For the standard CLT, one can easily estimate a rate of convergence if you assume that the random variables have a little more than two moments.

Let $S_n$ be the centered-scaled sum of $n$ iid random variables with three moments. Let $X$ be a standard normal. Using Taylor's theorem, for any smooth bounded function $h$ with bounded derivates, you get

$$ |\mathbb{E}[h(S_n)] - \mathbb{E}h(X)| \leq \frac{1}{n^{1/2}} $$

Is there an analogous result for Donsker's theorem? Let $F$ be some smooth functional on the continous functions. Let $S_n(t)$ be a scaled-centered and interpolated version of a discrete simple random walk. Let $B(t)$ be standard brownian motion on $[0,1]$. Is there a statement that says something like,

$$ |\mathbb{E}[F(S_n(t)] - \mathbb{E}[F(B(t)]| \leq n^{-\alpha} $$

given enough assumptions on $F$?