# pr.probability – How to prove the coupling version of the Donsker’s Invariance Principle?

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’?

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