Transition kernel explained

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

(S, l S) , (T, l T) be two measurable spaces. A function\kappa \colon S x l T \to [ 0 , +inf ty]

is called a (transition) kernel from

S to T if the following two conditions hold:

• For any fixed

B \in l T , the mappings \mapsto \kappa(s,B)

is

l S/ l B([ 0 , +inf ty]) -measurable;

• For every fixed

s \in S , the mappingB \mapsto \kappa(s, B)

is a measure on

(T, l T) .

Classification of transition kernels

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

\kappa _{s} \colon l T \to [ 0 , + inf ty] with\kappa _{s(B)=\kappa(s,B)} for all B \in l T and all s \in S . With this notation, the kernel \kappa is called

• a substochastic kernel, sub-probability kernel or a sub-Markov kernel if all

\kappa _{s} are sub-probability measures

• a Markov kernel, stochastic kernel or probability kernel if all

\kappa _{s} are probability measures

• a finite kernel if all

\kappa _{s} are finite measures

• a

\sigma -finite kernel if all \kappa _{s} are \sigma -finite measures

• a s-finite kernel if all

\kappa _{s} are s -finite measures, meaning it is a kernel that can be written as a countable sum of finite kernels

• a uniformly

\sigma -finite kernel if there are at most countably many measurable sets B _{1,} B _{2,} ... in T with \kappa _{s(B} _{i)} < inf ty for all s \in S and all i \in \N .

Operations

In this section, let

(S, l S) , (T, l T) and (U, l U) be measurable spaces and denote the product σ-algebra of l S and l T with l S ⊗ l T

Product of kernels

Definition

Let

\kappa ^{1} be a s-finite kernel from S to T and \kappa ^{2} be a s-finite kernel from S x T to U . Then the product \kappa ^{1} ⊗ \kappa ^{2} of the two kernels is defined as\kappa ^{1} ⊗ \kappa ^{2} \colon S x ( l T ⊗ l U) \to [ 0 , inf ty] \kappa ^{1} ⊗ \kappa ^{2(s,A)=} \int _{T} \kappa ^{1(s,} d t) \int _{U} \kappa ^{2((s,t),} du) 1 _{A(t,u)}

for all

A \in l T ⊗ l U .

Properties and comments

The product of two kernels is a kernel from

S to T x U . It is again a s-finite kernel and is a \sigma -finite kernel if \kappa ^{1} and \kappa ^{2} are \sigma -finite kernels. The product of kernels is also associative, meaning it satisfies (\kappa ^{1} ⊗ \kappa ^{2)} ⊗ \kappa ^{3=} \kappa ^{1} ⊗ (\kappa ^{2 ⊗} \kappa ^{3)} for any three suitable s-finite kernels \kappa ^{1,\kappa} ^{2,\kappa} ^{3} .

The product is also well-defined if

\kappa ^{2} is a kernel from T to U . In this case, it is treated like a kernel from S x T to U that is independent of S . This is equivalent to setting\kappa((s,t),A): = \kappa(t,A) for all A \in l U and all s \in S .

Composition of kernels

Definition

Let

\kappa ^{1} be a s-finite kernel from S to T and \kappa ^{2} a s-finite kernel from S x T to U . Then the composition \kappa ^{1} ⋅ \kappa ^{2} of the two kernels is defined as\kappa ^{1} ⋅ \kappa ^{2} \colon S x l U \to [ 0 , inf ty] (s, B) \mapsto \int _{T} \kappa ^{1(s,} dt) \int _{U} \kappa ^{2((s,t),} du) 1 _{B(u)}

for all

s \in S and all B \in l U .

Properties and comments

The composition is a kernel from

S to U that is again s-finite. The composition of kernels is associative, meaning it satisfies (\kappa ^{1} ⋅ \kappa ^{2)} ⋅ \kappa ^{3=} \kappa ^{1} ⋅ (\kappa ^{2} ⋅ \kappa ^{3)}

for any three suitable s-finite kernels

\kappa ^{1,\kappa} ^{2,\kappa} ^{3} . Just like the product of kernels, the composition is also well-defined if \kappa ^{2} is a kernel from T to U .

An alternative notation is for the composition is

\kappa ^{1} \kappa ^{2}

Kernels as operators

Let

l T ^{+,} l S ^{+} be the set of positive measurable functions on (S, l S), (T, l T) .

Every kernel

\kappa from S to T can be associated with a linear operatorA _{\kappa} \colon l T ^{+} \to l S ^{+}

given by

(A _{\kappa} f)(s) = \int _{T} \kappa (s, dt) f(t).

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

A _{\kappa ^{1}} A _{\kappa ^{2}} = A _{\kappa ^{1} ⋅ \kappa ^{2}}

References

^{[1] [2] [3] [4] [5] [6]}

Notes and References

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