We introduce the concept of an asymptotic e-process, which is a doubly indexed stochastic process $(E_{m,n})_{m,n\in\mathbb{N}}$ that approximates an e-process with monitoring time $n$ in terms of a suitable limiting behavior for an approximation parameter $m\to \infty$. This theory is motivated by practical applications in sequential hypothesis testing, in which e-variables can only be constructed approximately from observations due to model misspecification or estimation errors. We derive an asymptotic version of Ville's inequality, which bounds excursion probabilities of $(E_{m,n})_{m,n\in\mathbb{N}}$ over some threshold uniformly over $n$ up to a time horizon $r_m$ that is determined by the quality of process approximation over $m$. We investigate properties of asymptotic e-processes, their connections to asymptotic supermartingales, and provide examples of how they can be constructed from asymptotic e-variables.

翻译：本文介绍了渐近e-过程的概念，这是一个双重索引随机过程$(E_{m,n})_{m,n\in\mathbb{N}}$，它通过近似参数$m\to \infty$的适当极限行为来逼近具有监测时间$n$的e-过程。这一理论源于序贯假设检验的实际应用，其中由于模型误设或估计误差，e-变量只能通过观测值近似构建。我们推导了Ville不等式的渐近版本，该不等式在由过程近似质量$m$决定的时间范围$r_m$内，一致地限制了$(E_{m,n})_{m,n\in\mathbb{N}}$超过某个阈值的游程概率。我们研究了渐近e-过程的性质、它们与渐近鞅的联系，并提供了如何从渐近e-变量构建这些过程的示例。