Difference between revisions of "Instantaneous state"
From Encyclopedia of Mathematics
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''of a homogeneous [[Markov chain|Markov chain]] with a countable set of states'' | ''of a homogeneous [[Markov chain|Markov chain]] with a countable set of states'' | ||
| − | A state (say, | + | A state (say, $i$) for which the density of the transition probability, |
| − | + | $$a_{ii}=\lim_{h\downarrow0}\frac{p_{ii}(h)-1}{h},$$ | |
| − | where | + | where $p_{ii}(h)$ is the probability of transition from $i$ to $i$ in time $h$, is equal to $-\infty$. In the opposite case the state $i$ is called non-instantaneous, or retarded. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</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></table> | + | <table> |
| + | <TR><TD valign="top">[1]</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> | ||
| + | </table> | ||
| Line 16: | Line 19: | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Williams, "Diffusions, Markov processes, and martingales" , '''1''' , Wiley (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E.B. Dynkin, "Markov processes" , '''1''' , Springer (1965) (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Freedman, "Brownian motion and diffusion" , Holden-Day (1971)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Williams, "Diffusions, Markov processes, and martingales" , '''1''' , Wiley (1979)</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967)</TD></TR> | ||
| + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> E.B. Dynkin, "Markov processes" , '''1''' , Springer (1965) (Translated from Russian)</TD></TR> | ||
| + | <TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Freedman, "Brownian motion and diffusion" , Holden-Day (1971)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | [[Category:Markov processes]] | ||
Latest revision as of 18:24, 17 October 2014
of a homogeneous Markov chain with a countable set of states
A state (say, $i$) for which the density of the transition probability,
$$a_{ii}=\lim_{h\downarrow0}\frac{p_{ii}(h)-1}{h},$$
where $p_{ii}(h)$ is the probability of transition from $i$ to $i$ in time $h$, is equal to $-\infty$. In the opposite case the state $i$ is called non-instantaneous, or retarded.
References
| [1] | I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) |
Comments
References
| [a1] | D. Williams, "Diffusions, Markov processes, and martingales" , 1 , Wiley (1979) |
| [a2] | K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967) |
| [a3] | E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) |
| [a4] | D. Freedman, "Brownian motion and diffusion" , Holden-Day (1971) |
How to Cite This Entry:
Instantaneous state. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Instantaneous_state&oldid=15939
Instantaneous state. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Instantaneous_state&oldid=15939
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article