Examples[edit]

An invertible measure-preserving transformation on a standard probability space that obeys the 0-1 law is called a Kolmogorov automorphism. All Bernoulli automorphisms are Kolmogorov automorphisms but not vice versa. The presence of an infinite cluster in the context of percolation theory also obeys the 0-1 law.


Let ${\displaystyle \{X_{n}\}_{n}}$ be a sequence of independent random variables, then the event ${\displaystyle \left\{\lim _{n\rightarrow \infty }\sum _{k=1}^{n}X_{k}{\text{ exists }}\right\}}$ is a tail event. Thus by Kolmogorov 0-1 law, it has either probability 0 or 1 to happen. Note that independence is required for the tail event condition to hold. Without independence we can consider a sequence that's either ${\displaystyle (0,0,0,\dots )}$ or ${\displaystyle (1,1,1,\dots )}$ with probability ${\displaystyle {\frac {1}{2}}}$ each. In this case the sum converges with probability ${\displaystyle {\frac {1}{2}}}$.