# Strong Law Of Large Numbers For Bounded Observations

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

**Players**: Reality, Forecaster, Sceptic**Protocol**:`⚠ $\quad$`

`⚠ $\mathcal{K}_0:=1$`

.`⚠ $\quad$`

FOR `⚠ $n=1,2,...$`

:`⚠ $\quad$`

`⚠ $\quad$`

Forecaster announces `⚠ $m_n\in[-1,1]$`

`⚠ $\quad$`

`⚠ $\quad$`

Sceptic announces `⚠ $M_n\in\mathbb{R}$`

`⚠ $\quad$`

`⚠ $\quad$`

Reality announces `⚠ $x_n\in[-1,1]$`

`⚠ $\quad$`

`⚠ $\quad$`

`⚠ $\mathcal{K}_n:=\mathcal{K}_{n-1}+M_n(x_n-m_n)$`

**Winner**: Sceptic wins if `⚠ $\mathcal{K}_n$`

is never negative and either
`⚠ $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n (x_i-m_i) = 0$`

or `⚠ $\lim_{n\to\infty}\mathcal{K}_n=\infty$`

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 `⚠ $x_n$`

with means `⚠ $m_n$`

(more generally, for bounded random variables `⚠ $x_n$`

with conditional means `⚠ $m_n$`

). Indeed, if Reality produces `⚠ $x_n$`

stochastically from a distribution with the mean value (given the past) `⚠ $m_n$`

, `⚠ $\mathcal{K}_n$`

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.