Limits of sequences of events
Section of Probability spaces and events, part of Stochastic processes
Limits of sequences of events
| Type | Section |
| Taught in | Probability spaces and events |
| Part | Stochastic processes |
For a sequence of events A_n:
- at least one occurs on \bigcup_n A_n;
- all of them occur on \bigcap_n A_n;
- infinitely many occur on \limsup_n A_n=\bigcap_m\bigcup_{n\ge m} A_n;
- all but finitely many occur on \liminf_n A_n=\bigcup_m\bigcap_{n\ge m} A_n.
The limsup asks for an occurrence in every tail. The liminf asks for one tail in which every event occurs. Almost-sure convergence is a statement that a limsup of deviation events has probability 0, which is why the first Borel–Cantelli lemma is the bridge from a summable series of probabilities to a pathwise limit on Expectation and convergence.
A probability space is a sample space, a sigma-algebra of events, and a countably additive probability. Countable unions make tail events measurable. If the probabilities of A_n sum to a finite number, only finitely many occur, with probability 1. If the events are independent and the probabilities sum to infinity, infinitely many occur, with probability 1.
Used in
See also
Probability space · Borel–Cantelli lemmas · Generated and Borel sigma-algebras · Sigma-algebra · Countable additivity