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.