Originally Posted by

**puggles** 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!

Problem:

Deﬁne the sequence $\displaystyle X_{n}$ inductively by setting $\displaystyle X_{0} = 1$, and selecting $\displaystyle X_{n+1}$ randomly and uniformly from the interval $\displaystyle [0, X_{n}]$. Prove that $\displaystyle n^{-1}$$\displaystyle \log$$\displaystyle X_{n}$ 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):

let $\displaystyle \log$$\displaystyle X_{n}=$$\displaystyle \log$$\displaystyle X_{n} - $$\displaystyle \log$$\displaystyle X_{n-1} + $$\displaystyle \log$$\displaystyle X_{n-1} + ... + $$\displaystyle \log$$\displaystyle X_{1} + $$\displaystyle \log$$\displaystyle X_{0}$