Finding the limiting distribution using moment generation functions

Question

Question: Let $Z_{i}$ ~ $N(0,1)$ be independent random variables for i = 1,2,... . Obtain the limiting distribution of $$\frac{1}{\sqrt{n}}\Bigl(\sum_{i=1}^n\Bigl(Z_{i} + \frac{1}{n}\Bigr)\Bigr)$$ using moment generating functions.

So far I have: Let $Y_{n}=\frac{1}{\sqrt{n}}(\sum_{i=1}^n(Z_{i} + \frac{1}{n}))$

$\\M_{y_n}(t)=E(e^{ty_n})=E(exp(t[\frac{1}{\sqrt{n}}\sum_{i=1}^n(Z_{i} + \frac{1}{n})]))=...=e^\frac{t}{\sqrt{n}}\prod_{i=1}^nE(exp\frac{t}{\sqrt{n}}Z_{i})=e^\frac{t}{\sqrt{n}}(M_{Z_i}(\frac{t}{\sqrt{n}}))^n$

I'm not too sure where to go from here. I think I can apply Taylor expansion to $M_{Z_i}(\frac{t}{\sqrt{n}})$ at 0, but how do I work with the $\frac{t}{\sqrt{n}}$? Am I to take the derivative of that function and calculate it at zero? Likely a simple concept that I am not understanding, so any clarification would help, thanks!

Answer 1

By the properties of Gaussian RVs (unrelated to MGFs),

$$ Z_i \sim \mathcal{N}(0, 1) \Rightarrow (Z_i + \frac{1}{n}) \sim \mathcal{N}\left(\frac{1}{n}, 1\right). $$

The MGF of this distribution is

$$ e^{\frac{t}{n} + t^2} $$

Since the RVs are independent, the weighted sum has a simple form

$$ \left( e^{\frac{t}{n\sqrt{n}} + \left(\frac{t}{\sqrt{n}}\right)^2} \right)^n = e^{\frac{t}{\sqrt{n}} + t^2} . $$

At the limit, this is $$ e^{t^2}, $$

which is the MGF of $\mathcal{N}(0, 1)$.