Stochastics Seminar

Speaker: Olav Kallenberg

Title: Inner and outer conditioning in point processes

Abstract: General Palm measures extend the classical notion of regular conditional distributions to the context of point processes. Thus, for any pair of random variables \(X\) and \(Y\), we may form the conditional distributions \({\cal L}(Y|X)\) and \({\cal L}(X|Y)\) by dual disintegrations of the joint distribution \({\cal L}(X,Y)\) with respect to the marginals \({\cal L}(X)\) and \({\cal L}(Y)\). In a similar way, given a point process \(\xi\) on a Borel space, we may form the kernel of associated Palm measures by disintegration of the compound Campbell measure \(C_\xi\). Here the dual disintegration gives rise to the Gibbs kernel, of significance in statistical mechanics, and both kernels play fundamental roles in general random measure theory. In this talk, my modest aim is to highlight some crucial underlying ideas, avoiding when possible the technical and mathematical subtleties of the subject.