I don't see a SLLN here.
Why don't you obtain the distribution of and then take the log of that rv?
This was a test question from last semester that I stared at blankly for a long time and couldn't figure out where to begin. Obviously since I need to show convergence almost surely, its the Strong Law of Large Numbers, but beyond that I got really stuck. I'd love some input on at least how to get started. Thanks!
Deﬁne the sequence inductively by setting , and selecting randomly and uniformly from the interval . Prove that converges almost surely to a constant, and evaluate the limit.
Hint given (I'm having trouble with the /sum so I wrote out the sum expanded):