Markov chains: recurrent states

Dependencies:

1. Markov chain

Let $X=[X_0,X_1,\dots ]$ be a markov chain. Let $R_i=\underset{t=1}{\overset{\mathrm{\infty }}{\bigvee }}(X_t=i)$, i.e., $R_i$ is the event that we'll enter state $i$ at some time $t\ge 1$. Then state $i$ is said to be recurrent iff $Pr(R_i\mid X_0=i)=1$. Intuitively, this means state $i$ is recurrent iff we will always come back to state $i$ if we start from it.

Dependency for:

1. Markov chains: positive recurrence
2. Markov chains: recurrent state to acessible state (incomplete)
3. Markov chains: recurrent iff infinite visits
4. Markov chains: long run proportion is inverse of time to reenter (incomplete)
5. Markov chains: recurrence is a class property
6. Markov chains: finite sink is recurrent (incomplete)
7. Markov chains: recurrent class is sink
8. Markov chains: recurrent iff expected number of visits is infinite

Info:

• Depth: 7
• Number of transitive dependencies: 18

Transitive dependencies:

1. /analysis/topological-space
2. /sets-and-relations/countable-set
3. /sets-and-relations/de-morgan-laws
4. σ-algebra
5. Generated σ-algebra
6. Borel algebra
7. Measurable function
8. Generators of the real Borel algebra (incomplete)
9. Measure
10. σ-algebra is closed under countable intersections
11. Group
12. Ring
13. Semiring
14. Matrix
15. Probability
16. Conditional probability (incomplete)
17. Random variable
18. Markov chain