Hello all,

Here is my problem:

Let X_n be iid, $$E(X_n)=\mu$$, $$Var(X_n)=\sigma^2.

Show that \frac{1}{n}\sum_{i=1}^{n} (X_i-\bar{X})^2\rightarrow\sigma^2 in probability.

This seems like it should be easy. It's in the section from the book on the Weak Law of Large Numbers (among other things.) I thought I should show that  (X_i-\bar{X})^2 is an iid random variable with finite variance and expected value \sigma^2, so then I could use WLLN to say the sum converges in probability to \sigma^2. But when I try that, I find that  (X_i-\bar{X})^2 is not iid.

I feel like this is easy and I'm just missing something basic. Any thoughts?