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

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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

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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

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

Product of kernels

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Definition

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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

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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

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Definition

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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

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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

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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

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  1. ^ a b 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. ^ a b c d e 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. ^ a b 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.