Communicating states

Communicating states is a term from the theory of Markov chains , a sub-area of probability theory . Two states of a Markov chain clearly communicate when the probability of getting from one state to the other is really greater than zero. Communicating states are important because many important properties of Markov chains such as periodicity , recurrence and transience between communicating states are inherited.

definition

Let there be a homogeneous Markov chain in discrete time and with finite or countable state space.

A state is called accessible from the state or the state leads to a state if that applies to a ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle l}$ ${\displaystyle k}$ ${\displaystyle n\in \mathbb {N} _{0}}$

${\displaystyle p_{lk}^{n}:=P(X_{n}=k\mid X_{0}=l)>0}$

applies. The probability of getting from to in a finite number of steps must therefore be really positive. This is noted as or . ${\displaystyle l}$ ${\displaystyle k}$ ${\displaystyle l\rightarrow k}$ ${\displaystyle l\rightsquigarrow k}$

If can now be reached from and can be reached from , the states and communicate , which is often abbreviated as or . ${\displaystyle l}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle k\leftrightarrow l}$ ${\displaystyle k\leftrightsquigarrow l}$

A state means essential if the state can be reached again from every state that can be reached from . It is therefore essential if it always follows from. ${\displaystyle l}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle l}$ ${\displaystyle l}$ ${\displaystyle l\rightarrow k}$ ${\displaystyle k\rightarrow l}$

example

If one looks at the transition graph of a Markov chain above, the state space is . ${\displaystyle S=\{-2,-1,0,1,2\}}$

No other state can be reached from state −2, as is the case with state 2. On the other hand, from each of the states −1, 0 and 1, every further state of the Markov chain can be reached.

The state −2 only communicates with itself, as does the state 2. The states −1,0 and 1 communicate with each other, but not with the states −2 or 2, since no return is possible from these.

State −2 is essential, just like state 2. Because for these states only they can be reached, and thus also return from themselves. On the other hand, the other states are not essential, because state 2, for example, can be reached by each. A return is excluded from this.

properties

The relation of communicating is an equivalence relation , the equivalence classes are also called communication classes . In the example above, the states form a communication class. If there is only one communication class, one speaks of an irreducible Markov chain . ${\displaystyle -1,0,1}$

States that communicate with one another have the same period ; they are also always all transient or all zero-recurrent or all positive recurrent .

Trivially, no other state can be reached from an absorbing state . It follows directly from this that chains with absorbing states cannot be irreducible. Likewise, any absorbent condition is essential, as is any recurrent condition.

In the case of a finite state space of from attainable, one exists - path in the transition graph . Communicate and , accordingly, there is both a - -path and a - -path. ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle l}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle l}$ ${\displaystyle k}$

Web links

• BA Sevast'yanov: Markov chains . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Ulrich Krengel: Introduction to probability theory and statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 , doi : 10.1007 / 978-3-663-09885-0 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .

Individual evidence

1. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 241.
2. ↑ Krengel: Introduction to Probability Theory and Statistics. 2005, p. 207.
3. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 241.