# Transition with prohibitions

transition with taboo states, for a Markov chain

2010 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