6.1 Tail sigma-algebra

The tail sigma-algebra of an infinite random sequence contains those events which can be determined by observing the long-run behaviour of the sequence.

6.2 Example: Liminf and limsup of a random sequence

The liminf and limsup of a random sequence are measurable with respect to the tail sigma-algebra. The event that the sequence has a limit is a tail event.

6.3 Kolmogorov's 0-1 law

All events in the tail sigma-algebra of an independent random sequence are almost sure events, in the sense that the event either occurs with probability one, or does not occur with probability one.

6.4 Example: Supremum of a random walk

As a consequence of Kolmogorov's 0-1 law, the probability that a random walk attains arbitrarily large values is either 0 or 1.



Alternative reading material

  • [Jacod & Protter, Chapter 10]
  • [Williams, Chapters 4.9-4.12]
Last modified: Wednesday, 27 January 2021, 1:34 PM