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