# Determine the convergence of $\overline{X^2}_{n}$

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.

• Existence of finite moments is usually an assumption made here. – StubbornAtom Jan 29 at 6:01

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 $$\mathbb{E}[X_{i}^{2}]$$
• Need to assume that $\mathbb{E}X_1^2<\infty$. Also one may use the SLLN to get a.s. convergence. – d.k.o. Jan 29 at 7:36