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. Our results include a dimension-free CSS for log-concave random vectors, a dimension-free CSS for sub-Gaussian random vectors, and CSSs for sub-$\psi$ random vectors (which includes sub-gamma, sub-Poisson, and sub-exponential distributions). For sub-Gaussian distributions we also provide a CSS which tracks a time-varying mean, generalizing Robbins' mixture approach to the multivariate setting. Finally, we provide several CSSs for heavy-tailed random vectors (two moments only). Our bounds hold under a martingale assumption on the mean and do not require that the observations be iid. Our work is based on PAC-Bayesian theory and inspired by an approach of Catoni and Giulini.

翻译：本文推导并研究了一种时域一致置信球面——置信球面序列(CSSs)——该序列能以高概率同时覆盖所有样本量下的随机向量均值。我们的研究成果包括：针对对数凹随机向量的维度无关CSS、针对亚高斯随机向量的维度无关CSS，以及针对亚ψ随机向量（涵盖亚伽马、亚泊松和亚指数分布）的CSS。对于亚高斯分布，我们还提出了一种能追踪时变均值的CSS，将罗宾斯混合方法推广到多元场景。最后，我们为仅具有二阶矩的重尾随机向量提供了若干CSS。所得界在均值满足鞅假设的条件下成立，且不要求观测值独立同分布。本工作基于PAC-贝叶斯理论，并受到Catoni与Giulini研究方法的启发。