The non‐asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probabilities in literature, the lower bounds on tail probabilities are relatively fewer. In this paper, we introduce systematic and user‐friendly schemes for developing non‐asymptotic lower bounds of tail probabilities. In addition, we develop sharp lower tail bounds for the sum of independent sub‐Gaussian and sub‐exponential random variables, which match the classic Hoeffding‐type and Bernstein‐type concentration inequalities, respectively. We also provide non‐asymptotic matching upper and lower tail bounds for a suite of distributions, including gamma, beta, (regular, weighted, and noncentral) chi‐square, binomial, Poisson, Irwin–Hall, etc. We apply the result to establish the matching upper and lower bounds for extreme value expectation of the sum of independent sub‐Gaussian and sub‐exponential random variables. A statistical application of signal identification from sparse heterogeneous mixtures is finally considered.

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关于随机变量的非渐近和尖锐的下尾边界

随机变量的非渐近尾部边界在概率，统计和机器学习中起着至关重要的作用。尽管在文献中成功地建立了尾部概率的上限，但尾部概率的下限相对较少。在本文中，我们介绍了系统的和用户友好的方案，用于开发尾概率的非渐近下界。此外，我们为独立的亚高斯和亚指数随机变量的总和开发了尖锐的下尾边界，分别与经典的Hoeffding型和Bernstein型浓度不等式匹配。我们还提供了一系列分布的非渐近匹配上下限，包括伽马，β，（正态，加权和非中心）卡方，二项式，泊松，欧文·霍尔等。我们应用该结果为独立的亚高斯和亚指数随机变量之和的极值期望建立匹配的上限和下限。最后考虑从稀疏异质混合物中进行信号识别的统计应用。