It's on P91, probability theory and examples, version 5, by Rick Durrett. Although it's a section about large deviation, I think my question is universal for moment generating functions.
Define . Let and .
1. Why in ? I think the reason is my proof in the second question. But I want to confirm it.
2. Assume , can I prove ? My idea is the following. Let . Note that , for all . Thus .
If this is true, I am confused about the proof of Lemma 2.7.2, which discusses . Because it's unnecessary.