Some local approximation properties of simple point processes

For simple point processes xi on a Borel space S, we prove some approximations involving conditional distributions, given that xi hits a small set B. Beginning with general versions of some classical limit theorems, going back to the pioneering work of Palm and Khinchin, we proceed to prove that, under suitable regularity conditions, the contributions to B and B-c are asymptotically conditionally independent. We further derive approximations in total variation of reduced Palm distributions and show that, when xi hits some small sets B-1,..., B-n , the corresponding restrictions are asymptotically independent. Next we give general versions of the asymptotic relations P{xi B > 0} similar to E xi B and prove some ratio limit theorems for conditional expectations E[eta vertical bar xi B > 0], valid even when E xi is not sigma-finite and the Palm distributions may fail to exist.