# Asymptotic Theory for Random Forests

arXiv: https://arxiv.org/abs/1405.0352 · HTML: https://mlsanity.com/paper/1405.0352/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 May 2014.

## Built on by (most cited first, 4 of 4)

- [Meta-Learners for Estimation of Causal Effects: Finite Sample Cross-Fit Performance](https://mlsanity.com/paper/2201.12692/) (2022)
- [Modelling hetegeneous treatment effects by quantitle local polynomial decision tree and forest](https://mlsanity.com/paper/2111.15320/) (2021)
- [Confidence Band Estimation for Survival Random Forests](https://mlsanity.com/paper/2204.12038/) (2022)
- [Exact Finite-Sample Variance Decomposition of Subagging: A Spectral Filtering Perspective](https://mlsanity.com/paper/2604.10469/) (2026)

## Abstract (arXiv)

Random forests have proven to be reliable predictive algorithms in many application areas. Not much is known, however, about the statistical properties of random forests. Several authors have established conditions under which their predictions are consistent, but these results do not provide practical estimates of random forest errors. In this paper, we analyze a random forest model based on subsampling, and show that random forest predictions are asymptotically normal provided that the subsample size s scales as s(n)/n = o(log(n)^{-d}), where n is the number of training examples and d is the number of features. Moreover, we show that the asymptotic variance can consistently be estimated using an infinitesimal jackknife for bagged ensembles recently proposed by Efron (2014). In other words, our results let us both characterize and estimate the error-distribution of random forest predictions, thus taking a step towards making random forests tools for statistical inference instead of just black-box predictive algorithms.