\[asyGrowProp\].

We begin with part (i). Noting that for every $n\ge1$ the random variable $D^{(\alpha)}_{\inn,n}$ is stochastically dominated by the random variable $D^{(\alpha)}_{\inn}$ and that the random variables $\{D^{(\alpha)}_{\inn,n}\}$ converge to $D^{(\alpha)}_{\inn}$ in distribution, it suffices to show that $\E D^{(\alpha)}_{\inn}=d\kappa_d(\alpha+d)^{-1}\E R^{\alpha+d}$. Indeed, Campbell’s formula implies that $$\begin{aligned}
\E D^{(\alpha)}_{\inn}&=\int_{\R^d}|\xi|^\alpha \P(R>|\xi|)\d \xi= d\kappa_d \int_0^\infty r^{\alpha+d-1}\P(R>r)\d r=d\kappa_d(\alpha+d)^{-1}\E R^{\alpha+d}.\end{aligned}$$ For parts (ii) and (iii) we can also proceed by applying Campbell’s formula. Indeed, let $\varepsilon>0$ be arbitrary and choose $t_0=t_0(\varepsilon)$ as in Section \[htggSec\]. For part (ii), we first obtain that the expressions $$\begin{aligned}
(\log n)^{-1}\int_{ \BT_{t_0}(o)}|\xi|^{\alpha}\P(R>|\xi|)\d \xi\end{aligned}$$ and $$\begin{aligned}
(\log n)^{-1}\int_{\T_n\setminus \BT_{n/2}(o)}|\xi|^{\alpha}\P(R>|\xi|)\d \xi&\le 2\beta(\log n)^{-1} \int_{\T_1\setminus B^{\T_1}_{1/2}(o)}|\eta|^{\alpha-s}\d \eta\end{aligned}$$ tend to $0$ as $n\to\infty$. , $$\begin{aligned}
(\log n)^{-1}\int_{\BT_{n/2}(o)\setminus \BT_{t_0}(o)}|\xi|^{\alpha}\P(R>|\xi|)\d \xi&\le (1+\varepsilon)(\log n)^{-1}d\kappa_d\beta \int_{t_0}^{n/2}r^{-1}\d r\\
&=(1+\varepsilon)(\log n)^{-1}d\kappa_d\beta (\log n-\log 2 -\log t_0).\end{aligned}$$ Since an analogous argument gives the corresponding lower bound, this completes the proof of part (ii). It remains to deal with part (iii). Proceeding similarly to part (ii), we first note that $$\begin{aligned}
n^{s-\alpha-d}\int_{\BT_{t_0}(o)}|\xi|^{\alpha}\P(R>|\xi|)\d \xi\end{aligned}$$ tends to $0$ as $n\to\infty$. , $$\begin{aligned}
n^{s-\alpha-d}\int_{\T_n\setminus \BT_{t_0}(o)}|\xi|^{\alpha}\P(R>|\xi|)\d \xi&\le (1+\varepsilon)\beta\int_{\T_1\setminus B^{\T_1}_{t_0/n}(o)}|\eta|^{\alpha-s}\d \eta,\end{aligned}$$ and the right-hand side tends to $(1+\varepsilon)\beta\int_{[-1/2,1/2]^d}|\eta|^{\alpha-s}\d\eta$ as $n\to\infty$. Again, since the lower bound can be obtained using similar arguments, this completes the 