Asymptotic normality implies consistency? > Does asymptotic normality imply consistency? Update: Asymptotic normality implies consistency, as proven in this quesiton: Showing that asymptotic normality implies consistency.

Take any example of a consistent and asymptotically normal estimator $\hat\theta_n$ and spoil it like this: $\tilde\theta_n=\hat\theta_n+\frac{1}{n^{1/3}}$.