Borel–Cantelli Lemmas

Worked Example

Let's consider a sequence of independent events E_n where the probability of the n-th event is given by p_n = 1/(n+1) for n ≥ 1. We want to find the probability that no events occur in the first 4 trials (N=4), and the probability of at least one event in those trials.

We first calculate the individual probabilities:

• p₁ = 1/2
• p₂ = 1/3
• p₃ = 1/4
• p₄ = 1/5

The partial sum of these probabilities is:

Σ p_n = 1/2 + 1/3 + 1/4 + 1/5 = 77/60 ≈ 1.2833

Since the events are independent, the probability that none of these events occur is the product of their complements:

Π (1 - p_n) = (1 - 1/2) * (1 - 1/3) * (1 - 1/4) * (1 - 1/5) = (1/2) * (2/3) * (3/4) * (4/5) = 1/5 = 0.2

Therefore, the probability that at least one event occurs in the first 4 trials is:

1 - 0.2 = 0.8

Notice that as N goes to infinity, the harmonic series Σ 1/(n+1) diverges. By the second Borel-Cantelli lemma (since events are independent), the probability that infinitely many of these events occur is exactly 1.

Experiment

You can observe how finite cumulative sums compare with tail probabilities using the Borel–Cantelli Lemmas module.

1. Change the Event law to "Summable" (p_n = 2^-n).
2. Set Terms to 10. Notice that the Partial probability sum approaches 1, and the No-event probability is approximately 0.289.
3. Change the Event law to "Divergent" (p_n = 1/(n+1)).
4. Keep Terms at 10. The Partial probability sum is now over 2.01, and the No-event probability drops to about 0.09.
5. 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.

Limitations

The Borel-Cantelli lemmas deal with infinite limits. A finite illustration or simulation cannot strictly prove the behavior at infinity. For example, even if a sum grows very slowly for the first few million terms, we can only conclude its long-term behavior by proving whether its infinite series converges or diverges. Additionally, the first lemma (if the sum of probabilities is finite, then the probability of infinitely many events is 0) holds regardless of independence. However, the second lemma (if the sum diverges, the probability is 1) strictly requires the events to be mutually independent.

References

• Random Services: Independence - Detailed definitions and examples illustrating independent events, probability spaces, and the Borel-Cantelli lemmas.

These materials provide a rigorous mathematical foundation for the probability laws demonstrated in this interactive module.