In this post, we will pursue a approach to the strong law of large numbers based on the convergence of random variables.
Definition (Tail -algebra) Let be a sequence of random variables. Let be the smallest -algebra with respect to which all are measure. The tail -algebra is the intersection of all .
Theorem (Kolmogorov’s 0-1 law) If are independent and , then or .
Theorem (Kolmogorov’s maximal inequality) Suppose that are independent with and . If then
Theorem Suppose that are independent and have . If
then with probability one converges.