Simple conditions on the observed information ensure asymptotic normality of the conditional distributions of the randomly normed score statistic and maximum likelihood estimator given a suitable asymptotically ancillary statistic. In particular, asymptotic normality holds conditional on any asymptotically ancillary statistic asymptotically equivalent to observed information. The results apply to inference from a general stochastic process and are of particular relevance in the case of nonergodic models.
"Asymptotic Ancillarity and Conditional Inference for Stochastic Processes." Ann. Statist. 20 (1) 580 - 589, March, 1992. https://doi.org/10.1214/aos/1176348542