Suppose iid $X_1,\ldots, X_n\sim p(x;\theta)$ and suppose $T_n(X_1,\ldots, X_n)$ has pdf $$ p_{T_n(X)}(x) = \frac{1}{n}\sum_{i=1}^n \log \frac{p(x_i;\theta_1)}{p(x_i,\theta_2)} $$ where $\theta_1$ and $\theta_2$ are two true value of $\theta$. I wish to prove that $\sqrt{n}(T_n-\mu)\to N(0,\sigma)$ for some $\sigma$, where $\mu$ stands for the mean of $T_n$.
My try: I plan to use central limit theorem (CLT). I thought, in the beginning, this question is only a trivial application of $CLT$ but now I am a bit confused. Indeed, if I want to use CLT, I am actually assuming that the r.v $Y$ with pdf $p_Y(y) = \log\frac{p(y;\theta_1)}{p(y;\theta_2)}$ is iid. Is it true? and if yes, what is the mean value of $T(X)$ and what is the value of $\sigma$?
Thank you!