We study the tail of the null distribution of the $\log$ likelihood ratio statistic for testing sharp hypotheses about the parameters of an exponential family. We show that the classical chisquare approximation is of exactly the right order of magnitude, although it may be off by a constant factor. We then apply our results and techniques to find the error probabilities of a sequential version of the likelihood ratio test. The sequential version rejects if the likelihood ratio statistic crosses a given barrier by a given time. Our approach uses a local limit theorem which takes account of large deviations and integrates the local result by using the theory of Hausdorff measures.
"Large Deviations of Likelihood Ratio Statistics with Applications to Sequential Testing." Ann. Statist. 6 (1) 72 - 84, January, 1978. https://doi.org/10.1214/aos/1176344066