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A systematic approach to a class of problems in the theory of noise and other random phenomena--II: Examples

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The method of Part I is applied to the problem of finding the probability distribution ofu equiv int_0^t K(tau)x^2(tau) dtau, whereK(tau)is a given function andx(tau)is the Uhlenbeck process. The earlier methods of Kac and the author yielded the characteristic function of this distribution as the reciprocal square root of the Fredholm determinant D of an integral equation. The present method yields a second-order linear differential equation with initial condition only for D as function oft. For the special casesK(tau) = 1andK(tau)= e^{-alpha tau}the characteristic function is obtained in closed form. In Section III, we have verified directly from the integral equation the differential equation for D and some relations between D and the initial and end point values of the Volterra reciprocal kernel which appear in the joint characteristic function foru, x(0) and x(t).

Published in:

Information Theory, IRE Transactions on  (Volume:3 ,  Issue: 1 )

Date of Publication:

March 1957

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