ere inscribed in $D$ that is centred at $\rho_{n-1}$. Calling this sphere $S_n$, we have that $S_n = \{y\in \mathbb{R}^d \colon |y-\rho_{n-1}| =r_n\}$. We now select $\rho_n$ to be a point that is uniformly positioned on $S_n$. Once again, we note that if $\rho_{n-1}\in D$ almost surely, then the uniform distribution of both $\rho_{n-1}$ and $\rho_{n}$ ensures that $\mathbb{P}(\rho_{n}\in \partial D) = 0$. Consequently, the sequence $\rho_n$ continues for all $n\geq1$. In the case that $\rho_n$ approaches the boundary, the sequence of spheres $S_n$ become arbitrarily small in size.

Thanks to the strong Markov property and the stationary and independent increments of Brownian motion, it is straightforward to prove the following result.

Fix $x\in D$ and define $\rho'_1 = W_{\tau_{S'_1}}$, where $\tau_{S'_1}=\inf\{t>0 \colon
     W_t \in S'_1 \}$ and $S'_1$ is the largest sphere, centred at $x$, inscribed in $D$. For $n\geq 2$, given $\rho'_{n-1}\in D$, let $\rho'_n = W_{\tau_{S'_n}}$, where $\tau_{S'_n}=\inf\{t>0 \colon W_t \in S'_n \}$ and $S'_n$ is the largest sphere, centred at $\rho'_{n-1}$. Then the sequences $(\rho_n, n\geq 0)$ and $(\rho'_n, n\geq 0)$ have the same law.

As an immediate consequence, $\lim_{n\to\infty}\rho_n$ almost surely exists and, moreover, it it equal in distribution to $W_{\tau_D}$. The sequence $\rho\coloneqq (\rho_n, n\geq 0)$ may now replace the role of $(W_t, t\leq \tau_D)$ in , and hence in (\[FKMC\]), albeit that one must stop the sequence $\rho$ at some finite $N$. By picking a threshold $\varepsilon>0$, we can choose $N(\varepsilon)$ as a cutoff for the sequence $\rho$ such that $N(\varepsilon) = \min\{n\geq 0: \inf_{z\in\partial D}|\rho_n -z|\leq \varepsilon\}$. Intuitively, one is inwardly ‘thickening’ the boundary $\partial D$ with an ‘$\varepsilon$-skin’ and stopping once the walk-on-spheres hits the $\varepsilon$-skin. As the sequence $\rho$ is random, $N(\varepsilon)$ is also random. Starting with Theorem 6.6 of [@Mu] and the classical computations in [@Mo], it i