Improved criteria for distributional convergence of point processes

In earlier work by the author, the convergence in distribution of a sequence of point processes towards a simple limit has been characterized by three conditions, where the first ensures convergence of the supports, the second that the limit of every convergent subsequence is simple, and the third that the sequence is tight. The purpose of this note is to strengthen the mentioned result, by showing that the tightness follows from the other two conditions. Similar results are obtained for convergence of random measures towards a diffuse limit and of row-sums in null-arrays towards an infinitely divisible limit.