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. /sets-and-relations/poset
2. /measure-theory/linearity-of-lebesgue-integral
3. /measure-theory/lebesgue-integral
4. /sets-and-relations/equivalence-relation
5. /sets-and-relations/de-morgan-laws
6. /sets-and-relations/countable-set
7. /analysis/topological-space
8. Group
9. Ring
10. Field
11. Vector Space
12. Semiring
13. Matrix
14. Identity matrix
15. σ-algebra
16. σ-algebra is closed under countable intersections
17. Measure
18. Probability
19. Conditional probability (incomplete)
20. Generated σ-algebra
21. Measurable function
22. Borel algebra
23. Generators of the real Borel algebra (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