Palm measure duality and conditioning in regenerative sets. (English) Zbl 0953.60037

For a simple point process \(\Xi\) on a suitable topological space, the associated Palm distribution at a point \(s\) is a limit of the conditional distribution given that \(\Xi\) intersects a neighborhood \(I\) of \(s\), as \(I\) shrinks to \(s\). The author studies the corresponding approximation problem for more general random sets \(\Xi\), which, in particular, do not need to be locally finite. For this purpose he shows that the corresponding Palm distributions can be expressed in terms of certain conditional densities. These possess martingale properties which allow to treat the approximation problem successfully as a limiting problem involving conditional hitting probabilities. As an extensive application of this dual approach, Palm distributions of regenerative sets with respect to their local time random measures are studied.


60G57 Random measures
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