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 dmu_n rightarrow int F dmu

$$

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 $xin nS$. Let $X$ be a random walk starting from $x$ until it exits the square $S$ and let $B$ be a Brownian motion until it exits $S$. For $forall epsilon>0$, then there exists $N>0$ such that $nge N$, 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’?