Strong law of large numbers

Strong law of large numbers: Let $(X_n)_{n\ge 1}$ be an iid sequence of random variables such that $\mathbb{E}[|X_1|]<\infty$ , and let expectation $\mu=\mathbb{E}[X_1]$ . Define the sample mean

\overline{X}_n=\frac{1}{n}\sum_{k=1}^n X_k.

Then

\mathbb{P}\!\left(\lim_{n\to\infty}\overline{X}_n=\mu\right)=1,

i.e., $\overline{X}_n\to\mu$ almost surely.

This strengthens the weak law of large numbers by replacing convergence in probability with almost-sure convergence. The phrase “almost surely” is the probability-theory analogue of almost-everywhere convergence with respect to the probability measure.