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Second Borel-Cantelli lemma under pairwise independence

Statement

Let $(A_{n})_{n \in N}$ be pairwise independent events with $\sum_{n = 0}^{\infty} P (A_{n}) = + \infty .$ Then $P (A_{n} i . o .) = 1.$

Facts & Assumptions

Given: Pairwise independent events $(A_{n})_{n \in N}$ with $\sum_{n = 0}^{\infty} P (A_{n}) = + \infty$ .

[L1]

The frequency law gives $\frac{\sum_{k = 0}^{n} 1_{A_{k}}}{\sum_{k = 0}^{n} P (A_{k})} \to 1$ almost surely. (Pairwise-independent Borel-Cantelli frequency law)

[L2]

The event ${A_{n} i . o .}$ is the event that infinitely many of the $A_{n}$ occur. (Limsup and the infinitely often event)

Proof

technique · direct

1.1givenL1

Let $S_{n} : = \sum_{k = 0}^{n} 1_{A_{k}}$ and $a_{n} : = \sum_{k = 0}^{n} P (A_{k})$ . The divergence hypothesis makes $a_{n} \to + \infty$ , and [L1] gives $S_{n} / a_{n} \to 1$ almost surely. Therefore on a full-probability event there is $N$ such that $S_{n} \geq a_{n} / 2$ for every $n \geq N$ , hence $S_{n} \to + \infty$ .

2.1step 1.1L2∎

The partial counts $S_{n} (ω)$ diverge to $+ \infty$ exactly when the event $A_{n}$ occurs for infinitely many indices $n$ . By [L2], this is precisely the event $A_{n} i . o .$ . Since step 1.1 shows it has probability $1$ , the second Borel-Cantelli conclusion follows.

Depends on

Used by

Convergence in probability need not be almost sure Counterexample
Kolmogorov extension alone does not give a continuous version Counterexample
Weak law does not imply strong law Counterexample
The three series impose separate conditions Example
Necessity of the truncated mean and variance conditions Lemma
Brownian law of the iterated logarithm at infinity Theorem
Integrability is necessary for an iid finite mean strong law Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rick Durrett, Probability: Theory and Examples, 5th ed., Theorem 2.3.7 and Theorem 2.3.9 (standard reference, not scraped)