Extend HolderGauss approximations to other dependence concepts

Establish whether the HolderGauss approximations for the proposed high-dimensional multiscale scan statistics remain valid under dependence concepts other than \(\beta\)-mixing, such as physical dependence.

Background

Most of the paper assumes independent noise vectors, while the appendix establishes versions of the HolderGauss approximation for the logarithmically and polynomially weighted scan statistics under geometric $\beta$-mixing. The authors suggest that analogous results may hold under other dependence frameworks, specifically physical dependence, but no such extension is developed or proved. The problem concerns both the validity of the Gaussian approximation and the associated multiscale inference guarantees under these alternative dependence assumptions.