  In applications spaning from image analysis and speech recognition, to energy
dissipation in turbulence and time-to failure of fatigued materials,
researchers and engineers want to calculate how often a stochastic observable
crosses a specific level, such as zero. At first glance this problem looks
simple, but it is in fact theoretically very challenging. And therefore, few
exact results exist. One exception is the celebrated Rice formula that gives
the mean number of zero-crossings in a fixed time interval of a zero-mean
Gaussian stationary processes. In this study we use the so-called Independent
Interval Approximation to go beyond Rice's result and derive analytic
expressions for all higher-order zero-crossing cumulants and moments. Our
results agrees well with simulations for the non-Markovian autoregressive
model.
