I'm learning Kolmogorov's zero-one law in probability theory:
Let $(Ω,{\mathcal F},P)$ be a probability space and let $F_n$ be a sequence of mutually independent $\sigma$-algebras contained in $\mathcal{F}$. Let
$$G_n=\sigma\bigg(\bigcup_{k=n}^\infty F_k\bigg)$$
be the smallest $\sigma$-algebra containing $F_n, F_{n+1}, \dots$. Then Kolmogorov's zero-one law asserts that for any event
$$ F\in \bigcap_{n=1}^\infty G_n$$
one has either $P(F) = 0$ or $1$.
I've no idea how $G_n$ and $\bigcap_{n=1}^{\infty} G_n$ would look like. What's the point of such construction? Could any one come up with some concrete examples of how this theorem works?
Best Answer
The usual name for the sigma algebra you have there is the "tail" $\sigma-$algebra. It is all the stuff that does not depend on finitely many of the $\sigma-$ algebras.
I usually think about the Kolmogorov 0-1 law like this: if you have a sequence of independent random variables and an event that is invariant if you ignore finitely many of the variables, then the probability of that event is either 0 or 1.
A typical example of how you can actually use this is to show that the convergence in the classical central limit theorem cannot be almost sure.
Let $X_i$ be a sequence of $iid$ random variables with $EX_i = 0$ and $EX_i^2 = 1$ and let $$S_n = \sum_{i=1}^n X_i$$
It is easy to see that for any fixed $n$ and $M>0$ the event $$\{ \limsup_k \frac{S_k}{\sqrt{k}} >M\}$$ lies in $$\sigma(\cup_{k=n}^\infty F_k)$$ since it only depends on the "tail" of our sequence (this is clear if you write out the definition of $\limsup$). Therefore by independence, the probability of this event is either 0 or 1. Using the central limit theorem, it is not hard to see that this is probability is positive for any fixed $M$ and therefore it is $1$. We then obtain that $$P(\limsup \frac{S_k}{\sqrt{k}} = \infty) = 1$$
Symmetry gives that $$P(\liminf \frac{S_k}{\sqrt{k}} = - \infty) = 1$$ so we cannot have almost sure convergence (or convergence in probability).