Borel–Cantelli Lemmas
Contrast finite event windows with the probability of infinitely many occurrences.
Free theory articleContrast finite event windows with the probability of infinitely many occurrences.
Concept & interpretation
The Borel–Cantelli Lemmas are two fundamental theorems in probability theory that connect the probabilities of a sequence of events to the probability that infinitely many of these events occur. They are critical for establishing almost sure convergence and the strong law of large numbers.
Experiment
You can observe how finite cumulative sums compare with tail probabilities using the Borel–Cantelli Lemmas module.
- Change the Event law to "Summable" (pn = 2-n).
- Set Terms to 10. Notice that the Partial probability sum approaches 1, and the No-event probability is approximately 0.289.
- Change the Event law to "Divergent" (pn = 1/(n+1)).
- Keep Terms at 10. The Partial probability sum is now over 2.01, and the No-event probability drops to about 0.09.
- Increase Terms to 20. The divergent sum continues to grow, causing the no-event probability to shrink further, illustrating that it eventually goes to 0 as N → ∞, leading to an at-least-one probability of 1.