Let's say I sample $X_{1},X_{2},\dots,X_{n}$ from a random variable X with a distribution. It is not important to know what the distribution is at this point.
I am trying to determine whether $\overline{X^2}_{n}$ converges given that $\overline{X^2}_{n}:= \frac{1}{n}\sum_{i = 1}^{n}X^{2}_{i}$.
I am thinking of applying Law of Large number in this case, but I have not figured out exactly to determine whether it converges.
Maybe I do not even need LLN.
Any tip would be appreciated.
For each index $i$, define $y_{i} = X_{i}^{2}$. Then, the sequence $\{y_{i}\}$ forms a new sequence of i.i.d random variables. We can rewrite our sum as follows:
$$\overline{X^{2}_{n}} = \frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} = \frac{1}{n} \sum_{i = 1}^{n} y_{i},$$
which, by the Weak Law of Large Numbers converges to $\mathbb{E}[y_{i}].$ So, we have that our sequence converges to $\mathbb{E}[y_{i}] = \mathbb{E}[X_{i}^{2}].$
Thus, the series converges to $\mathbb{E}[X_{i}^{2}]$