# On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies

arXiv: https://arxiv.org/abs/2207.12382 · HTML: https://mlsanity.com/paper/2207.12382/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2022.

## Built on by (most cited first, 8 of 8)

- [E-Values Expand the Scope of Conformal Prediction](https://mlsanity.com/paper/2503.13050/) (2025)
- [Treatment of Statistical Estimation Problems in Randomized Smoothing for Adversarial Robustness](https://mlsanity.com/paper/2406.17830/) (2024)
- [Risk-limiting Financial Audits via Weighted Sampling without Replacement](https://mlsanity.com/paper/2305.06884/) (2023)
- [Improved Offline Contextual Bandits with Second-Order Bounds: Betting and Freezing](https://mlsanity.com/paper/2502.10826/) (2025)
- [Global Sequential Testing for Multi-Stream Auditing](https://mlsanity.com/paper/2602.21479/) (2026)
- [Multi-Armed Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2603.17925/) (2026)
- [Learning to Bet for Horizon-Aware Anytime-Valid Testing](https://mlsanity.com/paper/2603.19551/) (2026)
- [Always-On Experimentation](https://mlsanity.com/paper/2609.38695/) (2026)

## Abstract (arXiv)

This paper considers the problem of constructing a confidence sequence, which is a sequence of confidence intervals that hold uniformly over time, for estimating the mean of bounded real-valued random processes. This paper revisits the gambling-based approach established in the recent literature from a natural two-horse race perspective, and demonstrates new properties of the resulting algorithm induced by Cover (1991)'s universal portfolio. The main result of this paper is a new algorithm based on a mixture of lower bounds, which closely approximates the performance of Cover's universal portfolio with constant per-round time complexity. A higher-order generalization of a lower bound on a logarithmic function in (Fan et al., 2015), which is developed as a key technique for the proposed algorithm, may be of independent interest.