# 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.

### Bibliography

- Glenn Shafer and Vladimir Vovk.
*Probability and finance: It's only a game!*. New York: Wiley, 2001. Section 3.2.