
> 
> Donsker's invariance principle:
> Let $X\_1,X\_2,...$ be i.i.d. real-valued random variables with mean 0 and variance 1. We define $S\_0=0$ and $S\_n= X\_1+ ... + X\_n$ for $n \geq 1$. To get a process in continuous time, we interpolate linearly and define for all $t \geq 0$
> $$
> S\_t = S\_{[t]}+ (t-[t])(S\_{[t]+1}- S\_{[t]}).
> $$
> Then we define for all $t \in [0,1]$
> $$
> S^\*\_n(t)= \frac{S\_{nt}}{\sqrt{n}}.
> $$
> Let $C[0,1]$ be the space of real-valued continuous function defined on $[0,1]$ and endow space with the supremumnorm. Then $(S^\*\_n(t))\_{0 \leq t \leq 1}$ can be seen as a random variable taking values in $C[0,1]$. Now let $\mu\_n$ be its law on that space of continuous functions and let $\mu$ be the law of Brownian motion on $C[0,1]$. Then the following holds:
> 
> 
> 



> 
> **Theorem (Donsker):** The probability measure $\mu\_n$ converges weakly to $\mu$, i.e. for every $F: C([0,1]) \rightarrow \mathbb{R}$ bounded and continuous,
> $$
> \int F d\mu\_n \rightarrow \int F d\mu
> $$
> as $n \rightarrow \infty$.
> 
> 
> 


But for a two-dimensional case, the 'coupling version' is as following.


$\textbf{'coupling version'}$: Fix a square $S$ of size $s$. Fix $x\in nS$. Let $X$ be a random walk starting from $x$ until it exits the square $nS$ and let $B$ be a Brownian motion until it exits $nS$. For $\forall \epsilon>0$, then there exists $N>0$ such that $n\ge N$, one can couple $X$ and $B$ so that
$$d(X,B)\le \epsilon n.$$


My questions:
(1) Can we extended the classical Donsker's to the two-dimensional case?


(2) Is there any reference for the proof of the 'coupling version'?



How to prove the coupling version of the Donsker's Invariance Principle?
Yes, this can be done in any dimension but the two dimensional case is especially simple. Rotating a simple RW in two dimensions by 45 degrees yields a random walk where the $x$ and $y$ coordinates are independent. Couple each coordinate to a BM using Skorokhod embedding to obtain the desired coupling.


