Interpretation of the range space condition for countable discrete domains

Investigate and characterize the range space condition Ran(T_{P0}^θ) induced by the Stein kernel K0 and the integral operator T_{P0}: L^2(P0)→L^2(P0) when the domain is countable (so that L^2(P0) is isomorphic to ℓ^2), and develop a detailed interpretation of the corresponding smoothness class for KSD-based goodness-of-fit testing on countably supported discrete spaces.

Background

The paper’s minimax framework defines alternatives via the range space Ran(T_{P0}^θ), capturing smoothness through the spectrum of the operator built from the Stein kernel and the null distribution. While several continuous-domain examples are analyzed, the authors point out that for countably supported discrete domains, L^2(P0) becomes ℓ^2 and the structure of Ran(T_{P0}^θ) needs dedicated treatment.

This open problem seeks a thorough, domain-specific understanding of the range-space smoothness condition for countable discrete settings, which would clarify the alternative classes and guide KSD-based testing in such domains.