Is there a statistic such that for large sample sizes $a_n (\hat{\theta} - \theta) \sim N(0, \Sigma)$ approximately but $a_n \neq n^{1/2}$? Various central limit theorems are of the form $a_n(\hat{\theta}-\theta)\sim N(0, \Sigma)$ approximately as $n \to \infty$ and usually $a_n = n^{1/2}$. Are there central limit theorems for statistics used in real applications where $a_n \neq n^{1/2}$?

Let an i.i.d. sample from a Normal random variable with density
$$\frac{1}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right).$$
Assume you specify a Skew Normal likelihood and attempt to estimate its parameters based on the above sample. The Skew Normal density is
$$\frac{2}{s}\phi\left(\frac{x-\xi}{s}\right)\Phi\left(\lambda\frac{x-\xi}{s}\right)$$.
Then, in such a case where the true value of $\lambda =0$,
Chiogna, M. (2005). A note on the asymptotic distribution of the maximum likelihood estimator for the scalar skew-normal distribution. Statistical Methods and Applications, 14(3), 331-341,
has proven that
$$n^{1/6}\hat \lambda_{MLE} \rightarrow_d Z^{1/3},\;\;\; Z\sim N(0, V_z).$$
The interesting question here is, does that mean that
$$n^{1/2}\hat \lambda^3_{MLE} \rightarrow_d Z\;\;\; ???$$
