# Do third order asymptotics exist?

Most asymptotic results in statistics prove that as $n \rightarrow \infty$ an estimator (such as the MLE) converges to a normal distribution based on a second-order taylor expansion of the likelihood function. I believe there’s a similar result in Bayesian literature, the “Bayesian Central Limit Theorem”, which shows that the posterior converges asymptotically to a normal as $n \rightarrow \infty$

My question is – does the distribution converge to something “before” it becomes normal, based on the third term in the Taylor series? Or is this not possible to do in general?