Convergence of independent permutation compositions under nondegeneracy

Establish whether weakly convergent independent permutation sequences [?] satisfy composition convergence under an additional variance lower bound or comparable nondegeneracy condition; specifically, determine whether [?] implies [?] when the sequences are independent and nondegenerate.

Background

The paper proves that weak convergence of permutation sequences [?] does not generally imply weak convergence of their compositions [?]. The counterexample uses the strong dependence [?]. The authors explain that the obstruction is an asymptotic correlation between operator errors and identify independence, supplemented by a variance lower bound or similar condition, as a plausible setting in which composition convergence might hold. They explicitly note that no independent nondeterministic counterexample is known to them.

References

For instance, one might conjecture that, if $\sigma_N$ and $\tau_N$ are independent, then eq:continuity conjecture holds. This cannot quite be true, since our counterexample argument yields deterministic sequences $\sigma_N \to U, \tau_N \to U$ with $\sigma_N \tau_N \to \Delta \ne U$, and such sequences would technically be independent. However, we have not constructed any independent non-deterministic counterexamples. Hence it seems plausible to conjecture that eq:continuity conjecture holds if one assumes a variance lower bound or some similar nondegeneracy condition in addition to independence.

— On the composition of permutons  (2610.12945 - Gorski, 9 Oct 2026) in Section 7, Section "Future work"