利用e-过程对Ville鞅定理的复合推广
A composite generalization of Ville's martingale theorem using e-processes
AI总结：
该研究将Ville鞅定理推广至复合情形，引入e-过程与逆资本外测度，推导穿线不等式证明强大数定律的复合版本，表明SLLN违反事件外测度为零且e-过程可在该类序列上爆炸至无穷大。
摘要：
我们提供了Ville定理的复合版本：一个事件测度为零当且仅当存在一个非负鞅，该鞅在该事件发生时爆炸至无穷大。这是将测度论概率与Shafer和Vovk新近发展的逐序列博弈论概率联系起来的经典结果。我们对Ville结果的扩展涉及非负鞅和测度零事件的适当复合推广：分别由“e-过程”和一种新的逆资本外测度提供。随后，我们针对仅要求具有有限一阶矩的随机变量和，推导了一种新颖的穿线不等式，并用其证明了强大数定律（SLLN）的复合版本。这使我们能够表明，SLLN的违反是外测度为零的事件，且我们的e-过程在每一个此类违反序列上爆炸至无穷大，而这已被证明无法通过非负（上）鞅实现。
英文摘要：
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.
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