  This paper investigates predictive probability inference for classification
tasks using random forests in the context of imbalanced data. In this setting,
we analyze the asymptotic properties of simplified versions of the original
Breiman's algorithm, namely subsampling and under-sampling Infinite Random
Forests (IRFs), and establish the asymptotic normality of these two models. The
under-sampling IRFs, that tackle the challenge of the predicting the minority
class by a downsampling strategy to under-represent the majority class show
asymptotic bias. To address this problem, we introduce a new estimator based on
an Importance Sampling debiasing procedure built upon on odds ratios. We apply
our results considering 1-Nearest Neighbors (1-NN) as individual trees of the
IRFs. The resulting bagged 1-NN estimator achieves the same asymptotic rate in
the three configurations but with a different asymptotic variance. Finally, we
conduct simulations to validate the empirical relevance of our theoretical
findings.
