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

Author

Siegert, A.

Volume

3

Issue

1

fYear

1957

fDate

3/1/1957 12:00:00 AM

Firstpage

38

Lastpage

43

Abstract

The method of Part I is applied to the problem of finding the probability distribution of $u \\equiv \\int_0^t K(\\tau )x^2(\\tau ) d\\tau$ , where $K(\\tau )$ is a given function and $x(\\tau )$ 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 of $t$ . For the special cases $K(\\tau ) = 1$ and $K(\\tau )$ $= e^{-\\alpha \\tau }$ 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 for $u, x(0) and x(t)$ .

Keywords

Markov processes; Noise; Probability functions; Autocorrelation; Detectors; Differential equations; Gaussian processes; Information theory; Integral equations; Kernel; Noise reduction; Nonlinear equations; Partial differential equations; Physics; Probability distribution; Receivers;

fLanguage

English

Journal_Title

Information Theory, IRE Transactions on

Publisher

ieee

ISSN

0096-1000

Type

jour

DOI

10.1109/TIT.1957.1057391

Filename

1057391