  Regression trees and random forests are popular and effective non-parametric
estimators in practical applications. A recent paper by Athey and Wager shows
that the random forest estimate at any point is asymptotically Gaussian; in
this paper, we extend this result to the multivariate case and show that the
vector of estimates at multiple points is jointly normal. Specifically, the
covariance matrix of the limiting normal distribution is diagonal, so that the
estimates at any two points are independent in sufficiently deep trees.
Moreover, the off-diagonal term is bounded by quantities capturing how likely
two points belong to the same partition of the resulting tree. Our results
relies on certain a certain stability property when constructing splits, and we
give examples of splitting rules for which this assumption is and is not
satisfied. We test our proposed covariance bound and the associated coverage
rates of confidence intervals in numerical simulations.
