Minimal sufficiency of order statistics in convex models

Let $\mathcal{P}$ be a convex and dominated statistical model on the measurable space $(\mathcal{X},\mathcal{A})$, with $\mathcal{A}$ minimal sufficient, and let $n\in\mathbb{N}$. Then $\mathcal{A}^{\otimes n}_{\operatorname{sym}}$, the $\sigma$-algebra of all permutation invariant sets belonging to the $n$-fold product $\sigma$-algebra $\mathcal{A}^{\otimes n}$, is shown to be minimal sufficient for the corresponding model for $n$ independent observations, $\mathcal{P}^n = \left\{P^{\otimes n}:P\in\mathcal{P}\right\}$.

The main technical tool provided and used is a functional analogue of a theorem of Grzegorek (1982) concerning generators of $\mathcal{A}^{\otimes n}_{\operatorname{sym}}$.