Convergence in $L_2$ of sequence of iid variables

Question

$X_1,X_2,...$ is a sequence of independent random variables with density $f(x)=3x^21_{[0,1]}(x)$. I'm asked to find for what values of $a$ sequence $Z_n=n^{1/3}(S_n-a)$ is convergent in $L_2$, where $S_n=\frac{\sum\limits_{i=1}^nX_i}{n}$. I already know that $S_n\to\frac34$ almost surely, so for the $Z_n$ to be convergent we need $a=\frac34$. But how do I show that $Z_n$ converges in $L_2$ for this value of $a$?

Answer 1

As $\mathbb{E}(X_i)=3/4$ for all $i \geq 1$, we have

$$S_n - \frac{3}{4} = \frac{1}{n} \sum_{i=1}^n (X_i-\mathbb{E}(X_i)).$$

Since the random variables $X_i$ are independent, the variance of the sum equals the sum of the variances (Bienaymé formula). Hence,

$$\mathbb{E}((S_n-3/4)^2) = \text{var}(S_n) = \frac{1}{n^2} \sum_{i=1}^n \text{var}(X_i) = \frac{1}{n} \text{var}(X_1) \xrightarrow[]{n \to \infty} 0.$$