Strong Law Of Large Numbers For Bounded Observations
Consider the following forecasting game (the Bounded Forecasting Game):
Players: Reality, Forecaster, Sceptic
Winner: Sceptic wins if is never negative and either or holds.
Theorem Sceptic has a winning strategy.
This theorem easily implies the usual measure-theoretic strong law of large numbers for bounded independent random variables with means (more generally, for bounded random variables with conditional means ). Indeed, if Reality produces stochastically from a distribution with the mean value (given the past) , will be a martingale, and one can apply Ville's inequality.