Markov chains: recurrent states

Dependencies:

  1. Markov chain

Let $X = [X_0, X_1, \ldots]$ be a markov chain. Let $R_i = \bigvee_{t=1}^{\infty} (X_t = i)$, 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:

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