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。这些新变体从不会弱于原始不等式，且在典型情况下具有严格更强的效力。证明过程简洁且初等，并可轻松推广至其他以马尔可夫不等式为中间步骤的大量不等式，使其获得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，我们一致改进了普适推断的效力，并获得了更紧的基于赌博的非参数置信区间。模拟实验表明，在各种设定下均能获得非平凡的效力提升（且无任何损失）。