Markov chains: positive recurrence is a class property

Dependencies:

1. Markov chain
2. Markov chains: accessibility and state classes
3. Markov chains: positive recurrence
4. Markov chains: long run proportion is inverse of time to reenter (incomplete)

Let $X = [X_0, X_1, \ldots]$ be a markov chain. If state $i$ is positive recurrent and $i$ communicates with state $j$, then $j$ is also positive recurrent. This means that either all states in a class are positive recurrent, or no state in the class is positive recurrent.

Proof can be found in Proposition 4.5 of [intro-to-prob-models-book].

Dependency for:

1. Markov chains: finite sink is positive recurrent

Info:

• Depth: 10
• Number of transitive dependencies: 27

Transitive dependencies:

1. /analysis/topological-space
2. /sets-and-relations/countable-set
3. /sets-and-relations/poset
4. /sets-and-relations/equivalence-relation
5. /sets-and-relations/de-morgan-laws
6. σ-algebra
7. Generated σ-algebra
8. Borel algebra
9. Measurable function
10. Generators of the real Borel algebra (incomplete)
11. Measure
12. σ-algebra is closed under countable intersections
13. Group
14. Ring
15. Semiring
16. Matrix
17. Identity matrix
18. Probability
19. Conditional probability (incomplete)
20. Random variable
21. Markov chain
22. Markov chains: long run proportion of a state
23. Markov chains: recurrent states
24. Chapman-Kolmogorov equation
25. Markov chains: accessibility and state classes
26. Markov chains: long run proportion is inverse of time to reenter (incomplete)
27. Markov chains: positive recurrence