# Convergence in Distribution for a sequence of standardized chi square random variables

Let $\{X_n\}$ be a sequence of random variables where $X_n \sim \chi ^2_{_n} \forall n$ . The sequence $\{X_n\}$ has an associated sequence of MGFs given by $\{M_{x_n}(t)\}$ ,where $M_{x_n}(t)=(1-2t)^\tfrac{-n}{2} \forall n$ . Let the random sequence $\{Z_n\}$ be defined by $Z_n=\frac{X_n - n}{\sqrt{2n}}$ (standardization) . Prove that $$Z_n \overset{d}{\rightarrow} Z \sim N(0,1).$$

I know how can I prove that, but the confusion is related to the question itself

Why the associated sequence of MGFs of $Z_n$ is $(1-\sqrt{2/n})^\tfrac{-n}{2}exp(-\sqrt{n/2}t)$ ?

Note that by the properties of MGF, for a random variable $Z = aX + b$,
$$M_{Z}(t) = M_{aX+b}(t) = M_{X}(at)e^{bt}.$$