Gaussian approximation under weaker moment conditions

Determine whether comparable higher-order Wasserstein Gaussian approximation bounds hold for multivariate martingale sums generated by uniformly ergodic Markov chains under moment conditions weaker than the L^{(2+\eta)p} condition used in the main theorem.

Background

The main theorem assumes h belongs to Lq with q=(2+\eta)p, because the proof requires an intermediate exponent s satisfying p<s<q/2. The authors explain that, if explicit dependence on the Wasserstein order p is not required, their argument can be adapted to the weaker L{2p+\eta} condition. Whether comparable bounds can be obtained under still weaker moment assumptions is left unresolved.

References

It remains open whether comparable bounds hold under weaker moment conditions.

— Gaussian Approximation for Multivariate Martingale Sums from Uniformly Ergodic Markov Chains  (2609.09480 - Zhang et al., 8 Sep 2026) in Section 2, subsection “L^{(2+\eta)p-moment condition”