# Is there a rate of convergence for Donsker's theorem?

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 $F$?

• I think you want to look at strong approximation methods, which are closely related to the Skorohod embedding theorem AntL mentions in her/his answer. One book that covers this is Strong Approximations in Probability and Statistics by Csorgo and Revesz. I'm looking at Theorem 2.2.1 in it and it seems to be just what you need. – Paul Tupper May 24 '16 at 18:31

KMT coupling allows to construct $S(t)$ and $B(t)$ on the same probability space in such a way that the maximum of the difference is small. For example, see Lemma 17 in https://arxiv.org/pdf/1110.1254v3.pdf (doi:10.1214/13-AOP867) which follows from the results of Goetze and Zaitsev, This Lemma says that assuming $2+\delta$ moments we can define a random walk process $S(t)$ and and $B(t)$ on the same probability space in such a way that $$\mathbf P(\max_{u\le n}|S(u)-B(u)|\ge n^{1/2-\delta})\le Cn^{2\gamma+\gamma \delta -\delta/2}$$ for $0<\gamma<\frac{\delta}{2+\delta}$.