We derive and study time-uniform confidence spheres -- 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. Our results include an empirical-Bernstein CSS for bounded random vectors (resulting in a novel empirical-Bernstein confidence interval with asymptotic width scaling proportionally to the true unknown variance), CSSs for sub-$\psi$ random vectors (which includes sub-gamma, sub-Poisson, and sub-exponential), and CSSs for heavy-tailed random vectors (two moments only). Finally, we provide two CSSs that are robust to contamination by Huber noise. The first is a robust version of our empirical-Bernstein CSS, and the second extends recent work in the univariate setting to heavy-tailed multivariate distributions.

翻译：我们推导并研究了时间一致置信球（置信球序列，CSSs），该序列能够以高概率同时覆盖所有样本量下随机向量的均值。受Catoni与Giulini开创性工作的启发，我们统一并扩展了其分析框架，使其既适用于序贯设定，又能处理多种分布假设。我们的结果包括：有界随机向量的经验-伯恩斯坦CSS（由此得到一种渐近宽度与真实未知方差成比例的新型经验-伯恩斯坦置信区间）、子-$\psi$随机向量（涵盖子伽马、子泊松和子指数分布）的CSS，以及重尾随机向量（仅需二阶矩）的CSS。最后，我们提出两种对Huber噪声污染具有鲁棒性的CSS：第一种是经验-伯恩斯坦CSS的鲁棒版本，第二种将近期单变量情形下的工作扩展至重尾多变量分布。