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 $(1-\sqrt{2/n})^\tfrac{-n}{2}exp(-\sqrt{n/2}t)$ ?


1 Answer 1


Note that by the properties of MGF, for a random variable $Z = aX + b$,

$$M_{Z}(t) = M_{aX+b}(t) = M_{X}(at)e^{bt}.$$

Using this result \begin{align*} M_{Z_n}(t) & = M_{\frac{X_n - n}{\sqrt{2n}}}(t)\\ & = M_{\frac{X_n}{\sqrt{2n}}}(t) \exp\left(-\dfrac{nt}{\sqrt{2n} } \right)\\ & = M_{X_n}\left( \dfrac{t}{\sqrt{2n}}\right) \exp\left(-\dfrac{\sqrt{n}t}{\sqrt{2} } \right)\\ & = \left(1 - 2\dfrac{t}{\sqrt{2n}} \right)^{-n/2} \exp\left(-\sqrt{\dfrac{n}{2 }} t\right)\\ & = \left(1 - \sqrt{\dfrac{2}{n}}t \right)^{-n/2} \exp\left(-\sqrt{\dfrac{n}{2 }} t\right).\\ \end{align*}

Thus it is a straightforward application of the properties of the MGF.


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