The law of the iterated logarithm for the total length of the nearest neighbor graph (Q1827470)

Consider a Poisson point process \({\mathcal{P}}_n\) on \([-n,n]^d\) with intensity \(1\). For every Poisson point \(x=(x_1,\ldots, x_d)\in {\mathcal{P}}_n\) let \(y\in {\mathcal{P}}_n \setminus \{x\}\) denote the nearest neighbor to \(x\) (in Euclidean distance). Set \(L_n=\sum_{(x,y)\in{\mathcal{E}}_n} | x-y|\), where \({\mathcal{E}}_n\) denotes the collection of pairs \((x,y)\) of Poisson points and their respective nearest neighbors. The authors' main result proves that \[ \mathop{\limsup}\limits_{n\rightarrow\infty} {{L_n-EL_n}\over{\sqrt{2\sigma^2_n \log\log \sigma^2_n}}}=1\quad\text{a.s.,} \] where \(\sigma^2_n=\text{Var} (L_n)\). Since, by a standard argument, \(\sigma^2_n \sim n^d \sigma^2\) with some positive constant \(\sigma^2\), the above LIL can be reformulated as \[ \limsup_{n\rightarrow\infty} {{L_n-EL_n}\over{\sqrt{2\sigma^2 n^d \log\log n^d}}}=1\quad\text{a.s.} \]