Let
be a sequence of IID random variables with finite mean, µ, and let Sn be the sequence of sample means,
![](https://test.transtutors.com/qimg/d49a6222-bc26-4f7a-942b-3d7fd8adaeed.png)
(a) Show that the characteristic function of Sn can be written as
![](https://test.transtutors.com/qimg/870c2802-0ca0-4241-aaab-50660af2c426.png)
(b) Use Taylor's theorem to write the characteristic function of the XK
![](https://test.transtutors.com/qimg/5e4d84fe-e403-4060-a54c-35e66fdab53a.png)
In so doing, you have proved that the distribution of the sample mean is that of a constant in the limit as
Thus, the sample mean converges in distribution.