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 $n\rightarrow\infty$, \begin{equation}\ (X_{e,n})\Rightarrow (X_e), \end{equation} 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 $k\ge 2$.


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