The axis-crossing intervals of random functions

Abstract
For an arbitrary random process\xi(t)there exists a functionx(t)which may be obtained by infinite clipping. The axis crossings ofx(t)are identical with those of\xi(t). This paper relates the probability densityP(\tau)of axis-crossing intervals to\gamma(\tau), the autocorrelation function ofx(t), i.e., the autocorrelation after clipping. It is shown that the expected number of zeros per unit time is proportional to\gamma \prime (0+), i.e., the right-hand derivative of\gamma (\tau)at\tau = 0. Next a theorem is proved, stating thatP(\tau) = 0over a finite range0 \leq \tau < Tif and only if\gamma(\tau)is linear in\mid \tau \midover the corresponding range of\mid \tau \mid. If\gamma (\tau)is nearly linear for small\tau, then the initial behavior ofP(\tau)is like\gamma \prime \prime (\tau). These results are illustrated by some simple, random square-wave models and by a comparison with Rice's results for Gaussian noise.

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