
limiting distribution
Let X1,....,Xn be a random sample from a Poisson($\displaystyle \lambda$) distribution.
Find the limiting distribution of $\displaystyle \overline{X^2}$.
I found the mean to be E( $\displaystyle \overline{X^2}$) = $\displaystyle (1/n)*E(X_i)^2$
but then i don't know which method to use.
Thank you

The square is a continuous function.
Since $\displaystyle \bar X\to \lambda$
we have $\displaystyle \left(\bar X\right)^2\to \lambda^2$
and I don't understand what this.... $\displaystyle (1/n)*E(X_i)^2$ means
I guess zero as n goes to infinity