The Strong Law of Large Numbers (SLLN) is a deeper and stronger result than the weak law. It states that the sample average of a sequence of random variables converges almost surely (with probability 1) to the expected value. Our first result starts again with the version requiring finite second moments.
Given a collection of i.i.d. random variables \(X_{1}, \dots, X_{n}\) each with finite mean \(\mathbb{E}[X_{i}]=\mu\) and \(\mathbb{E}[X_{i}^2]<\infty\), then the sample mean converges almost surely to \(\mu\), i.e. \[ \bar{X}_{n}\stackrel{a.s.}{\to}\mu\quad (n \to \infty), \] that is, \(\mathbb{P}(\lim_{ n \to \infty }\bar{X}_{n}=\mu)=1\).
Proof:
Next we state that general result for the Strong Law of Large Numbers which has the condition of pairwise independence.
Let \(X_{1}, X_{2}, \dots\) be pairwise independent identically distributed random variables with \(\mathbb{E}\lvert X_{1} \rvert<\infty\). Let \(\mathbb{E}[X_{1}]=\mu\) then \[ \frac{S_{n}}{n}= \frac{\sum_{i=1}^nX_{i}}{n} \stackrel{a.s}{\to}\mu\quad \text{as}\quad n \to \infty. \]
Proof: