Suppose that $\sqrt{N} \left ( \widehat{\theta} - e \right ) \overset{d} N\left ( 0, \sqrt{2} \right )$,
If $Y_{n} \equiv \sqrt{N} \left ( log \widehat{\theta} - \beta_{1}\right )$, provide constants $\beta_{1}$ and $\beta_{2}$ such that $\beta_{2} \times Y_{n}^{2} \overset{d}{\rightarrow} \chi_{1}^{2}$.
My attempt: The delta-method in general means that if $$\sqrt{N}\left ( \widehat{\theta} -\theta \right ) \overset{d} {\rightarrow} N\left ( 0, \sigma^{2} \right ) \Rightarrow \sqrt{N}\left ( g(\widehat{\theta}) -g(\theta) \right ) \overset{d} {\rightarrow}N\left ( 0, \sigma^{2} \left [ g'(\theta) \right ]^{2} \right )$$
So $g(\theta) = log \theta = \beta_{1}$,
By the delta-method, $$\sqrt{N} \left ( log \widehat{\theta} - log \theta \right ) \overset{d}{\rightarrow} N (0,\sqrt{2})$$ $$\Rightarrow \sqrt{N} \left ( log \widehat{\theta} - \beta_{1}\right ) \overset{d}{\rightarrow} N(0,\sqrt{2})$$ $$\Rightarrow Y_{n} \overset{d}{\rightarrow} N(0,\sqrt{2})$$ $$\Rightarrow \frac{1}{\sqrt{2}} Y_{n} \overset{d}{\rightarrow} N(0,1)$$ $$\Rightarrow \left (\frac{1}{\sqrt{2}} \right )^{2} Y_{n}^{2} \overset{d}{\rightarrow} \chi_{1}^{2}$$ $$\Rightarrow \frac{1}{4} Y_{n}^{2} \overset{d}{\rightarrow} \chi_{1}^{2} $$
So $\beta_{2} = \frac{1}{4}$ and $\beta_{1} = loge$.
Does my solution look correct?
