I have the sequence of independent r.v. $(X_n)$ with distributions $$ \Pr\left(X_n = \frac1{\sqrt n}\right)=\Pr\left(X_n=-\frac1{\sqrt n}\right) = \frac12$$ I should find the distribution to which the following two processes converge to in distribution: $$Y_n = \sum_{i=1}^n X_i,\qquad Z_n = \sum_{i=n+1}^{2n} X_n.$$ From some experiments I got to the conclusion that this distribution must be a Gaussian. My idea was to use the Lindeberg principle and then the Portemanteau theorem. I have that $$E[X_n] = 0 \, \forall n, E[X_n^2]=\frac1n \quad\text{ and }\quad E[|X_n|^3]=O(n^{-3/2})$$
Now I choose the sequnce of iid Gaussian r.v. $W_n \sim \mathcal{N}(0,1/n)$. Using the principle I have that $$|E[g(X_1+\ldots+X_n)]-E[g(W_1+\ldots+W_n)] \leq \frac{C}{6}\sum_{i=1}^{n}(E(|X_i|^3)+E(|W_i|^3)) \text{ and } \\|E[g(X_{n+1}+\ldots+X_{2n})]-E[g(W_{n+1}+\ldots+W_{2n})] \leq \frac{C}{6}\sum_{i=n+1}^{2n}(E(|X_i|^3)+E(|W_i|^3))$$ Approximating $$\sum_{i=1}^{n}(E(|X_i|^3)+E(|W_i|^3)= \int_1^n(E(|X_i|^3)+E(|W_i|^3)$$ since both the third moments are $O(n^{-3/2})$ we get something $O(n^{-1/2}) \to 0$ as $n\to +\infty$.
Update: So far I can say that the sequence of r.v. $$Z_n = X_{n+1}+\dots+X_{2n} \to W_{n+1}+\dots+W_{2n}, W_i \sim \mathcal{N}(0,1/i)$$ in distribution, thanks to the Linderberg Principle.
Also, being Normal r.v. the $W_i$ their sum is itself a Normal r.v. $W$ with mean $0$ and $$Var(W)=\sum_{i=n+1}^{2n}(Var(W_i))\approx ln(2)$$ Can I conclude that $Z_n\to^d W\sim \mathcal{N}(0,ln(2))$? Can you help me understand why the sequence $(Y_n,n\geq 1)$ does not converge? Is it because the variance of the corresponding normal would be $\approx log(n)$ and not a constant?
Partial answer: let $R_n=\sqrt n X_n$. Then $\{R_n\}$ is i.i.d. taking values -1 and 1 with probability 1/2 each. It is well known that $\sum a_n R_n$ converges almost surely iff it converges in distribution iff it converges in probability iff $\sum a_n^{2} <\infty$. Since $\sum \frac 1 n =\infty$ it follows that $\{Y_n\}$ does not converge in any sense. (Reference: search for Khinichine's Inequalities and Ito -Nisio Theorem).