I post this post en MSE (link) but I think that is more suitable for this site.
I'm studying from Kurtz's book "Markov Processes Characterization and convergence" and I have a question about the convergence of processes in $\mathbb{Z}^d$ that are solution of some equation. (see chapter 6 section 4).
Let $X^n$ be the solution of the equation
$$X^n(t) = X(0) + \sum_{l\in \mathbb{Z}^d}l\cdot N_l\left(\int_0^t \beta^n_l(X^n(s))ds \right)$$ where $\beta^n_l \colon \mathbb{Z}^d \to [0,\infty)$, $N_l$ are independent real valued poisson processes with rate parameter $1$ and $X(0)$ is a random variable independent from $(N_l)$.
Suppose that $\beta^n_l \stackrel{n\to \infty}{\longrightarrow} \beta_l$ pointwise and that $(\beta^n_l)_n$ is uniformly bounded.
I want to prove that $$X^n \Rightarrow X \quad \text{ in}\quad D_{\mathbb{Z}^d}[0,\infty)$$ where $X$ is the solution of $$X(t) = X(0) + \sum_{l\in \mathbb{Z}^d}l\cdot N_l\left(\int_0^t \beta_l(X(s))ds \right)$$
The most common approach to do this is to prove tightness of the family $X^n$ and then prove that every limit point is $X$.
For the tightness part the only thing that I could manage to do is to prove tightness of every process $R^n_l(t)= N_l\left(\int_0^t \beta_l^n(X^n(s))ds\right)$. This is easy because $$w'(R_l^n,\delta,T) \leq w'(N_l,\delta\cdot C_l, T\cdot C_l )$$ where $C_l$ is a uniform bound of the functions $(\beta_l^n)_n$ But with this I don't really have tightness of the whole process, and if I had it I don't know how to prove that the unique limit point is $X$.
In chapter 6 section 1 of the book there is a theorem that proves convergence of solution of a similar equation but with only one element on its right side.
Any help will be appreciated.
