A confidence sequence (CS) is a sequence of confidence sets that contains a target parameter of an underlying stochastic process at any time step with high probability. This paper proposes a new approach to constructing CSs for means of bounded multivariate stochastic processes using a general gambling framework, extending the recently established coin toss framework for bounded random processes. The proposed gambling framework provides a general recipe for constructing CSs for categorical and probability-vector-valued observations, as well as for general bounded multidimensional observations through a simple reduction. This paper specifically explores the use of the mixture portfolio, akin to Cover's universal portfolio, in the proposed framework and investigates the properties of the resulting CSs. Simulations demonstrate the tightness of these confidence sequences compared to existing methods. When applied to the sampling without-replacement setting for finite categorical data, it is shown that the resulting CS based on a universal gambling strategy is provably tighter than that of the posterior-prior ratio martingale proposed by Waudby-Smith and Ramdas.

翻译：置信序列（CS）是指在任意时间步长内以高概率包含底层随机过程目标参数的置信集序列。本文提出一种基于通用赌博框架构建有界多元随机过程均值CS的新方法，将近期建立的有界随机过程的掷硬币框架进行扩展。该赌博框架为分类观测值和概率向量值观测值，以及通过简单降维处理的一般有界多维观测值提供了构建CS的通用方案。本文重点探索了类似Cover通用投资组合的混合投资组合在该框架中的应用，并研究了由此产生的CS的性质。仿真结果表明，与现有方法相比，这些置信序列具有更紧的界。当应用于有限分类数据的无放回抽样时，基于通用赌博策略的CS被证明在紧致性上严格优于Waudby-Smith和Ramdas提出的后验-先验比鞅方法。