Chapter 6 · Subchapter 3

Borel–Cantelli: What Happens Infinitely Often

Ask what happens not once but forever: which events recur without end? Two half-page lemmas answer — and between them they encode the entire logic of rare events, guarantees, and why repetition changes everything.

In this subchapter

Limsup of events: "infinitely often", precisely
Borel–Cantelli I: summable implies eventually never
Borel–Cantelli II: independence converts divergence into recurrence
Independence is load-bearing: the counterexample
The long run in practice
The tail σ-algebra, glimpsed

6.3.1Limsup of events: "infinitely often", precisely

Definition — limsup of events, i.o.

For events A₁, A₂, …, the event that A_n occurs infinitely often is

{ A_n i.o. } := limsup_n A_n = ⋂_n=1^∞ ⋃_k≥n A_k

Read it in two passes: ⋃_k≥nA_k = "at least one A happens at time n or later"; the intersection over n = "no matter how far out you stand, something is still coming". The outcome ω belongs to the limsup iff ω ∈ A_n for infinitely many n — the two readings are equal by a one-line unraveling, and the definition inherits measurability: it is a countable combination of countable combinations, so it is an event (closure clauses of 1.2). The dual event liminf A_n = ⋃_n⋂_k≥nA_k is "eventually always" — everything from some time on belongs.

Example 1 · Two sequences, read through the limsup

(a) On the infinite coin-flip space (6.1 Example 3): A_n = {n-th flip is H}. Any fixed sequence with infinitely many heads — by the law that will be proved — belongs to every tail union: limsup = "infinitely many heads". A sequence that is eventually all tails belongs to no tail union: it escapes the limsup. (b) On ([0,1], ℒ, λ): A_n = [0, 1/n]. Every ω > 0 eventually leaves — belongs to finitely many A_n; only ω = 0 is in infinitely many. The limsup is the singleton {0}: one point, null. Already here: a sequence of events of probabilities 1, 1/2, 1/3, … → 0, whose limsup has probability zero. The lemmas below quantify exactly this competition between shrinking probabilities and accumulating time.

6.3.2Borel–Cantelli I: summable implies eventually never

Theorem 1 (Borel–Cantelli I)

If Σ_n P(A_n) < ∞, then P(A_n i.o.) = 0. No independence, no structure, nothing: pure measure arithmetic.

Proof — three lines, all previously banked

P(A_n i.o.) = P(⋂_n⋃_k≥nA_k) ≤ P(⋃_k≥NA_k) ≤ Σ_k≥N P(A_k) for every N — the first step by monotonicity (the limsup is contained in every tail union), the second by subadditivity (2.1). The right side is the tail of a convergent series: send N → ∞, get 0. ∎

Example 2 · A rare-event guarantee, priced

A daily risk event with probability 10⁻⁴ — independent or not, correlated in any pattern whatsoever — is outside the lemma's reach: Σ 10⁻⁴ = ∞, so Theorem 1 is silent about it. That silence is itself the finding: a constant small probability, repeated forever, is not a summable schedule, and no convergence conclusion is available. The version the lemma does price: events with P(A_n) = 10⁻⁴/n² — summable — occur finitely often, a.s., regardless of any dependence structure. The practical shape: decaying rare-event schedules are eventually quiet; flat ones are not, and no correlation structure rescues either way. Theorem 1 is the stronger for its austerity: it converts summability into an a.s. guarantee with zero structural assumptions.

The expected-count bridge (from 5.2, now load-bearing)

5.2 Example 6 computed E[Σ 1_{A_n}] = Σ P(A_n) by MCT. Borel–Cantelli I is its contrapositive in spirit: if the expected total count is finite, the actual count is finite a.s. (a nonnegative integer-valued variable with finite expectation is finite a.s. — Markov's inequality, arriving in 7.2, makes this one line; for now the tail-sum bound above carries the same content). Both results teach one discipline: infinity in the average and infinity in the event are connected, in one direction, always.

6.3.3Borel–Cantelli II: independence converts divergence into recurrence

Theorem 2 (Borel–Cantelli II)

If the events A₁, A₂, … are independent (mutually, in the sense of 6.1) and Σ_n P(A_n) = ∞, then P(A_n i.o.) = 1: the events recur, almost surely, forever.

Proof — independence turns complements into products, and products die

By De Morgan, (limsup A_n)^c = ⋃_n ⋂_k≥n A_k^c: the escape from infinitely-many is "eventually none happen". Show each inner intersection is null: for fixed n,

P( ⋂_k=n^N A_k^c ) = ∏_k=n^N (1 − P(A_k)) ≤ exp( − Σ_k=n^N P(A_k) )

using independence for the product (complements of independent events are independent — a σ-algebra-level fact from 6.1 §2's note) and 1 − x ≤ e^−x (first-year calculus). As N → ∞ the partial sums Σ_k≥n P(A_k) diverge by hypothesis, so the bound → 0: the inner intersection has probability 0, and the countable union of null sets is null (2.1 continuity). So the escape event has probability 0; the limsup has probability 1. ∎

Example 3 · The canonical application: runs of heads

Flip a fair coin forever. Let A_n = {flips n, n+1, …, n+9 are all heads} — a run of 10 starting at n; P(A_n) = 2⁻¹⁰, constant. Independent? The A_n overlap (consecutive runs share flips), but the subfamily A₁, A₁₁, A₂₁, … uses disjoint flip-blocks, hence is independent (6.1), with divergent sum. BC II: runs of 10 occur infinitely often, a.s. — and the same argument for runs of any fixed length: the "law of runs" that gamblers find unbelievable is a two-lemma proof. Correlated-but-divergent events: run this argument's structure in reverse — independence of a subsequence is often enough, a trick worth remembering.

6.3.4Independence is load-bearing: the counterexample

Example 4 · Divergent, dependent, and dead

On ([0,1], ℒ, λ): A_n = [0, 1/n]. Then Σ P(A_n) = Σ 1/n = ∞ — the hypothesis of BC II's sum half holds — but the events are nested, maximally dependent, and A_n i.o. = {0}, probability 0 (Example 1b). Every one of the infinitely many chances lands on the same shrinking territory: repetition without independence is not repetition at all — it is the same event wearing successively smaller hats. BC II fails exactly here, and the failure is honest: the theorem's independence clause is not a decoration but the mechanism (the proof multiplied complements; dependent complements do not multiply). Contrast: shift to A_n = [x_n, x_n + 1/n mod 1] with independent-uniform placement — the sums diverge, the placement is independent, and BC II delivers: infinitely many hits, a.s. The same marginals, the same divergence, opposite verdicts — the dependence structure, invisible to any per-event statistic, decides everything. This is the course's most compact demonstration that "what the numbers are" does not determine "how the events behave together".

6.3.5The long run in practice

Reading the lemmas as a risk-management grammar

Events you can bound summably (decaying with n, whatever the dependence): eventually never, a.s. — the honest form of "this can only happen finitely many times before the design catches it", for error schedules that genuinely decay.
Independent, divergent: infinitely often, a.s. — the honest form of "given enough time, every tail scenario materializes". A daily 4σ event on a stationary light-tailed model is (if the model is true, and independence approximately holds) not a possibility but a schedule.
The forbidden inference: flat-probability events with heavy dependence — neither lemma applies; any confident claim here is unlicensed. Liquidity crises, correlated defaults: the exact region where Borel–Cantelli refuses to speak and where model error, not probability, is the binding constraint.

6.3.6The tail σ-algebra, glimpsed

One more idea, briefly, because 7.2 needs it. For independent X₁, X₂, …, the tail σ-algebra is 𝒯 = ⋂_n σ(X_n+1, X_n+2, …) — the questions answerable from the sequence after discarding any finite prefix. "Does X̄_n converge", "is the series Σ X_n/n convergent" — tail questions: changing finitely many variables changes neither answer. Kolmogorov's 0–1 law (proved in 7.2 alongside the strong law): every tail event for an independent sequence has probability 0 or 1 — no intermediate values. The limsup events of this subchapter are tail events (whether A_n recurs is unaffected by any finite prefix), and BC I/II already exhibit the law's conclusion in action: the answers arrived at were exactly 0 and 1, never anything between. That is not a coincidence; it is the 0–1 law's shadow falling forward, and it foreshadows a deep theme: for independent sequences, asymptotic questions have deterministic answers even when no individual step does.

Summable chances vanish forever, a.s. Divergent independent chances recur forever, a.s. Between those two theorems, the long run has no other moods.

Where we are, and what is next

The last subchapter of Chapter 6 is a cautionary classic: Bertrand's paradox — "a random chord of a circle: what is the probability it is longer than the side of the inscribed equilateral triangle?" — with three defensible answers (1/2, 1/3, 1/4), and the resolution that "at random" is meaningless until a measure is named. It is the discipline lesson for every modeling decision, and it closes the chapter on the course's favorite theme: probability questions are measure questions.