Málstofa í stærðfræði
Fyrirlesari: Hermann Þórisson, University of Iceland
Titill: What is typical?
Staðsetning: Naustið (Endurmenntun)
Tími: Föstudagur 30. nóvember kl. 11.40
The word “typical” is often used in a loose sense for events in a stationary random process (such as occurrences of heads in repeated coin tosses). This concept can be made precise using so-called Palm version of the process. It is however not well known that there are in fact two Palm versions and that it is the less known version that captures the typicality property. So using the well known version is flawed except in the special case when the two versions coincide.
In this talk the elementary example of repeated coin tosses (indexed by the integers) will be used to make transparent what the issue is.
In the latter half of the talk we shift drastically to random measures on a rather general class of Abelian groups (in the coin tossing example the group is the integers under addition and the random measure is formed by mass points of size one at the heads). After giving the formal definition of the two Palm versions we present a theorem motivating the claim that it is the less known Palm version that captures the typicality property. If time allows we skim through the proof which relies on the concepts of shift-coupling and mass-stationarity.