Strong law of large numbers

Apr 25, 2020 • Sihyung Park
probability  

$\newcommand{\argmin}{\mathop{\mathrm{argmin}}\limits}$ $\newcommand{\argmax}{\mathop{\mathrm{argmax}}\limits}$

$X_1, X_2, \cdots$ are pairwise independent and identically distributed random variables. If $E|X_1| = \mu < \infty$, then $\frac{X_1 + \cdots + X_n}{n} \to \mu \:\: a.s.$

SLLN is “strong” in two senses. First, it only requires pairwise independence which is much generous condition than mutual independence required in WLLN. Second, it implies almost sure convergence rather than convergence in probability.