Some graphical considerations in time series analysis

The pictorial information in a stationary time series as depicted by crossings of levels and crossings of random levels and related quantities is studied. It is shown that such graphical features are directly connected with the covariance function and hence with the spectral density. Many of these features can be actually applied in estimation and in the study of extremes. In the Gaussian case, the finite dimensional distributions are completely determined by the axis crossings and by the crossings of a random curve (to be defined) if the process is essentially bounded. Certain graphical patterns are suggested for a fast recognition of low-order ARMA models.