I am seeking to use the following theorem by Kallenberg (1975) (the version quoted below is from Brown, 1979)

> **Theorem**. Let $\{N_n\}$ be a sequence of point processes and $\{p_n\}$ a sequence in $(0,1]$ such that $p_n \to 0$. Let $N'_n$ denote the point process obtained by independently deleting each of the points of $N_n$ with probability $1 - p_n$. Then $N'_n \to$ some $N$, in distribution, if and only if $p_n N_n \to$ some $\eta$, in distribution, in which case $N$ is a Cox process directed by $\eta$.

I am having trouble interpreting $p_n N_n$:
- I have inferred that each $p_n N_n$ is a random measure, from their converging to what I think is a random measure $\eta$ [that directs the Cox process][1].
- Brown (1979) refers to Kallenberg (1976) for notation. I haven't made much headway in Kallenberg (2017).
- If $N_n$ is fixed as a Poisson point process of intensity $\lambda$ then each $N'_n$ is a Poisson point process of intensity $p_n \lambda$ ([independent thinning of a Poisson point process][2]).
- Based on related readings (eg Westcott, 1976), I'm guessing that it has something to do with rescaling?

So I'm guessing that $p_n N_n$ is something like "rescale the measure of $N_n$"?


(I'm seeking to apply the theorem to a fixed point process $N \equiv N_n$ which is not necessarily a Poisson process.)



<cite authors="Brown, Tim">_Brown, Tim_, [**Position dependent and stochastic thinning of point processes**](https://doi.org/10.1016/0304-4149(79)90030-9), Stochastic Processes and their Applications 9(2), 189-193 (1979).</cite>

<cite authors="Kallenberg, Olav">_Kallenberg, Olav_, [**Limits of compound and thinned point processes**](https://doi.org/10.2307/3212440), J. Appl. Probab. 12, 269-278 (1975). [ZBL0318.60050](https://zbmath.org/?q=an:0318.60050).</cite>

<cite authors="Kallenberg, Olav">_Kallenberg, Olav_, [**Random Measures, Theory and Applications**](https://doi.org/10.1007/978-3-319-41598-7) (2017).

<cite authors="Westcott, Mark">_Westcott, Mark_, [**Simple Proof of a Result on Thinned Point Processes**](https://doi.org/10.1214/aop/1176996183), Ann. Probab. 4(1): 89-90 (February, 1976).</cite>

 [1]: https://en.wikipedia.org/wiki/Cox_process
 [2]: https://en.wikipedia.org/wiki/Point_process_operation