Independence and the Borel-Cantelli Lemmas

Independence formalizes the idea that knowing some events tells you nothing about others, and the Borel-Cantelli lemmas turn summability of probabilities into sharp almost-sure statements about how often a sequence of events occurs.

Definition

Events are independent when the probability of their joint occurrence factorizes into the product of their probabilities, and the Borel-Cantelli lemmas relate the convergence or divergence of the sum of event probabilities to whether infinitely many of the events occur almost surely.

Scope

The topic covers independence of events, sigma-algebras, and random variables, the grouping and approximation lemmas that support it, the first and second Borel-Cantelli lemmas, Kolmogorov's zero-one law for tail events, and applications to almost-sure convergence and the recurrence of rare events.

Core questions

• What does independence mean for events, for sigma-algebras, and for random variables, and how are these notions related?
• When does a sequence of events occur only finitely often, and when does it recur infinitely often?
• Why must the converse Borel-Cantelli lemma assume independence?
• Why does a tail event of an independent sequence have probability either zero or one?

Key concepts

• independence of events
• independence of sigma-algebras
• tail sigma-algebra
• infinitely-often event
• almost-sure recurrence

Key theories

First Borel-Cantelli lemma

If the probabilities of a sequence of events have a finite sum, then with probability one only finitely many of the events occur; no independence is required, and the result underlies many almost-sure convergence arguments.

Second Borel-Cantelli lemma

If the events are independent and the sum of their probabilities diverges, then with probability one infinitely many of the events occur, giving a sharp converse to the first lemma under independence.

Kolmogorov zero-one law

Any event in the tail sigma-algebra of a sequence of independent random variables has probability either zero or one, so asymptotic properties such as convergence of a series of independent terms are deterministic in their truth value.

Clinical relevance

These results are the workhorses behind strong laws of large numbers and the analysis of records, runs, and rare events; in reliability and risk analysis they determine whether a recurring hazard strikes infinitely often, and in number theory and ergodic theory the zero-one law explains why many limiting properties hold either always or never.

History

Borel proved the convergence half in 1909 in his study of normal numbers, and Cantelli supplied the independence converse in 1917. Kolmogorov later subsumed both within his zero-one law for tail events, making them central tools of the measure-theoretic theory.

Key figures

• Emile Borel
• Francesco Paolo Cantelli
• Andrey Kolmogorov

Seminal works

• durrett2019

Frequently asked questions

Why does the second Borel-Cantelli lemma require independence but the first does not?

Without independence, divergent probabilities can still describe events that overlap so heavily that only finitely many distinct ones occur; independence rules out this conspiracy and forces infinitely many occurrences.

What is a tail event?

A tail event is one whose occurrence does not depend on any finite number of the underlying random variables, such as the convergence of an infinite series; Kolmogorov's law says such events have probability zero or one when the variables are independent.