Characterize distributions yielding vanishing binomial empirical process

Characterize the probability measures μ on {0,1}^N for which the binomial empirical process satisfies Δ_n(μ) = E sup_{j∈N} |(1/n)∑_{i=1}^n X_j^{(i)} − E[X_j]| → 0 as n→∞ for i.i.d. samples X^{(i)} ∼ μ; additionally, ascertain whether the sufficient condition described in Theorem more_general_suff_condition—namely, total boundedness of (N, ξ) together with the existence, for every ε>0, of events (E_k) with P(E_k) ≤ ε that cover each “bad event” {X_i ≠ X_j} using at most K such E_k— is also necessary for Δ_n(μ) → 0.

Background

The paper studies the binomial empirical process Δ_n(μ) over distributions μ on {0,1}^N, focusing on when Δ_n(μ) vanishes as n grows. For independent coordinates, sharp characterizations and rates are known, but with dependence the picture is subtler.

The authors establish that covariance (and even third moments) do not determine the behavior of Δ_n(μ), and provide necessary (total boundedness in (N, ξ)) and sufficient conditions (in terms of ξ- or ρ-covering numbers and a more general event-covering condition) for Δ_n(μ) → 0. However, a full characterization remains unresolved, and it is unclear whether their sufficient structural condition is also necessary.

This problem asks for a complete description of all μ for which Δ_n(μ) → 0, and whether the structural "event-covering" sufficient condition in Theorem more_general_suff_condition is in fact a necessary condition as well.