# Time-Uniform Self-Normalized Concentration for Vector-Valued Processes

arXiv: https://arxiv.org/abs/2310.09100 · HTML: https://mlsanity.com/paper/2310.09100/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2023.

## Built on by (most cited first, 9 of 9)

- [Vector-valued self-normalized concentration inequalities beyond sub-Gaussianity](https://mlsanity.com/paper/2511.03606/) (2025)
- [Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximation](https://mlsanity.com/paper/2410.15057/) (2024)
- [Low coordinate degree algorithms II: Categorical signals and generalized stochastic block models](https://mlsanity.com/paper/2412.21155/) (2024)
- [Eluder dimension: localise it!](https://mlsanity.com/paper/2601.09825/) (2026)
- [A variational approach to dimension-free self-normalized concentration](https://mlsanity.com/paper/2508.06483/) (2025)
- [Interactive Learning of Single-Index Models via Stochastic Gradient Descent](https://mlsanity.com/paper/2602.17876/) (2026)
- [Self-Normalized Martingales and Uniform Regret Bounds for Linear Regression](https://mlsanity.com/paper/2605.01628/) (2026)
- [On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequences](https://mlsanity.com/paper/2610.03727/) (2026)
- [High-probability zeroth-order online convex optimisation beyond Euclidean geometry](https://mlsanity.com/paper/2509.21484/) (2025)

## Abstract (arXiv)

Self-normalized processes arise naturally in many learning-related tasks. While self-normalized concentration has been extensively studied for scalar-valued processes, there are few results for multidimensional processes outside of the sub-Gaussian setting. In this work, we construct a general, self-normalized inequality for multivariate processes that satisfy a simple yet broad sub-psi tail condition, which generalizes assumptions based on cumulant generating functions. From this general inequality, we derive an upper law of the iterated logarithm for sub-psi vector-valued processes, which is tight up to small constants. We show how our inequality can be leveraged to derive a variety of novel, self-normalized concentration inequalities under both light and heavy-tailed observations. Further, we provide applications in prototypical statistical tasks, such as parameter estimation in online linear regression, autoregressive modeling, and bounded mean estimation via a new (multivariate) empirical Bernstein concentration inequality.