Campbell’s theorem

The function h(t ) i(t + r)dt is often referred to as the autocorrelation function of the Amotion h(t) however, the reader should be careful to note the difference between the autocorrelation function of h(t)—an integrable function—and the autocorrelation function of Y(t)—a function that is not integrable because it does not die out in time. With this distinction in mind, Campbell s theorem can be expressed by saying that the autocovariance function of a shot noise process is n times the autocorrelation function of the function h(t). 

The last equality in 3-270 follows from Campbell s theorem. 

As a further application of the Wiener-Khinchine theorem, we shall now calculate the power density spectrum of the shot noise process. The autocorrelation function for such a process is given by Campbell s theorem, Eq. (3-262), repeated below... 

Campbell s Theorem, 174 Cartwright, M. L., 388 Caywood, T. E., 313 C-coefficients, 404 formulas for, 406 recursion relations, 406 relation to spherical harmonics, 408 tabulations of, 408 Wigner s formula, 408 Central field Dirac equation in, 629 Central force law... 

These equations are called Campbell s theorem , and we shall refer to this process as Campbell s process . 

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