Kolmogorov's zero–one law

Statement

Let $X_1, X_2, \dots$ be independent random variables on a probability space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{T} = \bigcap_{n=1}^{\infty} \sigma(X_n, X_{n+1}, \dots)$ be the tail σ-algebra, where $\sigma(X_n, X_{n+1}, \dots)$ is the smallest σ-algebra that makes $X_m$ measurable for all $m \ge n$. Then $P(A) \in \{0, 1\}$ for each $A \in \mathcal{T}$.

Remarks

An event is in $\mathcal{T}$ if changing finitely many of the $X_n$ cannot change whether it occurs. Examples of tail events: "the series $\sum_n X_n$ converges" and "$\limsup_n (X_1 + \dots + X_n)/n \le c$" for a real $c$. Proof idea: $\mathcal{T}$ is independent of $\sigma(X_1, \dots, X_n)$ for each $n$, hence of $\sigma(X_1, X_2, \dots)$, which contains $\mathcal{T}$. So each $A \in \mathcal{T}$ is independent of itself, and $P(A) = P(A)^2$. The theorem says that a tail event is almost sure or almost impossible. It does not say which of the two cases holds.

History

The theorem is named after Andrey Kolmogorov.

Explanations

L1 human

Some events depend only on the "far end" of an infinite sequence of independent trials, such as "the averages converge". For such an event, the probability cannot be $1/2$ or any value strictly between $0$ and $1$: it is always $0$ or $1$.