Law of large numbers (${\overline{X}}_n\to \mathbb{E}{X}_1$)

Laws of large numbers give conditions under which sample means approach a deterministic expected value. For independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$, the strong law of large numbers gives almost sure convergence, while the weak law of large numbers gives convergence in probability. These conclusions specify different modes of convergence of random variables, rather than a universal numerical accuracy guarantee. The central limit theorem is a separate result about the remaining fluctuations after rescaling.