Hello can some one point me in the right direction with this question?
Let $ F(x) = \begin{cases} 1- x^{-4} & x>1 \\ 0 & elsewhere \\ \end{cases} $ be the CDF of a random variable $X$
a) Find the limit in probability of $(\bar X_n)^2$ where $\bar X_n$ is a sample mean of a sample of size $n$ from $F$. Justify your steps.
Compute the expectation and variance for $X$. Verify that the variance is finite. Apply the central limit theorem for sample sums.