论文标题
Ville的Martingale定理的综合概括
A composite generalization of Ville's martingale theorem
论文作者
论文摘要
我们提供了Ville定理的综合版本，即事件的量度为零，并且仅当存在一个非负Martingale时，该事件在发生该事件时会爆炸到无穷大。这是一个经典的结果，将测量理论概率与Shafer和Vovk最近开发的序列序列游戏理论概率联系起来。我们对Ville的结果的扩展涉及对非负MARTINGALES和MEATUR-ZERO事件的适当综合概括：这些事件分别由``E-Processes''提供，以及新的逆资本外部措施。然后，我们为随机变量的总和开发出一种新颖的线条不等式，仅需要一个有限的第一瞬间，我们用来证明这是大数字（SLLN）的强法的综合版本。这使我们能够证明违反SLLN是外部度量零的事件，并且我们的电子过程在每个违规序列上都会爆炸到无穷大的情况下，而这是通过非负（超级）the脚无法实现的。
We provide a composite version of Ville's theorem that an event has zero measure if and only if there exists a nonnegative martingale which explodes to infinity when that event occurs. This is a classic result connecting measure-theoretic probability to the sequence-by-sequence game-theoretic probability, recently developed by Shafer and Vovk. Our extension of Ville's result involves appropriate composite generalizations of nonnegative martingales and measure-zero events: these are respectively provided by ``e-processes'', and a new inverse capital outer measure. We then develop a novel line-crossing inequality for sums of random variables which are only required to have a finite first moment, which we use to prove a composite version of the strong law of large numbers (SLLN). This allows us to show that violation of the SLLN is an event of outer measure zero and that our e-process explodes to infinity on every such violating sequence, while this is provably not achievable with a nonnegative (super)martingale.
