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 ( r ) ( n = 3, 4, 5, ...) to the desired probability density p ( r ). The third approximation p 3 is already correct to order r 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 ( r ), less accurate for small values of r , but possibly more accurate for larger values. Numerical computation of both p 3 , p 4 , p 5 and p * 3 , p * 4 , p * 5 is carried out for a low-pass spectrum, and the results are compared with observation.