Let $X_1,...,X_n$ be independent random variables distributed according to $\Bbb P\{X_k=1\}=\frac{1}{\log_{2}(k+1)}$ and $\Bbb P\{X_k=0\}=1-\frac{1}{\log_{2}k}$ for $k=1,2,...,n$. Let $X=X_1+X_2+...+X_n$. Note $X_1,X_2,...,X_n$ are not identically distributed. Show that $\Bbb P\{\frac{X - \Bbb EX}{\sqrt {Var X} } \le x \} \to \int_{ - \infty }^x {\frac{{{e^{ - \frac{{{x^2}}}{2}}}}}{{\sqrt {2\pi } }}dx} $ as $n\to\infty$.

Anyone can help with this problem? There is a hint saying that $\Bbb EX \sim \frac{n}{\log_2 n}$ and $VarX \sim \Bbb E^2X $. Thank you!


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