Came across Kelly criterion with 李永樂老師’s video. The proof is indeed rather elementary. However, it also puzzled me when I tried to derive it myself. It seems that the argument is completely okay. Hopefully, someone can correct me.

Update: This is a good lesson learned. Yes. Have to be really careful in terms of expectation. I can compute the expectation but as N goes to infinity. The net-worth either goes to 0 or infinity. So the expected value didn’t say much. On the other hand, converge to something.