Delta method to find limiting distribution to $\sqrt{n}\left( (\overline{X_n})^2 -4 \right)$.

Question

I'm studying for comps and came across this problem. I'm having a tough time finding examples of how to solve problems such as this. If anyone knows of a text that has a fair amount of these problems solved out as examples please let me know.

Let $\overline{X_n}$ denote the mean of a random sample of size $n$ from a Poisson distribution with mean $\lambda = 2$. Use the delta method to find the limiting distribution of $\sqrt{n}\left( (\overline{X_n})^2 -4 \right)$.

I know that for a Poisson distribution $\lambda = \mu = \sigma^2$. Thus, $\lambda = \mu = \sigma^2 = 2$ for this problem. In addition the definition of the delta method is:

Let $\{X_n\}$ be a sequence of random variables such that $$\sqrt{n}(X_n - \theta) \xrightarrow{D} N(0,\sigma^2)$$ Suppose function $g(x)$ is differentiable at $\theta$ and $g'(\theta) \ne 0.$ Then $$\sqrt{n}(g(X_n) - g(\theta)) \xrightarrow{D} N(0,\sigma^2(g'(\theta))^2)$$

Any help would be much appreciated.

Answer 1

This looks like plain substitution into the delta method you are describing.

Pick $\theta = \mathbb{E}[X_k] = 2$ and notice that by CLT you have $$ \sqrt{n} (X_n - \theta) \to \mathcal{N}\left(0, \sigma^2\right). $$ For which value of $\sigma$ is this true?

Now let $g(x) = x^2$, notice that $g(x)$ is differentiable at $\theta = 2$ and $$g'(\theta) = g'(2) = 2\cdot 2 = 4 \ne 0.$$ Now you can apply the conclusion, what do you get?