independence implies integrability?!

Question

Let $(X_n)_n$ a sequence of independent random variables and identically distributed. Let $$Y_n=\frac{1}{n}\sum_{k=1}^nX_k \ \ \ \ and \ \ \ W_n=\frac{1}{n-1}\sum_{k=1}^n(X_k-Y_n)^2$$

If $Y_n$ and $W_n$ are independent, does this mean that $E(X_1^2)<+\infty?$

I am thankful for any ideas.

Answer 1

You can say even more. All the absolute moments of $X_1$ are finite. This is due to the fact that $Y_n$ and $W_n$ are independent only if $X_1\sim N(\mu,\sigma^2)$.