satisfy the conditions of Stone’s theorem (Stone, 1977), which guarantees the universal consistency of a large family of local
averaging methods. Devroye et al. (1996, Thm. 6.1) and Györﬁ et al. (2002, Thm. 4.2) show that partitioning estimators such
as decision trees qualify provided that (1) diam(X(cid:96)) →p 0 and (2) n(cid:96) →p ∞ for all (cid:96) as n → ∞, effectively creating leaves
of inﬁnite density. The former is derived by Meinshausen (2006, Lemma 2) under (A3).(iii) and (A3).(iv); the latter follows
trivially from (A3).(v). Thus RF discriminators weakly converge on the Bayes risk in the large sample limit, completing part
(a) of the proof.
Desideratum (b) effectively says that original and synthetic data become indistinguishable as n and t increase. Recall
n . Let θ(0)
denote the resulting splits once the discriminator has converged. In subsequent rounds, synthetic data are sampled according
Adversarial Random Forests for Density Estimation and Generative Modeling