Monotone Nonincreasing Random Fields on Partially Ordered Sets. II. Probability Distributions on Polyhedral Cones

In this part of the paper, we investigate the structure of an arbitrary measure μ supported by a polyhedral cone C in R ^d in the case where the decumulative distribution function g_μ of the measure μ satisfies certain continuity conditions. If a face Γ of the cone C satisfies appropriate conditions, the restriction μ|_Γint of the measure μ to the interior part of Γ is proved to be absolutely continuous with respect to the Lebesgue measure λ_Γ on the face Γ. Besides, the density of the measure μ|_Γint is expressed as the derivative of the function g_μ multipied by a constant. This result was used in the first part of the paper to find the finite-dimensional distributions of a monotone random field on a poset. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 5–56.