Palm Measure Duality and Conditioning in Regenerative Sets

Abstract

For a simple point process $\Xi$ on a suitable topological space, the associated Palm distribution at a point s may be approximated by the conditional distribution, given that $\Xi$ hits a small neighborhood of $s$. To study the corresponding approximation problem for more general random sets, we develop a general duality theory, which allows the Palm distributions with respect to an associated random measure to be expressed in terms of conditional densities with suitable martingale and continuity properties. The stated approximation property then becomes equivalent to a certain asymptotic relation involving conditional hitting probabilities. As an application, we consider the Palm distributions of regenerative sets with respect to their local time random measures.