On the convergence of point processes
Let \(N^n\) be a counting process with compensator \(A^n\), \(n\in \mathbb N\), and \(N\) a counting process with compensator \(A=A(N)\). It is well-known that (under certain continuity assumptions on \(A\)) the condition \[ \text{(a)}\quad P-\lim_n(A^n_t-A_t(N^n))=0\quad \text{ for all } t\text{ implies } \text{(b)}\quad N^n\to N\quad (\text{in law}) \] but (b) does not imply (a), even for \(A_ t=t\). It is shown here that (b) is equivalent to another condition on convergence of the compensators. This looks much more complicated and involves weak convergence in \(L^1\) of their values in certain stopping times.
scientific article; zbMATH DE number 4064174
scientific article