Weak Law of Large Numbers

Let $(X_m)_{m\in\mathbb{N}}$ be a sequence of independent and identically distributed random variables on a probability space $(\Omega,\mathcal{F},P)$ such that $X_1$ and $X_1^{2}$ have finite expectation, and write $\mu=\mathbb{E}[X_1]$. For $n\in\mathbb{N}$ let

Then

in the sense of Almost Sure Convergence, Convergence in Probability, and Convergence in Distribution. Explicitly, for every $\varepsilon>0$ and every $n$,

which tends to $0$ as $n\to\infty$; the bound combines Markov's and Chebyshev's Inequalities with the additivity of the variance over independent summands from Expectation of a Product of Independent Random Variables.