We introduce the first probabilistic framework tailored for sequential random projection, an approach rooted in the challenges of sequential decision-making under uncertainty. The analysis is complicated by the sequential dependence and high-dimensional nature of random variables, a byproduct of the adaptive mechanisms inherent in sequential decision processes. Our work features a novel construction of a stopped process, facilitating the analysis of a sequence of concentration events that are interconnected in a sequential manner. By employing the method of mixtures within a self-normalized process, derived from the stopped process, we achieve a desired non-asymptotic probability bound. This bound represents a non-trivial martingale extension of the Johnson-Lindenstrauss (JL) lemma, marking a pioneering contribution to the literature on random projection and sequential analysis.

翻译：我们首次引入了专为序列随机投影设计的概率框架，该方法根植于不确定性条件下序列决策中的挑战。分析过程因随机变量之间的序列依赖性和高维特性而变得复杂，这是序列决策过程中自适应机制产生的副产品。我们的工作创新性地构建了一个停时过程，有助于分析以序列方式相互关联的一系列浓度事件。通过将混合方法应用于源自停时过程的自归一化过程，我们得到了所需的非渐近概率界。该界是Johnson-Lindenstrauss (JL)引理的一个非平凡鞅扩展，标志着对随机投影和序列分析文献的开创性贡献。