Law of Large Numbers
The general idea behind the laws of large numbers is the combined effect of a large number of identical and independent random factors.
A sequence of random variables $X_1, X_2, …, X_n$ satisfies the law of large numbers if
$\cfrac{X_1, X_2, …, X_n}{n} - \cfrac{\mathbb{E}[X_1], \mathbb{E}[X_2], …, \mathbb{E}[X_n]}{n} \rightarrow_{p} 0$ as $n \rightarrow \infty$
Types of laws of large numbers:
- Weak Law of Large Numbers - the most general law of large numbers
- Bernoulli Theorem - the simplest law of large numbers
- Central Limit Theorem
Sources
- Gmurman V.E., Probability Theory and Mathematical Statistics – 9th edition. Moscow: Vyssh. shk., 2003.
- Law of Large Numbers on exponenta.ru