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.

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