Tight approximation rates under moment constraints for heavy-tailed mixing distributions

Determine the sharp (tight up to constants) polynomial rate in m of the best total variation approximation error (m, ℳ_α(β), TV) for mixing distributions with finite α-th moment (heavy-tailed families), closing the gap between the current upper and lower bounds presented for ℳ_α(β).

Background

The authors extend their analysis to mixing distributions controlled by moment constraints, defining ℳ_α(β) as the family with bounded α-th moments. They derive polynomial upper and lower bounds in m for the best approximation error in total variation but note a nontrivial gap between these bounds.

They suggest that characterizing the information geometry of Gaussian mixture families might help, but the exact rate remains unknown. The open problem is to close this gap by proving tight rates for heavy-tailed mixing distributions under moment constraints.