I want to show that the MLE of $N(\theta, \theta) $, namely : $$\theta_1 = \frac { \sqrt{1+\frac 4 n \sum^n x_i^2} } 2 $$ converges in probability towards the true parameter $ \theta$. I thought about showing that the mean square error converges to zero but I don't know the law of $\theta_1$.
What can I do? I want to show that it converges in probability and if possible also to find the law when $n\to \infty$ of $\theta_1$.
By the WLLN $$ 1/n \sum X_i^2 \xrightarrow{p} \mathbb{E}X^2=Var(X)+\mathbb{E}^2X=\theta+\theta^2. $$ as $n \to \infty$, and $$ g(x) = \frac{\sqrt{1 + 4 x}}{2} $$ it a continuous transformation. Hence by the continuous mapping theorem, $$ g\left( \sum X_i^2/n \right) \xrightarrow{p}g(\theta + \theta^2)=\frac{ \sqrt{(1+2\theta)^2}}{2} = 1/2+\theta, $$ as $n \to \infty$.