I would like a result along the following lines to be true, but haven't been able to locate it in the literature; pointers would be welcome.

Let $X_t$ be a finite-state, irreducible, aperiodic Markov chain. Let $f$ be a bounded functional, which is mean-zero with respect to the stationary distribution of $X$. Let $$ S_t = \frac{1}{\sqrt{n}} \sum_{s \leq t} f(X_s), \quad t \leq n $$ be the partial sum process. Then there exists $\epsilon > 0$, $N_0 > 0$, and a Brownian motion $B_t$ such that, for all $n > N0$, $$ g(n) := sup_{t \leq n} |B_t - S_t| \leq n^{1/2 - \epsilon}. $$ (Interpreted either as "$t$ is an integer less than $n$" or as "interpolate $S_t$ linearly between integers".)

In the case of partial sums of iid random variables, if I remember correctly, the Skorohod embedding gives $g(n) \sim n^{1/4}log(n)$, and the KMT approximation gives $g(n) \sim log(n)$. In the Markovian setting, I'm hoping for $g(n)$ to depend on the mixing time of the Markov chain.

It's possible that the correct generality for this is "ergodic sequence" rather than "Markov chain".