# Maximum nearest neighbor distance for a Poisson point process

Is the maximum nearest neighbor distance between points of the process, over all the infinitely many points of a stationary Poisson point process of intensity $$\lambda$$ in $$\mathbb{R}^d$$, almost surely finite? What is its distribution?

I am interested in the case $$d=1$$ and $$d=2$$. Also, if it is any easier, you can replace between points of the process by between an arbitrary location in $$\mathbb{R}^d$$ and the closest point of the process.

$$\newcommand\la\lambda\newcommand\de\delta\newcommand\R{\mathbb R}$$This maximum (or, rather, supremum) distance, say $$M$$, is $$\infty$$ almost surely (a.s.).
Indeed, recall that a (simple) Poisson point process of intensity $$\la\in(0,\infty)$$ on $$\R^d$$ is a random Borel measure $$m$$ over $$\R^d$$ such that, for any pairwise disjoint bounded Borel subsets $$A_1,\dots,A_k$$ of $$\R^d$$, the random variables (r.v.'s) $$m(A_1),\dots,m(A_k)$$ are independent Poisson r.v.'s with respective parameters $$\la|A_1|,\dots,\la|A_k|$$, where $$|\cdot|$$ is the Lebesgue measure. It is not hard to show (see e.g. Proposition 9.1.III (ii, iii), p. 4) that $$m=\sum_{i=1}^\infty\de_{X_i}$$, where $$\de_x$$ is the Dirac delta measure supported on $$\{x\}$$, for $$x\in\R^d$$, and the $$X_i$$'s are random points in $$\R^d$$ that are a.s. pairwise distinct. So, $$M=\sup_i\min\{\|X_k-X_i\|\colon k\ne i\}$$ a.s., where $$\|\cdot\|$$ is the Euclidean norm.
Now take any real $$a>0$$ and any natural $$n$$. Take the hypercube $$C_{a,n}:=[0,3na)^d$$ and partition it naturally into $$n^d$$ congruent smaller hypercubes each with edgelengths $$3a$$. In each of these $$n^d$$ hypercubes $$C_j$$ ($$j=1,\dots,n^d$$) each with edgelengths $$3a$$, take the central sub-hypercube, say $$B_j$$, with edgelengths $$a$$. Note that
$$p:=P\big(m(B_j)=1,m(C_j\setminus B_j)=0\big) \\ =P\big(m(B_1)=1\big)\,P\big(m(C_1\setminus B_1)=0\big)>0$$ for each $$j=1,\dots,n^d$$.
Then the probability $$P(M\ge a)$$ will be no less than the probability that at least one of the $$n^d$$ congruent smaller hypercubes $$C_j$$ contains exactly one point of the random points $$X_i$$ and that one point is in $$B_j$$. The latter probability is $$1-(1-p)^{n^d}\to1$$ as $$n\to\infty$$. So, $$P(M\ge a)\ge1$$ for all real $$a>0$$, and hence $$P(M=\infty)=1$$.
• Thank you! The reason I asked the question is because for Poisson-binomial point processes discussed in my previous questions, if the $F$ distribution is uniform, the maximum nearest neighbor distance is always finite regardless of the scaling factor $s$, and when $s\rightarrow\infty$, the Poisson-binomial process is a Poisson process. However, the maximum may not be bounded as $s\rightarrow\infty$, leaving open the possibility that for a Poisson process, $M=\infty$ (which you proved). Dec 31, 2021 at 20:19