The distribution of intervals between zeros of a stationary random function

Abstract

The probability density P _m of the spacing between the i th 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 r . It is shown that for fixed values of r , the density P _m ( r ) 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 ( 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 then P _m ( r ) is of order r ^{1/2 (m+2)(m+3)-2} at the origin, and in particular P ( r ) is of order r (adjacent zeros have a strong mutual repulsion). The first two terms of the series give the value of P ₀ ( t ) correct to r ¹⁸ . (2) In a singular case, the covariance function p( 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 ( r ) is shown to tend to a positive value P _m (0) as r -> 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 th at P ₀ (0) is in the neighbourhood of 1.155^ ^m ( - 6^ ') . The conjecture of Favreau, Low & Pfeffer (1956) according to which in one case P ₀ ( t ) 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.