The Strong Law of Large Numbers

The weak law of large numbers guarantees that for large $n$, the probability of a significant deviation between $M_n$ and $\mu$ is small — but it says nothing about whether such deviations, however individually unlikely, might still recur infinitely often over the entire infinite sequence. The strong law of large numbers closes this gap with a fundamentally different, stronger notion of convergence: almost-sure convergence, which guarantees that (with probability exactly $1$) $M_n$ genuinely converges to $\mu$ as a single infinite realisation unfolds — not just that deviations become improbable at each fixed $n$, but that the sequence, followed all the way out, actually settles down. The lesson closes with a sharp example distinguishing the two convergence notions: a sequence that converges in probability but not almost surely.

Learning objectives. By the end of this note you should be able to: