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.

  • $\begingroup$ Existence of finite moments is usually an assumption made here. $\endgroup$ – 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 $\mathbb{E}[X_{i}^{2}]$

  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.