The distribution of intervals between zeros of a stationary random function

Abstract
The probability densityPmof the spacing between theith zero and the (i+m+ 1)th zero of a stationary, random functionf(t) (not necessarily Gaussian) is expressed as a series, of a type similar to that given by Rice (1945) but more rapidly convergent. The partial sums of the series provide upper and lower bounds successively forPm. The series converges particularly rapidly for small spacingsr. It is shown that for fixed values ofr, the densityPm(r) diminishes more rapidly than any negative power ofm. The results are applied to Gaussian processes; then the first two terms of the series forPm(T) may be expressed in terms of known functions. Special attention is paid to two cases: (1) In the ‘regular’ case the covariance function i/r(t) is expressible as a power series in thenPm(r) is of order r1/2(m+2)(m+3)-2at the origin, and in particularP(r) is of orderr(adjacent zeros have a strong mutual repulsion). The first two terms of the series give the value ofP0(t) correct tor18. (2) In a singular case, the covariance function p(t) has a discontinuity in the third derivative. This happens whenever the frequency spectrum off(t) isO(frequency)-4at infinity. ThenPm(r) is shown to tend to a positive valuePm(0) as r -> 0 (neighbouring zeros are less strongly repelled). Upper and lower bounds forPm(0) (m= 0, 1, 2, 3) are given, and it is shown th atP0(0) is in the neighbourhood of 1.155^m( - 6^ ') . The conjecture of Favreau, Low & Pfeffer (1956) according to which in one caseP0(t) is a negative exponential, is disproved. In a final section, the accuracy of other approximations suggested by Rice (1945), McFadden (1958), Ehrenfeldet al. (1958) and the present author (1958) are compared and the results are illustrated by computations, the frequency spectrum off(t) being assumed to have certain ideal forms: a low-pass spectrum, band-pass spectrum, Butterworth spectrum, etc.

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