  The last decade has witnessed a growing interest in random forest models
which are recognized to exhibit good practical performance, especially in
high-dimensional settings. On the theoretical side, however, their predictive
power remains largely unexplained, thereby creating a gap between theory and
practice. The aim of this paper is twofold. Firstly, we provide theoretical
guarantees to link finite forests used in practice (with a finite number M of
trees) to their asymptotic counterparts. Using empirical process theory, we
prove a uniform central limit theorem for a large class of random forest
estimates, which holds in particular for Breiman's original forests. Secondly,
we show that infinite forest consistency implies finite forest consistency and
thus, we state the consistency of several infinite forests. In particular, we
prove that q quantile forests---close in spirit to Breiman's forests but easier
to study---are able to combine inconsistent trees to obtain a final consistent
prediction, thus highlighting the benefits of random forests compared to single
trees.
