Asymptotic e-processes

2026-04-21

AI Summary

This work addresses the challenge in sequential hypothesis testing where model misspecification or estimation error prevents exact construction of e-variables, thereby lacking finite-sample guarantees. We introduce, for the first time, the notion of an asymptotic e-process, defined as a doubly indexed stochastic process $(E_{m,n})$, whose limiting behavior as $m \to \infty$ approximates a standard e-process. We establish its connection to asymptotic supermartingales, derive a corresponding variant of Ville’s inequality, and provide practical construction methods. This framework unifies the theoretical foundation for approximate e-variables, offering sequential inference guarantees under controllable approximation error and explicitly quantifying the trade-off between approximation accuracy and the effective monitoring horizon $r_m$.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

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.

Problem

• asymptotic e-process
• sequential hypothesis testing
• e-variables
• model misspecification
• estimation errors

Innovation

• asymptotic e-process
• sequential hypothesis testing
• Ville's inequality
• asymptotic supermartingale
• e-variable