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

Question

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 $F$?

Answer 1

A possible approach is to use the Skorokhod embedding theorem to represent the random walk as a realization of a path of a Brownian motion sampled at random time, and then to use results on the modulus of continuity of the Brownian motion (however, this is rather a strong rate of convergence than a weak rate of convergence).

See e.g.

Pierre Étoré, MR 2217816 On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients, Electron. J. Probab. 11 (2006), no. 9, 249--275 (electronic).

Answer 2

As was suggested by Antl and Paul Tauper, have a look at the strong coupling (embedding). You can use Komlós–Major–Tusnády coupling, see doi: 10.1007/BF00533093 and doi:10.1007/BF00532688 More recent developments can be found in papers of Sakhanenko, Zaitsev and Goetze who consider non-identically distributed random variables and random vectors.

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 $$ \mathbf P(\max_{u\le n}|S(u)-B(u)|\ge n^{1/2-\gamma})\le Cn^{2\gamma+\gamma \delta -\delta/2} $$ for any $0<\gamma<\frac{\delta}{2(2+\delta)}$.

Answer 3

If we assume that $F$ is 1-Lipschitz, using Kantorovich-Rubinstein duality the desired estimates can be obtained from estimates of the Wasserstein distance between (the laws of) $S_n$ and $B$. Such convergence rates are available in the literature, see e.g. Theorem 3.4 in

Coutin, Laure; Decreusefond, Laurent, Donsker’s theorem in Wasserstein-1 distance, Electron. Commun. Probab. 25, Paper No. 27, 13 p. (2020). ZBL1434.60100.