Asymptotic Distribution by Central Limit Theorem

Question

Let $X_{1},X_{2},\ldots$ be i.i.d. exponential random variables with mean $1$ and variance $1$. Let $$Y_j=\sqrt{j}\left(X_j-1\right)$$ for all $j\in\mathbb{N}$. I want to find the asymptotic distribution of $$\overline{Y}_n=\frac{1}{n} \sum_{j=1}^n Y_j.$$

Answer 1

If the assumptions of the central limit theorem with Lindeberg condition hold then:

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

For each $ j\in\mathbb{N}$ the variance of $Y_{j}$ is $j$. We simply need to verify Lyapounov condition: $$ \frac{1}{\left(\frac{n\cdot\left(n+1\right)}{2}\right)^{2}}\cdot\sum_{k=1}^{n}E\left[\mid Y_{k}\mid^{4}\right]\to0$$

Since $\left(\frac{n\cdot\left(n+1\right)}{2}\right)^{2}$ grows like $n^{4}$ and $\sum\limits_{k=1}^{n}E\left[\mid Y_{k}\mid^{4}\right]$ is $O\left(n^{3}\right)$, the condition is satisfied.