Uniform lower bounds for approximating general Gaussian location–scale mixtures

Establish nontrivial uniform lower bounds on the total variation approximation error when approximating a general Gaussian location–scale mixture f_H(x) = E[φ((x − X)/S)/S] by an m-atomic location–scale mixture H_m, i.e., derive lower bounds on inf_{H_m ∈ 𝒫_m} TV(f_{H_m}, f_H) that hold uniformly over all m-atomic pairs (X_m, S_m) beyond special cases such as pure scale mixtures with X ≡ 0.

Background

In extending their framework to Gaussian location–scale mixtures, the authors present a variational lower bound in terms of characteristic functions but point out that obtaining uniform lower bounds over all m-atomic approximations is challenging. They provide partial results for the special case of pure scale mixtures (X ≡ 0) with bounded S, but a general theory is missing.

This open problem targets the development of uniform lower bounds (e.g., rates in m) for the approximation error in total variation for the broader class of location–scale mixtures, which would parallel the results established for pure location mixtures.