Propriety of the reference posterior distribution in Gaussian process modeling

In a seminal article, Berger, De Oliveira and Sansó [J. Amer. Statist. Assoc. 96 (2001) 1361–1374] compare several objective prior distributions for the parameters of Gaussian process models with isotropic correlation kernel. The reference distribution stands out among them insofar as it always leads to proper posterior. They prove this result rough kernels: Spherical, Exponential power ρ<2, Matérn smoothness ν<1. This paper provides proof smooth ρ=2, ν≥1, Rational Quadratic, along tail rates these kernels.