1.4 Law of Large Numbers

Theorem 1.14 Let \(X_i\), \(i\in \mathbb{N}\) be independent RVs such that \(\mathbb{E}(X_i) = \mu\) and \(\mathbb{V}(X_i) = \sigma^2\). Then, for each \(\epsilon > 0\), \[ \lim_{n\to \infty} \mathbb{P}\left( \left| \frac{1}{n} \sum_{i=1}^n X_i - \mu \right| > \epsilon \right) = 0 \,.\]

The above kind of convergence is sometimes called convergence in probability. There are other modes of convergence such as convergence almost surely and uniform convergence.