In lecture, I indicated that I would provide an optional video proving the central limit theorem, at least in the case that \(X\) has a moment-generating function. I have broken up the proof into three parts. The content of the proof is optional, however I felt that it would be a disservice to completely omit it from the class. Next time, we will talk about applications of the central limit theorem.
First we rephrase our goal, then find an expression for the moment-generating function of the scaled empirical average.
Next, we introduce the logarithm of the moment generating function of \(X\), and establish some key properties.
Lastly, we put together the prior parts to achieve our goal.