We investigate the concept of an asymptotic e-process, which is a doubly-indexed stochastic process $(E_{m,n})_{m,n\in\mathbb{N}}$ that possesses, asymptotically for an approximation index $m\to\infty$, the properties of an e-process along a monitoring time index $n$. This constitutes the first in-depth study of this recently introduced concept, which is relevant in asymptotic sequential anytime-valid inference. Our theory is motivated by practical applications in sequential hypothesis testing, in which e-variables and e-processes can only be constructed approximately from observations due to model misspecification or estimation errors. Technically, asymptotic e-processes satisfy an asymptotic version of Ville's inequality, which bounds excursion probabilities of $(E_{m,n})_{m,n\in\mathbb{N}}$ uniformly over $n$ up to a monitoring time horizon $r_m$. We show the necessity of allowing for finite values of $r_m$, recovering truly anytime-valid guarantees asymptotically if $r_m\to\infty$. We derive various properties of asymptotic e-processes, and study their connections to asymptotic supermartingales. We also investigate general methods for their construction such as calibration, the cumulative product of asymptotic e-variables, and the monitoring an of an e-process that depends on an estimated parameter. The latter construction constitutes a generalization of a recent approach within the context of asymptotic post-hoc inference.

翻译：我们研究了渐近e过程的概念，这是一个双重指标随机过程$(E_{m,n})_{m,n\in\mathbb{N}}$，其在近似指标$m\to\infty$的渐近意义下，沿监控时间指标$n$具有e过程的性质。这是对该近期引入概念（与渐近序贯即时有效推断相关）的首次深度研究。我们的理论源于序贯假设检验的实际应用，由于模型设定错误或估计误差，e变量和e过程只能通过观测数据近似构建。技术上，渐近e过程满足Ville不等式的渐近形式，该不等式在监控时域$r_m$内统一限制了$(E_{m,n})_{m,n\in\mathbb{N}}$对$n$的越界概率。我们证明了允许$r_m$取有限值的必要性，并在$r_m\to\infty$时恢复真正的即时有效保证。我们推导了渐近e过程的多种性质，并研究了其与渐近鞅的关系。同时探讨了构造方法，包括校准、渐近e变量的累积乘积，以及依赖于估计参数的e过程的监控。最后一种构造是对近期渐近事后推断方法的一种推广。