Kolmogorov's zero–one law

Kolmogorov's zero–one law, named after the Russian mathematician Andrey Kolmogorov, is a fundamental theorem in probability theory that characterizes certain events in a probability space as having a probability of either 0 or 1. This law is a key concept in understanding the nature of tail events and has significant implications in various fields such as statistics, mathematics, and computer science.

Overview

Kolmogorov's zero–one law applies to a specific type of event in probability theory known as a tail event. Tail events are defined with respect to a sequence of independent random variables and are events whose occurrence or non-occurrence is not affected by the outcome of a finite number of these variables. In simpler terms, the outcome of a tail event is determined by the collective behavior of all the variables in the sequence, rather than any finite subset of them.

The law states that if an event is a tail event in a sequence of independent random variables, then the probability of that event is either 0 or 1. This is a surprising result because it implies that certain events are either almost surely going to happen or almost surely not going to happen, even if it is not immediately apparent which is the case.

Formal Definition

Let \((X_n)_{n=1}^\infty\) be a sequence of independent random variables defined on a probability space \((\Omega, \mathcal{F}, P)\). A tail event \(T\) is an event in the tail sigma-algebra \(\mathcal{T}\), which is defined as: \[\mathcal{T} = \bigcap_{n=1}^\infty \sigma(X_n, X_{n+1}, X_{n+2}, \ldots)\] where \(\sigma(X_n, X_{n+1}, X_{n+2}, \ldots)\) denotes the sigma-algebra generated by the random variables \(X_n, X_{n+1}, X_{n+2}, \ldots\).

Kolmogorov's zero–one law asserts that for any tail event \(T\), the probability \(P(T)\) is either 0 or 1.

Implications

The implications of Kolmogorov's zero–one law are profound. It helps in understanding the long-term behavior of sequences of random variables. For example, it can be used to prove that certain properties of random sequences, such as convergence properties, have probabilities of either 0 or 1. This law also plays a crucial role in the theory of random walks and martingales.

Examples

One classic example of a tail event is the event that a sequence of coin tosses results in infinitely many heads. According to Kolmogorov's zero–one law, the probability of this event is either 0 or 1, even though intuitively it might seem to have some probability strictly between 0 and 1.

See Also

• Law of large numbers
• Borel-Cantelli lemma
• Probability theory
• Random variable
• Sigma-algebra

References

• Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition. Wiley.
• Billingsley, P. (1995). Probability and Measure, 3rd Edition. Wiley.