A Thinning Analogue of de Finetti's Theorem

We consider a notion of uniform thinning for a finite sequence of random variables $(X_1,...,X_n)$ obtained by removing one random variable, uniformly at random. If a triangular array of random variables $(X_{n,k} : n \in \mathbb{N}_+, 1 \le k \le n)$ satisfies that the law of $(X_{n,1},...,X_{n,n})$ is obtained by uniformly thinning $(X_{n+1,1},...,X_{n+1,n+1})$, then we call the array thinning-invariant. We give a representation for the Choquet simplex of all thinning-invariant triangular arrays of random variables, when all random variables take values in a compact metric space (with Borel measurable distributions). We give two applications: to long-ranged, asymmetric classical spin chains, and long-ranged, asymmetric simple exclusion processes.