Consistent family of stochastic kernels

A consistent family of stochastic kernels , also called a consistent family of Markov kernels , denotes in probability theory a family of stochastic kernels that are in a certain way stable with regard to the connection. For example, in the theory of stochastic processes, they are used to construct processes with specified properties from simpler structures such as transition semigroups or folding semigroups .

definition

Let an index set and a family of stochastic kernels be given by after . The family is called consistent , if for everyone out with always ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle (K_{s,t})_{s,t\in I,s<t}}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle r,s,t}$ ${\displaystyle I}$ ${\displaystyle r<s<t}$

${\displaystyle K_{r,s}\cdot K_{s,t}=K_{r,t}}$

applies. In this case, referred to the concatenation of the stochastic nuclei and . ${\displaystyle K_{r,s}\cdot K_{s,t}}$ ${\displaystyle K_{r,s}}$ ${\displaystyle K_{s,t}}$

example

Each transition semigroup defines a consistent family of stochastic kernels ${\displaystyle (K_{t}^{*})_{t\in I}}$

${\displaystyle K_{s,t}:=K_{s-t}^{*}}$.

Because of the semigroup property given by the Chapman-Kolmogorow equation then holds

${\displaystyle K_{r,s}\cdot K_{s,t}=K_{r-s}^{*}\cdot K_{s-t}^{*}=K_{r-t}^{*}=K_{r,t}}$.

Likewise, each convolution half-group defines a consistent family of stochastic kernels, because by ${\displaystyle (\mu _{t})_{t\in I}}$

${\displaystyle K_{t}(x,A):=\delta _{x}*\mu _{t}(A)}$

a transitional half-group is defined and thus a consistent family again. This follows from the compatibility of the convolution and concatenation of stochastic kernels. Here denotes the Dirac measure on and the folding of and . ${\displaystyle \delta _{x}}$ ${\displaystyle x}$ ${\displaystyle \nu _{1}*\nu _{2}}$ ${\displaystyle \nu _{1}}$ ${\displaystyle \nu _{2}}$

properties

Generation of projective families

Every consistent family of stochastic kernels with index set on a Polish space such as that generates a projective family of probability measures on the measurement space . To do this, you finally choose many and . Then a probability measure for each is given by ${\displaystyle I\subset [0,\infty ),0\in I}$ ${\displaystyle E}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle (E^{I},{\mathcal {B}}(E)^{\otimes I})}$ ${\displaystyle 0=n_{0}<n_{1}<\dots <n_{m}}$ ${\displaystyle J=\{n_{0},n_{1},\dots ,n_{m}\}}$ ${\displaystyle x\in X}$

${\displaystyle P_{J}=\delta _{x}\otimes \bigotimes _{i=0}^{m-1}K_{n_{i},n_{i+1}}}$

and they form a projective family. ${\displaystyle P_{J}}$

Generation of cores on product spaces

Each consistent family of stochastic kernels also creates a stochastic kernel of to . Because according to the above section there is a projective family of probability measures for each and thus also a unique probability measure according to Kolmogorov's extension theorem . The measurability in one shows by means of the finite square cylinder from . ${\displaystyle K}$ ${\displaystyle (E,{\mathcal {B}}(E))}$ ${\displaystyle (E^{I},{\mathcal {B}}(E)^{\otimes I})}$ ${\displaystyle x\in E}$ ${\displaystyle (E^{I},{\mathcal {B}}(E)^{\otimes I})}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {B}}(E)^{\otimes I}}$

Creation of dimensions on product spaces

Like any stochastic kernel, the above kernel and defines an arbitrary probability measure on through ${\displaystyle K}$ ${\displaystyle \nu }$ ${\displaystyle E}$

${\displaystyle \nu _{K}(A)=\int _{E}K(x,A)\nu (dx)}$

a measure of probability ${\displaystyle (E^{I},{\mathcal {B}}(E)^{\otimes I})}$

Applications

Consistent families of stochastic kernels are used in particular in the theory of stochastic processes , where they serve to define probability measures on large product spaces. Their projections onto the components can be understood as a so-called canonical stochastic process and then form the basis for further investigations.

They also make it possible to define stochastic processes with certain properties based on simple structures. Thus, according to the above example, each fatigue half-group defines a transition half-group and this in turn defines a consistent family of stochastic kernels. These can be continued into a stochastic process using the procedure outlined above. These are then precisely the processes with independent and steady growth , to which, for example, the Wiener process also belongs, but whose continuity is not yet self-evident in this construction.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 297-300 , doi : 10.1007 / 978-3-642-36018-3 .