Given a sequence of exchangeable random variables $X_1,\ldots,X_n$, and a measurable function $g: \mathbb{R}^n \to \mathbb{R}$. Let $S_n=g(X_1,\ldots,X_n)$. Then what is a natural construction of a Stein pair $(S_n,S_n')$ so that $(S_n,S_n')=_d(S_n',S_n)$ and satisfies $E[S_n'|\{X_i\}]=(1-\lambda)S_n$ for some $\lambda \in (0,1)$? My working example is the most straightforward form: $S_n=\sum_{i=1}^n X_i$.
In the simplest case for i.i.d. r.v.'s, the construction is by simple replacement: let $X_i'$ be an independent copy of $X_i$, $I$ be uniform over $\{1,\ldots,n\}$ independent of all other r.v.'s and $S_n'=g(X_1,\ldots,X_{I-1},X_I',X_{I+1},\ldots,X_n)$. Then the above condition is satisfied with $\lambda=1/n$.
Stein's method has found successful applications in proving quantitive bounds for CLT with weak/local dependence, but I am not spotting a reference or related literature on this seemingly easy(?) case. Any reference or comment is greatly appreciated.
