The distribution of intervals between zeros of a stationary random function

The probability density P$_{m}$ of the spacing between the ith zero and the (i+m+1)th zero of a stationary, random function f(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 for P$_{m}$. The series converges particularly rapidly for small spacings $\tau $. It is shown that for fixed values of $\tau $, the density P$_{m}$($\tau $) diminishes more rapidly than any negative power of m. The results are applied to Gaussian processes; then the first two terms of the series for P$_{m}$($\tau $) may be expressed in terms of known functions. Special attention is paid to two cases: (1) In the 'regular' case the covariance function $\psi $(t) is expressible as a power series in t; then P$_{m}$($\tau $) is of order $\tau ^{\frac{1}{2}(m+2)(m+3)-2}$ at the origin, and in particular P ($\tau $) is of order $\tau $ (adjacent zeros have a strong mutual repulsion). The first two terms of the series give the value of P$_{0}$($\tau $) correct to $\tau ^{18}$. (2) In a singular case, the covariance function $\psi $(t) has a discontinuity in the third derivative. This happens whenever the frequency spectrum of f(t) is O (frequency)$^{-4}$ at infinity. Then P$_{m}$($\tau $) is shown to tend to a positive value P$_{m}$(0) as $\tau \rightarrow $ 0 (neighbouring zeros are less strongly repelled). Upper and lower bounds for P$_{m}$(0) (m = 0, 1, 2, 3) are given, and it is shown that P$_{0}$(0) is in the neighbourhood of 1$\cdot $155$\psi ^{\prime \prime \prime}$/(-6$\psi ^{\prime \prime}$). The conjecture of Favreau, Low & Pfeffer (1956) according to which in one case P$_{0}$($\tau $) is a negative exponential, is disproved. In a final section, the accuracy of other approximations suggested by Rice (1945), McFadden (1958), Ehrenfeld et al. (1958) and the present author (1958) are compared and the results are illustrated by computations, the frequency spectrum of f(t) being assumed to have certain ideal forms: a low-pass spectrum, band-pass spectrum, Butterworth spectrum, etc.