How should I prove the normal approximation of beta distribution as follows:
Let $\mathrm B_{r_1, r_2}\sim \mathrm{Beta}(r_1, r_2)$, then prove that
$\sqrt{r_1+r_2} (\mathrm B_{r_1, r_2}- \dfrac{r_1}{r_1+r_2}) \to \mathrm N(0, \gamma(1-\gamma))$
where $r_1, r_2 \to \infty$ and $\frac{r_1}{r_1+r_2}\to \gamma$ $(0<\gamma<1)$.
My attempt: By some calculation, I figured out that $\mathbb E(\mathrm B_{r_1, r_2})=\dfrac{r_1}{r_1+r_2}$ and $\operatorname{Var}(\mathrm B_{r_1, r_2})=\dfrac{r_1 r_2}{(r_1 + r_2)^2 (r_1 +r_2 +1)}$.
Therefore, it remains to prove that $\sqrt{r_1 + r_2}\dfrac{\mathrm B_{r_1, r_2}-\mathbb E(\mathrm B_{r_1, r_2})}{\sqrt{\operatorname{Var}(\mathrm B_{r_1, r_2})}}$ converges to $Z \sim \mathrm N(0, 1)$.
Here I think I have to apply CLT, but I don't know how to because the given quantity does not contain sample mean. Does anyone have ideas?
Thanks for your help!
Best Answer
Represent your $B_{r_1,r_2}$ as $\Gamma_{r_1}/(\Gamma_{r_1}+\Gamma_{r_2}).$ The $\Gamma_r$ distributions are asymptotically normal as $r\to\infty$ by the CLT. Now get your result by applying the delta method.