I am learning martingale and related concepts recently and come across the following problem.
Suppose $D$ is a bounded, connected, open subset of $\mathbb{R}^2$ with boundary $\partial D$.
Consider a Markov chain $\{Z_n\}_{n\geq 0}$ on $D$ which evolves as follows: for each $n\geq 0$, conditional on $\sigma(Z_k)_{k\leq n}$, the random variable $Z_{n+1}$ is uniformly distributed on the circle of radius $R_n$ centered at $Z_n$, where $2R_n$ is the distance from $Z_n$ to $\partial D$.
Prove that $$Z_n \rightarrow Z_{\infty} \;a.s., Z_{\infty}\in \partial D. $$
It is not too hard to see that coordinates of $Z_n$ are bounded (as $D$ is). It is also easy to see that the coordinates of $Z_n$ are martingale. Then by the boundedness of the coordinates and the martingale convergence theorem, we know that $Z_n$ indeed converges to some $Z_{\infty}$.
But how to show $Z_\infty\in \partial D$? I am not able to produce a contradiction to $Z_\infty \in D$ with positive probability.