Question:
Let $Y_1,\cdots,Y_n $ ~$_{i.i.d} N(\mu_0，\sigma^2_0)$, now consider following 3 estimators for $\theta_0 = \mu^2_0$: $\hat\theta_1 = \bar Y^2, \hat\theta_2 = \bar Y^2 - \frac{\sum_{i=1}^n(Y_i - \bar{Y})^2}{n(n-1)}, \hat{\theta_3} = Max\{0,\hat{\theta_2}\}$.
Argue that all three estimators have a same asympotic distribution.
According to Asympotic Equivalence and Markov inequality, I've proved that $\hat{\theta_1}$ has an identical asympotic distribution with $\hat{\theta_2}$. But I don't know how to show that $\hat{\theta_3}$ has a same asympotic distribution as $\hat{\theta_1}$.
There is a random variable $G$, such that $\hat\theta_1, \hat\theta_2 \to G$ in law. By the Portmanteau lemma, convergence in law is preserved when we apply a continuous function. In particular, $\max(\hat\theta_2, 0) \to \max(G,0)$ in law. But since $\hat\theta_1 \geq 0$, we must have $G \geq 0$. Hence, $\max(G,0) = G$.