Moments do not suffice to characterize Δ_n: extend beyond covariance and third moments

Prove that for every fixed integer k ≥ 1, there exist probability measures μ and ν on {0,1}^N that share all moments up to order k but have different asymptotic behaviors of the binomial empirical process, namely one with Δ_n(·) → 0 and the other with Δ_n(·) not → 0, thereby showing that knowledge of kth-order moments does not suffice to characterize the decay of Δ_n.

Background

The authors construct two processes with identical pairwise covariances (and even identical third moments) but diverging behaviors of Δ_n(·): in one case Δ_n → 0, in the other Δ_n → 1/2. This disproves the hypothesis that covariance (or third-order moments) determine Δ_n’s convergence.

They conjecture that this phenomenon extends to any fixed moment order: for any k, matching moments up to order k still do not pin down whether Δ_n(μ) vanishes.