I'd like some assistant on the proof of the following Lemma:
If [MATH]X_n[/MATH] is [MATH]AN(\mu,\sigma_n^2)[/MATH], then also [MATH]X_n[/MATH] is [MATH]AN(\overline\mu,\overline\sigma_n^2)[/MATH] if and only if [MATH]\frac{\overline\sigma_n}{\sigma_n}\rightarrow1[/MATH], [MATH]\frac{\overline\mu_n-\mu_n}{\sigma_n}\rightarrow0[/MATH].
The hint says to use Pólya's Theorem, which relates weak convergence and uniform convergence, but I'm not sure about how to use it. Thanks in advance.
If [MATH]X_n[/MATH] is [MATH]AN(\mu,\sigma_n^2)[/MATH], then also [MATH]X_n[/MATH] is [MATH]AN(\overline\mu,\overline\sigma_n^2)[/MATH] if and only if [MATH]\frac{\overline\sigma_n}{\sigma_n}\rightarrow1[/MATH], [MATH]\frac{\overline\mu_n-\mu_n}{\sigma_n}\rightarrow0[/MATH].
The hint says to use Pólya's Theorem, which relates weak convergence and uniform convergence, but I'm not sure about how to use it. Thanks in advance.