Use Stein's Identity to Calculate Variance of X Bar Squared

Question

Suppose that I have a random sample where each random variable is iid normally distributed with mean $\mu$ and variance $\sigma^2$. Suppose I want to calculate the variance of $\bar{X}^2$ "using Stein's idendity." I'm not quite sure that I understand how Stein's identity applies to this situation. I know that the normal distribution is part of the linear exponential family, but I don't understand how that can help me find properties about $\bar{X}^2$. Can somebody please show how this is supposed to work? Thank you.

Answer 1

Write $V(\bar{X}^2)=E(\bar{X}^4)-[E(\bar{X}^2)]^2$ and apply Stein's lemma iteratively to get the $2$nd and $4$th non-centered moments of $\bar X$.