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Sep 20, 2016 at 23:32 comment added Anthony Quas I'm glad you got what you wanted. The Birkhoff ergodic theorem that is at the heart of my answer is a generalization of the Strong Law of Large Numbers, so the answers are, in fact, quite similar even though they look different.
Sep 20, 2016 at 23:26 comment added Anindya I swtiched to Nate's answer as this was a very direct argument which nails the result. I will follow up on your Ergodic Theory reference though. It's very interesting that a pretty simple and natural question is leading into quite deep waters. Thanks.
Sep 20, 2016 at 1:51 vote accept Anindya
Sep 20, 2016 at 23:20
Sep 20, 2016 at 1:33 history answered Anthony Quas CC BY-SA 3.0