Markov kernel: Difference between revisions

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Revision as of 23:31, 17 March 2022

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In [[probability theory]], a '''Markov kernel''' (also known as a '''stochastic kernel''' or '''probability kernel''') is a map that in the general theory of [[Markov process]]es plays the role that the [[Stochastic matrix|transition matrix]] does in the theory of Markov processes with a [[finite set|finite]] [[state space]].<ref>{{Cite journalbook | last1 = Reiss | first1 = R. D. | title = A Course on Point Processes | doi = 10.1007/978-1-4613-9308-5 | series = Springer Series in Statistics | year = 1993 | isbn = 978-1-4613-9310-8 }}</ref>

== Formal definition ==

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# For every (fixed) <math>B \in \mathcal B</math>, the map <math> x \mapsto \kappa(B, x)</math> is <math>\mathcal A</math>-measurable

# For every (fixed) <math> x \in X</math>, the map <math> B \mapsto \kappa(B, x)</math> is a [[probability measure]] on <math>(Y, \mathcal B)</math>

In other words it associates to each point <math>x \in X</math> a [[probability measure]] <math>\kappa(dy|x): B \mapsto \kappa(B, x)</math> on <math>(Y,\mathcal B)</math> such that, for every measurable set <math>B\in\mathcal B</math>, the map <math>x\mapsto \kappa(B, x)</math> is measurable with respect to the <math>\sigma</math>-algebra <math>\mathcal A</math>.<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability Theory: A Comprehensive Course|series=Universitext |year=2014 |publisher=Springer|page=180|edition=2|doi=10.1007/978-1-4471-5361-0|isbn=978-1-4471-5360-3 }}</ref>

== Examples ==