The idea of the restricted mean has been used to establish a significantly improved version of Markov's inequality that does not require any new assumptions. The result immediately extends on Chebyshev's inequalities and Chernoff's bound. The improved Markov inequality yields a bound that is hundreds or thousands of times more accurate than the original Markov bound for high quantiles in the most prevalent and diverse situations. The Markov inequality benefits from being model-independent, and the long-standing issue of its imprecision is solved. Practically speaking, avoidance of model risk is decisive when multiple competing models are present in a real-world situation.

翻译：受限均值概念被用于建立一种显著改进版的马尔可夫不等式，且无需引入任何新假设。该结果可直接推广至切比雪夫不等式和切尔诺夫界。在最具普遍性和多样性的场景下，改进后的马尔可夫不等式对高尾分位数给出的界值，其精度比原始马尔可夫界高出数百甚至数千倍。马尔可夫不等式具有模型无关性的优势，而其长期存在的精度不足问题由此得到解决。从实践角度看，当现实场景中存在多个竞争模型时，规避模型风险具有决定性意义。