We present simple randomized and exchangeable improvements of Markov's inequality, as well as Chebyshev's inequality and Chernoff bounds. Our variants are never worse and typically strictly more powerful than the original inequalities. The proofs are short and elementary, and can easily yield similarly randomized or exchangeable versions of a host of other inequalities that employ Markov's inequality as an intermediate step. We point out some simple statistical applications involving tests that combine dependent e-values. In particular, we uniformly improve the power of universal inference, and obtain tighter betting-based nonparametric confidence intervals. Simulations reveal nontrivial gains in power (and no losses) in a variety of settings.

翻译：我们提出了马尔可夫不等式、切比雪夫不等式以及切尔诺夫界的简单随机化与可交换改进版本。我们的变体从不逊于原始不等式，且通常具有严格更强的效能。证明过程简洁且初等，并能轻松推及大量以马尔可夫不等式为中间步骤的其他不等式，为其生成类似的随机化或可交换版本。我们指出了若干涉及依赖e值检验的简单统计应用。特别地，我们统一提升了通用推断的统计效能，并获得了更紧致的基于对赌的非参数置信区间。仿真实验表明，在多种场景下该方法能带来显著的效能增益（且无任何损失）。