Transition with prohibitions

From Encyclopedia of Mathematics
Revision as of 16:57, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

transition with taboo states, for a Markov chain

The set of trajectories of the Markov chain that never enters in a specified set of states in a given time interval. Let, for example, be a Markov chain with discrete time and set of states , while is the set of "taboo" states (the taboo set). Then the taboo probabilities are

The properties of the taboo probabilities are analogous to those of the ordinary transition probabilities , since the families of matrices and , , form multiplication semi-groups; however, while , . Different problems, e.g. the study of the distribution of the time to the first entrance of the Markov chain into a given set or limit theorems for branching processes (cf. Branching process) under conditions of non-extinction, in fact amount to the investigation of various properties of taboo probabilities.


[1] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960)



[a1] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 1 , Springer (1975) (Translated from Russian)
How to Cite This Entry:
Transition with prohibitions. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article