Extrema and level crossings of χ2 processes

We study the sample path behaviour of χ 2 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 χ 2 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 χ 2 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 χ 2 processes.

[1]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[2]  D. Slepian,et al.  Large Excursions of Gaussian Processes , 1959 .

[3]  Eugene Lukacs,et al.  Applications of characteristic functions , 1964 .

[4]  S. Berman.  Stationary and Related Stochastic Processes , 1967 .

[5]  harald Cramer,et al.  Stationary And Related Stochastic Processes , 1967 .

[6]  G. Lindgren,et al.  Some Properties of a Normal Process Near a Local Maximum , 1970 .

[7]  Richard M. Dudley,et al.  Sample Functions of the Gaussian Process , 1973 .

[8]  K. Sharpe,et al.  SOME PROPERTIES OF THE CROSSINGS PROCESS GENERATED BY A STATIONARY X2 PROCESS , 1978 .

[9]  Extremal and related properties of stationary processes , 1979 .

[10]  G. Lindgren.  Model processes in nonlinear prediction with application to detection and alarm , 1980 .

[11]  Georg Lindgren,et al.  Point processes of exits by bivariate Gaussian processes and extremal theory for the chi^2-process and its concomitants , 1980 .

[12]  Georg Lindgren,et al.  Extreme values and crossings for the X 2-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications , 1980, Advances in Applied Probability.

[13]  Kenneth S. Miller.  An introduction to vector stochastic processes , 1980 .

[14]  R. Adler,et al.  The Geometry of Random Fields , 1982 .

[15]  S. Berman.  Sojourns of Stationary Processes in Rare Sets , 1983 .

[16]  Georg Lindgren,et al.  Use and Structure of Slepian Model Processes for Prediction and Detection in Crossing and Extreme Value Theory , 1984 .

[17]  Simeon M. Berman,et al.  Sojourns of vector Gaussian processes inside and outside spheres , 1984 .

[18]  R. J. Adler,et al.  BEHAVIOUR OF x2 PROCESSES AT EXTREMA , 1985 .