chi \coloneqq {g}(\rho_{N}) + \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))
        , \qquad x\in D,$$ which is again justified by an obvious [strong law of large numbers]{}and the [central limit theorem]{}in the spirit of Corollary \[rate1\].

\[rate2\] When $D$ is bounded and convex, ${g}$ is continuous and in $ L^1_\alpha(D^\mathrm{c})$ and ${f}$ is a function in $ C^{\alpha +\varepsilon}(\overline{D})$ for some $\varepsilon>0$, then $$\label{WoSMC3}
            \lim_{n \to\infty} \frac{1}{n}\sum_{i = 1}^n \chi^{i} = \mathbb{E}_x\left[ {g}(\rho_{N}) + \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\right]             = u(x),$$ almost surely where $\chi^{i}$, $i\geq 1$ are [*iid*]{}copies of $\chi$ and $u(x)$ is the solution to (\[aDirichlet\_g\]). Moreover, when $$\int_{D^\mathrm{c}}\frac{{g}(x)^2}{1+|x|^{\alpha + d}}\,{\rm d}x<\infty.
            \label{f22}$$ then $\operatorname{Var}(\chi)<\infty$ and, in the sense of weak convergence, $$\lim_{n\to\infty}n^{1/2}\left(\frac 1n \sum_{i=1}^n  \chi^{i}- u(x)\right)= \operatorname{Normal}(0,  \operatorname{Var}(\chi)).$$

Theorem \[hasacorr\] and Lemma \[integral\] ensure that the [strong law of large numbers]{}may be invoked. For the [central limit theorem]{}, we need $\mathbb{E}_x[\chi^2]<\infty$. Taking account of the fact that $\chi$ is the sum of two terms, the Cauchy–Schwarz inequality ensures that $\mathbb{E}_x[\chi^2] $ is finite if $\mathbb{E}_x\left[{g}(\rho_{N})^2\right] $ and $ \mathbb{E}_x\left[\left( \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\right)^2\right]$ are finite. Recall that $\mathbb{E}_x\left[{g}(\rho_{N})^2\right] = \mathbb{E}_x[{g}(X_{\sigma_D})^2]$ and, from Corollary \[rate1\], that is sufficient to ensure that this expectation is bounded.

Now note that, on account of the fact that ${f}$ is bounded, there exists a constant $\kappa\in(0,1)$, such that, for each $n\leq N$, appealing to (\[timescale\]), we have $r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\leq \kappa \sigma_n$