Let $(X_i)_{i\in \mathbb N}$ be a sequence of independent random variable with density function $f(x)=\theta x^{\theta -1}\mathbb I_{(0,1)}(x)$, with $\theta>0$. $X_i$ are distributed as a Beta of parameters $\theta$ and $1$.
I would like to understand which is the approximative distribution of $$Z_n=-\frac{1}{\frac{1}{n}\sum_{i=1}^n\log X_i}.$$
How can I do? I was thinking to use the law of large numbers as follows. I define $Y_i:=-\log X_i$. I can prove that $Y_i$ is distributed as an Exponential of parameter $\theta$ and consequently $\mathbb E(Y_i)=\frac{1}{\theta}$. By the law of large numbers I can conclude that $$\frac{1}{n}\sum_{i=1}^nY_i\to\frac{1}{ \theta}\qquad\text{almost surely}$$ and therefore $Z_n\to \theta$ almost surely. Is this correct?
I doubt because the exercise ask me to compute the approximate distribution as $n$ is large.