTIATION AND INTEGRATION OF RANDOM FUNCTIONS We introduce t h e concept of t h e derivative of a random function.
In this case the function K(t) giving the value of the covariance between X(t2) and X(^) will tend to zero as | r | — oo. Hence K(t) may either decrease monotonically as represented in Fig. 4 or be of the form represented in Fig. 5. 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.
K(D _ "" -cr'oc FIG,. 6. "" 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.