Asymptotic analysis has always been very useful for deriving distributions in statistics in cases where the exact distribution is unavailable. More importantly, asymptotic analysis can also provide insight into the inference process itself, suggesting what information is available and how this information may be extracted. The development of likelihood inference over the past twenty-some years provides an illustration of the interplay between techniques of approximation and statistical theory.
"Asymptotics and the theory of inference." Ann. Statist. 31 (6) 1695 - 2095, December 2003. https://doi.org/10.1214/aos/1074290325