We study sequential mean estimation in $\mathbb{R}^d$. In particular, we derive 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). Many of our results are optimal. 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：我们研究 $\mathbb{R}^d$ 中的序贯均值估计问题。特别地，我们推导了时域一致置信球面——置信球面序列（CSSs）——该序列以高概率同时覆盖所有样本量下的随机向量均值。我们的结果包括：对数凹随机向量的维度无关CSS、亚高斯随机向量的维度无关CSS，以及亚-$\psi$ 随机向量（包含亚伽马、亚泊松和亚指数分布）的CSS。其中许多结果具有最优性。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。我们的边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。本研究基于PAC-贝叶斯理论，并受到卡托尼和朱利尼方法的启发。