Markov kernel

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In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that plays the role, in the general theory of Markov processes, that the transition matrix does in the theory of Markov processes with a finite state space.[1]

Formal definition[edit]

Let be measurable spaces. A Markov kernel with source and target is a map with the following properties:

  1. The map is -measurable for every
  2. The map is a probability measure on for every .

In other words it associates to each point a probability measure on such that, for every measurable set , the map is measurable with respect to the -algebra [2]

Examples[edit]

Simple random walk[edit]

Take (the power set of ), then the Markov kernel with

where is the indicator function, describes the transition rule for the random walk on

Galton-Watson process[edit]

Take then

with i.i.d. random variables .

General Markov processes with finite state space[edit]

Take and then the transition rule can be represented as a stochastic matrix with

In the convention of Markov kernels we write

.

Construction of a Markov kernel[edit]

If is a finite measure on and is a measurable function with respect to the product -algebra and has the property

then the mapping

defines a Markov kernel.[3]

Properties[edit]

Semidirect product[edit]

Let be a probability space and a Markov kernel from to some . Then there exists a unique measure on , such that:

.

Regular conditional distribution[edit]

Let be a Borel space, a -valued random variable on the measure space and a sub--algebra. Then there exists a Markov kernel from to , such that is a version of the conditional expectation for every , i.e.

It is called regular conditional distribution of given and is not uniquely defined.

References[edit]

  1. ^ Reiss, R. D. (1993). "A Course on Point Processes". Springer Series in Statistics. ISBN 978-1-4613-9310-8. doi:10.1007/978-1-4613-9308-5. 
  2. ^ Klenke, Achim. Probability Theory: A Comprehensive Course (2 ed.). Springer. p. 180. doi:10.1007/978-1-4471-5361-0. 
  3. ^ Erhan, Cinlar (2011). Probability and Stochastics. New York: Springer. pp. 37–38. ISBN 978-0-387-87858-4. 
§36. Kernels and semigroups of kernels