The Borel-Cantelli Lemmas

You run a trading system for years and occasionally it hits a rare fault — a data feed glitch, a fill that reports twice. Will that fault keep recurring forever, no matter how long you run the system, or will it — despite being possible at every single moment — eventually stop happening for good? Astonishingly, the answer depends on nothing but how fast the fault's probability shrinks over time, and the Borel-Cantelli lemmas give the exact dividing line, with a clean pass/fail test based on a single sum.

The analogy: an infinite row of lottery tickets with shrinking odds

Imagine buying one lottery ticket every day, forever, but the odds of winning get worse and worse each day — day $n$'s ticket wins with probability $p_n$, shrinking toward zero. Will you win infinitely often over your infinite lifetime, or only finitely many times before your luck permanently runs dry? It feels like it should depend on details of exactly how the odds shrink and how "generous" they are early on. It doesn't, in a very precise sense: it depends on one number — whether the infinite sum $p_1+p_2+p_3+\cdots$ is finite or infinite. Add up all your daily win-probabilities: if that total is finite, you almost surely win only finitely many times, ever; if that total diverges to infinity, you almost surely win infinitely many times. Nothing else about the sequence matters for this dichotomy.

Writing it down

Let $A_n$ be a sequence of events (say, "a fault occurs on day $n$"), and let "$A_n$ infinitely often" mean the event that infinitely many of the $A_n$ occur (not that all of them occur — just that no matter how far out you go, another one still happens eventually).

First Borel-Cantelli lemma. If $\displaystyle\sum_{n=1}^{\infty} P(A_n) < \infty$, then

$$P(A_n \text{ infinitely often}) = 0 .$$

In words: if the probabilities shrink fast enough that their infinite sum is finite, the event almost surely stops happening for good, eventually. No independence is needed for this direction — it holds for any sequence of events whatsoever.

Second Borel-Cantelli lemma. If the $A_n$ are independent and $\displaystyle\sum_{n=1}^{\infty} P(A_n) = \infty$, then

$$P(A_n \text{ infinitely often}) = 1 .$$

In words: if the events are independent and their probabilities sum to infinity — even if each individual probability is tiny — the event is almost certain to recur forever, no matter how far out you look. Independence is essential here; without it, a diverging sum does not guarantee recurrence.

Worked example 1: a fault that eventually stops for good

A system fault on day $n$ occurs with probability $P(A_n) = 1/n^2$. Check the sum: $\sum_{n=1}^{\infty} 1/n^2 = \pi^2/6 \approx 1.645$, a finite number (this is the classic Basel-problem sum). By the first Borel-Cantelli lemma — no independence needed — $P(A_n \text{ infinitely often}) = 0$: almost surely, after some finite day, the fault never occurs again. Concretely, the fault's probability drops fast: by day $n=1000$, $P(A_{1000}) = 0.000001$, and the tail sum from day 1000 onward, $\sum_{n=1000}^\infty 1/n^2 \approx 0.001$, is itself already tiny — the "remaining chance of ever seeing the fault again after day 1000" is bounded by that shrinking tail sum, which is the actual mechanism behind the almost-sure eventual stopping.

Worked example 2: a fault that recurs forever

Now suppose the fault's probability decays more slowly, $P(A_n) = 1/n$, and faults on different days are independent (a reasonable model if each day's fault is caused by an unrelated random glitch). Check the sum: $\sum_{n=1}^\infty 1/n$ is the harmonic series, which diverges to infinity — provably, since $\sum_{n=1}^{N} 1/n$ grows like $\ln N$, unbounded as $N\to\infty$. By the second Borel-Cantelli lemma (independence holds, sum diverges), $P(A_n \text{ infinitely often}) = 1$: almost surely, no matter how far you run the system, the fault will occur again eventually, over and over, forever. The only change from example 1 was $1/n^2$ to $1/n$ — a seemingly small difference in decay rate — and it completely flips the long-run outcome from "stops for good" to "recurs forever."

What this means in practice

Whether a rare risk event — a circuit-breaker trigger, a settlement fail, a strategy drawdown of a given severity — will keep recurring indefinitely as you trade for longer and longer depends precisely on how fast its probability decays with market conditions or time, exactly the dichotomy these lemmas formalise. The strong law of large numbers is itself proved using the first Borel-Cantelli lemma, applied to the event "the sample average deviates from the true mean by more than $\varepsilon$."