asymptotic distributions of the sum of all outgoing and of the sum of all incoming power-weighted edge lengths at a typical vertex. First, in Theorem \[outEdgeThm\], we consider the case of outgoing edges. The proof is essentially based on the observation that conditioned on $R^*$ the random variable $D^{(\alpha)}_{\out}$ concentrates around its conditional mean.

Using the canonical coupling between the Poisson point process $X^{(n)}$ on the torus $\T_n$ and the Poisson point process $X$ on $\R^d$ we deduce that $$\P\big(D^{(\alpha)}_{\mathsf{out},n}\ne D^{(\alpha)}_{\out}\big)\le \P(R^*>n/2),$$ and the latter probability tends to $0$ as $n\to\infty$. In order to determine the tail behavior of the random variable $D^{(\alpha)}_{\out}$, for any $r>0$ we put $$D_r=\sum_{(\xi,t)\in X}|\xi|^\alpha1_{B_r(o)}(\xi),$$ noting that $D_{R^*}=D^{(\alpha)}_{\out}$. , $$\begin{aligned}
\label{inEdgeEq1}
t^{s/(\alpha+d)}p^{(\alpha)}_{\out,t}=\int_0^\infty t^{s/(\alpha+d)}\P(D_r>t)\P_R(\d r),\end{aligned}$$ where $\P_R$ denotes the distribution of the random variable $R$. Let $\varepsilon>0$ be arbitrary. In order to analyze , we first consider the case where $t\ge (v(\varepsilon)^{-1}r)^{\alpha+d}$, i.e., where $r\le v(\varepsilon) t^{1/(\alpha+d)}$, writing $v(\varepsilon)=(1+\varepsilon)^{-1}(d\kappa_d/(\alpha+d))^{-1/(\alpha+d)}$. Let $N_r$ be a Poissonian random variable with mean $\kappa_dr^d$, where $r>0$. Additionally, $\{U_i\}_{i\ge1}$ be an iid sequence of random vectors that are independent of $N_r$ and uniformly distributed in $B_1(o)$, $D_r=r^\alpha\sum_{i=1}^{N_r} |U_i|^{\alpha}$. First, $$\P(D_r>t)\le \P(N_r\ge n_{r,t})+\P\Big(\sum_{i=1}^{n_{r,t}}r^{\alpha}|U_i|^{\alpha}\ge t\Big),$$ where $n_{r,t}=\lfloor tr^{-\alpha}\kappa_dv(\varepsilon)^{\alpha+d}(1+\varepsilon)^{(\alpha+d)/2}\rfloor$. Note that for sufficiently large values 