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, $ \xi ( t) $ be a Markov chain with discrete time and set of states $ S $, while $ H $ is the set of "taboo" states (the taboo set). Then the taboo probabilities $ {} _ {H} p _ {ij} ( t) $ are

$$ {} _ {H} p _ {ij} ( t) = {\mathsf P} \{ \xi ( k) \notin H ( k = 1 \dots t- 1 ),\ \xi ( t) = j \mid \xi ( 0) = i \} , $$

$$ i, j \in S. $$

The properties of the taboo probabilities $ {} _ {H} p _ {ij} ( t) $ are analogous to those of the ordinary transition probabilities $ p _ {ij} ( t) $, since the families of matrices $ P( t) = \| p _ {ij} ( t) \| _ {i,j \in S } $ and $ P _ {H} ( t) = \| {} _ {H} p _ {ij} ( t) \| _ {i,j \in S\setminus H } $, $ t \geq 0 $, form multiplication semi-groups; however, while $ \sum _ {j \in S } p _ {ij} ( t) = 1 $, $ \sum _ {j \in S } {} _ {H} p _ {ij} ( t) \leq 1 $. 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