Central limit theorem and integrability If $(Y_n)_n$ is a sequence of independent random variables and identically distributed, and if $\frac{\sum_{k=1}^nY_k}{\sqrt{n}}$ converges in distribution to a random variable Y, does this mean that $Y_1 \in L^2?$
I don't have any idea how to begin, and the result may appear as a converse for the central limit theorem. 
Do you have any idea or a reference for this theorem?
 A: Let $W_n=\frac{1}{\sqrt{n}}\sum_{k=1}^n(Y_{2k+1}-Y_{2k}),\forall x \in \mathbb{R},\varphi_{W_n}(x)=|\varphi_{Y_1}(\frac{x}{\sqrt{n}})|^{2n}$ and $\lim_n\varphi_{W_n}(x)=|\varphi_{Y}(x)|^2,$
$\exists \eta>0 $ such that $ \forall x \in [-\eta,\eta],|\varphi_{Y_1}(x)|\geq\frac{1}{2}$ and $|\varphi_Y{(x)}|\geq\frac{1}{2},$
$n(1-\Re(\varphi_{Y_3-Y_2}(\frac{\eta}{\sqrt{n}})))=E[n(1-\cos(\frac{\eta(Y_3-Y_2)}{\sqrt{n}}))]$ and $\lim_nn(1-\Re(\varphi_{Y_3-Y_2}(\frac{\eta}{\sqrt{n}})))=\lim_nn(1-|\varphi_{Y_1}(\frac{\eta}{\sqrt{n}})|^2)=\lim_n -n\ln(|\varphi_{Y_1}(\frac{\eta}{\sqrt{n}})|^2)=-\ln(|\varphi_Y(\eta)|^2),$ 
using Fatou's lemma we have $E[\liminf_n n(1-\cos(\frac{\eta(Y_3-Y_2)}{\sqrt{n}}))]\leq \liminf_nE[n(1-\cos(\frac{\eta(Y_3-Y_2)}{\sqrt{n}}))]$ which means that $E[(Y_3-Y_2)^2]\leq-\frac{4}{\eta^2}\ln(|\varphi_Y(\eta)|)<+\infty,$
which means that $Y_3-Y_2 \in L^2,$ $Y_3$ and $Y_2$ are independent, then $Y_2 \in L^2.$ 
Notice that we can apply the central limit theorem, and $Y$ will be normal $N(0,\sigma^2),$ where $\sigma^2=Var(Y_1),$ and we can deduce that $E[Y_1]=0.$ 
