I was reading about the law of large numbers, and under its strong formulation it says that the sample average converges almost surely. That means that it may exist a finite subset with measure $0$ whose sample average does not converge. I was wondering myself about infinite sequences with all elements $\{x_k \} \in [0, 1]$ and the average of all their terms. For me, this is a problem that can take advantage of the law of large numbers to reach a solution. But the fact that this almost surely is there, makes me think about the existence of a sequence whose all-terms-average does not exist.
Do this kind of sequences exist? Could you write an example?
Many thanks in advance!!
The sequence $100011111111100000000...$ with runs of length $1$, $3$, $9$, $27$, etc, is an example. Each run reverses the preponderance of all previous votes, so the running average fluctuates between limits but does not converge.