A regression approximation of wavelength and amplitude distribution in an almost surely continuous process η (t), is based on a successively more detailed decomposition, η (t) = η n(t) + Δn(t), into one regression term η n on n suitably chosen random quantities, and one residual process Δn. The distances between crossings, maxima, etc., are then approximated by the corresponding quantities in the regression term, and explicit expressions given for the densities of these quantities.

We study the sample path behaviour of χ2 processes in the neighbourhood of their level crossings and extrema via the development of Slepian model processes. The results, aside from being of particular interest in the study of χ2 processes, have a general interest insofar as they indicate which properties of Gaussian processes (which have been heavily researched in this regard) are mirrored or lost when the assumption of normality is not made. We place particular emphasis on the behaviour of χ2 processes at both high and low levels, these being of considerable practical importance. We also extend previous results on the asymptotic Poisson form of the point process of high maxima to include also low minima (which are in a different domain of attraction) thus closing a gap in the theory of χ2 processes.