Monotone nonincreasing random fields on partially ordered sets. I (Q937856)

Given any poset \(H\) and a due finitely additive positive measure \(\rho\) on \(H\times \mathbb{R}\), the (part I of the) paper provides a real random field \(\eta_{\rho}\) on \(H\) such that \(\eta_{\rho}(a) \geq \eta_{\rho}(b)\) whenever \(a<b\) in \(H\). The construction involves that of measures on ideals of posets by \textit{M. A. Antonets} and \textit{I. A. Shereshevsky} [St. Petersbg. Math. J. 5, No.~6, 1075--1097 (1994); translation from Algebra Anal. 5, No.~6, 39--68 (1993; Zbl 0833.60017)]. Moreover, under certain conditions on \(H\) and \(\rho\) the finite dimensional distributions (fidis) of the field \(\eta_{\rho}\) are computed along with the mean, variance and correlation function. Thus a method of producing positive definite functions on \(H\) arises. The study of probability measures supported by polyhedral cones in \(\mathbb{R}^d\) pertinent to description of the fidis is placed in part II [ibid. 131, No. 2, 5445--5470 (2005; Zbl 1102.60047)].