I know that, by Donsker's theorem, simple random walk on $\mathbb{Z}$ will converge to Brownian motion on $\mathbb{R}$. Here, simple random walk means that the Markov chain with probability from $n$ to $n+1$, and from $n$ to $n-1$ are equally $1/2$.
Then, we can consider similar random walk on $n$-cycle. More rigorously, consider $\mathbb{Z}_n$ and suppose probability from $n$ to $n+1$, and from $n$ to $n-1$ are equally $1/2$. Then, as $n$ increases this will also converges to Brownian motion on $S^1$? (Here, Brownian motion on $S^1$ is the Markov process induced by the heat kernel)
If it is true, probably we will need to rescale the original random walk. Since the simple random walk on $\mathbb{Z}_n$ does not consider geometry of $S^1$.
Maybe this is classical question, but I couldn't find good references. Is there any book about "convergence of Random walk to Brownian motion on manifolds(Or, just $S^1$)"?