We derive and study time-uniform confidence spheres - termed confidence sphere sequences (CSSs) - which contain the mean of random vectors with high probability simultaneously across all sample sizes. Inspired by the original work of Catoni and Giulini, we unify and extend their analysis to cover both the sequential setting and to handle a variety of distributional assumptions. More concretely, our results include an empirical-Bernstein CSS for bounded random vectors (resulting in a novel empirical-Bernstein confidence interval), a CSS for sub-$\psi$ random vectors, and a CSS for heavy-tailed random vectors based on a sequentially valid Catoni-Giulini estimator. Finally, we provide a version of our empirical-Bernstein CSS that is robust to contamination by Huber noise.

翻译：我们推导并研究了时间一致的置信球面——称为置信球面序列（CSS），这些球面以高概率在全体样本量下同时包含随机向量的均值。受Catoni和Giulini原创工作的启发，我们统一并扩展了他们的分析，使其涵盖序列设置，并处理多种分布假设。更具体地说，我们的结果包括：针对有界随机向量的经验-伯恩斯坦CSS（从而得到一种新颖的经验-伯恩斯坦置信区间）、针对子-ψ随机向量的CSS，以及基于序列有效的Catoni-Giulini估计量针对重尾随机向量的CSS。最后，我们提供了能够抵抗Huber噪声污染的经验-伯恩斯坦CSS版本。