In the mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications. Kernels can for example be used to define random measures or stochastic processes. The most important example of kernels are the Markov kernels.

Definition

Let , be two measurable spaces. A function

is called a (transition) kernel from to if the following two conditions hold:[1]

  • For any fixed , the mapping
is -measurable;
  • For every fixed , the mapping
is a measure on .

Classification of transition kernels

Transition kernels are usually classified by the measures they define. Those measures are defined as

with

for all and all . With this notation, the kernel is called[1][2]

  • a substochastic kernel, sub-probability kernel or a sub-Markov kernel if all are sub-probability measures
  • a Markov kernel, stochastic kernel or probability kernel if all are probability measures
  • a finite kernel if all are finite measures
  • a -finite kernel if all are -finite measures
  • a s-finite kernel if all are -finite measures, meaning it is a kernel that can be written as a countable sum of finite kernels
  • a uniformly -finite kernel if there are at most countably many measurable sets in with for all and all .

Operations

In this section, let , and be measurable spaces and denote the product σ-algebra of and with

Product of kernels

Definition

Let be a s-finite kernel from to and be a s-finite kernel from to . Then the product of the two kernels is defined as[3][4]

for all .

Properties and comments

The product of two kernels is a kernel from to . It is again a s-finite kernel and is a -finite kernel if and are -finite kernels. The product of kernels is also associative, meaning it satisfies

for any three suitable s-finite kernels .

The product is also well-defined if is a kernel from to . In this case, it is treated like a kernel from to that is independent of . This is equivalent to setting

for all and all .[4][3]

Composition of kernels

Definition

Let be a s-finite kernel from to and a s-finite kernel from to . Then the composition of the two kernels is defined as[5][3]

for all and all .

Properties and comments

The composition is a kernel from to that is again s-finite. The composition of kernels is associative, meaning it satisfies

for any three suitable s-finite kernels . Just like the product of kernels, the composition is also well-defined if is a kernel from to .

An alternative notation is for the composition is [3]

Kernels as operators

Let be the set of positive measurable functions on .

Every kernel from to can be associated with a linear operator

given by[6]

The composition of these operators is compatible with the composition of kernels, meaning[3]

References

  1. 1 2 Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 180. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  2. Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 30. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
  3. 1 2 3 4 5 Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 33. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
  4. 1 2 Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 279. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  5. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 281. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  6. Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. pp. 29–30. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
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