# Asymptotic Distributions and Rates of Convergence for Random Forests via Generalized U-statistics

arXiv: https://arxiv.org/abs/1905.10651 · HTML: https://mlsanity.com/paper/1905.10651/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2019.

## Built on by (most cited first, 18 of 18)

- [Adversarial random forests for density estimation and generative modeling](https://mlsanity.com/paper/2205.09435/) (2022)
- [Random Forest Weighted Local Fr'echet Regression with Random Objects](https://mlsanity.com/paper/2202.04912/) (2022)
- [MDA for random forests: inconsistency, and a practical solution via the Sobol-MDA](https://mlsanity.com/paper/2102.13347/) (2021)
- [Bridging Breiman's Brook: From Algorithmic Modeling to Statistical Learning](https://mlsanity.com/paper/2102.12328/) (2021)
- [On Variance Estimation of Random Forests with Infinite-Order U-statistics](https://mlsanity.com/paper/2202.09008/) (2022)
- [The Infinitesimal Jackknife and Combinations of Models](https://mlsanity.com/paper/2209.00147/) (2022)
- [Extrapolated cross-validation for randomized ensembles](https://mlsanity.com/paper/2302.13511/) (2023)
- [Forest Guided Smoothing](https://mlsanity.com/paper/2103.05092/) (2021)
- [On Differentially Private U Statistics](https://mlsanity.com/paper/2407.04945/) (2024)
- [Jackknife Variance Estimation for H'ajek-Dominated Generalized U-Statistics](https://mlsanity.com/paper/2509.12356/) (2025)
- [Confidence Band Estimation for Survival Random Forests](https://mlsanity.com/paper/2204.12038/) (2022)
- [Orthogonal Series Estimation for the Ratio of Conditional Expectation Functions](https://mlsanity.com/paper/2212.13145/) (2022)
- [Riemann-Lebesgue Forest for Regression](https://mlsanity.com/paper/2402.04550/) (2024)
- [Global Censored Quantile Random Forest](https://mlsanity.com/paper/2410.12209/) (2024)
- [A Central Limit Theorem for the permutation importance measure](https://mlsanity.com/paper/2412.13020/) (2024)
- [Empirical Likelihood for Random Forests and Ensembles](https://mlsanity.com/paper/2511.13934/) (2025)
- [Asymptotic Theory and Phase Transitions for Variable Importance in Quantile Regression Forests](https://mlsanity.com/paper/2511.23212/) (2025)
- [Consistency of Honest Decision Trees and Random Forests](https://mlsanity.com/paper/2601.14991/) (2026)

## Abstract (arXiv)

Random forests remain among the most popular off-the-shelf supervised learning algorithms. Despite their well-documented empirical success, however, until recently, few theoretical results were available to describe their performance and behavior. In this work we push beyond recent work on consistency and asymptotic normality by establishing rates of convergence for random forests and other supervised learning ensembles. We develop the notion of generalized U-statistics and show that within this framework, random forest predictions can potentially remain asymptotically normal for larger subsample sizes than previously established. We also provide Berry-Esseen bounds in order to quantify the rate at which this convergence occurs, making explicit the roles of the subsample size and the number of trees in determining the distribution of random forest predictions.