Asymptotic e-processes

We investigate the concept of an asymptotic e-process, which is a doubly-indexed stochastic process (Em,n)_m,n∈mathbbN that possesses, asymptotically for an approximation index m→∞, 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 (Em,n)_m,n∈mathbbN uniformly over n up to a monitoring time horizon rm. We show the necessity of allowing for finite values of rm, recovering truly anytime-valid guarantees asymptotically if rm→∞. 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.