We study distribution testing in the standard access model and the conditional access model when the memory available to the testing algorithm is bounded. In both scenarios, the samples appear in an online fashion and the goal is to test the properties of distribution using an optimal number of samples subject to a memory constraint on how many samples can be stored at a given time. First, we provide a trade-off between the sample complexity and the space complexity for testing identity when the samples are drawn according to the conditional access oracle. We then show that we can learn a succinct representation of a monotone distribution efficiently with a memory constraint on the number of samples that are stored that is almost optimal. We also show that the algorithm for monotone distributions can be extended to a larger class of decomposable distributions.

翻译：我们研究了在标准访问模型和条件访问模型中，当测试算法可用内存有限时的分布测试问题。在这两种场景下，样本以在线方式出现，目标是在给定时间点可存储样本数量的内存约束下，使用最优样本数测试分布的性质。首先，我们给出了当样本根据条件访问预言机抽取时，测试同一性所需的样本复杂度与空间复杂度之间的权衡。随后，我们证明能够以近乎最优的内存约束（针对存储样本数量）高效学习单调分布的简洁表示。我们还表明，针对单调分布的算法可以推广到更大类别的可分解分布。