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I am self-studying probability theory and currently the Poisson point process (PPP) gives me hell, firstly because the definition of a point process in general and PPP in particular seems rather cumbersome at first to tenth glance and secondly because there are no examples in the script I use and hardly any on the net. So I constructed my own example, regarding which I have some questions... Sorry for the long intro, here comes the math:

Suppose we distribute $n$ points randomly and uniformly on the interval $[0, n]$. Thus we have $X_1, ..., X_n \sim Unif([0, n])$. For each $\omega \in \Omega$ we have $N(\omega) = \sum_{i = 1}^{10} \delta_{X_i(\omega)}$ as a random counting measure on $[0, n]$. Obviously

$\mathbb{P}[N(\omega)[B] = k] = \binom{n}{k}(\frac{\lambda[B]}{n})^k(\frac{\lambda[B^c]}{n})^{n-k}\sim Bin(n, \frac{\lambda[B]}{n})$.

Now since $Bin(n, \frac{\lambda[B]}{n}) \rightarrow_{n\to\infty} Pois(\lambda[B])$, the limit process on $\mathbb{R}_+$ with infinitely many points, should be a PPP. Is this correct so far? If so, how to prove independence on disjoint subintervals?

Now the problem is, that there is no uniform distribution on $\mathbb{R}_+$, so the process with underlying random variables can only be defined as a limit process. I wondered if this may be the reason, why a point process is defined as a measurable map from $\Omega$ to the set of locally finite counting measures. Is this the case?

I would also greatly appreciate other examples of PPPs "built from scratch", ie with explicit definition of the map $N:\Omega\rightarrow\mathfrak{N}$, where $\mathfrak{N}$ is the set of locally finite counting measures. Thanks for help!

Edit: Maybe I found another way to do it, please correct, if there are any errors.

Let $A_n := [n, n+1)$, thus $\mathbb{R}_+ = \sqcup_{n \in \mathbb{N}_0} A_n$. Let $(X_{n, j})_{n, j \in \mathbb{N}_0} \sim Unif([0, 1))$ be i.i.d., $(Y_n)_{n \in \mathbb{N}_0} \sim Pois(1)$ be i.i.d.. Define

$N_n: \Omega \rightarrow \mathfrak{N}(A_n, \mathfrak{B}(A_n)), \omega \mapsto \sum_{j=1}^{Y_n(\omega)}\delta_{(X_{n, j})+n)}$, where again $\mathfrak{N}(A_n, \mathfrak{B}(A_n))$ is the set of locally finite counting measures on $(A_n, \mathfrak{B}(A_n))$. Then

$N:=\sum_{n\in\mathbb{N}_0}N_n = \sum_{n\in\mathbb{N}_0} \delta_{X_n}$ a.s.

does the job.

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