# Skorokhod representation for weak convergence of exchangeable arrays

Let $$(X_e)=(X_e)_{e\in \mathbb{N}^{(k)}}$$ be a $$k$$-dimensional exchangeable real random array (see this note for the definition), where $$k\in \{1,2,\ldots \}$$ is fixed and $$\mathbb{N}^{(k)}$$ denotes subsets of $$\mathbb{N}$$ of size $$k$$. Suppose $$(X_{e,n})=(X_{e,n})_{e\in \mathbb{N}^{(k)}}$$, $$n=1,2,\ldots$$ is a sequence of $$k$$-dimensional exchangeable array and as $$n\rightarrow\infty$$, $$$$\ (X_{e,n})\Rightarrow (X_e),$$$$ where $$\Rightarrow$$ denotes convergence of finite-dimensional distributions. According to Skorokhod Representation, there exist $$(X_{e,n}^*)\overset{d}{=}(X_{e,n})$$ and $$(X_e^*)\overset{d}{=}(X_e)$$, so that $$(X_{e,n}^*)\rightarrow (X_e^*)$$ a.s..

Question: is it possible to strengthen the conclusion of Skorokhod Representation to ensure that $$(X_{e,n}^*,X_e^*)_{e\in \mathbb{N}^{(k)}}$$ is an exchangeable array?

This seems true for $$k=1$$, where $$(X_e)_{e\in \mathbb{N}}$$ reduces to an exchangeable sequence.

Proof for $$k=1$$ case: By de Finetti's Theorem, Theorem 3.2 of "Kallenberg (2005) Probabilistic Symmetries and Invariance Principles" and Skorokhod Representation, one has $$(X_{e,n})\overset{d}{=}(X_{e,n}^*):=(F_n(U_e))$$ and $$(X_{e})\overset{d}{=}(X_e^*):=(F(U_e))$$ for some random quantile functions $$F_n$$ and $$F$$ satisfying a.s. $$F_n\rightarrow F$$ at continuity points of $$F$$, and $$(U_e)_{e\in \mathbb{N}}$$ are i.i.d. Uniform$$[0,1]$$ independent of $$F_n$$ and $$F$$.

However, there seems no analog of this strategy for $$k\ge 2$$.