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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.
  • 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).
  • 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 $N_n$"?

Edit 1: Each $N_n$ is a random counting measure: If $A$ is measurable then $N_n(A)$ is a random variable that counts the number of points in $A$. Meanwhile I think $\eta$ is a random measure. Therefore something is needed to squeeze the $N_n$ "counting over a measurable set" into $\eta$ supplying an "intensity" at points.

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

Brown, Tim, Position dependent and stochastic thinning of point processes, Stochastic Processes and their Applications 9(2), 189-193 (1979).

Kallenberg, Olav, Limits of compound and thinned point processes, J. Appl. Probab. 12, 269-278 (1975). ZBL0318.60050.

Kallenberg, Olav, Random Measures, Theory and Applications (2017).

Westcott, Mark, Simple Proof of a Result on Thinned Point Processes, Ann. Probab. 4(1): 89-90 (February, 1976).

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1 Answer 1

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If $N_n$ is fixed as a Poisson point process of intensity $\lambda$

The idea precisely is that $\lambda_n$ goes to infinity (as otherwise the limit vanishes). In fact, $\lambda_n \propto 1/p_n$.

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

You're overthinking it. This is merely a product. E.g. consider this (deterministic but still quite illustrative) example: $p_n = 1/n$ and $N_n = \sum_{k\in \mathbb Z} \delta_{k/n}$. Here, $p_n N_n$ converges to the Lebesgue measure.

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