_x[{g}(X_{\tau_D})^2]<\infty$. However, if we consider the computation in (\[worksforsquared\]) of the Appendix, which shows that $\mathbb{E}_x[{g}(X_{\tau_D})]<\infty$ when ${g}$ is continuous and in $L^1_{\alpha}(D^\mathrm{c})$, then it is easy to see that the same statement holds replacing ${g}$ by ${g}^2$. Under finiteness of the second moment, the [central limit theorem]{}completes the proof.

We now return to the proof of Theorem \[main\]. Our approach is to break it into several parts. For convenience, we shall henceforth write $X^{(x)}=(X^{(x)}(t)\colon t\geq 0)$ to indicate the dependency of $X$ on its initial position $X_0 = x$ (equivalent to writing $(X, \mathbb{P}_x)$). For any $x = (x_1, \dots, x_d)\in\mathbb{R}^d$ such that $x_1>0$, we have $V(x) = \{(z_1, \dots, z_d)\in\mathbb{R}^d \colon z_1>0\}$ for the open half-space containing $x$ and denote its boundary $\partial V(x) = \{(z_1, \dots, z_d)\in\mathbb{R}^d \colon z_1=0\}$. For any Borel set $A\subset\mathbb{R}^d$, we write $
\sigma_A = \inf\{t>0 \colon X_t\not\in A\}.
$ We will typically use in place of $A$ the set $V(x)$ as well as $B(x,1) = \{z\in\mathbb{R}^d\colon |z- x|<1\}$, the unit ball centred at $x\in\mathbb{R}^d$. Finally write ${\rm\bf i} = (1,0, \dots, 0)\in\mathbb{R}^d$.

\[scaled\] Without loss of generality (by appealing to the spatial homogeneity of $X$ which allows us to appropriately choose our coordinate system) suppose that $x= |x|\,{
        \rm\bf i}\in D$ is such that $\partial V(x)$ is a tangent hyperplane to both $D$ and $B_1$. Then $X^{(x)}_{\sigma_{B_1}}$ is equal in distribution to $|x|\, X^{(\rm\bf i)}_{\sigma_{B({\rm\bf i},1)}}$ and $X^{(x)}_{\sigma_{V(x)}}$ is equal in distribution to $|x|\,X^{(\rm\bf i)}_{\sigma_{V(\mathbf{i})
    }}$.

The scaling property of $X$ ensures that we can write $$X^{(x)}_{s} = |x|\hat{X}^{(\mathbf{i})}_{|x|^{-\alpha}s}, \qquad s\geq 0,
        \label{scaling}$$ where $\hat{X}^{(x)}$ is equal in law to $X^{(x)}$. Note that $$\begin{aligned}
        \sigma_{B_1} & = \inf\Bp{t> 0\colon {