The second-order distribution of integrated shot noise

Shot noise, represented by a series of impulses with Poisson distribution in time, and with arbitrary time-independent amplitude distribution, is sent through an RC integrator. Time-dependent statistics of the output are investigated by means of an integro-differential equation describing the statistical flow. The exact second-order probability density of the output is obtained. The time-dependent Edgeworth series for zero initial output is exhibited, and is seen to bear a simple relation to the familiar equilibrium Edgeworth series. Results are shown to reduce to those of the Fokker-Planck equation describing integrated "white" noise as the frequency with which impulses arrive becomes infinite,