# Instantaneous state

From Encyclopedia of Mathematics

*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=33724

This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article