Border-corrected estimator for Ripley K

Assume the point-process supplier's AC and Countable Choice convention. Let $\eta$ be the stationary locally square-integrable point process with finite positive intensity from Pair-correlation density and Ripley K-function. Given a compact convex observation window $W\subseteq {\mathbb{R}}^d$ with $0<\mathrm{\mid }W\mathrm{\mid }<\mathrm{\infty }$, where $\mathrm{\mid }\cdot \mathrm{\mid }$ is Borel Lebesgue measure as in Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, let $\xi (A):=\eta (A\cap W)$ for Borel $A$ be the complete observed counting measure in $W$ as in Random measures and locally finite point processes, and fix a radius $r\ge 0$. Define the border window

$W_{-r}:=\{x\in W:\bar{B}[x,r]\subseteq W\},$

where $\bar{B}[x,r]$ is the closed Euclidean ball under the Euclidean norm used in Pair-correlation density and Ripley K-function. Since $W$ is closed, $W_{-r}={\bigcap }_{\mathrm{\parallel }z{\mathrm{\parallel }}_2\le r}(W-z)$ is closed and hence Borel. Let $N_{\xi }(A)=\xi (A)$ count occurrences, including multiplicities, and put $\widehat{\rho }=N_{\xi }(W)\mathrm{/}\mathrm{\mid }W\mathrm{\mid }$. The border anchor weight is $w_{W,r}(x)=1$ for $x\in W_{-r}$ and $0$ otherwise.

If $N_{\xi }(W_{-r})=0$, return not defined. Otherwise $N_{\xi }(W)>0$ because $W_{-r}\subseteq W$, so $\widehat{\rho }{N}_{\xi }(W_{-r})>0$. For an occurrence representation $\xi ={\sum }_i{\delta }_{X_i}$, return

${\widehat{K}}_{\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}}(r):=\frac{1}{\widehat{\rho }{N}_{\xi }(W_{-r})}{\sum }_{i:X_i\in W_{-r}}{\sum }_{j\mathrm{\ne }i}\mathbf{1}\{\mathrm{\parallel }{X}_j-X_i{\mathrm{\parallel }}_2\le r\}\mathrm{.}$

Every eligible anchor has its full closed radius-$r$ neighborhood inside $W$, so all its radius-$r$ partner occurrences are observed. This defines a finite-data procedure only; it asserts no unbiasedness, consistency, or asymptotic guarantee.

Axiom convention. The procedure itself makes no choices; AC and Countable Choice are inherited through its point-process and K dependencies.