Penrose–Ong conjecture on central limit theorems for the on-line nearest-neighbour graph

Let $\tilde{\d835𝒪}^{d,\d835α}(\d835𝒰_n)$ and $\tilde{\d835𝒪}^{d,\d835α}(\d835𝒫_λ)$ denote the centered total power-weighted edge lengths of the on-line nearest-neighbour graph on, respectively, $n$ uniform points and a Poisson point process of intensity $λ$ in the unit cube. For $\d835\d\in\mathbb{N}$ and $\d835α\in(0,d/4)$, let $\d835σ_{d,α}^2$ and $\d835δ_{d,α}^2$ be the constants in the variance limits and central limit theorems of Proposition 3.1.

Penrose–Ong conjecture. Suppose $d\in\mathbb{N}$. The limit theorems for $\d835α\in(0,d/4)$, namely the variance limits and the corresponding Gaussian convergence for the binomial and Poisson models, are also valid for

$$α\in[d/4,d/2).$$

This extends the known central limit theorem range from $\d835α\in(0,d/4)$ to the full predicted Gaussian regime $\d835α\in(0,d/2)$. The cited results establish the theorem below $d/4$ and non-Gaussian limiting behavior above $d$, while the intermediate range remains open in the supplied text.