Weak law of large numbers

Sample averages of iid variables converge in probability to the mean.

Weak law of large numbers

Weak law of large numbers: Let $(X_n)_{n\ge 1}$ be an iid sequence of random variables with expectation $\mu=\mathbb{E}[X_1]$ and finite variance $\mathrm{Var}(X_1)<\infty$ . Define the sample mean

\overline{X}_n=\frac{1}{n}\sum_{k=1}^n X_k.

Then for every $\varepsilon>0$ ,

\mathbb{P}\bigl(|\overline{X}_n-\mu|>\varepsilon\bigr)\to 0 \quad\text{as } n\to\infty.

This is a convergence-in-probability statement on the underlying probability space . A standard proof uses Chebyshev's inequality applied to $\overline{X}_n$ , and the result is weaker than the strong law of large numbers , which upgrades the mode of convergence.