What's the intuition behind Kolmogorov zero-one law? Why can't the probability be a number in between?

Question

I'm trying to understand intuitively why the probability of a tail event must be zero or one. Why can't it be some real number? I'm sure that when Kolmogorov proved this theorem he must have postulated it using intuition first.

Answer 1

I don't know what Kolmogorov was thinking, but the theorem is an intuitive consequence of the idea that an infinite-dimensional random sequence $(X_i)_{i\geq0}$ is determined by the behavior of finite-dimensional subsets of elements $(X_{i_1},\dots,X_{i_n})$. That is, there's no "hidden behavior" of the sequence that you can't observe by looking at finite-dimensional subsets of the sequence.

So, if all the behavior depends on finite-dimensional subsets, then a "tail event" which, by definition, can't directly depend on any finite-dimensional subset, can only measure something "interesting" about the sequence through an indirect dependence on earlier finite subsets of elements. This can occur through inter-element dependence. For example, if every 100th element is the same, a "tail event" can depend indirectly on a finite subset of elements (specifically the single element $X_{100}$) through the dependence of infinitely many later elements on that earlier element.

In the case that the elements are independent, though, the possibility of such an indirect dependence is removed. A tail event can't depend, directly or indirectly, on any finite subset of elements, therefore it can't measure anything interesting about the sequence, so it's left to measures things that are trivially measurable by virtue of being always true or always false.

Does that help?