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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.

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  • $\begingroup$ Existence of finite moments is usually an assumption made here. $\endgroup$ – StubbornAtom Jan 29 at 6:01
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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 $\mathbb{E}[X_{i}^{2}]$

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  • $\begingroup$ Thanks there! I appreciate the input. $\endgroup$ – Joseph Jan 29 at 5:30
  • $\begingroup$ You're welcome @BolenRoss $\endgroup$ – Ekesh Kumar Jan 29 at 6:43
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    $\begingroup$ Need to assume that $\mathbb{E}X_1^2<\infty$. Also one may use the SLLN to get a.s. convergence. $\endgroup$ – d.k.o. Jan 29 at 7:36

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