Suppose we have a sequence of marked point processes $N_n$ on the same filtration space, $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ with $\mathcal{F}_t$-predictable intensities $\lambda_n(t,k)$. To keep things simple, we can assume we are working on a finite time span, $[0,T]$.
If we know that $\lambda_n \Rightarrow \lambda$ for some $\mathcal{F}_t$-predictable process, what do we know about $\{N_n\}$? For example,
- Do we need some sort of assumption on $\mathcal{F}$? Does it need to be a natural history?
- Can we say $N_n \Rightarrow N$ if $N$ is a point process with intensity $\lambda$ and $\{N_n\}$ is tight?
- Do we know something else? For example, do the finite-dimensional distributions of $N_n$ converge to $N$?