Probability Central Limit Application

Question

Let $X_{1},X_{2},\ldots$ be independent and suppose that $P\left(X_{j}=\sqrt{j}\right)=P\left(X_{j}=-\sqrt{j}\right)=\frac{1}{2}$ for all $j\in\mathbb{N} .$ We want to study the asymptotic distribution of the sample mean $$\overline{X_{n}}=\frac{1}{n}\underset{j=1}{\overset{n}{\sum}}X_{j} .$$

Answer 1

By the central limit theorem:

$$ \lim_{n\to\infty}\frac{\sum_{j=1}^{n}X_{j}}{n}=\lim_{n\to\infty}\frac{\sum_{j=1}^{n}X_{j}}{\sqrt{n\left(n+1\right)}}\overset{d}{\longrightarrow}N\left(0,\frac{1}{2}\right) $$