$\E\#(X\cap S)$. , this mean can be expressed as $$\begin{aligned}
\int_{\R^d}\P(R>|\xi|) \d \xi={d\kappa_d} \int_{0}^\infty r^{d-1}\P(R>r)\d r=\kappa_d \E R^d.\end{aligned}$$ It remains to determine the tail behavior of $D^{(\alpha)}_{\inn}$, for $\alpha>0$. The derivation of the lower bound is based on the elementary relation $$\begin{aligned}
\P(X\cap S\setminus( B_{t^{1/\alpha}}(o)\times[0,\infty))\ne\es)=\P\(\max_{(\xi,r)\in X\cap S}|\xi|^\alpha\ge t\)\le\P\big(D^{(\alpha)}_{\inn}\ge t\big).\end{aligned}$$ Let $\varepsilon>0$ be arbitrary. Note that the random variable $\#(X\cap S\setminus( B_{t^{1/\alpha}}(o)\times[0,\infty)))$ is Poissonian with mean $$\begin{aligned}
\int_{\R^d\setminus B_{t^{1/\alpha}}(o)}\P(R\ge|\xi|)\d\xi={d\kappa_d}\int_{t^{1/\alpha}}^\infty r^{d-1}\P(R\ge r)\d r,\end{aligned}$$ and observe that $t>t_0(\varepsilon)$ the latter expression is bounded from below by $$\begin{aligned}
(1-\varepsilon){d\kappa_d}\beta\int_{t^{1/\alpha}}^\infty r^{d-s-1}\d r&=(1-\varepsilon)(s-d)^{-1}{d\kappa_d}\beta  {t^{(d-s)/\alpha}}.\end{aligned}$$ Therefore, $$\begin{aligned}
\P\big(D^{(\alpha)}_{\inn}\ge t\big)&\ge1-\exp(-(1-\varepsilon)(s-d)^{-1}{d\kappa_d}\beta{t^{(d-s)/\alpha}})\\
&\ge(1-2\varepsilon)(s-d)^{-1}d\kappa_d\beta{t^{(d-s)/\alpha}},\end{aligned}$$ provided that $t>0$ is large. For the upper bound, we need more refined arguments, since in principle $D^{(\alpha)}_{\inn}$ could be larger than $t$, even if $\max_{(\xi,r)\in X\cap S}|\xi|^\alpha\le t$. To achieve the desired upper bound we show, loosely speaking, that the behavior of the sum $D^{(\alpha)}_{\inn}$ is already determined by its largest summand. Let $\varepsilon\in(0,1)$ be arbitrary. Then our upper bound is based on the inequality $$\begin{aligned}
\P(D^{(\alpha)}_{\inn}>t)&\le \P\(\max_{(\xi,r)\in X\cap S}|\xi|\ge(t(1-\varepsilon))^{1/\alpha}\)+\P(\#(X\cap S)\ge \varepsilon t^{1/4})\\
&+\P(\#(X\cap S\setminus (B_{t^{3/(4\alpha)}}(o)\times[0,\infty)))\ge 2),\end{aligned}$$ and in the rest of the proof the three summands on the right-hand side are considered separately. Using similar bounds as in the derivation of the lower bounds, we 