Markov chains: long run proportion is inverse of time to reenter (incomplete)

Dependencies: (incomplete)

  1. Markov chain
  2. Markov chains: recurrent states
  3. Markov chains: accessibility and state classes
  4. Markov chains: long run proportion of a state

Let $X = [X_0, X_1, \ldots]$ be a markov chain with state space $D$. Let $T_j = \min_{t \ge 1} (X_t = j)$. Let $m_j = \E(T_j \mid X_0 = j)$. Let $i$ be a state that communicates with state $j$. Then $\Pr(\pi_j = 1/m_j \mid X_0 = i) = 1$.

(Requires proof)

Dependency for:

  1. Markov chains: finite sink is positive recurrent
  2. Markov chains: positive recurrence is a class property

Info:

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. σ-algebra
  7. Generated σ-algebra
  8. Borel algebra
  9. Measurable function
  10. Generators of the real Borel algebra (incomplete)
  11. Measure
  12. σ-algebra is closed under countable intersections
  13. Group
  14. Ring
  15. Semiring
  16. Matrix
  17. Identity matrix
  18. Probability
  19. Conditional probability (incomplete)
  20. Random variable
  21. Markov chain
  22. Markov chains: long run proportion of a state
  23. Markov chains: recurrent states
  24. Chapman-Kolmogorov equation
  25. Markov chains: accessibility and state classes