$$D^{(\alpha)}_{\inn,n}=\sum_{(\xi,r)\in X^{(n)}}|\xi|^\alpha1_{\BT_r(o)}(\xi),$$ denote the sum of $\alpha$th powers of the lengths of all incoming edges at the node $(o,R^*)$, where $\alpha\ge0$, $n\ge1$. As in the case of $D^{(\alpha)}_{\out,n}$, we will see that the random variables $D^{(\alpha)}_{\inn,n}$ converge in distribution to a random variable $D^{(\alpha)}_{\inn}$ given by $$D^{(\alpha)}_{\inn}=\sum_{(\xi,r)\in X}|\xi|^\alpha1_{B_r(o)}(\xi),$$ where as before $X$ denotes an independently $[0,\infty)$-marked homogeneous Poisson point process in $\R^d$ with intensity $1$. The following result is devoted to the tail probabilities $p^{(\alpha)}_{\mathsf{in},t}=\P\big(D^{(\alpha)}_{\inn}>t\big)$

\[inEdgeThm\] For every $\alpha\ge0$ the random variables $\big\{D^{(\alpha)}_{\inn,n}\big\}_{n\ge1}$ converge to the random variable $D^{(\alpha)}_{\inn}$ in distribution. , $\P\big(D^{(\alpha)}_{\inn}=\infty\big)=1$ for $s\le d$, whereas if $s>d$, then $D^{(0)}_{\inn}$ is a Poissonian random variable with mean $\kappa_d\E R^d$. Finally, if $\alpha>0$ and $s>d$, then $$\lim_{t\to\infty} t^{(s-d)/\alpha}p^{(\alpha)}_{\inn,t}=d\kappa_d\beta (s-d)^{-1}.$$

Besides considering limit distributions, we also determine leading-order asymptotics for the expectations $\big\{\E D^{(\alpha)}_{\inn,n}\big\}_{n\ge1}$ as $n\to\infty$. Note that here it is not necessary to distinguish between ingoing and outgoing edges, since $\E D^{(\alpha)}_{\out,n}=\E D^{(\alpha)}_{\inn,n}$ for all $n\ge1$.

\[asyGrowProp\] Let $\alpha>0$ be arbitrary. Then one can distinguish between three limiting regimes for the random variable $\E D^{(\alpha)}_{\inn,n}$ depending on the sign of $\alpha+d-s$.

1.  If $s>\alpha+d$, then $\lim_{n\to\infty}\E D^{(\alpha)}_{\inn,n}=d\kappa_d(\alpha+d)^{-1}\E R^{\alpha+d}$.

2.  If $s=\alpha+d$, then $\lim_{n\to\infty}(\log n)^{-1}\E D^{(\alpha)}_{\inn,n}=d\kappa_d\beta$.

3.  If $s<\alpha+d$, then $\lim_{n\to\infty}n^{s-\alpha-d}\E 