Difference between revisions of "Transition function"
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A family of measures used in the theory of Markov processes for determining the distribution at future instants from known states at previous times. Let a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937601.png" /> be such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937602.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937603.png" /> contains all one-point subsets from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937604.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937605.png" /> be a subset of the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937606.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937607.png" /> given for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376011.png" /> is called a transition function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376012.png" /> if: a) for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376015.png" />, it is a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376016.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376017.png" />; b) for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376020.png" />, it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376021.png" />-measurable function of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376022.png" />; c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376023.png" /> and for all limit points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376025.png" /> from the right in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376026.png" />, | A family of measures used in the theory of Markov processes for determining the distribution at future instants from known states at previous times. Let a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937601.png" /> be such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937602.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937603.png" /> contains all one-point subsets from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937604.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937605.png" /> be a subset of the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937606.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937607.png" /> given for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t0937609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376011.png" /> is called a transition function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376012.png" /> if: a) for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376015.png" />, it is a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376016.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376017.png" />; b) for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376020.png" />, it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376021.png" />-measurable function of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376022.png" />; c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376023.png" /> and for all limit points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376025.png" /> from the right in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376026.png" />, |
Revision as of 15:17, 17 April 2012
transition probability
2020 Mathematics Subject Classification: Primary: 60J35 [MSN][ZBL]
A family of measures used in the theory of Markov processes for determining the distribution at future instants from known states at previous times. Let a measurable space be such that the -algebra contains all one-point subsets from , and let be a subset of the real line . A function given for , , and is called a transition function for if: a) for given , and , it is a measure on , with ; b) for given , and , it is a -measurable function of the point ; c) and for all limit points of from the right in the topology of ,
and d) for all , and from , the Kolmogorov–Chapman equation is fulfilled:
(*) |
(in some cases, requirement c) may be omitted or weakened). A transition function is called a Markov transition function if , and a subMarkov transition function otherwise. If is at most countable, then the transition function is specified by means of the matrix of transition probabilities
(see Transition probabilities; Matrix of transition probabilities). It often happens that for any admissible , and the measure has a density with respect to a certain measure. If in this case the following form of equation (*) is satisfied:
for any and from and from , then is called a transition density.
Under very general conditions (cf. [1], [2]), the transition function can be related to a Markov process for which (in the case of a Markov transition function, this process does not terminate, i.e. -a.s.). On the other hand, the Markov property for a random process enables one, as a rule, to put the process into correspondence with a transition function [3].
Let be homogeneous in the sense that the set of values of for from forms a semi-group in under addition (for example, , , ). If, moreover, the transition function depends only on the difference , i.e. if , where is a function of , , satisfying the corresponding form of conditions a)–d), then is called a homogeneous transition function. The latter name is also given to a function for which (*) takes the form
For certain purposes (such as regularizing transition functions) it is necessary to extend the definition. For example, one takes as given a family of measurable spaces , , while a transition function with respect to this family is defined as a function , where , , , , that satisfies a suitable modification of conditions a)–d).
References
[1] | J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) MR0272004 Zbl 0203.49901 |
[2] | I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027 |
[3] | S.E. Kuznetsov, "Any Markov process in a Borel space has a transition function" Theory Probab. Appl. , 25 : 2 (1980) pp. 384–388 Teor. Veroyatnost. i ee Primenen. , 25 : 2 (1980) pp. 389–393 MR0572574 Zbl 0456.60077 Zbl 0431.60071 |
Comments
For additional references see also Markov chain; Markov process.
References
[a1] | C. Dellacherie, P.A. Meyer, "Probabilities and potential" , 1–3 , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French) MR0939365 MR0898005 MR0727641 MR0745449 MR0566768 MR0521810 Zbl 0716.60001 Zbl 0494.60002 Zbl 0494.60001 |
[a2] | M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) MR0958914 Zbl 0649.60079 |
[a3] | S. Albeverio, Z.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" Forum Math. , 3 (1991) pp. 389–400 |
Transition function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transition_function&oldid=23670