  Random forests are a scheme proposed by Leo Breiman in the 2000's for
building a predictor ensemble with a set of decision trees that grow in
randomly selected subspaces of data. Despite growing interest and practical
use, there has been little exploration of the statistical properties of random
forests, and little is known about the mathematical forces driving the
algorithm. In this paper, we offer an in-depth analysis of a random forests
model suggested by Breiman in \cite{Bre04}, which is very close to the original
algorithm. We show in particular that the procedure is consistent and adapts to
sparsity, in the sense that its rate of convergence depends only on the number
of strong features and not on how many noise variables are present.
