I'm incredibly stuck on this problem.
Calculate the characteristic function (chf) of the sequence of random variables $$Y_n = \frac{(1)}{\sqrt n} \sum_{i=1}^n X_i$$ where $X_i \sim X$ and are independent. Also take the limit as n goes to positive infinity of the chf.
Got a hint to remember that if X and Y are independent, then
$$E(f(X)g(Y)) = E(f(X))E(g(Y))$$
To be honest, I'm completely lost on this. I know that the characteristic function of X for a given value u is $E_x(e^{iuX})$. Thanks!
$E(e^{iuY_n})=E(e^{iu\frac{1}{\sqrt{n}}\sum_1^n{X_i}})=\prod_1^n{E(e^{iu\frac{X_i}{\sqrt{n}}})}=(E(e^{iu\frac{X}{\sqrt{n}}}))^n$
note if $E(X)\neq 0,E(Y)\rightarrow\infty $. Assume $E(X)=0$
Also for generic chf, we have $\psi(u)=E(e^{iuX})=1+\psi'(0)u+\frac{1}{2}\psi''(0)u^2+o(u^2)$
following the properties of chf, $\psi'(0)=iE(X)=0, \psi''(0)=i^2E(X^2)=-\sigma^2$
$(\psi(\frac{u}{\sqrt{n}}))^n\sim (1-\frac{1}{2}\sigma^2\frac{u^2}{n}+o(\frac{u^2}{n}))^n\rightarrow e^{-\frac{1}{2}\sigma^2 u^2}$