Monotone nonincreasing random fields on posets. II.~Probability distributions on polyhedral cones

L. B. Beinenson

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 5-56

Résumé

In this part of the paper we investigate the structure of an arbitrary measure $\mu$ concentrated on a polyhedral cone $C$ in $\mathbf{R}^d$ in the case when the decumulative distribution function $g_\mu$ of the measure $\mu$ satisfies certain continuity conditions. If a face $\Gamma$ of the cone $C$ satisfies appropriate conditions, the restriction $\mu|_{\Gamma^{\operatorname{int}}}$ of the measure $\mu$ to the inner part of $\Gamma$ is proved to be absolutely continuous with respect to the Lebesgue measure $\lambda_\Gamma$ on the face $\Gamma$. Besides, the density of the measure $\mu|_{\Gamma^{\operatorname{int}}}$ is expressed as a derivative of the function $g_\mu$ 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.