Kolmogorov's 0-1 Law

Some events depend on the entire history of a random process, but oddly enough, their occurrence doesn’t depend on any finite piece of that history. These are called tail events, and for independent random variables, they are deterministic: they either always happen or never happen.

Tail Events and Tail σ-field

Let $X_1, X_2, \dots$ be a sequence of random variables. We define the tail $\sigma$-field $\mathcal{T}$ as representing information that is “infinitely far in the future.”

Formally, let $\cF_n' = \sigma(X_n, X_{n+1}, \dots)$ be the $\sigma$-field generated by the variables from index $n$ onwards. The tail $\sigma$-field is the intersection of all such future fields:

$$\mathcal{T} = \bigcap_{n=1}^\infty \cF_n' = \bigcap_{n=1}^\infty \sigma(X_n, X_{n+1}, \dots)$$

An event $A \in \mathcal{T}$ is called a tail event.

Intuition

A tail event is an event whose occurrence is unchanged if we arbitrarily alter values of $X_1, \dots, X_N$ for any finite $N$. It depends only on the asymptotic behavior of the sequence.

Examples

Tail Events

• $\lbrace \lim_{n \to \infty} X_n \text{ exists} \rbrace$
• $\lbrace \sum_{n=1}^\infty X_n \text{ converges} \rbrace$
• $\lbrace X_n > 5 \text{ infinitely often} \rbrace$
• $\lbrace \limsup_{n \to \infty} \frac{S_n}{n} = c \rbrace$

Non-Tail Events

• $\lbrace X_1 > 5 \rbrace$ (Depends on $X_1$)
• $\lbrace \sum_{n=1}^{100} X_n > 20 \rbrace$ (Depends on first 100 terms)
• $\lbrace S_n > 0 \text{ for all } n \rbrace$ (Depends on every partial sum, including $S_1 = X_1$)

Kolmogorov’s 0-1 Law

Theorem: Kolmogorov's 0-1 Law

Statement

If $X_1, X_2, \dots$ are independent random variables, then any tail event $A \in \mathcal{T}$ has probability 0 or 1.

$$\Pr(A) \in \{0, 1\}$$

Proof

1. Independence from Finite History Let $A \in \mathcal{T}$. By definition, $A \in \sigma(X_{n+1}, X_{n+2}, \dots)$ for any $n$. Since the sequence is independent, $\sigma(X_1, \dots, X_n)$ is independent of $\sigma(X_{n+1}, \dots)$. Therefore, $A$ is independent of $\sigma(X_1, \dots, X_n)$ for every $n$.

2. Independence from Infinite History Since $A$ is independent of $\sigma(X_1, \dots, X_n)$ for all $n$, it is independent of the field $\bigcup_{n=1}^\infty \sigma(X_1, \dots, X_n)$. By the standard $\pi$-$\lambda$ theorem argument, this independence extends to the generated $\sigma$-field:

   $$\cF_\infty = \sigma\left(\bigcup_{n=1}^\infty \sigma(X_1, \dots, X_n)\right) = \sigma(X_1, X_2, \dots).$$

   Thus, $A$ is independent of $\cF_\infty$.

3. Self-Independence Since $A$ is a tail event, $A \in \mathcal{T} \subseteq \cF_\infty$. This means $A$ must be independent of itself (as it is independent of every event in $\cF_\infty$, including itself).

4. Conclusion The independence condition gives:

   $$\Pr(A \cap A) = \Pr(A)\Pr(A) \implies \Pr(A) = \Pr(A)^2.$$

   The only solutions to $x = x^2$ for $x \in [0,1]$ are $x=0$ and $x=1$.

Implications

This law is incredibly powerful because it tells us that for independent sequences, asymptotic behaviors are deterministic.

• A random walk is either recurrent with probability 1 or transient with probability 1. It can’t be “50% chance recurrent”.
• A random series $\sum X_n$ either converges with probability 1 or diverges with probability 1.