Iterated Palm Conditioning and Some Slivnyak-Type Theorems for Cox and Cluster Processes

A huge literature is devoted to extensions and ramifications of Slivnyak’s celebrated theorem—the fact that a point process on a Borel space is Poisson if and only if its reduced Palm distributions agree almost everywhere with the original distribution. Here we prove some multivariate versions of the mentioned property for randomizations and Cox processes. This leads to Slivnyak-type theorems for certain Cox cluster processes, arising naturally both in exchangeability theory and in the context of superprocesses. Our proofs are based on a general iteration principle, which enables us to calculate Palm measures via a preliminary conditioning.