Second Skorokhod embedding in high dimensions 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 $n^{1/4}$.
 A: For higher dimensions there is a paper of J. R. Baxter, R. V. Chacon(1974) "Potentials of stopped distributions" giving the first embedding under some conditions, see it at https://doi.org/10.1215/ijm/1256051015 . In particular, under these conditions,  they show that $\mathbb E\tau<\infty$ when the second moment of $X$ is finite, which should imply the second embedding theorem. Conditions are weakened in reference [1] below. There is also a nice one-dimensional version of the paper, which explains the construction by Chacon and Walsh (1976) One-dimensional potential embedding.
Other 2 references I know are

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*N. Falkner. On Skorohod embedding in n-dimensional Brownian motion by means of natural stopping times. In Seminaire de Probabilitees, XIV, volume 784 of Lecture Notes in Math., pages 357–391. Springer, Berlin, 1980. MR580142

*D. Heath. Skorokhod stopping via potential theory. In Seminaire de Probabilites, VIII, pages 150–154. Lecture Notes in Math., Vol. 381. Springer, Berlin, 1974. MR368185

