##  [Borel-Cantelli Lemma](/borel-cantelli-lemma-0) 

 Definition

A pair of fundamental statements about sequences of events (En) in a probability space: (i) If the sum of probabilities Σ P(En) is finite then the probability that infinitely many of the En occur is zero (first lemma). (ii) If the En are independent and Σ P(En) diverges then infinitely many En occur with probability one (second lemma).

 

 

 

 

 

 





## Principle

Principle

Summability of event probabilities controls long‑term occurrence: finite sum forces eventual absence almost surely; divergence combined with independence forces persistent occurrence almost surely.

 

 

 

 

 





## Demonstration

Demonstration

For coin tosses let En be the event that toss n is heads with probability p_n. If Σ p_n < ∞ then with probability one only finitely many heads occur at the specified rare pattern. If p_n are constant positive and events independent, Σ p_n = ∞ and infinitely many occur a.s.

 

 

 

 

## Misapplication

Misapplication

Using the second lemma without an independence (or suitable weak dependence) hypothesis — divergence of Σ P(En) alone does not imply infinitely often in the presence of strong dependence.

 

 

 

 

 





## Consequence

Consequence

Gives a simple, widely used criterion for almost sure statements about limsup events; it underpins proofs of almost sure convergence, zero–one laws and probabilistic coverings.

 

 

 

 

## Reversal

Reversal

The converse of the first lemma fails in general: Σ P(En)=∞ need not imply infinitely often without independence. Dependencies can produce either fewer or more occurrences than suggested by sums alone.

 

 

 

 

 





## Boundary

Boundary

Requires a probability space and measurable events; the first lemma holds without independence, the second requires independence or strengthened hypotheses (pairwise independence is not sufficient in general).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension between deterministic summability criteria and stochastic dependence: the same numeric condition (Σ P(En)=∞) has different implications depending on independence assumptions, so 'Borel–Cantelli' may refer to either lemma and must be disambiguated.

 

 

 

 

 





## Synthesis

Synthesis

Borel‑Cantelli links the arithmetic of sums of probabilities to almost‑sure occurrence patterns: finiteness of sums forces eventual nonoccurrence a.s., while divergence implies almost sure infinite occurrence under independence, making it a basic tool for turning probabilistic size estimates into long‑run almost‑sure statements.