An asymptotically normal estimator is a consistent estimator whose distribution around the true parameter θ approaches a normal distribution with standard deviation shrinking in proportion to as the sample size n grows. Using to denote convergence in distribution, tn is asymptotically normal if
for some V, which is called the asymptotic variance of the estimator.
The central limit theorem implies asymptotic normality of the sample mean as an estimator of the true mean. More generally, maximum likelihood estimators are asymptotically normal under fairly weak regularity conditions — see the asymptotics section of the maximum likelihood article.