Two-sided order statistics of a boundary-singular Poisson process ($k{X}_{-k}\to 1,\log (1-X_n)/n\to -1$)

For intensity $x^{-2}(1-x{)}^{-1}$ on $(0,1)$, the cumulative-intensity maps based at $1/2$ areThe left and right images are independent unit-rate Poisson processes. Labeling the original points by $X_{-k}<1/2<X_n$, with $k\ge 1,n\ge 0$, gives $L(X_{-k})/k\to 1$ and $F(X_n)/n\to 1$. Since $xL(x)\to 1$ at zero and $F(x)=1-\log (1-x)+o(1)$ at one, $k{X}_{-k}\to 1$ and $\log (1-X_n)/n\to -1$.