# 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. May 24 '16 at 18:31

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).

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 $$0<\gamma<\frac{\delta}{2(2+\delta)}$$.

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.