Following Probability that a sequence of random variables converges to 0 or 1, I have proved an extension of Exercise 5.5.7 in Durrett's "Probability Theory and Examples": Given $0<\chi<1$, $0<\nu^*_1<\mu_0<\nu^*_2<1$, define $$ \nu'_k = \left. \begin{cases} \chi\nu'_{k-1}+(1-\chi)\nu^*_1, & \text{with probability } \frac{\nu^*_2-\nu'_{k-1}}{\nu^*_2-\nu^*_1}\\ \chi\nu'_{k-1}+(1-\chi)\nu^*_2, & \text{with probability } \frac{\nu'_{k-1}-\nu^*_1}{\nu^*_2-\nu^*_1} \end{cases}\right. $$ where $\nu'_0=\mu_0$, then $\Pr(\lim_{k\to \infty} \nu'_k = \nu^*_2) = \frac{\mu_0-\nu^*_1}{\nu^*_2-\nu^*_1}$ and $\Pr(\lim_{k\to \infty} \nu'_k = \nu^*_1) = \frac{\nu^*_2-\mu_0}{\nu^*_2-\nu^*_1}$.
I think this result can be extended to higher dimension: Given any state space $\Omega$ with $|\Omega|=N<\infty$, a full-support belief $\mu_0\in int(\Delta(\Omega))$ and a distribution of beliefs $\tau^*\in\Delta(\Delta(\Omega))$ such that $E_{\tau^*}[\nu]=\mu_0$, let $\{\nu^*_1, \nu^*_2, ..., \nu^*_S\}$ be the support of $\tau^*$, and define
$$ \nu'_k = \left. \begin{cases} \chi\nu'_{k-1}+(1-\chi)\nu^*_1, & \text{with probability } p_{1,k-1}\\ \chi\nu'_{k-1}+(1-\chi)\nu^*_2, & \text{with probability } p_{2,k-1}\\ \cdots \\ \chi\nu'_{k-1}+(1-\chi)\nu^*_S, & \text{with probability } p_{S,k-1}\\ \end{cases}\right. $$
for $k\geq 1$, where $\nu'_0=\mu_0$ and $\{p_{s,k-1}\}_s$ are such that $E(\nu'_k|\nu'_{k-1})=\nu'_{k-1}$.
$\{\nu'_k\}_k$ is a martingale, and I'd like to prove that $\Pr(\lim_{k\to \infty} \nu'_k = \nu^*_s) = \tau^*(\nu^*_s)$ for $s=1,2,...,S$. However, the method I used to prove the two-state case seems not to work for higher dimensions. I would be really thankful if anyone could help with the proof. Thanks in advance.