5Several authors have conjectured that RF consistency may not require honesty (A3).(i) or subsampling (A3).(ii) after all. Empirical
performance certainly seems unencumbered by these requirements. However, both come with major theoretical advantages—the former
by making predictions conditionally independent of the training data while preserving some form of adaptive splits, the latter by avoiding
thorny issues arising from duplicated samples when bootstrapping. See Biau (2012, Rmk. 8), Wager and Athey (2018, Appx. B), and Tang
et al. (2018) for a discussion.

Watson, Blesch, Kapar, & Wright

off-manifold data at each iteration t, no matter how small vol(X(cid:96)) becomes. However, this only shows that convergence can
fail for ﬁnite t. Since the discriminator is consistent, it will accurately identify synthetic points in round t + 1, pruning the
space still further.
Adversarial Random Forests for Density Estimation and Generative Modeling