# Strong Law Of Large Numbers For Bounded Observations

Consider the following forecasting game (the Bounded Forecasting Game):

Players: Reality, Forecaster, Sceptic
Protocol:
.
FOR :
Forecaster announces
Sceptic announces
Reality announces

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.