The first Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E X^2<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau$ is a stopping time.
The second Skorokhod embedding (which follows easily from the first one) says that a random walk $S_n:=\sum _{k=1}^n X_i$ where $X_i$ are normalized i.i.d. can be coupled with a Brownian motion such such that almost surely, for all large $n$ we have $|S_n-B_n| \le n^{1/4}\log ^2 n$.
In dimension $d\ge 2$ the first embedding theorem is not true but the second one is true. Is there a simple proof of this fact?
I am aware of the fact that there are better quantitative results like KMT. These results hold also in higher dimenstions and they say that we can couple the R.W. and the B.M. up to a logarithmic distance. The problem is that these results are very complicated. I wonder if there is a simpler proof if we only want to couple up to a distance of $n^{1/4}$.
