On the intervals between successive zeros of a random function

Abstract

A new approach is suggested to the problem of the statistical distribution of the intervals between successive zeros of a random, Gaussian function. Hence is derived a sequence of approximations p$_{n}$($\tau $) (n = 3, 4, 5,...) to the desired probability density p($\tau $). The third approximation p$_{3}$ is already correct to order $\tau ^{4}$, and has the correct limiting form in the case of a narrow spectrum. The analysis also gives rise to an alternative approximation p$_{n}^{\ast}$($\tau $), less accurate for small values of $\tau $, but possibly more accurate for larger values. Numerical computation of both p$_{3}$, p$_{4}$, p$_{5}$ and p$_{3}^{\ast}$, p$_{4}^{\ast}$, p$_{5}^{\ast}$ is carried out for a low-pass spectrum, and the results are compared with observation.