# Catoni-style confidence sequences for heavy-tailed mean estimation

arXiv: https://arxiv.org/abs/2202.01250 · HTML: https://mlsanity.com/paper/2202.01250/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2022.

## Built on by (most cited first, 17 of 17)

- [E-detectors: a nonparametric framework for sequential change detection](https://mlsanity.com/paper/2203.03532/) (2022)
- [A unified recipe for deriving (time-uniform) PAC-Bayes bounds](https://mlsanity.com/paper/2302.03421/) (2023)
- [Huber-Robust Confidence Sequences](https://mlsanity.com/paper/2301.09573/) (2023)
- [Risk-limiting Financial Audits via Weighted Sampling without Replacement](https://mlsanity.com/paper/2305.06884/) (2023)
- [Time-Uniform Confidence Spheres for Means of Random Vectors](https://mlsanity.com/paper/2311.08168/) (2023)
- [Improved Offline Contextual Bandits with Second-Order Bounds: Betting and Freezing](https://mlsanity.com/paper/2502.10826/) (2025)
- [Sequential change detection via backward confidence sequences](https://mlsanity.com/paper/2302.02544/) (2023)
- [Catoni-style Confidence Sequences under Infinite Variance](https://mlsanity.com/paper/2208.03185/) (2022)
- [A lower confidence sequence for the changing mean of non-negative right heavy-tailed observations with bounded mean](https://mlsanity.com/paper/2210.11133/) (2022)
- [Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance](https://mlsanity.com/paper/2310.03722/) (2023)
- [Segmented Confidence Sequences and Multi-Scale Adaptive Confidence Segments for Anomaly Detection in Nonstationary Time Series](https://mlsanity.com/paper/2508.06638/) (2025)
- [Online Heavy-tailed Change-point detection](https://mlsanity.com/paper/2306.09548/) (2023)
- [Positive Semidefinite Matrix Supermartingales](https://mlsanity.com/paper/2401.15567/) (2024)
- [Catoni-Style Change Point Detection for Regret Minimization in Non-Stationary Heavy-Tailed Bandits](https://mlsanity.com/paper/2505.20051/) (2025)
- [UCB for Large-Scale Pure Exploration: Beyond Sub-Gaussianity](https://mlsanity.com/paper/2511.22273/) (2025)
- [Bentkus-type asymptotic e-values](https://mlsanity.com/paper/2606.06332/) (2026)
- [Robust Multi-Agent Bandits with Heavy-Tailed Rewards and Information Asymmetry](https://mlsanity.com/paper/2608.10529/) (2026)

## Abstract (arXiv)

A confidence sequence (CS) is a sequence of confidence intervals that is valid at arbitrary data-dependent stopping times. These are useful in applications like A/B testing, multi-armed bandits, off-policy evaluation, election auditing, etc. We present three approaches to constructing a confidence sequence for the population mean, under the minimal assumption that only an upper bound σ2 on the variance is known. While previous works rely on light-tail assumptions like boundedness or subGaussianity (under which all moments of a distribution exist), the confidence sequences in our work are able to handle data from a wide range of heavy-tailed distributions. The best among our three methods -- the Catoni-style confidence sequence -- performs remarkably well in practice, essentially matching the state-of-the-art methods for σ2-subGaussian data, and provably attains the √log log t/t lower bound due to the law of the iterated logarithm. Our findings have important implications for sequential experimentation with unbounded observations, since the σ2-bounded-variance assumption is more realistic and easier to verify than σ2-subGaussianity (which implies the former). We also extend our methods to data with infinite variance, but having p-th central moment (1<p<2).