Strong Law of Large Numbers

Theorem
Let $P$ be a population.

Let $P$ have mean $\mu$ and finite variance.

Let $\sequence {X_n}_{n \ge 1}$ be a sequence of random variables forming a random sample from $P$.

Let:


 * $\ds {\overline X}_n = \frac 1 n \sum_{i \mathop = 1}^n X_i$

Then:


 * $\ds {\overline X}_n \xrightarrow {\text {a.s.} } \mu$

where $\xrightarrow {\text {a.s.} }$ denotes almost sure convergence.