Chapter 8
Limit Theorems
As a corollary we have the Chebyshev's inequality.
Chebyshev's inequality. Let be a random variable with finite mean and finite variance . Then for each
                                        
The proof of Chebyshev's inequality follows by replacing by in the Markov inequality.
There are two types of law of large numbers, weak law of large numbers and strong law of large numbers. Weak law of large numbers describe the asymptotic behavior of averages of random variables using convergence in probability and strong law of large numbers describe limit of averages of random variables using almost sure convergence.

Theorem 8.0.40 (Weak law of large numbers)   Let $ X_1, X_2, \dots $ be a sequence of independent and identically distributed random variables, each having finite mean $ \mu$ and finite variance $ \sigma^2$. Then $ \frac{S_n}{n}$ converges in probability to $ \mu$. i.e., for each $ \epsilon > 0$,

$\displaystyle \lim_{n \to \infty} P \Big\{ \Big\vert \frac{S_n}{n} \, - \, \mu \Big\vert \geq \epsilon \Big\} \, = \, 0 \, , $
where $ S_n \, = \, X_1 + \dots + X_n$.