Transition with prohibitions

transition with taboo states, for a Markov chain

2020 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J35 [MSN][ZBL]

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.

References

[C]	K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) MR0116388 Zbl 0092.34304

References

[GS]	I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 1 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027