----- > [!proposition] Proposition. ([[weak law of large numbers]]) > > Let $(X_{n})_{n \geq 1}$ be [[independent|i.]][[probability distribution|i.d.]] [[random variable|random variables]] having the same [[expectation]] $m$ and [[variance]] $\sigma^{2}<\infty$. Set $\overline{X}(n):= \frac{1}{n} \sum_{i=1}^{n}X_{i}.$ > Then $\overline{X}(n) \to m$ in [[convergence in measure|in probability]] as $n \to \infty$. > [!proof]- Proof. ([[weak law of large numbers]]) > We have $\mathbb{E}[\overline{X}(n)]=\frac{1}{n} \sum_{i=1}^{n}\mathbb{E}[X_{i}]=m$ and, by indepedence, $\operatorname{var }[\overline{X}(n)]=\frac{1}{n^{2}} \sum_{i=1}^{n} \operatorname{var }[X_{i}]=\frac{\sigma^{2}}{n}.$ Hence, by [[Chebyshev's Inequality]], $\mathbb{P} [ |\overline{X(n)}- m|> \varepsilon ] \leq \frac{\operatorname{var }[\overline{X}(n)]}{\varepsilon^{2}}=\frac{\sigma^{2}}{\varepsilon^{2}n}$ for all $\varepsilon>0$. Taking $n \to \infty$ yields the result. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```