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 $n > N\_{0}$,
$$
g(n) := \sup\_{t \leq n} |B\_{t} - S\_{t}| \leq n^{1/2 - \epsilon}.
$$
(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 i.i.d. 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".



Approximating Markov chains by Brownian motion
It turns out that to find a whole slew of results of this shape I needed to know that


(a) this sort of statement is called "strong invariance principle" more often than "strong approximation";


(b) the result as requested above is the subject of the book by Walter Philipp and William Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Memoirs of the AMS vol. 2 no. 161, July 1975.


