Let $X_1,\ldots,X_n,...$ be iid Poisson random variables with parameter $\theta$. We wish to estimate $f(\theta) = (\theta−1)^2$. For all values of $\theta$, find a limiting distribution for $g(X_n) = \left(\bar{X}_n−1\right)^2$ where $\bar{X}_n$ is the sample mean.
I have considered using the MGF approach. I assumed this: The MGF for the sample mean is just the MGF of the sum of Poisson RVs divided by $n.$ However, I was struggling to find the limit of this to form a conclusion.
I have also considered using the CDF approach but am unsure of how to calculate the CDF of $X_n$ in order to substitute the value that I need.
Any help is appreciated!
It is difficult to guess what is meant by the limit distribution of a sequence whose limit distribution is actually degenerate. Possibly, it is $$ \sqrt{n}\left( \left(\bar{X}_n−1\right)^2 - (\theta-1)^2\right) = \underbrace{\sqrt{n}\left(\bar{X}_n−\theta\right)}_{\xrightarrow{d}\, \mathcal N(0,\theta)}\cdot \underbrace{\left(\bar{X}_n+\theta-2\right)}_{\xrightarrow{p}\,2\theta-2} \xrightarrow{d}\, \mathcal N(0,\theta(2\theta-2)^2) $$ if $2\theta-2\neq 0$, i.e. $\theta\neq 1$. The last convergence is due to Slutsky's theorem.
If $\theta=1$, then $$n\left(\bar{X}_n-1\right)^2\xrightarrow{d}\chi^2_1$$ as d.k.o. stated in the comment.