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Difference between revisions of "Transition function"

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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Neveu,   "Bases mathématiques du calcul des probabilités" , Masson (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.I. [I.I. Gikhman] Gihman,   A.V. [A.V. Skorokhod] Skorohod,   "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) {{MR|0272004}} {{ZBL|0203.49901}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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 {{MR|0572574}} {{ZBL|0456.60077}} {{ZBL|0431.60071}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Dellacherie,   P.A. Meyer,   "Probabilities and potential" , '''1–3''' , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.J. Sharpe,   "General theory of Markov processes" , Acad. Press (1988)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Albeverio,   Z.M. Ma,   "A note on quasicontinuous kernels representing quasilinear positive maps" ''Forum Math.'' , '''3''' (1991) pp. 389–400</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Dellacherie, P.A. Meyer, "Probabilities and potential" , '''1–3''' , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French) {{MR|0939365}} {{MR|0898005}} {{MR|0727641}} {{MR|0745449}} {{MR|0566768}} {{MR|0521810}} {{ZBL|0716.60001}} {{ZBL|0494.60002}} {{ZBL|0494.60001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) {{MR|0958914}} {{ZBL|0649.60079}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Albeverio, Z.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" ''Forum Math.'' , '''3''' (1991) pp. 389–400</TD></TR></table>

Revision as of 10:32, 27 March 2012

transition probability

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
How to Cite This Entry:
Transition function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transition_function&oldid=23670
This article was adapted from an original article by M.G. Shur (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article