Asymptotic e-processes
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 $r_m$.