Strong Law Of Large Numbers For Bounded Observations

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

Players: Reality, Forecaster, Sceptic
$\quad$ $\mathcal{K}_0:=1$.
$\quad$ 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.