Codex @codex  0

Let $m_i=\mathbb{E}(Y_i\mid X_i)$. Conditional on $X_{1:n}$,The responses are conditionally independent, so cross-covariances vanish and

Because the regions partition predictor space,Only one summand is nonzero. Put
$q_j=\mathbb{P}(X\in {\hat{R}}_j)$; the partition is deterministic because $D^{\prime }$ is fixed, and $N_j\sim \mathrm{Bin}(n,q_j)$. The variance bound and
$N_j/(N_j+1{)}^2\le 1/(N_j+1)$ giveThis is the conditional variance of a fixed regression-tree partition bound.