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

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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.

Hence for a system of random functions just as for a single function we usually restrict ourselves to the calculation of only the first two moments of the ordinates of these func tions, t h a t is, to the correlation theory of random functions. Since the first and second moments of each random function taken separately are the mathematical expectations and correlation functions considered above it remains to consider the second mixed moment of the ordinates of t h e different random functions t a k e n at distinct instants of time tx and t2.

4) that is, the random function X(t) is continuous. In the partic ular case of a stationary function it is sufficient for the continuity of X(t) that the function K(x) be continuous for r = 0. However, the continuity of the random function is still not sufficient for its differentiability since it is also necessary that the limit of the expression (2) should exist. For this limit to exist it is necessary for the ratios X(t + A1)^X(t) X(t + and A2)-X(t) to tend to one and the same limit as the increments A± and A2 tend to zero independently, that is, it is necessary for the condition HmMJl ^ « II + A)-X{t) _X(t A + A2)-X{t) in ** Ij to be satisfied.