MS-E1600 - Probability Theory D, 11.01.2021-22.02.2021
Kurssiasetusten perusteella kurssi on päättynyt 22.02.2021 Etsi kursseja: MS-E1600
Lecture 6: Events of the infinite horizon
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]