Markov chains: recurrence is a class property

Dependencies:

1. Markov chain
2. Markov chains: accessibility and state classes
3. Markov chains: recurrent states
4. Markov chains: recurrent iff expected number of visits is infinite

Let $X = [X_0, X_1, \ldots]$ be a markov chain with transition function $P$. Suppose state $i$ is transient and states $i$ and $j$ communicate. Then $j$ is also recurrent.

This implies that either all states of a state class are recurrent, or no state in the class is recurrent. We call a class recurrent iff all states in the class are recurrent.

Proof can be found in Corollary 4.2 of [intro-to-prob-models-book].

Dependency for:

1. Markov chains: finite sink is recurrent (incomplete)
2. Markov chains: recurrent class is sink

Info:

• Depth: 9
• Number of transitive dependencies: 31

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. /measure-theory/linearity-of-lebesgue-integral
7. /measure-theory/lebesgue-integral
8. σ-algebra
9. Generated σ-algebra
10. Borel algebra
11. Measurable function
12. Generators of the real Borel algebra (incomplete)
13. Measure
14. σ-algebra is closed under countable intersections
15. Group
16. Ring
17. Field
18. Vector Space
19. Semiring
20. Matrix
21. Identity matrix
22. Probability
23. Conditional probability (incomplete)
24. Random variable
25. Markov chain
26. Markov chains: recurrent states
27. Chapman-Kolmogorov equation
28. Markov chains: accessibility and state classes
29. Expected value of a random variable
30. Linearity of expectation
31. Markov chains: recurrent iff expected number of visits is infinite