Editing Random forest (section)

=== Consistency results ===
Assume that <math>Y = m(\mathbf{X}) + \varepsilon</math>, where <math>\varepsilon</math> is a centered Gaussian noise, independent of <math>\mathbf{X}</math>, with finite variance <math>\sigma^2<\infty</math>. Moreover, <math>\mathbf{X}</math> is uniformly distributed on <math>[0,1]^d</math> and <math>m</math> is [[Lipschitz_continuity|Lipschitz]]. Scornet<ref name="scornet2015random"/> proved upper bounds on the rates of consistency for centered KeRF and uniform KeRF.

==== Consistency of centered KeRF ====
Providing <math>k\rightarrow\infty</math> and <math>n/2^k\rightarrow\infty</math>, there exists a constant <math>C_1>0</math> such that, for all <math>n</math>,
<math> \mathbb{E}[\tilde{m}_n^{cc}(\mathbf{X}) - m(\mathbf{X})]^2 \le C_1 n^{-1/(3+d\log 2)}(\log n)^2</math>.

==== Consistency of uniform KeRF ====
Providing <math>k\rightarrow\infty</math> and <math>n/2^k\rightarrow\infty</math>, there exists a constant <math>C>0</math> such that,
<math>\mathbb{E}[\tilde{m}_n^{uf}(\mathbf{X})-m(\mathbf{X})]^2\le Cn^{-2/(6+3d\log2)}(\log n)^2</math>.