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

Info:

Transitive dependencies:

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