 Limsup of events

ID: limsup-of-events

A_{n}

in a probability space, the infinitely-often event is

lim sup_{n \to \infty} A_{n} = ⋂_{m = 1}^{\infty} ⋃_{n = m}^{\infty} A_{n} = {ω : for every m there is n \geq m with ω \in A_{n}} .

⋃_{m \geq 1} ⋂_{n \geq m} A_{n}^{c}

: after some outcome-dependent index, no more events occur.^[ref]

Deleting finitely many events leaves the limsup of events unchanged. It is therefore a tail event of the sequence of indicator random variables

1_{A_{n}}

, whether or not those random variables are independent. For independent events, the Kolmogorov zero-one law restricts its probability to zero or one; the Borel-Cantelli lemmas determine which alternative holds from the series of probabilities. Also

1_{A_{n}} \to 0

precisely when only finitely many events occur, connecting the First Borel-Cantelli lemma to almost sure convergence and the existing complete convergence criterion.

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