One of the most commonly used methods for forming confidence intervals for statistical inference is the empirical bootstrap, which is especially expedient when the limiting distribution of the estimator is unknown. However, despite its ubiquitous role, its theoretical properties are still not well understood for non-asymptotically normal estimators. In this paper, under stability conditions, we establish the limiting distribution of the empirical bootstrap estimator, derive tight conditions for it to be asymptotically consistent, and quantify the speed of convergence. Moreover, we propose three alternative ways to use the bootstrap method to build confidence intervals with coverage guarantees. Finally, we illustrate the generality and tightness of our results by a series of examples, including uniform confidence bands, two-sample kernel tests, minmax stochastic programs and the empirical risk of stacked estimators.

翻译：在统计推论中,最常用的形成信任间隔的方法之一是经验型陷阱,在限制测量员的分布不为人知时,这种陷阱特别方便,然而,尽管其普遍存在的作用,其理论属性对于非随机正常估测员仍不十分清楚。在本文中,我们在稳定的条件下,确定了经验型靴测算仪的有限分布,得出了尽可能一致的严格条件,并量化了趋同速度。此外,我们提出了三种替代方法,用靴式陷阱方法来建立具有覆盖保证的信任间隔。最后,我们通过一系列例子,包括统一信任带、两类模子内核测试、微轴透视程序以及堆叠测算员的经验风险,来说明我们结果的普遍性和紧凑性。