= Two-sided order statistics of a boundary-singular Poisson process
{title2=$kX_{-k}\to1,\quad\log(1-X_n)/n\to-1$}
For intensity $x^{-2}(1-x)^{-1}$ on $(0,1)$, the cumulative-intensity maps based at $1/2$ are
$$
L(x)=x^{-1}-2-\log\frac{x}{1-x},\qquad F(x)=2-x^{-1}+\log\frac{x}{1-x}.
$$
The left and right images are independent unit-rate <Poisson processes>. Labeling the original points by $X_{-k}<1/2<X_n$, with $k\ge1,n\ge0$, gives $L(X_{-k})/k\to1$ and $F(X_n)/n\to1$. Since $xL(x)\to1$ at zero and $F(x)=1-\log(1-x)+o(1)$ at one, $kX_{-k}\to1$ and $\log(1-X_n)/n\to-1$.
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