Many important results in probability theory concern sums or averages of independent random variables. Some of these are dubbed “laws of averages” among the lay community, but go by the name “laws of large numbers” to probabilists and statisticians. Proposition 6.33 was a very weak version, stating that the sample mean of independent and identically distributed (iid) random variables with finite variance converges in probability to E(X). It turns out that only a finite mean is required for convergence in probability and the stronger, almost sure convergence.