Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Transition kernel
open-in-new
<h2 id="definition">Definition</h2> <p>Let ( S , S ) {\displaystyle (S,{\mathcal {S}})} , ( T , T ) {\displaystyle (T,{\mathcal {T}})} be two <a href="/facts/Measurable_space/AJ98lhT4">measurable spaces</a>. A function </p> κ : S × T → [ 0 , + ∞ ] {\displaystyle \kappa \colon S\times {\mathcal {T}}\to [0,+\infty ]} <p>is called a (transition) kernel from S {\displaystyle S} to T {\displaystyle T} if the following two conditions hold:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <ul><li>For any fixed B ∈ T {\displaystyle B\in {\mathcal {T}}} , the mapping</li></ul> s ↦ κ ( s , B ) {\displaystyle s\mapsto \kappa (s,B)} is S / B ( [ 0 , + ∞ ] ) {\displaystyle {\mathcal {S}}/{\mathcal {B}}([0,+\infty ])} -<a href="/facts/Measurable/yCq7zlI4">measurable</a>; <ul><li>For every fixed s ∈ S {\displaystyle s\in S} , the mapping</li></ul> B ↦ κ ( s , B ) {\displaystyle B\mapsto \kappa (s,B)} is a <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> on ( T , T ) {\displaystyle (T,{\mathcal {T}})} . <h2 id="classification-of-transition-kernels">Classification of transition kernels</h2> <p>Transition kernels are usually classified by the measures they define. Those measures are defined as </p> κ s : T → [ 0 , + ∞ ] {\displaystyle \kappa _{s}\colon {\mathcal {T}}\to [0,+\infty ]} <p>with </p> κ s ( B ) = κ ( s , B ) {\displaystyle \kappa _{s}(B)=\kappa (s,B)} <p>for all B ∈ T {\displaystyle B\in {\mathcal {T}}} and all s ∈ S {\displaystyle s\in S} . With this notation, the kernel κ {\displaystyle \kappa } is called<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <ul><li>a substochastic kernel, sub-probability kernel or a sub-Markov kernel if all κ s {\displaystyle \kappa _{s}} are <a href="/facts/Sub-probability_measure/lRa89F8T">sub-probability measures</a></li> <li>a <a href="/facts/Markov_kernel/7HTE5nYm">Markov kernel</a>, stochastic kernel or probability kernel if all κ s {\displaystyle \kappa _{s}} are <a href="/facts/Probability_measure/8BjhgoPR">probability measures</a></li> <li>a finite kernel if all κ s {\displaystyle \kappa _{s}} are <a href="/facts/Finite_measure/GrjjuSXt">finite measures</a></li> <li>a σ {\displaystyle \sigma } -finite kernel if all κ s {\displaystyle \kappa _{s}} are <a href="/facts/Sigma-finite_measure/5PXwoAX7"> σ {\displaystyle \sigma } -finite measures</a></li> <li>a s {\displaystyle s} -finite kernel if κ {\displaystyle \kappa } can be written as a countable sum of finite kernels (so that in particular, all κ s {\displaystyle \kappa _{s}} are <a href="/facts/S-finite_measure/aSrExF41"> s {\displaystyle s} -finite measures</a>).</li> <li>a uniformly σ {\displaystyle \sigma } -finite kernel if there are at most countably many measurable sets B 1 , B 2 , … {\displaystyle B_{1},B_{2},\dots } in T {\displaystyle T} with κ s ( B i ) < ∞ {\displaystyle \kappa _{s}(B_{i})<\infty } for all s ∈ S {\displaystyle s\in S} and all i ∈ N {\displaystyle i\in \mathbb {N} } .</li></ul> <h2 id="operations">Operations</h2> <p>In this section, let ( S , S ) {\displaystyle (S,{\mathcal {S}})} , ( T , T ) {\displaystyle (T,{\mathcal {T}})} and ( U , U ) {\displaystyle (U,{\mathcal {U}})} be measurable spaces and denote the <a href="/facts/Product_%25CF%2583-algebra/8LJINLNc">product σ-algebra</a> of S {\displaystyle {\mathcal {S}}} and T {\displaystyle {\mathcal {T}}} with S ⊗ T {\displaystyle {\mathcal {S}}\otimes {\mathcal {T}}} </p> <h3>Product of kernels</h3> <h4>Definition</h4> <p>Let κ 1 {\displaystyle \kappa ^{1}} be a s-finite kernel from S {\displaystyle S} to T {\displaystyle T} and κ 2 {\displaystyle \kappa ^{2}} be a s-finite kernel from S × T {\displaystyle S\times T} to U {\displaystyle U} . Then the product κ 1 ⊗ κ 2 {\displaystyle \kappa ^{1}\otimes \kappa ^{2}} of the two kernels is defined as<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> κ 1 ⊗ κ 2 : S × ( T ⊗ U ) → [ 0 , ∞ ] {\displaystyle \kappa ^{1}\otimes \kappa ^{2}\colon S\times ({\mathcal {T}}\otimes {\mathcal {U}})\to [0,\infty ]} κ 1 ⊗ κ 2 ( s , A ) = ∫ T κ 1 ( s , d t ) ∫ U κ 2 ( ( s , t ) , d u ) 1 A ( t , u ) {\displaystyle \kappa ^{1}\otimes \kappa ^{2}(s,A)=\int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{A}(t,u)} <p>for all A ∈ T ⊗ U {\displaystyle A\in {\mathcal {T}}\otimes {\mathcal {U}}} . </p> <h4>Properties and comments</h4> <p>The product of two kernels is a kernel from S {\displaystyle S} to T × U {\displaystyle T\times U} . It is again a s-finite kernel and is a σ {\displaystyle \sigma } -finite kernel if κ 1 {\displaystyle \kappa ^{1}} and κ 2 {\displaystyle \kappa ^{2}} are σ {\displaystyle \sigma } -finite kernels. The product of kernels is also <a href="/facts/Associative/EQX8lV6r">associative</a>, meaning it satisfies </p> ( κ 1 ⊗ κ 2 ) ⊗ κ 3 = κ 1 ⊗ ( κ 2 ⊗ κ 3 ) {\displaystyle (\kappa ^{1}\otimes \kappa ^{2})\otimes \kappa ^{3}=\kappa ^{1}\otimes (\kappa ^{2}\otimes \kappa ^{3})} <p>for any three suitable s-finite kernels κ 1 , κ 2 , κ 3 {\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}} . </p><p>The product is also well-defined if κ 2 {\displaystyle \kappa ^{2}} is a kernel from T {\displaystyle T} to U {\displaystyle U} . In this case, it is treated like a kernel from S × T {\displaystyle S\times T} to U {\displaystyle U} that is independent of S {\displaystyle S} . This is equivalent to setting </p> κ ( ( s , t ) , A ) := κ ( t , A ) {\displaystyle \kappa ((s,t),A):=\kappa (t,A)} <p>for all A ∈ U {\displaystyle A\in {\mathcal {U}}} and all s ∈ S {\displaystyle s\in S} .<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h3>Composition of kernels</h3> <h4>Definition</h4> <p>Let κ 1 {\displaystyle \kappa ^{1}} be a s-finite kernel from S {\displaystyle S} to T {\displaystyle T} and κ 2 {\displaystyle \kappa ^{2}} a s-finite kernel from S × T {\displaystyle S\times T} to U {\displaystyle U} . Then the composition κ 1 ⋅ κ 2 {\displaystyle \kappa ^{1}\cdot \kappa ^{2}} of the two kernels is defined as<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> κ 1 ⋅ κ 2 : S × U → [ 0 , ∞ ] {\displaystyle \kappa ^{1}\cdot \kappa ^{2}\colon S\times {\mathcal {U}}\to [0,\infty ]} ( s , B ) ↦ ∫ T κ 1 ( s , d t ) ∫ U κ 2 ( ( s , t ) , d u ) 1 B ( u ) {\displaystyle (s,B)\mapsto \int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{B}(u)} <p>for all s ∈ S {\displaystyle s\in S} and all B ∈ U {\displaystyle B\in {\mathcal {U}}} . </p> <h4>Properties and comments</h4> <p>The composition is a kernel from S {\displaystyle S} to U {\displaystyle U} that is again s-finite. The composition of kernels is <a href="/facts/Associative/EQX8lV6r">associative</a>, meaning it satisfies </p> ( κ 1 ⋅ κ 2 ) ⋅ κ 3 = κ 1 ⋅ ( κ 2 ⋅ κ 3 ) {\displaystyle (\kappa ^{1}\cdot \kappa ^{2})\cdot \kappa ^{3}=\kappa ^{1}\cdot (\kappa ^{2}\cdot \kappa ^{3})} <p>for any three suitable s-finite kernels κ 1 , κ 2 , κ 3 {\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}} . Just like the product of kernels, the composition is also well-defined if κ 2 {\displaystyle \kappa ^{2}} is a kernel from T {\displaystyle T} to U {\displaystyle U} . </p><p>An alternative notation is for the composition is κ 1 κ 2 {\displaystyle \kappa ^{1}\kappa ^{2}} <a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="kernels-as-operators">Kernels as operators</h2> <p>Let T + , S + {\displaystyle {\mathcal {T}}^{+},{\mathcal {S}}^{+}} be the set of positive measurable functions on ( S , S ) , ( T , T ) {\displaystyle (S,{\mathcal {S}}),(T,{\mathcal {T}})} . </p><p>Every kernel κ {\displaystyle \kappa } from S {\displaystyle S} to T {\displaystyle T} can be associated with a <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> </p> A κ : T + → S + {\displaystyle A_{\kappa }\colon {\mathcal {T}}^{+}\to {\mathcal {S}}^{+}} <p>given by<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> ( A κ f ) ( s ) = ∫ T κ ( s , d t ) f ( t ) . {\displaystyle (A_{\kappa }f)(s)=\int _{T}\kappa (s,\mathrm {d} t)\;f(t).} <p>The composition of these operators is compatible with the composition of kernels, meaning<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> A κ 1 A κ 2 = A κ 1 ⋅ κ 2 {\displaystyle A_{\kappa ^{1}}A_{\kappa ^{2}}=A_{\kappa ^{1}\cdot \kappa ^{2}}} <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 180. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. <a href="978-1-84800-047-6" target="_blank">978-1-84800-047-6</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 180. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. <a href="978-1-84800-047-6" target="_blank">978-1-84800-047-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="978-3-319-41596-3" target="_blank">978-3-319-41596-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="978-3-319-41596-3" target="_blank">978-3-319-41596-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 279. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. <a href="978-1-84800-047-6" target="_blank">978-1-84800-047-6</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 279. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. <a href="978-1-84800-047-6" target="_blank">978-1-84800-047-6</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>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. <a href="978-3-319-41596-3" target="_blank">978-3-319-41596-3</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 281. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. <a href="978-1-84800-047-6" target="_blank">978-1-84800-047-6</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>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. <a href="978-3-319-41596-3" target="_blank">978-3-319-41596-3</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>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. <a href="978-3-319-41596-3" target="_blank">978-3-319-41596-3</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>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. <a href="978-3-319-41596-3" target="_blank">978-3-319-41596-3</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>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. <a href="978-3-319-41596-3" target="_blank">978-3-319-41596-3</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> </ol>